lesson 5 problem solving practice graph ratio tables

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• Grade 6 McGraw Hill Glencoe - Answer Keys

Explanation:

Reason Inductively the distance around a circle is also called the circumference. The diameter of a circle is the distance across the circle through its center.

a. The table shows the circumference and diameter for different circles. Complete the table by estimating the ratio \(\frac{c}{d}\) for each circle.

b. The ratio \(\frac{c}{d}\) is representing by the Greek letter \(\pi\), which is called pi. Describe the ratio \(\frac{c}{d}\) for the value you calculated.

c. Make a Prediction Find two circular objects. Measure the diameter and then make a prediction about the circumference of each circle. How could you check your prediction?

• Type below:

d. Use the Internet or another media source to research pi. Write a few sentence about you learned.

H.O.T. Problems Higher Order Thinking

Model with Mathematics Write a real-world problem using ratios or rates that could be represented on the coordinate plane.

Persevere with Problems Give the coordinates of the point located halfway between (2, 1) and (2, 4).

Persevere with Problems The graph below shows the cost of purchasing pencils from the school office. The graph is missing a point to indicate the cost of 12 pencils. Complete the graph by plotting the missing information. Explain your answer.

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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

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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

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• Resource Library
• 6th Grade Mathematics
• Ratio Tables

Education Standards

Wyoming standards for mathematics.

Learning Domain: Ratios and Proportional Relationships

Standard: Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.

Standard: Use ratio and rate reasoning to solve real-world and mathematical problems.

Standard: Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.

Maryland College and Career Ready Math Standards

Standard: Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, "The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak."ť "For every vote candidate A received, candidate C received nearly three votes."ť

Standard: Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

Common Core State Standards Math

Cluster: Understand ratio concepts and use ratio reasoning to solve problems

Standard: Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”

Ratio Tables and Graphs

Relate ratio tables to graphs.

Students focus on interpreting, creating, and using ratio tables to solve problems. They also relate ratio tables to graphs as two ways of representing a relationship between quantities.

Key Concepts

Ratio tables and graphs are two ways of representing relationships between variable quantities. The values shown in a ratio table give possible pairs of values for the quantities represented and define ordered pairs of coordinates of points on the graph representing the relationship. The additive and multiplicative structure of each representation can be connected, as shown:

Goals and Learning Objectives

• Complete ratio tables.
• Use ratio tables to compare ratios and solve problems.
• Plot values from a ratio table on a graph.
• Understand the connection between the structure of ratio tables and graphs.

Lesson Guide

Have students read the prompt, examine the ratio table, and graph the points.

• Why do you think the table is called a ratio table?
• How can you complete the table?
• How is the information in the ratio table represented in the graph?
• How can you plot the points on the graph?

Use any disagreement to prompt a discussion.

Mathematics

The start of the lesson focuses students’ attention on the connection between the pairs of values in the table and the points on the graph and asks them to add information to each representation. Incorrect attempts at adding values or points to the representations are especially productive at this point in the unit; use them to surface misconceptions about the connections between tables and graphs and to provide opportunities for revising student thinking.

Note: You may need to point out that the packages are all the same size if students raise the question.

Ms. Lopez sells muffins at her bakery. She sells them in packages.

She uses a ratio table to determine the number of packages she will need for different numbers of muffins. The table she uses shows equivalent ratios of number of packages to number of muffins.

She can also use a graph to show this information.

Complete the ratio table with your class.

Using the graph on the handout, graph paper, or a graphing tool, plot all the points in the ratio table on a graph.

Examine the graph. How is the information in the ratio table represented in the graph?

HANDOUT: Ratio Tables and Graphs

Math Mission

Discuss the Math Mission. Students will explore the relationship between ratio tables and graphs and use them to solve problems.

Explore the relationship between ratio tables and graphs and use them to solve problems.

Have students work in pairs on the problem.

SWD: Students with disabilities may have difficulty graphing information accurately. Provide direct instruction on graphing skills and then allow students to complete the graphs during guided practice.

Mathematical Practices

Mathematical Practice 7: Look for and make use of structure.

In making connections between the tables and graphs, students will be working with the additive and multiplicative structure of linear relationships.

Mathematical Practice 8: Look for and express regularity in repeated reasoning.

The regularity in the relationships between values often becomes clearer when students move between tables and graphs.

Interventions

Student looks for a relationship between values within a row.

• What do you notice about the relationship between values in each column?
• Focus on the ratio between values in each column, and then compare this ratio across the columns.
• Are the ratios between the two values in each column equivalent?

Student thinks ratios in a table are not equivalent.

• [Other pair of students] said they are equivalent. How can we figure out who is right?
• What strategies have you used in other lessons to compare two ratios?
• Does it make sense that the ratios would be different in this situation?

Student doesn’t adequately relate the ratio table to the graph.

• How can you see the ratio on the graph? How can you see the ratio in the table?
• Look at the coordinates of the points on the graph. Where did those coordinates come from?
• In 39 packages, there will be 936 biscuits. 720 + 144 + 72 = 936

Ms. Lopez also sells biscuits in packages.

She uses this ratio table to determine the number of packages she will need for different numbers of biscuits.

• Complete the ratio table.
• Make a graph that shows the information in the ratio table.
• Use the ratio table to find the number of biscuits that will fill 39 packages.

Ask yourself:

• The second column of the ratio table (with a 1 in the top cell) tells you the number of biscuits in 1 package.
• Each pair of numbers in the columns of the ratio table represents an ordered pair that defines the coordinates of a point on the graph.
• How could you add to or subtract from numbers in the table to find the number of biscuits in 39 packages?

ELL: Allow ELLs to write up parts of their answers. It can be hard for ELLs to explain the whole problem, but they can either draw what they mean or show a graph or a table. This will help them prepare for the Ways of Thinking section of the lesson.

• Mr. Lee’s granola has more almonds. Explanations will vary, but should include a comparison of the ratios shown in each table.
• Explanations will vary. Possible answer: The graph of Mr. Lee’s ratio is steeper than Ms. Lopez’s ratio. The greater the ratio, the steeper the graph will be.

Ms. Lopez makes granola in her bakery. Her friend, Mr. Lee, has a bakery on the other side of town, and he makes granola, too.

Both Ms. Lopez and Mr. Lee use ratio tables to show equivalent ratios of almonds to oats in their granola.

• Use the ratio tables to determine whose granola has a higher ratio of almonds to oats. Explain your answer.
• Using a piece of graph paper or a graphing tool, make a graph of the information from each granola ratio table using the same coordinate grid.
• Use the two graphs to justify your answer as to whose granola has the higher ratio of almonds to oats.
• To compare the amounts of oats and almonds in the two types of granola, find a row from each table that has either the same number for oats or the same number for almonds.
• The steepness of each graph—called the slope —gives you a visual way to compare the ratios of almonds to oats for the two types of granola.

Prepare a Presentation

Preparing for ways of thinking.

Listen and look for the following student work to highlight during the Ways of Thinking discussion:

• Students who understand the connection between the ratio table and the graph
• Students who disagree—and discuss to resolve it—about which granola has more almonds and about how to interpret the tables
• Students who attend to the role of the ratio in determining the points of the graph

ELL: When eliciting answers, be cognizant of the difficulties some ELLs encounter when they have to express themselves in a foreign language. Allow them to use graphs or drawings to show what they mean at all times.

Challenge Problem

Possible answers.

• Yes, Anna is correct. Explanations will vary. In comparing the graphs in this context, the graph that is “lower” (i.e., has greater horizontal change per unit of vertical change) shows a greater ratio of oats to almonds.

Be prepared to show:

How you completed the ratio table of packages to biscuits

How you used the information in the table to make a graph

How you found the number of biscuits in 39 packages

Also be prepared to demonstrate and explain:

How you used the granola ratio tables to determine whose granola has a higher ratio of almonds to oats

How you graphed the information from the two tables

How the graphs provide support for your answer as to whose granola has the higher ratio of almonds to oats

Anna graphed the ratios of cups of almonds to cups of oats, using the x -axis to represent cups of oats and the y -axis to represent cups of almonds. She says that if you make this type of graph for any two types of granola mixtures, the graph of the granola that has the higher ratio of almonds to oats will always be steeper.

Is she right? Explain and give examples.

Make Connections

Begin the Ways of Thinking discussion with presentations of at least two different ways to use the ratio table to find the number of biscuits that will fill 39 packages. If no student addresses it, invite students to think about what can be added together from the table in order to solve the problem.

Then focus on different strategies for solving the granola problem. Call on students to get a range of approaches. For example: 

• Students who compare the ratios where the amount of almonds is the same (3 cups) 

1 + 2 = 3 cups almonds

3 + 6 = 9 cups of oats

Ratio of almonds to oats is 3:9.

Ratio of almonds to oats is 3:5.

There is the same amount of almonds. Ms. Lopez has more oats, so Mr. Lee’s granola is “almondier.”

• Students who compare the ratios where the amount of oats is the same (15 cups)

Ratio of almonds to oats is 5:15.

Ratio of almonds to oats is 9:15.

There is the same amount of oats. Mr. Lee has more almonds.

Students who compare the ratios where the total amount of cups is the same (8 cups: the 2:6 ratio in Ms. Lopez’s table has 8 cups total and the 3:5 ratio in Mr. Lee’s table has 8 cups total). 

8 cups of granola with a almond to oat ratio of 2:6

8 cups of granola with a almond to oat ratio of 3:5 So, Ms. Lopez’s granola has more oats and Mr. Lee’s granola has more almonds.

Close the discussion with a presentation from at least one pair of students who attempted the Challenge Problem. If students are very clear on interpreting the graphs, ask how their interpretation would change if the quantities on the axes switched positions (almonds on the horizontal axis and oats on the vertical axis). In comparing the graphs in this context, the graph that is “lower” (i.e., has greater horizontal change per unit of vertical change) shows a greater ratio of almonds to oats.

As students present their work, you may want to point out that any time they are creating or filling in a ratio table they are comparing ratios to see if they are equivalent. Prompt students to express this regularity in repeated reasoning themselves by asking how they know the values in their table are correct and how they know the points on the graph are correct.

Ask students to highlight the additive structure implicit in the tables, where appropriate, particularly in finding the number of biscuits that will fill 39 packages, and to connect this structure to the graphs as students share their work on the Challenge Problem.

Performance Task

Ways of thinking: make connections.

Take notes about your classmates' work with ratio tables and graphs.

As your classmates present, ask questions such as:

• Can you say more about how the ratio table relates to the corresponding graph? For example, can you show how an increase in the number of packages would be reflected in the table and on the graph?
• How could you use the graph you made from the ratio table of biscuits to packages to find the number of biscuits in 39 packages? Compare this to how you would use the ratio table.
• How do the graphs of the two granola mixtures tell you which mixture has the higher ratio of almonds to oats?

Representing Ratios

Have pairs quietly discuss the relationship between the ratio table, the double number line, and the graph, and explain how each shows the ratio of cups of milk to cups of flour. As student pairs work together, listen for students who may still have misconceptions so you can address them in the class discussion. After a few minutes, discuss the Summary as a class. Talk about how all three models represent the same relationship between the quantities.

Formative Assessment

Summary of the math: representing ratios.

Read and Discuss

• You can show this ratio with a ratio table, a double number line, or a graph.
• For each pair of values in a ratio table, you can place a corresponding lined-up pair of numbers on a double number line and make a graph by plotting a corresponding point (using an ordered pair) on a coordinate grid.
• Explain what a ratio table is?
• Describe how to make a graph that shows the information from a ratio table?
• Describe how to make a graph that shows the information from a double number line?

Reflect On Your Work

Have each student write a brief reflection before the end of class. Review the reflections to find out what about ratios still confuses students.

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

One thing that still confuses me about ratios is …

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Lesson5 Homework Practice Graph Ratio Tables

Lesson5 Homework Practice Graph Ratio Tables - Displaying top 8 worksheets found for this concept.

Some of the worksheets for this concept are Tteeaacchheerrss gguuiiddee, Lesson 6 comparing ratios using ratio tables, Name date period lesson 7 homework practice, Ratios rates unit rates, Homework practice and problem solving practice workbook, Writing linear equations module 5, Ratios and proportional relationships lessons 1 6, Just the maths.

Found worksheet you are looking for? To download/print, click on pop-out icon or print icon to worksheet to print or download. Worksheet will open in a new window. You can & download or print using the browser document reader options.

1. TTEEAACCHHEERRSS GGUUIIDDEE

2. lesson 6: comparing ratios using ratio tables, 3. name date period lesson 7 homework practice, 4. ratios, rates & unit rates, 5. homework practice and problem-solving practice workbook, 6. writing linear equations module 5, 7. ratios and proportional relationships: lessons 1-6, 8. ''just the maths'' -.

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Grade 5 Math - Ratio tables

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Tags: Ratio tables worksheets with answers , Grade 5 Math Ratio table word problems , Ratios and rates worksheets , Understanding ratios 5th grade , Ratios and proportional relationships 5th grade worksheets , Equivalent ratios and graphs

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Lesson5 Homework Practice Graph Ratio Tables

Displaying top 8 worksheets found for - Lesson5 Homework Practice Graph Ratio Tables .

Some of the worksheets for this concept are Tteeaacchheerrss gguuiiddee, Lesson 6 comparing ratios using ratio tables, Name date period lesson 7 homework practice, Ratios rates unit rates, Homework practice and problem solving practice workbook, Writing linear equations module 5, Ratios and proportional relationships lessons 1 6, Just the maths.

Found worksheet you are looking for? To download/print, click on pop-out icon or print icon to worksheet to print or download. Worksheet will open in a new window. You can & download or print using the browser document reader options.

1. TTEEAACCHHEERRSS GGUUIIDDEE

2. lesson 6: comparing ratios using ratio tables, 3. name date period lesson 7 homework practice, 4. ratios, rates & unit rates, 5. homework practice and problem-solving practice workbook, 6. writing linear equations module 5, 7. ratios and proportional relationships: lessons 1-6, 8. ''just the maths'' -.