Mathematics · Grade 6

Active learning ideas

Ratio Tables and Equivalent Ratios

Active learning transforms ratio tables from abstract symbols into concrete tools students can manipulate and explain. When students physically build tables with real quantities, they see how multiplication and division preserve relationships, making equivalence visible and meaningful.

• Ontario Curriculum Expectations
• 6.RP.A.3.A

• 20–40 min
• Pairs → Whole Class
• 4 activities

Activity 01

• Collaborative Problem-Solving
• 35 min · Pairs

Pairs: Recipe Scaling Challenge

Provide recipes for 4 servings. Pairs create ratio tables to scale for 10 or 6 servings, filling missing ingredient amounts. They swap tables with another pair to verify equivalence using cross-multiplication. Discuss which scaling factor was easiest.

Explain how to determine if two ratios are equivalent using different mathematical strategies.

Facilitation Tip

During Recipe Scaling Challenge, circulate to listen for students discussing how scaling one ingredient affects others, prompting them to articulate the multiplicative relationship.

What to look for

Provide students with a partially filled ratio table for a recipe, such as cups of flour to cups of sugar. Ask them to complete the table for a different number of servings and explain the pattern they used to find the missing values.

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Activity 02

• Collaborative Problem-Solving
• 40 min · Small Groups

Small Groups: Ratio Sort and Table

Give cards with ratio pairs like 2:3, 4:6, 5:7. Groups sort equivalent sets into ratio tables, then extend tables to find missing values. Present one table to class and explain the pattern.

Construct a ratio table to find missing values in a proportional relationship.

Facilitation Tip

In Ratio Sort and Table, provide colored index cards so groups can visually group equivalent ratios before building tables, reinforcing the connection between sorting and scaling.

What to look for

Present a scenario: 'For every 3 apples, there are 2 oranges. If there are 12 apples, how many oranges are there?' Students must show their work using a ratio table and write one sentence explaining how they knew their answer was correct.

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Activity 03

• Collaborative Problem-Solving
• 25 min · Whole Class

Whole Class: Proportional Relay

Divide class into teams. Project a ratio problem; one student per team runs to board, adds a row to the table, tags next teammate. First accurate table wins. Review patterns as class.

Analyze the patterns within a ratio table that indicate proportionality.

Facilitation Tip

For Proportional Relay, assign roles so students practice explaining their scaling steps aloud, building both accuracy and communication skills.

What to look for

Pose the question: 'How can you tell if two ratios, like 4:6 and 10:15, are equivalent without using cross-multiplication?' Students should discuss strategies involving creating ratio tables or finding unit rates.

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Activity 04

• Collaborative Problem-Solving
• 20 min · Individual

Individual: Missing Value Hunt

Students receive worksheets with incomplete ratio tables from scenarios like bike gears. They solve independently, then pair to check work. Share one tricky solution with class.

Explain how to determine if two ratios are equivalent using different mathematical strategies.

Facilitation Tip

In Missing Value Hunt, ask students to record not just answers but also the operation they used, which helps reveal whether they rely on multiplication, division, or a mix.

What to look for

Provide students with a partially filled ratio table for a recipe, such as cups of flour to cups of sugar. Ask them to complete the table for a different number of servings and explain the pattern they used to find the missing values.

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

A few notes on teaching this unit

Start with concrete materials to build the concept of equivalent ratios before moving to symbolic tables. Avoid teaching cross-multiplication as a rule until students have internalized the idea of scaling. Research shows students grasp proportionality better when they experience it through real-world contexts like recipes or sharing problems. Move from hands-on to abstract gradually, using ratio tables as a bridge between the two.

Students will confidently construct ratio tables, identify equivalent ratios through multiple methods, and use tables to solve proportional problems. They will explain their reasoning clearly, using both multiplicative patterns and unit rates to justify their answers.

---

Watch Out for These Misconceptions

• During Recipe Scaling Challenge, watch for students adding the same number to each ingredient instead of multiplying by the same factor.

  Pause the activity and ask groups to compare their scaled recipes side by side with the original, prompting them to notice that adding changes the taste (ratio) while multiplying keeps it consistent.

• During Ratio Sort and Table, watch for students assuming any two ratios with the same simplified fraction are equivalent without checking the scaling factor.

  Have groups sort cards first by simplified form, then by scaling factor, forcing them to confront cases like 2:4 and 3:6 versus 2:4 and 5:10 to see the difference in scaling paths.

• During Proportional Relay, watch for students assuming ratio tables must always show increasing values.

  Introduce a 'shrinking' problem in the relay, such as splitting 12 cookies among fewer friends, and ask teams to build a table that includes values less than the original to normalize bidirectional scaling.

---

Methods used in this brief

• Collaborative Problem-Solving