# Equivalent Ratios and Proportional Relationships

## Lesson Objectives

- Understand the concept of equivalent ratios
- Generate equivalent ratios
- Use ratio tables to represent equivalent ratios
- Identify proportional relationships in tables and graphs
- Solve real-world problems involving ratios and proportions

## Common Core Standards

6.RP.A.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.

6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

6.RP.A.3a Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.

6.RP.A.3d Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

## Prerequisite Skills

- Understanding of ratios
- Multiplication and division skills

## Key Vocabulary

- Equivalent ratios
- Ratio table
- Scale factor
- Proportional relationship
- Proportion

## Warm-up Activity (5 minutes)

Review the following definitions:

- Ratio:
__https://www.media4math.com/library/22157/asset-preview__ - Visualizing ratios:
__https://www.media4math.com/library/43385/asset-preview__

Show two sets of objects with the same ratio (e.g., 2 apples to 3 oranges, and 4 apples to 6 oranges). Ask students to describe the relationship between the two sets. Encourage them to use ratio language and explain why these ratios are equivalent.

## Teach (25 minutes)

### Definitions

Introduce the following definitions:

- Equivalent ratios:
__https://www.media4math.com/library/43384/asset-preview__ - Visualizing equivalent ratios:
__https://www.media4math.com/library/43386/asset-preview__ - Proportion:
__https://www.media4math.com/library/43387/asset-preview__ - Proportional relationship:
__https://www.media4math.com/library/22139/asset-preview__ - Scale factor:
__https://www.media4math.com/library/43390/asset-preview__

Define equivalent ratios as ratios that represent the same relationship between two quantities.

Connect equivalent ratios to equivalent fractions. For example, the ratio 2:3 can be written as the fraction 2/3. Multiplying both the numerator and the denominator by the same number (e.g., 2) gives 4/6, which is equivalent to 2/3.

Therefore, 2:3 and 4:6 are equivalent ratios.

### Generate Equivalent Ratios

Use this video to introduce the concept of equivalent ratios. In the video are three math examples that involve equivalent ratios.

__https://www.media4math.com/library/1795/asset-preview__

Show that multiplying or dividing both terms of a ratio by the same non-zero number creates an equivalent ratio.

Example:

2:3 = (2×2):(3×2) = 4:6.

Emphasize that this is similar to finding equivalent fractions.

A useful application of equivalent raios is simplifying ratios that include fractions. Use this slide show to demonstrate examples of this technique, but reinforce that this is an application of equivalent fractions:

__https://www.media4math.com/library/slideshow/math-examples-ratios-fractions__

### Deepen Understanding of Ratio Tables

Review the concept of ratio tables introduced in Lesson 1. Explain that ratio tables help organize and compare equivalent ratios systematically.

Use this slide show to explore an application of equivalent fractions and ratio tables in the context of cooking:

__https://www.media4math.com/library/slideshow/applications-equivalent-ratios-cooking__

Demonstrate how to fill in missing values in a ratio table by scaling up or down. Show how to use the table to find equivalent ratios and solve problems.

### Real-World Applications

Show this video, which focuses on ratios with fractions, to continue working with equivalent ratios:

__https://www.media4math.com/library/1792/asset-preview__

### Proportions

Explain that proportional relationships show a constant ratio between two quantities. Introduce the concept of a proportion, which is an equation stating that two ratios are equivalent (e.g., 2/3 = 4/6).

Use this video to introduce proportions:

__https://www.media4math.com/library/1798/asset-preview__

Demonstrate how to recognize proportional relationships in tables (constant ratio) and graphs (straight line through the origin).

## Review (10 minutes)

Use this video to review ratio problems:

__https://www.media4math.com/library/1794/asset-preview__

Guide students through practice exercises:

- Create equivalent ratios for 3:4
- Complete a ratio table for the relationship "for every 2 blue marbles, there are 5 red marbles"
- Identify whether given relationships are proportional using tables and graphs
- Solve a problem involving equivalent ratios in a real-world context
- Convert measurements using ratio reasoning (e.g., convert 12 inches to feet using the ratio 12 inches : 1 foot)

## Assess

Administer a 10-question quiz to evaluate student understanding.

## Quiz

- What is an equivalent ratio to 2:5?

- If 3 pizzas cost $24, how much would 5 pizzas cost in this proportional relationship?

- Complete the ratio table: 2:3, 4:?, 6:9, ?:15

- Is the relationship between x and y proportional if y = 2x + 1?

- What is the constant of proportionality in the ratio 12:3?

- If a recipe calls for 2 cups of flour for every 3 cups of milk, how many cups of milk are needed for 8 cups of flour?

- Which pair of ratios is equivalent: 4:7 and 8:14, or 3:5 and 9:16?

- In a proportional relationship, if x increases by a factor of 3, what happens to y?

- In a fruit basket with a 3:2 ratio of apples to oranges, how many oranges are there if there are 15 apples?

- If the ratio of white to brown eggs in a carton is 5:3, how many white eggs are there if there are 12 brown eggs?

## Answers

- 4:10 (or any ratio where the second number is 2.5 times the first)
- $40
- 2:3, 4:6, 6:9, 10:15
- No (y-intercept is not 0)
- 4
- 12 cups
- 4:7 and 8:14
- y also increases by a factor of 3
- 10 oranges
- 20 white eggs

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