Proportional or Not?

Overview

Students determine whether two quantities form a proportional relationship by testing equivalent ratios in tables and examining graphs. They build on prior work with ratios and unit rates, then justify conclusions using the constant of proportionality and the origin.

Learning intentions

• Students will be able to recognize proportional relationships in real-world situations.
• Students will be able to test whether ratios in a table are equivalent.
• Students will be able to identify whether a graph is a straight line through the origin.
• Students will be able to explain their reasoning using mathematical evidence.

Success criteria

• I can compare ratios in a table to decide whether they are equivalent.
• I can identify a proportional graph as a straight line through the origin.
• I can explain what the origin and another point mean in the situation.
• I can justify my conclusion with calculations or graph evidence.

Curriculum links

• Recognize and represent proportional relationships between quantities.
• Decide whether two quantities are proportional by testing equivalent ratios in a table.
• Decide whether two quantities are proportional by graphing and checking for a straight line through the origin.
• Explain points on a graph of a proportional relationship, including the origin and the unit-rate point.

Lesson structure (60 minutes)

1. 0–7 min · Do Now and hook. Teacher writes two situations on the board: “3 notebooks cost $6” and “5 notebooks cost $12,” then asks, “Do these represent the same price per notebook?” Students calculate or estimate each unit rate, record a decision, and briefly share their reasoning with a partner.

2. 7–17 min · Direct instruction. Teacher models how to test a ratio table by dividing corresponding quantities, emphasizing that a proportional relationship has one constant unit rate. Model a table such as (x: 1, 2, 4, 6) and (y: 3, 6, 12, 18), then contrast it with (y: 3, 6, 13, 18). Students identify the unit rate and explain why the second table fails the test.

3. 17–30 min · Partner investigation. Teacher places students in pairs and distributes the ratio table and double number line mat for students to use as a visual organizer. Provide each pair with two tables on the board: A: (x: 1, 2, 3, 5); (y: 4, 8, 12, 20) B: (x: 1, 2, 3, 5); (y: 4, 8, 13, 20). Students complete the organizer, calculate (y/x) for each row, classify each relationship as proportional or nonproportional, and write one evidence-based sentence.

4. 30–42 min · Graphing connection. Teacher models plotting ((0,0), (1,3), (2,6), (3,9)), highlighting that the points form a straight line through the origin. Then model a graph with points ((1,3), (2,6), (3,8)), explaining that it does not represent a proportional relationship. Students sketch both examples on graph paper, label the axes, and identify which graph is proportional. Discuss that ((1,r)) represents the unit rate and ((0,0)) means zero of one quantity corresponds to zero of the other.

5. 42–54 min · Independent practice and check. Teacher distributes the proportional relationships practice worksheet and circulates, asking students to explain their tests rather than relying on visual appearance. Students complete four tasks: classify two ratio tables, analyze two graphs, complete one missing table value, and write a justification using either equivalent ratios or the graph’s relationship to the origin. Pause midway for a quick whole-class check of one table and one graph.

6. 54–60 min · Exit ticket and debrief. Teacher presents the relationship “A taxi charges $4 for 1 mile, $8 for 2 miles, and $11 for 3 miles” and asks students to decide whether it is proportional, show one calculation, and state what point ((1,4)) would mean if the relationship were graphed. Students submit their response and share one reliable way to test proportionality.

Resources

• Student graph paper
• Pencils, rulers, and colored pencils
• Whiteboard and markers
• Calculators for students who need computational support
• the ratio table and double number line mat
• the proportional relationships practice worksheet
• Prepared ratio tables and graph examples
• Exit-ticket slips or half-sheets of paper

Assessment

• During partner work, check whether students calculate a consistent (y/x) value and distinguish equivalent ratios from matching differences.
• During graphing and independent practice, ask students to explain why a line must pass through the origin, not merely whether it looks straight.
• Use the exit ticket to identify students who need additional support with unit rates, graph interpretation, or justification.

Differentiation

• Support students with a ratio table containing a completed first row, the sentence frame “The relationship is proportional/nonproportional because…,” and a reminder to calculate (y \div x) for each row.
• Allow students to use the double number line mat, graph paper with labeled axes, and a calculator when computation interferes with reasoning.
• For multilingual learners, preview “proportional,” “constant,” “equivalent,” “origin,” and “unit rate” with short examples and display the sentence frames orally and in writing.
• Challenge early finishers to create a proportional and a nonproportional table with the same first two rows, then explain how a third row reveals the difference.