Equivalent ratios describe the same relationship using different numbers. In this topic, we focus on how to solve ratio and proportion problems by keeping the relationship between quantities the same. Rather than jumping straight to cross-multiplication, we use visual models that help show what the numbers mean.

One useful model is a strip diagram. A strip diagram shows the quantities as parts of a bar, making it easier to see how the values are grouped, scaled, or compared. Strip diagrams are especially helpful when one quantity, the other quantity, or the total is missing. They help students decide what is known, what is unknown, and how the quantities are related.

We will also use double-number lines to solve ratio and proportion problems. A double-number line shows two related quantities increasing together. For example, if 3 tickets cost $12, a double-number line can show how the number of tickets and the total cost grow at the same rate. This helps students see equivalent ratios as scaled copies of the same relationship.

As future teachers, it is important to help students understand that ratio and proportion problems are about relationships, not just procedures. Strip diagrams and double-number lines help students reason through the problem before choosing a calculation.

 

Student Learning Goals

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

  • Define equivalent ratios.

  • Use strip diagrams to represent ratio and proportion problems.

  • Use double-number lines to represent related quantities.

  • Solve ratio problems when one quantity is unknown.

  • Solve ratio problems when the total is unknown.

  • Solve ratio problems when the total is known but the parts are unknown.

  • Explain how scaling creates equivalent ratios.

  • Compare visual strategies for solving proportion problems.

Key Vocabulary

  • Ratio - A comparison of two quantities.

  • Equivalent Ratios - Ratios that describe the same relationship.

  • Proportion - A statement showing that two ratios are equivalent.

  • Strip Diagram - A bar model used to show related quantities and parts.

  • Double-Number Line - A model with two number lines showing how related quantities grow together.

  • Scale Factor - The number used to multiply quantities to create an equivalent ratio.

  • Total - The whole amount made by combining the parts.

  • Unknown - The missing value in a problem.

Strip Diagrams

  • One part missing - The ratio of red to blue beads is 2:5. There are 10 red beads. How many blue beads are there? - Shows that 2 parts became 10, so each part is worth 5.

  • Total missing - The ratio of cats to dogs is 3:4. There are 12 cats. How many animals are there total? - Shows 3 parts equals 12, so each part is 4, and 7 total parts equals 28.

  • Total known - The ratio of boys to girls is 2:3. There are 25 students total. How many boys and girls are there? - Shows 5 total parts equals 25, so each part is 5.

Teacher Connection

Ratio and proportion problems can be challenging because students often try to apply memorized procedures before understanding the situation. Visual models help students slow down and identify the relationship between the quantities.

Strip diagrams are useful when students need to see parts and totals. Double-number lines are useful when students need to see two quantities growing together. Both models help students understand why equivalent ratios are made by multiplying or dividing both quantities by the same factor.

Helpful teacher questions include:

  • What two quantities are being compared?

  • What does each part represent?

  • Are we given one part, both parts, or the total?

  • What is the value of one ratio unit?

  • How are the quantities scaling?

  • Would a strip diagram or double-number line make this problem clearer?

  • How do you know the ratios are equivalent?

Quick Reflection Question

Why might a strip diagram be more helpful than cross-multiplication when students are first learning to solve ratio and proportion problems?