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What Happens When We Nudge a Factor?
Some Properties of Multiplication · Inductive
Students commonly assume multiplying each factor by the same scale yields the same increment — the area-diagram partition keeps this visible and concrete. Watch for students who add the increment to the product rather than recognising it as a multiplicative contribution from each factor. Example 1, section 6.1 p.140 is a useful independent read for students who want to see three-term distribution worked in full.
0–10Pose the opening question from section 6.1: what changes when each factor of 23×27 is increased by 1? Students compute 23×27 and 24×28 independently, then compare and articulate in words what they notice.
10–25Introduce the distributive property a(b+c) = ab+ac through the area-diagram interpretation from section 6.1. Walk through how increasing b by 1 adds a to the product, and how increasing both a and b by 1 adds a+b+1, using students' own computed pairs as evidence.
25–38Work through Figure it Out Q1, section 6.1 p.142 ("Observe the 10×10 multiplication grid; if the middle entry of a 3×3 frame is pq, write algebraic expressions for the other 8 entries using Identity 1."), reading Identity 1: (a+m)(b+n) = ab+mb+an+mn from the board and labelling each of the nine cells.
38–45Teacher models Example 1, section 6.1 p.140 ("Expand (3a/2)(a − b + 1/5) using distributivity over three terms; collect like terms using exponent notation."), emphasising how each term of the bracket is multiplied in turn and how exponent notation records repeated factors.
Figure it Out Q1, section 6.1 p.142 — Observe the 10×10 multiplication grid; if the middle entry of a 3×3 frame is pq, write algebraic expressions for the other 8 entries using Identity 1.
The distributive property states that a(b + c) = ab + ac. Using this property, which of the following correctly expands the product (a + 3)(b + 5)?
- A ab + 15
- B ab + 5a + 3b + 15
- C ab + 5a + 3b + 8
- D ab + 8
B. ab + 5a + 3b + 15
What each wrong option reveals
- A The student multiplied only the two variable terms and the two constant terms (a × b and 3 × 5), omitting the cross-product terms 5a and 3b.
- C The student added the two constants (3 + 5 = 8) for the final term rather than multiplying them (3 × 5 = 15), indicating a partial understanding of the four-term expansion.
- D The student added the constants (3 + 5 = 8) instead of forming all four product terms, suggesting confusion between addition and the full distributive expansion.
Also saved as 10 periods or 13 periods.
- 2Expanding Products, Naming Every Part
Some Properties of Multiplication - 3Fast Multiplication: Digits Dancing with Distributivity
Some Properties of Multiplication - 4Applying Identity 1: Patterns, Pairs, Predictions
Some Properties of Multiplication - 5Squares from Squares: Seeing (a+b)² in a Diagram
Special Cases of the Distributive Property - 6Shrinking the Square: (a−b)² and the Difference of Squares
Special Cases of the Distributive Property - 7Patterns from Identities: When Algebra Explains Numbers
Special Cases of the Distributive Property - 8Error Hunt: Finding and Fixing Wrong Algebra
Mind the Mistake, Mend the Mistake - 9Counting Circles Four Ways: One Answer, Many Expressions
This Way or That Way, All Ways Lead to the Bay - 10Border Tiles and Area Regions: Equivalent Expressions from Geometry
This Way or That Way, All Ways Lead to the Bay - 11Aditya's Half-Region and the L-Shape: More Methods, Same Truth
This Way or That Way, All Ways Lead to the Bay - 12Calendar Diagonals and Consecutive Numbers: Algebra as Proof
This Way or That Way, All Ways Lead to the Bay - 13Number-Theoretic Claims: Using Identities to Verify Divisibility
This Way or That Way, All Ways Lead to the Bay - 14Area Puzzles, Tile Patterns, and Identities at Work
This Way or That Way, All Ways Lead to the Bay - 15All Paths Converge: A Consolidation Through Multi-Method Problems
This Way or That Way, All Ways Lead to the Bay - 16One Principle, Many Faces: A Working Closure
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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