Proportion: Equivalent Ratios

Active learning works well here because students often confuse scaling with adding. By handling physical strips and real recipes, they feel the difference between keeping a relationship intact and breaking it. Movement and talk turn abstract ratios into something they can see and discuss.

• Class 6
• Mathematics
• 4 activities
• 25 min–45 min

Learning Objectives

1. Calculate the missing term in a proportion using cross-multiplication.
2. Compare two given ratios to determine if they are in proportion.
3. Generate equivalent ratios by scaling up or down a given ratio.
4. Explain the concept of proportion as the equality of two ratios.

Ready-to-Use Activities

Collaborative Problem-Solving

⏱ 35 min·Pairs

Bar Model Building: Ratio Strips

Cut paper strips into ratio lengths, like 2 units red and 3 blue for 2:3. Pairs create equivalent strips by doubling to 4:6, then halve back. They verify using cross-multiplication and record findings.

Prepare & details

What defines two ratios as being in proportion to one another?

Facilitation Tip: During Bar Model Building, insist every pair records the scale factor they used on a sticky note so the class can compare strategies later.

Setup: Flexible seating that allows clusters of 5-6 students; desks can be grouped in rows of three facing each other if fixed furniture limits rearrangement. Wall or board space for displaying group norm charts and the session agenda is helpful.

Materials: Printed problem brief cards (one per group), Role cards: Facilitator, Questioner, Recorder, Devil's Advocate, Communicator, Group norm chart (printable poster format), Individual reflection sheet and exit ticket, Timer visible to the class (board countdown or projected timer)

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

Collaborative Problem-Solving

⏱ 45 min·Small Groups

Recipe Scaling: Dosa Batter Mix

Provide a basic dosa recipe ratio of 3:1 rice to urad dal. Small groups scale for 10 or 20 dosas, calculate quantities, mix samples, and test proportions by taste and texture.

Prepare & details

How do we maintain the relationship between two numbers when scaling them up or down?

Facilitation Tip: While scaling dosa batter, give each group only one measuring cup so they must agree on a common unit before mixing.

Setup: Flexible seating that allows clusters of 5-6 students; desks can be grouped in rows of three facing each other if fixed furniture limits rearrangement. Wall or board space for displaying group norm charts and the session agenda is helpful.

Materials: Printed problem brief cards (one per group), Role cards: Facilitator, Questioner, Recorder, Devil's Advocate, Communicator, Group norm chart (printable poster format), Individual reflection sheet and exit ticket, Timer visible to the class (board countdown or projected timer)

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

Collaborative Problem-Solving

⏱ 30 min·Pairs

Card Matching: Proportion Puzzles

Prepare cards with ratios and missing values, like 3:4 = 9:?. Students in pairs match equivalents and solve unknowns by scaling or cross-multiplying, then explain to the class.

Prepare & details

Predict the missing value in a proportion using cross-multiplication.

Facilitation Tip: For Card Matching, set a 90-second timer so students quickly test pairs and discard wrong matches instead of over-thinking.

Setup: Flexible seating that allows clusters of 5-6 students; desks can be grouped in rows of three facing each other if fixed furniture limits rearrangement. Wall or board space for displaying group norm charts and the session agenda is helpful.

Materials: Printed problem brief cards (one per group), Role cards: Facilitator, Questioner, Recorder, Devil's Advocate, Communicator, Group norm chart (printable poster format), Individual reflection sheet and exit ticket, Timer visible to the class (board countdown or projected timer)

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

Collaborative Problem-Solving

⏱ 25 min·Individual

Map Scale Hunt: Classroom Proportions

Draw a class map with a 1:10 scale. Individuals measure distances, scale to real schoolyard lengths, and predict missing map values using proportion rules.

Prepare & details

What defines two ratios as being in proportion to one another?

Facilitation Tip: On the Map Scale Hunt, let students measure twice but record once to avoid confusion between centimetres and grid squares.

Setup: Flexible seating that allows clusters of 5-6 students; desks can be grouped in rows of three facing each other if fixed furniture limits rearrangement. Wall or board space for displaying group norm charts and the session agenda is helpful.

Materials: Printed problem brief cards (one per group), Role cards: Facilitator, Questioner, Recorder, Devil's Advocate, Communicator, Group norm chart (printable poster format), Individual reflection sheet and exit ticket, Timer visible to the class (board countdown or projected timer)

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

Teaching This Topic

Start with a quick human bar: ask two students to stand apart, then double the distance while keeping the ratio of heights the same. Use this image repeatedly in every activity to anchor the idea of multiplication, not addition. Avoid worksheets early on; let students draw, cut, and mix before they write formal methods. Research shows concrete-manipulative time shortens the jump to symbolic cross-multiplication.

What to Expect

Students will confidently check if two ratios are equivalent and scale them correctly without reversing the order. They will explain their steps aloud and justify their choices using bar models or recipe measures. Struggling pairs will still show their working so you can spot the exact step that needs support.

Watch Out for These Misconceptions

Common Misconception

During Bar Model Building, watch for students who add equal lengths instead of scaling both parts. Ask them to place their strips side by side and compare the total lengths—if the totals differ, the ratio is not maintained.

What to Teach Instead

Prompt them to measure one strip, then double both segments individually. Have them label each segment with the scale factor so the relationship stays visible.

Common Misconception

During Card Matching, watch for students who match ratios like 2:3 and 4:5 because the numbers look similar. Ask them to simplify both pairs and compare the final forms aloud.

What to Teach Instead

Have them colour-code matching parts on the cards so wrong pairs stand out immediately; then they must justify each match to a partner.

Common Misconception

During Recipe Scaling, watch for students who reverse the order when scaling, e.g., 2 cups to 3 cups becomes 3 cups to 2 cups. Ask them to taste-test a small spoonful of the incorrectly scaled batter.

What to Teach Instead

Ask them to write the original ratio above the scaled one and draw arrows showing the scale factor from left to right before mixing again.

Assessment Ideas

Quick Check

After Bar Model Building, display pairs of ratios such as 5:8 and 10:16 on the board. Ask students to hold up green cards if proportional, red if not, and write the scale factor below. Then show 3:7 = ?:14 and have them solve individually before pairing to compare answers.

Exit Ticket

During Recipe Scaling, give each student a pre-printed dosa ratio card (e.g., 1:2). Ask them to write two equivalent ratios on the card and, on the back, explain which ingredient they scaled first and why that order matters in cooking.

Discussion Prompt

After Map Scale Hunt, pose a bus scenario: 'A bus travels 80 km in 2 hours. How far in 5 hours?' Ask pairs to use the scale factor they discovered on the classroom hunt to set up their proportion, then share their method with another pair before writing the final answer.

Extensions & Scaffolding

• Challenge: Give students a ratio like 7:11 and ask them to create a 15-step scale drawing of a classroom object that matches this ratio.
• Scaffolding: Provide pre-cut ratio strips for Bar Model Building with scale factors already marked in faint pencil so students focus on grouping, not measuring.
• Deeper: Ask students to find a real-world advertisement that uses a ratio, bring it to class, and write a two-sentence justification proving it is an equivalent ratio.

Key Vocabulary

Ratio	A comparison of two quantities, often written as a:b or a/b.
Proportion	A statement that two ratios are equal. For example, a:b = c:d.
Equivalent Ratios	Ratios that represent the same relationship or value, even if the numbers are different. For example, 1:2 and 2:4 are equivalent ratios.
Cross-multiplication	A method used to check if two ratios are equal or to find a missing value in a proportion by multiplying the numerator of one ratio by the denominator of the other.