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Cross-Products and the Proportionality Test
Proportionality—A Quick Recap · Inductive
The idli-batter context makes proportionality tangible before notation takes over; students who rush to simplify both ratios without checking cross-products sometimes miss cases where simplification is not straightforward — watch for that shortcut being applied incorrectly. A useful self-study pointer: the formal definition connecting equivalent fractions to proportional ratios in section 3.1 is worth re-reading alongside any worked examples students encountered in their earlier ratio work.
0–10Students write three pairs of ratios from everyday contexts — ingredients, distances, speeds — and test each pair using the cross-multiplication rule a×d = b×c to judge whether they are proportional, sharing results with a neighbour.
10–25The class examines the idli-batter scenario from section 3.1 (Viswanath 6:3 vs Puneet 4:2) to confirm both simplify to 2:1; students articulate in their own words why equal simplified ratios guarantee proportionality, and record the definition a/c = b/d.
25–38Pairs generate their own counter-example — a pair of ratios that looks similar but fails the cross-product test — and explain to the class why the relationship breaks. Teacher collects two or three examples on the board.
38–45Quick whole-class round: teacher reads a ratio pair aloud; students signal thumbs-up or thumbs-down for proportional/not, then give the cross-product check as justification.
Two ratios 3:4 and 9:12 are said to be in proportion. Which of the following correctly states the cross-product rule that confirms this?
- A 3 + 12 = 4 + 9
- B 3 × 12 = 4 × 9
- C 3 × 4 = 9 × 12
- D 3 ÷ 9 = 12 ÷ 4
B. 3 × 12 = 4 × 9
What each wrong option reveals
- A The student is adding outer and inner terms instead of multiplying them, confusing cross-products with sums.
- C The student is multiplying the two means together and the two extremes together, reversing the positions in the cross-product.
- D The student is attempting a ratio comparison but dividing first term by third and fourth by second, not forming the standard cross-product pairs.
Also saved as 7 periods or 10 periods.
- 2Measuring Cities with a Representative Fraction
Ratios in Maps - 3From Spice Mixes to the General Multi-Term Rule
Ratios with More than 2 Terms - 4Splitting Totals: The Division Algorithm in Action
Dividing a Whole in a Given Ratio - 5Triangles and Coins: Ratio Division Beyond Measurement
Dividing a Whole in a Given Ratio - 6From Grade Ratios to Pie Chart Sectors
A Slice of the Pie - 7Gathering and Charting Real Data
A Slice of the Pie - 8When One Goes Up, the Other Comes Down
Inverse Proportions - 9Workers, Pumps and Provisions: Inverse Proportion at Work
Inverse Proportions - 10Solving Inverse Proportion Problems Across Contexts
Inverse Proportions - 11Combined Rates, Rows and Speeds: Extended Inverse Problems
Inverse Proportions - 12Four Problems, Four Methods: Putting the Chapter to Work
Synthesis
- 1. A Square and A Cube
- 2. Power Play
- 3. A Story of Numbers
- 4. Quadrilaterals
- 5. Number Play
- 6. We Distribute, Yet Things Multiply
- 7. Proportional Reasoning-1
- 8. Fractions in Disguise
- 9. The Baudhāyana-Pythagoras Theorem
- 10. Proportional Reasoning–2
- 11. Exploring Some Geometric Themes
- 12. Tales by Dots and Lines
- 13. Algebra Play
- 14. Area
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