# Introduction To Ratio Tables

## Overview

Students use ratio cards to find equivalencies and form partnerships for the week. As a class, students discuss and decide on classroom norms.

Give each student a ratio card. Instruct students to find a classmate whose card has a ratio that is equivalent to theirs. Classmates with equivalent ratios are now partners for the week. With the class, discuss and decide on classroom norms, or rules. Tell students how to access the application they will use this year.

# Key Concepts

Students understand that ratio relationships are multiplicative. They use ratio tables to show ratio relationships.

# Goals and Learning Objectives

- Distinguish between ratio tables and tables that do no show equivalent ratios.
- Understand how ratio tables are used to solve ratio problems.
- Use the basic features of the application.
- Create and understand the classroom norms.
- Use mathematical reasoning to justify an answer.

## Preparation

- Print and cut a Ratio Icebreaker Activity Card for each student in the class. Shuffle the cards before distributing them to students.
- Write on the board or on chart paper:

Find a classmate whose card matches yours.

- Choose a hand signal or phrase for common activities, such as putting technology away and focusing on the teacher.

# Justify Equivalent Ratios

# Lesson Guide

Hand out one Ratio Icebreaker Activity card to each student. Tell students to find another classmate who has a ratio that is equivalent to the ratio on their card. Explain that the two students who have equivalent ratios will be partners.

Check that all students have successfully sent you a note, and project the Justify Equivalent Ratios image. Ask students:

- How did you know that your ratio card was equivalent to your partner’s ratio card?

Use this discussion to assess students’ ability to explain their reasoning. Do not spend much time correcting reasoning errors at this point in the class.

# Possible Answers

- Answers will vary. Possible answers: Students might talk about simplifying the ratio on their card, identifying ratios that are not equivalent, and the usefulness of finding the decimal equivalent of their ratio.
- Answers will vary.
- Both 21:35 and 12:20 are equivalent to 3:5.

Ratio Icebreaker Activity

Partner 1 | Partner 2 |
---|---|

6:9 | Note: Use the “Partner 1” Icebreaker Card at left only if you have an odd number of students in your class; it will form a group of three with the pair of cards below. If you have an even number of students, do not cut out or use the card at left. |

14:21 | 10:15 |

12:30 | 10:25 |

14:49 | 6:21 |

21:35 | 12:20 |

Partner 1 | Partner 2 |

--- | --- |

15:35 | 21:49 |

12:32 | 27:72 |

36:45 | 20:25 |

12:21 | 24:42 |

15:18 | 30:36 |

Partner 1 | Partner 2 |

--- | --- |

15:21 | 10:14 |

25:40 | 40:64 |

18:21 | 30:35 |

21:24 | 49:56 |

24:16 | 6:4 |

Partner 1 | Partner 2 |

--- | --- |

15:6 | 35:14 |

28:8 | 42:12 |

10:6 | 30:18 |

21:9 | 14:6 |

48:18 | 24:9 |

Partner 1 | Partner 2 |

--- | --- |

10:8 | 30:24 |

42:24 | 28:16 |

30:25 | 18:15 |

42:30 | 14:10 |

16:10 | 72:45 |

## Opening

# Justify Equivalent Ratios

Discuss the following with your classmates.

- How did you know that your ratio card was equivalent to your partner’s ratio card?
- What is another ratio that is also equivalent to your ratio cards?
- What is another ratio that is also equivalent to the two ratios shown in the image?

# Classroom Norms

# Lesson Guide

Explain to students that classroom norms help them to do their best work and meet their learning goals. Today students will spend some time answering these questions as a class:

- What do you need to do your best work?
- What ways can you work with your classmates to make the class an effective group?
- What classroom norms, or rules, will allow each member of the class the opportunity to grow?

Have students spend a few minutes thinking about these three questions on their own. Then have students talk about their ideas with their partner. After a few minutes, have students share their ideas with the class. It is best for students to first think of the ideas themselves—they will take more ownership of the ideas as a result. Record the students’ ideas for classroom norms on the board.

These Hints for students are ideas for classroom norms:

- Ask questions and think creatively.
- Talk about your ideas and questions with your classmates.
- Use mathematics vocabulary when you talk or write about math.
- Make connections between what you know and what you are learning.
- If you do not know how to solve a problem, write what you do know.
- Think about what you are learning from a task, instead of just trying to finish it quickly.
- Think of a mistake as a chance to learn, not as something to hide.
- Check your work and think about what caused you to make a mistake. Learn how to correct your work.
- Work together with your classmates so that everyone learns.

Tell students about “6-inch voices and sound levels.” This classroom norm means that the volume of students’ voices and technology should travel less than 6 inches. Practice this norm a few times and check with students about how well it is working. This norm is helpful to keep the classroom noise level at an acceptable volume during partner work, especially if students watch Opening videos.

After all ideas about classroom norms are listed, review each idea for clarity and class agreement. Then record the final list and project the list for the class. Focus students’ attention on the classroom norms instead of other aspects of the technology. You will introduce some application features during the beginning of Work Time.

To help the class run efficiently, demonstrate your class’s technology routines. For example, show students that clapping your hands is a signal for students to stop their work and to focus attention on the teacher.

# Possible Answers

- Answers will vary.
- Answers will vary.
- Answers will vary. The class will create their own list of classroom norms together.

## Opening

# Classroom Norms

Classroom norms help us do our best work and meet our learning goals.

Discuss the following with your classmates.

- What do you need to do your best work?
- What ways can you work with your classmates to make the class an effective group?
- What classroom norms, or rules, will allow each member of the class the opportunity to grow?

## Hint:

Here are ideas for classroom norms:

- Ask questions and think creatively.
- Talk about your ideas and questions with your classmates.
- Use mathematics vocabulary when you talk or write about math.
- Make connections between what you know and what you are learning.
- If you do not know how to solve a problem, write down what you do know.
- Think about what you are learning from a task, instead of just trying to finish it quickly.
- Think of a mistake as a chance to learn, not as something to hide.
- Check your work and think about what caused you to make a mistake. Learn how to correct your work.
- Work together with your classmates so that everyone learns.
- Be prepared to explain your thinking about your approach and mathematics.

# Math Mission

# Lesson Guide

Discuss the Math Mission. Students will detect mistakes in ratio tables.

## Opening

Detect mistakes in ratio tables.

# Compare Ratio Tables

# Lesson Guide

Ask one student to read all parts of the problem aloud. Check that students understand the problem is asking them to check Lucy and Sophie’s work and describe their reasoning. While discussing the problem, focus on reading comprehension and language issues rather than on strategies and solutions.

Spend some time discussing with students the most helpful ways to react to incorrect work:

- What is going to be most helpful when you notice an error?
- Would laughing be helpful to the student?
- What about bragging that you know the answer?
- What about describing your reasons for disagreeing with the answer?

Emphasize that it is important for students to notice their mistakes and be accepting of them, because students can learn from them. Tell students not to be hesitant about sharing their mistakes with the whole class—when a student shares mistakes, everyone learns.

Then demonstrate how you will share information with the class. Students will see the list of classroom norms the class agreed on.

Have students start working on the problem on their own. After a few minutes, have them check in with their partner to complete their work.

# Mathematical Practices

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

Students think about Lucy’s and Sophie’s work and compare the two different ratio tables used to solve a problem. Students identify mistakes to better understand the importance of preserving the ratio relationship in their ratio tables.

# Interventions

**Student has difficulty with the technology.**

- Ask [student name] to help you figure out how to use the technology.

**Student is confused about how Lucy and Sophie could create different ratio tables.**

- Remember ratio tables show several equivalent ratios.
- How can you check that these ratios are equivalent?

**Student can’t describe the steps that Lucy used.**

- What calculation did Lucy use to change the ratio 17 to 4 to the ratio 170 to 40?
- Do you know the calculations used to make the rest of the ratio table?
- Did you notice any mistakes in her calculations?

**Student can’t describe the steps that Sophie used.**

- What calculation did Sophie use to change the ratio 17 to 4 to the ratio 4.25 to 1?
- Do you know the calculations used to make the rest of the ratio table?
- Did you notice any mistakes in her calculations?

**Student does not notice Sophie’s mistake.**

- Look closely at all the table entries. Can you see a calculation that relates it to the original ratio or another equivalent ratio in the table?
- When you add the same number to both rows, does it result in an equivalent ratio? Explain why.

# Possible Answers

- It will cost Mr. Washington $136 to buy brushes for all his students. Use either Lucy’s table or a direct solution such as the following:

$174 brushes = $4.25 per brush; $4.25 per brush × 32 brushes = $136 - Answers will vary. Possible answer: Lucy started with $17 for 4 brushes and multiplied both amounts by 10. After entering the result in the column to the right, she also multiplied 17 and 4 by 2 and entered the result in the next column. She then found the difference between the two new columns, giving her the result of $136 for 32 brushes.

Sophie started by finding the unit price for 1 brush as $4.25 by dividing both 17 and 4 by 4. She then found the price for 10 brushes by shifting the decimal one place to the right, which multiplies them by 10. To find the price of 30 brushes, she multiplied the previous column by 3. For 32 brushes, she needed to add the cost of 2 brushes to the price (e.g., multiplying $4.25 by 2), but she made the mistake of simply adding 2 to both rows in the column. - Sophie made a mistake where she incorrectly added 2 to $127.50 and 30. Sophie could check her answer by dividing the price by 32.

# Challenge Problem

## Answer

- Mr. Washington can buy 56 brushes. Students may use a ratio table to figure out the answer, but allow students to use other thinking tools as appropriate.

## Work Time

# Compare Ratio Tables

Read the problem. Copy the chart into your Notebook.

Mr. Washington wants to purchase a new paint brush for each of his 32 students. The brushes are in packs of 4. Each pack of 4 brushes sells for $17.

Lucy and Sophie made different ratio tables to solve the problem.

- How much will it cost Mr. Washington to buy paint brushes for all his students?
- Describe the steps that Lucy and Sophie used.
- Explain any mistakes you notice. Write advice so the student does not make the same type of mistake next time.

# Challenge Problem

- How many paint brushes can Mr. Washington buy for $238?

## Hint:

How do you know the steps that Lucy and Sophie used?

What operation is done to the quantities in one column to get the quantities in the next column?

Why is it an error to add 2 to both quantities in a column to get the quantities in the next column?

# Reflect on Your Work

# Lesson Guide

Explain to students that they will be writing a brief reflection at the end of most classes. They will be given sentences they can complete, or they can write something else about their learning today.

Demonstrate how students will share their work with the teacher. Tell students what they should do with the technology at the end of the lesson and how they should begin the next lesson.

Review the reflections to see the confusion students still have about ratio tables. Share any confusions with the class and discuss.

## Work Time

# Reflection

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

**Something that still confuses me about ratio tables is…**