# Refining Problem Solving Skills

## Overview

Students critique the diagrams of other students from the previous lesson and receive feedback about their own diagrams. Students revise their diagrams from the first part of the lesson based on the feedback they receive.

# Key Concepts

Students are expected to use the mathematical skills they have acquired in previous lessons or in previous math courses. The lessons in this unit focus on developing and refining problem-solving skills. Students will:

- Try a variety of strategies to approaching different types of problems.
- Devise a problem-solving plan and implement their plan systematically.
- Become aware that problems can be solved in more than one way.
- See the value of approaching problems in a systematic manner.
- Communicate their approaches with precision and articulate why their strategies and solutions are reasonable.
- Make connections between previous learning and real-world problems.
- Create efficacy and confidence in solving challenging problems in a real-world setting.

# Goals and Learning Objectives

- Read and interpret maps, graphs, and diagrams.
- Solve problems that involve linear measurement.
- Estimate length.
- Critique a diagram.

SWD: Some students with disabilities will benefit from a preview of the goals in each lesson. Students can highlight the critical features and/or concepts and will help them to pay close attention to salient information. Students need to know their goal is to develop and refine their problem solving skills.

# Math Mission

# Lesson Guide

Discuss the Math Mission. Students will critique their classmates' diagrams of a ship's journey through the Welland Canal and revise their own diagram.

SWD: Model for students how to give constructive feedback to their peers. They should comment on one thing that their peer did well and one specific area that could be improved. Remind them to be specific, objective, and kind with their comments.

## Opening

# Math Mission

Critique diagrams of a ship’s journey through the Welland Canal, and revise your own diagram.

# Critique and Revise

# Lesson Guide

Begin the lesson by asking students:

What do you think is a strong point of your diagram?

Give students an opportunity to share a strong point of their diagram with their partner. Have partners trade diagrams with another pair of students and critique their diagrams. Students should analyze the diagrams for clarity, reasonableness, and accuracy.

Guide students to ask themselves the following questions as they critique the diagrams:

- Is the diagram understandable?
- Are there ways the diagram could show the information more clearly?
- Is there anything in the diagram that is not correct?

After students have critiqued the diagrams, have them refine their own diagrams independently based on the critique of others. While students are critiquing and revising their diagrams, circulate around the room asking guiding questions and looking for strategies and thought processes to highlight during the class discussion.

ELL: Show these questions in writing and provide sentence frames to support ELLs such as the following:

Post these sentence frames as a reference for future lessons.

- “I think the results were....(very, somewhat, relatively, extremely)....accurate because....”
- “The image on the screen is different/similar from the actual drawing or model in that.....”
- “It shows...”
- “This diagram makes sense to me because.....”
- “The structure of the mathematics was brought up by.....”

# Mathematical Practices

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

- Students provide focused, specific critiques on diagrams created by their classmates to help them refine their work.

**Mathematical Practice 4: Model with mathematics.**

- Students model the mathematics through the creation of their diagrams.

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

- Students attend to precision while refining their diagrams ensuring that their revisions depict the correct data in a way that others can comprehend.

# Interventions

**Student has difficulty getting started.**

- What are the strengths and weaknesses of the diagram you are critiquing?
- How can you improve upon your diagram?
- What items are essential in every diagram?

**Student has an answer, but it is incorrect.**

- Does your diagram depict the correct information?
- Can others understand your diagram?

**Student has an answer but is having difficulty articulating their reasoning.**

- How did you improve upon your diagram?
- How did the critique of your diagram help guide your thinking when revising it?
- What were some mistakes you found in your diagram?

**Student has the correct answer and is waiting on others to finish.**

- What other diagrams can you create to show the journey through the canal?
- Compare and contrast your diagram to the diagram shown on the tablet. How do they compare?

# Possible Answers

- Answers will vary.

## Work Time

Trade diagrams and critique each other’s work. Use the following questions to help guide your critique of another student's work.

- Is the diagram understandable?
- Are there ways that the diagram could show the information more clearly?
- Is there anything in the diagram that is not correct?

# Diagram of Welland Canal

# Lesson Guide

Guide students to ask themselves the following questions as they critique the diagram:

- Is the diagram understandable?
- Are there ways the diagram could show the information more clearly?
- Is there anything in the diagram that is not correct?

# Possible Answers

- Answers will vary. Possible answer: The diagram shows the vertical distance gained and the horizontal distance traveled for a ship traveling from Lake Ontario to Lake Erie through the locks along the Welland Canal.

## Work Time

# Diagram of Welland Canal

When you have finished critiquing your partner’s diagram, look at this diagram of the Welland Canal.

- What does the diagram show?
- Critique the diagram.
- What does the title of the given diagram tell you?
- What do the numbers along the vertical axis represent?
- What do the numbers along the horizontal axis represent?
- What does 0 mean?

# Revise Your Work

# Lesson Guide

After students have critiqued the diagrams, have them refine their own diagrams independently based on the critique of others.

While students are critiquing and revising their diagrams, circulate around the room asking guiding questions and looking for strategies and thought processes to highlight during the class discussion.

ELL: For this task, encourage students to explain their ideas to one another. Math language must be used. Encourage the use of English without discouraging students from using their first language(s). Students may have questions when they are making revisions for their diagrams. Make sure students are expressing the horizontal and vertical distances accurately for the Welland Canal.

# Possible Answers

- Revisions will vary.

## Work Time

# Revise Your Work

- Revise your diagram based on your partner’s feedback and on any ideas you may have from looking at other diagrams of the Welland Canal.

# Prepare a Presentation

# Preparing for Ways of Thinking

As students give feedback and revise their diagrams, look for students who:

- Excel at providing specific, useful feedback in their critiques. (Ask students to replay this discussion for the class later.)
- Attend to precision while revising their diagrams.
- Students who get stuck trying to revise their diagrams according to the feedback they received.

# Challenge Problem

## Answers

- Answers will vary.

## Work Time

# Prepare a Presentation

- Prepare a presentation that explains the wonder of the the Welland Canal mathematically.

# Challenge Problem

- Critique the diagram of a different classmate. Take notes.

# Make Connections

# Lesson Guide

Have students share their work on their revisions.

Ask guiding questions, for example:

- How did the critique help you refine your diagram?
- What was helpful about analyzing the diagrams of your classmates?
- How did your diagram compare with the diagram of the Welland Canal shown in task 3?
- How did you label your horizontal and vertical axes? Why was this important?

As a class, critique the diagram of the Welland Canal shown in task 3. What are the strengths and weaknesses of this diagram? Ask students to compare and contrast their diagram with the one provided.

Allow students who gave focused feedback the opportunity to “replay” their critique to the class. Have students share their revisions and provide a critique as an entire class.

ELL: As ELLs explain their reasons verbally and in writing, their answers may have language errors. Remember, language mistakes are natural.Focus on the content being communicated

- Allow processing time
- Do not emphasize grammar
- Model standard English

## Performance Task

# Ways of thinking: Make Connections

- Take notes about other classmates’ approaches to revising their diagrams.

As your classmates present, ask questions such as:

- What was the most helpful advice your partner gave you when critiquing your diagram? How did you use this advice to help you improve your diagram?
- What elements were lacking in your original diagram? How did you incorporate these elements into your new diagram?
- What is a strength of your diagram?
- How might you still improve your diagram?
- What advice would you give to help others improve their diagrams?

# Reflect on Your Work

# Lesson Guide

Have each student write a brief reflection before the end of class. Review the reflections to find out what students learned from looking at other students' diagrams and what revisions they made to their own diagrams.

## Work Time

# Reflect on Your Work

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

**From looking at other diagrams, I learned…**

**A revision I made to my diagram is…**