Identifying and Finding Equivalent Ratios

Introduction

This lesson plan will introduce your learner to equivalent ratios by exploring how ratios can be expressed differently while representing the same relationship. By the end of the lesson, your learner will know several strategies for identifying and finding equivalent ratios.

Before beginning the lesson, your learner should be able to recognize multiples and factors of numbers.

Key Concepts

Here are a few concepts that are important to understand about equivalent ratios:

• Ratio: A ratio represents the relationship between two quantities and is typically expressed as a fraction, with one quantity over the other. For example, a ratio of oranges to apples in a fruit bowl could be represented as 2/5.
• Ratio Vocabulary: Ratio vocabulary includes terms such as "per," "each," "for every," and others, which are used to describe the relationship between the quantities being compared in a ratio. They can help students identify ratios in problems.
• Equivalent Ratios: Equivalent ratios are ratios that have the same value but may have different numbers. They represent the same relationship between quantities but are expressed differently. For instance, the ratios 2/5, 4/10, and 6/15 are all equivalent because they represent the same relationship between oranges and apples.
• Identifying Equivalent Ratios: Students can identify equivalent ratios with visual tools such as tables, tape diagrams, and arrays. With practice, they will learn to apply their understanding of multiples and equivalent fractions.

Teaching Plan

The following activities will help your learner become confident with identifying and finding equivalent ratios. Remember to go at a pace that is comfortable for your learner.

Examples and visuals to support the lesson:

1. Adjusting Quantities in a Recipe

This activity will introduce your learner to the concept of equivalent ratios by exploring various adjustments to a recipe.

• Start by presenting a scenario involving ratios, such as ingredients for a recipe, and ask your learner to determine how the quantities change when the recipe is doubled, tripled, and so forth. Encourage them to explore different strategies, such as drawing pictures or counting objects.
• Next, have your learner organize the results in a table, detailing the quantities of each ingredient and the total number of ingredients. Encourage them to identify any patterns they observe, such as the new quantities being multiples of the original quantities with each recipe change.
• If your learner is familiar with using arrays for multiplication, suggest creating an array to visually represent how the quantities change. The array can help them recognize the multiplication patterns formed by equivalent ratios. Tape diagrams can also be used to illustrate these patterns.

Skill Check
I can use pictures, tables, and other models to find multiples of ratios.

2. Finding Equivalent Ratios

After your learner has identified patterns in the recipe example, guide them in forming a concrete definition of equivalent ratios.

• Ask them to write ratios depicting the relationships between the ingredients. For instance, if the recipe calls for 2 oranges and 5 apples, they can express the ratio of oranges to apples as 2/5, 4/10, 6/15, and so on. Encourage them to represent the ratios as fractions, which will help in future calculations.
• If they haven't already noticed, highlight that all the ratios are equal to each other, hence termed equivalent ratios. You can add "equivalent ratio" to their vocabulary list or word wall.
• Have them extend the list of equivalent ratios for the recipe example. Once they demonstrate that they can effectively find equivalent ratios, have them compile a list of various other equivalent ratios for the recipe. This can include part-to-total ratios like oranges to the total fruits.
• Allow your learner to discover equivalent ratios using their preferred method, whether through drawings, concrete objects, or multiplication. For instance, to derive a ratio equivalent to 2/7, they can multiply both parts of the ratio by 5, resulting in 10/35. As long as they multiply (or divide) both parts of the ratio by the same number, the new ratio remains equivalent.

Skill Check
I can define and recognize equivalent ratios. I can use multiplication to find equivalent ratios.

3. Using Equivalent Ratios to Solve Problems

After your learner has demonstrated comprehension of equivalent ratios and can find them on their own, they can apply their knowledge to problem-solving.

• Present them with word problems featuring ratios in various contexts, such as comparing the weekly allowance of two siblings or the colors of beads on a necklace.
• Make sure they carefully read the word problems. Encourage them to sketch a picture or diagram to model the ratios if one isn't provided. This will show that they understand the context of the problem.
• Next, have them write the ratios as fractions or organize them in a table. Observe their problem-solving strategies and encourage them to explain their methods.
• Note that initially, they may prefer extending their drawings or diagrams to find equivalent ratios. However, the ultimate goal is for them to proficiently write the ratios as fractions and use multiplication or division to find equivalent fractions.

Skill Check
I can use equivalent ratios to solve problems.

Summary

In this lesson plan, your learner explored equivalent ratios through a practical scenario involving recipes. They investigated how quantities changed when recipes were doubled, tripled, or adjusted in other ways. Through various activities, they identified patterns and developed an understanding of equivalent ratios. They then applied this knowledge to solve word problems, enhancing their problem-solving skills and mathematical reasoning.

Teaching Plan adapted from Utah Middle School Math Project under CC BY 4.0.