# Diagrams & Problem Solving Strategies

## Overview

# Lesson Overview

Students revise their packing plans based on teacher feedback and then take a quiz.

Students will use their knowledge of volume, area, and linear measurements to solve problems. They will draw diagrams to help them solve a problem and track and review their choice of problem-solving strategies.

# Key Concepts

Concepts from previous lessons are integrated into this assessment task: finding the volume of rectangular prisms. Students apply their knowledge, review their work, and make revisions based on feedback from the teacher and their peers. This process creates a deeper understanding of the concepts.

# Goals and Learning Objectives

- Apply your knowledge of the volume of rectangular prisms.
- Track and review your choice of strategy when problem-solving.

# Critique

# Lesson Guide

Return students' solutions to the Self Check task. If you have not added questions to individual pieces of work, write your list of questions on the board now. Students can then select questions appropriate to their own work.

Give students a few minutes to read over the feedback.

SWD: Students with disabilities may have a more challenging time identifying areas of improvement to target from the Intervention questions. Model explicitly for students (using an example project or student sample) how to self-assess areas of growth by using the appropriate Interventions in order to plan for revisions.

## Opening

# Critique

Review your work on the Self Check problem and think about the following questions.

- Can you make a diagram of your packing plan?
- Do you think there’s a way you could pack the truck to get even more bananas in?
- What reasons can you give for saying that your packing plan is the best method for packing the most bananas?

# Math Mission

# Lesson Guide

Discuss the Math Mission. Students will review their work and make revisions to find the best packing plan.

## Opening

Review your work based on the feedback to find the best packing plan.

# Revise Your Work

# Lesson Guide

Put students in pairs to revise their work. Encourage students to incorporate ideas from their partners into their work.

Support student problem solving.

- Try not to make suggestions that move students toward a particular approach to this task. Instead, ask questions that help students to clarify their thinking.
- If students find it difficult to get started, these questions might be useful:
- What questions were you asked in the feedback?
- How could you and your partner work together to address one of those feedback questions?

If several students in the class are struggling with the same issue, you could write a relevant question on the board. You might also ask a student who has performed well on a particular part of the task to help a struggling student.

SWD: Make sure that all students have access to and can comprehend the information in the Interventions so that they can accurately interpret your assessment of their work.

Students with disabilities may benefit from support in understanding the expectations. Allow multiple means of representing the information in the Interventions (visual presentation of text, TTS, visual supports, etc.).

Create and provide an enhanced version of the Interventions with embedded text structures (labels, highlights, words in bold) to cue students to pay closer attention to particular terms.

ELL: Student partner work provides students with opportunities to negotiate meaning, share ideas, and communicate with others during problem solving. This type of language-rich environment encourages students to work together and learn from one another, which is essential for ELLs' learning.

# Mathematical Practices

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

Students must make sense of the task. They must plan and carry out a solution strategy. After they complete the task, they must go back and check that their solution makes sense. Students also have the opportunity to work with another student and try to understand his or her approach to the task.

**Mathematical Practice 6: Attend to precision.**

Students are asked to present their work in a clear, precise way. They should try to use correct mathematical language and symbols and to label their work carefully so it is easy to follow.

# Interventions

**Student has difficulty getting started.**

- What feedback did you get?
- How can you use the feedback to revise your work?

**Student works unsystematically.**

- How can you check that you addressed all the feedback?

**Student presents his or her work poorly.**

- Is your work clear?
- Have you given enough explanation?

**Student has a correct solution.**

- Can you find a different packing plan?

# Possible Answers

- Pack the boxes in the crate so that the 75 cm sides of the boxes are parallel to the 1.5 m side of the crate, and the 40 cm sides of the boxes are parallel to the 2.4 m side of the crate. This makes a 2 box by 6 box by 3 box prism, for a total of 2 ⋅ 6 ⋅ 3, or 36 boxes. The boxes will fit perfectly along the 2.4 m and 1.5 m sides of the crate, and there will be only a 5 cm gap along the other side.
- Pack the crates in the truck so that the 1.5 m sides are parallel to the 3 m side of the truck, and the 2.4 m sides of the crates are parallel to the 2.5 m side of the truck. This makes a 2 crate by 1 crate by 7 crate prism, for a total of 2 ⋅ 1 ⋅ 7, or 14 crates. In all, there will be 14 ⋅ 36, or 504 boxes of bananas in the truck.
- The crates inside the truck form a 2.4 m by 3 m by 9.8 m prism, which has a volume of 70.56 m
^{3}. The truck bed is a 2.5 m by 3 m by 10 m prism, which has a volume of 75 m^{3}. Only 4.44 m^{3}of the truck will be empty. - Explanations will vary. Possible answer: This is the best plan because it minimizes the amount of unused space. Other ways of packing the crates (as listed below) will leave bigger gaps, and you will not be able to ship the maximum amount of product in each crate or truck:

- Put the 2.4 m sides of the crates parallel to the 2.5 m side of the truck and the 1.5 m sides of the crates parallel to the 10 m side of the truck. This would fit a 2 crate by 1 crate by 6 crate prism, for a total of only 12 crates.
- Put the 1.4 m sides of the crates parallel to the 2.5 m side of the truck. Only 1 crate would fit in this direction, leaving a 1.1 m by 10 m by 3 m gap, which is 33 m
^{3}. - Put the 1.5 m sides of the crates parallel to the 2.5 m side of the truck. Only 1 crate will fit in this direction, leaving a 1 m by 10 m by 3 m gap, which is 30 m
^{3}, plus a 0.2 m by 1.5 m by 10 m gap on the top, which is 3 m^{3}, for a total of 33 m^{3}.

## Work Time

# Revise Your Work

Work with your partner to revise your work on the Self Check problem based on the questions from the Opening and feedback from your partner and teacher.

Self Check Problem:

You are working for a trucking company that transports bananas. You want to transport the greatest number of bananas in each truckload. The bananas fill boxes, which are packed into crates. The crates are then packed into a truck.

- How can you pack each crate to fit the greatest number of boxes of bananas?
- How can you pack the truck to fit the greatest number of crates?
- Based on your packing plan, how much space in the truck will not be filled?
- Why is your packing plan the best possible plan?

# Prepare a Presentation

# Preparing for Ways of Thinking

While students work with their partners, note different student approaches to the task.

- How do they organize their work?
- Do they notice if they have chosen a strategy that does not seem to be productive? If so, what do they do?

## Work Time

# Prepare a Presentation

- Describe the feedback that you received from your teacher and your partner.
- Explain how you revised your work based on that feedback.

# Make Connections

# Lesson Guide

Organize a whole class discussion to consider issues arising from the work students did to revise their work. You may not have time to address all these issues, so focus discussion on the issues most important to your students.

# Mathematics

Have students give their presentations. Include students whose strategies did not work so they can talk about how and when they realized their strategy did not work and what they did about it. Have students share the feedback they received and how they addressed those comments. Have students ask questions and make observations as they view each other's presentations.

Ask questions such as the following:

- What was the most difficult part of this task? How did you approach finding a solution?
- Did you and your partner ever disagree about the best packing plan? Did you resolve your differences of opinion? How?

SWD: Some students with disabilities may struggle with self-assessment; use your knowledge of student strengths and vulnerabilities to inform and create interventions you will put into place for this period of class time.

## Performance Task

# Ways of Thinking: Make Connections

- Take notes as your classmates present their work and discuss how they revised it.

## Hint:

As your classmates present, ask questions such as:

- Can you explain your strategy for solving the problem?
- Did you revise your work or your explanation based on any of the questions in the Opening? How?
- What part of the problem was the most difficult for you? Why?
- Was it difficult for you to revise your work? In what ways?

# Reflect On Your Work

# Lesson Guide

Have each student write a brief reflection before the end of the class. Review the reflections to find out what students would do differently if they started the Self Check task over again.

ELL: Asking students to reflect on their learning 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. In addition, make sure students use their rubric to help them identify areas of improvement.

## Work Time

# Reflect On Your Work

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

**What I would do differently if I started the Self Check problem now is …**