Subject:
Ratios and Proportions
Material Type:
Lesson Plan
Level:
Middle School
7
Provider:
Pearson
Tags:
Language:
English
Media Formats:
Text/HTML

# Identifying Verbal Descriptions ## Overview

Students interpret verbal descriptions of situations and determine whether the situations represent proportional relationships.

# Key Concepts

In a proportional relationship, there has to be some value that is constant.

There are some relationships in some situations that can never be proportional.

# Goals and Learning Objectives

• Identify verbal descriptions of situations as being proportional relationships or not
• Understand that some relationships can never be proportional
• Understand that for two variable quantities to be proportional to one another, something in the situation has to be constant

# Lesson Guide

• Have students view the image.
• Have partners share ideas about the question “In which set of stairs is the height of your location on the stairs proportional to the number of stairs you have climbed?”

ELL: Be sure that your pace is appropriate when posing these questions, especially when interacting with ELLs.

# Teacher Demonstration

If there are stairs near the classroom, have students observe someone going up the stairs to illustrate what is meant by “the height of your location,” which could be taken at any point on a person’s body (e.g., bottom of foot, hip, or head).

# Mathematics

This situation requires students to think about the constant of proportionality in a new way: as the constant height of a stair in a staircase of stairs with identical heights, which is similar to the stacks of books and cups situations. Students may have trouble understanding the quantities involved in the situation, particularly “the height of your location.” Ask students the following:

• What are some different ways you can measure someone’s height at a certain point as she climbs the stairs?
• What if you measured someone’s height at the moment when she lifted one foot off the fourth step? The number of steps would be 4. Can you find her height knowing just that she climbed four stairs in the first picture? In the second picture?

ELL: Oral instructions can be hard for ELLs to follow. To ensure that all students understand the directions, ask a few students to repeat what they are being asked to do. If you think it is clear to all ELLs, then proceed. When checking for understanding, say something like: "In a scale from 1 to 4, where 4 means ‘I am really clear about what I have to do,' and 1 is 'I am not clear,' show me what number represents your level of understanding." If you see a few 3s or 2s, explain the instructions again.

# Stairs

Discuss the following with your classmates.

• In which set of stairs is the height of your location on the stairs proportional to the number of stairs you have climbed? # Lesson Guide

Discuss the Math Mission. Students will determine whether a situation represents a proportional relationship.

## Opening

Determine whether a situation represents a proportional relationship.

# Lesson Guide

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

ELL: Word problems tend to be difficult for ELLs because they are highly language dependent. In addition, many word problems require formal operations or the ability to think abstractly and to manipulate concepts through language. Have students work with a partner so they can share and exchange ideas as they are working through word problems. Consider pairing ELLs with students who have strong English skills.

# Mathematics

Students must interpret verbal descriptions of situations with two variable quantities and a set of general conditions on those quantities and decide:

• Whether the two quantities can be proportional to one another
• Under what specific condition(s) they would be proportional to one another

SWD: Students with disabilities may not make these observations about identifying the constant of proportionality as readily and may need this information pointed out during direct instruction or guided practice.

# Interventions

• Sketch a diagram of the situation, and label the quantities involved.
• There are two variable quantities in the situation. Can you identify what they are?
• Does one quantity depend on the other quantity in this situation? If so, use words to describe how one depends on the other, and then use mathematics to describe how one depends on the other.
• Assign a letter to each of the variable quantities, and then pick some values for each quantity that make sense. Make a table of possible values. What would be true about these values if the quantities were proportional to one another? What would be true about the situation if the values were like this?

[common error] Student misinterprets what he or she is supposed to do by adding variables to the situation or by changing the conditions of the variables in inappropriate ways.

• Use the description to identify the two quantities that vary in the situation.
• What happens to one quantity when the other quantity increases in value?
• Make up some reasonable values for the two variable quantities, and put them in a table so you can examine their relationship. What would be true about the values of these quantities if they were proportional to one another?
• Is there anything about the situation that could create a constant rate of change between the two quantities?

Student doesn’t recognize the quantities in the situation.

• Sketch a diagram of the situation to represent the quantities that vary in relation to one another.
• Sketch a graph that shows one possible relationship between the quantities in the situation. First, decide which quantity goes on which axis.

Student thinking is hindered by the absence of given values.

• Make up some values that would make sense for each quantity, and use those to think about the relationship between them.
• Make a table of possible values that you make up for each quantity. Think of values that would make sense in the situation.
• Graph some points that would make sense to represent the quantities in the situation. What would need to be true about the graph if this was a proportional relationship? What would need to be true about the values of each quantity if that were true about the graph?

• This is true if the steps she takes are the same length.

# Proportional or Not Proportional?

A girl is walking along a road and counting the number of steps she takes as she walks.

The distance she covers is proportional to the number of steps she takes.

• When is this statement true? If it is never true, explain why. ## Hint:

What would have to be true about the girl’s steps for the statement to be true?

# Lesson Guide

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

# Mathematics

Students must interpret verbal descriptions of situations with two variable quantities and a set of general conditions on those quantities and decide:

• Whether the two quantities can be proportional to one another
• Under what specific condition(s) they would be proportional to one another

# Interventions

• Sketch a diagram of the situation, and label the quantities involved.
• There are two variable quantities in the situation. Can you identify what they are?
• Does one quantity depend on the other quantity in this situation? If so, use words to describe how one depends on the other, and then use mathematics to describe how one depends on the other.
• Assign a letter to each of the variable quantities, and then pick some values for each quantity that make sense. Make a table of possible values. What would be true about these values if the quantities were proportional to one another? What would be true about the situation if the values were like this?

[common error] Student misinterprets what he or she is supposed to do by adding variables to the situation or by changing the conditions of the variables in inappropriate ways.

• Use the description to identify the two quantities that vary in the situation.
• What happens to one quantity when the other quantity increases in value?
• Make up some reasonable values for the two variable quantities, and put them in a table so you can examine their relationship. What would be true about the values of these quantities if they were proportional to one another?
• Is there anything about the situation that could create a constant rate of change between the two quantities?

Student doesn’t recognize the quantities in the situation.

• Sketch a diagram of the situation to represent the quantities that vary in relation to one another.
• Sketch a graph that shows one possible relationship between the quantities in the situation. First, decide which quantity goes on which axis.

Student thinking is hindered by the absence of given values.

• Make up some values that would make sense for each quantity, and use those to think about the relationship between them.
• Make a table of possible values that you make up for each quantity. Think of values that would make sense in the situation.
• Graph some points that would make sense to represent the quantities in the situation. What would need to be true about the graph if this was a proportional relationship? What would need to be true about the values of each quantity if that were true about the graph?

• This is true if the faucet is dripping at a constant rate.

# Leaky Faucet

Water is dripping rapidly from a faucet into a tub.
You plug the drain to see how quickly the water accumulates in the tub.

The volume of water in the tub is proportional to the time that has passed since you plugged the drain.

• When is this statement true? If it is never true, explain why. ## Hint:

What would have to be true about the drips of water for the statement to be true?

# Lesson Guide

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

# Interventions

• Sketch a diagram of the situation, and label the quantities involved.
• There are two variable quantities in the situation. Can you identify what they are?
• Does one quantity depend on the other quantity in this situation? If so, use words to describe how one depends on the other, and then use mathematics to describe how one depends on the other.
• Assign a letter to each of the variable quantities, and then pick some values for each quantity that make sense. Make a table of possible values. What would be true about these values if the quantities were proportional to one another? What would be true about the situation if the values were like this?

[common error] Student misinterprets what he or she is supposed to do by adding variables to the situation or by changing the conditions of the variables in inappropriate ways.

• Use the description to identify the two quantities that vary in the situation.
• What happens to one quantity when the other quantity increases in value?
• Make up some reasonable values for the two variable quantities, and put them in a table so you can examine their relationship. What would be true about the values of these quantities if they were proportional to one another?
• Is there anything about the situation that could create a constant rate of change between the two quantities?

Student doesn’t recognize the quantities in the situation.

• Sketch a diagram of the situation to represent the quantities that vary in relation to one another.
• Sketch a graph that shows one possible relationship between the quantities in the situation. First, decide which quantity goes on which axis.

Student thinking is hindered by the absence of given values.

• Make up some values that would make sense for each quantity, and use those to think about the relationship between them.
• Make a table of possible values that you make up for each quantity. Think of values that would make sense in the situation.
• Graph some points that would make sense to represent the quantities in the situation. What would need to be true about the graph if this was a proportional relationship? What would need to be true about the values of each quantity if that were true about the graph?

• Your height off the ground is never proportional to the amount of time that has passed since the Ferris wheel started moving. Your height does vary as the Ferris wheel moves; it is not constant.

# Riding the Ferris Wheel

You are riding a Ferris wheel at an amusement park.

Your height from the ground is proportional to the amount of time that has passed since the Ferris wheel started moving.

• When is this statement true? If it is never true, explain why. ## Hint:

Sketch a picture of a Ferris wheel. Indicate the distance between each Ferris wheel rider and the ground.

# Lesson Guide

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

# Mathematics

Students must interpret verbal descriptions of situations with two variable quantities and a set of general conditions on those quantities and decide:

• Whether the two quantities can be proportional to one another
• Under what specific condition(s) they would be proportional to one another

# Interventions

• Sketch a diagram of the situation, and label the quantities involved.
• There are two variable quantities in the situation. Can you identify what they are?
• Does one quantity depend on the other quantity in this situation? If so, use words to describe how one depends on the other, and then use mathematics to describe how one depends on the other.
• Assign a letter to each of the variable quantities, and then pick some values for each quantity that make sense. Make a table of possible values. What would be true about these values if the quantities were proportional to one another? What would be true about the situation if the values were like this?

[common error] Student misinterprets what he or she is supposed to do by adding variables to the situation or by changing the conditions of the variables in inappropriate ways.

• Use the description to identify the two quantities that vary in the situation.
• What happens to one quantity when the other quantity increases in value?
• Make up some reasonable values for the two variable quantities, and put them in a table so you can examine their relationship. What would be true about the values of these quantities if they were proportional to one another?
• Is there anything about the situation that could create a constant rate of change between the two quantities?

Student doesn’t recognize the quantities in the situation.

• Sketch a diagram of the situation to represent the quantities that vary in relation to one another.
• Sketch a graph that shows one possible relationship between the quantities in the situation. First, decide which quantity goes on which axis.

Student thinking is hindered by the absence of given values.

• Make up some values that would make sense for each quantity, and use those to think about the relationship between them.
• Make a table of possible values that you make up for each quantity. Think of values that would make sense in the situation.
• Graph some points that would make sense to represent the quantities in the situation. What would need to be true about the graph if this was a proportional relationship? What would need to be true about the values of each quantity if that were true about the graph?

• This is true if the tram is moving at a constant rate.

# Trams in the Park

At an amusement park, a tram moves along a track.

The total distance the tram travels is proportional to the time that it travels.

• When is this statement true? If it is never true, explain why. # Lesson Guide

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

# Mathematics

Students must interpret verbal descriptions of situations with two variable quantities and a set of general conditions on those quantities and decide:

• Whether the two quantities can be proportional to one another
• Under what specific condition(s) they would be proportional to one another

# Interventions

• Sketch a diagram of the situation, and label the quantities involved.
• There are two variable quantities in the situation. Can you identify what they are?
• Does one quantity depend on the other quantity in this situation? If so, use words to describe how one depends on the other, and then use mathematics to describe how one depends on the other.
• Assign a letter to each of the variable quantities, and then pick some values for each quantity that make sense. Make a table of possible values. What would be true about these values if the quantities were proportional to one another? What would be true about the situation if the values were like this?

[common error] Student misinterprets what he or she is supposed to do by adding variables to the situation or by changing the conditions of the variables in inappropriate ways.

• Use the description to identify the two quantities that vary in the situation.
• What happens to one quantity when the other quantity increases in value?
• Make up some reasonable values for the two variable quantities, and put them in a table so you can examine their relationship. What would be true about the values of these quantities if they were proportional to one another?
• Is there anything about the situation that could create a constant rate of change between the two quantities?

Student doesn’t recognize the quantities in the situation.

• Sketch a diagram of the situation to represent the quantities that vary in relation to one another.
• Sketch a graph that shows one possible relationship between the quantities in the situation. First, decide which quantity goes on which axis.

Student thinking is hindered by the absence of given values.

• Make up some values that would make sense for each quantity, and use those to think about the relationship between them.
• Make a table of possible values that you make up for each quantity. Think of values that would make sense in the situation.
• Graph some points that would make sense to represent the quantities in the situation. What would need to be true about the graph if this was a proportional relationship? What would need to be true about the values of each quantity if that were true about the graph?

• This is true if each piece of paper in the stack has the same thickness.

# Stack of Paper

There is a stack of paper.

The height of the stack of paper is proportional to the number of pieces of paper in the stack.

• When is this statement true? If it is never true, explain why. # Preparing for Ways of Thinking

Listen and look for students who:

• Understand the situation, with recognition of needing to identify a constant of proportionality
• Make sense of the variable quantities (e.g., naming them, discussing their behavior, discussing what they are determined by, and proposing reasonable values for them)
• Graph reasonable examples of the relationships to explore their possible behaviors
• [common error] Appear hindered by the absence of given values and make mistakes as a result

# Challenge Problem

Problems will vary. Students’ problems should involve an object with a constant measure that when multiplied creates another measure proportional to the single object’s measure.

# Prepare a Presentation

Prepare a presentation that explains and analyzes your findings.

1. How did you determine whether a situation represents a proportional relationship?
2. What do the situations that represent proportional relationships have in common?
3. What is true about the situations that do not represent proportional relationships?

# Challenge Problem

Create your own problem similar to the Work Time problems. Come up with a situation and write a statement that makes a claim about proportional relationships in the situation. Trade problems with another student.

# Mathematical Practices

Mathematical Practice 2: Reason abstractly and quantitatively.

• Have the first presenter be a student who drew a rudimentary diagram. Ask the class to develop it collectively over the course of a discussion so that it eventually includes clear representations of the variable quantities and of the element that defines a constant of proportionality.
• Draw on student work that included making a table and making a graph for possible values (not proportional), or build a new table and graph collectively with the class. Begin with values (and points on a graph) that do not exhibit a proportional relationship, and ask what would need to be the case in each—the table and the graph—if the relationship was proportional.

SWD: For students with disabilities, participating in a whole class discussion such as Ways of Thinking can be intimidating for a variety of reasons. However, it is important for students to work on the speaking and listening skills implicit to this portion of the lesson. Possible supports for students include:

• Give students a few minutes to discuss their ideas with a partner or small group before sharing in the whole class setting. Have students discuss the questions posed and what they learned in the lesson.
• Conference with individual students prior to the discussion to ascertain what they might be able to successfully contribute to the discussion. Give students time to rehearse their contribution or to write notes for themselves to refer to as they speak. This will support students with expressive language difficulties or students who are anxious or reluctant to participate in class discussions.

# Ways of Thinking: Make Connections

Take notes about your classmates’ explanations and strategies for determining whether a situation represents a proportional relationship.

## Hint:

• How did you know that the relationship is proportional?
• Is there a particular word that you looked for to help you determine whether a situation is proportional?
• In the problem you wrote, what are the variables in the situation?
• How would you describe the unit rate (for example, miles per hour)?

# A Possible Summary

In real-world situations where objects or events have a constant measure (e.g., length, width, or height, or a constant increase over time or a constant distance per time period), there can be variable quantities that are proportional to each other, with this constant measure as the constant of proportionality. In order for two variable quantities to be proportional to one another, something in the situation has to be constant.

Be sure to emphasize the following points:

• In a proportional relationship, there has to be something that is constant.
• There are some relationships in some situations that can never be proportional.

# Summary of the Math: Proportional Situations

Write a summary about what makes a situation proportional.

## Hint:

• Do you explain that in order for a situation to represent a proportional relationship, some value in the situation must be constant?
• Do you discuss the fact that there are some relationships that can never be proportional, and provide an example?

# Lesson Guide

Have each student write a brief reflection before the end of the class. Review the reflections to discover what students learned about proportional relationships.

SWD: Some students with disabilities may struggle to explain their mathematical reasoning in words. Provide sentence starters or paragraph frames to support students with this task.

# Reflection

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

One thing I learned about proportional relationships is …