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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 MathematicsUnit 1Rational 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 MathematicsUnit 1Rational 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 MathematicsUnit 1Rational 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 MathematicsUnit 1Rational 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 MathematicsUnit 1Rational 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 MathematicsUnit 1Rational 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 MathematicsUnit 1Rational 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 MathematicsUnit 1Rational 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 MathematicsUnit 1Rational 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 MathematicsUnit 1Rational 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 MathematicsUnit 2Rates, 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 MathematicsUnit 2Rates, 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 MathematicsUnit 2Rates, 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 MathematicsUnit 2Rates, 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 MathematicsUnit 2Rates, 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 MathematicsUnit 2Rates, 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 MathematicsUnit 2Rates, 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 MathematicsUnit 2Rates, 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 MathematicsUnit 2Rates, 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 MathematicsUnit 2Rates, 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 MathematicsUnit 2Rates, 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 MathematicsUnit 2Rates, 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 MathematicsUnit 2Rates, 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 MathematicsUnit 2Rates, 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 MathematicsUnit 3Geometry 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 MathematicsUnit 3Geometry and Measurement 26 Louisiana Comprehensive Curriculum, Revised 2008 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 MathematicsUnit 3Geometry and Measurement 27 Louisiana Comprehensive Curriculum, Revised 2008 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 Grade 8 MathematicsUnit 3Geometry and Measurement 28 Louisiana Comprehensive Curriculum, Revised 2008 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 MathematicsUnit 3Geometry and Measurement 29 Louisiana Comprehensive Curriculum, Revised 2008 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 MathematicsUnit 3Geometry and Measurement 30 Louisiana Comprehensive Curriculum, Revised 2008 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 MathematicsUnit 3Geometry 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 MathematicsUnit 3Geometry 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 MathematicsUnit 3Geometry 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 MathematicsUnit 3Geometry 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 MathematicsUnit 3Geometry 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 MathematicsUnit 3Geometry 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 MathematicsUnit 3Geometry 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? Grade 8 MathematicsUnit 4Measurement and Geometry 38 Louisiana Comprehensive Curriculum, Revised 2008 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 Grade 8 MathematicsUnit 4Measurement and Geometry 39 Louisiana Comprehensive Curriculum, Revised 2008 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 Grade 8 MathematicsUnit 4Measurement and Geometry 40 Louisiana Comprehensive Curriculum, Revised 2008 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 Grade 8 MathematicsUnit 4Measurement and Geometry 41 Louisiana Comprehensive Curriculum, Revised 2008 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 Grade 8 MathematicsUnit 4Measurement and Geometry 42 Louisiana Comprehensive Curriculum, Revised 2008 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 Grade 8 MathematicsUnit 4Measurement and Geometry 43 Louisiana Comprehensive Curriculum, Revised 2008 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/ Grade 8 MathematicsUnit 4Measurement and Geometry 44 Louisiana Comprehensive Curriculum, Revised 2008 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 Grade 8 MathematicsUnit 4Measurement and Geometry 45 Louisiana Comprehensive Curriculum, Revised 2008 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 Grade 8 MathematicsUnit 4Measurement and Geometry 46 Louisiana Comprehensive Curriculum, Revised 2008 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. Grade 8 MathematicsUnit 4Measurement and Geometry 47 Louisiana Comprehensive Curriculum, Revised 2008 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. Grade 8 MathematicsUnit 4Measurement and Geometry 48 Louisiana Comprehensive Curriculum, Revised 2008 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. Grade 8 MathematicsUnit 4Measurement and Geometry 49 Louisiana Comprehensive Curriculum, Revised 2008 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 Grade 8 MathematicsUnit 4Measurement and Geometry 50 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.‖ Grade 8 MathematicsUnit 4Measurement and Geometry 51 Louisiana Comprehensive Curriculum, Revised 2008 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. Grade 8 MathematicsUnit 4Measurement and Geometry 52 Louisiana Comprehensive Curriculum, Revised 2008 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. Grade 8 MathematicsUnit 4Measurement and Geometry 53 Louisiana Comprehensive Curriculum, Revised 2008 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. Grade 8 MathematicsUnit 4Measurement and Geometry 54 Louisiana Comprehensive Curriculum, Revised 2008 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. Grade 8 MathematicsUnit 4Measurement and Geometry 55 Louisiana Comprehensive Curriculum, Revised 2008 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. Grade 8 MathematicsUnit 4Measurement and Geometry 56 Louisiana Comprehensive Curriculum, Revised 2008 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? Grade 8 MathematicsUnit 5Algebra, Integers, and Graphing 57 Louisiana Comprehensive Curriculum, Revised 2008 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) Grade 8 MathematicsUnit 5Algebra, Integers, and Graphing 58 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. Grade 8 MathematicsUnit 5Algebra, Integers, and Graphing 59 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 Grade 8 MathematicsUnit 5Algebra, Integers, and Graphing 60 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 Grade 8 MathematicsUnit 5Algebra, Integers, and Graphing 61 Louisiana Comprehensive Curriculum, Revised 2008 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. Grade 8 MathematicsUnit 5Algebra, Integers, and Graphing 62 Louisiana Comprehensive Curriculum, Revised 2008 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 Grade 8 MathematicsUnit 5Algebra, Integers, and Graphing 63 Louisiana Comprehensive Curriculum, Revised 2008 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. Grade 8 MathematicsUnit 5Algebra, Integers, and Graphing 64 Louisiana Comprehensive Curriculum, Revised 2008 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. Grade 8 MathematicsUnit 5Algebra, Integers, and Graphing 65 Louisiana Comprehensive Curriculum, Revised 2008 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. Grade 8 MathematicsUnit 5Algebra, Integers, and Graphing 66 Louisiana Comprehensive Curriculum, Revised 2008 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 Grade 8 MathematicsUnit 5Algebra, Integers, and Graphing 67 Louisiana Comprehensive Curriculum, Revised 2008 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 Grade 8 MathematicsUnit 5Algebra, Integers, and Graphing 68 Louisiana Comprehensive Curriculum, Revised 2008 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. Grade 8 MathematicsUnit 5Algebra, Integers, and Graphing 69 Louisiana Comprehensive Curriculum, Revised 2008 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 Grade 8 MathematicsUnit 5Algebra, Integers, and Graphing 70 Louisiana Comprehensive Curriculum, Revised 2008 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. Grade 8 MathematicsUnit 5Algebra, Integers, and Graphing 71 Louisiana Comprehensive Curriculum, Revised 2008 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) Grade 8 MathematicsUnit 6Growth and Patterns 72 Louisiana Comprehensive Curriculum, Revised 2008 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. Grade 8 MathematicsUnit 6Growth and Patterns 73 Louisiana Comprehensive Curriculum, Revised 2008 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: Grade 8 MathematicsUnit 6Growth and Patterns 74 Louisiana Comprehensive Curriculum, Revised 2008 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. Grade 8 MathematicsUnit 6Growth and Patterns 75 Louisiana Comprehensive Curriculum, Revised 2008 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. Grade 8 MathematicsUnit 6Growth and Patterns 76 Louisiana Comprehensive Curriculum, Revised 2008 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. Grade 8 MathematicsUnit 6Growth and Patterns 77 Louisiana Comprehensive Curriculum, Revised 2008 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. Grade 8 MathematicsUnit 6Growth and Patterns 78 Louisiana Comprehensive Curriculum, Revised 2008 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. Grade 8 MathematicsUnit 6Growth and Patterns 79 Louisiana Comprehensive Curriculum, Revised 2008 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. Grade 8 MathematicsUnit 6Growth and Patterns 80 Louisiana Comprehensive Curriculum, Revised 2008 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? Grade 8 MathematicsUnit 7What Are the Data? 81 Louisiana Comprehensive Curriculum, Revised 2008 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. Grade 8 MathematicsUnit 7What Are the Data? 82 Louisiana Comprehensive Curriculum, Revised 2008 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. Grade 8 MathematicsUnit 7What Are the Data? 83 Louisiana Comprehensive Curriculum, Revised 2008 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. Grade 8 MathematicsUnit 7What Are the Data? 84 Louisiana Comprehensive Curriculum, Revised 2008 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. Grade 8 MathematicsUnit 7What Are the Data? 85 Louisiana Comprehensive Curriculum, Revised 2008 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. Grade 8 MathematicsUnit 7What Are the Data? 86 Louisiana Comprehensive Curriculum, Revised 2008 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. Grade 8 MathematicsUnit 7What Are the Data? 87 Louisiana Comprehensive Curriculum, Revised 2008 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 Grade 8 MathematicsUnit 7What Are the Data? 88 Louisiana Comprehensive Curriculum, Revised 2008 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. Grade 8 MathematicsUnit 7What Are the Data? 89 Louisiana Comprehensive Curriculum, Revised 2008 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. Grade 8 MathematicsUnit 7What Are the Data? 90 Louisiana Comprehensive Curriculum, Revised 2008 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. Grade 8 MathematicsUnit 7What Are the Data? 91 Louisiana Comprehensive Curriculum, Revised 2008 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? Grade 8 MathematicsUnit 8Examining Chances 92 Louisiana Comprehensive Curriculum, Revised 2008 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 Grade 8 MathematicsUnit 8Examining Chances 93 Louisiana Comprehensive Curriculum, Revised 2008 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. Grade 8 MathematicsUnit 8Examining Chances 94 Louisiana Comprehensive Curriculum, Revised 2008 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. Grade 8 MathematicsUnit 8Examining Chances 95 Louisiana Comprehensive Curriculum, Revised 2008 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 Grade 8 MathematicsUnit 8Examining Chances 96 Louisiana Comprehensive Curriculum, Revised 2008 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. Grade 8 MathematicsUnit 8Examining Chances 97 Louisiana Comprehensive Curriculum, Revised 2008 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 Grade 8 MathematicsUnit 8Examining Chances 98 Louisiana Comprehensive Curriculum, Revised 2008 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 Grade 8 MathematicsUnit 8Examining Chances 99 Louisiana Comprehensive Curriculum, Revised 2008 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. Grade 8 MathematicsUnit 8Examining Chances 100 Louisiana Comprehensive Curriculum, Revised 2008 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). Grade 8 MathematicsUnit 8Examining Chances 101 Louisiana Comprehensive Curriculum, Revised 2008 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 MathematicsUnit 8Examining 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 MathematicsUnit 8Examining 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 MathematicsUnit 8Examining Chances 104