# Comparing and Scaling - GMS by linxiaoqin

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```									                                    Instructional Unit Plan
School: CMS, GMS, and STMS

Content Area(s)                                     Implementation Dates:
Mathematics: Comparing and Scaling

Unit Focus: Ratio, Proportion and Percent.

Investigation 1
   To explore several ways to make comparisons
   To begin to understand how to determine when comparisons can be made using
multiplication or division versus addition or subtraction
   To begin to develop ways to use ratios, fractions, rates, and unit rates to answer
questions involving proportional reasoning
Investigation 2
   To further develop the ability to make sensible comparisons of data using ratios,
fractions, and decimal rates, with a focus on percents
   To develop the ability to make judgments about rounding data to estimate ratio
comparisons
   To observe what is common about situations that call for a certain type of ratio
comparison
Investigation 3
   To recognize situations in which ratios are a useful form of comparison
   To form, label, and interpret ratios from numbers given or implied in a situation
   To explore several informal strategies for solving scaling problems involving ratios
(which is equivalent to solving proportions)
Investigation 4
   To find unit rates
   To represent data in tables and graphs
   To look for patterns in tables in order to make predictions beyond the tables
   To connect unit rates with the rule describing a situation
   To begin to recognize that constant growth in a table will give a straight line graph
   To find the missing value in a proportion
Investigation 5
   To use geometric scaling to estimate population counts
   To apply proportional reasoning to situations in which capture-tag-recapture methods
are appropriate for estimating population counts
   To use ratios and scaling up or down (finding equivalent ratios) to find the missing value
in a proportion
   To use rates to describe population and traffic density (space per person or car)
Investigation 6
   To select and apply appropriate strategies to make comparisons
   To review when ratio and difference strategies are useful in solving problems
   To use proportional reasoning to fairly apportion available space so that the group is
representative of the larger community

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New Mexico Content Standards, Benchmarks and Performances Addressed:
Mathematics

Strand 1: Number And Operations
Standard: Students will understand numerical concepts and mathematical operations.
5-8 Benchmark 1: Understand numbers, ways of representing numbers, relationships among numbers, and number
systems.
 Illustrate the relationships among natural (i.e., counting) numbers, whole numbers, integers, rational and irrational
numbers.
 Use properties of the real-number system to explain reasoning and to formulate and solve real-world problems.
 Simplify numerical expressions using order of operations.
5-8 Benchmark 2: Understands the meaning of operations and how they relate to one another.
 Add, subtract, multiply, and divide rational numbers (e.g., integers, fractions, terminating decimals) and take
positive rational numbers to whole-number powers.
 Convert terminating decimals into reduced fractions.
 Calculate given percentages of quantities and use them to solve problems (e.g., discounts of sales, interest
earned, tips, markups, commission, profit, simple interest).
 Add and subtract fractions with unlike denominators.
5-8 Benchmark 3: Compute fluently and make reasonable estimates.
 Use estimation to check reasonableness of results, and use this information to make predictions in situations
involving rational numbers, pi, and simple algebraic equations.
 Convert fractions to decimals and percents and use these representations in estimations, computations, and
applications.
 Calculate the percentage of increases and decreases of a quantity.
 Add and subtract fractions with unlike denominators.

Strand 2: Algebra
Standard: Students will understand algebraic concepts and applications.
5-8 Benchmark 1: Understand patterns, relations, and functions
 Identify and continue patterns presented in a variety of formats.
 Represent a variety of relationships using tables, graphs, verbal rules, and possible symbolic notation, and
recognize the same general pattern presented in different representations.
 Simplify numerical expressions by applying properties of rational numbers, and justify the process used .
 Solve problems involving rate, average speed, distance, and time.

Strand 4: Measurement
Standard: Students will understand measurement systems and applications.
5-8 Benchmark 1: Understand measurable attributes of objects and the units, systems, and process of measurement.
 Choose appropriate units of measure and ratios to recognize new equivalences (e.g., 1 square yard equals 9
square feet) to solve problems.
 Select and use the appropriate size and type of unit for a given measurement situation.
 Use measures expressed as rates and measures expressed as products to solve problems, check the units of the
solutions, and analyze the reasonableness of the answer.
5-8 Benchmark 2: Apply appropriate techniques, tools, and formulas to determine measurements.
 Solve problems involving scale factors, ratios, and proportions.

Strand 5: Data Analysis And Probability
Standard: Students will understand how to formulate questions, analyze data, and determine probabilities.
5-8 Benchmark 1: Formulate questions that can be addressed with data and collect, organize, and display relevant
 Describe how data representations influences interpretation.
 Select and use appropriate representation for presenting collected data and justify the selection.
 Identify and explain the misleading representations of data.
 Collect, organize, and represent data sets that have one or more variables and identify relationships among
variables within a data set.

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   Compute the minimum, lower quartile, median, upper quartile, and maximum of a data set.
   Use and explain sampling techniques (e.g., observations, surveys, and random sampling) for gathering data.
   Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, identifying
missing information, and selecting, collecting, and displaying appropriate data to address the problem.
5-8 Benchmark 2: Select and use appropriate statistical methods to analyze data.
 Use the analysis of data to make convincing arguments.
 Use appropriate technology to gather and display data sets and identify the relationships that exist among
variables within the data set.
 Use data samples of a population and describe the characteristics and limitations of the sample.
 Identify data that represent sampling errors and explain why the sample and the display might be biased.
 Identify claims based on statistical data and evaluate the validity of the claims.
5-8 Benchmark 4: Understand and apply basic concepts of probability.
 Describe the probability of events using fractions, decimals, and percents.
 Use probability to generate convincing arguments, draw conclusions, and make decisions in a variety of
situations.
 Determine the probability of a simple event or a compound event composed of a simple, independent events.

Paraphrase knowledge and skills required by the Standards:
   Use informal language to ask comparison questions, such as:
o “What fraction of the class is going to the picnic?”
o “What percent of the girls play basketball?”
o “Which model of car has the best fuel economy?”
   Decide when the most informative comparison is the difference between two quantities and when it is
ratios between pairs of quantities.
   Develop the ability to make judgments about rounding data to estimate ratio comparisons.
   Find equivalent ratios to make more accurate and insightful comparisons.
   Scale a ratio or fraction up or down so that a larger or smaller object or population has the same
relative characteristics as the original.
   Represent data in tables and graphs.
   Apply proportional reasoning to situations in which capture-tag-recapture methods are appropriate for
estimating population counts.
   Set up and solve proportions that arise in applications.
   Look for patterns in tables that will allow predictions.
   Connect unit rates with a rule describing the situation.
   Begin to recognize that constant growth in a table will give a straight-line graph.
   Use rates to describe population and traffic density.

Vocabulary:
   Comparison
   Concentrate
   Context
   Decimal
   Denominator
   Difference
   Equation
   Equivalent, Equivalent Ratio
   Fraction
   Numerator
   Part to Part Ratio
   Part to Whole Ratio
   Percent
   Population density
   Proportion

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   Rate
   Ratio
   Scale, Scaling
   Unit rate

Description of the Assessments that will be used to provide evidence of student learning that
targets the standards:

Within the unit, ACE questions, Checkups, and Quizzes may be used as ongoing student
assessment.

Investigation I
 ACE questions: #1-5 and #14
 Mathematical Reflections on page 15

Investigation 2
 Check up I, pages 86-87
 Mathematical Reflections on page 15

Investigation 3
 Mathematical Reflections on page 36

Investigation 4
 Check up 2, page 88
 Quiz, page 89-90
 Mathematical Reflections on page 51

Investigation 5
 Mathematical Reflections on page 64

Investigation 6
 Mathematical Reflections on page 81
 Unit Test, pages 96-97
 Paper Pool Project, pages 98-99

Description of the rubrics or criteria that will be used to assess student performance:
Students will be assessed according to their
 Understanding
 Procedures, Strategies, Reasoning
 Communications

Unit-Test Rubric—Teacher formatted

   Mathematical Reflection: (19 pts total)
o A point value will be assigned to each of the 5 questions:
 #1: 3pts total; 1 for each correct rate.
 #2-5: 4 pts. each; 2 for appropriate answer and 2 for
appropriate explanation.

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   Check-Up 2:

Key for the test is provided.

Learning Activities – Description of the learning activities that will develop the knowledge
and skills required by the performance standard:

Investigation 1: Making Comparisons

Objective:
 SWBAT to explore several ways to make comparisons using ratios, fractions and percents.

Activities:
Problem 1.1 and follow –up page 6.

1.2       Targeting an Audience
Problem 1.2 page 7
Problem 1.2 follow-up page 8. Can be done as a whole group activity.

1.3       Getting the Message Across
Problem 1.3 and follow-up page 9

Homework/Practice Problems:
ACE questions: #1-5 and #14.
Quiz: #6-11 and the Mathematical Reflections on page 15.

Investigation 1 Resources/Materials:
   Pencils, paper
   Textbook
   Transparencies 1.1 – 1.3
   Large sheets of paper
   Calculator (optional)

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Key Links Practice for Investigation 1, Comparing and Scaling

1.Which one of the following is not equivalent?

7
a)    8

b) 0.875

c) 88%

875
d)    1000

1a. Write or illustrate three different ways of
expressing the same number. (Fraction,
decimal and percent)

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Key Links Practice for Investigation 1, Comparing and Scaling

2a. If you combine the total number of students in Ms.
Ochoa’s and Mr. Moreno’s classrooms there are 45
students. One-third of the students are boys. How
many are girls?
a)       30 girls
b)       15 girls
c)       8 girls
d)       5 girls

2b. The ratio of boys to girls in Ms. Ochoa’s class is 7 to
12. What is the ratio of boys to girls in Mr. Moreno’s

2c. Write three possible combinations that will
produce a group of students of boys to girls, with a
ratio of 1 to 5.

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Investigation 2: Comparing and Finding Percents

Objectives:
 To further develop the ability to make sensible comparisons of data using ratios,
fractions, and decimals rates with a focus on percents.
 To develop the ability to make judgments about rounding data to estimate ratio
comparisons.
 To observe what is common about situations that call for a certain type of ratio
comparison.

Activities:
2.1     Comparing Leisure Activities
2.1     Follow-up
A.C.E. questions 1-8, 17-22

2.2:    Comparing Your Class to the Nation
2.2     Follow-up
A.C.E. questions 9-16, 23-26

Homework/Practice Problems:
Mathematical Reflections, page 25

Assessment
Check up 1, pages 86-87

Investigation 2 Resources/Materials:
 Graphing calculators
 Centimeter grid paper.

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Key Links Practice for Investigation 2, Comparing and Scaling

1.     A homeroom class of 32 eighth graders at Santa Teresa
Middle School completed a survey about their participation
in team sports. Each student was asked to list any sport he
or she liked to play. The results for four of the most popular
sports are given in the following table.
Sport                              Female                Male
Track and Field                              7                    13
Softball                                     10                   8
Football                                     4                    11
Total Surveyed                               17                   15

1a. Which of the following best represents the fraction of the
class that is female?
15                     17                    32
a.   32               b.    32               c.   17     d. none of
these.

1b. In which sport does the greatest percent of the class
participate?
b.     Track and Field
c.     Softball
d.     Football

question 1b, give details.

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Key Links Practice for Investigation 2, Comparing and Scaling

2.
12 ft                                   9 ft

16 ft                                                                           12 ft

Above are the floor plans for two college dorm rooms. One is
for two students, and the other is for one student.

1.     Are the floor plans similar rectangles? Explain why or why
not.

2.     Select the ratio that best represents a comparison of the
two dorm rooms:
2                      9                           3           3
a.    3               b.    16                      c.   4      d.   2

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Investigation 3: Comparing by Using Ratios

Objectives
   To recognize situations in which ratios are a useful form of comparison
   To form, label, and interpret ratios from numbers given or implied in a situation
   To explore several informal strategies for solving scaling problems involving ratios
(which is equivalent to solving proportions)

Pacing Table: 2-3 Days

Activities
3.1       Mixing Juice pg. 28 A-C
Follow-Up, pg. 28 1-2
ACE Questions: pg. 31-35: 1,2,12

3.2       Helping the Cook pg. 29 A-B
Follow-Up pg. 29 1-2
ACE Questions: pg. 31-35: 3,6,13-23

3.3       Sharing Pizza pg. 30 A-B
Follow-Up pg. 30 1-2
ACE Questions: pg. 31-35: 4, 8, 25

Reflections:
Problems: pg. 36 3-4
Big Math Idea (Summary)

Investigation 3 Resources/Materials:
 Centimeter grid paper
 Transparencies 3.1-3.3
 Orange and White chips or
 Orange and white square of paper (about 25 orange and 100 white per group)
 Orange juice concentrate
 Pitcher

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Key Links Practice for Investigation 3, Comparing and Scaling

1.     You are a guest at a Pizza Party. There are 3 tables,
set up for guests. You many sit anywhere you choose.
The pizzas at each table must be shared equally.
Since you are especially hungry and pizza is your
favorite food, which of the following tables would you
sit at in order to get the most pizza possible?

a) Table1: 5 seats and 2 pizzas
b) Table 2: 7seats and 3 pizzas
c) Table 3: 12 seats and 5 pizzas
d) All tables will allow for the same amount of pizza

more pizza than the other options. You may use pictures,
numbers and words to demonstrate your understanding.

Extension:
Another table was added at the pizza party, with 10
seats and 4 pizzas. Will this option allow you to have
more pizza than any of the other three tables? Explain
why or why not.

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Key Links Practice for Investigation 3, Comparing and Scaling

2.     Alex and Dora are making punch for a class party.
The directions on the liquid punch mix say to use 3
cups of mix for every 7 cups of water. They must
make a total of 50 cups of punch for the party. How
many cups of mix and how many cups of water will
they need in order to serve 50 cups of punch?

a) 20 cups of mix and 30 cups of water
b) 25 cups of mix and 25 cups of water
c) 15 cups of mix and 35 cups of water
d) 12 cups of mix and 38 cups of water

If 20 unexpected guests arrived at the party and each
drank 2 cups of punch, how much mix and water would be
needed?

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Investigation 4: Comparing by Finding Rates

Objective:

Activities:
4.1     Comparing Fuel Economy, page 38-39
4.1 Follow-Up
ACE questions 4 & 5 (page 46)

4.2     Using Unit Rates, page 40-41
4.2 Follow-Up [optional for practice in graphing]
ACE questions 1, 6, 10, & 11

4.3     Solving Problems with Rates, page 42
4.3 Follow-Up as a journal entry
ACE question 8, 9, 17 & 18
ACE question #2 if you need to retouch skills required for #1.

4.4 Follow-Up
ACE questions 7, 12, 13, 14, 15 & 16

Investigation 4 Resources/Materials:
 Comparing and Scaling textbooks
 Centimeter graph paper
 Centimeter graph transparency (for students)
 Markers
 Transparencies 4.1-4.4 (for teacher)

Vocabulary:
     rates
     scaling up and down (higher and lower terms)
     unit rates

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Key Links Practice for Investigation 4, Comparing and Scaling

Tony can type at a constant rate of 55 words per minute.

1a.        Write an equation for the number of words, W,
Tony can type in T minutes.

1b.        How many words can Tony type in 20 minutes?

a) 75 words
b) 1100 words
c)      75 words per minute
d) 1100 words per minute

1c.        If Tony has a half hour to type a 1600 word
essay, will he have time to type the entire essay?

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Key Links Practice for Investigation 4, Comparing and Scaling

A veterinarian clinic has a patient load of 150 cats and
dogs. The ratio of cats to dogs is 4 to 8.

2a.         How many patients are cats?

a)     100
b)     80
c)     50
d)     40

2b.        How many patients are dogs?

a)     100
b)     80
c)     50
d)     40

2c.        If the ratio of cats to dogs would have been
1 to 2, would answers for questions a & b
change? Explain why or why not.

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Investigation 4: Comparing by Finding Rates

Objective:

Activities:
5.1     Estimating Populations and Population Densities
Ace questions: 3 and 10

5.2     Estimating a Deer Population using beans
Ace questions: 1,4,11,13

5.3     Finding Population Densities
Ace questions: 6,12

5.4     Comparing the Dakotas (if time allows)
Ace questions: 12

5.5     Predicating Traffic Jams (if time allows)
Ace questions: 12
Mathematical Reflections pg. 64

Investigation 5 Resources/Materials:
 Transparencies
 centimeter & inch grids
 containers
 white beans (700-800)
 markers

Comparing and Scaling Unit Materials:
 Graphing calculators
 Containers (large enough so students can mix contents) of 300–800 white beans, with lid
 Scoops for sampling (optional)
 Marker
 Large sheets of paper
 Centimeter and inch grid paper (provided as BLM)
 Can of orange juice concentrate and pitcher (optional)
 News article that reports an estimate of crowd size (optional)

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Key Links Practice for Investigation 5, Comparing and Scaling

1. Michael plays a lot of basketball, and he keeps a
record of his free throw attempts in practices and
games. Michael has made 48 out of 100 free-
throw attempts in one week. What fraction
below is closest to his free-throw attempts?
a.     ¾
b.     ½
c.     9/10
d.     5/8

How would his success rate change if he made the
next 10 free throw attempts?

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Key Link Practice For Investigation 5, Comparing And Scaling

2.      At Gadsden Middle School, Mr. Hamel’s
students conducted a class survey about their
favorite football teams. Of the 27 students in the
class, 18 picked the Dallas Cowboys, 6 picked
the New England Patriots, and 3 picked the
Arizona Cardinals. If you randomly selected a
person in the hallway and asked them to choose
between the Cowboys, Patriots, or Cardinals
what are the chances that the person will pick
the Patriots as the favorite team?

a.   2/3
b.   1/9
c.   2/9
d.   ½

If there were1000 students in Gadsden Middle
School what percent of the students would choose
the Arizona Cardinals as their favorite team?