Lesson 14

Solving Equivalent Ratio Problems

14.1: What Do You Want to Know? (5 minutes)

Warm-up

The warm-up prepares students for the next info gap activity by first asking them to brainstorm what information they would need to know to solve an equivalent ratio problem. Next, the teacher demonstrate asking students to share what they want to know and why they want to know it before giving them the information.

Launch

Give students 2 minutes of quiet think time.

Student Facing

Consider the problem: A red car and a blue car enter the highway at the same time and travel at a constant speed. How far apart are they after 4 hours?

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

Demonstrate asking students the questions they will use in the Info Gap in the next activity. Ask them, “What specific information do you need?” As students pose questions, write them down and ask, “Why do you need that information?”

When students explain why they need the information, provide it to them. After sharing each piece of information, ask the class whether they have enough information 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 red car is traveling faster than the blue car.
  • One car is traveling 5 miles per hour faster than the other car.
  • The slower car is traveling at 60 miles per hour.
  • The blue car is traveling at 60 miles per hour.
  • The faster car is traveling at 65 miles per hour.
  • The red car is traveling as 65 miles per hour.
  • Both cars entered the highway at the same location.
  • Both cars are traveling in the same direction.

14.2: Info Gap: Hot Chocolate and Potatoes (30 minutes)

Activity

In this info gap activity, students solve problems involving equivalent ratios. If students use a table, it may take different forms. Some students may produce a table that has many rows that require repeated multiplication. Others may create a more abbreviated table and use more efficient multipliers. Though some approaches may be more direct or efficient than others, it is important for students to choose their own method for solving them, and to explain their method so that their partner can understand (MP3).

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:

Problem Card 1

Jada mixes milk and cocoa powder to make hot chocolate. She wants to use all the cocoa powder she has left. How much milk should Jada use?

Data Card 1

  • One batch of Jada’s recipe calls for 3 cups of milk.
  • One batch of Jada’s recipe calls for 2 tablespoons of cocoa powder.
  • Jada has 2 gallons of milk left.
  • Jada has 9 tablespoons of cocoa powder left.
  • There are 16 cups in 1 gallon.

Problem Card 2

Noah needs to peel a lot of potatoes before a dinner party. He has already peeled some potatoes. If he keeps peeling at the same rate, will he finish all the potatoes in time?

Data Card 2

  • Noah has already been peeling potatoes for 10 minutes.
  • Noah has already peeled 8 potatoes.
  • Noah needs to peel 60 more potatoes.
  • Noah needs to be finished peeling potatoes in 1 hour and 10 minutes.
  • There are 60 minutes in 1 hour.

Launch

Arrange students in groups of 2. In each group, distribute a problem card to one student and a data card to the other student.

Action and Expression: Internalize Executive Functions. Begin with a small-group or whole-class demonstration and think aloud of a sample situation to remind students how to use the info gap structure.  Keep the worked-out table and double number lines on display for students to reference as they work.
Supports accessibility for: Memory; Conceptual processing
Conversing: This activity uses MLR4 Information Gap to give students a purpose for discussing information necessary to solve problems involving equivalent ratios. 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

Anticipated Misconceptions

Students may misinterpret the meaning of the numbers or associate quantities incorrectly and multiply 8 by 6 (because \(10 \boldcdot 6\) is 60). Encourage them to organize the given information in a table or a double number line.

Activity Synthesis

Select one student to explain each distinct approach. Highlight how multiplicative reasoning and using the table are similar or different in each case.

When all approaches have been discussed, ask students: “When might it be helpful to first find the amount that corresponds to 1 unit of one quantity and scale that amount up to any value we want?” Encourage students to refer to all examples seen in this lesson so far.

14.3: Comparing Reading Rates (10 minutes)

Optional activity

This activity provides an opportunity for additional practice in solving equivalent ratio problems. Monitor for students solving the problems in different ways.

Launch

Give students 4 minutes of quiet work time and then have them discuss their solutions with a partner. 

Representation: Internalize Comprehension. Activate or supply background knowledge about equivalent ratios describing situations that happen at the same rate which can be displayed on double number lines. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing
Writing: MLR5 Co-Craft Questions. Before asking students to solve the given question, display only Lin's, Diego's and Elena's reading rates. Invite students think of possible mathematical questions about this situation. Keep in mind students do not need to answer their created questions. Students share and revise their questions with a partner and then with the whole class. Record questions shared with the class in a public space. This helps students connect mathematical language to get into the context of the problem as they reason about the three different quantities (i.e., number of pages read, number of pages in the book, and number of days reading).
Design Principle(s): Maximize meta-awareness​

Student Facing

  • Lin read the first 54 pages from a 270-page book in the last 3 days.
  • Diego read the first 100 pages from a 320-page book in the last 4 days.
  • Elena read the first 160 pages from a 480-page book in the last 5 days.

If they continue to read every day at these rates, who will finish first, second, and third? Explain or show your reasoning.

Student Response

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

Student Facing

Are you ready for more?

The ratio of cats to dogs in a room is \(2:3\). Five more cats enter the room, and then the ratio of cats to dogs is \(9:11\). How many cats and dogs were in the room to begin with?

Student Response

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

Activity Synthesis

Select students to present their solutions. Sequence solutions with diagrams first and then tables. Make sure students see connections between the different representations and ways of solving the problems.

Lesson Synthesis

Lesson Synthesis

When solving problems involving equivalent ratios, we often have three pieces of information and need to find a fourth. For example:

  • If you eat 12 strawberries in 3 minutes, how long will it take to eat 8 strawberries at that rate?

We can use a table to solve this problem very quickly. For example:

number of strawberries number of minutes
12 3
1 \(\frac14\)
8 2
  • If you jump 8 times in 10 seconds, how many jumps can you make in 45 seconds at that rate? 

Where would you put the one in this table? What is the answer to the question?

number of jumps number of seconds
8 10

14.4: Cool-down - Water Faucet (5 minutes)

Cool-Down

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

Student Lesson Summary

Student Facing

To solve problems about something happening at the same rate, we often need:

  • Two pieces of information that allow us to write a ratio that describes the situation.

  • A third piece of information that gives us one number of an equivalent ratio. Solving the problem often involves finding the other number in the equivalent ratio.

Suppose we are making a large batch of fizzy juice and the recipe says, “Mix 5 cups of cranberry juice with 2 cups of soda water.” We know that the ratio of cranberry juice to soda water is \(5:2\), and that we need 2.5 cups of cranberry juice per cup of soda water.

We still need to know something about the size of the large batch. If we use 16 cups of soda water, what number goes with 16 to make a ratio that is equivalent to \(5:2\)?

To make this large batch taste the same as the original recipe, we would need to use 40 cups of cranberry juice.

cranberry juice (cups) soda water (cups)
5 2
2.5 1
40 16