Unit 3: Ratios & Proportional...

CCGPS

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7thGrade

Unit3:Ratios&ProportionalRelationships

Mathematics

Georgia Department of Education Common Core Georgia Performance Standards Framework Teacher Edition

Sixth Grade Mathematics Unit 3

MATHEMATICS GRADE 7 UNIT 3: Ratios and Proportional Relationships Georgia Department of Education

Dr. John D. Barge, State School Superintendent April 2012 of 56

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Unit 3 Ratios & Proportional Relationships

TABLE OF CONTENTS

Overview .......................................................................................................................................3

Standards Addressed in this Unit ..................................................................................................3

Key Standards & Related Standards .................................................................................3

Standards for Mathematical Practice ................................................................................5

Enduring Understandings..............................................................................................................7

Essential Questions .......................................................................................................................8

Concepts & Skills to Maintain ......................................................................................................8

Selected Terms and Symbols ........................................................................................................8

Classroom Routines ......................................................................................................................9

Strategies for Teaching and Learning .........................................................................................10

Evidence of Learning ..................................................................................................................11

Tasks ...........................................................................................................................................12

What is Unit Rate? ..........................................................................................................13

Orange Fizz Experiment .................................................................................................23

Creating A Scale Map .........................................................................................………31

Which Is The Better Deal? ..............................................................................................37

Patterns & Percents .........................................................................................................41

Nate & Natalie’s Walk ....................................................................................................51

FAL: Developing a Sense of Scale ................................................................................55





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OVERVIEW

In Grade 7, students will analyze proportional relationships and use them to solve real-world and mathematical problems. Students will do this by completing the following:

Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.

Recognize and represent proportional relationships between quantities.

Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.

Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.

Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.

Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.

STANDARDS ADDRESSED IN THIS UNIT KEY STANDARDS MCC7.RP.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 12 mile in each 1

4 hour, compute the unit rate as the complex fraction ∙ miles per hour,

equivalently 2 miles per hour. MCC7.RP.2 Recognize and represent proportional relationships between quantities.





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MCC7.RP.2a Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. MCC7.RP.2b Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. MCC7.RP.2c Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn. MCC7.RP.2d Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.

MCC7.RP.3 Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. MCC7.G.1 Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.

RELATED STANDARDS MCC7.EE.3 Solve multi‐step real‐life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations as strategies to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1 10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3 4inches long in the center of a door that is 27 1 2inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation. MCC7.NS.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.

MCC7.NS.1a Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.





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MCC7.NS.1b Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real‐world contexts. MCC7.NS.1c Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real‐world contexts. MCC7.NS.1d Apply properties of operations as strategies to add and subtract rational numbers.

MCC7.NS.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

MCC7.NS.2a Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real‐world contexts. MCC7.NS.2b Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non‐zero divisor) is a rational number. If p and q are integers then – (p/q) = (-p)/q = p/(-q). Interpret quotients of rational numbers by describing real‐world contexts. MCC7.NS.2c Apply properties of operations as strategies to multiply and divide rational numbers. MCC7.NS.2d Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0’s or eventually repeats.

MCC7.NS.3 Solve real‐world and mathematical problems involving the four operations with rational numbers. STANDARDS FOR MATHEMATICAL PRACTICE

The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive





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reasoning , strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately) and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).

1. Make sense of problems and persevere in solving them.

In grade 7, students solve problems involving ratios and rates and discuss how they solved them. Students solve real world problems through the application of algebraic and geometric concepts. Students seek the meaning of a problem and look for efficient ways to represent and solve it. They may check their thinking by asking themselves, “What is the most efficient way to solve the problem?”, “Does this make sense?”, and “Can I solve the problem in a different way?”

2. Reason abstractly and quantitatively.

In grade 7, students represent a wide variety of real world contexts through the use of real numbers and variables in mathematical expressions, equations, and inequalities. Students contextualize to understand the meaning of the number or variable as related to the problem and decontextualize to manipulate symbolic representations by applying properties of operations.

1. Construct viable arguments and critique the reasoning of others.

In grade 7, students construct arguments using verbal or written explanations accompanied by expressions, equations, inequalities, models, and graphs, tables, and other data displays (i.e. box plots, dot plots, histograms, etc.). They further refine their mathematical communication skills through mathematical discussions in which they critically evaluate their own thinking and the thinking of other students. They pose questions like “How did you get that?”, “Why is that true?” “Does that always work?”. They explain their thinking to others and respond to others’ thinking.

4. Model with mathematics.

In grade 7, students model problem situations symbolically, graphically, tabularly, and contextually. Students form expressions, equations, or inequalities from real world contexts and connect symbolic and graphical representations. Students explore covariance and represent two quantities simultaneously. They use measures of center and variability and data displays (i.e. box plots and histograms) to draw inferences, make comparisons and formulate predictions. Students need many opportunities to connect and explain the connections between the different representations. They should be able to use all of these representations as appropriate to a problem context.

5. Use appropriate tools strategically.

Students consider available tools (including estimation and technology) when solving a mathematical problem and decide when certain tools might be helpful. For instance, students in grade 7 may decide to represent similar data sets using dot plots with the same scale to visually compare the center and variability of the data.





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6. Attend to precision. In grade 7, students continue to refine their mathematical communication skills by using clear and precise language in their discussions with others and in their own reasoning. Students define variables, specify units of measure, and label axes accurately. Students use appropriate terminology when referring to rates, ratios, and components of expressions, equations or inequalities.

7. Look for and make use of structure.

Students routinely seek patterns or structures to model and solve problems. For instance, students recognize patterns that exist in ratio tables making connections between the constant of proportionality in a table with the slope of a graph. Students apply properties to generate equivalent expressions (i.e. 6 + 2x = 3 (2 + x) by distributive property) and solve equations (i.e. 2c + 3 = 15, 2c = 12 by subtraction property of equality), c = 6 by division property of equality). Students compose and decompose two‐ and three‐dimensional figures to solve real world problems involving scale drawings, surface area, and volume.

8. Look for and express regularity in repeated reasoning. In grade 7, students use repeated reasoning to understand algorithms and make generalizations about patterns. During multiple opportunities to solve and model problems, they may notice that a/b ÷ c/d = ad/bc and construct other examples and models that confirm their generalization. They extend their thinking to include complex fractions and rational numbers. Students formally begin to make connections between covariance, rates, and representations showing the relationships between quantities.

ENDURING UNDERSTANDINGS

Fractions, decimals, and percents can be used interchangeably

The relationships and rules that govern whole numbers, govern all rational numbers

In order to add or subtract fractions, we must have like denominators

When we multiply one number by another, we may get a product that is bigger than the original number, smaller than the original number or equal to the original number

When we divide one number by another, we may get a quotient that is bigger than the original number, smaller than the original number or equal to the original number

Ratios use division to represent relationships between two quantities





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ESSENTIAL QUESTIONS

What information do I get when I compare two numbers using a ratio?

What kinds of problems can I solve by using ratios?

How do I compute unit rate in tables, graphs, equations and diagrams?

How do I compute unit rate in real-world problems?

How do I use ratios and their relationships to solve real world problems?

How do I recognize proportional relationships between quantities?

How do I represent proportional relationships between quantities?

How do I solve multistep ratio problems using proportional relationships?

How do I solve multistep percent problems using proportional relationships?

How do I represent proportional relationships by equations?

How do I solve problems involving scale drawings of geometric figures?

How do I compute actual lengths and areas from a scale drawing?

How do I reproduce a scale drawing at a different scale? CONCEPTS AND SKILLS TO MAINTAIN

It is expected that students will have prior knowledge/experience related to the concepts and skills identified below. It may be necessary to pre-assess in order to determine if time needs to be spent on conceptual activities that help students develop a deeper understanding of these ideas.

number sense

computation with whole numbers and decimals, including application of order of operations

addition and subtraction of common fractions with like denominators

measuring length and finding perimeter and area of rectangles and squares

characteristics of 2-D and 3-D shapes

data usage and representations SELECTED TERMS AND SYMBOLS

The following terms and symbols are often misunderstood. These concepts are not an inclusive list and should not be taught in isolation. However, due to evidence of frequent difficulty and misunderstanding associated with these concepts, instructors should pay particular attention to them and how their students are able to explain and apply them.

The definitions below are for teacher reference only and are not to be memorized by the students. Students should explore these concepts using models and real life examples. Students should understand the concepts involved and be able to recognize and/or demonstrate them with words, models, pictures, or numbers.





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The websites below are interactive and include a math glossary suitable for elementary children. Note – At the elementary level, different sources use different definitions. Please preview any website for alignment to the definitions given in the frameworks.

http://www.amathsdictionaryforkids.com/

This web site has activities to help students more fully understand and retain new vocabulary (i.e. the definition page for dice actually generates rolls of the dice and gives students an opportunity to add them). This dictionary is for all levels of students and provides links to sample questions.

http://intermath.coe.uga.edu/dictnary/homepg.asp

Definitions and activities for these and other terms can be found on the Intermath website. Intermath is geared towards middle and high school students.

Fraction: A number expressed in the form a/b where a is a whole number and b is a positive whole number.

Multiplicative inverse: Two numbers whose product is 1r. Example: 3 4 and 4 3 are multiplicative inverses of one another because 3 4 × 4 3 = 4 3 × 3 4 = 1.

Percent rate of change: A rate of change expressed as a percent. Example: if a

population grows from 50 to 55 in a year, it grows by 5 50 = 10% per year

Ratio: A comparison of two numbers using division. The ratio of a to b (where b ≠ 0) can be written as a to b, as , or as a:b.

Proportion: An equation stating that two ratios are equivalent. Scale factor: A ratio between two sets of measurements.

CLASSROOM ROUTINES

The importance of continuing the established classroom routines cannot be overstated. Daily routines must include such obvious activities as estimating, analyzing data, describing patterns, and answering daily questions. They should also include less obvious routines, such as how to select materials, how to use materials in a productive manner, how to put materials away, how to access classroom technology such as computers and calculators. An additional routine is to allow plenty of time for children to explore new materials before attempting any directed activity with these new materials. The regular use of routines is important to the development of students' number sense, flexibility, fluency, collaborative skills and communication. These





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routines contribute to a rich, hands-on standards based classroom and will support students’ performances on the tasks in this unit and throughout the school year. STRATEGIES FOR TEACHING AND LEARNING

Students should build on their knowledge and understandings of rate and unit concepts in Grade 6.

Proportional relationships should be further developed through the analysis of graphs, tables, equations, and diagrams.

Grade 7 needs to push for the students’ deep understanding of the characteristics of a proportional relationship.

Percents are now being introduced in Grade 6, so percent problems should continue to follow the thinking involved with rates and proportions. Examples of providing the students with the opportunity to solve problems based within contexts are: researching newspaper ads, constructing their own questions, keeping a log of prices (particularly sales) and determining savings by purchasing items on sale.

Students should be actively engaged by developing their own understanding.

Mathematics should be represented in as many ways as possible by using graphs, tables, pictures, symbols and words.

Interdisciplinary and cross curricular strategies should be used to reinforce and extend the learning activities.

Appropriate manipulatives and technology should be used to enhance student learning.

Students should be given opportunities to revise their work based on teacher feedback, peer feedback, and metacognition which includes self-assessment and reflection.

Students should write about the mathematical ideas and concepts they are learning.

Consideration of all students should be made during the planning and instruction of this unit. Teachers need to consider the following:

What level of support do my struggling students need in order to be successful with this unit?

In what way can I deepen the understanding of those students who are competent in this unit?

What real life connections can I make that will help my students utilize the skills practiced in this unit?





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EVIDENCE OF LEARNING By the conclusion of this unit, students should be able to demonstrate the following competencies:

Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.

Recognize and represent proportional relationships between two quantities.

Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions in proportional relationships.

Represent proportional relationships by equations.

Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.

Solve multistep ratio and percent problems using proportional relationships.

Solve problems involving scale drawings of geometric figures and compute actual lengths and areas from a scale drawing.

Reproduce a scale drawing at a different scale.





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TASKS

The following tasks represent the level of depth, rigor, and complexity expected of all seventh grade students. These tasks or tasks of similar depth and rigor should be used to demonstrate evidence of learning. It is important that all elements of a task be addressed throughout the learning process so that students understand what is expected of them.

Task Name Task Type

Grouping Strategy Standards Addressed Content Addressed

What is Unit Rate? Learning Task

Individual or Small Group

Introduction and application of writing rates numerically and verbally.

Orange Fizz Experiment Learning Task

Individual or Small Group

Task requires use of ratios, proportions and proportional reasoning.

Creating a Scale Map Performance Task

Small Group

Through mapping an area students will create a scale drawing with an interpretive scale.

Which Is The Better Deal? Learning Task

Individual or Pairs

Unit rates are used to determine the most cost effective products.

Patterns & Percents Learning Task Small Group

Introduction to percent problems using bar models, ratios & proportional reasoning.

Developing a Sense of Scale

Formative Assessment Lesson Partner/Small Group

Solving real-world problems involving proportional relationships.





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Learning Task: What is Unit Rate? ESSENTIAL QUESTIONS:

How do we write unit rates? How does a unit rate represent a real-world situation? How do I interpret a unit rate (using words and mathematically)?

MATERIALS:

Activity sheets 1-4 for each student active-board or transparencies of these sheets for class discussion

TASK COMMENTS:

Students will develop an understanding of the unit rates associated with a proportional relationship. Students will also develop the ability to determine the appropriate rate to use in solving a problem and to use the corresponding unit rate to solve missing-value problems. STANDARD ADDRESSED IN THIS TASK: MCC7.RP.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction (1/2)/(1/4) miles per hour, equivalently 2 miles per hour. INTRODUCTION:

The lesson begins with a teacher –led lesson or mini-lesson introducing unit rates. Sheet 1 displays a teacher-led lesson. The pictures help students see that division is used to calculate a unit rate, as well as help students determine which of the two parts of the rate relationship should be the divisor in the division problem.

The teacher should model how to use the picture to find the unit rate. In problem 1, the teacher points out that to find the cost of one apple, the money has to be divided into three equal parts.





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A simple mapping, such as that in figure 1, can show the cost of one apple. The teacher should record the information depicted by the mapping (30 cents/1 apple), then write a sentence interpreting this rate.

As the teacher continues the lesson, students should find the unit rate for problem 2 in a similar manner and then write the interpretive statement. In this instance, the cost of one orange does not equal an integral amount.

The less familiar contexts of problems 3 and 4 can be addressed in a similar manner. Two possible rates are generated for the same relationship (2 British pounds = 3 U.S. dollars). Each results in its own diagram.

The following questions can be used to help students solve the problems on sheet 1:

1. What operation was modeled in each solution? 2. For each problem, what quantity was divided and into how many equal groups? Can you

explain why? 3. What computation can be set up to solve each problem? 4. How can each division problem be written using fractional notation?

Sheet 1 can be accessed at

http://www.cehd.umn.edu/ci/rationalnumberproject/images/89_4/sheet_1.pdf

On Sheet 2, students are asked to generate two possible rates for each relationship, compute the equivalent unit rates, and write a short sentence interpreting each of the unit rates. The rates should be written in the form "a/b." Be sure that students do not "drop" the labels. They are necessary to interpret the rate. Students must understand that a unit rate always describes "how many for one." Some unit rates have strange or nonfunctional meanings, For example, 3 tennis balls per can is easier to envision than (1/3 can) per 1 tennis ball.

Sheet 2 can be accessed at http://www.cehd.umn.edu/ci/rationalnumberproject/images/89_4/sheet_2.pdf





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On Sheet 3, students learn how to select an appropriate unit rate and use the unit rate to solve missing-value problems. Additional practice in selecting the appropriate rate can be furnished by posing missing value questions for each of the situations presented on sheet 2. This additional experience may be necessary for many students before they attempt the applications on Sheet 4.

Sheet 3 and Sheet 4 can be accessed at http://www.cehd.umn.edu/ci/rationalnumberproject/images/89_4/sheet_3.pdf http://www.cehd.umn.edu/ci/rationalnumberproject/images/89_4/sheet_4.pdf

Resource used for this task: Cramer, K.., Behr, M., & Bezuk, N. (1989). Proportional relationships and unit rates.

Mathematics Teacher, 82 (7), 537-544. http://www.cehd.umn.edu/ci/rationalnumberproject/89_4.html

Sheet 1





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6 bags/30 pounds 30 pounds/6 bags

(1/5) bag/1 pound

5 pounds/1 bag

One pound of flour fills 1/5 of a bag

One bag holds 5 pounds of flour

3 cans/9 balls

One can contains 3 balls

One ball fills 1/3 of a can

9 balls/3 cans

3 balls/1 can

(1/3)can/1 ball

$6.50/5 gallons

One dollar will purchase 0.769 gallons

One gallon costs $1.30

5 gallons/$6.50

0.769 gallons/$1

$1.30/1 gallon

10 laps/25 minutes

One lap is run in 2.5 minutes

In one minute 0.4 laps are run

25 minutes/10 laps

2.5 minutes/1 lap

0.4 laps/1 minute





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3 apples/90 cents 90 cents/3 apples

(1/30) apple/1 cent 30 cents/1 apple

The amount of apple for one cent; the cost of one apple

30 cents/1 apple

2 pounds/3 dollars 3 dollars/2 pounds

2/3 pound/1 dollar 1.5 dollars/1 pound

The number of British pounds that equal one dollar; the number of dollars that equal one British pound

2/3 pounds/1 dollar

.60

.90

1.20

1.50

2/3

4/3

2

8/3

10/3

The price increase by thirty cents for each apple

The number of pounds can be found by multiplying the number of dollars by 2/3

2/3/ pounds/1 dollar x 20 dollars = 40/3 pounds; Unit price x number of dollars = number of pounds

30 cents/1 apple x 7 apples =$2.10 Unit price x number of apples = cost





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12 laps/4 minutes

3 laps/1 minute

Mark’s train travels 3 laps each minute

(3 laps/1 minute) x 9 minutes = 27 laps

10 minutes/6 laps

5/3 minutes/1 lap

Donna runs 1 lap in 5/3 minutes

(5/3 minutes/1 lap) x 5 laps = 8 1/3 minutes

5 cups water/2 scoops mix

2.5 cups water/1 scoop mix

Ryan should use 2.5 cups of water for each scoop of mix

(2.5 cups water/1 scoop mix) x 9 scoops mix = 22.5 cups water

3 cans white/2 cans blue

1.5 cans white/1 can blue

Anne should mix 1.5 cans of white paint with each can of blue.

(1.5 cans white/1 can blue) x 6 cans blue = 9 cans white





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Learning Task: What is Unit Rate?





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Learning Task: Orange Fizz Experiment ESSENTIAL QUESTIONS:

How can representing mathematical ideas in different ways (graphs, tables, equations, diagrams, words) help me solve problems?

What strategies can be used to compare ratios? How do I interpret a unit rate (using words and mathematically)?

MATERIALS:

Warm-up Problems for discussion Student Task sheet

TASK COMMENTS:

In the task, students analyze and solve an open-ended problem where they are asked to compare the strength of the orange taste of three drink mixes. Approaches to the task may include a range of different strategies. Students will investigate how ratios can be formed and scaled up to find equivalent ratios. In addition, students will use proportional reasoning to decide how to use the different mixes to make drinks for 200 people. The task is a challenging problem for students and at least 1 to 2 class periods should be allowed for students to explore and have a whole-class discussion on the task.

STANDARD ADDRESSED IN THIS TASK: MCC7.RP.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. MCC7.RP.2 Recognize and represent proportional relationships between quantities. MCC7.RP.3 Use proportional relationships to solve multistep ratio and percent problems. INTRODUCTION:

Prior to the task students need to discuss and explore making comparisons with ratios, percents, and fractions. Warm-up problems are provided to lead the discussion of multiple ways to make comparisons. Allow 15-20 minutes of discussion. Models and drawings, as illustrated below may facilitate student understanding. Relate the strategies from the warm-up problems to the task where students will be comparing the mixes. Make these connections during the whole-group discussion. Students should be able





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to see how each form, ratios, percents, and fraction, provides information needed to derive one of the other forms. Warm-Up Problems: A useful way to compare numbers is to form ratios. Talk to your classmates about what is the same and what is different about these ratio statements. (a) Write the ratio in the problem in multiple ways; (b) write an equivalent ratio; and (c) compare each pair of ratios- what is alike or different about each?

1. The ratio of boys to girls in Ms. Dade’s class is 12 boys to 18 girls. 2. The ratio of boys to the class in Mr. Hill’s class is 14 boys to 30 students.

Solution: Students may write the ratios as:

(1) 12:18,

, 12 to 18 (2) 14: 30,

, 14 to 30

There are several possibilities for equivalent ratios; examples include: (1) 6:9 or 24: 36 (2) 7: 15 or 28: 60

Comparison: #1 is comparing part-to-part and #2 is comparing part-to-whole. Discuss how to compare the two classes using tables, percents, common denominators or models. When #1 is set up as part-to-whole (12: 30), a comparison can be made easily using common denominators. The model below also illustrates the comparison.

3. The ratio of cats to dogs in our house is .

4. The ratio of cats to animals in Darla’s house is 2:6. Solution: Students may write the ratios as:

(3) 1: 4,

, 1 to 4 (4) 2: 6,

, 2 to 6

There are several possibilities for equivalent ratios; examples include: (3) 2: 8 or 4: 16 (4) 1: 3 or 4: 12





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Comparison: #1 is comparing part-to-part and #2 is comparing part-to-whole. Discuss how to compare the two classes using tables, percents, common denominators or models. The model below compares the ratios using a bar model and percents.

(3) Using the bar model, students will see the relationship between the parts of the set

and the whole set. Additionally, students can “see” how percent can be determined. The whole set represents 100% of the animals, however, cats make-up 20 % of the animals and dogs make-up 80%.

(4) The second bar model shows us that there are 2 cats and 4 dogs. We are given that there are 6 animals altogether. We can easily estimate to compare percentages among the two households. We know that there is more than 25% of the population at Darla’s house that is made up of cats. We could also discuss how to calculate the percentage of cats more precisely.

Task Directions: A famous cola company is trying to decide how to change their Orange Fizz formula to produce the best tasting orange soft drink on the market. The company has three different Orange Fizz formulas to test with the public. The formula consists of two ingredients: orange juice concentrate and carbonated water. Using the company’s new formulas, answer the following questions. Formula A: 1 cup of orange concentrate to 2 cups of carbonated water Formula B: 2 cups of orange concentrate to 5 cups of carbonated water Formula C: 2 cups of orange concentrate to 3 cups of carbonated water Part A:

1. Which formula will make the drink that has the strongest orange taste? Show your work and explain.





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Solution: The approaches to the solution may vary. Students should be able to show and explain that Formula C will have the strongest orange taste since the ratio of concentrate to orange fizz (part-to-whole) will be 2:5. Formulas A and B have ratios of concentrate to juice of 1:3 and 2:7 respectively. Students may justify their findings by comparing fractions through like denominators, using models, and/or percents or decimals. For example:

They could compare two ratios at a time: (Formula A)

(Formula B )

Then compare Formula A to Formula C, since A was larger than B. (Formula A)

(Formula C )

Therefore, the drink with the highest ratio of concentrate to Orange Fizz would have the strongest orange taste.

2. Which formula has the highest percentage of carbonated water in the mixture? Estimations may be used. Show your work and explain.

Solution: Students could use bar models to compare through estimation the highest percentage of carbonated water in the mixture. Solution approaches may vary. Students may realize that they can use the information from #1 to help them solve the problem. For example, since Formula B had the lowest concentrate then it must have the highest amount of carbonated water among the three formulas. Formula B has 5 to 7 ratio of carbonated water to juice. Students may estimate percentages using a bar model or students may calculate decimals or precise percentages using proportions.





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Part B:

3. For researchers to test their product, they will need to produce enough of each Orange Fizz formula to take to various locations around the area for taste testing. Researchers would like for at least 200 people to sample each formula.

Each sample will contain of a cup of Orange Fizz.

Calculate the amount of orange concentrate and carbonated water that would be needed to make enough Orange Fizz for the survey. Fill in the table below.

Solution: Students may use a variety of approaches to calculate the answer. Students may find it easier to set up a table to arrive at their final answers. Sample:

Formula A: Orange Concentrate Carbonated Water Total Amount

1 2 3 cups (Six ½c. servings) 2 4 6 cups (12- ½ c. servings) 3 6 9 cups (18- ½ c. servings) 4 8 12 cups (24- ½ c. servings)

… … … 17 34 51cups (102- ½c. servings)

Students may continue to add 1:2:3 to each row until they reach 17:34:51, or they may notice that they can multiply a row by a scale factor to get to their result more quickly.


1 2 3 cups (Six ½c. servings) 17 34 51cups (102- ½c. servings)

Formula B: Orange Concentrate Carbonated Water Total Amount

2 5 7 cups (14- ½ c. servings) 16 40 56 cups (112- ½ c. servings)

Formula C: Orange Concentrate Carbonated Water Total Amount

2 3 5 cups (10- ½ c. servings) 20 30 50 cups (100- ½ c. servings)





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Learning Task: Orange Fizz Experiment Warm-up Problems: A useful way to compare numbers is to form ratios. Talk to your classmates about what is the same and what is different about these ratio statements. (a) Write the ratio in the problem in multiple ways; (b) write an equivalent ratio; and (c) compare each pair of ratios- what is alike or different about each?

1. The ratio of boys to girls in Ms. Dade’s class is 12 boys to 18 girls. 2. The ratio of boys to the class in Mr. Hill’s class is 14 boys to 30 students.

3. The ratio of cats to dogs in our house is .

4. The ratio of cats to animals in Darla’s house is 2:6.





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Learning Task: Orange Fizz Experiment A famous cola company is trying to decide how to change their Orange Fizz formula to produce the best tasting orange drink on the market. The company has three different Orange Fizz formulas to test with the public. The formula consists of two ingredients: orange juice concentrate and carbonated water. Using the company’s new formulas, answer the following questions. Formula A: 1 cup of orange concentrate to 2 cups of carbonated water Formula B: 2 cups of orange concentrate to 5 cups of carbonated water Formula C: 2 cups of orange concentrate to 3 cups of carbonated water Part A:

1. Which formula will make a drink that has the strongest orange taste? Show your work and explain.

2. Which formula has the highest percentage of carbonated water in the mixture? Estimations may be used. Show your work and explain.





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Part B:

3. For researchers to test their product, they will need to produce enough of each Orange Fizz formula to take to various locations around the area for taste testing. Researchers would like for at least 200 people to sample each formula.

Each sample will contain of a cup of Orange Fizz.

Calculate the amount of orange concentrate and carbonated water that would be needed to make enough Orange Fizz for the survey. Fill in the table below.


Formula B: Orange Concentrate Carbonated Water Total Amount

Formula C: Orange Concentrate Carbonated Water Total Amount





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Learning Task: Creating a Scale Map ESSENTIAL QUESTIONS:

How are distances and measurements translated into a map or scale drawing?

How do I use a legend of a map?

How do I create a legend for a map, including an accurate scale?

How do I determine the an appropriate scale for the area (such as my yard or school) that I am measuring and mapping?

MATERIALS:

Student Guide 3-1 Student Guide 3-2 Measuring tapes Yardsticks Meter sticks; or trundle wheels; compasses (for finding directions) rulers crayons or colored pencils or markers poster paper (approximately 22 x 28 inches).

TASK COMMENTS:

Students have many opportunities to copy and read maps, however not to create one. When students have to create a map, they realize that mathematics plays a major role in map making. Students can work in pairs or groups of 3. Students will create a scale map of their school, school grounds, or their yard at home. They will include landmarks, important details, legends, and an accurate scale. Suggested time – 2 to 3 class periods. Math Skills to Highlight:

Measuring and rounding distances Determining an appropriate scale (ratio and proportions) Making a scale drawing





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STANDARDS ADDRESSED IN THIS TASK: MCC7.RP.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction (1/2)/(1/4) miles per hour, equivalently 2 miles per hour. MCC7.RP.2 Recognize and represent proportional relationships between quantities. MCC7.G.1 Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.

INTRODUCTION:

Give students options regarding the maps they would like to draw. Students who have a large yard may wish to measure and create a map of that; students who live in apartments might prefer to measure and create a map of their school or school grounds. If students are to draw a map of their school, you might decide to limit the map to a section or wing.

Begin this project by explaining that maps use a scale to ensure accurate distances. On a

world map, one inch might equal 1,500 miles, while on the map of a small town, an inch might equal one-fourth of a mile. On a map of a yard or block, an inch might only equal several feet. Scales are created with mathematics.

Distribute copies of Student Guide 3-1 and review it with your students.

Creating a Scale Map

If you wish, you may simplify the project by instructing students to create a scale map of your school or school grounds. In this way, you may take your class to obtain the necessary measurements together.

In measuring distances, remind students to be as accurate as possible. We suggest that distances be rounded to the nearest foot, nearest yard, or nearest meter.

Suggest that students sketch a rough copy of their map first. This will help them to “see” relationships and estimate where things should be.

Distribute copies of Student Guide 3-2, “How to Make a Map.” Go over it with your students, making sure they understand how to create a scale for their maps. You may want to check the scales students select before they start making the final copies of their maps.





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After students draw their maps, they should label and color them. Suggest that they provide legends, label the direction, and, of course, note the scale.

Extension:

Encourage students to pursue a study of an instrument called the architect’s scale, and use it to make a scale drawing of their house. If their house has two stories, they may need to create more than one drawing.

Resource used for this task: Muschia, G. R. &., Muschia, J.A. (1996). Hands-on math projects with real-life applications.

(pp. 112-115). New Jersey: Jossey-Bass.

Solution:

Solutions to student’s maps will vary. To assess student understanding, make sure that measured distances are reasonable and that students have an accurate and appropriate scale for their drawing.





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Student Guide 3-1

Name: Project Due Date:

Creating A Scale Map

Situation/Problem:

You and your partner(s) are to create a scale map of a familiar place such as your school, school grounds, or the yard of your home.

Possible Strategies:

1. Accurately measure distances (rounding to the nearest foot, yard, or meter). 2. Note landmarks. In case of a yard, this might include things like trees, woodpiles, sheds,

etc. You might include such things in a legend on your map. 3. Create a rough sketch of your map before drawing a final copy. A “rough sketch” will

help you to visualize perspectives and landmarks. Special Considerations:

Use a measuring tape, yardstick, meter stick or trundle wheel for measuring distances.

Use a pad and pencil to record distances. Don’t try to remember the distances; this may cause mistakes in your map and your scale.

As you record distances, sketch your map, placing landmarks “about” where they would be. Record the distances in feet, yards, or meters. It’s a good idea to locate landmarks using the measurements from two boundaries.

Use a compass to find directions. Be sure to label the directions correctly on your map.

Consult Student Guide 3-2 for information about working with scale drawings.

Be sure the final copy of your map is accurate. Label distances and landmarks, add color, and include a legend and directions. You may want to compare your map to the original area and check it for accuracy.

To Be Submitted:

1. Your scale map 2. Your records of measurements and calculations





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Student Guide 3-2


How To Make A Map

1. Decide upon the boundaries of your map. 2. Make a sketch of the area you will include on your map. Note the approximate position

of any landmarks. In a school, land marks might include stairwells, display cases, or water fountains. Landmarks in a yard might include trees, flowerbeds, decks, sheds, or woodpiles.

3. Accurately measure the boundaries (length and width) of the area. Locate the position of landmarks by obtaining at least two measurements from boundaries.

4. Select the scale by considering your longest measurement, and how to “fit” it on the paper. Remember that the scale should be as long as possible so that your map will look good on the paper.

5. To choose the best scale, divide the longest length of your paper in inches (or centimeters) by the longest dimension of the boundary in feet (or meters).

Round your quotient down to the nearest quarter or eighth inch (or centimeter). Here’s an example: The longest boundary (longest length) on your map is 80 feet.

The longest dimension of your paper is 28 inches. .35. Since .35 is between

.25 (one fourth inch) and 0.375 (three-eighths inch), you must round down so that

your scale will be inch = 1 foot.

Now take the other dimension of the boundary and the other dimension of the paper, and divide the length of the paper by the length of the boundary.

Round your quotient down to the nearest quarter or eighth inch (or centimeter).

Compare the scales. If they are the same, great! If they are different, use the smaller scale.





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6. To place items on your map, use your measurements and the scale you have chosen. For example, suppose an apple tree is 21 feet from the fence on the eastern side of the yard,

and 16 feet from the fence on the northern side. If your scale is inch = 1 foot, multiply

the number of inches by the number of feet to determine the number of inches the actual distance would be on your map. Note the example of the math below.

14

211

214

514

14

161

4

Place the tree 5 inches from the fence on the eastern side, and 4 inches from the

northern side.





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Learning Task: Which Is The Better Deal? ESSENTIAL QUESTIONS:

How does my understanding of unit rate save me money? How can I determine the unit rate for a product that I might

purchase? MATERIALS:

Student Guide 3-5 Student Data Sheet 3-6 White poster paper Rulers markers and felt-tipped pens for making charts Samples of products (Students can clip advertisements or print products off the internet.

They need to have a picture of the product, explanation/details, price) TASK COMMENTS:

Students will evaluate and compare the packaging and prices of products they select from advertisements or the internet. For example, if a student selects canned colas, they will compare the price of a12-pack of canned coke to a 6-pack of canned coke and determine which buy would be the better deal. Working individually or in pairs, students will select three different products.

Math Skills to Highlight:

1. Gathering and analyzing data 2. Comparing prices and features of products 3. Making decisions based on collected data 4. Creating charts to illustrate conclusions 5. Computing unit rate

STANDARDS ADDRESSED IN THIS TASK: MCC7.RP.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction (1/2)/(1/4) miles per hour, equivalently 2 miles per hour. MCC7.RP.3 Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.





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INTRODUCTION:

Prior to the task, the teacher will introduce the relationship between unit rate and purchasing products. We live in a society where we have a choice of purchasing countless consumer products. Just consider how many types of sneakers we can buy, or how many brands of potato chips we can choose from. To help us select the best package of products, we are going to look at how to figure unit price. Ask your students what rationale they use to buy products. For example, why did they buy the drinks they have at home? Most likely they buy the same type drinks every time. They either purchase the same drinks due to price, taste, or some other reason. Explain that people usually have reasons for buying one product instead of another.

Begin the project by explaining that students will select a type of product – see Student Data Sheet 3-6 for some examples.

Distribute copies of Student Guide 3-5 and review it with your students. Emphasize that they should compare three different products and each product needs to be packaged in three different sizes. For example, if the student chooses coke then they need to price it as a 6-pack, 12-pack, and 24-pack (and each package needs to have the same number of ounces). Teachers can model this with bringing in examples of products with the same ounces or use ads.

Remind students to create a rough chart on scrap paper before attempting to draw their final copy. Encourage them to design a chart that will be informative as well as easy to read.

As a wrap-up to the task, students may present oral presentations using charts and /or other evidence to support their findings. You may also wish to display the charts.

Resource used for this task:

Muschia, G. R. &., Muschia, J.A. (1996). Hands-on math projects with real-life applications. (pp. 126-130). New Jersey: Jossey-Bass.

Solution: Key elements to look for in the student’s work are: (1) accurate calculations of unit rate for the products selected, (2) reasonableness in their comparisons and unit rate, and (3) correct use of terminology when orally presenting their findings.





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Student Guide 3-5


Which Is The Better Deal Situation/Problem: You and your partner(s) are to select a product, and compare the size of package and price (three different sizes/prices). You are to trying to determine which is the best deal by finding their unit price. After you have reached your conclusions, design a chart to support your findings and present your data to the class through an oral report. Possible Strategies:

1. Look in sales papers for groceries/retail stores. 2. Brainstorm with your partner(s) which products you might like to compare.

Special Considerations:

After selecting your product, decide which size or quantity in a package you will compare. Write these categories on a sheet of paper, then compare the price.

After obtaining your data, analyze it and make decisions comparing quantity/size to price. Compute the unit rate.

Create a chart illustrating your results. Sketch a rough copy of your chart first. This enables you to revise the chart before starting the final copy. Arrange the design so it presents the data clearly. List your products by brand name and show your comparison of quantity/size to price. If there is room on your chart, you may wish to provide a brief summary of your results and why you chose that quantity/size product for that price.

Before presenting your findings to the class, write notes so that you don’t forget to mention any important information. Rehearse your presentation.

To Be Submitted:

1. Research/Comparison Notes 2. Chart





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Student Data Sheet 3-6


Which Is The Better Deal

Popular products are compared regularly. Many educated consumers rely on unit pricing to make sure they are getting the best deal to fit their needs and budgets.

Some products and quantity/size to compare:

Soda

Potato Chips

Ice Cream

Milk

Paper products

Snack crackers

Any product that is packaged in more than one size can be compared. For example, you could compare the unit price of a 6-pack of Coke to the unit price of a 6-pack of Pepsi.





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Learning Task: Patterns & Percents ESSENTIAL QUESTIONS:

How can models be used to solve percent problems? What patterns can be found in tables and graphs when working with percents? How do I apply mental math strategies to solve percent problems?

MATERIALS:

Warm-up Problems- Models & Patterns Patterns & Percents Task sheet

TASK COMMENTS:

The learning task is focused on using models and patterns to develop number sense for solving percent problems. The lesson will link models, tables, graphs and proportional reasoning to percent problems. STANDARD ADDRESSED IN THIS TASK: MCC7.RP.2 Recognize and represent proportional relationships between quantities.

MCC7.RP.2a Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. MCC7.RP.2b Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

MCC7.RP.3 Use proportional relationships to solve multistep ratio and percent problems. INTRODUCTION:

Percent problems are specific kinds of proportion problems. Students began their work with percent problems in 6th Grade Math. 7th Grade Math expands their understanding and fluidity with percent problems. Students should be reminded that a percent is a part-to-whole relationship wherein the whole is always 100. Models are excellent tools to help students connect concrete to algebraic representations. To introduce the Patterns & Percents task, first work through the warm-up problems provided with the students. Discuss the patterns that can be found when working with percents. Use vocabulary such as ratio and proportion to continue making connections to prior learning experiences.





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Emphasize the application of mental math strategies. This may help students determine whether or not their answers to problems are reasonable. Warm-Up Problems: Let’s begin with 10%.

How would you write 10% as a decimal? ________________

How would you write 10% as a fraction? ________________ Let’s find a pattern using 10%. If the bar below represents a whole number amount, then what percent would it equal?

Solution: 100%

Now let’s give the bar a value. The bar now represents $10 which is 100%. Label the bar.

Solution: Solutions are in blue on the bar model.

Divide the bar up into ten equal parts. Label each part with its correct percent value and the correct money value. (Hint: divide $10 by 10)

Solution: Solutions are in blue on the bar model.

Let’s look at this relationship. Is this a proportional relationship? How do you know? Fill in the following table and graph the relationship. We will use the values from the bar model above.





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Percent Amount based on $10

0 0 10 1 20 2 30 3 40 4 50 5 60 6 70 7 80 8 90 9

100 10

Solution: Solutions are in blue in the table and graph above. The relationship of 10% of 10 is a proportional relationship. You may show students the relationship as a proportion. For

example, . We usually have students set up the proportion as .

Answer the following problems based on what we have learned about percents and patterns. You may use the model, table or graph above to help you. (Hint: Use what you know about 10% of 10 to help you answer each problem.)

1. What is 15% of 10? Solution: 1.5: Use the graph or bar model to help students understand the answer. 2. Jane went to the store to do some shopping. The sign in the window read, Big Sale Today

Only- 20% off of everything in the store!! Jane bought headphones for her I-Pod that were regularly priced for $10. What did Jane pay for the I-Pod headphones before tax?

Solution: $8.00: Use the graph or bar model to help students see 20% of $10. Discuss discounts. 3. Walt makes $10 an hour and gets a 15% raise. How much will Walt make an hour after his

raise? Solution: $11.50: Use the graph or bar model to help students see 15% of $10. Discuss the impact of a raise on someone’s salary.





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Task Directions: For problems 1 – 5, label the percent bar with its appropriate dollar values and percent values. Divide the bar into ten equal parts. Then find 10% of the total. Write a proportion to represent the relationship. 1] The total amount is $200.

Solution:

Solutions are in blue on the bar model. %

%

2] The total amount is $800.

Solution:


%


Solution:


%


Solution:


%

.





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Solution:


%

.

Instead of always drawing a bar to help us find 10% of any value, let’s use our knowledge of patterns and proportions to find 10% of a number. Let’s look at the proportions you set up. What patterns do you see? Explain.

10%100%

$20$200

10%100%

$80$800

10%100%

$48$480

10%100%

$4.80$48

10%100%

$6.40$64

Solution: Students should be able to see that 10% of a number is of the amount of the original number. Students may notice that the place value changes- “move over one space to the left”. Point out to students that every proportion is equivalent and has a common factor of .

Answer the following problems based on your knowledge of 10% of a number. Set up a proportion to help you get started. Use this knowledge to help you determine each answer.

1. Julia wants to buy a dress that is on sale for 20% off. The original price was $84. What is the sale price?

Solution: Students may set up the proportion %

%

.. If students double 10% or $8.40, the discount is

$16.80; therefore the sale price is $67.20.





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2. Shawn earned $220 at his summer job. Shawn put 60% of his money in savings. How much money did Shawn put into savings?

Solution: Students may set up the proportion %

%. If students calculate 6 times the 10% amount or

$22.00 then the amount that Shawn puts into savings is $132.00. Students may prefer to make a table to find the amount. For example,


0 0 10 22 20 44 30 66 40 88 50 110 60 132 70 154 80 176 90 198 100 220





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Learning Task: Patterns & Percents Warm-Up Problems: Let’s begin with 10%.

How would you write 10% as a decimal? ________________

How would you write 10% as a fraction? ________________ Let’s find a pattern using 10%. If the bar below represents a whole number amount, then what percent would it equal?

Now let’s give the bar a value. The bar now represents $10 which is 100%. Label the bar.

Divide the bar up into ten equal parts. Label each part with its correct percent value and the correct money value. (Hint: divide $10 by 10)

Let’s look at this relationship. Is this a proportional relationship? How do you know? Fill in the following table and graph the relationship. We will use the values from the bar model above.






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Answer the following problems based on what we have learned about percents and patterns. You may use the model, table or graph above to help you. (Hint: Use what you know about 10% of 10 to help you answer each problem.)

1. What is 15% of 10?

2. Jane went to the store to do some shopping. The sign in the window read, Big Sale Today Only- 20% off of everything in the store!! Jane bought headphones for her I-Pod that were regularly priced for $10. What did Jane pay for the I-Pod headphones before tax?

3. Walt makes $10 an hour and gets a 15% raise. How much will Walt make an hour after his raise?





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Learning Task: Patterns & Percents For problems 1 – 5, label the percent bar with its appropriate dollar values and percent values. Divide the bar into ten equal parts. Then find 10% of the total. Write a proportion to represent the relationship. 1] The total amount is $200.





Instead of always drawing a bar to help us find 10% of any value, let’s use our knowledge of patterns and proportions to find 10% of a number. Let’s look at the proportions you set up. What patterns do you see? Explain.





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10%100%

$20$200

10%100%

$80$800

10%100%

$48$480

10%100%

$4.80$48

10%100%

$6.40$64

Answer the following problems based on your knowledge of 10% of a number. Set up a proportion to help you get started. Use this knowledge to help you determine each answer.

1. Julia wants to buy a dress that is on sale for 20% off. The original price was $84. What is the sale price?

2. Shawn earned $220 at his summer job. Shawn put 60% of his money in savings. How much money did Shawn put into savings?





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Performance Task: Nate & Natalie’s Walk ESSENTIAL QUESTIONS:

How can we make predictions from graphs? What makes a relationship “proportional”? How can I tell if a proportional relationship

exists? How can I use tables, graphs or equations to determine whether a relationship is

proportional?

MATERIALS:

Warm-up Problems- Models & Patterns Patterns & Percents Task sheet

TASK COMMENTS:

Students will use proportional reasoning to compare the distance that a brother and sister walk. In the performance task, students will use tables and graphs to represent a proportional relationship. Prior to doing this performance task, students should understand that graphing is a way to visually represent ratios and proportional relationships. This visual is a tool that can be used to determine the reasonableness of an equation and to draw conclusions about proportional relationships. STANDARD ADDRESSED IN THIS TASK: MCC7.RP.2 Recognize and represent proportional relationships between quantities.


MCC7.RP.2c Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn. MCC7.RP.2d Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.





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INTRODUCTION: Hand out the task and allow students to work individually for 3-5 minutes without intervening. Circulate around the classroom to get an idea about what strategies are being used to solve the problem. After 3-5 minutes, students may work with their partner or in their small groups. Support students problem-solving by:

asking questions related to the mathematical ideas, problem-solving strategies, and connections between representations.

asking students to explain their thinking and reasoning. Task Directions: Nate and his sister Natalie are walking around the track at school. Nate and Natalie walk at a steady rate and Nate walks 5 feet in the same time that Natalie walks 2 feet.

a) Draw a diagram or picture that represents Nate and Natalie’s walk around the track.

Solution: Students may use a variety of representations. For example, bar models or a number line as shown below could be used to represent the walk around the track.

Questions to encourage thinking:

Are Nate and Natalie walking at the same rate? Can you explain what your diagram shows about Nate and Natalie’s walk?

b) Set up a table and draw a graph to represent this situation. Let the x-axis represent the

number of feet that Nate walks and the y-axis represents the number of feet that Natalie walks.





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Solution: Table:

Nate (x) Natalie (y) 0 0 5 2

10 4 15 6 20 8

c) What patterns do you see in the table? Explain the pattern. Express this as an equation. Solution: Students should be able to explain that for every 5 steps Nate makes, Natalie makes 2. So the ratio of Nate’s steps to Natalie’s is 5:2. The equation that represents the situation is

y = x

d) How do you read the graph? Explain what the coordinate (20, 8) means in the context of

Nate and Natalie’s walk?

Solution: The coordinate represents that when Nate makes his 20th step that Natalie has made 8 steps.

e) When Nate walks 45 feet, how far will Natalie walk? Explain in writing or show how you

found your answer.

Solution: Students may use the equation, table, or graph as a way to answer the problem. They may also

set up a proportion to solve the problem. For example, . When solved the answer is 18.

Natalie will have taken 18 steps when Nate has taken 45 steps. Comment: For further discussion, ask, “Can you predict how far Natalie will walk if Nate walks 1000 feet?” This discussion should focus on the most efficient methods for solving the problem (equation or proportion) versus using a table or a graph.





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Performance Task: Nate & Natalie’s Walk Nate and his sister Natalie are walking around the track at school. Nate and Natalie walk at a steady rate and Nate walks 5 feet in the same time that Natalie walks 2 feet.

a) Draw a diagram or picture that represents Nate and Natalie’s walk around the track.

b) Set up a table and draw a graph to represent this situation. Let the x-axis represent the number of feet that Nate walks and the y-axis represents the number of feet that Natalie walks.

c) What patterns do you see in the table? Explain the pattern. Express this as an equation.

d) How do you read the graph? Explain what the coordinate (20, 8) means in the context of Nate and Natalie’s walk?

e) When Nate walks 45 feet, how far will Natalie walk? Explain in writing or show how you found your answer.





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Formative Assessment Lesson: Developing a Sense of Scale Source: Formative Assessment Lesson Materials from Mathematics Assessment Project https://www.georgiastandards.org/Common-Core/Documents/l63_developing_a_sense_of_scale_complete.pdf

ESSENTIAL QUESTIONS:

What strategies are appropriate in scaling problems? How do I solve a proportionality problem? How do I solve real-world problems that involve proportional relationships?

TASK COMMENTS:

Tasks and lessons from the Mathematics Assessment Project are specifically designed to help teachers effectively formatively assess their students. The way the tasks and lessons are designed gives the teacher a clear understanding of what the students are able to do and not do. Within the lesson, teachers will find suggestions and question prompts that will help guide students towards understanding. For more information access the MAP website: http://www.map.mathshell.org/materials/background.php?subpage=formative The PDF version of the task can be found at the link below: https://www.georgiastandards.org/Common-Core/Documents/l63_developing_a_sense_of_scale_complete.pdf

STANDARDS ADDRESSED IN THIS TASK: MCC7.RP.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 12 mile in each 1

4 hour, compute the unit rate as the complex fraction ∙ miles per hour,

equivalently 2 miles per hour. MCC7.RP.2 Recognize and represent proportional relationships between quantities.






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MCC7.RP.2c Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn. MCC7.RP.2d Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.

MCC7.RP.3 Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.

Unit 3: Ratios & Proportional...

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