Curriculum  /  Math  /  7th Grade  /  Unit 1: Proportional Relationships  /  Lesson 5

Proportional Relationships

Lesson 5 of 18

Criteria for Success

Tips for teachers, anchor problems, problem set, target task, additional practice.

Write equations for proportional relationships from word problems.

Common Core Standards

Core standards.

The core standards covered in this lesson

Ratios and Proportional Relationships

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

7.RP.A.2.C — 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.

Foundational Standards

The foundational standards covered in this lesson

Expressions and Equations

6.EE.B.7 — Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.

6.EE.C.9 — Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.

The essential concepts students need to demonstrate or understand to achieve the lesson objective

  • Recall that a ratio relationship $$A:B$$  has two unit rates, $$A/B$$  and $$B/A$$ .
  • Find the constant of proportionality from a situation (not a table) by finding the value of $$y/x$$ ,  where $$y$$  is the dependent variable and $$x$$  is the independent variable. 
  • Write an equation using the constant of proportionality for a proportional relationship.
  • Use an equation to solve problems.
  • Decontextualize situations to represent them as equations and re-contextualize equations to explain their meanings as they relate to situations (MP.2).

Suggestions for teachers to help them teach this lesson

Lesson Materials

  • Calculators (1 per student)

Unlock features to optimize your prep time, plan engaging lessons, and monitor student progress.

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

A repair technician replaces cracked screens on phones. He can replace 5 screens in 3 hours. 

a.   Write an equation you can use to determine how long it takes to replace any number of screens.

b.   Write an equation you can use to determine how many screens can be replaced in a certain number of hours.

c.   Use one of your equations to determine how long it would take to replace 30 screens.

d.   Use one of your equations to determine how many screens could be replaced in 12 hours.

Guiding Questions

A train traveling from Boston to New York moves at a constant speed. The train covers 260 miles in 3.25 hours. 

Write an equation that represents this relationship. Explain what your equation tells you about the train.

Deli turkey meat is on sale at two different grocery stores, as represented below.

What is the unit price of turkey per pound at each store? Which store has the better sale?

A set of suggested resources or problem types that teachers can turn into a problem set

Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

The 7th graders at Wilson Middle School are taking a field trip to the science museum. So far, 12 students have turned in a total of $81 to pay for their ticket.

a.   Write an equation to represent the cost, $$c$$ , for  $$t$$  tickets.

b.   If all 72 students turn in their ticket money, how much money will be collected?

Student Response

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

  • Challenge: The average person uses about 90 gallons of water per day. At this rate, how long would it take a family of 4 to use enough water to fill an average-size swimming pool? An olympic-size swimming pool?
  • EngageNY Mathematics Grade 7 Mathematics > Module 1 > Topic B > Lesson 8 — Problem Set 1-4; Do not include any questions with graphs as these will be addressed in upcoming lessons.
  • Open Up Resources Grade 7 Unit 2 Practice Problems — Lesson 5
  • EngageNY Mathematics Grade 7 Mathematics > Module 1 > Topic B > Lesson 9 — Example 1, Problem Set 1 and 3
  • Illustrative Mathematics Sore Throats, Variation 1
  • RDA Performance Task Bank Grade 7 Mathematics Sample SR Item

Topic A: Representing Proportional Relationships in Tables, Equations, and Graphs

Solve ratio and rate problems using double number lines, tables, and unit rate.

7.RP.A.1 7.RP.A.2

Represent proportional relationships in tables, and define the constant of proportionality.

7.RP.A.2 7.RP.A.2.B

Determine the constant of proportionality in tables, and use it to find missing values.

7.RP.A.2.A 7.RP.A.2.B

Write equations for proportional relationships presented in tables.

7.RP.A.2.B 7.RP.A.2.C

7.RP.A.2 7.RP.A.2.C

Represent proportional relationships in graphs.

7.RP.A.2 7.RP.A.2.A 7.RP.A.2.D

Interpret proportional relationships represented in graphs.

7.RP.A.2 7.RP.A.2.D

Create a free account to access thousands of lesson plans.

Already have an account? Sign In

Topic B: Non-Proportional Relationships

Compare proportional and non-proportional relationships.

Determine if relationships are proportional or non-proportional.

Topic C: Connecting Everything Together

Make connections between the four representations of proportional relationships (Part 1).

7.RP.A.2 7.RP.A.2.A 7.RP.A.2.B 7.RP.A.2.C 7.RP.A.2.D

Make connections between the four representations of proportional relationships (Part 2).

Use different strategies to represent and recognize proportional relationships.

Topic D: Solving Ratio & Rate Problems with Fractions

Find the unit rate of ratios involving fractions.

Find the unit rate and use it to solve problems.

7.RP.A.1 7.RP.A.3

Solve ratio and rate problems by setting up a proportion.

Solve ratio and rate problems by setting up a proportion, including part-part-whole problems.

Solve multi-step ratio and rate problems using proportional reasoning, including fractional price increase and decrease, commissions, and fees.

Use proportional reasoning to solve real-world, multi-step problems.

7.RP.A.1 7.RP.A.2 7.RP.A.3

Request a Demo

See all of the features of Fishtank in action and begin the conversation about adoption.

Learn more about Fishtank Learning School Adoption.

Contact Information

School information, what courses are you interested in, are you interested in onboarding professional learning for your teachers and instructional leaders, any other information you would like to provide about your school.

Effective Instruction Made Easy

Effective Instruction Made Easy

Access rigorous, relevant, and adaptable math lesson plans for free

lesson 5 problem solving practice graph proportional relationships

Math Worksheets Land

Math Worksheets For All Ages

  • Math Topics
  • Grade Levels

Graphing Proportional Relationships Worksheets

How do you graph proportional relationships? Suppose your teacher asks you to graph a proportional relationship between two variables (x and y) with the unit rate of 0.4, which is a change in a single unit of x will cause a change of 0.4 units in y. Let's think about some potential values between x and y. Remember that here we have an independent value x and an independent value y. What that means is that some value of y will be equal to a constant multiplied with the value of x. If we were to express this using an expression, it would be: Y = kx (K equals to any constant number) X, Y = (0, 0), (1, 0.4), (2, 0.8), (3, 1.2), (4, 1.6), (5, 2) If you were to draw a graph of these values, you would know that it is going to make a straight line. It shows that the graph has a proportional relationship. This series of lessons and worksheets will help students learn how to graph and identify graphs of proportional relationships.

Aligned Standard: Grade 8 Expressions and Equations - 8.EE.B.5

  • Faster Paces Step-by-Step Lesson - Two boys are riding bikes. Which one moves at a faster pace?
  • Guided Lesson - All the problems compare a graph to an equation. So kids find it easier to graph the equation. Yet others find it easier to determine the equation of the graph.
  • Guided Lesson Explanation - I like equations and for the answers I put everything into an equation. I will come back and work on this one to have a graph comparison version later this month.
  • Independent Practice - It's a big battle royale of graphs versus equations.
  • Matching Worksheet -This one is pretty easy because it's matching. I would insist on everyone showing their work.
  • Answer Keys - These are for all the unlocked materials above.

Homework Sheets

Compare the graph and the equation without any change.

  • Homework 1 - The graph below represents the number of cakes that Eric makes in a day. The equation (right) represents how many cakes Lewis makes in a day. Who makes cakes at a faster rate?
  • Homework 2 - When both equations are constant. This makes it very easy to compare.
  • Homework 3 - The graph below represents the number of trips made by Truck A over 7 days. The equation below represents the number of trips made by Truck B over 7 days. Who takes more trips over the course of a week?

Practice Worksheets

The color of the line dot here was requested by many teachers. It shows up well on Smart boards.

  • Practice 1 - The equation represents the rate at which Jackson sells books. Who sells more books over 5 hours?
  • Practice 2 - The equation represents the rate at which Sarah travels on her scooter. If both Jessie and Sarah were to travel for 7 straight days, who would you predict to travel further?
  • Practice 3 - Over a day, who watches more movies?

Math Skill Quizzes

Because of the size of the graph, I can only get two questions on each page.

  • Quiz 1 - Who uses strawberries at a faster rate?
  • Quiz 2 - Who eats more cookies over the course of a week?
  • Quiz 3 - The equation represents the number of greeting cards made by the Kim and the number of sheets used. Who uses fewer sheets?

How to Spot Proportional Relationships on Graphs

When two variables have a proportional relationship, they will exist in a ratio to one another that is constant. This means that when they are visualized on a line graph, first they will form a line that passes through the origin (0, 0). They will exist in some form of the equation y = kx. To translate this for you as x increase, so does y. If the y value decreases, then so does the x value. As a result, they are relentless proportion with one another. Regardless of how big or small their values are, these variables will always be relative to one another.

Get Access to Answers, Tests, and Worksheets

Become a paid member and get:

  • Answer keys to everything
  • Unlimited access - All Grades
  • 64,000 printable Common Core worksheets, quizzes, and tests
  • Used by 1000s of teachers!

Worksheets By Email:

Get Our Free Email Now!

We send out a monthly email of all our new free worksheets. Just tell us your email above. We hate spam! We will never sell or rent your email.

Thanks and Don't Forget To Tell Your Friends!

I would appreciate everyone letting me know if you find any errors. I'm getting a little older these days and my eyes are going. Please contact me, to let me know. I'll fix it ASAP.

  • Privacy Policy
  • Other Education Resource

© MathWorksheetsLand.com, All Rights Reserved

Please log in to save materials. Log in

  • 7th Grade Mathematics
  • Percentages

Education Standards

Wyoming standards for mathematics.

Learning Domain: Ratios and Proportional Relationships

Standard: Compute unit rates, including those involving complex fractions, with like or different units.

Standard: Recognize and represent proportional relationships between quantities.

Standard: Decide whether two quantities in a table or graph are in a proportional relationship.

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

Standard: Represent proportional relationships with equations.

Standard: 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.

Maryland College and Career Ready Math Standards

Standard: 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.

Standard: 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.

Standard: 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.

Common Core State Standards Math

Cluster: Analyze proportional relationships and use them to solve real-world and mathematical problems

Analyzing Proportional Relationship Graphs

Analyzing Proportional Relationship Graphs

Students analyze the graph of a proportional relationship in order to find the approximate constant of proportionality, to write the related formula, and to create a table of values that lie on the graph.

Key Concepts

  • The constant of proportionality determines the steepness of the straight-line graph that represents a proportional relationship. The steeper the line is, the greater the constant of proportionality.
  • On the graph of a proportional relationship, the constant of proportionality is the constant ratio of y to x , or the slope of the line.
  • A proportional relationship can be represented in different ways: a ratio table, a graph of a straight line through the origin, or an equation of the form y = kx , where k is the constant of proportionality.

Goals and Learning Objectives

  • Identify the constant of proportionality from a graph that represents a proportional relationship.
  • Write a formula for a graph that represents a proportional relationship.
  • Make a table for a graph that represents a proportional relationship.
  • Relate the constant of proportionality to the steepness of a graph that represents a proportional relationship (i.e., the steeper the line is, the greater the constant of proportionality).

The Price of Tomatoes

Lesson guide.

  • Have students examine each representation (i.e., verbal description, unit rate, formula, table, and graph), drawing connections between them.

Mathematics

The task allows students to synthesize what they have learned about proportional relationships represented in different forms: verbal descriptions, unit rates, tables, formulas, and graphs. The representations build on one another and should be connected as you move from one to the next. As students focus on each, ask:

  • How does each unit price show up in the tables? (Answer: The unit price is shown in a single pair of values in each table—the pair (1, k )—and in the constant ratio between every pair of values.)
  • How does each unit price show up in the formulas? (Answer: as the constant of proportionality)

ELL: When working with ELLs, it is important to link new concepts to students’ prior knowledge and experiences. This allows students to make more meaningful connections with the new mathematical concepts being learned. Remind students to review their Notebook if they are unsure how to make connections between each representation.

These ratio tables show the relationship between the cost and weight of tomatoes at two different stores.

  • At Fruit World, the price for tomatoes is $0.60 per pound. The ratio is 0.6 : 1.
  • At Veggie Mart, the price for tomatoes is $1.15 per pound. The ratio is 1.15 : 1.
  • The formula for the cost of tomatoes at Fruit World is c = 0.6 w .

lesson 5 problem solving practice graph proportional relationships

  • Have partners explain how they know which graph represents each market.

The representations build on one another and should be connected as you move from one to the next. As students focus on each, ask:

  • How do the unit prices help you identify which graph is which? (Answer: The unit price is the constant ratio of the y-value to the x-value for the pair of values that represent a point on the line. This value can be quickly found because it is the y-value when the x-value equals 1 .)
  • The line that represents the cost of the tomatoes at Veggie Mart is the line containing the point (1, 1.15).
  • The line that represents the cost of the tomatoes at Fruit World is the line containing the point (1, 0.6).
  • Answers will vary.

The graph shows both proportional relationships.

Fruit World: c = 0.6 w

Veggie Mart: c = 1.15 w

  • Which line represents the cost of the tomatoes at Veggie Mart?
  • Which line represents the cost of the tomatoes at Fruit World?

lesson 5 problem solving practice graph proportional relationships

Math Mission

Discuss the Math Mission. Students will analyze the graph of a proportional relationship to find the approximate constant of proportionality, write the related formula, and create a table of values that lie on the graph.

SWD: Struggling students may still need explicit instruction and guided scaffolding to recognize the relationships between the constant of proportionality, table, graph, and equation. Provide small group instruction to help them make connections between the constant of proportionality and the other representations given.

Analyze the graph, write the related formula, and create a table of values for a proportional relationship.

Graphs of Proportional Relationships

Have students work in pairs for all problems and the presentation.

SWD: Prompt students with disabilities to refer to the visual representations of information in the graphs in order to make visual connections about the steepness of the lines and the value of the constant of proportionality.

Interventions

Student does not know how to approximate the constant of proportionality from the graphs.

  • Are you comparing the y -value of a particular point to the x -value of that same point to find the constant of proportionality?
  • Pick any point on a graph and identify the y - and x -coordinates of that point. What is the ratio between the y - and x -coordinates? Pick another point on the same graph and do the same thing. Do you get the same ratio?

Student does not see the connection between corresponding values in a relationship and points on the graph.

  • Values on the vertical axis are the y -values. Values on the horizontal axis are the x -values.
  • The coordinates of each point on the graph ( x , y ) is a pair of corresponding values that fits the proportional relationship between the quantities represented by x and y .

Student does not see the connection between the constant of proportionality and the steepness of the graph.

  • Look at the graph that is steepest. What is its constant of proportionality? Now look at the graph that is least steep. What is its constant of proportionality? Compare the two ratios.

Student thinks the endpoints of each graph determine each constant of proportionality, but does not know how to interpret them correctly.

  • Pick any point on a graph and identify the x - and y -coordinates of that point. What is the ratio between the x - and y -coordinates? Now look at the endpoint shown on the same graph and do the same thing. Do you get the same ratio? Try it with another point on the same graph.

Possible Answers

Answers may vary slightly since students are approximating from the graphs, not from specific values. Possible answers:

Line a : 5 2

Line b : 5 4

Line c : 5 6

Line d : 5 11

Line e : 1 3

The graph of a proportional relationship with a constant of 1 would lie between lines b and c . Explanations will vary. Possible explanation: The constant of proportionality gives the steepness of the line and 1 is between 5 4 and 5 6 , which are the constants of proportionality of lines b and c .

The coordinate grid shows the graphs of five proportional relationships.

  • Approximate, as closely as possible, the constant of proportionality for each line.
  • The graph of a proportional relationship with a constant of 1 would lie between two of these lines. Explain why.

lesson 5 problem solving practice graph proportional relationships

  • If you can find one point on the line, how can you determine the constant of proportionality?
  • What would a line with a constant of proportionality of 1 look like? What are some examples of ordered pairs that have a ratio of 1?

Write a Formula

If student does not know how to get started.

  • Pick two points on the graph. How would you determine the constant of proportionality?
  • What might be a good point to choose.
  • What is the general form of a proportional relationships?

Answers will vary depending on the graph students choose.  

Line a : y = ( 5 2 ) x  

Line b : y = ( 5 4 ) x  

Line c : y = ( 5 6 ) x  

Line d : y = ( 5 11 ) x  

Line e : y = ( 1 3 ) x

Choose one of the lines and write the formula of the line. Use x and y to name the variables.

Once you know the constant of proportionality, how can you use it to write the formula?

Make a Table

  • How could you determine one point on the graph?
  • How could you determine a second point?
  • What relationship can you use to generate as many points as you want?
  • Answers will vary depending on the line students choose. Check that students’ tables include only positive values for x and y and include five corresponding pairs of values with the correct constant of proportionality.

Choose a different line. Make a table that lists five pairs of values that define points on the graph.

Prepare a Presentation

Preparing for ways of thinking.

Look and listen for students who:

  • Have a range of ways of talking about the steepness of the graphs and who express uncertainty about the significance of the steepness of a graph.
  • Discuss what a steeper graph means in terms of the relationship between x and y .
  • Discuss what a steeper graph means in terms of the constant of proportionality.
  • Observe that all of the graphs intersect at the point (0, 0).

Challenge Problem

Possible answer.

Graphs will vary. Check that students’ graphs have a slope greater than 1.

Possible answer:

lesson 5 problem solving practice graph proportional relationships

Prepare a presentation about what you learned about graphs and the constant of proportionality. Be sure to talk about how the constant of proportionality relates to steepness.

Make a graph with a constant of proportionality that is greater than 1.

Make Connections

Discuss the different formulas and tables students wrote for the different graphs.

Examine at least one case for which students have created distinct tables (with at least one distinct pair of values) for the same graph. Have students confirm that distinct tables for the same graph show the same constant of proportionality. Then make explicit the fact that many more pairs of values are possible than can be included in a table by asking:

  • If you combined all of the tables created in this class for this graph, would you have all of the values on the graph? (Answer: no)
  • What if you added 1,000 more pairs of values to the table? Would you have most of the values on the graph then? (Answer: no)
  • Would the corresponding values all have the same constant of proportionality? (Answer: yes)

ELL: When giving directions for this discussion, be sure that students understand that they can complement their oral explanation with graphs, pictures, and drawings. Provide sentence frames that you think will be helpful for ELLs.

Mathematical Practices

Mathematical Practice 3: Construct viable arguments and critique the reasoning of others.

  • Have the class sort out any discrepancies among their constants of proportionality for each graph by giving students a chance to revise their approximations in light of others’ work. Use any discrepancies that aren’t resolved quickly as an opportunity to support students in critiquing the reasoning presented and constructing arguments about the correct constant of proportionality. Ask students to refer both to their table and to the graph as they make their cases to one another.

Performance Task

Ways of thinking: make connections.

Take notes about your classmates’ explanations for how they found the constant of proportionality, formula, table, and to explain how the constant of proportionality relates to the steepness of a line.

As classmates present, ask questions such as:

  • How did you find the constant of proportionality based on that graph?
  • How did you figure out how to write the formula for that graph?
  • What is the ratio of any two points on a line with a constant of proportionality of 1?
  • How does the constant of proportionality relate to the steepness of the graph
  • Have pairs quietly read and then discuss the information shown.
  • As student pairs work together, make a note to clarify any misunderstandings in the class discussion.
  • After a few minutes, discuss the summary as a class. Ask a volunteer to explain how the constant of proportionality represents the slope of the graph that represents a proportional relationship.

Then discuss the following information with the class:

  • How to find the constant of proportionality from a graph
  • How to write a formula for the graph of a proportional relationship
  • How to make a table to show the values that appear on a graph

SWD: When students discuss their thoughts, it provides them with opportunities to practice the thinking process involved in solving a set of problems. Listen to the students’ thought processes, correct their misconceptions, and fill in incomplete processes.

Formative Assessment

Summary of the math: the slope.

Read and Discuss

  • In general, if quantity p is proportional to quantity q according to the formula p = kq , then the slope of the graph that represents the relationship between p and q will be equal to k , the constant of proportionality.
  • Explain how to find the constant of proportionality of a graph?
  • Explain how to write the formula for the graph of a proportional relationship?
  • Explain how to identify points on a graph, and create a table of values for a graph?
  • Explain the relationship between the constant of proportionality and the steepness of the graph?

The Cost of Paint

This task allows you to assess students’ work and determine what difficulties they are having. The results of the Self Check will help you determine which students should work on the Gallery and which students would benefit from review before the assessment. Have students work on the Self Check individually.

Have students submit their work to you. Make notes on what their work reveals about their current levels of understanding and their different problem-solving approaches.

Do not score students’ work. Share with each student the most appropriate Interventions to guide their thought process. Also note students with a particular issue so that you can work with them in the Putting It Together lesson that follows.

SWD: Provide clear feedback to students as they attempt to solve problems or articulate concepts. This type of feedback guides students explicitly as they develop their thinking about mathematics, and gives them ideas about their next steps.

Student only partially records his or her computation strategies.

  • Explain in more detail how you figured out your solution.

Student uses different strategies to find the missing values.

  • Can you think of a solution strategy that would work for all of the missing values?

Student interprets the problem structure as additive rather than multiplicative.

  • What are the two quantities that vary in relation to one another in this situation?
  • Is there a constant of proportionality between the two quantities in this situation?

Student divides instead of multiplying or multiplies instead of dividing.

  • Make an estimate of the answer based on how big or small you expect it to be. Does your answer fit your estimate?
  • Use words to write down how to calculate the answer and read them aloud. Do your calculations fit your words?

Student correctly uses a unit rate to find the answers.

  • Can you write a formula using the unit rate that works for finding all the missing values?
  • What would you expect the graph of the relationship in this problem to look like? Identify three features you expect the graph to have.
  • Sketch a graph that shows the relationship.

Student uses cross-multiplication incorrectly to find the answers.

  • Identify which values in this computation are prices and which are amounts of paint: [insert a student’s calculation using cross-multiplication].
  • Can you describe, in words, the relationship between the two quantities of paint in the calculation: [insert a student’s calculation using cross-multiplication]?
  • Can you describe, in words, the relationship between the price and the amount of paint in this ratio: [insert one ratio from student’s work]?

Student correctly uses cross-multiplication to find the answers.

  • Can you explain why this method works?
  • Can you find a simpler way of calculating the missing values that works the same way in every case?

Student finds the answers correctly and efficiently.

  • Find at least two different methods for finding the missing values.
  • Missing prices: $9.00, $11.25, $37.50, $68.10 Missing amount of paint: 5.1 liters

Check the Cost of Paint

The prices are proportional to the amount of paint in the cans.

Calculate the missing prices of the paint cans and the amount of paint in the last can.

lesson 5 problem solving practice graph proportional relationships

Reflect On Your Work

Have each student write a brief reflection before the end of the class. Review the reflections to find out if students can determine the constant of proportionality of a graph.

Write a reflection about the ideas discussed in class today. Use the sentence starter below if you find it to be helpful.

An easy way to find the constant of proportionality of a graph is …

Version History

  • → Resources
  • → 7th Grade
  • → Slope & Rate of Change

Graphing and Interpreting Graphs of Proportional Relationships Lesson Plan

Get the lesson materials.

Represent Proportional Relationships Graphs & Equations Guided Notes w/ Doodles

Represent Proportional Relationships Graphs & Equations Guided Notes w/ Doodles

Graphing and Interpreting Graphs of Proportional Relationships Lesson Plan

Ever wondered how to teach proportional relationships in an engaging way to your 7th and 8th grade students?

In this lesson plan, students will learn about representing proportional relationships through graphs and equations and explore their real-life applications. Then, they will also practice interpreting graphs of proportional relationships and understanding what the points represent in the context of the problem. Through artistic and interactive guided notes, checks for understanding, practice activities including a doodle & color by number worksheet, and a maze worksheet, students will gain a comprehensive understanding of proportional relationships.

The lesson culminates with a real-life application example that explores how proportional relationships are used in a practical context. Students will learn how to apply their knowledge of proportional relationships to solve real-world problems .

  • Standards : CCSS 8.EE.B.5 , CCSS 7.RP.A.2.a , CCSS 7.RP.A.2.b , CCSS 7.RP.A.2.c , CCSS 7.RP.A.2.d
  • Topics : Slope & Rate of Change , Ratio & Rates
  • Grades : 7th Grade , 8th Grade
  • Type : Lesson Plans

Learning Objectives

After this lesson, students will be able to:

Represent proportional relationships using graphs and equations

Interpret graphs and equations of proportional relationships

Calculate the constant of proportionality from a graph or equation

Calculate unit rates from a graph or equation

Explain the meaning of a point on the graph of a proportional relationship in terms of the situation

Explain how graphing proportional relationships is useful in real life

Prerequisites

Before this lesson, students should be familiar with:

H ow to plot points on a coordinate plane

Basic algebraic skills, including operations with decimals and whole numbers

Colored pencils or markers

Guided notes

Key Vocabulary

Proportional relationship

Constant of proportionality

Introduction

Proportional Relationships Equations, Graphs, Constant of Proportionality Guided Notes Image Introduction

As a hook, ask students why understanding proportional relationships is important in real life. You can provide examples such as determining the amount of time it takes to complete a task, or comparing prices at the grocery store. Refer to the last page of the guided notes as well as the FAQs below for more ideas.

Use the first page of the guided notes to introduce the concept of representing proportional relationships with graphs and equations. Walk through the key points such as understanding what each point on the graph represents in terms of the situation, how to calculate the constant of proportionality & unit rate, and how to construct equations based on the constant of proportionality. You can model using the example of Charlie in space, as shown in the guided notes (page 1).

Based on student responses and understanding, reteach any concepts that students need extra help with. Then, have the students move onto the next page of guided notes (page 2) to practice constructing equations, graphing, interpreting graphs of proportional relationships. If your class has a wide range of proficiency levels, you can pull out students for reteaching, while more advanced students can work on the practice exercises independently.

Proportional Relationships Equations, Graphs, Constant of Proportionality Guided Notes Image Practice

Have students practice graphing proportional relationships, interpreting graphs, and constructing equations using the maze activity (page 3), and color by number activity (page 4). Students will also be asked to solve for constant of proportionality in the practice. Walk around to answer student questions. You can also assign it as homework.

Real-Life Application

Proportional Relationships Equations, Graphs, Constant of Proportionality Guided Notes Image Real Life Applications

Use the last page of the guided notes (page 5) to bring the class back together, and introduce the concept of real-world application of proportional relationships. Explain to students that proportional relationships can be found in various real-life situations. These situations involve two quantities that change together at a constant rate or ratio.

Give examples of real-world scenarios where proportional relationships can be observed, such as:

Distance and Time: Discuss how the distance travelled is directly proportional to the time spent on the road.

Job Wages: Explain how some jobs pay employees based on the number of hours they work. For example, if a person earns $18 per hour, the amount of money they earn is directly proportional to the number of hours they work. As the number of hours worked increases, their earnings also increase.

Recipes: Share how recipes often require proportional measurements of ingredients. For example, if a recipe calls for 2 cups of flour and 1 cup of sugar, the ratio between flour and sugar remains constant even if the total amount of ingredients increases or decreases.

Ask students if they can think of any other real-life situations that involve proportional relationships. Encourage them to share their ideas with the class.

Refer to the FAQ section in the resource for more ideas on how to teach real-life applications of proportional relationships.

Additional Self-Checking Digital Practice

If you’re looking for digital practice for Representing Proportional Relationships Graphs & Equations, try my Pixel Art activities in Google Sheets. Every answer is automatically checked, and correct answers unlock parts of a mystery picture. It’s incredibly fun, and a powerful tool for differentiation.

Here are some activities to explore:

Graphs of Proportional Relationships Pixel Art

Constant of Proportionality Pixel Art

Proportional vs. Non-Proportional Relationships Pixel Art

Solving Proportions Pixel Art Google Sheets

Additional Print Practice

A fun, no-prep way to practice Representing Proportional Relationships Graphs & Equations is Doodle Math — they’re a fresh take on color by number or color by code. It includes multiple levels levels of practice, perfect for a review day or sub plan.

Here are some activities to try:

Constant of Proportionality Doodle Math (Winter Themed)

Constant of Proportionality Doodle Math (Thanksgiving Themed)

1. What is a proportional relationship? Open

A proportional relationship is a relationship between two quantities where the ratio of one quantity to the other remains constant. In other words, as one quantity increases or decreases, the other quantity increases or decreases by the same factor.

2. How do you represent a proportional relationship graphically? Open

To represent a proportional relationship graphically, you can plot the ordered pairs on a coordinate plane. Each point on the graph represents a pair of values from the proportional relationship. The graph will show a straight line passing through the origin (0, 0), indicating a constant ratio between the two quantities.

3. How do you write an equation for a proportional relationship? Open

To write an equation for a proportional relationship, you can use the formula y = kx, where y represents one quantity, x represents the other quantity, and k represents the constant of proportionality. The constant of proportionality is the ratio between the two quantities that remains constant.

4. What does a point (x, y) on the graph of a proportional relationship mean in terms of the situation? Open

In terms of the situation, a point (x, y) on the graph of a proportional relationship represents a particular combination of values for the two quantities involved. The x-coordinate represents the input or independent variable, while the y-coordinate represents the output or dependent variable. The point shows how the two quantities are related and how they change together.

5. How do you calculate the constant of proportionality? Open

To calculate the constant of proportionality, you can choose any two corresponding values from the proportional relationship. Divide the y-value by the x-value to find the ratio between the two quantities. The resulting ratio will be the constant of proportionality.

6. What is a unit rate? Open

A unit rate is a rate that compares a quantity to one unit of another quantity. It tells you how much of one quantity is associated with one unit of another quantity. For example, if the constant of proportionality is 3, it means that for every 1 unit increase in the independent variable (x), the dependent variable (y) increases by 3 units.

7. What are some real-life applications of proportional relationships? Open

Proportional relationships can be found in various real-life situations. Some examples include:

The relationship between distance and time in speed measurements.

The relationship between the number of workers and the amount of work completed in a fixed amount of time.

The relationship between the number of items purchased and the total cost at a fixed price per item.

8. How can guided notes and doodles help students understand graphing and equations of proportional relationships? Open

Guided notes and doodles are effective teaching tools that engage students and promote active learning. They provide structured notes and visual representations that support students in understanding and retaining the concepts. By including interactive elements like coloring and problem-solving activities, guided notes and doodles make the learning experience enjoyable and memorable for students. They also serve as valuable references for students to refer back to when practicing or reviewing the topic.

Want more ideas and freebies?

Get my free resource library with digital & print activities—plus tips over email.

Library homepage

  • school Campus Bookshelves
  • menu_book Bookshelves
  • perm_media Learning Objects
  • login Login
  • how_to_reg Request Instructor Account
  • hub Instructor Commons
  • Download Page (PDF)
  • Download Full Book (PDF)
  • Periodic Table
  • Physics Constants
  • Scientific Calculator
  • Reference & Cite
  • Tools expand_more
  • Readability

selected template will load here

This action is not available.

Mathematics LibreTexts

2.1.2: Introducing Proportional Relationships with Tables

  • Last updated
  • Save as PDF
  • Page ID 38104

  • Illustrative Mathematics
  • OpenUp Resources

Let's solve problems involving proportional relationships using tables.

Exercise \(\PageIndex{1}\): NOtice and Wonder: Paper Towels by The Case

Here is a table that shows how many rolls of paper towels a store receives when they order different numbers of cases.

clipboard_e1c7f67f56a69e5d6a86607ae24e596b0.png

What do you notice about the table? What do you wonder?

Exercise \(\PageIndex{2}\): Feeding a Crowd

  • How many people will 10 cups of rice serve?

Exercise \(\PageIndex{3}\): Making Bread Dough

A bakery uses 8 tablespoons of honey for every 10 cups of flour to make bread dough. Some days they bake bigger batches and some days they bake smaller batches, but they always use the same ratio of honey to flour. Complete the table as you answer the questions. Be prepared to explain your reasoning.

  • How many cups of flour do they use with 20 tablespoons of honey?
  • How many cups of flour do they use with 13 tablespoons of honey?
  • How many tablespoons of honey do they use with 20 cups of flour?
  • What is the proportional relationship represented by this table?

Exercise \(\PageIndex{4}\): Quarters and Dimes

4 quarters are equal in value to 10 dimes.

  • How many dimes equal the value of 6 quarters?
  • How many dimes equal the value of 14 quarters?
  • What value belongs next to the 1 in the table? What does it mean in this context?

Are you ready for more?

Pennies made before 1982 are 95% copper and weigh about 3.11 grams each. (Pennies made after that date are primarily made of zinc). Some people claim that the value of the copper in one of these pennies is greater than the face value of the penny. Find out how much copper is worth right now, and decide if this claim is true.

If the ratios between two corresponding quantities are always equivalent, the relationship between the quantities is called a proportional relationship .

This table shows different amounts of milk and chocolate syrup. The ingredients in each row, when mixed together, would make a different total amount of chocolate milk, but these mixtures would all taste the same.

Notice that each row in the table shows a ratio of tablespoons of chocolate syrup to cups of milk that is equivalent to \(4:1\).

About the relationship between these quantities, we could say:

  • The relationship between amount of chocolate syrup and amount of milk is proportional.
  • The relationship between the amount of chocolate syrup and the amount of milk is a proportional relationship.
  • The table represents a proportional relationship between the amount of chocolate syrup and amount of milk.
  • The amount of milk is proportional to the amount of chocolate syrup.

We could multiply any value in the chocolate syrup column by \(\frac{1}{4}\) to get the value in the milk column. We might call \(\frac{1}{4}\) a unit rate , because \(\frac{1}{4}\) cups of milk are needed for 1 tablespoon of chocolate syrup. We also say that \(\frac{1}{4}\) is the constant of proportionality for this relationship. It tells us how many cups of milk we would need to mix with 1 tablespoon of chocolate syrup.

Glossary Entries

Definition: Constant of Proportionality

In a proportional relationship, the values for one quantity are each multiplied by the same number to get the values for the other quantity. This number is called the constant of proportionality.

In this example, the constant of proportionality is 3, because \(2\cdot 3=6\), \(3\cdot 3=9\), and \(5\cdot 3=15\). This means that there are 3 apples for every 1 orange in the fruit salad.

Definition: Equivalent Ratios

Two ratios are equivalent if you can multiply each of the numbers in the first ratio by the same factor to get the numbers in the second ratio. For example, \(8:6\) is equivalent to \(4:3\), because \(8\cdot\frac{1}{2}=4\) and \(6\cdot\frac{1}{2}=3\).

A recipe for lemonade says to use 8 cups of water and 6 lemons. If we use 4 cups of water and 3 lemons, it will make half as much lemonade. Both recipes taste the same, because and are equivalent ratios.

Definition: Proportional Relationship

In a proportional relationship, the values for one quantity are each multiplied by the same number to get the values for the other quantity.

For example, in this table every value of \(p\) is equal to 4 times the value of \(s\) on the same row.

We can write this relationship as \(p=4s\). This equation shows that \(s\) is proportional to \(p\).

Exercise \(\PageIndex{5}\)

When Han makes chocolate milk, he mixes 2 cups of milk with 3 tablespoons of chocolate syrup. Here is a table that shows how to make batches of different sizes. Use the information in the table to complete the statements. Some terms are used more than once.

clipboard_e9568cafb1213ff763e38f30ddeed45ff.png

  • The table shows a proportional relationship between ______________ and ______________.
  • The scale factor shown is ______________.
  • The constant of proportionality for this relationship is______________.
  • The units for the constant of proportionality are ______________ per ______________.

Bank of Terms: tablespoons of chocolate syrup, 4, cups of milk, cup of milk, \(\frac{3}{2}\)

Exercise \(\PageIndex{6}\)

A certain shade of pink is created by adding 3 cups of red paint to 7 cups of white paint.

  • What is the constant of proportionality?

Exercise \(\PageIndex{7}\)

A map of a rectangular park has a length of 4 inches and a width of 6 inches. It uses a scale of 1 inch for every 30 miles.

  • What is the actual area of the park? Show how you know.
  • The map needs to be reproduced at a different scale so that it has an area of 6 square inches and can fit in a brochure. At what scale should the map be reproduced so that it fits on the brochure? Show your reasoning.

(From Unit 1.2.6)

Exercise \(\PageIndex{8}\)

Noah drew a scaled copy of Polygon P and labeled it Polygon Q.

clipboard_e93e95974ec1b75bb0960bb0cb0e3f2a9.png

If the area of Polygon P is 5 square units, what scale factor did Noah apply to Polygon P to create Polygon Q? Explain or show how you know.

(From Unit 1.1.6)

Exercise \(\PageIndex{9}\)

Select all the ratios that are equivalent to each other.

Skip to Main Content

  • My Assessments
  • My Curriculum Maps
  • Communities
  • Workshop Evaluation

Share Suggestion

Representing proportional relationships to solve problems.

  • Printer Friendly Version
  • Grade Levels 7th Grade
  • Related Academic Standards CC.2.1.7.D.1 Analyze proportional relationships and use them to model and solve real-world and mathematical problems.
  • Assessment Anchors M07.A-R.1 Demonstrate an understanding of proportional relationships.
  • Eligible Content M07.A-R.1.1.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas, and other quantities measured in like or different units. 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. M07.A-R.1.1.3 Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. M07.A-R.1.1.4 Represent proportional relationships by equations. 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.
  • Competencies

This lesson asks students to use proportional reasoning to solve problems. Students will:

  • understand the connection of rates to proportions.
  • set up proportions and use them to solve problems.
  • write proportions as equations, in the form, y = kx , and use the equations to find other converted measurements.
  • identify the constant of variation from different representations.

Essential Questions

  • How is mathematics used to quantify, compare, represent, and model numbers?
  • How are relationships represented mathematically?
  • How can expressions, equations and inequalities be used to quantify, solve, model and/or analyze mathematical situations?
  • Ratio: A comparison of two numbers by division.
  • Unit Rate: A rate simplified so that it has a denominator of 1.

60–90 minutes

Prerequisite Skills

  • one Conversion Chart ( M-7-3-1_Conversion Chart.docx ) per student
  • one Constant of Proportionality worksheet ( M-7-3-1_Constant of Proportionality Practice and KEY.docx ) per student
  • one Proportion Practice activity sheet ( M-7-3-1_Proportion Practice and KEY.docx ) per student
  • one Lesson 1 Exit Ticket ( M-7-3-1_Lesson 1 Exit Ticket and KEY.docx ) per student
  • one Lesson 1 Small Group Practice sheet ( M-7-3-1_Small Group Practice and KEY.docx ) as needed
  • one Expansion Work sheet ( M-7-3-1_Expansion Work and KEY.docx ) as needed

Related Unit and Lesson Plans

  • Proportional Reasoning
  • Recognizing Proportional Relationships in Various Forms
  • Interpreting Points on the Graph of a Proportional Relationship

Related Materials & Resources

The possible inclusion of commercial websites below is not an implied endorsement of their products, which are not free, and are not required for this lesson plan.

  • This activity allows students to solve different proportion problems. http://ca.ixl.com/math/grade-7/solve-proportions
  • This activity allows students to solve different word problems, involving proportions.

            http://ca.ixl.com/math/grade-7/solve-proportions-word-problems

  • This activity allows students to find the constant of proportionality from different graphs.

            http://ca.ixl.com/math/grade-7/constant-rate-of-change

  • This Web page provides examples of solving proportions, as well as how to set up proportions to solve problems. The page also explains how to check for proportionality.

            https://www.purplemath.com/modules/ratio2.htm

Formative Assessment

  • The Write-Pair-Share activities may be used to pre-assess students’ understanding of ratios, rates, and unit rates.
  • The Constant of Proportionality worksheet ( M-7-3-1_Constant of Proportionality Practice and KEY.docx ) may be used to assess students’ ability to identify the constant, k , in different proportional representations.
  • The Proportion Practice activity sheet ( M-7-3-1_Proportion Practice and KEY.docx ) and Guided Practice activity may be used to assess students’ ability to apply proportional reasoning to problem-solving situations.
  • Use the Lesson 1 Exit Ticket ( M-7-3-1_Lesson 1 Exit Ticket and KEY.docx ) to quickly evaluate student mastery.

Suggested Instructional Supports

Instructional procedures.

Write-Pair-Share Activity 1

As an introduction to thinking about proportional relationships, ask students to define and provide examples of ratios , rates , and unit rates . Students should also discuss how the three terms are similar and different. Give students 2–3 minutes to record their ideas. Then, have partners share their ideas. After about 5 minutes, have one member from each pair share definitions and examples of the terms. Encourage discussion and debate.  

“ Rates and unit rates are types of ratios. A ratio is simply a comparison of one value to another value. A rate is a comparison of two values, measured in different units. A unit rate is a type of rate in which the denominator is 1; in other words, a unit rate compares the value of one measurement to 1 of another type of measurement. Examples of ratios, rates, and unit rates are represented in the Venn diagram:”

Write-Pair-Share Activity 2

Ask students to think about customary and metric unit conversions they know, such as the fact that there are 12 inches in 1 foot. Give students 2–3 minutes to make a list of some conversions. Pair each student with a partner and give students time to share their lists. After about 5 minutes, one member from each group should come to the front of the room and write the conversions under appropriate categories, such as length/distance, weight, capacity, and time. When students have listed all they can think of, suggest any others they missed. Ask students to record the list of conversions in their notes or provide them with a prepared copy ( M-7-3-1_Conversion Chart.docx ).

Encourage a discussion on conversions and rates. Students should understand that the list of conversions represents rates, where a measurement in one unit is compared to a measurement in another unit. It is important that students understand that these rates may be used in proportional thinking.

Converting Values within the Customary System

After students have discussed ratios, rates, and unit rates, they may convert some values using proportional reasoning.

“We are now going to apply proportional reasoning to convert measurements within the Customary System.”

Present the following problems:

  • 5 yd = ____ft              ( 15 ft )
  • 48 oz. = ___lb             ( 3 lbs )
  • 4.5 lbs = ___oz.           ( 72 oz )

For each example, the following ideas should be presented:

  • Setting up the proportion.
  • Fractional reasoning
  • Inverse operations
  • Cross-products
  • Identifying the constant of proportionality.
  • Writing the proportional relationship as an equation in the form y = kx .
  • Using the equation to identify more converted measurements.

“Let’s look at the first problem:

5 yd = ____ ft

1. Setting up the proportion

2a. Solving the proportion using fractional reasoning

2b. Solving the proportion using inverse operations

“Sometimes a proportion may be too complex to get the right answer simply by reasoning. Fortunately, there are other ways to determine the unknown value in a proportion. A proportion is really just an algebraic equation, and you know by now that algebraic equations can be solved for a variable by using inverse operations. We can apply this method here. Follow along as I show the solution steps.

2c. Solving the proportion using cross-products

3. Identifying the constant of proportionality

“In this proportion, can we say that either side of the equation represents a unit rate?” ( No, because none of the denominators is 1. ) “But there is an easy fix.  Watch what happens if we flip both ratios in the proportion upside down:

4. Writing the proportional relationship as an equation in the form of y = kx

“Once we know the constant of proportionality, we can write our proportion in the form    y = kx , where k represents the constant of proportionality, and x and y represent our independent and dependent variables, as usual. In our example, the equation would be:

“So how does knowledge of the equation y = 3 x help us? What can we do with this equation?” Provide time for discussion and debate.

“We can use the equation to find other converted measurements. We can state in words: y feet equals 3 times x yards. Alternatively, if we wish to find the number of yards in a given number of feet, we will substitute the number of feet for y , and solve for x .”

“For example, consider the problem:

___ yd = 30 ft

“Substituting 30 for y into the equation y = 3 x , gives 30 = 3 x . Solving for x gives x = 10. Thus:

10 yd = 30 ft

“Now let’s look at the next problem:

48 oz = ____ lbs

“First, we will construct a proportion using the ratio we were given:

“Now, we must think of another ratio that compares ounces and pounds. There are 16 ounces in every 1 pound, so we can use this as the second ratio:

2. Solving the proportion

“Now that we have our proportion, we can solve it for x . If we use fractional reasoning, we may realize that 48 is 3 times more than 16. This implies that x must be 3 times more than 1 (to preserve the equality of the ratios). Therefore, x = 3.”

“We might also choose to use cross-products to solve the proportion. To do so, we would rewrite the proportion as a statement showing that the cross-products are equal:

“Now we solve the new equation for x:

“No matter the method we use to solve the proportion, it is clear that x = 3.

48 oz = 3 lbs

“With our proportion completed, we can now find the constant of proportionality, k , and write an equation in the form y = kx . Remember, to find the constant of proportionality, determine the value of each ratio in the proportion, given that there is a denominator of 1. Here, the second ratio is already written as a unit rate, and the value of each ratio is 16.”

“Thus:

“We can state in words:

y ounces equals 16 times x pounds

“Now, you will try some examples with a partner.” Instruct pairs of students to follow the same steps as listed above (#1–4) with each of the next examples. The answers are provided in the table below. As students work, circulate the classroom to assess understanding and answer any questions.

Note: In each case, point out that the unit rate is the constant of proportionality. This idea will be revisited in Lesson 3.

Provide 5–10 additional examples for practice. Monitor students as they work to check for understanding.

Identifying the Constant of Proportionality in Tables, Graphs, Equations, and Verbal Descriptions

Now that students have had an opportunity to see the applicability of the constant of proportionality and develop a conceptual understanding of the term, provide various representations of proportional relationships and ask students to identify the constant.

“Consider the following proportional relationship: A driver drives at a speed of 65 miles per hour. The constant of proportionality is the described rate, which is 65. Let’s look at this relationship in a table:

“The constant of proportionality is represented by the ratio of the change in y -values per change in corresponding x -values. Since this table shows x -values that increase by 1, the change in consecutive y -values represents the rate of change, k . The constant of proportionality, or unit rate, is also represented by the y -value given for the x -value of 1.”

“Because we have identified that our constant of proportionality is 65, this proportional relationship can be represented by the equation y = 65 x .”

“Now let’s look at a graph of this relationship:

Provide students with a few more examples of tables and graphs, and ask them to identify the constant of proportionality from each representation. All representations do not need to be related. In other words, show a table, representing a particular proportion, a graph representing another proportion, an equation representing yet another proportion, and so on. The first example included representations of the same proportion, so students could easily make comparisons, regarding the presence of the constant in a description, table, equation, and graph. Include tables that do not show consecutive x -values. Provide students with the Constant of Proportionality worksheet ( M-7-3-1_Constant of Proportionality Practice and KEY.docx ).

Solving with Proportions

“We’ve used proportional reasoning to find converted measurements. Now, let’s look at some other applications of proportional reasoning.”

Go through examples similar to the following. Ask students to write a proportion for each and solve.

Go through additional examples if more instruction and practice is needed. When you are satisfied that students understand how to accurately set up and solve proportions of this type, introduce similar figures.

Similar Figures and Proportional Reasoning

“Another important use of proportional reasoning is to solve for missing values in mathematically similar figures. Similar figures have corresponding congruent angles and proportional corresponding sides. Thus, all pairs of corresponding sides have the same ratio. You can use the ratios to form a proportion to solve for missing side lengths.”

“These figures are similar. Find the missing side length using the ratio of sides to form a proportion.”

Give students time to write the proportion and find the solution.

“The following proportion may be used to solve for the missing height:

“Using cross-products, this proportion simplifies to 24 x = 1008, where x = 42. Thus, the height of the larger triangle is 42 inches.”

“The proportional relationship may be represented by the equation, y = 1.5 x , where 1.5 represents the constant of proportionality and x represents the dimension of the smaller triangle. Thus, the base of the smaller triangle, multiplied by 1.5, gives the base of the larger triangle. The height of the smaller triangle multiplied by 1.5 gives the height of the larger triangle.”

Guided Practice

Write 2–3 additional examples of similar figures on the board. Ask students to find the missing side length(s) in each example using proportions formed with the side length ratios. Monitor students as they work. Ask each student or small group questions about the process being used in order to gauge the level of understanding. Assist as needed.

Have students work with a partner to complete the Proportion Practice activity sheet ( M-7-3-1_Proportion Practice and KEY.docx ).

Proportional Reasoning and Scales

“Proportional relationships are also represented in scale drawings and maps. Suppose a map has a key where 1 inch = 60 miles.”

Have students answer the following questions.

  • What do you think a 2-inch segment on the map would represent in real life? ( 120 miles )
  • …a segment of 7.5 inches? ( 450 miles )
  • How long would the segment need to be to represent the width of a state that is 300 miles across? ( 5 inches )

Have students complete the Lesson 1 Exit Ticket ( M-7-3-1_Lesson 1 Exit Ticket and KEY.docx ) at the close of the lesson to evaluate their level of understanding.

Use the Routine section for suggestions on ways to review lesson concepts throughout the school year. The Small Group section provides ideas for giving additional learning opportunities to students who may benefit from them. The Expansion section includes a challenge for students who are prepared to move beyond the requirements of the standard.

  • Routine: During the school year, have students identify proportional relationships in the real world. For example, cumulative savings that increase by a constant rate each month represent a proportional relationship. In addition, when working with patterns, students should recognize sequences that are proportional. The connection of patterns to proportionality is very important. Such a discussion may be substituted for any part of this lesson or added as an extension. Students will get additional practice with examining proportional relationships when they determine proportionality in lesson 2, and interpret the meaning of points on a graph of a proportional relationship in lesson 3.
  • Small Groups: Students who need additional practice may by pulled into small groups to work on the following activity: Small Group Practice ( M-7-3-1_Small Group Practice and KEY.docx ). Students can work on the problems together or work individually and compare answers when done.
  • Expansion: Students who are prepared for a greater challenge may be given the Expansion Work worksheet ( M-7-3-1_Expansion Work and KEY.docx ). The worksheet includes more problems related to proportionality and the constant of proportionality, while also asking students to create their own representations of proportional relationships.

Related Instructional Videos

IMAGES

  1. Lesson 5 Homework Practice Graph Proportional Relationships

    lesson 5 problem solving practice graph proportional relationships

  2. Lesson #5 Graph Proportional Relationships

    lesson 5 problem solving practice graph proportional relationships

  3. Graphing Proportional Relationships Worksheet with Answer KEY

    lesson 5 problem solving practice graph proportional relationships

  4. Solved Graphing Proportional Relationships

    lesson 5 problem solving practice graph proportional relationships

  5. Proportional Relationship Worksheet

    lesson 5 problem solving practice graph proportional relationships

  6. Graphing Proportional Relationships Worksheet with Answer KEY

    lesson 5 problem solving practice graph proportional relationships

VIDEO

  1. SOLVING PROBLEMS INVOLVING PROPORTION

  2. APPLICATION OF FUNDAMENTAL THEOREMS OF PROPORTIONALITY Grade 9 Week 5 Learning Task 2 LEAP Part 1

  3. Solving Problems Involving Proportions |Math 6 || Teacher Jhaniz

  4. Solving Proportions

  5. Connect Proportional Relationships and Slope

  6. 7th Grade STAAR Practice Solving Scale Problems (7.5C

COMMENTS

  1. Lesson 5

    Unit 1 7th Grade Lesson 5 of 18 Objective Write equations for proportional relationships from word problems. Common Core Standards Core Standards 7.RP.A.2 — Recognize and represent proportional relationships between quantities. 7.RP.A.2.C — Represent proportional relationships by equations.

  2. Introducing proportional relationships

    Quiz Unit test Lesson 2: Introducing proportional relationships with tables Learn No videos or articles available in this lesson Practice Up next for you: Proportional relationships Get 5 of 7 questions to level up! Start Not started Lesson 3: More about constant of proportionality Learn Constant of proportionality from tables Practice

  3. Interpreting graphs of proportional relationships (practice ...

    Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

  4. PDF Lesson 5: Identifying Proportional and Non-Proportional Relationships

    1. Determine whether or not the following graphs represent two quantities that are proportional to each other. Give reasoning. a. b. c. 2. Create a table and a graph for the ratios 2:22, 3 to 15 and 1/11. Does the graph show that the two quantities are proportional to each other? Explain why or why not. 3.

  5. Proportional relationships (practice)

    Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

  6. PDF 5.3 Graphing Proportional Relationships

    Section 5.3 Graphing Proportional Relationships 245 Exploration 2 Graphing Proportional Relationships 5.3 Algebraic Reasoning MA.8.AR.3.1 Determine if a linear relationship is also a proportional relationship. MA.8.AR.3.4 Given a mathematical or real-world context, graph a two-variable linear equation from a written description, a table or an equation in slope-intercept form.

  7. PDF Problem Solving Practice Graph Proportional Relationships

    Lesson 5 Problem-Solving Practice Graph Proportional Relationships 1. BAKING Rachel baked 3 cakes in 2 hours, 4 cakes in 3 hours, and 5 cakes in 4 hours. Determine whether the number of cakes baked is proportional to the number of hours. 2. RAINFALL It rained 2 inches in one hour, then after two hours, it had rained a total of 3 inches. After ...

  8. Lesson 5: Graph Proportional Relationships

    Lesson 5: Graph Proportional Relationships ... homework_practice_graph_proportional_relationships_answers.pdf: File Size: 293 kb: File Type: pdf: Download File. INTERACTIVE Self-Check Quiz. Reteach Documents: See it again for the first time! reteach_graph_proportional_relationships.pdf:

  9. PDF NAME DATE PERIOD Lesson 5 Problem-Solving Practice

    Lesson 5 Problem-Solving Practice Graph Proportional Relationships 1. BAKING Rachel baked 3 cakes in 2 hours, 4 cakes in 3 hours, and 5 cakes in 4 hours. Determine whether the number of cakes baked is proportional to the number of hours. 2. RAINFALL It rained 2 inches in one hour, then after two hours, it had rained a total of 3 inches.

  10. 2.4.2: Interpreting Graphs of Proportional Relationships

    Practice. Exercise 2.4.2. 4. There is a proportional relationship between the number of months a person has had a streaming movie subscription and the total amount of money they have paid for the subscription. The cost for 6 months is $47.94. The point ( 6, 47.94) is shown on the graph below. Figure 2.4.2. 6.

  11. Graphing Proportional Relationships Worksheets

    When two variables have a proportional relationship, they will exist in a ratio to one another that is constant. This means that when they are visualized on a line graph, first they will form a line that passes through the origin (0, 0). They will exist in some form of the equation y = kx. To translate this for you as x increase, so does y.

  12. Math, Grade 7, Proportional Relationships, Analyzing Proportional

    Possible answers: Line a: 5 2 5 2. Line b: 5 4 5 4. Line c: 5 6 5 6. Line d: 5 11 5 11. Line e: 1 3 1 3. The graph of a proportional relationship with a constant of 1 would lie between lines b and c. Explanations will vary. Possible explanation: The constant of proportionality gives the steepness of the line and 1 is between 5 4 5 4 and 5 6 5 6 ...

  13. Proportional relationships: graphs (video)

    Jacob Miller. 7 years ago. A proportional relationship is one where there is multiplying or dividing between the two numbers. A linear relationship can be a proportional one (for example y=3x is proportional), but usually a linear equation has a proportional component plus some constant number (for example y=3x +4).

  14. Graphing and Interpreting Graphs of Proportional Relationships Lesson

    In this lesson plan, students will learn about representing proportional relationships through graphs and equations and explore their real-life applications. Then, they will also practice interpreting graphs of proportional relationships and understanding what the points represent in the context of the problem.

  15. 2.3.3: Solving Problems about Proportional Relationships

    Read the problem card and solve the problem independently. Share the data card and discuss your reasoning. Pause here so your teacher can review your work. Ask your teacher for a new set of cards and repeat the activity, trading roles with your partner. Exercise 2.3.3.3 2.3.3. 3: Moderating Comments.

  16. BetterLesson Coaching

    Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. 7.RP.A.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.

  17. 2.1.2: Introducing Proportional Relationships with Tables

    Definition: Proportional Relationship. In a proportional relationship, the values for one quantity are each multiplied by the same number to get the values for the other quantity. For example, in this table every value of p is equal to 4 times the value of s on the same row. We can write this relationship as p = 4s.

  18. PDF LESSON Proportional Relationships and Graphs 4-3 Reteach

    Proportional Relationships and Graphs Practice and Problem Solving: D Tell whether the relationship is a proportional relationship. Explain your answer. The first one is done for you. 1. Each shirt costs $10. 2. There are 50 crayons in each box. _____ _____ _____ _____ 3. A person walks 5 feet per second. 4. A gym costs $20 per month plus a fee.

  19. Proportional relationships

    Unit 1 Factors and multiples. Unit 2 Patterns. Unit 3 Ratios and rates. Unit 4 Percentages. Unit 5 Exponents intro and order of operations. Unit 6 Variables & expressions. Unit 7 Equations & inequalities introduction. Unit 8 Percent & rational number word problems. Unit 9 Proportional relationships.

  20. Representing Proportional Relationships to Solve Problems

    This lesson asks students to use proportional reasoning to solve problems. Students will: understand the connection of rates to proportions. set up proportions and use them to solve problems. write proportions as equations, in the form, y = kx, and use the equations to find other converted measurements.

  21. Graphing Proportional Relationships

    Skills Practice; Lesson Plans; ... Graphs of a proportional relationship always start at the (0, 0) point and are always a straight line. ... How to Solve a One-Step Problem Involving Rates 5:06 ...

  22. Interpreting graphs of proportional relationships

    Video transcript. - [Voiceover] Let's get some practice interpreting graphs of proportional relationships. This says the proportion relationship between the distance driven and the amount of time driving shown in the following graph, so we have the distance driven on the vertical axis, it's measured in kilometers, then we have the time driving ...

  23. Graphing Proportional Relationships Practice

    This relationship can be represented by the equation y = 3 x. Use this information to graph the proportional relationship. 3. Consider the table below that represents how many miles a particular ...