Exploring Proportions

Overview

Students build on prior ratio work to understand proportions as equivalent ratio relationships. They then use tables and equations (cross-multiplication reasoning) to find missing values in real-world contexts.

Learning intentions

• Students will be able to explain what a proportion is using ratio language.
• Students will be able to determine whether two ratios form a proportion by finding equivalent ratios.
• Students will be able to complete tables of equivalent ratios with whole-number measurements.
• Students will be able to solve proportion problems to find unknown quantities, using cross-multiplication reasoning.

Success criteria

• I can define a proportion as two equivalent ratios.
• I can show equivalent ratios in a table and decide if two ratios are proportional.
• I can write a proportion equation from a problem situation.
• I can find the missing value correctly and justify my method using cross-multiplication.

Curriculum links

• Ratios and proportional relationships: understand ratio language and compare quantities using equivalent ratios.
• Solve real-world ratio problems using reasoning about tables of equivalent ratios and equations.
• Make tables of equivalent ratios and use them to compare ratios (including missing values).
• Use ratio and rate reasoning to solve problems, including plotting or using coordinate pairs as needed.
• Unit-level reasoning supporting percent context is part of the larger unit (but this lesson focuses on proportions).

Lesson structure (60 minutes)

1. 0–5 min · Activate prior learning. Teacher displays two ratio statements (e.g., 2:3 and 4:6) and asks students to decide “equal or not equal?”; students quick-write their reasoning and share with a partner.
2. 5–12 min · Mini-lesson: What proportions mean. Teacher explains that a proportion is an equation stating two ratios are equivalent (a/b = c/d with b and d not zero), and models how equivalence can be checked using a table of equivalent ratios. Students take a brief note template: “Proportion means the ratios describe the same relationship.”
3. 12–22 min · Guided practice: Tables of equivalent ratios. Teacher gives groups a worksheet set with two ratio pairs and a partial table (whole-number measurements). Students fill missing entries and determine whether the ratios are proportional, showing their multiplication/division steps.
4. 22–35 min · Direct instruction: From ratio to equation (cross-multiplication reasoning). Teacher presents a context problem (e.g., “Wings to beaks is 2:1. If there are 10 wings, how many beaks?”). Teacher writes the proportion as a fraction equation, then demonstrates cross-multiplying to check and solve for the missing value, emphasizing reasoning (“if ratios are equal, cross products must match”). Students practice writing the proportion equation for one similar problem and predict the next step before solving.
5. 35–48 min · Independent problem solving (real-world). Students solve 3 problems independently:

• one where they decide if two ratios form a proportion,
• one where they find a missing value in a proportion,
• one where they use a table to confirm the answer. Teacher circulates, prompting students to explain using ratio language (“for every ___, there are ___”).

6. 48–55 min · Small-group share and correction. Teacher selects 2 student strategies (one fully correct, one common misconception such as adding instead of multiplying). Students discuss “What makes this a proportion?” and revise answers or explanations.
7. 55–60 min · Exit ticket. Students complete one short problem: choose whether two ratios are proportional and find one missing value if they are (with work shown). Teacher collects for quick review.

Resources

• Printed proportion practice worksheet (tables + equation problems)
• Proportion “definition” anchor chart (two equivalent ratios)
• Small whiteboards or student response cards (for quick checks)
• Sticky notes for strategy sharing
• Counters or ratio tiles (optional) for visualization
• Graph paper or coordinate plane sheets (only if needed for extension within the lesson—kept minimal)
• Timer for transitions

Assessment

• Formative check during Step 1: student justifications (equal vs not equal).
• Formative check during Step 3: teacher reviews tables for correct equivalence and missing values.
• Formative check during Step 5: teacher listens for correct proportion language and cross-multiplication reasoning.
• Exit ticket in Step 7: accurate decision and missing value with brief explanation.

Differentiation

• Support: Provide sentence starters for ratio language (“The ratios are proportional because…”) and proportion justification (“Cross-multiply because…”). Offer a partially completed table example on the worksheet.
• Support: Allow students to use multiplication tables or “scale factor” notes (multiply/divide both parts by the same number) before using cross-multiplication.
• Extension: For the final independent problem, ask students to create their own two ratios that are proportional and then find an unknown and verify using a table.
• EAL/SEN: Keep numbers manageable, reduce reading load by underlining key quantities in word problems, and allow oral explanation recorded by the teacher if needed.