Author:
Chris Adcock
Subject:
Ratios and Proportions
Material Type:
Lesson Plan
Level:
Middle School
Grade:
6
Provider:
Pearson
Tags:
6th Grade Mathematics, Price, Problem-solving, Size
License:
Creative Commons Attribution Non-Commercial
Language:
English
Media Formats:
Text/HTML

Education Standards

The Relationship Between Size & Price

The Relationship Between Size & Price

Overview

During this two-day lesson, students work with a partner to create and implement a problem-solving plan based on the mathematical concepts of rates, ratios, and proportionality. Students analyze the relationship between different-sized gummy bears to solve problems involving size and price.

Key Concepts

Throughout this unit, students are encouraged to apply the mathematical concepts they have learned over the course of this year to new settings. Help students develop and refine these problem-solving skills:

  • Creating a problem solving plan and implementing their plan systematically
  • Persevering through challenging problems to find solutions
  • Recalling prior knowledge and applying that knowledge to new situations
  • Making connections between previous learning and real-world problems
  • Communicating their approaches with precision and articulating why their strategies and solutions are reasonable
  • Creating efficacy and confidence in solving challenging problems in a real world

Goals and Learning Objectives

  • Create and implement a problem-solving plan.
  • Organize and interpret data presented in a problem situation.
  • Analyze the relationship between two variables.
  • Use ratios.
  • Write and solve proportions.
  • Create rate tables to organize data and make predictions.
  • Use multiple representations—including tables, graphs, and equations—to organize and communicate data.
  • Articulate strategies, thought processes, and approaches to solving a problem, and defend why the solution is reasonable.

Three Bears

Lesson Guide

Have students watch the video.

After watching the video, pose these questions:

  • How many Mini Bears would you have to eat in order to eat the equivalent of one Super Bear? How many Regular Bears would you have to eat?
  • What information would help you solve the problem?
  • Make a prediction about the solution to the problem.

Encourage students to discuss the questions with their partner and to make a prediction about the number of Mini Bears and Regular Bears it would take to equal a Super Bear.

ELL: Some ELLs may not be familiar with gummy bears. Have a sample of gummy bears for them to feel, or provide life-size images of each type of bear so that students can understand the context of the problem.

Opening

Three Bears

Watch the video.

Think about these questions:

  • How many Mini Bears would you have to eat in order to eat the equivalent of 1 Super Bear? How many Regular Bears would you have to eat?
  • What information do you need to solve this problem?
  • Make a prediction about the solution to the problem.

VIDEO: Three Bears

Math Mission

Lesson Guide

Discuss the Math Mission. Students will determine the number of Mini Bears and Regular Bears one would have to eat in order to eat the equivalent of 1 Super Bear.

Opening

Find the number of Mini Bears and Regular Bears you would have to eat in order to eat the equivalent of 1 Super Bear.

Compare Gummy Bears

Lesson Guide

This problem provides you with an opportunity to observe students' problem-solving skills. It gives students a chance to develop and implement a problem-solving plan to arrive at a solution.

Before beginning this problem, review the Problem-Solving Steps 1–5 with students. Encourage students to open their Problem-Solving Steps notes to help them review the process.

Problem-Solving Steps

  1. Understand the situation.

  2. Represent the situation.

  3. Answer questions about the situation.

  4. Check that your answer makes sense.

  5. Prepare a presentation.

Have students watch the two videos. Then give students 3–5 minutes to work solo. During solo time, refrain from giving students hints and pointers or asking guiding questions. Provide students with the opportunity to begin developing their own unique problem-solving plan, as well as time to organize the information presented in the videos. If more than half of the class is struggling with the topic, give a hint such as, “Begin by understanding the problem and organizing the information from the videos.”

After solo work, allow students 10–15 minutes to work with a partner to solve the problems. Have students share their problem-solving plans. Carefully monitor students for understanding by asking probing questions. Modify the amount of time provided based on the class performance. When most of the class has completed the work, move on to Ways of Thinking.

  • What is the problem asking you to find out?
  • What information is given to solve the problem?
  • What are the quantities that vary?
  • How are the variables related?
  • Which variable is independent? Dependent?
  • Can you represent your problem with a table?
  • Can you represent your problem with a graph?
  • Can you represent your problem with an equation?
  • Write your answer as a complete sentence.
  • Support your answer with words, pictures, numbers, diagrams, equations, and so on.
  • Explain why your answer makes sense.

Mathematical Practices

Mathematical Practice 1: Make sense of problems and persevere in solving them.

Students must make sense of the relationship between the number of gummy bears and the weight from the information presented in the videos.

Mathematical Practice 2: Reason abstractly and quantitatively.

Students will abstract the given situation involving the relationship between the number of gummy bears and their weight symbolically using equations, rates, ratios, and proportions to quantitatively contextualize the situation.

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

Students will construct a viable argument for the reasonableness of their solution as they prepare their presentation. They will have the opportunity to critique the reasoning of the presenters to help clarify confusion and correct misconceptions.

Mathematical Practice 4: Model with mathematics.

Students will model the situation through the use of tables, graphs, and equations.

Mathematical Practice 5: Use appropriate tools strategically.

Students will select and use appropriate tools, including calculators, to help them find the number of Mini Bears and Regular Bears it takes to make up one Super Bear.

Mathematical Practice 6: Attend to precision.

Students must be precise when writing and using an equation to solve the problem, when specifying units of measure, when labeling the axes on their graph, and when communicating their reasoning during their presentation.

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

Students will use the structure of a problem-solving plan to help them solve the problems. Through these problems, students will explore the structure of the problem-solving model so they can apply it to other situations throughout this unit.

Interventions

Student has difficulty getting started.

  • Restate the problem to your partner in your own words.
  • What are the quantities and how do they vary?
  • Substitute easier numbers in the problem and solve it. How can you use those same strategies to solve the same problem with more difficult numbers?
  • What problem-solving strategies or tools can you use to help you solve the problem?
  • What information do you know? What are you trying to find?
  • What tools are available to help you get started on the problem?
  • How is the number of Mini Bears related to the weight?

Student has an incorrect solution.

  • How can you use your Problem-Solving Steps notes to confirm that your solution is correct?
  • Is your answer reasonable?
  • How can you use a table to decide if your answer is reasonable?
  • Does the information in your graph match your solution?
  • How can you label the units (types of bears) in the problems to check for reasonableness?

Student has a solution but is having difficulty articulating his or her thinking.

  • How can you describe your strategy to somebody who is struggling?
  • How did you use the information from the videos to solve the problem?
  • How can you write your answer as a complete sentence?

Student has a correct solution.

  • Write a Challenge Problem and trade problems with your partner.
  • What are some common errors you must watch out for when solving these types of problems?
  • Can you use a different tool (e.g., table, graph, or equation) to find the solution?

Possible Answers

  • Solutions will vary. Sample answer:
    10 Mini Bears = 12.1 grams
    1 Super Bear = 2,250 grams
    (2,250 ÷ 12.1) ⋅ 10 = 1,859.5041...
    You would have to eat 1,860 Mini Bears to equal to 1 Super Bear.

    Check: I know that 1,000 Mini Bears would weigh 1,210 grams, and since a Super Bear weighs 2,250 grams, I know the answer should be somewhere between 1,000 and 2,000. The answer should be closer to 2,000, and my answer is in that range.

Work Time

Compare Gummy Bears

Consider the first question: How many Mini Bears would you have to eat in order to eat the equivalent of 1 Super Bear?

Watch the videos.

  • Make note of any information in the videos that can help you solve this question.
  • Then use the problem-solving process to answer the question. You can approach this as a series of steps:
    • Understand the situation.
    • Represent the situation.
    • Answer questions about the situation.
    • Check that your answer makes sense.

VIDEO: Mini Gummy Bears

VIDEO: Super Bear

Hint:

  • Use ratios to help you organize the information from the videos. Be sure to include the appropriate units.
  • How can you use ratios to write and solve proportions?
  • Follow the steps in the problem-solving process to ensure that your solution is complete.

Regular Bears and Super Bears

Lesson Guide

Students apply strategies to solve the problem of how many Regular Bears are equivalent to a Super Bear.

SWD: Students with disabilities may not readily recognize relationships between measurements and weights. Instruct students on the nature of these relationships, perhaps color coding related weights to promote understanding.

Possible Answers

  • Solutions will vary. Sample answer:
    10 Regular Bears = 23.1 grams
    1 Super Bear = 2,250 grams
    (2,250 ÷ 23.1) ⋅ 10 = 974.02597
    You woul have to eat approximately 974 Regular Bears to equal  1 Super Bear.

    Check: I know that 1,000 Regular Bears would weigh 2,310 grams, and since a Super Bear weighs 2,250 grams, it must be slightly less than 1,000 Regular Bears. My answer is slightly less than 1,000, so I know it is reasonable.

Work Time

Regular Bears and Super Bears

Consider the second question: How many Regular Bears would you have to eat in order to eat the equivalent of 1 Super Bear?

Watch the video.

  • Make note of any information in the video that can help you solve this question. What information do you already know about the situation?
  • Then use the problem-solving process to answer the question.

VIDEO: Regular Bears

Prepare a Presentation

Preparing for Ways of Thinking

As pairs work, identify students who use different strategies and have them share during Ways of Thinking. Look for strategies to highlight; look for students who:

  • Write and solve proportions to answer the question.
  • Find a unit rate and write an equation to answer the question.
  • Create a ratio table to answer the question.
  • Create a graph to answer the question.

Work Time

Prepare a Presentation

Prepare a presentation about how you solved the problem. Include any calculations, formulas, graphs, and/or diagrams that you used.

Make Connections

Lesson Guide

Highlight the different strategies students used to solve the problems. Focus on how students created and implemented a problem-solving plan. Encourage classroom discourse regarding the approaches for solving problems and the validity of the answers. Prompt students to give the presenters positive feedback as well as opportunities for improvement. Students should be refining their own strategies, correcting solutions, and taking notes during the presentations.

Ask guiding questions:

  • How did you begin solving this problem? What was the first step?
  • How did you create a problem-solving plan?
  • What connections can you make between this problem and the lightning problems you solved earlier in this unit?
  • How can you use a unit rate to help you find the number of Mini Bears (or Regular Bears) in each Super Bear?
  • How did you organize the information from the videos?
  • What was the most challenging aspect of this problem? How did you overcome this challenge?
  • Which tool did you prefer to use to solve this problem? Why?
  • How do you know your answer is reasonable?

SWD: Students with disabilities may need help seeing the differences and similarities between various solution methods. Find two students that solved the problem using different strategies. Have both students present; then compare and contrast their solutions.

Performance Task

Ways of Thinking: Make Connections

Take notes about the approaches and representations your classmates used to solve the problem.

As your classmates present, think about the problem-solving process. How did the process help you to answer these questions?

Hint:

As your classmates present, ask questions such as:

  • How did you use ratios to organize the information provided in the videos?
  • How can you use the steps from the problem-solving process to ensure that your solution is complete?
  • Can you write a formula to represent this situation?
  • How is this problem similar to the lightning problems? How is it different?
  • What are the variables in the problem? How are they related?
  • What mathematical strategies are the most useful in solving a problem in which you are given a relationship between variables and are asked to make predictions?
  • How did you decide what units to use? Why is it important to include the units?
  • Did you have any errors in your thinking? If so, what were they and how can you correct them?
  • Can you show this relationship graphically? If so, how could you use your graph to find the solution to the problem?

Reflect On Your Work

Lesson Guide

Have each student write a brief reflection before the end of class. Review the reflections to learn how students' predictions of the solution compared to the actual solution.

ELL: The Reflect On Your Work exercise provides opportunities for ELLs to develop literacy in English and proficiency in mathematics. Make sure students use both academic and specialized mathematical language when reflecting on their learning at the end of each session.

Work Time

Reflection

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

Compared to the actual solution to the problem, the prediction I made in the beginning of the lesson was …