# Using Proportional Reasoning

## Mathematical goals

This lesson unit is intended to help you assess how well students are able to reason proportionally when comparing the relationship between two quantities expressed as unit rates and/or part-to-part ratios. In particular, it will help you assess how well students are able to:

- Describe a ratio relationship between two quantities.
- Compare ratios expressed in different ways.
- Use proportional reasoning to solve a real-world problem.

## Introduction

The lesson unit is structured in the following way:

- Before the lesson, students work individually on an assessment task designed to reveal their current understanding and difficulties. You then review their solutions and create questions for students to consider, in order to improve their work.
- After a whole-class introduction, students work in groups, putting diagrams and descriptions of orange and soda mixtures into strength order. Students then compare their work with their peers.
- Next, in a whole-class discussion, students critique some sample work stating reasons why two mixtures would or wouldn’t taste the same. Students then revise and correct any misplaced cards.
- After a final whole-class discussion, students work individually either on a new assessment task, or return to the original task and try to improve their responses.

## Materials Required

- Each student will need a mini-whiteboard, pen, and eraser, and a copy of
*Mixing Drinks*and*Mixing Drinks (revisited)*. - Each small group of students will need the cut-up
*Card Set: Orange and Soda Mixtures*and*Card Set: Blank Cards*, a sheet of poster paper and a glue stick. - You may wish to have some orange juice and soda for mixing/tasting but this is not essential.

## Time needed

15 minutes before the lesson, a 100-minute lesson (or two 55-minute lessons), and 15 minutes in a follow-up lesson. Timings given are approximate and will depend on the needs of your class.

## Lesson Type

## Mathematical Practices

This lesson involves a range of *mathematical practices* from the standards, with emphasis on:

- MP1: Make sense of problems and persevere in solving them
- MP2: Reason abstractly and quantitatively
- MP3: Construct viable arguments and critique the reasoning of others
- MP4: Model with mathematics
- MP5: Use appropriate tools strategically
- MP6: Attend to precision
- MP7: Look for and make use of structure
- MP8: Look for and express regularity in repeated reasoning

## Mathematical Content Standards

This lesson asks students to select and apply mathematical content from across the grades, including the *content standards:*