Before students formally learn the word ratio, they often compare quantities in everyday ways. They might compare the number of boys and girls in a class, cups of water and scoops of mix in a recipe, red blocks and blue blocks in a pattern, or miles traveled and hours driven. In each case, the important idea is that we are comparing two quantities at the same time.

In this topic, we begin by exploring these comparisons informally. Students use drawings, tables, and repeated groups to decide whether two comparisons describe the same relationship. For example, if one mixture uses 2 cups of juice for every 3 cups of water, then a mixture with 4 cups of juice and 6 cups of water has the same relationship because both quantities were doubled.

After students reason with examples, we introduce the word ratio. A ratio is a comparison between two quantities. Ratios can describe part-to-part comparisons, such as red blocks to blue blocks, or part-to-whole comparisons, such as red blocks to total blocks. Understanding ratios helps students build toward proportions, scaling, percent, similarity, and many real-world applications.

As future teachers, it is important to help students see ratios as relationships, not just pairs of numbers. The numbers matter because of how they compare to each other.

 

Student Learning Goals

By the end of this topic, students should be able to:

  • Compare two quantities using words, drawings, and tables.

  • Recognize when two quantity comparisons describe the same relationship.

  • Use scaling to create equivalent comparisons.

  • Define a ratio as a comparison of two quantities.

  • Distinguish between part-to-part and part-to-whole comparisons.

  • Explain why equivalent ratios represent the same relationship.

  • Use student-friendly language to describe ratio situations.

Key Vocabulary

  • Compare - To describe how two quantities are related.

  • Quantity - An amount of something.

  • Ratio - A comparison of two quantities.

  • Equivalent Ratios - Ratios that describe the same relationship.

  • Part-to-Part Ratio - A comparison between two parts of a group.

  • Part-to-Whole Ratio - A comparison between one part and the total group.

  • Scaling - Multiplying both quantities by the same number to keep the relationship the same.

  • Table - An organized way to show related quantities.

Teacher Connection

Ratios are often difficult because students may try to use addition instead of multiplication. For example, if 2 red blocks and 3 blue blocks are in one group, students may think 3 red blocks and 4 blue blocks is equivalent because both numbers increased by 1. But the relationship changed. Equivalent ratios come from multiplying or dividing both quantities by the same factor.

As a future teacher, it is helpful to begin with concrete examples before introducing formal notation. Drawings, tables, recipes, color mixtures, and pattern blocks can all help students understand that ratios compare quantities in a structured way.

Helpful teacher questions include:

  • What two quantities are being compared?

  • What does each number represent?

  • Is this a part-to-part or part-to-whole comparison?

  • How did both quantities change?

  • Were both quantities multiplied by the same number?

  • Does the relationship stay the same?

  • How could we show this with a drawing or table?

Quick Reflection Question

Why might students think that 2 to 3 and 3 to 4 are equivalent comparisons? How could a drawing or table help show that the relationship has changed?