Lesson 9

Solving Problems about Proportional Relationships

9.1: What Do You Want to Know? (10 minutes)

Warm-up

This warm-up prepares students for the Info Gap activity that follows. First, students brainstorm what information they would need to know to solve a problem that involves constant speed. Then, the teacher demonstrates the process of asking a student why they need a specific piece of information before sharing it with them, in preparation for students following this procedure with their partner in the next activity.

Launch

Give students 1 minutes of quiet think time to brainstorm what information they would need to know to solve the problem, followed by a whole-class discussion demonstrating the procedure for the Info Gap activity.

Student Facing

Consider the problem: A person is running a distance race at a constant rate. What time will they finish the race?

What information would you need to be able to solve the problem?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Ask students:

  • "What specific information do you need?" 
  • "Why do you need that information?"

Share each piece of information with the class after a student specifically asks for it (and explains why they need to know it).

  • The race is 10,000 meters long.
  • The race started at 9:15 a.m.
  • In 1 minute, the person ran $156\frac{1}{4}$ meters.
  • An equation relating distance and time is given by $d = 156\frac{1}{4} \boldcdot t$ where $d$ represents distance in meters and $t$ represents time in minutes.
  • It takes 32 minutes for the person to run 5,000 meters.
  • The person runs at a pace of 6.4 minutes per kilometer (or 1,000 meters).

After you share each piece of information, ask the class whether they have enough information to be able to solve the problem. When they think they do, give them 2 minutes to solve the problem and then have them share their strategies. (The person should finish the race at 10:19 a.m.)

Tell students that they will be working in groups of two in the next activity and that they will be using the same procedure that you just demonstrated to solve a problem.

9.2: Info Gap: Biking and Rain (30 minutes)

Activity

In this info gap activity, students write equations for several proportional relationships given in the contexts of a bike ride and steady rainfall. They use the equations to make predictions.

The info gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).

Here is the text of the cards for reference and planning:

Image of an info gap card

Launch

Engagement: Develop Effort and Persistence. Display or provide students with a physical copy of the written directions. Check for understanding by inviting students to rephrase directions in their own words. Keep the display of directions visible throughout the activity.
Supports accessibility for: Memory; Organization
Conversing: This activity uses MLR4 Information Gap to give students a purpose for discussing information necessary to write equations for proportional relationships. Display questions or question starters for students who need a starting point such as: “Can you tell me . . . (specific piece of information)”, and “Why do you need to know . . . (that piece of information)?"
Design Principle(s): Cultivate Conversation

Student Facing

Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.

If your teacher gives you the problem card:

  1. Silently read your card and think about what information you need to be able to answer the question.

  2. Ask your partner for the specific information that you need.

  3. Explain how you are using the information to solve the problem.

    Continue to ask questions until you have enough information to solve the problem.

  4. Share the problem card and solve the problem independently.

  5. Read the data card and discuss your reasoning.

If your teacher gives you the data card:

  1. Silently read your card.

  2. Ask your partner “What specific information do you need?” and wait for them to ask for information.

    If your partner asks for information that is not on the card, do not do the calculations for them. Tell them you don’t have that information.

  3. Before sharing the information, ask “Why do you need that information?” Listen to your partner’s reasoning and ask clarifying questions.

  4. Read the problem card and solve the problem independently.

  5. 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.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Invite students to share their equations and predictions. Record their equations displayed for all to see. Ask them to explain how they determined the constant of proportionality as well as why it made sense to represent the situation with a proportional relationship.

9.3: Moderating Comments (10 minutes)

Optional activity

In this activity students compute rates to decide which job applicant is working the fastest checking online comments. They compare rates and total number of comments checked, then see that using rates is the more useful information in this situation. 

Launch

Keep students in the same groups of 2.

Representation: Internalize Comprehension. Activate or supply background knowledge. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing

Student Facing

A company is hiring people to read through all the comments posted on their website to make sure they are appropriate. Four people applied for the job and were given one day to show how quickly they could check comments.

  • Person 1 worked for 210 minutes and checked a total of 50,000 comments.
  • Person 2 worked for 200 minutes and checked 1,325 comments every 5 minutes.
  • Person 3 worked for 120 minutes, at a rate represented by \(c = 331t\),
    where \(c\) is the number of comments checked and \(t\) is the time in minutes.
  • Person 4 worked for 150 minutes, at a rate represented by \(t = \left( \frac{3}{800} \right) c\).
  1. Order the people from greatest to least in terms of total number of comments checked.
  2. Order the people from greatest to least in terms of how fast they checked the comments.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

  1. Write equations for each job applicant that allow you to easily decide who is working the fastest.
  2. Make a table that allows you to easily compare how many comments the four job applicants can check.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Ask students which job applicant should get the job and why. If time permits, consider using MLR 1 (Stronger and Clearer Each Time).

Representing, Conversing: MLR7 Compare and Connect. Use this routine to prepare students for the whole-class discussion. At the appropriate time, invite students to create a visual display showing which job applicant should get the job and why.  Allow students time to quietly circulate and analyze the selections and justifications in at least 2 other displays in the room. Give students quiet think time to consider what is the same and what is different and whether or not they agree. Next, ask students to find a partner to discuss what they noticed. Listen for and amplify observations that highlight advantages and disadvantages to each method of determining the top job applicant. This will help students identify situation when comparing rates is more effective than the sum of quantities.
Design Principle(s): Optimize output (for justification); Cultivate conversation

Lesson Synthesis

Lesson Synthesis

Whenever we have a situation involving constant rates, we are likely to have a proportional relationship between the quantities of interest.

  • "What are some situations that we have seen where quantities were proportional to each other?"
  • "When we are in a situation where we have a proportional relationship between two quantities, what information do we need to find an equation?"
  • "How can we decide if a proportional relationship is a good representation of a particular situation?"
  • "Equations are good tools to make predictions or decisions. When and how did we use an equation to make a prediction or a decision today?"

9.4: Cool-down - Steel Beams (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

Whenever we have a situation involving constant rates, we are likely to have a proportional relationship between quantities of interest.

  • When a bird is flying at a constant speed, then there is a proportional relationship between the flying time and distance flown.
  • If water is filling a tub at a constant rate, then there is a proportional relationship between the amount of water in the tub and the time the tub has been filling up.
  • If an aardvark is eating termites at a constant rate, then there is proportional relationship between the number of termites the aardvark has eaten and the time since it started eating.

Sometimes we are presented with a situation, and it is not so clear whether a proportional relationship is a good model. How can we decide if a proportional relationship is a good representation of a particular situation?

  • If you aren’t sure where to start, look at the quotients of corresponding values. If they are not always the same, then the relationship is definitely not a proportional relationship.
  • If you can see that there is a single value that we always multiply one quantity by to get the other quantity, it is definitely a proportional relationship.

After establishing that it is a proportional relationship, setting up an equation is often the most efficient way to solve problems related to the situation.