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Grade 8

VIEWS: 63 PAGES: 107

									 Grade 8
Mathematics
                                                         Grade 8
                                                        Mathematics

                                                    Table of Contents

Unit 1: Rational Numbers, Measures, and Models .........................................................1

Unit 2: Rates, Ratios, and Proportions ..........................................................................11

Unit 3: Geometry and Measurement..............................................................................25

Unit 4: Measurement and Geometry..............................................................................38

Unit 5: Algebra, Integers and Graphing ........................................................................57

Unit 6: Growth and Patterns ..........................................................................................72

Unit 7: What Are the Data? ............................................................................................81

Unit 8: Examining Chances.............................................................................................92
                     Louisiana Comprehensive Curriculum, Revised 2008
                                   Course Introduction

The Louisiana Department of Education issued the Comprehensive Curriculum in 2005. The
curriculum has been revised based on teacher feedback, an external review by a team of content
experts from outside the state, and input from course writers. As in the first edition, the
Louisiana Comprehensive Curriculum, revised 2008 is aligned with state content standards, as
defined by Grade-Level Expectations (GLEs), and organized into coherent, time-bound units
with sample activities and classroom assessments to guide teaching and learning. The order of
the units ensures that all GLEs to be tested are addressed prior to the administration of iLEAP
assessments.

District Implementation Guidelines
Local districts are responsible for implementation and monitoring of the Louisiana
Comprehensive Curriculum and have been delegated the responsibility to decide if
         units are to be taught in the order presented
         substitutions of equivalent activities are allowed
         GLES can be adequately addressed using fewer activities than presented
         permitted changes are to be made at the district, school, or teacher level
Districts have been requested to inform teachers of decisions made.

Implementation of Activities in the Classroom
Incorporation of activities into lesson plans is critical to the successful implementation of the
Louisiana Comprehensive Curriculum. Lesson plans should be designed to introduce students to
one or more of the activities, to provide background information and follow-up, and to prepare
students for success in mastering the Grade-Level Expectations associated with the activities.
Lesson plans should address individual needs of students and should include processes for re-
teaching concepts or skills for students who need additional instruction. Appropriate
accommodations must be made for students with disabilities.

New Features
Content Area Literacy Strategies are an integral part of approximately one-third of the activities.
Strategy names are italicized. The link (view literacy strategy descriptions) opens a document
containing detailed descriptions and examples of the literacy strategies. This document can also
be accessed directly at http://www.louisianaschools.net/lde/uploads/11056.doc.

A Materials List is provided for each activity and Blackline Masters (BLMs) are provided to
assist in the delivery of activities or to assess student learning. A separate Blackline Master
document is provided for each course.

The Access Guide to the Comprehensive Curriculum is an online database of
suggested strategies, accommodations, assistive technology, and assessment
options that may provide greater access to the curriculum activities. The Access
Guide will be piloted during the 2008-2009 school year in Grades 4 and 8, with
other grades to be added over time. Click on the Access Guide icon found on the first page of
each unit or by going directly to the url http://mconn.doe.state.la.us/accessguide/default.aspx.
                       Louisiana Comprehensive Curriculum, Revised 2008


                                      Grade 8
                                     Mathematics
                   Unit 1: Rational Numbers, Measures, and Models


Time Frame: Approximately three weeks


Unit Description

This unit focuses on number theory and the use of rational numbers in problem-solving
contexts. Order of operations is reviewed in situations involving fractions, decimals, and
integers. Circle graphs are created based on the central angle measurements to display
data sets.


Student Understandings

The student uses fractions, decimals, and integers in the context of problem-solving
settings. Students also revisit the order of operations while working with rational
numbers. They use the measurement of the central angle to calculate fractional parts of
the circle for circle graphs.


Guiding Questions

       1. Can students compare rational numbers using symbolic notation as well as use
          position on a number line?
       2. Can students recognize, interpret, and evaluate problem-solving contexts with
          rational numbers?
       3. Can students use the order of operations correctly in interpreting the values of
          expressions with parentheses?
       4. Can students identify the measurement of angles from given fractions based
          on the central angle of a circle to create a circle graph?


Unit 1 Grade-Level Expectations (GLEs)

GLE # GLE Text and Benchmarks
Number and Number Relations
1.      Compare rational numbers using symbols (i.e., <, <, =, >, >) and position on a
        number line (N-1-M) (N-2-M)
3.      Estimate the answer to an operation involving rational numbers based on the
        original numbers (N-2-M) (N-6-M)
5.      Simplify expressions involving operations on integers, grouping symbols, and
        whole number exponents using order of operations (N-4-M)


Grade 8 MathematicsUnit 1Rational Numbers, Measures, and Models                            1
                       Louisiana Comprehensive Curriculum, Revised 2008


6.        Identify missing information or suggest a strategy for solving a real-life,
          rational-number problem (N-5-M)
36.       Organize and display data using circle graphs (D-1-M)


                                    Sample Activities


Activity 1: Compare and Order (GLE: 1)

Materials List: Rational Number Line Cards - student 1 BLM, Rational Number Line
Cards - student 2 BLM, Rational Number BLM, Compare and Order Word Grid BLM,
calculators, paper, pencil

Have students work in pairs. Provide a number line showing only the integers –1, 0, and
1. (Teachers may choose to use additional integers on the number line, but the activity
emphasizes values within this range.) Give each student a deck of cards containing
rational numbers including some negative rational numbers. Use the Rational Number
Line Cards - Student 1 BLM and the Rational Number Line Cards – Student 2 BLM to
make both sets of cards for each pair of students. Student 1 should get a deck of rational
numbers in fraction form, made by using Rational Number Line Cards - Student 1 BLM,
and Student 2 should get a deck of rational numbers in decimal form, made using
Rational Number Line Cards - Student 2 BLM. Have each student select a card from
his/her deck and compare the cards. The comparison can be done using a calculator,
mental math, or paper/pencil. Ask students to correctly place both rational numbers on
the number line and then write a correct comparison statement using symbols. For
example, if the two rational numbers were 1 and 0.05 , they would place a mark at the 1
                                            2                                           2

point and the .05 point on their number line; then they would write a correct statement
like ―0.05< 1 ‖ or ―0.05≤ 1 .‖ Continue the activity having students place these on the
            2             2
number line. Distribute the Rational Number Line BLM to students for additional
practice with comparing and ordering rational numbers.

A modified word grid (view literacy strategy descriptions) will be used to encourage
higher order thinking through comparing and contrasting mathematical characteristics of
numbers. The purpose of the Compare and Order Word Grid BLM is to develop an
understanding of the relative size of a number when using the four operations as they
make comparisons of the numbers. The Compare and Order Word Grid BLM can be
given as a homework assignment and returned the following day for discussion.

A question such as the following can lead to rich classroom discourse and could be
responded to in their math learning log (view literacy strategy descriptions): Is the rule
you discovered the same for any two numbers? Why or why not? (Encourage students to
think of cases in which they can challenge the answer). Explain to the students that their
math log will be used all year to record new learning, and they should write questions
that they want to understand through math class. This math learning log should be kept



Grade 8 MathematicsUnit 1Rational Numbers, Measures, and Models                            2
                       Louisiana Comprehensive Curriculum, Revised 2008


either in a separate notebook or a section in the binder used for reflection of mathematical
concepts throughout the year.


Activity 2: Grouping Dilemma (GLE: 5)

Materials List: Grouping Dilemma BLM, paper, pencil

         Display the tile pattern at the left on the overhead using tiles or a sketch.
         Distribute the Grouping Dilemma BLM and give the students directions to find
         the total number of tiles without counting each one. Have the students sketch the
pattern and ―loop‖ groups of tiles that help them determine the total number of tiles.
Examples: (There are many more groupings.)
        3×4+3 .


               32 +  3×2                     some students even see  4×4  -1


          3+3+3+3+3 or 3×5


Give the students time to explain their method of determining the total number of tiles.
Have them write the correct statement representing their groupings as this provides
evidence of their understanding of order of operations. Ask students to use information
from the classroom discussions to determine how many square tiles would be needed if
tiles were arranged in this same manner with the top right tile missing but there were 8
rows and 8 columns. Lead discussion as the students determine which of the methods
used earlier make it possible to find the number of tiles. (63)


Activity 3: Target (GLEs: 1, 5)

Materials List: playing cards minus the face cards, paper, pencil

Provide each group of four students a set of playing cards minus the face cards. Ask
students to shuffle the cards and tell them that the red cards represent negative numbers
and the black cards represent positive numbers. Have Player 1 place the first four cards in
the deck face up and identify one of the four numbers as the target number. Allow
Players 2, 3 and 4 about 45 seconds to build a number sentence using the three cards that
are not the target number as well as two different operation symbols. Have the players
compete to be the first to build a sentence that results in the target number as the answer.
If the sentence results in the target, award the player two points. If no one gets the target
number, give the player closest to the target number one point. Ask all players to write
the winning number sentence and their individual number sentences using the correct
order of operations. When the winner of the round has been determined, have group


Grade 8 MathematicsUnit 1Rational Numbers, Measures, and Models                           3
                        Louisiana Comprehensive Curriculum, Revised 2008


members compare their answers, writing them as a repeated inequality. After each round
of play, have the player to the right of the last player turn the cards over and determine
the target number.

Example: Suppose the four cards turned up are red 4, red 8, black 3, and red 7. Player 1
selects the black 3 as the target because there are 3 reds or negatives. One student writes
 4 7  8 and gets 3.5 , the second student writes 8  4  7 and gets 5 , and the
third student writes 4  8   7  and gets 7 1 . The students should write the inequality
                                                 2

7 1  3  3.5  5 . The player with the answer of 7 1 is closest to 3 and receives 1 point.
  2                                                   2




Activity 4: Target Story Chain (GLEs 1, 5)

Materials List: paper, pencil

Once the students have completed one game of Target, Activity 3, show them a model of
a math story chain (view literacy strategy descriptions) that demonstrates their
understanding of inequalities made in the Target game. The example below uses the
Target Number of 10. The object of the story chain is to represent the inequality with a
real-life situation. The first person starts the story and the paper is passed to the right.
The next person writes the second sentence, the third person writes the third sentence,
and the fourth person writes the question. It goes back to the first person to check that it
all is clear and easily understood. Have students list each of the possible sentences that
could be written in the mathematical story that would illustrate the inequality (or have the
students give other suggestions).

       Example of model and how to use the Story Chain:
       Target number: 10
       Closest Target equation or inequality 10 < 2 x 9 – 7
           (person 1) Sam has nine times as many marbles as Bill.
           (person 2) Bill has 2 marbles.
           (person 3) Jane has seven marbles less than Sam.
           (person 4 writes the question) Does Jane have more or less marbles than
              the target number?

Each of the students should have saved their last winning equation or inequality which
should be written at the top of a sheet of paper. Each student starts a word problem to
represent the equation or inequality, and the other group members will each add to the
word problem. The fourth person will write the question. Give the students about 30
seconds and when time is called pass their story to the person on the right. Pass the paper
to the original writer, who will check to make sure the number sentence and the word
problem match. The original writer will complete or correct any parts that he/she feels do
not match. Allow time for the last student to provide feedback with revision suggestions
to the other group members. Students‘ word problems can be passed to the teacher for use
on an assessment.



Grade 8 MathematicsUnit 1Rational Numbers, Measures, and Models                               4
                       Louisiana Comprehensive Curriculum, Revised 2008


Activity 5: Missing (GLE: 6)

Materials List: math learning log, Missing BLM, paper, pencil

Lead a class discussion about problem solving strategies that the students have used.
Have students make a list of basic steps involved in problem solving by responding in
their math learning log (view literacy strategy descriptions).When explaining the basic
steps involved in problem solving, make sure the students work through the entire
process. This a good opportunity for the teacher to provide academic feedback to the
students on methods used in problem solving. Go over how important it is to first read the
entire problem before following the steps of problem solving that they will follow.
Distribute the Missing BLM. Make sure the students: a) understand the problem; b) make
a plan, sketch or diagram of the problem; c) carry out the plan (do the computation); and
d) determine that the solution makes sense. Discuss the different problem solving
strategies. Put a situation like one of the problems on the Missing BLM on the overhead
and have students write a plan for solving the problem. Ask, ―Is there more than one way
to solve the problem?‖

Have students use their math learning logs a second time during this lesson and make any
additions or corrections to their initial entry about problem solving procedures.


Activity 6: London, Paris, Rome or . . .? (GLE: 36)

Materials List: compass, protractor, paper, pencil, Practice Reading Circle Graphs BLM

Give students the following data of vacation travelers during the month of July. There
were 1500 travelers that flew out of New Orleans, LA to cities outside the country during
the week of July 21, 2004. 25% of these travelers flew to London, 28% flew to Rome,
25% flew to Paris, 11% flew to Madrid and 11% flew to other places.

Have the students create a circle graph to display the given data. Challenge the students
to give the fraction and decimal equivalent for each city‘s visitors and determine a
reasonable estimate in fraction form for the ratio of visitors that went to Rome, London
and Paris to the total number of travelers. Have the students prove why the ratio they
wrote for the combined travelers is reasonable. Ask students to determine the total
number of vacationers to each city and to set up proportions to justify their thinking. This
is the first time they are required to use the central angle as part of the circle graph. In
seventh grade the students used fractional divisions without determining the degree
measure. Ask the students to determine the number of degrees in 25% of a circle (90
degrees). Have student(s) explain the method of determining the correct angle measure.
Then, have students set up proportions to determine the measures of the central angles to
use in the circle graph. When assessing student progress on these circle graphs, make sure
students are determining the correct degree measure of the fractional part of the circle
they are working with. Distribute the Practice Reading Circle Graphs BLM to assess
student understanding of the information found in the circle graph.



Grade 8 MathematicsUnit 1Rational Numbers, Measures, and Models                         5
                       Louisiana Comprehensive Curriculum, Revised 2008


Activity 7: Bull’s Eye (GLEs: 1, 3)

Materials List: Bull‘s Eye Chart BLM, Bull‘s Eye Target BLM, calculators

The Bull‘s Eye Chart BLM and Bull‘s Eye Target BLM are used for this activity. Give
students 5 minutes to complete the estimation column of the Bull‘s Eye Chart. Then, ask
the students to find the exact answers and fill in exact column (calculators can be used for
the exact and the estimated/exact columns). Have students calculate the values for the
estimated/exact answer column which provide ratios of the estimates to the exact
answers. Have students use the Bull‘s Eye Chart BLM to determine the points earned
based on the difference of their estimate and exact answer. Then they will write this
difference in the Points Scored column. Lead a discussion as to why the best answers are
those closest to one.


Activity 8: How good were my estimates? (GLE: 3, 36)

Materials List: Bull‘s Eye Chart BLM completed by students in Activity 7, pencil, paper,
compass, protractor

Using the data from the Bull‘s Eye Chart BLM, have the students determine the percent
of 10 point, 5 point, 2 point and 1 point answers they have given. Instruct students to
create a circle graph based on the fraction of their estimates that resulted in each of the
point values. Ask students to explain to their partners or group members what the data
shows them about their estimated answers.


Activity 9: How Much . . . About? (GLEs: 3, 6)

Materials List: How Much . . . About? BLM, paper, pencil

Provide students with several advertisements for sales or the How Much . . .About?
BLM. Have students determine the approximate final cost of several items by estimating
and discuss how to determine the final cost by using the fraction off versus using the
                                      1
fraction remaining. For example, at   4   -off sale, students could determine how much the
                                                                                         3
discount is and then subtract from the original price. They could also determine that 4
                                                                     3
of the original price still has to be paid, and thus they could find 4 of the original price.
Have students determine the final total price by including the calculation of any taxes.
Discuss strategies for determining the presale price of several ―on-sale‖ amounts, and
show them in examples. For example, if the sale price of an item is $40 and this reflects a
20% discount, have students determine the original price.




Grade 8 MathematicsUnit 1Rational Numbers, Measures, and Models                             6
                       Louisiana Comprehensive Curriculum, Revised 2008


Activity 10: Order (GLE: 1, 6)

Materials List: Order Cut Apart Cards BLM, Order Recording Sheet BLM, paper, pencil

Provide pairs of students with a stack of cards with fractions using the Order Cut Apart
Cards BLM as master. (More fractional numbers than provided on the sheet may be
needed). Have Student One select a starting point by selecting a card from the deck and
writing the number on the card on Order Recording Sheet BLM in the first blank. Next,
have Student Two indicate the solution to the equation by selecting a second card and
writing this number after the equal sign on the activity sheet. Student One will then
determine at least three numbers and operations that will result in the solution given.
Student One will write these steps on the activity sheet as an equation, using a different
function each time. The object is for the students to write a correct equation by
                                                  1
completing the missing terms. For example, if         is the starting point, and the ending
                                                  4
                       1                               1              3
point or solution is 1 4 then the student might add , divide by          and multiply by 5 to
                                                       2              4
arrive at the solution. Repeat this activity several times, having the students change roles.
Later have students use at least one negative number in their equation. Students should
record their equations on the activity sheet. Have students explain their strategy to their
partner.


Activity 11: How Much did I Start With? (GLE: 6)

Materials List: paper, pencil

Provide the students with the following situation:
       James was given a large sum of money by his mom for his 8th grade field trip.
       James counted the money left in his wallet on the trip home, and he had $12.00.
       He could not remember how much money his mom had given him, so he started
       going through what he had spent. He first spent 10% of his money on breakfast.
       He then remembered that he spent 1 of what he had left after breakfast on a
                                           2
       watch. James spent $6 for the cost of admission to the museum at the end of the
       day.
Determine how much money James‘ mom gave him for the field trip. Explain the method
you used.
                                                                                1
Solution: Working backwards, add the $12 and the $6 to get $18. This $18 is 2 of what
he had left after breakfast so he had $36 after breakfast. If the $36 was 90% of what he
started with, then 10% is $4 and he therefore started with $40.




Grade 8 MathematicsUnit 1Rational Numbers, Measures, and Models                          7
                       Louisiana Comprehensive Curriculum, Revised 2008


Activity 12: My Dream House (GLE: 6)

Materials List: Home Buyers Guide for students, My Dream House BLM, paper, pencil

Begin this lesson with the literacy strategy student questions for purposeful learning
(SQPL) (view literacy strategy descriptions) by writing the statement ―I can buy the
house of my dreams if I make $45,000/year.‖ This strategy is used to encourage the
students to generate questions that they would need to answer to verify the statement.
Have students in groups of four, and brainstorm different questions that they might need
to answer to determine whether this statement is true or false. Have each group of
students highlight 2 or 3 questions that their group has come up with for use with whole
class discussion. Write these questions on a sheet of newsprint for use as closure when
the students have completed this activity. These questions might include the following:
What kind of job pays $45,000? How much would I have to make an hour to make
$45,000 a year? How many hours a week must I work? If the questions you want them
to discover are not on their list, you might take an opportunity to put your own question
on the list.

Tell the students that they will now do a project in mathematics that will help them
answer the questions that they have about how much money it will cost to buy a house.
Distribute the My Dream House BLM and provide students with Home Buyers Guides or
the local real estate guide found at a local grocery store. Have the students look through
the Home Buyers‟ Guide and find a house that they would like to purchase. Students
should calculate the payback of a 30 year, 6.5% simple interest loan and determine the
monthly house payment and enter the information on the BLM. Have students determine
what their annual salaries must be for them to be able to afford their dream houses if the
monthly note cannot exceed 25% of their monthly salaries. If it becomes rather difficult
to assess 30 different houses, you might want to have the students come to a consensus
on 5 of the houses and do the activity with only these five.

Once the activity is completed, have the students reread the list of questions that were
generated at the beginning of the activity. As a class, discuss whether or not these
questions were answered as they completed the activity. Have the students write the ways
the SQPL strategy helped them with the problem solving involved in the Dream House
activity in their math learning logs.




Grade 8 MathematicsUnit 1Rational Numbers, Measures, and Models                           8
                      Louisiana Comprehensive Curriculum, Revised 2008


                                  Sample Assessments


Performance assessments can be used to ascertain student achievement.


General Assessments

    Provide sales papers to the student. The student will prepare a back-to-school
     brochure for parents with sketches and prices showing school supplies that the
     student might need for school. The list will have at least 8 different items priced
     individually, and the brochure will include items that total close to $20.00. The
     student will include a price list and a total, using 8% sales tax for all of the items
     listed in the brochure. The student will make a list of inequality statements from
     prices in the brochure by comparing prices of the different items.
    Give the student a list of about fifteen rational numbers including fractions,
     decimals and percents, making sure that some of the values are equivalent (i.e.
      1
      4 and 25%). The students will make a number line and place all fifteen rational
     numbers along the number line in the correct position. To complete the
     assessment, the student will write at least 10 inequality statements using the
     symbols <, >, , and .
    Provide the student with an advertisement and a budget. The student will purchase
     as many items as possible and stay within the budget.
    Whenever possible, create extensions to an activity by increasing the difficulty or
     by asking ―what if‖ questions.
    The student will create a portfolio containing samples of the experiments and
     activities.


Activity-Specific Assessments

    Activity 2: The students will respond to the following situation in his/her math
     learning log: Ms. Fields put the problem 5   3  3  2   2 10 on the board.
                                                4      2             3


       Erica got an answer of 0 and Sammy got an answer of 4 1 . Explain which of the
                                                                 2
       students is correct and justify for your answer using correct mathematical
       language.
              Solution: Sammy is correct. Erica performed the order of operations
              within the parentheses incorrectly. She divided nine by six and got one
              and a half.

    Activity 3: The student will discuss strategies that could be used by Player 1 when
       choosing the target number from the four cards turned up. The teacher will ask
       group members to choose one strategy that they think is best and share it with the
       class. The GLE indicates that the student will use the order of operations to solve


Grade 8 MathematicsUnit 1Rational Numbers, Measures, and Models                        9
                      Louisiana Comprehensive Curriculum, Revised 2008


      problems. Make sure that the students express how the order of operations can
      help them get closer to their ‗target‘ answers.

    Activity 7: The student will respond to the following prompt: Using your data
      from the Bull‘s Eye activity, explain when the estimated answer gave a value
      greater than one on the Bull‘s Eye and why.
              Solution: When the estimated answer is greater than the exact answer, the
              value was greater than one because the estimate was divided by the exact
              answer. The students should give evidence in their response that there is
              an understanding of how the size of rational numbers affects the outcome
              of mathematical operations.

    Activity 8: Have the students create a circle graph from their data as an
      assessment of their understanding of circle graphs and central angle
      measurements.

    Activity 11: The student will prepare a chart and an explanation to the problem
      below:

      A local store has a sale rack for clearance merchandise. The sign on the rack says,
      “Marked down an additional 20% each day!” James has been thinking about
      buying a jacket that costs $100. The clerk tells him it will be moved to the sale
      rack tomorrow. James is happy about this and decides he‘ll go back to the store in
      five days when the jacket will be free. When he gets to the store five days later, he
      sees that the jacket is not actually free. What price is really marked on the jacket?
      Why did James think the jacket would be free? Explain your thinking.

               Solution: Day 1 - $80; Day 2 - $64; Day 3 - $51.20; Day 4 - $40.96; Day
               5 $32.77 (this is a rounded answer). On day 5 the jacket is marked $32.77.
               James thought that 20% of $100 would be subtracted each day thus
               leaving a balance of 0 on day 5.




Grade 8 MathematicsUnit 1Rational Numbers, Measures, and Models                       10
                       Louisiana Comprehensive Curriculum, Revised 2008


                                       Grade 8
                                    Mathematics
                        Unit 2: Rates, Ratios, and Proportions


Time Frame: Approximately four weeks


Unit Description

This unit focuses on proportional relationships and solutions of problems involving rates,
ratios and percentages. This level of proficiency includes work with similar triangles and
the lengths of corresponding sides. There is some exploration of combinations and
permutations in this unit.


Student Understandings

Students have a full understanding of percents, including those greater than 100 and less
than 1, as well as percent of increase and decrease. They will find rates and apply them in
solving real-life problems involving proportions, including those involving fractions and
integers. Students will use lists, diagrams and tables to solve problems involving
combinations and permutations.


Guiding Questions

       1. Can students set up and solve percentage problems including those with
          percentages less than 1% and greater than 100%?
       2. Can students set up and solve percent of change problems (% increase, %
          decrease)?
       3. Can students set up and solve proportions representing real-life problems,
          including those with fractions, decimals, and integers?
       4. Can students interpret, model, set up, and solve proportions linking the
          measures of sides of similar triangles?
       5. Can students apply concepts of combinations and permutations and identify
          when order is important?




Grade 8 MathematicsUnit 2Rates, Ratios, and Proportions                               11
                      Louisiana Comprehensive Curriculum, Revised 2008


Unit 2 Grade-Level Expectations (GLEs)
GLE # GLE Text and Benchmarks
Number and Number Relations
7.       Use proportional reasoning to model and solve real-life problems (N-8-M)
8.       Solve real-life problems involving percentages, including percentages less than
         1 or greater than 100 (N-8-M) (N-5-M)
9.       Find unit/cost rates and apply them in real-life problems (N-8-M) (N-5-M) (A-
         5-M)
Geometry
29.      Solve problems involving lengths of sides of similar triangles (G-5-M) (A-5-
         M)
Data Analysis, Probability, and Discrete Math
39.      Analyze and make predictions from discovered data patterns (D-2-M)
42.      Use lists, tree diagrams, and tables to apply the concept of permutations to
         represent an ordering with and without replacement (D-4-M)
43.      Use lists and tables to apply the concept of combinations to represent the
         number of possible ways a set of objects can be selected from a group (D-4-M)


                                   Sample Activities


Activity 1: Representing Percents (GLE: 8)

Materials List: Percent Grid BLM, grid paper for students, Practice with Percents BLM,
paper, pencil, Internet access

Provide students with the Percent Grid BLM which contains10 x 10 grids that represent
100%. Have students shade in different percents such as 50%, 10%, 12 1 % , 150%, 2.5%,
                                                                        2
      1     1
75%, 2 % , 4 % . Do not say the percentages aloud. Have the students shade them from
the written representation as the discussion that surfaces between the 50% and 1 % can
                                                                               2
uncover some misconceptions.

Circulate around the room to check for understanding as students shade these different
percents. Have students share their ideas with the class and discuss different
representations.

Distribute the Practice with Percent BLM, and have students work individually to solve
and illustrate their solutions of these situations.

Have students work with a partner and verify their solutions to these problems. Discuss
results as a class.




Grade 8 MathematicsUnit 2Rates, Ratios, and Proportions                                12
                       Louisiana Comprehensive Curriculum, Revised 2008


Activity 2: How Much Improvement? (GLE: 8)

Materials List: How Much Improvement? BLM, paper, pencil, calculator

Distribute Percent Grid BLM (Activity 1) and have the students label the first ten by ten
grid as ‗the cost of a jacket is $50‘. Discuss that this $50 on the grid represents 100% of
the regular cost of the jacket. Ask the students to shade the part of the grid that would
represent a 25% decrease of the cost of the jacket. Get student feedback on what part of
the grid represents the price they would have to pay (75%). Using the grid, have the
students determine the cost of the jacket if the price were decreased 25% ($37.50).
Students may determine that if the entire grid represents $50 then each single cell would
represent $0.50. If the jacket is on sale for 75% or 75 cells worth $0.50 each, the sale
price of the jacket would be 75 x $0.50 or $37.50. Discuss the idea that 75% of $50
means to multiply .75 x 50 to get the price of the jacket as the students begin to
conceptually understand the discount.

Tell the students that on the same day a chair was marked as $120.00 which was 20% off
the regular price. Have the students shade the 20% off
                                                          discount cost 100  % discount
of the second grid on the Percent Grid BLM. Discuss                       
the value of the 20% if the 80% has a value of $120 and original cost           100%
how they might determine this. Eighty cells have a        120 80
value of $120 so each cell, which is 1% of the cost, has       
                                                            x    100
a value of $1.50. The original price of the chair would
                                                          80 x  12000
have been $150. Students should be ready to set up
proportions to solve percent problems as shown on the     x  150
right.                                                    original cos t  $150

Have the students discuss when percent of increase might be used (pay raises, markup of
retail prices over wholesale). Tell the students that the next grid of the Percent Grid BLM
will represent $13.50 which is the price that the store pays for a new pair of jeans.
Suppose the store will increase the price by 150% before the sale price is marked on the
jeans. Challenge the students to use the grids and determine the retail cost of the jeans
(price increase is $20.25 making the retail price $33.75). Have students discuss their
methods for finding the solution. One method might be as follows: If the first grid
represents $13.50, the second grid would also represent $13.50 and 100% more than the
price paid by the store. The third grid would
                                                           discount cost 100  discount %
represent $6.75 or ½ of the $13.50. $13.50 + 13.50                        
+ 6.75 = $33.75.                                           original cost         100%
                                                           48     x
Tell the students that this time $60 is the original           
                                                           60 100
price of a pair of shoes. The shoes are discounted to
                                                           60 x  4800
$48. Give students time to determine a method of
finding the percent of discount (at right).                x  80%
                                                          therefore 100 - 80  percent discount
                                                          the percent of discount is 20%



Grade 8 MathematicsUnit 2Rates, Ratios, and Proportions                                  13
                        Louisiana Comprehensive Curriculum, Revised 2008


Discuss the percent of change ratio as a comparison of the change in quantity to the
                                        amount of change
original amount. percent of change                       . Discuss the concept of percent
                                         original amount
of increase as being an amount more than the original and the percent of decrease as
being an amount less then the original.
                                                                            amount of change
Provide students with the How Much                    percent of change 
Improvement? BLM. Have students calculate                                    original amount
the percent increase/decrease for each                         x
                                                      1.5% 
student‘s scores on the pre and post tests.                   85
Guide a discussion that includes a variety of                   x
                                                      0.015 
problems resulting from the data (e.g., what                   85
percentage of the students earned a C?). Ask          x  1.275
questions such as the following: If Sam               85  1.275  86.275
scores 85% on one test and increases his score
                                                      86%
by 1 1 % on the next test, what is the score on
     2

the second test?

If Joanne scored 70% on the second test and this was a 2% decrease of her score on the
first test, what was her score on the first test?
(A 2% decrease means that 70 represents 98%                            amount of change
of her old score. Thus her first test was         percent of change 
                                                                        original amount
71.4% or 71%.)
                                                              x
                                                      2% 
Jack has an average of 83% after the first four              70
tests; he needs to have at least an 85% average                x
                                                      0.02 
after the fifth test. What is the lowest score he             70
can make on the fifth test? Have students             x  1.4
explain their thinking.                               70  1.4  71.4
                                                      71%
       Solution: An average of 83% on four
       tests gives a total of 332 points and an
       average of 85% on five tests will need a total of 425 points, leaving a score of at
       least 93% for Jack on the fifth test.


Activity 3: Real-life Percent Situations (GLE: 8)

Materials List: Percent Grid BLM, pencil, paper

Distribute a copy of the Percent Grid BLM to students. Give students the following
situations and have them use the grids to represent each one and solve using a proportion.
When discussing these situations, have the students indicate how their sketch of the
percent relates to the situation.
    a) Four hundred eighth grade students worked for a nursing home one Saturday and
        they received $200 for the yard work that was done. Joe was excited about getting


Grade 8 MathematicsUnit 2Rates, Ratios, and Proportions                                14
                       Louisiana Comprehensive Curriculum, Revised 2008


      his equal percent of the $200 the 8th graders collected. Show on a percent grid
      Joe‘s equal share and give the correct percent. (answer: Joe received ¼ % of the
      $200 or $.50. The grid would represent $200 and each 1 square would represent
      $2, so ¼ % would be represented by one-fourth of a square and ¼ of $2 is $.50.)
   b) Bill saved $600 from his summer earnings of $800. Show on a grid the percent of
      money that Bill saved and give the correct percent. (Bill saved 75% of his summer
      earnings. The grid would represent $800, each square representing $8 and 75%
      of this would be 75 squares times 8 or $600.)
   c) Sally saved $200 and Betty saved $300. Show on a percent grid the amount of
      money that Betty saved comparing her savings to Sally‘s savings. (Betty saved
      150% of what Sally saved. The grid would represent Sally‟s $200, each square
      representing $2, and it would take one entire grid and ½ of a second grid to
      represent Betty‟s $300.)

Once the students understand the meaning of percent, assign each pair of students a
percent and have them write a real-life scenario representing the use of this percent.
These scenarios will be used as the student pairs present their scenarios using a modified
questioning the author (QtA) (view literacy strategy descriptions). QtA is a strategy that
encourages students to interact with information read and to build understanding by
asking clarifying questions. The student pairs have authored their scenarios and will
present these to the class. As the authors of the scenario, the pair will be involved in a
collaborative process of building understanding with percent situations through reading
and explaining their situations and solutions to the class. Students should develop a
model of their percent situation and represent the situation mathematically. Once the
student pairs have developed their scenario, the pair answers questions from classmates
about their scenario and solution. The teacher strives to elicit students‘ thinking while
keeping them focused in their discussion. The pair will answer questions that the class
asks about their scenario and justify the mathematics involved in the solution.


Activity 4: Four’s a Winner (GLE: 8)

Materials List: Four‘s a Winner Game Card BLM, 2 paper clips per pair of students,
marker chips and/or two different colored markers, pencil, paper

Provide students with a sample game card as shown on the Four‘s a Winner Game Card
BLM. Distribute supplies to each pair of students. Each pair of students will need two
paper clips, two different color markers, Four‘s a Winner Game Card BLM, and paper
and pencil.

To play the game,
    Have the tallest student go first by placing one paper clip on a percent expression
       and the other paper clip on a number in the row below the expressions. For
       example, Student 1 places a paper clip on ―25% of‖ and a paper clip on ―160‖ in
       the bottom row of numbers. Student 1 should then mark his/her answer for 25%




Grade 8 MathematicsUnit 2Rates, Ratios, and Proportions                               15
                       Louisiana Comprehensive Curriculum, Revised 2008


       of 160 (40) either by placing a chip over the correct answer or by marking it with
       a colored marker.
      Next, instruct Student 2 to move either (only one) paper clip to create a new
       problem and find the answer on the game board. For example, Student 2 might
       move the paper clip on ―25% of‖ to ―50% decrease‖ and Student 2 would then
       place his marker on ―80.‖
      Continue play until one player gets four in a row, horizontally, vertically, or
       diagonally.
      Have students record the problems and answers on paper so that wins can be
       verified.


Activity 5: The Better Buy? (GLE: 9)

Materials List: The Better Buy BLM, Choose the Better Buy? BLM, pencils, paper, math
learning log, grocery ads (optional)

Begin this activity by putting a transparency of The Better Buy? BLM on the overhead.
Cover the bottom portion that gives group directions. Using a modified SQPL, (view
literacy strategy descriptions) have students independently write questions that this
statement (One potato chip costs $0.15.) might suggest to them. After about one minute,
have the students get into pairs, compare questions and write at least two of their
questions to post on the class list.

Once the class questions are posted, give the students ten minutes and have the pairs of
students determine method(s) of answering at least three of the class questions. Circulate
as students are answering their questions, and be sure that any misconceptions are
addressed before they begin independent work. Have the students answer the initial
question after they have completed work with their partners.

Next, provide students with Choose the Better Buy? BLM. Have students work
individually to find the unit rates to determine the better buy in each situation. Students
should verify results with a partner. Give opportunities for questions if students have a
problem that they do not agree upon.

Extend the activity by giving grocery ads from different stores carrying the same items to
each group of four students. Give students a list of items to purchase and have student
groups of four make projections about savings on groceries by shopping at store A versus
store B over a year. Have students present their findings to another group or the class.

Have students record in their math learning log (view literacy strategy descriptions) what
they understand about unit prices.




Grade 8 MathematicsUnit 2Rates, Ratios, and Proportions                                 16
                       Louisiana Comprehensive Curriculum, Revised 2008


Activity 6: Refreshing Dance (GLE: 9)

Materials List: Refreshing Dance BLM, pencils, paper

Have students work in groups of four to prepare a cost-per-student estimate for
refreshments at an 8th grade party. Distribute Refreshing Dance BLM. Have students
complete the chart and determine the total cost of refreshments for each student and the
total cost of the dance if they plan for 200 students.

Students will present their proposals and the answers to the questions to the class using a
modified professor know-it-all (view literacy strategy descriptions) strategy. Students
will answer questions about their proposal from the class. Using professor know-it-all,
the teacher will call on groups of students randomly to come to the front of the room and
provide ―expert‖ answers to questions from their peers about their proposal. The teacher
should remind the students to listen to the questions and to think carefully about the
answers received so that they can challenge or correct the professor know-it-alls if the
answers the ―experts‖ give are not correct or need elaboration and amending. Students
should be able to justify not only the cost of the refreshments but also the amount that
needs to be ordered.


Activity 7: My Future Salary (GLE: 8, 9, 39)

Materials List: grid paper for students, My Future Salary BLM, paper, pencil, Internet
access

Introduce SQPL (view literacy strategy descriptions) by posting the statement ―An
electrical engineer earns more money in one year than a person making minimum wage
earns working for 5 years.‖ Have students work in pairs to generate questions that they
would like to have answered about this statement. Have students share questions with the
class and make a class list of questions. Students must make sure that a question relating
to a comparison of job salaries is asked. Give students time to research the information
needed to answer the question. A site that has recent top salaries can be found at
http://money.cnn.com/2005/04/15/pf/college/starting_salaries/

The students can share their information with the class by using professor know-it-all
(view literacy strategy descriptions). The research group will go to the head of the class
and report their findings to the class and answer questions from the group about their
findings. Give other groups time to share their findings, also.

Ask the students why the minimum hourly wage is considered a unit rate (amount of
money paid per hour of work). Distribute the My Future Salary BLM and have students
make observations about what has happened to the minimum wage in the years since
1960. Lead a discussion with students about how the minimum wage has changed
through the years. Have students create a graph of the minimum wage from the
information in the chart and predict the minimum wage for the year 2010.



Grade 8 MathematicsUnit 2Rates, Ratios, and Proportions                                  17
                       Louisiana Comprehensive Curriculum, Revised 2008


Have the students calculate what a person working a minimum wage job working 40
hours per week made in 2003 and what that person would make using their prediction for
the year 2010. Discuss how the graph helps with making predictions.

The information on the My Future Salary BLM is also found on the following website:
http://www.workinglife.org/wiki/Wages+and+Benefits%3A+Value+of+the+Minimum+
Wage+%281960-Current%29. The web site http://www.bls.gov/bls/blswage.htm gives
current wages of jobs listed with the labor division of the U. S. government. The
Louisiana Board of Regents has an e-portal designed specifically for Louisiana students:
https://www.laeportal.com/main.aspx. This portal was designed to be used by eighth
grade students as they make a five year academic plan. There is a teacher section which
provides links to careers, salaries and other information that would be applicable to this
activity.


Activity 8: Similar Triangles (GLEs: 7, 29)

Materials List: 6 drinking straws for each pair of students, scissors, pencils, paper, math
learning log, ruler

Have students work in pairs to create an equilateral triangle using drinking straws for
sides. Ask students to explain how they know they have created an equilateral triangle.
(they have three straws the same length). Have them measure and record the side length.

Instruct students to make a second equilateral triangle with sides of different length than
those of triangle one. Have students measure with rulers the sides of their new triangle.
Ask them to determine a way to prove that the two triangles are similar using what they
have learned about proportions. Students should understand that the triangles are similar
because the sides are of proportionate lengths. Triangle one has sides twice as long as
triangle two, and the angles measure the same because they are equilateral triangles.
Equilateral triangles are also equiangular. Lead students to write a conjecture about the
relationship of proportionate sides and equal angles in two equilateral triangles. Ask
them if it seems possible that this relationship will hold true with other triangle types.

Next, have students construct or draw a triangle with all three sides of different lengths
(scalene). Have students label the triangle with the measure of each of the side lengths
and each angle measure. Instruct students to select one vertex of their new triangle and
label the vertex A. Have students extend the sides of the triangle from vertex A so that
            3
the side is the length of the
            2                                A

original side. Repeat this with the
other side from vertex A. Instruct
students to connect the two                                                 C


endpoints of the new sides for                                                                D

their triangle. Have students
                                         B
make some observations about

                                       E

Grade 8 MathematicsUnit 2Rates, Ratios, and Proportions                                    18
                       Louisiana Comprehensive Curriculum, Revised 2008


the two triangles that they have formed. Challenge students to use proportions to prove
that the two triangles are proportional.

Discuss how the angles of these two triangles are congruent but the side lengths are
proportionate. Tell the students that the symbol to show similarity is ‗‘. We call the two
triangles ‗similar triangles‘ because the angles are congruent and the side lengths are
proportionate.

Next, have them construct or draw a triangle using a ratio provided to them, perhaps a
         3
ratio of and determine if the same conjecture holds true for triangles with sides of
         4
different lengths. For example, if they create a triangle with side lengths of 3 inches, 4
inches, and 5 inches, a triangle with sides of 2.25 inches, 3 inches, and 3.75 inches would
meet the requirement. Once they have constructed the triangle, the students should set up
a proportion to verify proportionality. Be sure to look for and clear up any
misconceptions about using the correct angles when the figures are not oriented the same
way.

Have students record in their math learning log (view literacy strategy descriptions) what
they know about similar triangles.


Activity 9: Proportional Reasoning (GLEs: 7, 29)

Materials List: Proportional Reasoning BLM, meter sticks, objects to measure outside,
pencils, paper, calculator

This activity allows students to apply the concept of similar triangles. Distribute the
Proportional Reasoning BLM. Students will calculate the height of various objects by
measuring the object‘s shadow and the shadow of a meter stick placed vertically on the
ground. Following the directions on the BLM, lead students to understand that they can
solve the problems by creating a proportion between the corresponding parts of the right
triangle formed by the object and its shadow with the right triangle formed by the meter
stick and its shadow. Have students sketch and label dimensions of the corresponding
parts of the similar triangles formed with these objects.




Grade 8 MathematicsUnit 2Rates, Ratios, and Proportions                                 19
                        Louisiana Comprehensive Curriculum, Revised 2008




                                                  x/1 = 25/2.5
                                                  2.5x = 25
                                                  x = 25/2.5
                                                  x = 10 meters
       object
     x meters




                                shadow

                                25 meters


 1 meter

           2.5 meters




Once the students have returned to the classroom, have different groups put their
proportions on the board and make observations. Students should be able to see that the
ratios found by the groups should be close to the same.


Activity 10: Scaling the Trail (GLE: 7)
                                                                              C             E

Materials List: Scaling the Trail BLM, pencils, paper, ruler                            D


Provide each student with a Scaling the Trail BLM. Have the students
find the length of the trail using the information given on the BLM.
Discuss segment notation ( AB ) so that students record information              A            B
accurately. Challenge the students to add another 1 1 miles to the trail
                                                      4
by extending the trail in any direction from point A so that the trail leads closest to point
C. This will require students to determine the length of the segment that needs to be
added to the diagram in inches.


Activity 11: How Many Outfits are on Sale? (GLEs: 43)

Materials List: How many Outfits are on Sale? BLM, paper, pencil

Provide groups of four students with a copy of the How Many Outfits are on Sale? BLM,
a one page clothing sales brochure which depicts pants, shoes and shirts. Have the
students sketch a diagram to illustrate the different outfits that could be made from the
items on the brochure. The outfits should include pants, shirt, and shoes. Ask students to
determine which of these outfits would cost the least. Have students write a summary
showing all of their mathematical thinking and give the total number of possible
combinations that could be made from the items listed.




Grade 8 MathematicsUnit 2Rates, Ratios, and Proportions                                       20
                        Louisiana Comprehensive Curriculum, Revised 2008


Have groups prepare a presentation to use professor know-it all (view literacy strategy
descriptions) to justify their thinking about the possible combinations and which of the
combinations could be purchased for the least amount of money.


Activity 12: Combination or Permutation? (GLEs: 42, 43)

Materials List: index cards or slips of paper (one per student), paper, pencil

Have student groups of six write their names on a slip of paper or an index card. Have
students determine the total number of combinations of 3 students by making a list or
diagram. If students need help, let them use letters of their first names (if all are different)
or use A, B, C, D, E, and F to represent the six students. Make sure the students
understand that combinations involve an arrangement or listing where order is not
important (i.e., ABC is the same as BCA as these would be the same group of people
even though the order in which they are listed is different).

Show students how to make an organized list by modeling a tree diagram graphic
organizer (view literacy strategy descriptions). The student will make an organized list
with a tree diagram and, when complete, the last row will give the student the
arrangements that will lead to the answer. After giving students ample time to make the
list of combinations, lead a class discussion in which the class agrees on the list of
combinations that can be made. Then, have each student determine the ratio of the
number of times his/her name appears in a combination compared to the total number of
combinations. How would this ratio change if 4 of the 6 students were selected? Have
students discuss the change and any conjectures that can be made at this time.

Next, tell students that these same six names are now in a race, which changes the
problem to a permutation because order is important (i.e., ABC means A came in first,
but BCA, means B came in first). Ask how many arrangements there are for 1st, 2nd, or 3rd
place. Ask students to determine whether the number (120) is the same as it was in the
previous problem (60) and to explain why or why not? Ask students to determine the
number of permutations if 4 people were to be recognized for finishing 1st, 2nd, 3rd or 4th.

Have students discuss the difference in the two concepts and discuss when order is
important. Have students determine the ratio and the percent of times they would be in 1st
place, 2nd place, and 3rd place out of the total number of possible outcomes and write the
solution in their math learning log (view literacy strategy descriptions).




Grade 8 MathematicsUnit 2Rates, Ratios, and Proportions                                    21
                       Louisiana Comprehensive Curriculum, Revised 2008


Activity 13: Tour Cost (GLEs: 7, 42)

Materials List: Tour Cost! BLM for each group of four students, paper, pencil

Distribute Tour Cost! BLM and have students work with a partner to answer the
questions. The students should use a tree diagram graphic organizer (view literacy
strategy descriptions) to determine the possible routes that could be selected for the tour.
Have students prepare a presentation indicating which three-city tour would be most cost-
efficient for travel expenses.

As an extension, have students research costs of plane fare, bus fare and train fare.
Determine which of the methods of transportation will be acceptable to the sponsors.



                                   Sample Assessments


Performance assessments can be used to ascertain student achievement. For example:


General Assessments:

    The teacher will provide groups of four students with different real-life situations
     involving percents and unit rates. The students will prepare a presentation to
     explain their method of solution to the class. The teacher will evaluate the work of
     the group based on the use of a cooperative group rubric similar to one found at
     http://www.phschool.com/professional_development/assessment/rub_coop_proce
     ss.html. Some possible real-life situations might include the following:
     1. George bought six identical pairs of jeans for a total of $240 not including
         8.75% tax. How much would four pairs of jeans cost? How much would 20
         pair cost? What would the tax be on the six pairs? What would be the cost of
         one pair of jeans plus tax?
     2. At the end of 21 days, a company received 270 complaints. How many
         complaints can they expect during the next week? The next eight weeks? In
         one day? The company must show a 10% decrease in the number of
         complaints during the next 21 days. If it is to be successful, how many
         complaints will be acceptable?
     3. Sam worked one week to save a total of $156. If Sam worked a total of 24
         hours during the week, how many hours would he have to work to make a
         total of $1500? If the minimum wage is $6.65/hour, was Sam‘s pay minimum
         wage? What percent of increase or decrease would Sam need to be paid
         exactly $6.65/hour?
    The teacher will provide the student with the following facts about water usage.
     The student will make a booklet of word problems involving the facts about water
     usage. The students will incorporate the information to write six word problems


Grade 8 MathematicsUnit 2Rates, Ratios, and Proportions                                22
                      Louisiana Comprehensive Curriculum, Revised 2008


       using percentages, ratios and rates and solve them stating a justification for the
       answers given.
              Americans use a great deal of water. Below are some interesting
              facts about water usage: a) Each person, on average, uses 168
              gallons/day. b) It takes two gallons to brush your teeth with the
              water running. c) It takes twenty gallons to wash dishes by hand. d)
              It takes ten gallons to wash dishes in the dishwasher. e) It takes
              thirty gallons to take a 10 minute shower. f) Each household uses
              about 107,000 gallons per year.
    The student will research the cost of materials needed to remodel a bedroom. The
       student will be given the dimensions of a fictitious bedroom and an amount that
       can be spent on the remodeling. The student will prepare a paper for his/her
       parents, showing the remodeling desired and the prices of the different materials.
       The student will determine the approximate amounts of the different materials to
       be used and give the unit costs. The student will factor in the cost of the tax for
       the renovations to the bedroom and not go over his/her budgeted amount.
      The student will determine the height of a known landmark (e.g., water tower)
       using similar triangles and proportional reasoning.
      Provide the student with his/her actual grades for two tests and the student will
       calculate the percent of increase/decrease from the first test to the second test.
      Provide the student with a list of products, and the student will determine the best
       buy based on unit cost.
      The student will determine the scale ratio when given two similar triangles.
      Whenever possible, create extensions to an activity by increasing the difficulty or
       by asking ―what if‖ questions.
      The student will create a portfolio containing samples of experiments and
       activities.


Activity-Specific Assessments

      Activity 4: The student will prepare a presentation explaining preferred strategies
       for playing the Four‘s a Winner! game to share with the class. Students should
       explain procedures for determining percentages as strategies are discussed.

      Activity 9: The student will make a sketch of a hiking trail that has five straight
       segments with endpoints named A (beginning) through E (ending). The student
       will state a scale to use which is different than the scale used in the activity. The
       teacher will give the student specifications for the trail such as segment AB
       measures 1 1 miles and segment BC measures 3 miles. The student will prove that
                    2

       the proportions used to determine length of the segments in the sketch match the
       scale chosen.




Grade 8 MathematicsUnit 2Rates, Ratios, and Proportions                                   23
                       Louisiana Comprehensive Curriculum, Revised 2008


      Activity 10: The student will use the dietary information from fast-food
       restaurants to write at least ten unit rates and explain the unit rate with appropriate
       labels (e.g., calories per ounce of meat or grams of fat per ounce of meat or fries).

      Activity 12: The student will explain the similarities and differences between
       permutations and combinations and how order affects the solution as an entry in a
       math journal.




Grade 8 MathematicsUnit 2Rates, Ratios, and Proportions                                  24
                      Louisiana Comprehensive Curriculum, Revised 2008


                                     Grade 8
                                    Mathematics
                         Unit 3: Geometry and Measurement


Time Frame: Approximately four weeks


Unit Description

The content of this unit focuses on the properties of transformations on the coordinate
grid; the relationships among angles formed by parallel lines; the use of nets to help
students visualize three dimensional solids; and applications of the Pythagorean Theorem
and its converse.


Student Understandings

Students grasp the meaning of congruence and measurement. They can apply
transformations and identify properties that remain the same as figures undergo
transformations in the plane. Students see the links between planar nets and their
corresponding 3-D figures and can explain relationships between vertices, edges, and
faces of polyhedra. Students can provide one justification of the Pythagorean theorem
and its converse and apply both in real-life applications.


Guiding Questions

       1. Can students use transformations (reflections, translations, rotations) to match
          figures and note the properties of the figures that remain invariant under
          transformations?
       2. Can students define and apply the terms measure, distance, bisector, angle
          bisector, and perpendicular bisector appropriately and use them in discussing
          figures synthetically and with reference to coordinates as well?
       3. Can students draw and use planar nets to construct polyhedra, noting the
          relationships of sides, edges, and vertices?
       4. Can students discuss similar and congruent figures and make and interpret
          scale drawings of figures?
       5. Can students state and apply the Pythagorean theorem and its converse in
          finding the lengths of missing sides of right triangles and showing triangles
          are right respectively?
       6. Can students use the coordinate plane to represent models of real-life
          problems?




Grade 8 MathematicsUnit 3Geometry and Measurement                                     25
                       Louisiana Comprehensive Curriculum, Revised 2008


Unit 3 Grade-Level Expectations (GLEs)

GLE # GLE Text and Benchmarks
Geometry
23.     Define and apply the terms measure, distance, midpoint, bisect, bisector, and
        perpendicular bisector (G-2-M)
24.     Demonstrate conceptual and practical understanding of symmetry, similarity,
        and congruence and identify similar and congruent figures (G-2-M)
25.     Predict, draw, and discuss the resulting changes in lengths, orientation, angle
        measures, and coordinates when figures are translated, reflected across
        horizontal or vertical lines, and rotated on a grid (G-3-M) (G-6-M)
26.     Predict, draw, and discuss the resulting changes in lengths, orientation, and
        angle measures that occur in figures under a similarity transformation (dilation)
        (G-3-M) (G-6-M)
27.     Construct polyhedra using 2-dimensional patterns (nets) (G-4-M)
28.     Apply concepts, properties, and relationships of adjacent, corresponding,
        vertical, alternate interior, complementary, and supplementary angles (G-5-M)
30.     Construct, interpret, and use scale drawings in real-life situations (G-5-M) (M-
        6-M) (N-8-M)
31.     Use area to justify the Pythagorean theorem and apply the Pythagorean
        theorem and its converse in real-life problems (G-5-M) (G-7-M)
33.     Graph solutions to real-life problems on the coordinate plane (G-6-M)


                                    Sample Activities


Activity 1: Transformations! (GLEs: 23, 24, 25)

Materials list: One Inch Grid BLM, Index Card Shapes BLM, ¼ Inch Grid BLM,
Transformations BLM, Transformation Review BLM, pencils, paper, scissors, ruler,
unlined 3‖ x 5‖ index cards, large sheet of newsprint

Have students work in cooperative groups of 4 for this activity. Give each student in the
group a copy of One Inch Grid BLM. Have students cut off the edges around their grid
paper and tape the four sheets together to form a large coordinate grid. Tell students to
draw the x and y axes in the center of the large coordinate plane. Each sheet will represent
one quadrant of the coordinate plane.

Have students label the origin. Ask them to label both the x- and y-axes, indicating the
locations of –10 to 10 on each axis.

Distribute four 3‖ x 5‖ index cards to each group. Make sure the students
have assigned tasks as they prepare these cards. Have students follow the    A                  B
steps listed below, the results of which are shown on the Index Card
Shapes BLM:                                                                  C                  D
1. Index card #1 - Label the vertices of the index card with A, B, C, and D.


Grade 8 MathematicsUnit 3Geometry and Measurement                                        26
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2. Index card #2 - Mark the midpoint of one of the 3‖ sides. Draw segments
   connecting this midpoint to each of the vertices on the opposite side. Cut out the
   isosceles triangle that is formed. Label the vertices of the triangle E, F, and G.
3. Index card #3 - Put points on one of the 5‖ sides at 2‖ and 4‖ (i.e., 2 inches from one
   vertex and 1 inch from the other vertex). The segment between these two
   points forms the top of a trapezoid. Connect these points to the vertices on
   the opposite side. Cut out the trapezoid. Label the vertices of the trapezoid
   K, L, M, N.
4. Index card #4 – Measure 2 inches along one of the 5‖ sides and mark a              2 in
   point. Connect this point to the vertex on the opposite side to form an
   isosceles right triangle. Cut out this triangle. Label the vertices of the
   triangle formed H, R, J.

Post a large sheet of newsprint on the wall for the new vocabulary used. As each new
geometry term is discussed, have a student add the word to the word wall poster.

Distribute Transformations BLM and ¼ Inch Grid BLM
    Have students place the rectangle in the first quadrant of the ¼ Inch Grid BLM
       with vertices A and B on the coordinates given on the table on the
       Transformations BLM. Record the coordinates of all four vertices of the rectangle
       in its original position in column one of the table.
    Have the students translate the rectangle up (or down) and right (or left), making
       sure to move the rectangle to the position that is given for vertex B and then
       record the new coordinates in column two.
    Have students return the rectangle to its original location and record coordinates
       of each vertex after a 90 clockwise rotation. Discuss rotational symmetry as
       students begin to rotate their shapes. If students
       have difficulty visualizing what to do, have them            8




       use a small piece of tape to hold the rectangle on           6
                                                                              B                     C




       the grid. Students can place a sheet of patty paper                               original
                                                                    4

       or a transparency sheet over the grid and trace the                    A                     D



       x and y axes and the rectangle, being sure to label          2




       the coordinates on the copy. While holding the                    tracing paper
       transparency at the origin with their pencil points,
                                             -10         -5                                    5        10




       students can rotate the copy until the y-axis ends           -2




       up on the x-axis. This will result in a 90 degree            -4
                                                                                         rotation
       rotation. Have students discuss the new
       coordinates and identify the quadrant in which the           -6




       rotated rectangle lies.                                      -8



    Have students return the rectangle to its original
       location and then perform a reflection of the rectangle across the x-axis. Be sure
       to discuss line of symmetry as the rectangle is reflected. Model lifting the
       rectangle from the plane and flipping the rectangle over the x-axis, if needed.
       Have students record coordinates of the four vertices.
    Have students return the rectangle to its original position, perform a reflection
       across the y-axis, and then record the new coordinates.


Grade 8 MathematicsUnit 3Geometry and Measurement                                                      27
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      Have the students complete the same actions using their trapezoid, right triangle,
       and isosceles triangle, recording all of the new coordinates on the chart. Remind
       them always to return their shapes to the original position before making a
       transformation.

After the class has had time to complete the transformations of all four shapes, have the
groups make some conjectures about how they might be able to determine the positions
of polygons after a transformation from the information in the chart. Have the groups
share their conjectures with the class by using the professor know-it all (view literacy
strategy descriptions). The group that is sharing conjectures will be selected by the
teacher; therefore, all groups should be ready to go first. The group will go to the front of
the class, and using its conjectures, justify its thinking and answer questions from the
class about one of its conjectures. The teacher will then select a second group to share
another conjecture and continue until all conjectures and thinking are clearly understood
by the class.

Have the students use the Transformation Review BLM as a graphic organizer (view
literacy strategy descriptions) to guide them as they review the results of the different
transformations. Go through the example as a class. Allow students to discuss the answer
with a partner. Students should write that the result of reflecting a polygon across the y-
axis is that the x-coordinates are opposites of the originals and the y-coordinates stay the
same. The BLM gives them either the initial position with the transformation used or the
result of a transformation, and the student should give the other. As a result, some bridges
have more than one solution. Problem 4 presents a new situation for students.


Activity 2: Dilations (GLEs: 24, 26)

Materials list: Dilations BLM, Quadrant I Grid BLM, protractor, pencil, paper, ruler

Discuss dilations as another transformation. Ask if anyone has an idea about what a
dilation might be. Students will relate to the eye doctor dilating their eyes, but very few
of them relate a dilation to being an enlargement or a reduction.

Provide students with copies of the Quadrant I Grid BLM and the Dilations BLM. Have
students plot the vertices of the polygon given on the Dilations BLM on a coordinate grid
and then connect the points to form the polygon. Have students find the measure of each
angle, and find the distance from vertex to vertex (i.e., length) for each side. (Teacher
Note: Have students use rulers to measure lengths of sides which are not vertical or
horizontal.)                                              35



                                                                       A'                 B'


                                                          30
                                                                                                     C'




Next, have students use a ruler and draw a dotted         25




line from the origin and extend the line through          20




                                                               A            B
                                                                                     E'         D'




Vertex A of the polygon, continue to do this by
                                                          15
                                                                                      C




                                                          10



drawing lines from the origin through each of the
                                                                   E
                                                                                 D




                                                          5




other four vertices (see diagram).
                                                                                10             20         30   40   50




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Instruct the students to follow the steps on the Dilations BLM and then discuss their
conjectures about dilations and their effect upon angle measures, side lengths, and
coordinates of the original figure. Make sure the students understand that the dilation is
different from the reflections, translations and rotations because it is the only one that
produces similar figures – the other transformations produce congruent figures.

As a real-life connection, lead a discussion about when dilations that are used in everyday
life: using a projector to show an image to an entire class, enlarging a picture from the
image stored in a digital camera, projecting a video on large screens at sporting events, or
making a scale drawing of a large object.


Activity 3: The Bisection (GLE: 23)

Materials list: grid paper, ruler, pencil, math learning log

Provide students with the coordinates of the end points of a horizontal line segment and
have them draw the line segment on a coordinate system. Next, have students determine
the coordinates of the point that bisects the line segment. Discuss the length of the line
segment. Have the students determine how the coordinates can be used to determine the
length of the segment. After the midpoint is determined, discuss the coordinates of the
midpoint and how these coordinates relate to the coordinates for the endpoints of the
segment. Have students draw a line perpendicular to the line segment through the
midpoint, thus illustrating a perpendicular bisector. Repeat this activity with a vertical
line segment and line segments of positive or negative slope. Have students verbalize a
method for finding the coordinates of the midpoint of a segment if the endpoints are
known. (Average the x-coordinates and average the y-coordinates to find the x and y
coordinates of the midpoint).

As a real-life connection, have the students design a tile pattern for a rectangular room
with dimensions of 10 feet x 13 feet. The owner of the house has one request: a design
in the floor tiles should be in the center of the room. Students should use their
understanding of finding midpoint to determine where to place the design with the tile.

Students should record their method of finding the coordinates of the midpoint of a
segment in their math learning log (view literacy strategy descriptions). Remind the
students that their math learning log should reflect how they are thinking about the
procedure so that they can use their thinking later when reviewing the concept.




Grade 8 MathematicsUnit 3Geometry and Measurement                                          29
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Activity 4: Developing the Theorem (GLE: 31)

Materials list: grid paper, straight edge, scissors, paper, pencil

Have students draw a right triangle on grid paper with the two perpendicular sides having
lengths of 3 and 4 units. Have students draw a square using one of the legs of the triangle
as the side of the square (i.e., draw a 3 x 3 square). Repeat using the other leg as a side of
a square (i.e., draw a 4 x 4 square). Have students find the area of each square. Ask
students to determine a method for finding the area of the square of the hypotenuse of
their right triangle and to note how the areas of the three squares relate to one another.
(Some students may remember the Pythagorean theorem from previous years and use that
information to determine the length of the hypotenuse. Others may compare the length of
the hypotenuse to the units on the grid paper. The process used is not important, but all
students should eventually see that the hypotenuse length is 5 and the area of the
corresponding square is 25 square units.) Have students show that the sum of the areas of
the two smaller squares is the same as the area of the square formed by the hypotenuse by
cutting and rearranging the small squares inside the larger squares. Many texts and
websites show how to do this. Two websites which use animations to develop the
Pythagorean theorem are http://www.nadn.navy.mil/MathDept/mdm/pyth.html and
http://www.pbs.org/wgbh/nova/proof/puzzle/theorem.html.

Have students practice finding side lengths of various right triangles using the
Pythagorean theorem.

Using a modified version of reciprocal teaching (view literacy strategy descriptions),
have students brainstorm predictions as to whether or not the Pythagorean theorem will
work when finding side lengths of triangles that do not have a right angle. Reciprocal
teaching is used to move instruction from delivery to discovery. Have groups write their
predictions about the use of the theorem in these other triangles on paper. The prediction
is the first part of a reciprocal teaching lesson.

Assign the roles of questioner, clarifier, predictor and conjecturer to groups of four
students as they experiment with these other triangles. The ‗questioner‘ will begin by
asking the group to restate how it thinks its prediction relates to the triangles without
right angles. The clarifier should make sure that the answers that the questioner gets to
the questions are clear and understood by all group members. Have students draw a
triangle on the grid that is not a right triangle, and have the questioner ask the group
questions that will help it determine whether it gets the same results. The ‗clarifier‘ will
offer input, and the group will then work with the ‗conjecturer‘ to write its summary
statement. The predictor might make other predictions as other triangles are drawn to test
the conjectures made by the conjecturer. As a class, discuss conjectures that students
develop about the results of their explorations.




Grade 8 MathematicsUnit 3Geometry and Measurement                                        30
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Activity 5: The Theorem (GLE: 30, 31)

Materials list: The Theorem BLM, pencils, paper, calculators, graph paper

Provide students with the side lengths of several right triangles missing the length of one
of the sides. Discuss the use of the formula as it applies to the missing lengths in the
triangles. Extend this activity to include real-life situations that require students to find
the length of one of the sides of a right triangle with situations by distributing The
Theorem BLM. Have students verify their solutions to the BLM by comparing answers
with another student and discussing any results that differ.


Activity 6: How Big is This Room Anyway? (GLE: 30)

Materials list: meter sticks or tape measures, newsprint or other large paper for blueprint,
rulers, scissors, pencil, paper

Assign different groups of students the task of measuring the classroom dimensions.
Have the class determine a scale that would fit on a piece of newsprint or poster board,
and then have someone draw the room dimensions to scale on the poster. Tell students
that the class will make a classroom blueprint.

Divide students into groups of three to five. Assign each group a different object in the
classroom to measure (file cabinets, book shelves, trash can, etc. - remember only length
and width of the top of the object is needed for the blueprint). Have students convert
actual measurements using the scale measurements determined earlier. Instruct students
to measure, draw and cut out models from an index card. Have each student measure
his/her own desktop and make a scale model for the classroom blueprint. Remind
students to write their names on the desktop model. Ask, ―What is the actual area of your
desktop? What is the scale area of your desktop? What comparisons do you see as you
make observations of the areas of your room and desktop? List your observations.‖

Have groups submit their scale models of the classroom objects (not desks at this time)
for the blueprint. Discuss methods used to determine the measurements of the models,
and then glue the models in the correct position on the classroom blueprint. Have
students, one group at a time, place their desktop models on the classroom blueprint,
working so that those who sit in the center of the room can add their models first. Post
blueprints/scale models on the wall for all classes to compare.

Using the class scale model of the classroom, have students make predictions about
distance from various points in the room (i.e., If the distance from the teacher‟s desk to
the board is 5 inches on the scale model and the scale is 1 inch represents 4 feet, then the
that the actual distance is 20 feet.) Have student measure the actual distance(s) to check
for accuracy of the scale model of the classroom.




Grade 8 MathematicsUnit 3Geometry and Measurement                                        31
                       Louisiana Comprehensive Curriculum, Revised 2008


Activity 7: Netting the Cubes (GLE: 27)

Materials list: snap cubes, 2 cm Grid BLM, scissors, pencils, paper, math learning log

Provide pairs of students with 2 cm Grid BLM. Discuss as a class what they know about
a cube (i.e., 6 faces, 8 vertices, 12 edges). Challenge the students to cut different
connected patterns or nets to form a one unit cube from centimeter grid paper with no
gaps or overlaps (gaps occur when there is a part of the cube not covered, and overlaps
occur when the net folds on top of itself). Have students place patterns on the overhead,
and give the students time to check that all patterns cut will fold into a cube.

Have students take two cubes that the groups have formed and tape them together or use
snap cubes if available. Challenge the students to cut a net that will fold into a 2 unit x 1
unit x 1 unit prism. Have students determine the number of faces, edges and vertices of
this rectangular prism and compare the numbers to the 1 x 1 x 1 cube. Have them put nets
on the overhead, and allow students to challenge any that are questionable.


Have the students predict what a net will look like for a rectangular prism with
dimensions of 3 units x 1 unit x 1 unit. Discuss how the number of faces or surface area is
changing as cubes are added to the rectangular prism leaving two dimensions of 1 unit.
(As the length increases by one unit, the surface area increases by four square units.)
Challenge the students to cut a net for a rectangular prism with dimensions of 2 x 2 x 1
units. Discuss changes in the net for this rectangular prism as compared to the previously
constructed prism.

SPAWN (view literacy strategy descriptions) is used to have students reflect on content.
There are different prompts defined in SPAWN (S – students write with special powers to
change an aspect of the topic; P – students write solutions to problems posed; A –
students use unique viewpoint to explain; W – similar to S but the teacher presents the
aspects that have changed; and N – students anticipate what happens next.) Have students
record the SPAWN prompt in their math learning logs (view literacy strategy
descriptions) and give them time to respond. This prompt could be one using the ‗P‘ or
special Problem Solving prompt:
         We have been constructing nets to form different rectangular prisms. I want you
         to determine how many square units a net for a rectangular prism with dimensions
         of 3 x 2 x 2 cubes would have and make a sketch of this net in your math log with
         dimensions labeled. What do you think would have to be done to the net if cubes
         were added to the rectangular prism so that the dimensions became 3 x 3 x 2
         units?
Allow students time to share their responses with a partner or the class. Students should
listen for accuracy and logic.




Grade 8 MathematicsUnit 3Geometry and Measurement                                       32
                       Louisiana Comprehensive Curriculum, Revised 2008


Activity 8: The Net! (GLEs: 27, 31)

Materials list: boxes from home, rulers, Rectangular Prism BLM, Triangular Prism
BLM, Right-Triangular Prism BLM, tape, scissors, pencils, paper

Using a shoe box from home and one other rectangular prism box, have students discuss
the number and location of faces, vertices and edges. Have measurements of the boxes
used for modeling written on the boxes and the board. Lead a discussion about how
measures are involved when finding surface area.

Provide students with the Rectangular Prism BLM and have them fold and tape it
together to form a rectangular prism. If time is a factor, have students cut out and tape
together these nets at home the day before this activity begins. Ask student to determine
the number of faces, edges, and vertices. Have students find the area of one face of the
prism. Have students determine which other faces of the box would have the same area.
Have the students work in pairs to determine the surface area of the rectangular prism,
and then discuss method(s) used. Have the students list methods used to find surface area
on the board so that comparisons of methods can be made. Make comparisons of these
methods and the formula used on the LEAP Reference Sheet.

Next, provide students with the Triangular Prism BLM and have students construct the
prism by appropriately folding and taping it together. Determine faces, edges, and
vertices. Have students discuss shapes that make up each face of the triangular prism.
Determine a method of finding the area of each face. Provide rulers for measuring lengths
so that the groups can find the areas. The Triangular Prism BLM is an equilateral
triangular prism. Make sure the students realize that this is not a right triangle and that
they have to find the height of the equilateral triangles. These triangles are located at
either end of the center rectangle region of the net and the students will discover that they
can fold it in congruent parts to find the height of the triangles. This is also a good time
for a discussion about congruency.

A Right Triangular Prism BLM has been included. Have students identify faces, edges,
and vertices. Have students use the Pythagorean theorem to determine the area of the
right triangular ends of the prism as they find the surface area of the right-triangular
prism. Have students share methods by putting different methods on the board for
discussion.

Extend this activity by having the students bring a box from home which is cut so that the
six faces are clearly distinguishable. Students should find the surface area of the box that
they brought from home but keep the results secret. Have students place their ‗nets‘ on a
table in the room. Label these with letters and have the students rank the ‗nets‘ from
largest to smallest surface area without measuring. Discuss results.




Grade 8 MathematicsUnit 3Geometry and Measurement                                        33
                       Louisiana Comprehensive Curriculum, Revised 2008


Activity 9: The Converse of the Pythagorean Theorem (GLE: 31)

Materials list: grid paper, protractors, pencil, paper

Have student pairs cut out squares from grid paper that are 9, 16, 25, 36, 49, 64, 81, 100,
121, 144, and 169 square units. Then have them create triangles using the sides of any
three squares. Have students use a protractor to determine the measures of each angle in
the triangles formed. Next, have students determine the relationship between the sum of
the areas of the two smaller squares and the area of the largest square (i.e., are they the
same or different?) Have students make a conjecture about the relationship between the
areas of the squares when one of the angle measures of the triangle is 90 degrees. Remind
students that these relationships are those of the Pythagorean theorem and its converse
(studied in earlier activities). Lead a discussion of applications of the converse of the
Pythagorean theorem to real-life situations. For example, a carpenter goes to the corner of
a frame wall that he is building and marks off a 3 foot length on one board and a 4 foot
length on the adjacent board. He then nails a 5 foot brace to connect the two marks.
What is the purpose of his work? (He is making sure that the two boards are
perpendicular(that his wall is „ square‟) because a triangle with sides of 3-4-5 is a right
triangle.)


Activity 10: Angle Relationships (GLEs: 23, 28)

Materials list: paper, pencil, protractor

Have students investigate the relationship among the angles that are formed by
intersecting two parallel line segments with a transversal. Have students determine pairs
of angles that are complementary, supplementary, congruent, corresponding, adjacent,
and alternate interior. Using a protractor, have students determine the measure of each of
these pairs of angles. As an application, pose the following problem to students: As a
class project, you are going to build a picnic table with legs that form an ―X.‖ Of course,
the top of the table must be parallel to the floor. If one of the legs is attached so that it
forms a 40o angle with the top of the table, what measure should the leg form with the
ground to ensure the tabletop is parallel to the floor? Ask students to explain their
reasoning.


Activity 11: Folding squares (GLEs: 23,28)

Materials list: paper cut into squares for each student, pencil, paper

Provide square sheets of paper to each student. Have the students fold the paper in half
with a horizontal fold (fold 1), make a good crease, and open the paper up again. Then
instruct students to fold each half in half again using a second horizontal fold (fold 2),
make a good crease, and open the paper up again. Have students make a vertical fold
(fold 3), make a good crease, and open the paper up again. Ask students to make



Grade 8 MathematicsUnit 3Geometry and Measurement                                          34
                       Louisiana Comprehensive Curriculum, Revised 2008


observations about the relationships of length of the line segments formed by the folds.
Have students identify these as a bisector and a perpendicular bisector. Instruct students
to take the top right corner and fold it so that the vertex meets the intersection of their
center folds, rotate their paper 180, and repeat this fold with the opposite corner. Have
them open their paper and outline the folds. In their groups students should find methods
to determine the measures of all angles formed by the different folds.

Have students outline the hexagon that is formed after the folds
have been made (see diagram) and use what they know about
angle measures to determine the number of degrees in the angles
                                                                                        fol ds # 2
of a hexagon.                                                             fol d # 1



Have groups prepare a presentation to the class and justify their
angle measurements of all angles formed by the folds (i.e.,                fol d # 3

complementary, supplementary, vertical angles).


Activity 12: Scale Drawings (GLE: 30)

Materials list: Scale Drawings BLM, pencil, paper

Provide the students with the problems to practice scale drawing problems by distributing
Scale Drawing BLM. Give students time to work through these situations and then divide
students into groups of four to discuss these situations. Have students in groups come to
consensus on the solutions to these problems and then have them prepare for a discussion
using professor know-it-all (view literacy strategy descriptions). With this strategy, the
teacher selects a group to become the ―experts‖ on scale drawing required in the situation
that is selected. The group should be able to justify its thinking as it explains its
proportions or solution strategies to the class. All groups must prepare to be the
―experts‖ because they are not told prior to the beginning of the strategy which group(s)
will be the ―experts‖ and ask questions about scale drawings.


Activity 13: A Numberless Graph (GLE: 33)

Materials list: paper, pencil

Have students draw x-and y-axes and illustrate the situations below in quadrant I of a
coordinate plane.
 Label the x-axis, height, and the y-axis, weight. Place points A and B so that the
   person represented by point B is taller and heavier than the person represented by
   point A.
 Label the x-axis, size, and the y-axis, price. Place points A and B so that the object
   represented by point A is larger than the object represented by point B and the object
   represented by point B costs less the object represented by point A.



Grade 8 MathematicsUnit 3Geometry and Measurement                                      35
                      Louisiana Comprehensive Curriculum, Revised 2008



 Label the x-axis age, and the-y axis speed. Place points A and B so that the person
   represented by point B is the youngest and the person represented by point A is the
   fastest.

                                  Sample Assessments


Performance assessments can be used to ascertain student achievement. For example:


General Assessments

        Provide the students with paper and the scale of 0.25 inches to represent 2
         feet. The student will a) draw a model of a rectangular swimming pool
         measuring 16 feet by 36 feet; b) draw a 2 foot by 6 foot diving board so that it
         bisects one of the short ends of the pool; c) find the perimeter and area of the
         pool; and d) put a walk around the perimeter of the pool with a width of 4 feet
         and find the area and the outer perimeter of the walk.
        Provide the student with unlined paper and rulers. The student will design a
         stained-glass window to show understanding of the terms midpoint, bisector,
         perpendicular bisector, symmetry, similar, complementary, supplementary,
         vertical angles, corresponding angles, and congruent angles. The student will
         label the different components of his/her stained-glass window to assure that
         examples have been included for each of the vocabulary words from the unit.
         The student will present his/her stained-glass sketch to his/her group and
         justify examples to the group members. The teacher will provide the student
         with a rubric to self-assess his/her work prior to presentations and teacher
         evaluation.
        Provide the student with a sketch of a baseball diamond showing that there are
         90 feet between the bases. The student will prepare a presentation explaining
         how to determine the distance the catcher must throw the baseball to the 2nd
         baseman if he needs to get the runner on second base out.
        Provide the student with several right triangles that have a missing side
         measure. The student will find the lengths of the missing sides.
        Distribute a piece of grid paper which shows a polygon and a transformation
         of the polygon (the second polygon). Students determine a transformation or
         tranformations that would produce the second polygon.
        Whenever possible, create extensions to an activity by increasing the
         difficulty or by asking ―what if‖ questions.
        Students produce a portfolio containing samples of experiments and activities.
        Students create a scale drawing. A rubric that assesses the appropriateness of
         the scale factor, as well as the accuracy of the drawing, will be used to
         determine student understanding.




Grade 8 MathematicsUnit 3Geometry and Measurement                                      36
                      Louisiana Comprehensive Curriculum, Revised 2008


Activity-Specific Assessments

         Activity 5: Assign students these problems as journal prompts and have them
          explain the answers.
             a) Washington, DC, is 494 miles east of Indianapolis, Indiana.
             Birmingham, Alabama is 433 miles south of Indianapolis. Determine the
             distance from Birmingham to Washington D.C.

             b) The ladder of a water slide is 8 ft. high, and the length of the slide is 17
             ft. Determine the length of the horizontal base of the slide. Justify all of
             your thinking using valid mathematical reasoning.

         Activity 8: Provide a journal prompt such as: Will every rectangular prism
          have the same number of faces, vertices and edges? Explain.

         Activity 9: Provide the student with a list of number triples that represent the
          side lengths for triangles. Challenge students to determine which triples
          represent the side lengths of a right triangle.

         Activity 10: Provide the student with a sketch of an ironing board. In a math
          learning log entry, the student will explain the relationships of the angles
          formed by the legs of the ironing board.

         Activity 11: Provide the student with a list of vocabulary (bisector,
          perpendicular bisector, complementary angles, supplementary angles, vertical
          angles, adjacent angles, corresponding angles, corresponding angles) used in
          the unit. The student will write the vocabulary on the folded square used in the
          activity.




Grade 8 MathematicsUnit 3Geometry and Measurement                                       37
                       Louisiana Comprehensive Curriculum, Revised 2008


                                       Grade 8
                                     Mathematics
                          Unit 4: Measurement and Geometry


Time Frame: Approximately four weeks


Unit Description

In this unit, basic 2- and 3-dimensional shapes, their surface areas, and their volumes are
explored. Conversions of volume within the same system and comparisons of relative
sizes of units of volume across systems are made. Density, velocity, and monetary
conversions are connected to algebraic relationships. Analyses of rates of change of sides,
areas, and volumes of similar figures are also revisited. Such analyses are also applied to
the lengths of sides, areas, and volumes of similar figures due to changes in one or more
of the dimensions. Predictions based on data patterns are made, and single and multiple
event probabilities are explored.


Student Understandings

Students develop, understand, and apply the surface area and volume formulas for
prisms, cylinders, and pyramids. Students begin to understand and apply these concepts
to the cone, but they are not mastered at this level. They also select units and estimate the
surface area and volumes/capacity of specified figures. They are able to compare and
contrast the relative measures of objects or quantities measured in the metric and
customary systems, as well as convert between units of volume in the same system.
Working with derived units, such as density, velocity, and international monetary
conversion rates, the students can discuss the nature of rates of change within such units.
Students also find single and multiple event probabilities. Students can identify data
patterns and make predictions from these patterns.


Guiding Questions

       1. Can students describe the nature of surface area, volume, and capacity as
          measures of size?
       2. Can students apply and interpret the results of surface area and volume
          considerations applied to prisms, cylinders, pyramids, and cones?
       3. Can students make appropriate estimates of volume and capacity and use
          these in applications?
       4. Can students determine the effect of a change in linear scale on perimeter,
          area, and volume in similar figures?
       5. Can students discuss the rate of change of velocity in terms of speed and
          direction?



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       6. Can students find the density of a substance?
       7. Can students make predictions from data patterns?
       8. Can students find single and multiple event probabilities?


Unit 4 Grade-Level Expectations (GLEs)

GLE # GLE Text and Benchmarks
Measurement
17.       Determine the volume and surface area of prisms and cylinders (M-1-M) (G-7-M)
18.       Apply rate of change in real-life problems, including density, velocity, and
          international monetary conversions (M-1-M) (N-8-M) (M-6-M)
19.       Demonstrate an intuitive sense of the relative sizes of common units of volume in
          relation to real-life applications and use this sense when estimating (M-2-M) (G-1-
          M)
20.       Identify and select appropriate units for measuring volume (M-3-M)
21.       Compare and estimate measurements of volume and capacity within and between
          the U.S. and metric systems (M-4-M) (G-1-M)
22.       Convert units of volume/capacity within systems for U.S. and metric units (M-5-M)
Geometry
32.       Model and explain the relationship between the dimensions of a rectangular prism
          and its volume (i.e., how scale change in linear dimension(s) affects volume) (G-5-
          M)
33.       Graph solutions to real-life problems on the coordinate plane (G-6-M)
Data Analysis, Probability, and Discrete Math
39.       Analyze and make predictions from discovered data patterns (D-2-M)
43.       Use lists and tables to apply the concept of combinations to represent the number of
               possible ways a set of objects can be selected from a group (D-4-M)
45.       Calculate, illustrate, and apply single- and multiple-event probabilities, including
               mutually exclusive, independent events and non-mutually exclusive, dependent
               events (D-5-M)
Patterns, Relations, and Functions
48.       Illustrate patterns of change in dimension(s) and corresponding changes in volumes
          of rectangular solids (P-3-M)


                                   Sample Activities


Activity 1: Volume and Surface Area (GLE: 17)

Materials List: Volume and Surface Area BLM, 16 cubes for each pair of students,
paper, pencil, calculators, math learning log

Give student pairs a given set of 16 one-inch cubes or centimeter cubes, and ask
them to build all possible rectangular solids. Have students count the number of


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cubes to determine the volume of the solids and count the number of exposed
faces to calculate the surface area of each solid built, recording information on the
Volume and Surface Area BLM. Have students make sketches of the solids and
label the dimensions of the rectangular prisms built.

Have students repeat the exercise using a different number of cubes and record
the information in the chart.

Ask students to study their findings and list their observations. Make sure
observations include the relationship of the surface area and the shape of the
rectangular solid (i.e. the closer to the shape of a cube, the smaller the surface
area).

Have students respond to the prompt in their math learning log (view literacy
strategy descriptions).
        Measurements of rectangular solids can be linear, square and cubic units.
        These units refer to . . .‘


Activity 2: Rectangular prisms (GLEs: 17, 21)

Materials List: Volume and Surface Area BLM (from Activity 1), cm Grid BLM, LEAP
Reference Sheet BLM, scissors, tape, pencil, paper, colored pencils or markers

Have the students refer to the Volume and Surface Area BLM used in Activity 1.
Review observations made with shapes formed in Activity 1. Have each student
construct a net for a one-centimeter cube from centimeter grid paper, fold the net to form
a cube, and tape the cube to hold its shape. Ask students to make and record a prediction
as to how many of these one centimeter cubes it will take to make a cube with
dimensions of 2 cm x 2 cm x 2 cm. Have them work in groups of four to make enough
centimeter cubes to form the 2 cm x 2 cm x 2 cm cube. Discuss the concept that the
number of centimeter cubes that it takes to make a 2 cm x 2 cm x 2 cm cube is the
volume of the new cube and is recorded as cm3.

Distribute the LEAP Reference Sheet BLM and have groups of four use their centimeter
cubes and make the connection between the formula for volume as stated on the LEAP
Reference Sheet BLM and the 2 cm x 2 cm x 2 cm cube that they formed. Discuss the
dimensions of the cube, the concept that three edges meet at a vertex of a cube and that
there are 8 vertices. Each edge is a dimension and there are12 edges in the cube.

Have students construct a net 3 cm x 3 cm x 3 cm. Have students predict how many of
the centimeter cubes would fit inside of a 3 cm x 3 cm x 3 cm cube. Have students fold
their net into a cube and again relate the volume to the formula on the LEAP Reference
Sheet. Ask, “How many cubic centimeter blocks would fit into a 4 x 4 x 4 cube?‖ Have
students state the rule to use for finding volume of a rectangular prism, and make sure the




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students are relating the number of cubic units needed to form the cube to the volume and
are recording asnwers in cubic units as they multiply length, width, and height.

As an extension, have students determine the surface areas of the 2 cm, 3 cm, and 4 cm
cubes to help with understanding the difference between surface area and volume.

Use the SQPL strategy (view literacy strategy descriptions) to challenge the students to
further explore volume measurements. Put the following statement on the board or
overhead for students to read: ―It would take more than 10,000 one inch cubes to fill a
cube that is 8 ft on each side.‖ Have students work with a partner and brainstorm 2-3
questions that would have to be answered to prove or disprove the statement. As a whole
class have each pair of students present one of their questions and write this question on
chart paper or the board. Give the class time to read each of the questions presented.
Give pairs of students time to select the ideas that they would use to prove or disprove the
statement. It is important that the students understand that when changing units of
volume, all three dimensions have to be changed.

Using the professor-know-it-all strategy (view literacy strategy descriptions), have
different pairs of students explain their proof and answer questions from the class. Using
this strategy, the teacher randomly selects pairs of students, not volunteers.

Next, have the students construct and cut out a net for a rectangular prism with
dimensions of 1 cm x 2 cm x 2 cm. Ask students to determine the number of cubic
centimeter cubes that will fit inside of the rectangular prism. Ask the students what this is
called (volume). Have students fold their net to form a rectangular prism. Ask someone to
show the class the three dimensions of the rectangular prism. Challenge students to find
the surface area of the prism.

Have students mark the edges, vertices and faces with colored pencils or markers.
Instruct students to work in their groups to find a rule for determining the number of
centimeter cubes that will fit into a rectangular prism (volume). Discuss conjectures that
students make. Use the LEAP Reference Sheet BLM, and have the students explain how
the formulas for volume and surface area of a rectangular prism relate to the models they
have built in Activities 1 and 2 of this unit. Challenge groups to test their conjectures.


Activity 3: What’s the Probability? (GLE: 45)

Materials List: What‘s the Probability BLM, paper, pencil, calculator, computer access
or printout of basketball court and dimensions for each student, newsprint, markers

Use brainstorming (view literacy strategy descriptions) and have the students recall all
they know about probability. Write their ideas on a chart or the board. Brainstorming is
used in this lesson to pre-assess what the students recall from 7th grade and earlier grades
about probability. Model the use of a graphic organizer (view literacy strategy
descriptions) to organize the ideas that students have brainstormed. A circle map is a



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good one to use to gather this information and pre-assess the students‘ knowledge of
probability. An example of one possible graphic organizer is shown below.


                                         c hanc e




             perc ent

                                                         part of whol e




                               Probability


     frac tion


                                                                          mi ght or
                                                                          mi ght not
                                                                          happen




                                 2 out of 3
                                 c hanc e




Distribute the What‘s the Probability BLM and give the students time to work
independently on the probability situations given. Discuss results briefly prior to the next
situation.

Provide the students with this website to view a basketball court and its dimensions
(http://www.betterbasketball.com/basketball-court-dimensions/basketballcourt2.html ) or
use the printout from this site and make copies for each student. Give them the following
situation:
        Jason‘s basketball coach told the starters that they must score at least 10 points
        during the next four games. Jason practiced shooting only two point shots at the
        gym. When he left, he realized that he had dropped his house key and his practice
        schedule. He knew that the objects could be anywhere on the court, but he thought
        that the practice schedule would most likely be on the half of the court where he
        was practicing. Find the probability that the practice schedule will be found on
        this end of the court. Find the probability that his key is inside the free-throw area
        of his end of the court. What is the probability that both the key and practice
        schedule are inside of the free-throw area?
Since probability has not been extensively covered at this point (it was covered in the 7th
grade curriculum), have students work through a similar situation prior to assigning this
activity with groups of students. Draw any geometric figure, inscribe a second figure
inside (shade one part), and work with the students to determine the probability that an


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object falling randomly will land in the shaded area. It is a good time to use figures that
the students will subdivide to compute the area. Have students prepare a poster to show
how they figured the probability. Use the professor know-it all strategy (view literacy
strategy descriptions) to discuss results as a class. Randomly select groups to share and
justify their thinking.


Activity 4: Odd Volumes, is it Fair? (GLEs: 17, 43)

Materials List: Spinner BLM, math learning log, paper, pencil

Distribute the Spinner BLM and have students use a paper clip held in the center of the
circle with their pencil point, so the paper clip can spin around the tip of the pencil, as a
spinner. Model for the students how they will spin the spinner three different times,
gathering the three dimensions of a rectangular prism, then have them discuss how these
numbers can be used to find the volume. Have students work in pairs to collect data and
record the spins as dimensions of rectangular prisms. Have students complete the table at
the bottom of the Spinner BLM recording their spins under the headings of length, width,
height, and volume. Instruct students to spin three times and record each of the three
spins as a dimension of a rectangular prism. Ask students to recall how the volume of a
rectangular prism is determined (lwh).

Instruct students to determine the volume of each rectangular prism. Player 1 gets a point
if the volume is odd, and Player 2 gets a point if the volume is even. Have students
continue to spin until each has determined the volumes of 5 rectangular prisms. Once the
students have gone through the activity, challenge them to determine the number of
possible combinations of odd and even volumes using a chart or table. Discuss as a class
whether the rules of the game were fair. Have students record their thinking about
whether the game is fair or not in their math learning log (view literacy strategy
descriptions).


Activity 5: Cylinders (GLEs: 17, 21)

Materials List: LEAP Reference Sheet BLM, pencil, paper, math learning log, compass,
ruler

SPAWN writing (view literacy strategy descriptions) is an informal writing with students
responding to a given prompt. Begin by explaining to the students that they will reflect on
what they know about volume and surface area using the ‗P‖ or Problem Solving of
SPAWN writing. Write the following prompt on the board:
       We have been studying volume and surface area of rectangular prisms and
       cubes. What measurements will be necessary when finding the volume of
       a cylinder?
Ask students to record their thoughts in their math learning logs (view literacy strategy
descriptions). This is informal SPAWN writing and should not be taken as a grade. It is



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important that the students know the importance of communicating mathematically (give
points for completion if necessary). Use the students‘ SPAWN writing responses for
whole class discussion.

Have students create a cylinder from standard 8 1 inch x 11 inch paper. Engage the class
                                                    2

in a discussion about how to make a cylinder with a circumference of 8 1 inches without
                                                                            2
cutting or tearing the paper. Have students roll paper to form a cylinder shape. Ask what
the height of this cylinder is. (The height of the cylinder will be 11 inches). Ask the
students to predict whether the cylinder with a circumference of 8 1 inches or the
                                                                       2
cylinder with a circumference of 11 inches made from a full sheet of this paper would
have the larger volume.

Show students the circles formed when the paper is rolled. Ask them again what the
circumferences of the circles are and how they know. (The circumference is formed by
the side of the sheet of paper which is 8½ inches long.) Ask students if they remember
the formula for the circumference of the circle. Then ask how they can find the diameter
if the circumference is known. This is a good time to use the LEAP Reference Sheet
BLM and have them substitute values into the circumference formula to find the diameter
of the cylinder using 3.14 for pi.

Once the diameter‘s length is determined, have students use a ruler and a compass to
measure the radius needed to construct the circles for the cylinder. Have the students
draw circles with their compasses and cut them out with scissors. Before the cylinder is
assembled, have students determine the surface area of the cylinder. After the cylinder is
assembled, have students determine its volume using the formula on the LEAP Reference
Sheet. Have students take out their SPAWN writing notes and read them. Students should
make any additions or deletions needed after constructing the cylinder and using the
formula.

Have students determine volumes and surface areas by measuring cylinders with U.S.
system units and then repeat the activity using metric measures. Have students make a
table of the surface area dimensions given in metric and put the U.S. measurements in the
column beside the metric units for comparison. Have students make observations of the
comparisons.


Activity 6: Pyramids and Cones (GLE: 17)

Materials List: Volume Comparison of Pyramids and Rectangular Prisms BLM, Models
of Rectangular Prism and Pyramid BLM, Models of Cylinder and Cone BLM, scissors,
tape, paper, pencil, rice, beans or un-popped popcorn

Have the students explore only cones and pyramids at this level. (At this grade, the
concepts are explored, but not mastered.) Have students research the use of pyramids as
buildings. Some possible websites for research are
               www.pbs.org/wgbh/nova/pyramid/


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               http://interoz.com/egypt/construction/construc.htm
               www.maya-archaeology.org/
Have students include The Pyramid Arena located in Memphis, TN, and the Great
Pyramids of Egypt in their research. Once students have gained an understanding of what
a pyramid is, have them determine the surface area and volume of some of these
structures by making paper models. Instruct students to make a square bottom with a 4
cm base and then create four isosceles triangles with a base of 4 cm and a height of 6 cm.
Next, have students create a rectangular prism in which the pyramid will fit (it will be
close to a 4 cm x 4 cm x 6 cm rectangular prism). Have students compare the volume of
the pyramid with the volume of the rectangular prism by filling the pyramid with beans,
rice, or unpopped corn and pouring them into the prism. The pyramid should be about
one-third the volume of the rectangular prism.

Distribute the Volume Comparison of Pyramids and Rectangular Prisms BLM and the
Model for Rectangular Prism and Pyramid BLM. Have students work with a partner to
construct the models and complete the chart. The chart leads the students to discover the
                                             1
conjecture that the volume of the pyramid is the volume of the rectangular prisms.
                                             3

Next, have students find examples of cones used in everyday life and determine the
surface areas and volumes for some of the examples. Since the students have previously
created a cylinder with an 8 1 inch by 11 inch sheet of paper and discussed the
                              2
circumference, have them cut out the models for the cone and cylinder on Models of
Cylinder and Cone BLM. Have students compare the volumes of the cone and the
cylinder by filling the cone with rice (beans, corn kernels) and pouring it into the cylinder
until the cylinder is filled.

Have students work in groups of four to make a graphic organizer (view literacy strategy
descriptions) comparing the relationship of the cone to the cylinder, and the rectangular
prism to the pyramid. Have student groups share their findings using the professor-know-
it-all (view literacy strategy descriptions) strategy, selecting two or three groups of
students as the professor.


Activity 7: Comparing Cones: (GLE: 17)

Materials List: Comparing Cones BLM, Model for Cone BLM, pencil, paper, ruler

Distribute the Comparing Cones BLM and the Model for Cone BLM. Have students cut
out the circle leaving the points so that they can use them for the activity.

Have students cut along the radius and form a cone by moving point L so that it lies on
top of point A. Instruct students to measure the diameter, circumference, and height of
the cone and to record the measures in the Comparing Cones BLM. Have students
calculate the volume of the cone. Next, have students form a second cone by sliding point
L so that it lies on top of point B. Each time a new sized cone is formed, have students


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record the diameter, circumference and height. After the students have completed
forming cones by moving point L to at least 5 different locations, have them find the
volume of each of the cones and develop a conjecture about how the change in
circumference affects the volume in their math learning log (view literacy strategy
descriptions).


Activity 8: Common Containers (GLEs: 17, 19, 20, 21, 22)

Materials List: Common Containers BLM, LEAP Reference Sheet BLM, 6 containers
(boxes from cereal, cans, etc) with measurement labels removed for each group, ruler,
tape measure, pencil, paper

Prior to class put letters A, B, C, D, E, F, on the containers for each group.

Provide student pairs with several common containers (rectangular solids and cylinders)
found in the grocery or hardware store (with the labels removed or volume information
covered) and a copy of the Common Containers BLM. Label the containers with numbers
or letters. If the labels have information relative to volume in cubic units, save the labels
for later use and mark the labels with the same letter as the container. Have students
estimate the volume of each container, record this in the correct column on the Common
Containers BLM, and arrange the containers in order from smallest to largest volume.

Once the students have estimated the volumes, provide measuring instruments and have
students determine the volume of each container using U.S. units and record the volume
on the Common Containers BLM. Remind the students that the formulas needed are
found on their LEAP Reference Sheet BLM. It would be a good idea to have students
first write the formula, show substitution of values so that use of correct values can be
determined if an error is found, and then give the answer. Make sure students give the
correct unit on their answers.

Finally, have students repeat the process using a metric measurement tool.

Lead a discussion comparing measurements within and between systems. Once the
volumes of the containers have been determined, have students convert their answers to
another unit in the same system (i.e., convert from cubic inches to cubic feet and vice
versa—include conversions with metric units, also). If there were labels which had
volume written in cubic units, then allow students to compare their results with the
information provided on the labels.

Repeat the activity with larger containers. For example, show a picture of a silo to the
students. Explain to them that silos have been used for many years to store grain. Provide
the dimensions of a silo, and have students determine its volume. Buildings can also be
used as examples. The Superdome is in a cylindrical shape with a domed roof. Find the
actual dimensions of the Superdome at www.superdome.com. Go to the About Us section




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and scroll down to the table of facts. Assuming the roof is flat, have students approximate
the volume of the Superdome.

A basketball gym is typically a rectangular solid. Have students determine the volume of
their school‘s gym. In each case, have students express their answers in both U.S. and
metric units. Discuss selecting appropriate units for measuring volume and capacity.


Activity 9: Using Algebra to Make Conversions (GLE: 18)

Materials List: paper, pencil, computer access or printout of current currency rates, sale
ads from newspaper

Provide students with a listing of currency exchanges. Current information can be
obtained at http://moneycentral.msn.com/investor/market/rates.asp. Using the current
exchange rate for the dollar, have students write a proportion for converting U.S.
currency to a specified foreign currency. Next, have students convert from one foreign
currency to another foreign currency by developing a formula. Make sure students notice
that the formula will always involve the rate of change in the currencies. For example, if
one U.S. dollar is equal to 0.809 Euro dollars then US = 0.809E is the formula for
converting Euros to U.S. dollars. The rate of change is the current rate of exchange.

Distribute sales papers to the students. Review one of the sale items with the students.
Example: A sale paper shows a backpack that costs $5.95. Have students set up a
proportion so that they can figure the cost of the item in Euros. Based on the same
                                                   .829 Euros x Euros
conversion factor given above, the proportion is                        so the backpack
                                                      $1.00        $5.95
would cost 4.93 Euros. Give students a budget of $300 in US dollars and have them use
the sale brochure to spend the money and then find the cost of each item in Euros.


Activity 10: Changing Areas and Volumes (GLEs: 18, 19, 32, 48)

Materials List: cubes, Changing Volumes BLM, Real Life Volume Situations BLM,
paper, pencil

Draw a picture of a rectangle on the board and label with dimensions. Have students find
the area of the rectangle making sure that they give answers with correct units. Have
students draw a diagram showing that the length of one side of the rectangle is doubled
and have them calculate the area of the new rectangle. Have the students draw another
rectangle with the length of the original rectangle tripled and then find the area. Ask
students how changing one dimension affected the area. Students should see that
doubling one dimension of the rectangle doubles the area, or tripling one dimension
triples the area.




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 Next, have the students take the original rectangle and multiply one side by 2 and the
second side by 3 and calculate the new area. Ask students how the new area compares to
the original area. They should see that the new area is 6 times as great as the original.
Provide a few more examples of this type. Lead students to the conclusion that the
product of the factors used when changing the dimensions of the rectangle is the factor
that should be used to multiply by the original area to determine the area of the new
rectangle.

Begin the next part of the activity by having the students make predictions as to how
changing one, two or three dimensions of a rectangular prism will affect the volume of
the prism. Give students time to make predictions in their math learning log (view
literacy strategy descriptions). Next, have students explore the change in volume when
changes are made in one, two, or three linear dimensions of a rectangular prism to test
their predictions.

 Distribute Changing Volumes BLM. Give student pairs a supply of cubes. Have students
build a rectangular solid that is 4 units long, 3 units wide, and 2 units tall and determine
its volume. Instruct students to record the volume in Part 1 of the table on the Changing
Volume BLM. Ask students to brainstorm (view literacy strategy descriptions) what
objects might fit into a container with these dimensions. Next, have students double the
width of the solid to 6 units, but keep the original length and height. Record the
dimensions and volume in the table. Have students compare the volume of the new solid
with that of the old. Have students predict what objects might fit into containers with
these dimensions.

Have the students work in pairs to complete Part 2 of the volume explorations on the
Changing Volume BLM. Have students record their dimensions of a cube with a volume
of 8 cubic units. As they work through the BLM, students will double one dimension, two
dimensions and then all three dimensions of the cube. They are then given the volume of
another cube and asked to find its dimensions and will repeat the same process of
changing the dimensions as described above. When Part 2 is completed, students are
asked to describe the effect of the side lengths on the volume (i.e., how one can calculate
the new volume if the old volume and the dimension change are known). Explain to the
class that Part 3 of the Changing Volume BLM involves increasing fractional
dimensions. Challenge the students to complete this part of the BLM working with a
partner.

Reciprocal teaching (view literacy strategy descriptions) is a strategy in which the
teacher models and the students use summarizing, questioning, clarifying, and predicting
to better understand content text. Divide the class into groups and have one group explain
the result when doubling one dimension, another group two dimensions, and the third
group all three dimensions. The students should justify their conjectures to using the
information in their tables. Give student groups time to explain and justify their
conjectures to the other two groups. If there are two groups of students doing each
conjecture, they might move from one group to the other so that each group has an
opportunity to discuss conjectures of each scenario.



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As closure, have volunteers explain what happened in each case. Repeat this activity with
rectangular solids of other dimensions. Lead a discussion about the rate of change in the
volume compared to the other changes made. Distribute the Real Life Volume Situations
BLM to assess the student level of understanding of volume.


Activity 11: Density (GLE: 18)

Materials List: Finding Density BLM, triple beam balance (borrow from science), cm
rulers, 3 rectangular prisms of different sizes made of same substance, paper, pencil

The purpose of this activity is to have students learn how to calculate density.

Have students fold a sheet of paper into thirds (lengthwise folds), open the paper, and
draw a line down the paper marking the one-third line. Students will use split-page
notetaking (view literacy strategy descriptions) to organize their notes on density during
lesson discussion. Have the students write the following in the date/hour column: a)
what is density; b) density ratio; c) how density is used to identify solids. Have the
students take notes under the ‗topic‘ column of their note page. Give notes about each of
the ideas written in the left-hand column as the discussion continues. This note-taking
page will continue with velocity and can be used for the unit assessment.

                                                   folds




                          Date/Hour
                                             Topic: Density


                                           Density is the ratio of mass to volume.
                       What is density?    The density of something remains the
                                           same.




                       Density r atio      mass/volume



                                           Since the density of substances remains
                       How is density
                                           the same, scientists can deter mine the
                       used to identify
                                           density of a substance and match
                       solids?
                                           this to the substance in a density list.




Density is the ratio of mass to volume. Density is a measure of how tightly packed the
particles are in a substance. The density of a solid stays constant. Knowing this fact
allows us to identify unknown substances. For example, water has the density of 1 gram
per cubic centimeter and scientists do not adjust density for temperature or altitude unless
extreme accuracy is needed. It is important to give units when stating density.



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Explain to the students that they will rotate through three different density activities. At
each station, students will need centimeter rulers and a triple beam balance. Each station
should have a rectangular prism of a different size but of the same substance. For
example, there can be three blocks of wood that have different dimensions but are cut
from the same piece of wood. Another possibility is to use two rectangular bars of the
same brand of soap. Use one bar as is and cut the other into two prisms of two different
sizes. It will work best if the students measure the dimensions of the rectangular prisms to
the nearest tenth of a centimeter.

The students will calculate density of the object at each station. They need to find the
mass using the triple beam balance. Students should measure the dimensions and then
calculate the volume of the prism. As students do the work at each station, they should
record their measurements and the density on the Finding Density BLM. The students
should get values that show the three rectangular prisms have the same density (or very
close values) since they are all from the same material. Finding the average density will
give a better value for the density of the substance. Students should be able to state that
the density is the same if the same substance is used.


Activity 12: Different Densities (GLE 18)

Materials List: Station 1 – 1 Three Musketeers® candy bar, 1 Snickers® candy bar (fun or
regular size), cm ruler, triple beam balance; Station 2 – 1 bar of soap and 1 rectangular or
square pumice stone, cm ruler, triple beam balance; Station 3 – two small spheres such as
a large glass marble and a rubber ball, formula for volume of a sphere, cm ruler, triple
beam balance; calculators, pencils, transparencies of Class Data Charts BLM for Stations
1 - 3; copies of charts drawn on chart paper, one copy of the Density Experiments BLM
per group, extra fun-sized Musketeer and Snickers bar (at least 1 of each per group)

This activity is designed for students to practice finding density and to determine how the
densities of two materials compare to one another and to the density of water (1 gram per
cubic cm). Depending on the length of the class, it may be necessary to do the
experiments one day and have the summary discussion the next day. Prior to the
beginning of the class, make a transparency of the Class Data Charts BLMs or draw the
charts on chart paper and post them in the room.

Show students a sphere and ask students to indicate how they could find the length of the
sphere‘s radius with a ruler. After coming to a consensus on the process for doing this,
give students the formula for the volume of a sphere and a number to use as the radius.
Have students find the volume based the radius length provided. Students will need to
use the formula to do one of the experiments in this activity.

Divide students into groups of three or four. Assign each group a group number that
corresponds to one of the group numbers on the Class Data Charts. There may be a need
to have multiple setups for each station (2 or 3 with candy, 2 or 3 with soap/pumice, and




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                       Louisiana Comprehensive Curriculum, Revised 2008


2 or 3 with spheres) depending on the number of students in the class. Using multiple
setups of each station lessens the time needed for all groups to do the experiments.

Distribute the Density Experiments BLM and explain to the students that there are three
stations that each group is to visit. Indicate that students will do an experiment at each
station and then record information on the Density Experiments BLM. Remind students
that they need to post some of their results on a Class Data Chart. Indicate the location of
the Class Data Charts (i.e., indicate where the transparencies have been placed or show
them where the charts are posted). Inform students that they will be given 10 minutes at
each station and when time is up, they will rotate clockwise to the next station until all
three stations are visited. It will be necessary to monitor the work of groups as they go
from station to station and to direct them to the next station when time is up. Most
groups will need reminders to post their group results before moving to the next station.
In general, it is best to have all groups rotate at the same time.

When the experiments are completed, discuss the data collected at Station 1 by showing
the transparency on the overhead projector or having students direct their attention to the
posted chart. Have students check for consistency in data values from the various groups
making sure that it appears that the information for each item was posted in the correct
column. It may be necessary to have groups whose data are extreme outliers to
recalculate or possibly discard some data as erroneous. Have students find the average of
the densities calculated by each group for each item. Remind students that the density of
water is 1 gram per cubic cm. Have someone unwrap the candy bars and drop them into
a clear container of water. Ask students to compare the densities of the candy bars to the
density of water and conjecture about the results of placing each in water.

Give each group of students at least one Three Musketeers® and one Snickers® fun-sized
candy bars and a plastic knife. Have students cut the candy bars in half and view the cross
section. Ask the students to discuss why the densities are different based on what they
see. Depending on school policies, students may be allowed to eat the candy at this
point.

Repeat the same process with the data collected for Stations 2 and 3. Students should
realize that a substance with a density less than the density of water should float.

Have students respond in their math learning log (view literacy strategy descriptions) to
the prompt ―A mystery object is tightly wrapped in dark plastic. Mary contends that the
object is made of gold. Indicate how the truth of Mary‘s statement could be confirmed
without unwrapping the object.‖




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Activity 13: Velocity (GLE: 18)

Materials List: split-page notes from density, paper, pencil, Internet access

Have students write in the left hand column of their split-page notes (view literacy
strategy descriptions) a) What is velocity? b) What ratio do we use to find speed? c) How
does speed relate to velocity? Remind them that during the activity they will need to put
notes in the right-hand column to help them understand velocity. Indicate that velocity is
an indication of the speed and the direction that an object is moving. Remind the students
that during hurricane season, the meteorologists will state how fast the hurricane is
moving in miles per hour along with the direction that the hurricane is moving. The
national weather center website http://www.nhc.noaa.gov/ will show paths of storms
selected. To help students understand the meaning of velocity, type in the name of a
familiar hurricane in the search bar of the site. The storm track will be traced on the
screen with velocity shown along the track, helping the students understand that speed is
not the entire picture; we must also know the direction the hurricane is traveling.

Measure the time it takes to get to the gym or office in the school from your room. Tell
the students that you traveled ___ minutes and went to another location on the school
grounds. Ask them how they might be able to determine the final destination. Allow
students to ask only yes or no questions and record these questions on the board.
Questions such as ―Did you go right or left at the door of the room? Did you go straight
or turn down the hall to the right or left? Did you walk fast or slow?‖ After the students
have determined your destination, have them go back to the questions they asked and
relate them to the idea that the information they needed included direction and speed.
Again, reinforce the idea that velocity involves both speed and direction.

Ask students to predict where they would end up if they got on I-10 at Lake Charles and
drove 4 hours at 60 miles per hour. Students should recognize that they need a direction
to answer this question as this interstate runs east and west. A person going east would
stop in Alabama; a person going west would end in Texas. (Note: adjust the scenario
based on highways familiar to students or provide students with a map.)

Summarize by having students indicate other situations in which direction and speed are
important.


Activity 14: Calculating Velocity (GLE: 18)

Materials List: per group - balls, toy cars, or other things that roll, 1 stop watch, 1
compass, masking tape

Students should work in small groups of three or four. Have the students put down two
parallel strips of masking tape on the floor. Discuss how to find the shortest distance
between the lines and have the students measure that distance.




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Distribute balls, toy cars, or something that rolls and a stopwatch to each group. Have
students attempt to roll their objects along a perpendicular path from one strip to another.
After practicing a few times, have students roll the object three times and record the
number of seconds it takes for each trial. They should try to roll the object with the same
force each time. Have students determine the speed for each trial and find the average
speed for the three trials. Students should then place the compass on the floor to
determine the direction in which their objects traveled and record the velocity at which
their object moved.

Next, have students randomly place a third piece of masking tape so that it intersects the
two parallel lines (no right angles). Ask students to predict how the velocities of their
objects should change if they are rolled with the same force along the new path. Have
students repeat the exercise above.

Allow groups to share the velocities from each set of trials. Ask students to explain why
their speeds for each set of trials should have been different if the same force was used.
There should have been a change in time and direction for the rolls when the second path
was used.

Write the following information and have the groups determine the velocities of each
explaining that each of these objects is traveling southwest. Have students put the
resulting velocities in order from highest to lowest. (Answer: 10m/sec southwest; 8
m/sec southwest; 5 m/sec southwest; 4 m/sec southwest)
                  60 meters in 15 seconds
                  80 meters in 16 seconds
                  50 meters in 5 seconds
                  88 meters in 11 seconds


Activity 15: Data Patterns (GLE: 19, 39)

Materials List: Alligator BLM, paper, pencil

Lead a class discussion about data patterns and about when the students remember seeing
some kind of data pattern. Distribute Alligator BLM. Have the students work in pairs to
determine the pattern and make a prediction from the data. A picture of an alligator has
been included to allow students to determine the scale used in the picture. Students are
asked to give dimensions of a rectangular solid that would contain the gator pictured.
This activity might be discussed using professor-know-it-all (view literacy strategy
descriptions) to review solutions to Alligator BLM.




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Activity 16: Packaging Costs (GLEs: 17, 22, 32)

Materials List: paper, pencil

Have the students justify the most cost-efficient method of packaging 24 pieces of
chocolates that are each 1 inch cubes into a cardboard box. Students should determine all
possible dimensions for the box and which dimensions will give the most cost-efficient
package. The cardboard box is to be coated with red metallic foil that costs $0.68 per
square inch. Have students present their data in a table with the possible dimensions of
the boxes that can be used to pack the 24 candies with no wasted space. Have students
justify their decision of the best dimensions for the box with the cost differences a part of
their justification. Instruct students to give the dimensions of their box in inches and feet
and volume of their box in cubic inches and cubic feet.


Activity 17: Dimensions and Surface Area (GLEs: 20, 32, 33)

Materials List: paper, pencil, graph paper

Provide the students with the following situation to respond to in their math learning log
(view literacy strategy descriptions).
        Demetria is planning to make boxes for candy at Christmas. She wants to leave
        the base the same and see what would happen if the height is changed. She
        decided that the base of her box would be a three-inch square, and she would
        make a table to compare the change that the height has on the volume. Demetria
        needs a box that will hold 135 in3. Determine the dimensions that she will need to
        use. Would the volume of this rectangular prism be more accurately reported in
        cubic feet? Explain.
Have students complete a table like the one below and determine how changes in the
height of an object affect the volume.

       Surface area of the       Height        Volume
              base              (inches)        (in3)
              9 in2                 1
              9 in2                 2
              9 in2                 3
              9 in2
              9 in2
              9 in2


Have students plot the height and volume of the box with a base of 9 in2. Next, have
students calculate the cost of the box needed, plotting the height and volume on a
coordinate grid to verify their prediction.




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Have students make a list of different items that would be rectangular prisms and
determine whether the volume would be best recorded as cubic inches or cubic feet.


                                  Sample Assessments


Performance assessments can be used to ascertain student achievement.


General Assessment

        The student will prepare a proposal for a new playground for the children at
         the elementary school. The proposal should include a scale model of the
         playground proposed and an explanation of the geometry terms and shapes
         that are required. It must include a piece of equipment with a cone, cylinder,
         and a rectangular prism as part of the design. The proposal should give the
         scale that was used and the actual size of the different equipment. The same
         scale should be used for all pieces of equipment. The student will find
         examples of vertical angles, corresponding angles, supplementary and/or
         complementary angles and perpendicular and parallel lines on the playground
         models. These examples should be marked with a marker and identified
         within the proposal.
        The student will prepare a presentation with models of rectangular and
         triangular prisms that prove how doubling the dimensions affect the volume.
         The presentation will include measurements of prisms and an explanation of
         the method used to find the volume. The student will write his/her conjecture
         about the effect the changing of dimensions has on the volume of a prism.
        The student will prepare a brochure comparing US and metric volume
         measurement, using pictures of objects and listing a US volume measure and
         an approximate metric volume measurement or a metric volume measurement
         and an approximate US volume measure.
        Provide the student with a set of rectangular solids. The student will measure
         dimensions and find the volumes.
        Provide the student with two similar solids. The student will measure
         dimensions and find the volume of one solid, determine the scale factor, and
         use the scale factor to help calculate the volume of the other solid.
        Whenever possible, create extensions to an activity by increasing the
         difficulty or by asking ―what if‖ questions.
        The student will create portfolios containing samples of experiments and
         activities.




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Activity-Specific Assessments

         Activity 2: The students will work in small groups to prepare presentations to
          prove the groups‘ conjectures or rules on how to find volume of a rectangular
          prism to the class.
              Solution: The student should determine through these constructions that
              the volume of a rectangular prism can be found by multiplying the
              dimensions together and that the unit of measurement is a cube.

         Activity 5: The student will construct a cylinder that will hold three tennis
          balls. The student will be given a tennis ball to use to gather the
          measurements. If enough tennis balls are not available, the measurements are -
          - diameter about 6.5 cm and circumference about 20.5 cm. After the cylinders
          are completed, the student will compare his/her container dimensions to the
          dimensions of an actual tennis ball can. The student will give the volume and
          surface area of the cylinder.
              Solution: The circumference of the cylinder should be about 20.5 cm and
              the height of the can should be 3×6.5 cm or about 19.5 cm. The volume is
              about 63.375 cm3 and the surface area is about 409.955 cm2

         Activity 8: The student will measure dimensions and determine the volume of
          several containers (at least 5) in both U.S. and metric units, recording
          measurement of volume and surface area in both systems.

         Activity 9: The student will prepare a justification that explains the procedure
          for converting money from U. S. dollars to another currency and the total
          amount spent on four different items. The student will then prepare a brief
          explanation as to whether it would be sensible to purchase the item in another
          country using a currency other than U. S. dollars. The poster will show the
          proportions for the four different items and how the proportion was used to
          calculate the cost of each item purchased.




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                                      Grade 8
                                    Mathematics
                       Unit 5: Algebra, Integers, and Graphing


Time Frame: Approximately four weeks


Unit Description

The unit focus is on determining relationships of patterns. Representations of these
relationships are made using tables, graphs and equations. Equation solutions and
descriptions of how rates of change in one variable affect the rate of change in the other
variable are also explored as graphs are analyzed and slopes are discussed. The collecting
and analyzing of data into appropriate displays including box-and-whiskers plots are
explored. The unit includes explanations of factors that affect measures of central
tendency.


Student Understandings

Students show a strong command of working with positive whole number exponents in
evaluating expressions, in computing with scientific notation, or in representing
quantities in exponential growth settings. Students are able to use formulas for
perimeter/circumference, area, surface area, and volume settings flexibly and solve for
missing values in linear formulas, such as temperature conversion formulas. They can
discuss rates of change, such as found in the graphs of linear relationships. Students
develop an intuitive grasp of slope and will be able to compare and contrast slope in
linear settings. They are capable of shifting among representations and discussing the
nature of such representations for functions as tables, graphs, equations, and in verbal and
written formats. Students determine which display is appropriate for given situations and
find the information from a data set that is needed to make a box-and-whiskers plot. They
also determine how various factors affect measures of central tendency.


Guiding Questions

       1. Can students apply positive whole number exponents in evaluating
          expressions and in computing with scientific notation?
       2. Can students apply the order of operations in evaluating expressions involving
          fractions, decimals, integers, and real numbers along with parentheses and
          exponents?
       3. Can students shift among written, verbal, numerical, symbolic, and graphical
          representations of functions?
       4. Can students solve and graph solutions of multi-step linear equations and
          inequalities?



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       5. Can students explain and form generalizations about how rates of change
           work in linear and exponential settings?
       6. Can students describe and compare rates of change for situations where
           change is constant or varying?
       7. Can students construct a table of values for a given equation and graph it on
           the coordinate plane?
       8. Can students determine which display is appropriate for a given situation?
       9. Can students create a box-and-whiskers plot and explain the information that
           it shows?
       10. Can students take a data set and determine the effect an added number will
           have on the different measures of central tendency?


Unit 5 Grade-Level Expectations (GLEs)

GLE # GLE Text and Benchmarks
Number and Number Relations
2.      Use whole number exponents (0-3) in problem-solving contexts (N-1-M) (N-5-
        M)
4.      Read and write numbers in scientific notation with positive exponents (N-3-M)
5.      Simplify expressions involving operations on integers, grouping symbols, and
        whole number exponents using order of operations (N-4-M)
Algebra
10.     Write real-life meanings of expressions and equations involving rational
        numbers and variables (A-1-M) (A-5-M)
11.     Translate real-life situations that can be modeled by linear or exponential
        relationships to algebraic expressions, equations, and inequalities (A-1-M) (A-
        4-M) (A-5-M)
12.     Solve and graph solutions of multi-step linear equations and inequalities (A-2-
        M)
13.     Switch between functions represented as tables, equations, graphs, and verbal
        representations, with and without technology (A-3-M) (P-2-M) (A-4-M)
14.     Construct a table of x- and y-values satisfying a linear equation and construct a
        graph of the line on the coordinate plane (A-3-M) (A-2-M)
15.     Describe and compare situations with constant or varying rates of change (A-4-
        M)
16.     Explain and formulate generalizations about how a change in one variable
        results in a change in another variable (A-4-M)
Data Analysis, Probability, and Discrete Math
34.     Determine what kind of data display is appropriate for a given situation (D-1-
             M)
35.     Match a data set or graph to a described situation, and vice versa (D-1-M)
37.     Collect and organize data using box-and-whisker plots and use the plots to
        interpret quartiles and range (D-1-M) (D-2-M)
39.     Analyze and make predictions from discovered data patterns (D-2-M)



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                        Louisiana Comprehensive Curriculum, Revised 2008




GLE #      GLE Text and Benchmarks
40.        Explain factors in a data set that would affect measures of central tendency
           (e.g., impact of extreme values) and discuss which measure is most appropriate
           for a given situation (D-2-M)



                                      Sample Activities


Activity 1: Camping Sounds (GLEs: 12, 13)

Materials List: Grid BLM, Camping Sounds! BLM, Grid for Questions 5 and 6 BLM,
pencils, paper

Have the students work in pairs for this activity. Put the table of        n um be r
                                                                                         n um be r o f
                                                                                         a ni ma l s o u nd s
                                                                           o f n ig ht s h ea r d
values on the board or overhead. Students will begin this activity
by looking at the table of values. Have students write in their math               1 4
learning log (view literacy strategy descriptions) a short paragraph
describing a situation that the table represents. Ask students to                  2 7
predict the number of animal sounds they would hear if they
                                                                                   3 10
camped out ten nights. Have students explain their predictions and
encourage them to make some rule for the data in the chart                         4 13
( y = 3x+1 ).Distribute the Grid BLM and have students plot these
four ordered pair on a coordinate grid and explain the relationship
that is shown.

Distribute Camping Sounds! BLM and Grid for Questions 5 & 6 BLM. Give students
time to solve each of the problems. Discuss results.


Activity 2: Beaming Buildings! (GLEs: 10, 11, 12, 13, 14, 39)

Materials List: Patterns and Graphing BLM, More Practice with Patterns BLM, Grid
BLM, Patterns and Graphing Practice BLM, pencils, paper, toothpicks (15-20 per student
pair)

Distribute 15 – 20 toothpicks to each pair of students. Have the students place the
toothpicks in the arrangement shown below for buildings one through three. Have them
create a table of values showing the
building number represented by the x-
value and the number of support beams it
takes to build the building as the y-value                    bui ld in g 2      bui ld in g 3
                                                bui ld in g 1
for this pattern of buildings through
building #6.



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                                       Louisiana Comprehensive Curriculum, Revised 2008


For example, building one takes 7 beams, building 2 takes 12 beams, etc.

   x = building #        y = # beams

                    1   7

                    2 12
                    3   17
                    4   22
                    5
                                                          Figure 1    Figure 2      Figure 3
                        27
                                          y = 5x + 2
                 6       32
                 7      37
                 8      42
                 9      47
                10      52


Encourage students to match the x value as seen in the diagram above (the loops show the
five beams added each time a new building is developed). Ask students to use their table
values to predict the number of beams it will take for building #10.

At this point the students might notice that the number of beams a) increase by 5 and, b)
the ones digit alternates between 7 and 2. How often students have used tables to
develop ‗rules‘ for patterns may influence what they see in the pattern. Use leading
questions if necessary to help them develop these skills.

Engage the class in a discussion about these predictions and what they based their
predictions upon. Ask the students to work with a partner and use RAFT writing (view
literacy strategy descriptions) to determine which building would take 62 beams and have
the student(s) explain their thinking. Have them begin making the connection between
the data in the chart to the linear equation. RAFT writing is used once students have
gained new content so that they have opportunities to rework, apply and extend their
knowledge. The R is used to describe the role of the writer; the A refers to audience or to
whom the RAFT is being written; the F is used to give students a form to follow in their
writing; and the T is the subject matter or topic of the writing. In today‘s assignment,
         R = (a rule to determine the number of beams)
         A = (prove to the reader that your rule works for 62 beams)
         F = (write in the form of a paragraph)
         T = (why it is important to look for rules or ‗short cuts‘ in math)
Pairs of students should share their writing with other pairs of students. Students should
listen for accuracy and logic in each others‘ RAFTs.

Next, instruct students to plot the ordered pairs of building numbers and number of
beams on the Grid BLM to determine if the relationship is linear. Challenge pairs of
students to create a ‗what-if‘ question for another pair of students and be able to justify
their answers. Give students time to share questions with other pairs of students.

Distribute Patterns and Graphing BLM and have students work independently to
complete the questions about the patterns on this activity sheet. Once students have had
time to complete the activity sheet, have them work with a partner to discuss their
answers. Provide time for students to ask questions of other students if needed. If



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                       Louisiana Comprehensive Curriculum, Revised 2008


students still need more practice distribute Patterns and Graphing Practice BLM. This
BLM might also be used for homework practice.


Activity 3: From Table to Graph to Conjecture (GLEs: 2, 11, 13, 16)

Materials List: Circles and Patterns BLM, Grid BLM, pencils, paper

Have students create a table of values for the area of a circle using Circles and Patterns
BLM. Have students complete the table for circles with radii of 1 through 5 units. Have
students plot the ordered pairs  r,  r 2  on the Grid BLM and compare the formula for
finding the area of a circle with the graph of the points from the table of points. Have
students make observations about the shape of the graph (linear or non linear). Lead
students to make a conjecture about the effects of doubling or tripling, the radius on the
area of the circle by examining the table of values and the graph. Have students share
their conjectures and justify their reasoning as to why they think their conjectures are true
using the professor know-it-all strategy (view literacy strategy descriptions). Select
students at random to share their thinking and justify their reasoning as to why their
conjectures are true. Discuss the shape of the graph and whether the relationship is linear
or not.


Activity 4: Speed, Time and Distance (GLEs: 10, 14, 15, 34, 40)

Materials List: one stop watch per group, tape measures or meter sticks, paper, pencils,
Grid BLM, colored pencils

Instruct students to work in groups of four. One student in each group should have a stop
watch or second hand on a watch to be used as a timer. If possible, borrow stop watches
from the science or P. E. department. Have students mark off a distance of 10 meters and
take turns walking the distance and gathering data about the time it takes each student in
the group to walk the distance.

Students should then each take their time from the 10 meter walk and work independently
to create a table of values representing the time it takes him/her to walk distances of 15,
20, 25, and 30 meters at the same rate that was determined at 10 meters.

Have students determine the equation that represents their speed (unit rate). Next, have
students plot the coordinates from their tables on a coordinate grid with their three other
group members so that the values for 10, 15, 20, 25 and 30 meters are used for each of
the group members. Each student‘s graph should be done with a different color pencil for
comparison. Challenge groups of students to develop a conjecture as to the relationship of
time and distance shown on the graph. This is a good time to have the students think
about the independent and dependent variables and where these are placed on the graph.
Discuss graphs from the different students‘ data, and discuss whether the graphs are
linear and why they are or not. Students should calculate their speed and relate this to the



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rate of change on the graph. Lead a discussion to help the students begin to see the
connection between their speed, the coefficient of x and the slope of the line. Have
students decide as a class how they can best represent the class average walking speed.
Assist groups in collecting class data, and challenge groups to prepare at least two
different graph representations of the class data.


Activity 5: Linear Equations—Fuel Consumption (GLEs: 10, 11, 13, 14)

Materials List: paper, pencil, newsprint, markers, Internet, spreadsheet, graphing
calculator (optional)

Have students research the fuel (natural gas, electricity, gasoline) consumption for
various types of furnaces, refrigerators, water heaters, and automobiles. Using the fuel
consumption data for automobiles, have students use a spreadsheet to construct a table of
x- and y-values where x represents, for example, the gallons of gasoline and y represents
miles driven. This website gives many models of cars and their fuel consumption ratings:
http://www.autosite.com/content/research/index.cfm?Action=search&id=22041%3BASI
TE&Search=fuel+economy&Go=Go . Make sure to have some fuel consumption
ratings to use in class if the computer lab is not available for research. Once the table of
values are complete, have students plot the (x, y) coordinates from the table on the
coordinate plane. If graphing calculators are available, have the students set up a table of
values and plot the points to determine if the points are linear.

Next, have students write an equation for the fuel consumption. For example, if a
refrigerator uses 180 kwh per month, then an equation that depicts this situation is
 y  180 x where x is the number of months and y is the total kilowatts used. A website for
kilowatt rates of usage is http://www.ucemc.com/kwh%20usage%20chart.htm. Lead a
discussion about the equations developed for the data so that students understand the
applicability of their use of algebra. Have students brainstorm (view literacy strategy
descriptions) other situations with constant or varying rates of change (i.e. altitude and
barometric pressure, number of rotations made with the pencil sharpener and the length
of the pencil). After generating a list of these situations, have the students use a graphic
organizer (view literacy strategy descriptions) to organize their thinking. A Venn
diagram would be a method to use or classifying these situations representing constant or
varying rates of change. To use the Venn diagram, have students sketch two large
overlapping circles on newsprint or other large sheet of paper. Have students write above
one of the circles constant rates of change, and above the other varying rates of change.
Give students time to classify these situations. Some of their situations might be able to
be classified into both categories, such as driving or riding in a car (in town it would
probably be varying and on the highway it might be constant). Give time for groups to
justify their diagrams using professor know-it-all (view literacy strategy descriptions).
Using this strategy, do not take volunteer groups but randomly select groups of students
to justify their classification.




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Activity 6: Graphs to Situations! (GLEs: 11, 15, 16, 35)

Materials List: paper, pencil, newsprint, markers, Graph Situations BLM (cut these apart
prior to class), Graph Situations for Students BLM, Graph Situation with Possible Graph
Sketches BLM, Graph Situations Process Guide BLM

Print the Graph Situation BLM, and cut these problems apart so that each group of 4
students gets one situation. These problems show situations that can be graphed in the
first quadrant on a coordinate graph.

Distribute newsprint. Explain to the students that they should sketch a graph with axis
labels but without numbers and a title so that it can be used in a matching activity. Have
each pair of students create a graph that matches the situation they were given,
identifying them only with the letter printed with the problem. These graphs should be
sketches to illustrate the rates of change in each situation. Remind students to work
quietly and speak softly so that the other groups cannot hear what is being graphed. Each
graph will be posted on the wall and the other students will try to match each graph to a
situation so they don‘t want to give the answer away by talking too loudly.

When the graphs have been made, give students the Graph Situations for Students BLM
which is a reordered list of all situations with blanks rather than letters beside each one.
Have students work in pairs to match the graphs on the wall with the situations. Students
should also write whether the relationship shows a constant or varying rate of change
next to the situation as they fill in the letter.

Lead a discussion about student conclusions after they have matched the graphs. Possible
graph representations from the situations are given on Graph Situation with Possible
Graph Sketches BLM.

As an assessment of student understanding of the shapes of graphs, use the process guide
(view literacy strategy descriptions) to help students further process their understanding
of the various graph situations. Process guides scaffold students‘ comprehension within
unique formats. Provide each student with the Graph Situations Process Guide BLM.
This process guide is provided for use as an assessment of student understanding of the
relationship of graphs and real-life situations. Have students work with a partner to
illustrate the situations requested on the process guide and then have pairs of students
compare their sketches with those of other student pairs. If there are wide discrepancies
in the sketches, take this time to discuss results as a class.


Activity 7: Real-Life Inequalities (GLEs: 11, 12)

Materials List: paper, pencil, Inequality Situations and Graphs BLM

Have students write an inequality that represents a situation where a person has an
allowance of $25 a month and must spend no more than 60% of this amount ($15) on



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snacks and entertainment. Have students find solutions to the inequality. Sketch a number
line on the board and have the students determine a method of plotting their solutions.
Lead a discussion about the meaning of the inequality and its solutions. Have the students
work in pairs to complete the following inequality.

       There is a building code in some states that requires at least twenty square feet of
       space for each person in the classroom. Suppose a classroom is 28 feet long and
       18 feet wide. How many people can be in the classroom? Explain your answer
       and write an inequality to represent the solution. ( 20p  504; p  25.2 ; no more
       than 25 people)

Distribute Inequality Situations and Graphs BLM to students. Have the students write the
inequality that matches the situation, solve the inequality, sketch a graph to represent the
solutions, and be ready to justify their solutions to other groups.

Have students share solutions with other groups and discuss as a class any solutions that
they would like to challenge.


Activity 8 : Making a Box-and-Whisker Plot (GLE : 37)

Put the following numbers on the board: 10, 24, 16, 23, 20, 22, 14, 25, 19, 17, 18.
Indicate to students that these numbers represent the points Joe scored during the
basketball season last year and that a box-and-whisker plot will be made using this data.
Tell the students that a box-and-whisker plot requires five data points: low value, lower
quartile, median, upper quartile, high value.

Ask the students to put the scores in order from lowest to highest. Students should then
find the median of the data set (19). Show students how to find the lower quartile (16.5)
by finding the median of all values less than, but not including, the median. The students
should then find the upper quartile by determining the median of the upper half of the
data set (22.5).

Have students draw a number line across the bottom of their paper and number
sequentially along the number line. Once they have established their scale, the students
should then plot each of the data point above the corresponding value on the number line.
The ‗whiskers‘ extend from the high and low values to the upper quartile value and lower
quartile value, respectively. The ‗box‘ is drawn between the upper and lower quartile
values. The median should be marked with a vertical line.

Discuss the information the students can gather by looking at the box-and-whiskers plot.
Students should see that 50% of the games Joe scored between 16.5 and 22.5 points. It
also shows that in the lower scoring games (3 of them), Joe‘s scores varied more than in
the higher scoring games. Students may see many other things from the data. Be sure
that they understand that there are equal numbers of data points in each quarter of the
graph, each representing 25% of the total list.


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                              Joe's Basketball Points




           10    12     14     16     18      20        22   24   26



Activity 9: T Shirt Auction (GLEs: 2, 37)

Materials List: paper, pencil, T-Shirt Auction Word Grid BLM

Use a math modified word grid (view literacy strategy descriptions) to begin this activity.
Provide the students with the T-Shirt Auction Word Grid BLM. Have students complete
the T-Shirt Auction Word Grid by substituting the cost of the T-shirt into the situations
given. Explain that they should calculate the costs of the T-shirts using the operational
directions and record results on the chart.

 Have the students work in groups of four to create a box-and-whisker plot. Assign each
group one of the T-shirt prices from the chart from the T-Shirt Auction Word Grid BLM,
either the $10, $9, or $11.50 t-shirts. As students prepare their box-and-whisker plot,
groups should use all amounts in the column for their T-Shirt. This should give students
enough different auctioned prices so that the student groups can find the five data points
needed to make a box-and-whisker plot. Groups should find the five data points
necessary and then sketch a box-and-whisker plot of the auctioned prices of the T-Shirt.

The students should then post their box-and-whisker plots so that the class can compare
the median, upper quartile and lower quartile of each of the prices. Students should also
find the range of the prices and which operational direction gave the largest range.
Challenge the students to explain why the range was larger with some of the operations.


Activity 10: Reporting Results of the T-Shirt Auction! (GLEs: 10)

Materials List: paper, pencil, Reporting Results BLM, T-Shirt Auction Word Grid BLM
from Activity 8, chart paper, markers

Provide students with Reporting Results BLM. Have them complete the chart using
information from the T-Shirt Auction Word Grid BLM and then determine total amount
made at the auction on the T-Shirts that cost $11.50.

Have the students prepare a graph representing the data from the T-Shirt sales that best
illustrates the percent of T-Shirts sold at the auction at the different prices.




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Have students make a comparison of the effect that the exponents have on the auctioned
price and how this affected the range of the prices. Have the students present their
representations to the class for discussion.


Activity 11: Rate of Change (GLEs: 10, 14, 15, 16)

Materials List: paper, pencil, Rate of Change Grid BLM, colored pencils, graphing
calculators (optional)

Provide students with Rate of Change Grid or graphing calculators. Have them create a
table of at least five values, including x  0 and then two opposite x values, and plot
coordinates for y  x 2 , y  2 x, y  x  2, y  x 3 on a coordinate grid.

x                  x2                  2x                  x-2               x3

Have them plot and connect the points using different colors for the lines on their graphs.
Pair students and give them time to create conjectures about the relationships of these
equations and share conjectures with the class. Use the professor know- it- all strategy
(view literacy strategy descriptions) as students are required to describe their conjectures
to the class. Pairs of students should develop situations that represent at least two of the
equations graphed. Have students discuss the rate of change and whether or not this rate
of change is constant. During the professor know-it-all discussion, ask the students
questions such as the following: Which of the equations appears to have a linear
relationship? How can you tell? Does one of the linear relationships look as if it changes
at a faster rate than another? Discussion should evolve to the slope or slant of y = 2x is
steeper than y = x - 2 .

Have students go to their tables of values for their linear equations and compare the
changes in x and y. Ask questions that will lead the students to discover the move it takes
to get from one point on y = 2x to the next point going up or down first, and then right or
left (up 2 right 1), then begin looking at the slope as rise over run. Repeat this with
 y = x - 2 . Help students to see that that the counting process (rise/run) indicates that the
rates of change are constant in the linear graphs, but not in the non-linear graphs. Lead a
discussion about the differences in the non-linear equations. Have students go to their
table of values and compare the changes in their x- and y-values and how these changes
differ from the changes in the linear equations. Assign groups of four students one of the
four equations to describe a real-life situation that could be modeled with the equation.
Assist students in developing these real-life situations. It might help make sense to them
if they think about lengths, areas, and volumes.




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Activity 12: Computing Using Scientific Notation (GLEs: 2, 4, 5)

Materials List: paper, pencil, Scientific Notation BLM, calculators

Begin the activity with a discussion including the fact that scientific notation is a method
of recording very large and very small numbers using powers of ten. Have the students
write one hundred twenty-three billion (123,000,000,000). Lead a discussion about how
numbers this large become cumbersome and not easy to record accurately. This is why
scientific notation is used. This same number can be written as:
                                        1.23 x 1011

The first number 1.23 is called the coefficient. It must be greater than or equal to 1 and
less than 10.

The second number is called the base. It must always be 10 in scientific notation. The
base number 10 is always written in exponent form. In the number 1.23 x 1011 the
number 11 is referred to as the exponent or power of ten.

To write a number in scientific notation, place the decimal point in the original number so
that a number between 0 and 10 is created. Moving the decimal to the left is the same as
dividing by a power of 10. To determine what power of 10 was used in the division,
count the number of decimal places that the decimal would have to be moved to get back
to the original number. Use this number of decimal places as the power of 10. The zeros
to the right of 3 are no longer needed as they would be eliminated when the division was
made.

       Example: 1.23000000000 x 1011 becomes 1.23 x 1011 after the zeros are dropped.

Distribute Scientific Notation BLM and give students time to complete the situations.

Lead class discussion, having students explain their results and how they represented the
numbers in scientific notation. Have students make some comparisons of these large
numbers and develop conjectures as to how the power of ten helps determine the size of
the number.


Activity 13: Make My Answer Correct! (GLE: 5)

Materials List: paper, pencil

Have students work in pairs to create a set of problems that involve integers, multiple
operations, exponents of 0, 1, 2, and 3, and grouping symbols [i.e., (4+ 3) 52 = 175].
Have students rewrite their problems without any grouping symbols [ 4 + 3 x 52 = 175)
and exchange them with another group. The second group will determine where to place
grouping symbols so that the equation is true. Students can also determine what the




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answer would be if the problem were worked without the insertion of grouping symbols
[4 + 3 x 52 = 79] .


Activity 14: Graphing Solutions to Linear Inequalities (GLE: 12)

Materials List: paper, pencil, Inequality Cards BLM (cut cards apart prior to activity)

Prepare sets of Inequality Cards BLM to be used by groups of four students. Each set of
cards contains multi-step linear inequalities (i.e., Jacob wants to give his brother at least
25 baseball cards; he knows he can get five cards in one pack. Jacob has 3 baseball cards
that were purchased separately to give his brother). Write an inequality to show how
many packs of cards Jason must buy); solve the inequality and write the answers in terms
of the variable (e.g., 5x + 3 ≥ 25; x ≥ 4.4 or he will need to purchase 5 packs of cards);
graph the solution on a number line, and write the solution to the problem in a complete
sentence.

Have students play a ―Go Fish‖ type card game using the situation, inequality, solution to
the inequality and the graph of the inequality as a book of 4 cards. The goal is to make as
many books as time allows. For example a book might be one card that says ―Jacob has
no more than 8 baseball cards,‖ one card that says ―y  8 baseball cards,‖ one card that
says ― y = 8 and the fourth card should be the number line below. .



                                6   7    8   9    10




Activity 15: Formula Madness (GLEs: 2, 5, 12)

Materials List: paper, pencil, LEAP Reference Sheet BLM from unit 4, Formula Madness
BLM

Provide students with various types of common formulas such as those on the LEAP
Reference Sheet BLM. Reciprocal teaching (view literacy strategy descriptions) is a
strategy in which the teacher models and the students use summarizing, questioning,
clarifying, and predicting to better understand the content. Select a word problem
involving the surface area of a rectangular prism and model for the students how to use
that formula to solve the word problem by substituting the values given into the formula.

Have students read Situation 1 on the Formula Madness BLM. Have one or more
students explain to the class what the situation states. Question the students as to the size
of the sand box and have them give an example of another object to relate to the size of
the sandbox (i.e. double doors coming into the hallway at school). Ask students what
dimensions were used for the sandbox. Students need to make the connection between



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the term dimension and the actual dimensions of the objects in these situations. Have
them make a prediction about the volume of sand needed before solving the problem.

Distribute the Formula Madness BLM and have students work either individually or with
a partner to solve the problems. Have students use reciprocal teaching and explain the
remaining situations to another group, modeling the four processes that the teacher
modeled with problem 1.


Activity 16: Celsius to Fahrenheit to Formula (GLEs: 15, 16)

Materials List: paper, pencil, strips of paper (about 1.5‖ x 8.5‖) for each student

Provide students with strips of paper about 1.5‖ x 8.5‖. Have the students fold their strip
of paper into fourths. Next, have students draw a vertical number line 8.5‖ long. Instruct
students to label the bottom of the number line 0 C and 32 F. Discuss the freezing point
of water and the degree measurements. Ask if students know the boiling point of water;
many will remember this from science class. Have them label the top of their number line
100 C and 212 F. Since the strip of paper has been divided into fourths, the students
have marks for 25 C, 50 C, and 75 C. Challenge the students to determine the
equivalent degree measurement for Fahrenheit for these four values, and discuss student
results and strategies used.

Have pairs or small groups of students determine equivalent measures for 5 C, 10 C,
15 C and 20 C. Once the students have determined these equivalencies, have them plot
the coordinates for C and F degrees on a coordinate grid, and determine whether the
relationship is linear. Discuss the rate of change of when converting Celsius to Fahrenheit
degrees, and have the students determine this relationship by using the table data or the
graph. Using values in the chart, 90 C   32  F  which will simplify to 9 C   32  F  .
                                    50                                     5
(If the students use the midpoint of (50, 122) and the boiling point of (100, 212) the slope
is 90 .)
   50




Activity 17: Constant and Varying Rates of Change (GLE: 15)

Materials List: paper, pencil, Constant and Varying Rates of Change BLM, Situations
with Constant and Varying Rates of Change BLM, Grid BLM

Distribute Constant and Varying Rates of Change BLM and have students complete the
table of functional values that depict a constant rate of change of a specified amount. For
example, the BLM situation sets the value of y as 2 x. That is, for each increase of 1 unit
                                                     3

in the x-coordinate, there is an increase of 2 units in the y-coordinate. So, the constant
                                             3
                  2
rate of change is 3 to 1. Have students complete the tables and graph the coordinates.
Discuss the slope of the graph and how this relates to the rate of change.



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Distribute Situations with Constant and Varying Rates of Change BLM, and give
students time to set up their tables of values and graph these situations. Discuss results,
identifying situations that are constant and those that are varying.

Lead a discussion assuring students they are making the connection between the rate of
change and the slope of the line and the constant of the equation that is the y-intercept.


                                    Sample Assessments


Performance assessments can be used to ascertain student achievement. For example:

General Assessments
       The students will prepare a brochure comparing mileage of different cars. The
         student will include graphs of at least three cars and their mileage and explain
         the relationship of the mileage and the slope of the line. A website that the
         students can use to find different mileage comparisons is
         http://www.fueleconomy.gov/feg/FEG2004_GasolineVehicles.pdf
       The student will prepare a presentation using number sequences or pattern
         sequences and describe when the sequence results in a linear relationship and
         how they determine this.
       Provide students with a list of situations that can be represented with an
         algebraic expression. The student will write the expression that represents the
         situation.
       Provide the student with a list of expressions involving variables with whole
         number exponents up to three. The student will evaluate the expressions using
         a given set of values for the variables.
       Provide the student with a table of values that describe a linear situation (i.e.,
         a constant rate of change) and the student will determine the rate of change.
       Whenever possible, create extensions to an activity by increasing the
         difficulty or by asking ―what if‖ questions.
       The student will create portfolios containing samples of experiments and
         activities.


Activity-Specific Assessments

        Activity 5: Provide graphs of situations to the student. The student will match
           a set of graphs with a set of linear equations or inequalities.

        Activity 6: Provide the student with the Process Guide BLM to be used as an
           Activity Specific Assessment.

        Activity 9: Each pair of students will prepare a presentation for the class with
           their data plotted in a box-and-whiskers plot about the auctioned prices of the


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          movie star T-Shirts. Presentations will include the median auctioned price for
          their data and any outlier data that was gathered. Students will also explain
          why the range is larger with some of the operational directions than with
          others.

       Activity 17: The student will write a situation with a constant rate of change
          and create a question that could be answered from the situation. The student
          will write a situation with a varying rate of change and create a question that
          could be answered from the situation.




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                                        Grade 8
                                      Mathematics
                              Unit 6: Growth and Patterns


Time Frame: Approximately three weeks


Unit Description

This unit examines the nature of changes to the input variables in function settings
through the use of tables and sequences. There is emphasis on recognizing and
differentiating between linear and exponential change and developing the expression for
the nth term for a given arithmetic or geometric sequence.


Student Understandings

Students recognize the nature of linear growth and exponential growth in terms of
constant or multiplicative rates of change and can use this to test their generalizations.
They understand that a table, a graph, an algebraic expression, or a verbal description can
be used as different representations of the same sequence of numbers.


Guiding Questions

       1. Can students differentiate between linear and exponential growth patterns and
          discuss each verbally, numerically, graphically, and symbolically?
       2. Can students develop and generalize the rule for finding the nth term for a
          sequence of numbers?
       3. Can students sketch and interpret a trend line?


Unit 6 Grade-Level Expectations (GLEs)
GLE     GLE Text and Benchmarks
#
Algebra
13.     Switch between functions represented as tables, equations, graphs, and verbal
        representations, with and without technology (A-3-M) (P-2-M) (A-4-M)
14.     Construct a table of x- and y-values satisfying a linear equation and construct a
        graph of the line on the coordinate plane (A-3-M) (A-2-M)
Data Analysis, Probability, and Discrete Math
38. Sketch and interpret a trend line (i.e., line of best fit) on a scatterplot (D-2-M) (A-
      4-M) (A-5-M)
39. Analyze and make predictions from discovered data patterns. (D-2-M)



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Patterns, Relations, and Functions
46. Distinguish between and explain when real-life numerical patterns are
      linear/arithmetic (i.e., grows by addition) or exponential/geometric (i.e., grows by
      multiplication) (P-1-M)
47. Represent the nth term in a pattern as a formula and test the representation (P-1-M),
      (P-2-M), (P-3-M) (A-5-M)

                                     Sample Activities

Activity 1: Find that Rule (GLEs: 13, 39, 46, 47)

Materials List: Find that Rule BLM, More Patterns and Rules BLM, pencil, paper, math
learning log, poster paper, markers

Write the following questions on the board or overhead and have the students copy them
in the left column of a sheet of paper which has been formatted for split-page noteaking
(view literacy strategy descriptions). Students will complete the split-page notetaking as
they complete the activity.

                                     Arithmetic and Geometric Number Patterns
1) Notice that the consecutive x-
values in your tables change by
1. What do you notice about the
difference in the y values in
consecutive patterns in A – E on
Find the Rule BLM?
2) Define: arithmetic sequence
3a) What do you notice about
the differences in y values of the
More Patterns and Rules BLM
when the x values are
consecutive values?
3b) Divide each y value by the
preceding y value and determine
if there is a pattern.
4) Define: Geometric Sequence


Divide the students into groups of four. Distribute Find that Rule BLM and give the
students time to find the perimeters and then the areas of the arrangements, recording
each in the appropriate tables. Students should find the ―rule‖ for finding the perimeters
and areas in the summary chart on the second page of the BLM. Lead the class in a
discussion about the rules, having students explain how their rule would help them
determine the perimeter or area of the 100th or 150th arrangement.




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Define arithmetic sequences as sequences in which the difference between two
consecutive terms is the same and geometric sequence as sequences in which the quotient
between two consecutive terms is the same. Discuss whether the rules illustrate an
arithmetic or geometric sequence. Make sure the students understand that all of the
perimeter patterns show a linear relationship. The area relationships A-C show linear
patterns. Pattern E is an area relationship. It is not a linear or geometric; it is a quadratic
(power of 2).

Once the students have completed the Find that Rule BLM, instruct them to answer
question 1 in their split-page notetaking foldable.

Distribute the More Patterns and Rules BLM and have the groups work to complete the
BLM. Once the students have completed the questions, have groups of four get with
another group of four and discuss their answers. Circulate and redirect student thinking
as any questions or misconceptions arise.

Have students explain in their own words the difference between arithmetic and
geometric patterns in their math learning log (view literacy strategy descriptions).


Activity 2: Use That Rule! (GLE: 13, 46)

Materials List: Use That Rule! BLM, paper, pencil

Before class begins, write on the board before class:
       R – The role of the writer is from the perspective of the tile pattern;
       A – The audience is another math student
       F – The format is a summary of known facts
       T – The topic is the relationship of the number pattern and the graph that
       represents the number pattern.

Talk briefly about RAFT writing (view literacy strategy descriptions) indicating the
purpose of the writing is to help clarify, recall and question further ideas. This is the first
time RAFT writing is used and it will work better if everyone uses the same topic this
time for comparison and clarification.

For example: I will use the pattern “four times x +1”. My audience is my partner.

Partner, the arrangements
in my pattern indicate that
the area of any
arrangement in my pattern
can be expressed by the            1             2                    3

rule 4x + 1. All I need to do                 R u l e: 4x + 1 = Ar ea

is multiply the arrangement number by 4 and add 1. One way to determine the pattern is
to make a table like the one shown below:



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x (arrangement        y ( area)
number)
1                     5
2                     9
3                     13

There is a difference of 4 between two consecutive y values. The rule is y= 4x +1, so the
20th arrangement would have an area of 81 square units. I also know that this is an
arithmetic sequence, and if I graphed the points, I would get a line. The slope of the line
is 4, and the y-intercept is 1.

Distribute Use That Rule BLM and have students generate a mathematical representation
of the rule and sketch the first three arrangements of the pattern. Then generate ten terms
of the sequence and create a table of values for the arrangement number and the area
and/or perimeter relationship of the pattern that was developed.

Have students generate the sequence of values for the rule ―start with $1 and double your
money each day.‖ Then have them generate the values for the rule. Start with $1 and add
$2 each day. Lead a discussion as to the difference in these two examples and how an
arithmetic sequence is different from a geometric sequence.


Activity 3: Make Up a Rule (GLE: 13, 46)

Materials List: Practice with Rules BLM, paper, pencil

Have students work in pairs to generate an arithmetic or geometric sequence of numbers.
Have pairs exchange their created sequences without providing the rule. The challenge is
to generate the rules for the sequences. Give student pairs time to determine the rule for
the sequences, write the sequence and rule on their paper, and pass the sequence to
another group. Have student pairs continue this until they have generated rules for
different sequences from at least two other students. When students have completed the
rules for the sequences, have them determine if the sequence is arithmetic or geometric.

Distribute the Practice with Rules BLM and give students time to complete the sequence
and write the rules. During discussion of the BLM, challenge the students to explain why
they know if the sequence of numbers is linear or not.




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Activity 4: Real Rules (GLEs: 46, 47)

Materials List: Real Rules Car Mileage Chart BLM , newsprint, markers, Real Situations
with Sequences BLM, paper, pencil, graph paper

Distribute Real Rules Car Mileage Chart BLM. Discuss what information students can
gather from the chart. Model how to use the information from the charts and develop a
rule.

Examples:
                                                             11
        If I use the Ford F150 2WD pickup, the chart shows       miles per gallon of
                                                             15
       gasoline. A pattern showing the cost for the number of gallons of gasoline
       pumped and the cost of the gasoline (1, $2.55; 2, $5.10, 3, 7.65, etc). My rule
       might be y = 2.55x.

       If Joe is driving a C15 Sierra Hybrid on the highway and Bill is driving the Ford
       F150 on the highway, use the chart to develop a rule that would compare the
       difference in miles traveled per gallon of gasoline (1 gallon, 5 miles difference; 2
       gallons, 10 miles difference; 3, 15; etc). Question: If Joe and Bill each buy 20
       gallons of gasoline, how many more miles will Joe be able to travel? Explain
       your mathematical rule (Joe‘s mileage (j) = 5(Number of gallons of gasoline
       purchased) and write the equation (j = 5g).

Distribute newsprint and explain to the students that they will use the Real Rules Car
Mileage Chart BLM to create a linear representation of the mileage differences of the
two vehicles showing gallons of gas and miles traveled. Students should determine the
equation for the mileage graphed, then describe the rate of change and how this relates to
the slope. Have students create questions that could be answered once the rule for their
line has been determined using the nth term in their sequence. Instruct students to work in
pairs and discuss their graphs and questions before beginning professor know-it- all (view
literacy strategy descriptions). Remember, when one of the pairs of students is professor
know- it-all, the selection of him/her is random not voluntary. This assures that all
students are actively involved in understanding their rule and sequence. A site that gives
city/hwy mileage and average cost of fuel is available at
http://www.fueleconomy.gov/feg/FEG2004_GasolineVehicles.pdf .

Have students investigate the cost of mailing letters first class in the U.S. The first term
would be the cost of mailing a first-class letter not exceeding 1 ounce in weight. The next
term in the sequence is given by the sum of the first term and the constant rate increase
used by the postal service. Have students determine the nth term of the ―postal‖ sequence.
The US Postal rates are found at the following address:
http://www.usps.com/consumers/domestic.htm#first .

Distribute Real Situations with Sequences BLM, and give students time to work through
each of these relationships prior to class discussion.


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Activity 5: Name That Term! (GLEs: 46, 47)

Materials List: Name That Term BLM, pencil, paper

Provide students with Name That Term BLM and have them explore the patterns, then
answer the questions, either independently or with a partner. This activity works best if
students are not working in groups of four. Have students determine whether the
sequence is arithmetic or geometric and justify their choice using a table, graph or
explanation.

Have students develop at least two patterns and write a real-life situation that could be
illustrated with the sequence.

Students should exchange patterns with another pair of students or another individual and
discuss the sequences, justifying their table and graph relationships to the patterns.


Activity 6: How Much Do I Get? (GLE: 13, 46)

Materials List: paper, pencil, graph paper, colored pencils

Pose the following to the students and have them explore which is the better salary
option. You have been hired to do yard work for the summer, and you will be paid every
day for 15 days. But first you have to choose your salary option as (1) get paid $10 the
1st day, $11 the 2nd day, $12 the 3rd day, and so on, or (2) get paid $.01 the first day,
$.02 the 2nd day, $.04 the 3rd day, $.08 the 4th day, and so on. Have students create a
table or chart of their values and a rule that explains the relationship in the chart. Have
students decide which type of sequence each of the salary options illustrates and generate
the 15 terms in each sequence. Next, have students determine the amount they would get
paid at the end of the 15 days to determine the best salary option. Have the students graph
each of these salary options on the same graph in different colors and make observations
about the relationship of the two options. Lead the class in a discussion about how the
information in the chart, graph or rule all relates to the situation. Have students explain
in the mathematics learning log (view literacy strategy descriptions) whether either of
these situations could appropriately start at 0.

Solutions: The first situation increases by $1 a day and on day 15 would be paid $24,
while the second situation starts at $.01 and this amount doubles every day so that on the
15th day the person would be paid $163.84.

In the first situation, starting at 0 makes sense as far as how much he would get paid, but
it doesn‟t state whether or not he started with $9, so this would be important to determine
whether or not to start at 0. In the second situation, starting at 0 makes sense and could
also be illustrated as his having no money before starting work.




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Activity 7: Generally Speaking (GLE: 47)

Materials List: Generally Speaking BLM, paper, pencil

Provide students with Generally Speaking BLM. This is a variation of the word grid
(view literacy strategy descriptions) literacy strategy to help students with vocabulary.
The understanding of the algebraic language is very important to the students as they
work to master the algebra GLEs.

Divide the sequences among pairs of students and instruct them to describe a real-life
situation that matches each sequence. Many of these will work with money situations.

Next, provide students with a word description of a sequence and then have them write
the nth term as an equation. Examples of word descriptions might be as follows: a) Pat
had $4 on June 1 and each month he saved $25. y = 25x + 4; b) Mary makes $6.75 an
hour for babysitting. y = $6.75x


Activity 8: Are You Sure? (GLE: 47)

Materials List: Are You Sure? BLM, transparency of Are You Sure? Directions,
newsprint, markers, paper, pencils

Before class, make copies of Are You Sure? BLM and cut apart the cards. Distribute one
or two number sequences to each pair of students.

Make a transparency from Are You Sure? Directions BLM and place it on the overhead.
Have the students follow the directions for the sequences they were given. Give them
time to follow the instructions.

Have students write their questions related to their sequence on newsprint, including
some questions in which students are to find the value if the term or arrangement number
is known and others in which the term or arrangement number is to be calculated.
Example: For the sequence 8, 10, 12, a student determines the rule to be y = 2x + 6 and
writes one question: What will be the value of the 50th term in this pattern? Another
question that could be asked: Which term in the sequence will have a value of 26? Have
student pairs present their pattern and pose their questions to other groups or to the entire
class. Encourage class discussion as different questions are posed.




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Activity 9: From Collection of Data to Equations (GLEs: 14, 38, 39)

Materials List: bathroom cups (one per group of four), paper clips (2 per group), 40 – 80
pennies per group, 10 – 20 pieces of uncooked spaghetti per group, paper, pencil, graph
paper

Provide groups of four students with spaghetti, paper clips, small paper
bathroom cups and pennies for weight. Direct the students to make a bridge
with strands of spaghetti by suspending the spaghetti from the back of two
desks or chairs. Suspend the paper bathroom cup from the center of the spaghetti bridge
by using a paperclip and pushing it through the top of the cup.

Have students experiment to determine the weight that can be suspended from different
numbers of strands of spaghetti. Have them begin with one strand of spaghetti suspended
from two chair backs with the small bathroom cup hanging from the center of their strand
of spaghetti. Ask them to gently drop or place pennies into the cup until the single strand
of spaghetti bridge breaks, then record the number of pennies that were in the cup before
the fall.

Next, have them repeat the same process with two strands of spaghetti, record the
breaking weight, make a chart to record their data, and continue collecting breaking
weight with two, three, four and five pieces of spaghetti. Then have students plot their
data on a scatterplot and find a line of best fit. The data pattern shows a linear
relationship, and students can determine the rate of change from the information on their
scatterplot. Have the students predict the amount of weight that can be suspended from a
bridge with 10 pieces of spaghetti, 20 pieces, and describe in words why their prediction
will work.

Discuss the line of best fit for the scatterplot. This line should go through the origin and
the slope will be defined by the trend line. Students might find it easier to use a piece of
spaghetti to manipulate on their scatterplot and find the line of best fit. Once this line is
determined, students should determine the equation for the line.

Have students post their results, graph, and predictions for class discussion.




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                          Sample Assessments


Performance assessments can be used to ascertain student achievement.

General Assessments

          The student will make a concentration game matching sequences and rules
           that describe the sequence. The student will prepare at least 15 matching sets
           to complete the game.
          The student will generate at least three different patterns of area and perimeter
           and determine the rule that describes the pattern. The student will also label
           each rule as either an arithmetic or geometric sequence.
          The student will research and find a real-life situation that demonstrates an
           arithmetic sequence and another that demonstrates a geometric sequence. The
           students will present these situations to the class.
          Provide the student with a list of numbers and have the student explain in
           writing how to determine whether the list of numbers is an arithmetic
           sequence or a geometric sequence.
          Provide students with an arithmetic or geometric sequence that describes a
           real-world situation. The student will determine specific terms of the
           sequence.
          The student will determine whether a specific number is a term in a sequence
           whose nth term is given. For example, is 24 is a term in the sequence whose nth
           term is an  5  2(n  1) .
          Provide the student with several terms of an arithmetic or geometric sequence.
           The student will generate the rule and the nth term in the sequence.
          Whenever possible, create extensions to an activity by increasing the
           difficulty or by asking ―what if‖ questions.
          The student will create portfolios containing samples of their experiments and
           activities.


Activity-Specific Assessments

        Activity 2: The student will explain whether his/her sequence is arithmetic or
           geometric and why. The student will should make a class presentation of the
           pattern and graph.

        Activity 4: The students will research shipping rates from other companies
           and compare costs. The student will prepare an advertisement comparing the
           costs of the different shipping rates using some type of graphic representation.

        Activity 6: The student will explain in his/her math learning log if there is a
           time when both jobs pay the same amount and how he/she knows this.


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                                       Grade 8
                                      Mathematics
                              Unit 7: What Are the Data?


Time Frame: Approximately four weeks


Unit Description

This unit focuses on representations of data using appropriate graphs and displays.
Concepts of range, quartiles and shapes of distributions are explored as appropriate
graphic displays are explored.


Student Understandings

Students can represent and interpret one or two variable data and make graphs by hand or
using technology, where available. Students can discuss variability in data through the
nature of its spread, using range and quartiles, and illustrate the data with stem-and-leaf
and box-and-whisker plots. For two variable data, students can graph the data on the
coordinate plane and draw and interpret trend lines for the data set. In discussing
distributions, students should be able to note the effect that the shapes of different
distributions have on measures of central tendency (mean, median, and mode). Finally,
students should be able to analyze the validity projections and generalizations made about
patterns in different data sets.


Guiding Questions

       1. Can students select and defend their choice of graphs to represent data sets for
          one or two variable data?
       2. Can students discuss the nature of variability and graphically illustrate it with
          stem-and-leaf and box-and-whisker plots, as well as through the use of range
          and quartiles?
       3. Can students graph two variable data on a coordinate graph and draw and
          discuss trend lines for its pattern, if any?
       4. Can students describe the effect that various shapes of distributions have on
          the values of their mean, median, and mode(s)?
       5. Can students analyze generalizations and claims made on the basis of data
          analyses and offer discussions of the relative validity of such claims?




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Unit 7 Grade-Level Expectations (GLEs)

GLE # GLE Text and Benchmarks
Data Analysis, Probability, and Discrete Math
34.     Determine what kind of data display is appropriate for a given situation (D-1-
        M)
35.     Match a data set or graph to a described situation, and vice versa (D-1-M)
36.     Organize and display data using circle graphs (D-1-M)
37.     Collect and organize data using box-and-whisker plots and use the plots to
        interpret quartiles and range (D-1-M) (D-2-M)
38.     Sketch and interpret a trend line (i.e., line of best fit) on a scatterplot (D-2-M)
        (A-4-M) (A-5-M)
39.     Analyze and make predictions from discovered data patterns (D-2-M)
40.     Explain factors in a data set that would affect measures of central tendency
        (e.g., impact of extreme values) and discuss which measure is most appropriate
        for a given situation (D-2-M)


                                    Sample Activities


Activity 1: Getting to Know You (GLEs: 34, 39)

Materials List: Family Data BLM, transparency of completed Family Data BLM,
newsprint, markers, paper, pencil

Write the following column headings on the board Student Initials, Number of Family
Members, Age of the Oldest Child in the Family in Months, Number of Pets, and Number
of Hours you watch TV in a week. Tell the students to determine what number they
should write for each of these topics according to the people and pets in their household.
Having students determine what numbers they will write in each cell of the spreadsheet
will expedite the data collection in the classroom. Have students then complete his/her
information on the Family Data BLM by passing one sheet around the room. If a
computer is available, this data could be collected in a spreadsheet with each person
entering his/her own data. Once everyone in the class has entered his/her personal data,
copy the information on a transparency, and make a copy for each student for use with
different activities in this unit.

Have the students work in small groups to create an appropriate display of the data in
column 2 (Number of Family Members). Stress the idea that the display should give a
clear picture of the number of family members in the homes of their eighth grade
classmates. Distribute newsprint and markers and have each group prepare a bar graph or
a stem-and-leaf plot of the data. Have students hang their graphs around the room and
make observations of the different graphs of the same data. Using the information
provided on the graphs, have them predict the number of family members of most
students in the entire eighth grade.


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Using the data from Family Data BLM, have students predict the average number of
hours of TV that all of the boys in 8th grade watch in a week and the number of pets that
someone in the next hour class might have. Have students develop an argument to justify
their prediction. Have students explain how they might be able to predict the average
number of pets of all boys in the eighth grade as an entry in their math learning logs
(view literacy strategy descriptions)

Have students save their data in the chart for later use.


Activity 2: Scattered, Clustered, or Common? (GLEs: 34, 36, 38, 39)

Materials List: Family Data BLM from activity 1, newsprint or other paper for graphs,
pencils, paper

Have students refer to their Family Data BLM. Have groups of four students use data
from the spreadsheet for three different data displays. Each group should prepare three
displays from the data on the spreadsheet. Students should use data from the different
columns of information on the spreadsheet. Tell them that one of the graphs must be a
scatterplot. Have the students use a double stem and leaf plot to compare the number of
pets for the number of boys and girls in the class.

Have the students determine which lists can give them data involving two variables
needed to make a scatterplot. For example, they might consider: How do the number of
pets someone has and the number of hours they watch TV relate to each other?

Have students make the scatterplot and draw the trend line that represents the relationship
of the data. Any other comparison that students see that they find interesting can be
made. Post the student scatterplots and discuss the trend lines. Have different groups of
students explain their displays to the class. Leave these posted for observation by other
classes. These displays can be made with newsprint or any unlined or graph paper.

Instruct the students to create a circle graph to show percentages of the numbers of family
members in the households of the class (e.g., what percent of students have 3 family
members, 4 members). Students have used circle graphs in previous units and should be
able to construct circle graphs with angle measures corresponding to the percentages
found in the data.




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Activity 3: How Old Are Your Siblings? (GLEs: 37, 39, 40)

Materials List: Family Data BLM from Activity 1, Graph Characteristic Word Grid
BLM, paper, pencil

Use data from column 3, Age of Oldest Child in Family in Months, on the Family Data
BLM. Have the students put the data in ascending order. Once the students have the data
in order, ask for observations. Questions like the ones below can help the students learn
to analyze data and make predictions.
     Is there a relationship between the number of family members and the age of the
        oldest child?
     Is there a relationship between the number of children in the family and the age of
        the oldest child?

Using only the data about the oldest child in each family, have the students list the ages in
months in order from lowest to highest. There will probably be either a low or high
number in the list that will elicit discussion about what these extremes do to the data. If
there is no data extreme, make up one and have the students find the mean of the data
with and without the extreme data entry. Discuss outcome. Have students find the five
data points needed for a box-and-whiskers plot (median, low, high, upper quartile and
lower quartile). Have students create a box-and-whiskers plot using these data points. The
plots should contain a ―box‖ extending from the lower to the upper quartile, a line inside
the box showing the median, and two ―whiskers‖—one emanating from each end of the
box, extending to the low and high values of the data. Lead discussion about what the
plot shows about the data. Ask students to answer the following question using the data
examined in this class: If you were to ask any eighth grader on the playground the age in
months of the oldest child in his/her family, what would you predict his/her answer to be?
Why?

Now that students have used data in all columns of their Family Data BLM, have them
complete the Graph Characteristic Word Grid BLM using the modified word grid (view
literacy strategy descriptions) literacy strategy. Students should complete the table using
―yes‖ or ―no‖ to define characteristics of the different types of graphs. As the unit is
completed, students will make changes to their modified word grid if they have any
mistakes.


Activity 4: Who’s got the quickest REACTION? (GLEs: 36, 39)

Materials List: Reaction BLM, timer with tenths of seconds, math learning log,
protractors, pencil, paper, Internet (optional), meter sticks

Students should work in groups of four for this activity. If Internet is available, have the
students go to the website http://faculty.washington.edu/chudler/chreflex.html and read
through different reaction time activities.




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Students will complete a meter stick drop-and-catch activity. Students should mark with
masking tape, a three foot mark from the floor on a desk chair or wall and another mark
at one foot from the floor. For the meter stick drop-and-catch activity, one student will
hold the meter stick three feet above the ground with the one centimeter end of the meter
stick closest to the ground. The second student gets ready to catch the meter stick as it
drops vertically by placing his/her open hand one foot from the floor. The student who is
catching the meter stick should have his/her hand ready to catch the stick from the bottom
so that as the stick drops, the student can grab the stick. The centimeter mark on the
meter stick where the catcher grabs the stick is recorded on the Reaction BLM. Students
should collect three trial sets of data from each other and get an average reaction time for
each student. Each group member should make a prediction as to the reaction time of a
fourth trial from the data based on their first three trials and their average reaction time. If
time allows after the activity, let them try a fourth time to check their prediction.

Have students summarize how they determined their prediction in their math learning log
(view literacy strategy descriptions). To help students understand how to summarize their
method of predicting, ask them to respond by answering these questions: How did you
determine your predictions? What information did you use to make your prediction?

With the data from the activity, have students complete a histogram on the Reaction
BLM. To do this the students will need to gather information from the other groups of
students. Students should compare class data of the number of students that have average
reaction times in the different intervals: 0 - 10 cm, 11 - 20 cm, 21 - 30 cm, 31 - 40 cm,
41 - 50 cm, 51 - 60 cm, 61 – 70 cm, 71 – 80 cm or > 80 cm.

Once the histogram has been completed, lead the class in a discussion concerning the
information known about the reaction times. Next, instruct the students to determine the
percent or fraction of students in each interval and create a circle graph representing the
numbers and percent of the class in each interval. Compare circle graphs within the
groups and discuss how the percent that represents a section of the circle compares to the
degrees in a circle. Have students prove that their percent values are correct by setting up
and explaining the proportional relationship.


Activity 5: Circle Graphs (GLE: 36)

Materials List: High Cost of College BLM, pencil, paper, protractors

A circle graph is used when data are partitioned so that a ratio or percentage of each part
to the whole is needed. Have students collect data on the costs associated with attending
college. Include costs such as housing, food, transportation, books, clothes, and tuition.
Have students create a circle graph for their data using a spreadsheet program or by hand.
Use the High Cost of College BLM and have students prepare a circle graph representing
the different percentages of each category. Students should then determine which
category listed in the chart shows the greatest percent of increase from 1994. Discuss the
graphs and their method of determining the category with the greatest percent of increase.



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Have students take out their modified word grid (view literacy strategy descriptions),
check their responses for the circle graph, and make any changes that need to be made.


Activity 6: Stem-and-Leaf (GLEs: 40)

Materials List: Test Score Data BLM, pencil, paper, math learning log

Provide students the Test Score Data BLM. Ask students to develop a stem-and-leaf plot
for the data using the 10s for the stem and the 1s for the leaves. Have students determine
the three measures of central tendency (mean, median, and mode) for the set of data.
Lead a discussion about the characteristics of the data that affect the measures of central
tendency. Have students discuss how the addition of high or low values in the data affect
the mean, median, or mode.

Next, have students research the number of games won by each of the sports teams in a
certain division (e.g., the National League in baseball) during a past season. The
following websites might help with the data needed: http://www.nfl.com/stats ;
http://www.nbl.com.au/ ; and http://mlb.mlb.com/NASApp/mlb/mlb/stats/index.jsp .
Again, have students construct a stem-and-leaf plot for their data and determine the
mean, median, and mode for the number of wins. As an extension, assign students to
construct stem-and-leaf plots for opposing divisions and then compare the plots.

Have students take out their modified word grid (view literacy strategy descriptions),
check their responses for the stem-and-leaf plot, and make any changes that need to be
made.

Have students respond in their math learning log (view literacy strategy descriptions) to a
prompt about how to find different measures of central tendency from stem and leaf
plots.


Activity 7: Box and Whiskers (GLEs: 37, 40)

Materials List: Reading Box and Whiskers Plots BLM, 2 colors of sticky note paper,
pencil, paper

Distribute one sticky note paper to each student. Give girls one color and boys a second
color. Have each student write his/her age in months on a small sticky note. Once all ages
in months are on a sticky note, have students arrange their ages numerically from
youngest to oldest on the board. Once these are arranged on the board, it may be
necessary to write the numbers larger on the board for students to see from different areas
in the room. Students should first make three stem-and-leaf plots. One of the boys ages,
girls ages and the classroom ages. From the stem-and-leaf plots, have them generate box-
and-whisker plots.



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Repeat this activity with other sets of data. Have students discuss the spread of the data
by examining the plots. Have students discuss the effects of the extreme values on the
median and the mean. To generate a bigger spread in the original data, have students
include the ages of siblings in months. Distribute Reading Box and Whiskers Plots BLM
and give students time to work independently answering the questions relating to box and
whiskers plots. Discuss results as a class when students have had time to complete the
questions.

Have students take out their modified word grid (view literacy strategy descriptions) and
check their responses for the box-and-whiskers plot and make any changes that need to
be made.


Activity 8: Line of Best Fit (GLEs: 38, 39)

Materials List: Internet, paper, pencils, uncooked spaghetti

Provide students with pictures of 20 people that they are familiar with. Find pictures of
people like Tiger Woods by going to an Internet search engine and typing names of
individuals that students would recognize. Prepare a list of these people and their ages.
Have students prepare a table with two columns. The first column should be labeled
―Guess‖ and the second column should be labeled ―Actual Age.‖ Show the students
pictures of these people one at a time and have them guess their age, writing the ages in
the column labeled Guess. Next, provide students with the actual ages and have them
complete the second column labeled Actual Ages. Have students plot the guess, x, and
actual age, y, data in a scatterplot.

Next, have them examine the scatterplot to see if a trend or relationship seems to exist.
Have students use pieces of spaghetti or their pencils to approximate a line of best fit.
The line of best fit should be placed so that as many points are on one side of the line as
the other and so that as many of the points lie as close to the line as possible. Ask
students to sketch the line of best fit. Questions such as these will lead to a rich classroom
discussion:
     How close was your guess to the actual age?
     How can you tell this by looking at the graph?
     What does the graph tell you about your guesses that are above the line of best fit?

The line of best fit may be a rapidly increasing line (i.e., the slope is large) indicating a
large change in the vertical scale and only a small change along the horizontal scale.

Discuss what the results would look like if each guess was the same as the actual age (the
slope of the line would be one, and all points would lie along the line). Ask what the
equation would look like.




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Have students draw the line that would represent this line on their scatterplot. Ask them
to now look at their guesses and determine what the points below this exact line tell them
about their guesses? The points below this line show that they guessed the people were
older than they really were.

Have students take out their modified word grid (view literacy strategy descriptions) and
check their responses for the scatterplot and make any changes that need to be made.


Activity 9: Making Appropriate Graphs (GLEs: 34, 35,)

Materials List: Which Display is Appropriate? BLM, pencil, paper

Have students prepare a page for
split-page notetaking (view literacy            circle
strategy descriptions). Have                    graph
students write the names of the
different data displays in the column
on the left, and then as the teacher          scatterplot

reviews these displays, the students
should take notes on the right side of
their paper. Discuss when the                   bar graph

different displays are best used.
Students should put the notes on
their notetaking page.                         histogram


      Use circle graphs to display
       data that are partitioned so
       that a ratio or percentage of each part to the whole can be determined.
      Use a scatterplot to display data that show a trend over time.
      Use a bar graph to display data that have no numeric ordering.
      Use histograms when the data are grouped in categories along a numeric scale
       (e.g., ages of presidents when elected).

Next distribute Which Display is Appropriate? BLM and have students work individually
to complete the questions, using their notes. When students have completed the
questions, give the students time to justify their choices to a partner.


Activity 10: Match That Data! (GLE: 35)

Materials List: Match the Data and Situation - Set A BLM, Match the Data and Situation
- Set B BLM, pencil, paper

Have students work in groups of two or three to match a given data set to a display, and
to match a given display to the appropriate data set. Distribute Match the Data and


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Situation - Set A BLM and Match the Data and Situation - Set B BLM, which contain a
set of six cards with graphs that match a set of six cards with data set situations. If
possible, print Set A and Set B cards on different colored paper so they can be separated
easily should they get mixed up. These cards should be cut apart prior to the activity, and
each group will be given Set A and Set B. Each card in Set A can be matched with a card
in Set B. Lead a discussion asking students to explain what information helped match the
cards.


Activity 11: Situations to Graphs (GLE: 35)

Materials List: Situations to Graphs BLM, Graphing Situations Opinionnaire BLM,
pencil, paper

Distribute Graphing Situations Opinionnaire BLM to individual students and have them
complete the opinionnaire (view literacy strategy descriptions). An opinionnaire is used
for this activity to give students an opportunity to make decisions about how displays can
be used to represent data in different ways.

Provide students with six different situations from Situations to Graphs BLM (1-6) or
Situations to Graphs BLM (7-12). Each group of students creates six graphs that match
the situations. Caution the students not to label their graphs with the number of the
situation or any other label that will give the situation away. The students will then
switch graphs with a group that has the different six situations from the Situations to
Graphs BLM and match the group‘s graphs to the situations. Be sure to make note of
misconceptions as these discussions take place so that you can lead a discussion when
they are finished matching the cards and situations. Groups then match data sets with the
appropriate graph from the selection of graphs. Some additional sample graphs and
situations can be viewed at the following websites:
http://www.mste.uiuc.edu/presentations/motionStory.htm
http://www.nottingham.ac.uk/education/shell/graphs.htm

Have students take out their responses from the Graphing Situations Opinionnaire BLM
and make any changes that they need to make. As the final opinionnaire responses are
reviewed, they can be used to clarify any misconceptions.


Activity 12: A Stable Measure (GLE: 40)

Materials List: Data Extremes BLM, pencil, paper

Provide students with a set of numerical data containing some extreme values. Numerical
data that involve test scores provides real-life meaning to students. Have students
determine the mean, median, and mode for the data. Next, have students throw out the
extreme values and recalculate the mean, median, and mode.




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Lead a discussion about how different measures of central tendency can lead to different
conclusions. For example, a store manager kept a record of the sizes of dresses sold last
month in the formal dress department. The sizes were 8, 8, 10, 12, 14, 16, 8, 14, 12, 10, 8,
and 6. He found the mean of the dress sizes sold last month to be 10.5, the median was 10
and the mode was 8. Explain why these measures of central tendency show different
results. Have students determine which central measure would best represent the data set?

Another example: The cheerleaders were buying new tennis shoes for their next season.
The sizes needed were 4, 8, 10, 9, 11, 12, 10 and 9. Explain how the 4 in the list will
affect the mean, median and mode.

Distribute Data Extremes BLM and have students discover that the mean is most affected
by the extreme values and the median is most stable. Discuss results of BLM. Have
students use their math learning logs (view literacy strategy descriptions) to justify which
measure should be used to report such things as a class-average test score, batting
average, and most common shoe size.


                                   Sample Assessments

Performance assessments can be used to ascertain student achievement.

General Assessments
       The student will write situations that can be illustrated with a graph and then
         will exchange these situations with a partner. The partner will sketch a graph
         to represent the situation.
       Provide the student with a mean, median and mode of a set of data. The
         student will create a set of data that would result in the given mean, median
         and mode.
       The student will find an example of a graph in the newspaper or a magazine
         and explain in his/her journal what information is gained from the graph.
       Provide the student with data and have him/her develop all possible
         appropriate displays for the data. Rubrics will be used to assess the displays.
       Provide the student with a data display and have him/her interpret it.
       Provide the student with a scatterplot of data and have him/her provide the
         line of best fit and then interpret what that line represents.
       Whenever possible, create extensions to an activity by increasing the
         difficulty or by asking ―what if‖ questions.
       The student will create portfolios containing samples of his/her experiments
         and activities.
       The student will complete a data project (collection, organization,
         conclusions/predictions) and present the results on a poster to be displayed in
         the classroom.




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Activity-Specific Assessments

       Activity 2: The student will explain in his/her journal what the graphs tell
          them about characteristics of the class.

       Activity 9: Provide the student with a box-and-whiskers plot that shows ages
          of people at some event. The student will write a paragraph explaining the
          information that can be gathered from the plot.

       Activity 12: The student will explain in his/her journal how extreme values
          affect the mean, median, and mode of a data set.




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                                       Grade 8
                                     Mathematics
                              Unit 8: Examining Chances


Time Frame: Approximately three weeks


Unit Description

This unit examines sampling with and without replacement and the need for randomness
in statistical situations and how this affects games of chance. Permutations and
combinations are used in situations that describe counts for elementary ordering and
grouping. Single- and multiple-event probability situations explore the role of mutually
exclusive, independent, and non-mutually exclusive, dependent events.


Student Understandings

Students‘ understanding of choices and chances extends to include the role of
randomness in sampling and surveys, as well as for games of chance. They can analyze
the nature of independent, mutually exclusive and dependent, non-mutually exclusive
events. They can apply permutations to analyze orderings with and without replacements
and combinations and to examine the number of r-sized groups that can be formed from
n-objects or individuals. They can calculate, illustrate, and apply single- and multiple-
event probabilities for a wide variety of events.


Guiding Questions

       1. Can students recognize and discuss ways that randomness contributes to
          surveys, experiments, and games of chance?
       2. Can students determine the number of orderings (permutations) or
          combinations (groupings) that can occur under given conditions?
       3. Can students calculate and interpret single- and multiple-event probabilities in
          a wide variety of situations, including independent, mutually exclusive, and
          dependent, non-mutually exclusive settings?
       4. Can students suggest ways of minimizing bias in sampling or surveys through
          the use of random samples?




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Unit 8 Grade-Level Expectations (GLEs)

GLE # GLE Text and Benchmarks
Data Analysis, Probability, and Discrete Math
41.     Select random samples that are representative of the population, including
        sampling with and without replacement, and explain the effect of sampling on
        bias (D-2-M) (D-4-M)
42.     Use lists, tree diagrams, and tables to apply the concept of permutations to
        represent an ordering with and without replacement (D-4-M)
43.     Use lists and tables to apply the concept of combinations to represent the
        number of possible ways a set of objects can be selected from a group (D-4-M)
44.     Use experimental data presented in tables and graphs to make outcome
        predictions of independent events (D-5-M)
45.     Calculate, illustrate, and apply single- and multiple-event probabilities,
        including mutually exclusive, independent events and non-mutually exclusive,
        dependent events (D-5-M)


                                    Sample Activities


Activity 1: Selecting a Sample (GLE: 41)

Materials List: Random or Biased Sampling Opinionnaire BLM, Random or Biased
Sampling BLM, pencil, paper, brown paper bags (1 per group), color tiles (10 in each
bag: 5 of one color, 3 of another color, 1 of a third color, and 1 red)

Begin class by having students work in pairs to complete the Random or Biased
Sampling Opinionnaire BLM. Opinionnaires (view literacy strategy descriptions) are
tools used to elicit attitudes about a topic. A modified opinionnaire is being used to
generate some thinking about biased and unbiased sampling. Once the pairs of students
have completed the survey, have the pairs of students get into groups of four and discuss
their answers and reasons. Have the groups of four students write a summary statement
giving their idea(s) about random sampling. Have students share their ideas with the class
prior to the discussion of surveys.

To help students see how a random sample is selected, provide them with a brown lunch
bag containing 10 cubes or color tiles of 4 colors (5 of one color, 3 of another color, 1 of
a third color, and 1 red) and have them shake the bag. Then have one student in each
group remove a cube or tile and note its color (do not allow the students to look in the
bag prior to their data collection). To simulate a large population, replace the cube drawn,
shake the bag and draw another cube, note its color, then replace it. Have each student
determine the number of times that the process should be repeated to allow them to make
a good guess as to what the colors of the tile in the bag are and how many of each color
are in the bag, if there are 10 total cubes or tiles in the bag. Have students make
predictions as to what the next randomly selected sample color will be from their


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collection. Discuss how certain they are about their prediction, and then have them
collect the sample. Discuss why their predictions were accurate or not. Instruct the
students to use their data and make a prediction of how many tiles or cubes of each color
are in the bag if there are a total of 10 tiles or cubes in the bag. They should do this
without looking inside the bags. Have students write these predictions and then open the
bag and count the number of each color of tile in the bag. Ask how closely each
student‘s sample of 10 matched the population - this is a good time to discuss the
importance of sample size. Combine all the results in the class and then determine how
closely the aggregated data match the actual color proportions in the population. A
website with an interactive ‗Let‘s Make A Deal‘ probability page is available at
http://matti.usu.edu/nlvm/nav/frames_asid_117_g_4_t_2.html

Tell students that they will design a survey that is based on a random sampling
population. Lead a discussion about the need for surveys. When would a survey be done?
What would be gained from the survey? What can be found from a survey? Does it
matter who is surveyed? Ask student how they could determine which color T-Shirt to
order for the 8th grade party without asking each and every student in the 8th grade.
Discuss how the sample population affects the results. Distribute the Random or Biased
Sampling BLM. Have the students complete the questions independently prior to
assigning them their survey to assure understanding of biased and random sampling.

State that it would be better to survey all students; however, sometimes it is impossible to
survey all members of the population. In such cases, a sample must be taken. Help
students understand that their sampling population should be random and discuss how to
ensure this randomness. Have students determine a way to randomly select a sample from
the population of the entire 8th grade student body. Have students design a survey about
an issue that interests them and survey a sample from the 8th grade student body. Lead a
discussion about the pros and cons of just surveying their own class. Lead a discussion
about why random selection will help keep out any bias and provide a sample that is
representative of the entire 8th grade student body. Student groups should conduct their
surveys after the teacher has verified their survey question and bring results to class.

Have students complete their survey and prepare a presentation for the class by writing a
paragraph explaining their survey question, the sample population and the results of their
survey. Student groups should present these paragraphs to the class.

Have the students review their Random or Biased Sampling Opinionnaire BLM and make
any changes to their responses that can now be answered with a better understanding of
random and biased sampling.




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Activity 2: Let Me Count the Ways (GLE: 42)

Materials List: pencil, paper, math learning log, How Many Ways? BLM

Have students work in groups of four and determine how many ways they could possibly
line up in a single-file line. Ask them to record each of the ways that this could occur.
Discuss the student results and have the groups make observations about the relationship
of the results and the number of ways they could line up.
        Answer: There are four students, so there are four possible students eligible for
        position 1; 3 possible for position 2; 2 possible for position 3; and only 1possible
        for position 4 giving a total of 6 different ways for them to line up with student 1
        first. Therefore, there are a total of 24 different ways for the students to line up
        when students 2 – 4 are included in first place. The following are ways to show
        students an organized manner to determine the answer.

       The following lists are all possible ways the four students can line up:
       ABCD ACDB BACD BADC CABD CBDA DABC DBCA
       ABDC ADBC BDCA BDAC CADB CDAB DACB DCAB
       ACBD ADCB BCAD BCDA CBAD CDBA DBAC DCBA

The diagram below illustrates the same part of the problem as the list above, but in a tree
diagram. This is only for Student A, and there will be the same number of arrangements
for students B, C, and D.

                                                  3rd S tudent C   4th S tudent D

                                 2nd S tudent B
                                                                   4th S tudent C
                                                  3rd S tudent D


                                                  3rd S tudent C    4th S tudent D

           1st S tudent A        2nd S tudent B
                                                  3rd S tudent D    4th S tudent C




                                                  3rd S tudent C   4th S tudent D
                                 2nd S tudent B
                                                  3rd S tudent D   4th S tudent C


Stress that there are four student positions possible when the first person lines up, then
there are only 3 people left for the second spot, then two people left for the 3rd spot and
at this point only 1 person for the last spot.

A permutation is an arrangement or listing in which order is important. A combination is
an arrangement or listing in which order is not important. As in this example, the 1st, 2nd,
3rd, and 4th place in line is different as determined by which student is in each position,
making this a permutation. If we were forming a group of four students for a project, it
would not matter which order the students were picked, making it a combination.




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The number of permutations possible when all members of the initial set are used without
replacement can also be found mathematically by multiplying the number of members
available for each place in the order.

Example:
4 x 3 x 2 x 1 = 24 4 people for 1st place, 3 people for 2nd place, 2 people for 3rd place
and 1 person for 4th place. This is represented by factorial notation. A factorial (n!) is the
product of a whole number and every positive whole number less than itself
        Write: 4! = 4 x 3 x 2x 1
        Say: Four factorial equals four times three times two times one.

Challenge students to use what they know about permutations and determine the number
of ways that Pepperoni Pizza, Hamburger Pizza, Canadian Bacon Pizza, Vegetable Pizza
and Extra Cheese Pizza (order) can be listed on a menu for the local restaurant. Allow
students to use lists, tables, or tree diagrams to aid them in determining the number of
permutations. Have them share the diagram they used with others.
       Answer: There are 5 possible choices for the first pizza listed, 4 possible
       choices for the second pizza listed, 3 possible choices for the third pizza
       listed, 2 possible choices fore the fourth pizza listed and then only one will
       be left for the last position. 5! (5x4x3x2x1 = 120 ways)

Explain that the problems done thus far are permutations without replacement and all the
members of the initial set are used. Tell students that it is also possible to find
permutations without replacement using only some members of the initial set. For
example, if there are four students, it is possible to find all the different ways only 2 of
the students can line up.. Put students in groups. Have half of the groups create a list and
the others a tree diagram to find the different ways 2 out of the 4 students can line up.
There are 4 students so 4 students are possible for position 1, and 3 students possible for
position 2. This gives 3 possible ways to line up with student 1 first. Therefore there are a
total of 12 different ways for 2 out of 4 students to line up.

List:            AB BC CD DA
                 AC BD CA DB
                 AD BA CB DC

Tree Diagram:
The diagram below is only for Student A, and there will be the same number of
arrangements for students B, C and D. Stress that there are four students who can take
the first position when lining up and then there are only 3 people left for the second spot.
Reminder, only 2 of the 4 students available are being lined up.



                              2ndstudent B

 1st student A                  2nd student C

                               2nd student D
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Mathematically: The number of permutations possible when some members of the initial
set are used without replacement can be found mathematically by multiplying the number
of members available for each place in the order.
        Example:
         4 x 3 = 12 4 people for 1st place and 3 people for 2nd place make possible
        permutations or arrangements.

Distribute How Many Ways? BLM and have the students work individually or in pairs to
determine the number of possible outcomes for the different situations given. Have
students discuss answers with larger groups or have a class discussion.

Have students use their math learning logs (view literacy strategy descriptions) to explain
in their own words the difference in determining the possible number of combinations for
placing 3 pictures out of a set of 5 pictures in a certain position on a wall and the possible
number of ways three people can finish running a race when six people are running.
They should use a tree diagram, chart or list, or mathematical way to justify their answers
in their math learning logs.

FYI – You can‘t use the factorial notation because you are not using all members of the
set for your line up, only two of them at one time.
As you monitor students working on this problem, question them about the similarities
and differences of these situations to the previous situation.

This website can be used as an introduction to probability and has an interactive spinner,
die, and a collection of colored marbles.
http://www.mathgoodies.com/lessons/vol6/intro_probability.html


Activity 3: How Many Ways? (GLEs: 42, 43)

Materials List: paper, pencil, chart paper, marker, Which is it? BLM, calculators

Begin class using SQPL (view literacy strategy descriptions) by having partners
brainstorm (view literacy strategy descriptions) two to three questions they would like
answered about the following statement.
       There would be more possible combinations of officers for a class
       (President, Vice President, Secretary, and Treasurer) than there would be
       combinations of four-person committees from a class of ten students.

Write the SQPL statement on the board or overhead for students to see. Have pairs then
share their questions with the class. The class will make a list of questions that it hopes
to be answered during the lesson. Post this list of questions as the lesson progresses. An
internal summary can be made by pointing out to the students that they can now answer
certain questions that they had.




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As the lesson begins, pose a situation where a five-person committee must be formed
from seven individuals to plan for an upcoming event. Challenge students to determine
the number of different committees that could be formed from these seven students. This
will be different from those problems done
previously, because in these, order is not             1,2,3,4,5  1,2,5,6,7   2,3,5,6,7
important. The combinations are shown in the list 1,2,3,4,6       1,3,4,5,6   2,4,5,6,7
at the right.                                          1,2,3,4,7  1,3,4,5,7   3,4,5,6,7
                                                         1,2,3,5,6     1,3,4,6,7
                                                         1,2,3,5,7     1,3,5,6,7
Next, pose a scenario where five individuals must        1,2,3,6,7     1,4,5,6,7
fill the five roles of officers: one person is the       1,2,4,5,6     2,3,4,5,6
president, one is the vice president, one is the         1,2,4,5,7     2,3,4,5,7
                                                         1,2,4,6,7     2,3,4,6,7
secretary, one is the treasurer, and one is the
historian. Ask if any one of the five students could
serve in any of the positions, then ask how many different ways this group of five officers
could be selected. Lead discussion about the similarities and differences in these
situations and whether or not order is important.

Answer: First scenario – 21 different 5 person committees. Order does not change the
make up of the committees.

Answer: Second scenario – 120 different ways. 5×4×3×2×1 (Order is important
because if the person is selected for President, it is different than if that person is chosen
for Secretary.)

Make sure the discussion of these scenarios involves some brainstorming by students of
situations in which order is important (permutations) and not important (combinations).

Distribute Which Is It? BLM and have students practice determining whether the
situation involves a combination or a permutation and provides practice for the students
in solving these problems. Discuss student responses on the BLM as a class to clarify
any misconceptions.


Activity 4: What does the Cookie Thief Look Like? (GLE: 43)

Materials List: Who Stole the Cookies? BLM, paper, pencil, newsprint or chart paper,
markers

Provide students with Who Stole the Cookies? BLM and read the situation aloud to the
class.

       Jackie worked at a restaurant in the evening. She had a locker in the back where
       she put all of her personal belongings. One night she bought a big box of cookies
       to take to her grandmother the next day. She put this box of cookies in her locker
       so that she could take it home after work. When she went back to the locker at
       10:00 P.M. after work, the cookies were gone! One of her friends saw a stranger



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       at the lockers about 9:30 P.M. Jackie and her friend talked to the store manager,
       and they were given a list of possible characteristics to help in identification.

The characteristics were given to the friends in a chart like the one on Who Stole the
Cookies? BLM. Challenge pairs of students to come up with all the different descriptions
possible for the cookie thief. Have the pairs of students determine the different
combinations of descriptions that could have described the thief. Then have them display
their findings using some type of chart or graph.

Once the student pairs have completed their description, randomly select one group to be
professor-know-it-all (view literacy strategy descriptions) and have it explain to the class
the different descriptions and its method of organizing their descriptions. Allow class
members to ask questions of the group that is professor-know-it-all.


Activity 5: Experimental Probabilities (GLE: 44)

Materials List: styrofoam plates, paper clips, pencil, paper

Have students make a prediction based on the results of spinning a spinner that has been
divided into equal sections of three colors. These spinners can be made with the foam
plates that have the thumbprints around the circumference. Hefty plates have 36
thumbprints and are easy to divide into equivalent sections. Have students determine the
theoretical probability on any given spin. The chances of getting any one of the three
colors is one-third. Have students perform the experiment by spinning the spinner twenty
to thirty times. Once all spins have been completed, ask students to use their
experimental results to make the prediction of the next spin. This will work best if the
students are not told they will make a prediction until all 20-30 spins have been
completed.

Use a real-life example:

The local mall is having a grand opening celebration. They are using a spinner like the
one used in the experiment to determine the prizewinners every fifteen minutes. They
display the results throughout the day. When you get to the mall, the spinner result
display looks like the one below. The mall official will randomly select an audience
member to call a color and if that color wins, the member will win a prize.




                                red           blue       yellow



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Have the students work in pairs to determine what their choice would be if they were
selected as the next lucky person to spin. Students should explain answers with a sketch
or diagram.


Activity 6: Independent Events (GLE: 45)

Materials List: number cubes, pencil, paper

Have groups of four students create a game of chance like Yahtzee® using number cubes.
Have students determine the rules for their game, the materials (number cubes, spinner,
and cards) and then justify how the theoretical probability of winning makes their game a
fair game. After playing the game several times, students should determine the
experimental probabilities of obtaining each of the required outcomes. For example,
explore the possibility of rolling all number cubes and getting the same number on each.
The roll of each number cube is independent. Have students exchange games with
another group and follow the rules determined by the game‘s creator. Compare
experimental results with the theoretical results. Lead classroom discussion about the
independent events involved in each of the games created.


Activity 7: Dependent Events (GLE: 45)

Materials List: styrofoam plates, paper clips, Dependent Events BLM, pencil, paper

Create a multiple-event experiment where the events are dependent, and have the
students determine the probability of a result. Have each group of four students make two
spinners with sturdy plates that have the thumbprints or dimples around the edge such as
the Hefty® brand of plate. Secure a paper clip on the spinner by using a second paper clip
through the bottom of the plate. These plates are already divided into 36 thumbprints so
the students can easily divide the plate into thirds or fourths. Divide one of the plates into
thirds and let this plate represent the number of coins. It will make it easier for class
discussion if the groups use the same three numbers (suggestion: 2, 5, 10). Divide the
second plate into fourths and write the name of a coin in each of the four sections (penny,
nickel, dime, quarter). Distribute Dependent Events BLM and explain to the students that
they must figure the theoretical probability of spinning less than, more than or exactly
fifty cents. The groups will then collect experimental data.

Discuss how the results might be different if the spinners were not fair spinners. Sample
size should also be part of the discussion. Relate the situation to a possible game at the
fair or some other carnival. Discuss the probability of winning prizes at the fair.




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Activity 8: Is It Fair? (GLE: 45)

Materials List: two number cubes of different color, paper, pencil

Provide pairs of students with two number cubes of different colors. Ask students to roll
the number cubes and find the product of the two cubes. Player 1 will be the tallest
person, and will roll both number cubes first. If the product of the numbers rolled is odd,
Player 1 will receive two points. If the product is even, Player 2 will get 1 point. Have
play continue until one of the players reaches 20 points. Repeat the game exchanging
positions of Player 1 and Player 2. After the students have played the game at least two
times, have the students create a table showing the theoretical probability of the
occurrence of each product. Ask students then to determine the probability of an odd or
even product and whether the game rules were fair.

Challenge the groups to determine rules that would create a fair game using number cube
products. To do this, have the students form groups of four to make a modified story
chain (view literacy strategy descriptions). Student 1 will be the person closest to the
teacher, and the students will be numbered clockwise from Student 1. Student 1 will
write the first in a set of rules for making a fair game with the number cubes, pass the
paper to Student 2 who will in turn write the second rule or step, Student 3 and then
Student 4. This will continue until the group has completed their rules for the game.
Each student in the group should then get a chance to read and challenge any of the rules
or steps written so that their game is fair. Have groups follow the steps or rules that have
been written to play the game and determine if each player has an equal chance of
winning. An exit ticket is a student summary of the lesson as they respond to a prompt or
questions from the teacher. Have the students use an exit ticket to provide written
individual explanations of why their rules created a fair game.

 Lead discussion with the class about whether the events involved in the game were
independent or dependent events.


Activity 9: Who Did It? (GLEs: 41, 44)

Materials List: Who Did It? BLM, brown lunch bags (4 for each group), 10 color tiles of
four different colors (in each of 4 bags for each group), pencil, paper

Begin class with a discussion about sampling. Tell the students that today they will
collect results without replacement. Discuss what this means. Distribute the Who Did It?
BLM to each student, and give four brown lunch bags filled with the following (unknown
to them) tile or cube combinations to each group of four students. Bags should be labeled
A – D and should each contain a total of 10 tiles or cubes of four different colors. Let the
students know that there are 10 tiles in each bag and four different colors. Bag A and one
of the other bags should be identical (have the same number of each color of cubes).




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Tell the students that the sample in Bag A was found at the scene of a crime. CSI
investigators explored the contents of Bag A. When the investigators bagged the contents
of Bag A, they duplicated the items from bag A and made a second bag. A new CSI
trainee was also on the scene, and he took these bags back to the lab without labeling the
second one. These bags were placed in a box with bags from another crime scene. When
the CSI went to get the bags, there were four bags and only Bag A was labeled. The CSI
knew that there were two identical bags, and these bags are very important to their case.
They want to try to determine which are the identical bags and not touch the items any
more than necessary.

Discuss as a class that when samples are examined without replacement, the sample size
is constantly changing. Suppose this is what happens when the first tile is selected from
the bags: Bag A – red; Bag B – red; Bag C – green; Bag D – red. Now there are only
nine tiles in each bag to select from. Challenge the students to devise a plan to sample
contents of the bags without replacement so that they can make the best prediction based
on experimental probability without looking at the contents of the bags. Have students
record their results and make a prediction after the 6th selection from each bag, justifying
why this is their selection. Lead a discussion about whether the selections give enough
information to make the prediction. Ask if all four bags have to be completely empty to
make a valid prediction. Have students explain their thinking and their results.

Teacher Note: Student results will be different, and they will have to use some logical
reasoning as they compare the results they gather.


Activity 10: Replacement to Sample Set (GLEs: 41, 44)

Materials List: Who Did It? BLM (Activity 9), brown lunch bags (4 for each group), 10
color tiles of four different colors (in each of 4 bags for each group), pencil, paper

Begin class with a discussion about sampling without replacement that was done in
Activity 9. Tell the students that today they will collect results of the same problem with
replacement. Have students determine whether or not they think this method will be a
better way to make a prediction as to contents and why they made that choice. Have
students take out their Who Did It? BLM (Activity 9), and distribute the brown bags to
each group of four. Have student groups make a plan for determining the contents with
replacement. They should write this plan on the back of their Who Did It? BLM sheet.
Once they have devised a plan, they should collect their data and answer questions 4 and
5 on the Who Did It? BLM. Discuss as a class the results and the difference in sampling
with and without replacement.




Grade 8 MathematicsUnit 8Examining Chances                                              102
                      Louisiana Comprehensive Curriculum, Revised 2008


                                  Sample Assessments


Performance assessments can be used to ascertain student achievement. For example:


General Assessments

          The student will create a game of chance in which Player 1 has twice the
           chance of winning as Player 2.
          The student will prepare a presentation to explain how theoretical probability
           is used to make predictions like the weather forecast.
          The student will make four different sketches of polygons with a shaded area
           inside or outside of the polygon that would illustrate a 25%, 50%, 75% and
           60% probability of an object falling randomly on each figure and landing on
           the shaded area.
           Example: the figure at the right would represent a 50% probability
           of a randomly dropped object that would fall on the figure landing
           on the shaded area.
          The student will play several different games of chance and then analyze the
           probabilities of winning.
          The student will develop an experiment and then determine the experimental
           probability associated with the event taking place.
          Whenever possible, create extensions to an activity by increasing the
           difficulty or by asking ―what if‖ questions.
          The student will create portfolios containing samples of their experiments and
           activities.
          The student will complete a probability project assessed by a teacher-created
           rubric.


Activity-Specific Assessments

          Activity 1: Provide the student with a survey topic and the student will
           describe in his/her journal what population will be surveyed, the sample size,
           and the sample questions. The student will also explain how the survey will be
           used. The student will conduct the survey and prepare their results with an
           explanation as to how the survey results will be used.

          Activity 3: Secure menus from a restaurant that advertises several ways its
           product can be purchased (e.g., Burger King, Baskin-Robbins Ice Cream), and
           the student will determine the validity of the claim.

          Activity 7: The student will prepare directions and make a game that involves
           dependent events. The student will describe the game using the theoretical
           probability of outcomes to describe how the game is won.



Grade 8 MathematicsUnit 8Examining Chances                                         103
                     Louisiana Comprehensive Curriculum, Revised 2008




         Activity 9: The student will prepare a poster proving that his/her prediction is
          based on experimental probability after the 6th selection. The student will also
          use the actual contents of the bag to compare the theoretical probability of
          his/her prediction after the 6th selection. The student will include an
          explanation of how sampling without replacement affected his/her prediction.




Grade 8 MathematicsUnit 8Examining Chances                                           104

								
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