# Understanding A Gallery Problem

## Lesson Overview

Review the ways classroom habits and routines can strengthen students’ mathematical character. Explain what a Gallery is and how to choose a Gallery problem to solve. Direct students to choose one of three Gallery problems that introduce the unit’s technology resources. The three Gallery problems combine working with ratios and rates with the application resources available with this unit.

# Key Concepts

Students understand that a Gallery gives them a choice of problems to solve. Students think about the features of the problems to use when choosing a problem. Students know how to work on a Gallery problem and present a solution.

# Goals and Learning Objectives

- Know how to choose a problem from a Gallery.

# Task 1: Strengthen Mathematical Character

# Lesson Guide

Explain to students that during the past four lessons, they have used different rituals and routines, such as critiquing partner reasoning, using “ask myself” questions, and applying qualities of effective presentations. Practicing these rituals and routines strengthens students’ mathematical character by providing them with strategies for approaching mathematical situations and for working together to build a mathematical community.

Ask students:

- What other things can you do to be active in class and take responsibility for your own learning?

Have students think about the question for a minute, and then ask them to discuss their ideas with their partner.

If students are having trouble thinking of ideas, suggest that they think about the different parts of class time: working individually, working with partners, Ways of Thinking discussion, and so on. Ask students:

- What kinds of things do you already do to contribute to the class mathematical community?
- Can you use those ideas to think of other things you could do?

Then, as a class, have students share their ideas of ways to strengthen mathematical character. Record each idea and share with the class. When all the ideas have been listed, review each one for clarity.

These Hints provide students with ideas to strengthen their mathematical character when working with others:

- Present ideas to your classmates for their feedback.
- Ask questions of your classmates about their work.
- Write your classmates' successful, innovative strategies in your Notebook.
- Take notes about different ways of thinking about a problem.

These Hints provide students with ideas to strengthen their mathematical character when working on their own:

- Estimate an answer before solving a problem.
- Check if your answer is reasonable.
- Include visual representations of problems and solutions in your work.
- Use complete sentences in the solution.
- Ask yourself questions about your work to improve it.
- Solve the same problem using different strategies.
- Solve the same problem again using different numbers and look for patterns.
- Revise your work.

## Opening

# Strengthen Mathematical Character

Your habits and routines will help you approach mathematical situations and help you work with your classmates to build a mathematical community.

Discuss the following with your classmates.

- What other things can you do to be active in class and take responsibility for your own learning?

## Hint:

Here are ideas to strengthen your mathematical character when working with others:

- Present ideas to your classmates for their feedback.
- Ask your classmates questions about their work.
- Write your classmates' successful, innovative strategies in your Notebook.
- Take notes about different ways of thinking about a problem.

Here are ideas to strengthen your mathematical character when working on your own:

- Estimate an answer before solving a problem.
- Check if your answer is reasonable.
- Include visual representations of problems and solutions in your work.
- Use complete sentences in the solution.
- Ask yourself questions about your work to improve it.
- Solve the same problem using different strategies.
- Solve the same problem again using different numbers and look for patterns.
- Revise your work.

# Task 2: What Is a Gallery?

# Lesson Guide

Explain that most units will have a Gallery of problems for students to choose from and to work on over several days. Students may also work with their teacher to review concepts in the unit or pursue the wonderings that were recorded and shared with the class over the week.

In most units, students will complete a Self Check task before they work on the Gallery problems. The Self Check task provides an assessment of students’ understanding of the unit concepts. Today, students will go straight to the Gallery.

Ask students:

- What things should you consider when you choose which Gallery problem to work on?

Have students think about the question on their own, and then have them discuss their ideas with a partner. Ask students to share their ideas with the class, record each idea, and post/share with the class. When you have listed all the ideas, review each one for clarity.

These Hints provide students with things to consider when choosing a Gallery problem:

- Look at each problem before choosing one.
- Use your understanding of the unit’s math to help you decide on a problem.
- Choose the problem you find most interesting.

## Opening

# What Is a Gallery?

During Gallery lessons, you will be working independently, with a partner, or with your teacher.

- You might be working with your teacher to review concepts introduced in the unit.
- You might be working on a Gallery problem.

Gallery problems give you a chance to work on different kinds of problems.

- You choose which problems to solve.
- You decide when you are ready to share your work.
- You might present your work to the class.

Discuss the following with your classmates.

- What things should you consider when choosing a Gallery problem?

## Hint:

Here are ideas of things to consider when choosing a problem:

- Look at each problem before choosing one.
- Use your understanding of the unit’s math to help you decide on a problem.
- Choose the problem you find most interesting.

# Task 3: Math Mission

# Lesson Guide

Discuss the Math Mission. Students will learn about how the Gallery works.

## Opening

Learn about the Gallery and how it works.

# Task 4: Go to the Gallery

# Lesson Guide

Have each student read the instructions first, and then go to the Gallery lesson (Lesson 6) to choose a Gallery problem to work on. Give students time to work. When they are finished, have students return to Task 5 in this lesson to discuss their work.

# Preparing for Ways of Thinking

Have several students present their work. Make sure the three different types of Gallery problems are represented in the student presentations.

Encourage students to incorporate the mathematics they used as well as information about the unit’s application tools and resources.

# Mathematical Practices

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

- Students spend time understanding the problem before starting their work, and they persevere through difficulty to successfully complete the problem they choose.

## Work Time

# Go to the Gallery

Next you will experience a Gallery.

Read all the instructions before you begin.

- Exit from this lesson.
- Select Lesson 6.
- Choose a Gallery problem to work on.
- When done, save your work.
- If you have time, try another Gallery problem.
- When the class is done, return to this lesson and go to Task 5.

# Task 5: Make Connections

# Lesson Guide

Depending on the class’s level of understanding of ratios and rates, spend more time or less time discussing proportional relationships.

Where appropriate, guide the class discussion toward examples of making sense of real-world problems and persevering in solving them.

## Performance Task

# Ways of Thinking: Make Connections

Take notes about your classmates’ work on the Participating in a Gallery lesson.

## Hint:

As your classmates present about the math tools, ask questions such as:

- What tool did you choose and when is it helpful?

As your classmates present about what they learned about ratios and rates, ask questions such as:

- Are there specific reasons to research a concept?
- What can you find as resources?
- How did you search those resources?

# Task 6: Reflect On Your Work

# Lesson Guide

Have each student write a brief reflection before the end of class. Review the reflections to see how students feel about the opportunity to make choices as to which tasks to work on.

## 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.

**One thing I like about choosing a Gallery problem to work on is...**