Chapter 4 Proportional Relationships, Solving Problems (6 Weeks)

Table of Contents

Chapter 4 Proportional Relationships, Solving Problems (6 Weeks)

4.0 Anchor Problem: Purchasing Tunes-Town Music Company—What’s the Better Deal?

Section 4.1: Understand and Apply Unit Rates

4.1a Class Activity: Model Ratios (Review)

4.1a Homework: Model Ratios (Review)

4.1b Class Activity: Equivalent Ratios and Rate Tables

4.1b Homework: Equivalent Ratios and Rate Tables

4.1c Class Activity: Model and Understand Unit Rates

4.1c Homework: Model and Understand Unit Rates

4.1d Classwork: More with Unit Rates

4.1d Homework: More with Unit Rates

4.1e Extra Practice: Complex Fraction Unit Rates

4.1f Class Activity: Compare Unit Rates

4.1f Homework: Compare Unit Rates

4.1g Classroom Activity: Using Unit Rates

4.1h Unit Rate Projects:

4.1i Unit Rate Review

4.1j Self-Assessment: Section 4.1

Section 4.2: Identify and Communicate Unit Rates in Tables and Graphs.

4.2a Classroom Activity: Determine Proportionality (Review)

4.2a Homework: Determine Proportionality (Review)

4.2b Classroom Activity: Mixing Lemonade—Expressing Proportionality

4.2b Homework: Expressing Proportionality

4.2c Classwork: Proportions from Tables and Graphs

4.2c Homework: Proportions (Unit Rates) from Tables and Graphs

4.2d Classwork: Review Proportional Tables, Stories, and Graphs and Compare Rates

4.2d Homework: Compare Unit Rates Using Tables and Graphs

4.2e Classroom Activity: Jon’s Marathon—Find Unit Rate (Proportional Constant) in Tables and Graphs

4.2e Homework Activity: Revisit Making Lemonade

4.2f Review: Representing a Proportional Relationship with a Table, Graph, and Equation

4.2g Self-Assessment: Section 4.2

Section 4.3: Analyze Proportional Relationships Using Unit Rates, Tables, Graphs and Equations.

4.3a Classroom Activity: Writing Equations from Patterns

4.3a Homework: Writing Equations from Patterns

4.3b Classwork: Writing Equations from Graphs

4.3b Homework: Writing Equations from Graphs

4.3c Classroom Activity: Convert Units—Proportion in Tables and Graphs

4.3c Homework: Convert Units—Proportion in Tables and Graphs

4.3d Extra Assignment: Revisit Unit Rate Situations, Write Equations

4.3e Classroom Activity: Compare Proportional, Non-Proportional Patterns and Equations

4.3e Homework: Write Equations from Tables, Patterns, Graphs

4.3f Classroom Activity and Homework: Time Trials (Rates of Speed)

4.3g Class Activity and Homework: Jet Ski Rentals—Proportional vs. Non-proportional Equations

4.3h Self-Assessment: Section 4.3

Section 4.4: Analyze and Use Proportional Relationships and Models to Solve Real-World and Mathematical Problems.

4.4a Classroom Activity: Use Models to Solve Proportional Problems

4.4a Homework: Use Models to Solve Proportional Problems

4.4b Classroom Activity: Writing Proportions

4.4b Homework: Writing Proportions

4.4c Class Activity: Solving Proportions

4.4c Homework: Solving Proportions

4.4d Class Activity: PART-TO-PART and PART-TO-WHOLE Proportion Problems

4.4d Homework: PART-TO-PART and PART-TO-WHOLE Proportion Problems

4.4e Extra Assignment: Use Models or Proportion Equations

4.4f Classroom Activity: Odds and Probability: Chance Proportions

4.4f Homework: Odds and Probability: Chance Proportions

4.4g Classroom Activity: Percent Proportions

4.4g Homework: Write and Solve Three Percent Problems:

4.4h Classroom Activity: Unit Rates and Proportions in Markups and Markdowns

4.4h Homework activity: Unit Rates in Markups and Markdowns

4.4i Anchor Problem Revisited: Purchasing Tunes-Town Music Company

4.4j Self-Assessment: Section 4.4

Chapter 4 Proportional Relationships, Solving Problems (6 Weeks)

UTAH CORE Standard(s)

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. 7.RP.1

2.Recognize and represent proportional relationships between quantities. 7.RP.2

1. 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. 7.RP.2a
2. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. 7.RP.2b
3. 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. 7.RP.2c
4. 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. 7.RP.2d

3.Use proportional relationships to solve multi-step ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. 7.RP.3

CHAPTER OVERVIEW:

This chapter focuses on extending understanding of ratio to the development of an understanding of proportionality in order to solve one-step and multi-step problems. The chapter begins by reviewing ideas from 6th grade as well as 7th grade chapters 1 – 3 and transitioning students to algebraic representations. Students will rely on understandings developed in previous chapters and grades to finding unit rates, proportional constants, comparing rates and situations in multiple forms, writing expressions and equations, and analyzing tables and graphs. The goal is that students develop a flexible understanding of different representations of ratio and proportion to solve a variety of problems.

One important thing for teachers to note (and help students understand) is that a ratio can be written in many different ways, including part:part and part:whole. Fractions represent a part:whole relationship. Because of the overlap in expressing part:whole relationships, we often have to rely on the context to determine if a given value a/b is a fraction or a ratio.

Many ratio and proportion concepts are interrelated. Therefore, at the beginning of each section, we list the key points from the 7th grade ratio and proportion standards and,below them, list the concepts and skills that will be the primary focus for the section.

VOCABULARY:

Bar model, constant of proportionality, equation, part-to-part ratio, part-to-whole ratio, percent change, proportion, proportional constant, rate table, and ratio.

CONNECTIONS TO CONTENT:

Prior Knowledge

Student should be able to draw models of part-to-part and part-to-whole relationships. From these models, students should be able to operate fluently with fractions and decimals, especially reducing fractions, representing division as a fraction, converting between mixed numbers and improper fractions, and multiplication and division with fractions. Many of these concepts were reviewed in Chapter 1 and will be reviewed briefly in this chapter. Students worked extensively on ratio in 6th grade where they learned to write ratios as fractions, using a colon, or using words.

At the beginning of this chapter, it is assumed that students can use models to solve percent, fraction and ratio problems with models. Students will connect ratio and proportional thinking to a variety of multi-step problems.

Future Knowledge

A strong foundation in proportion is key to success throughout traditional middle and high school mathematics. Further, it is essential in Trigonometry and Calculus.In the next chapter, students will use proportions as a basis for understanding scaling. In 8th grade, proportions form the basis for understanding the concept of constant rate of change (slope) and the basic measures of statistics. Also in 8th grade students will finalize their understanding of linear relationships and functions; proportional relationships studied in this chapter are a subset of these relationships. Later, insecondary math, students will solve rational equations (such as ), apply ratio and proportion to similarity and then to trigonometric relationships, andlearn about change that is not linear.

MATHEMATICAL PRACTICE STANDARDS (emphasized):

/ Make sense of problems and persevere
in solving them. / By building from simpler contexts of ratio and proportion, students develop strategies for working withmulti-step problems without a clear path to a solutionor containing inconvenient numbers. Additionally, students should think flexibly about the relationship between fractions, ratio and proportion and their multiple representations to solve problems. After solving a problem students should be encouraged to think about the reasonableness of the answer.
/ Reason abstractly and
quantitatively. / Students will reason abstractly throughout the chapter particularly when they begin to write proportion equationsand solve them algebraically. Students shouldbe encouraged to think about the relationship between equations, tables and graphs and the contexts they represent. Students should also be encouraged to reason quantitatively throughout the chapter with the emphasis on units. When an answer is obtained, students should be able to state what the number represents. When students state a unit rate they should be able to give the units represented therein. Further, students should be able to draw models of the quantities and connect various representations to tables, graphs and equations.
/ Construct viable arguments and critique the reasoning of others. / Students should construct arguments and critique those of others’ throughout. With both, students should refer to representations of ideas and/or arithmetic properties in the construction of their arguments. Arguments should be made orally or in writing.
/ Model with mathematics. / As students study proportional relationships, they begin to see examples in their lives: traveling on the freeway at a constant rate, swimming laps, scaling a recipe, shopping, etc. Most proportional relationship from the real world can be modeled using the representations students learn in this chapter. Appropriate models include bar/tape models (either part-part-whole or comparison) or other similar representations, tables or graphs. Models selected should help to reveal relationships and structure. Further, students should be able to connect models to their algebraic representation.
/ Attend to precision. / Students should be encouraged throughout the chapter to express unit rates exactlywith appropriate units of measure by using fractions and mixed numbers rather than rounded decimals.When graphing or creating tables, labeling is crucial as it is when making an argument. Vocabulary should also be used appropriately in communicating ideas.
/ Look for and make use of structure. / Students should make use of structure as they identify the unit rate from tables by looking at the relationship of values both vertically and horizontally. Additionally, students will discover that graphs of proportional linear relationships cross through the origin. When comparing proportional relationships, students should note the structure of equations and how it relates to its table and the steepness of its lines. Lastly, students should solidify their understanding of the difference between ratio and fraction in this chapter. This is an important distinction and one that will allow them to think more flexibly with either.
/ Use appropriate tools strategically. / Students continue their transition in this chapter to more abstract representations of ideas. As such, students will be working towards using symbolic representation of ideas rather than models exclusively. This transition will occur at different rates for different students. Students should be encouraged to use mental math strategically throughout. For example, if students are given a rate of 3 pounds per $2, there is no need for a calculator to determine that the unit rate is 3/2 pounds per $1 or $ per pound.
/ Look for and express regularity in repeated reasoning. / Students should note repeated reasoning in rate tables and graphs throughout the chapter. They will also use repeated reasoning to uncover the unit rate associated with percentages of increase and decrease and in the relationship of ratios and fractions.

4.0 Anchor Problem: Purchasing Tunes-Town Music Company—What’s the Better Deal?

Problem # 1:

Discounts and Markups

(Adapted from illustrative mathematics.org)

In June,Tunes-Town wasn’t making enough money to stay in business, so in July,they reduced all prices by 20% to attract more business.

They found that even at July’s prices they weren’t making enough money, so they decided they had to sell their store to Beat-Street. In August,Beat-Street took over the store and the manager immediately increased all the current (July) prices by 20%, thinking this would go back to the original prices.

1) Track the price of a $100 car stereo throughout the story.

2) By what percentage should Beat-Street have raised prices to make August’sprices revert back to the original June prices? Use any strategy to solve.

Problem # 2:

What’s the Better Deal?

(From illustrative mathematics.org)

Beat-Street, Tunes-Town, and Music-Mind are music companies. Beat-Street offers to buy 1.5 million shares of Tunes-Town for $561 million. At the same time, Music-Mind offers to buy 1.5 million shares of Tunes-Town at $373 per share.

1.Who would get the better deal, Beat-Street or Music-Mind? Explain.

2.What is the total price difference?

Section 4.1: Understand and Apply Unit Rates

Section Overview:

The purpose of this section is to solidify the concept of unit rates. This includes finding unit rates from contexts, tables, graphs and/or proportional relationships written as equations. This section will begin the work of helping students to move fluidly among these representations to find missing quantities. Students will visualize ratios using models—both part-part-whole and comparison tape models. Particular attention should be paid to models of rates with unlike units (for example miles and hours). Students will identify the unit rate for both units (i.e. miles per hour and hours per mile). Students should then be able to move away from using models and be able to recognize the reciprocal relationship of the two unit rates to solve problems.

Along the way, this section will practice operations with rational numbers, with particular attention paid to precision with “inconvenient” division problems that come up from ratios. Students should be encouraged to find exact unit rates; i.e. they should be encouraged to use fractions to divide numbers, rather than using long division or a calculator and getting decimals approximations. Students should understand that using a rounded unit rate will give inaccurate results when finding missing quantities.

Up until this chapter, students worked primarily with part:whole relationships (percent, fractional portions, and probability). In this section, students will begin to connect how part:whole relationships give information about part:part relationships and vice versa. In 4.4 this idea will be solidified.

Rate tables will be introduced but will be further explored and connected to graphs in the next section.

Key Ratio and Proportion Concepts from Utah Core Standards

RP Standard 1:

1. Extend the concept of a unit rate to include ratios of fractions.
2. Compute a unit rate, involving quantities measured in like or different units.

RP Standard 2:

3.Determine if two quantities expressed in a table or in a graph are in a proportional relationship.

4.Determine a unit rate from a table, graph, equation, diagram or verbal description and relate it to the constant of proportionality.

5.Write an equation for a proportional relationship in the form .

6.Explain the meaning of the point in the context of a proportional relationship.

7.Explain the significance of and in a graph of a proportional relationship, where is the unit rate.

RP Standard 3:

8.Solve multistep problems involving percent using proportional reasoning

9.Find the percent of a number and extend the concept to solving real life percent applications.

10.Calculate percent, percent increase, decrease, and error.

Primary Concepts and Skills to be Mastered in This Section

1. Compute unit rate from a context.
2. Compute a unit rate from a table of values.
3. Compare two rates to determine equivalence or to contrast differences.
4. Find the unit rate for BOTH units (i.e. miles per hour and hours per mile).
5. Use unit rate to find a missing quantity.

4.1a Class Activity: Model Ratios (Review)

What is a ratio?

In the space below, write as many ratios as you can find in the picture shown above. Be sure to label what you are comparing.

Review Reducing Ratios and Relating Ratios to Fractions and Percents

Example 1: Use a model to find ratios that are equivalent to the ratio 2sunflowers to 6 roses.

2 sunflowers to 6 roses

We can place two flowers into each of four groups; this reveals a 1 (group) sunflower to 3 (groups) roses

The reduced ratio is1 sunflower to 3 roses. In other words, there is 1 sunflower for every 3 roses.

We can write other equivalent ratios by increasing the number of flowers in each group as long as all the groups have the same number of flowers.

For example, 5 sunflowers to 15 roses, notice that all the groupings have five flowers.

The ratio is also equivalent to ½ sunflower to 1 ½ roses.

Example 2: What percent of the flowers in Example 1 are sunflowers?

Because there are a total of 4 parts (1 part sunflowers and 3 parts roses) and sunflowers are 1 part of all the flowers, we know that 25% of the flowers are sunflowers. We could also say 1/4 of all the flowers aresunflowers and then covert 1/4 to 25%.

Working with a partner use a model to answer each question.

1. The high school has a new ski club this year. In the club there are 4 girls and 16 boys.
2. Use a model to find the ratio of girls to boys in the skiing club.

1. What percent of the students in the ski club are girls?