CLASS 8 MATHS · PART 1 · CHAPTER 6 · STUDY NOTES
We Distribute, Yet Things Multiply
Understand we distribute, yet things multiply through short explanations, labelled models and original practice. Read one topic, explain it in your own words, then check.
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Split a rectangle; multiply each part
One multiplication can be broken into easier multiplications without changing the total.Two areas, one total
y (units)
x (units)
- 10 by 8 · Area 80.
- 3 by 8 · Area 24.
- 180 · (5, 4)
- 224 · (11.5, 4)
A positive-length area model of 8 × 13. The split is vertical and both regions have height 8. Equal units on both axes.
Read the plotted values
| x | y |
|---|---|
| 0 | 0 |
| 10 | 0 |
| 10 | 8 |
| 0 | 8 |
| 0 | 0 |
| x | y |
|---|---|
| 10 | 0 |
| 13 | 0 |
| 13 | 8 |
| 10 | 8 |
| 10 | 0 |
- Distributive property
- a(b + c) = ab + ac; multiplication acts on every term inside the bracket.
8 × 13 = 8 × (10 + 3) = 80 + 24 = 104. Both parts keep the common height 8.
The rule also works with subtraction: a(b − c) = ab − ac. Signed numbers use the same algebra, even when a literal positive-area diagram is no longer suitable.
Take a moment. What do you think?
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Question
Calculate 7 × 19 by splitting 19 as 20 − 1.
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Increasing two sides adds two strips and a corner
When both factors change, there is a small cross-product term as well as the two main changes.Do not forget the corner
y (units)
x (units)
- Old rectangle · 5 × 3 = 15.
- Top strip · 5 × 1 = 5.
- Right strip · 1 × 3 = 3.
- Corner · 1 × 1 = 1.
- 115 · (2.5, 1.5)
- 25 · (2.5, 3.5)
- 33 · (5.5, 1.5)
- 41 · (5.5, 3.5)
The old 5 by 3 rectangle grows to 6 by 4. New area is 5 + 3 + 1 = 9. Equal units on both axes.
Read the plotted values
| x | y |
|---|---|
| 0 | 0 |
| 5 | 0 |
| 5 | 3 |
| 0 | 3 |
| 0 | 0 |
| x | y |
|---|---|
| 0 | 3 |
| 5 | 3 |
| 5 | 4 |
| 0 | 4 |
| 0 | 3 |
| x | y |
|---|---|
| 5 | 0 |
| 6 | 0 |
| 6 | 3 |
| 5 | 3 |
| 5 | 0 |
| x | y |
|---|---|
| 5 | 3 |
| 6 | 3 |
| 6 | 4 |
| 5 | 4 |
| 5 | 3 |
- Identity
- An equation true for all allowed values of its variables.
(a + 1)(b + 1) = ab + a + b + 1. The extra 1 comes from the new corner.
More generally (a + m)(b + n) = ab + an + bm + mn. The change from ab is an + bm + mn, which may be negative when changes are negative.
Let’s look a little closer.
Go deeper: One factor up, the other down
(a + 1)(b − 1) − ab = b − a − 1. The change can be positive, zero or negative. For 8 × 5, changing to 9 × 4 decreases the product by 4 because 5 − 8 − 1 = −4.
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Question
How much does 9 × 6 change when both factors increase by 1?
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Every term meets every term
Multiply each term in one bracket by each term in the other.Four pairings are required
- Term
- A part separated from other parts by addition or subtraction.
- Like terms
- Terms with the same variable part and the same powers.
(x + 2)(x + 3) gives x², 3x, 2x and 6. Combine the like terms to get x² + 5x + 6.
(x − 2)(x + 3) = x² + 3x − 2x − 6 = x² + x − 6. Keep each minus sign attached to its term.
- 1
Write all four products: x·x, x·3, 2·x and 2·3.
- 2
Simplify products to x² + 3x + 2x + 6.
- 3
Combine only 3x and 2x, giving x² + 5x + 6.
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Question
Expand (p + 4)(p − 5).
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The square of a sum includes two rectangles
Squaring the combined side makes four regions, not only two squares.Four regions in one square
y (units)
x (units)
- a² · Here a = 4, so area 16.
- ab · One 4 by 2 rectangle.
- ba · Another 2 by 4 rectangle.
- b² · Here b = 2, so area 4.
- 1a² · (2, 2)
- abab · (5, 2)
- abab · (2, 5)
- 4b² · (5, 5)
The full square has side 6. Its area 36 is 16 + 8 + 8 + 4. Equal units on both axes.
Read the plotted values
| x | y |
|---|---|
| 0 | 0 |
| 4 | 0 |
| 4 | 4 |
| 0 | 4 |
| 0 | 0 |
| x | y |
|---|---|
| 4 | 0 |
| 6 | 0 |
| 6 | 4 |
| 4 | 4 |
| 4 | 0 |
| x | y |
|---|---|
| 0 | 4 |
| 4 | 4 |
| 4 | 6 |
| 0 | 6 |
| 0 | 4 |
| x | y |
|---|---|
| 4 | 4 |
| 6 | 4 |
| 6 | 6 |
| 4 | 6 |
| 4 | 4 |
- Square-of-a-sum identity
- (a + b)² = a² + 2ab + b².
The two off-diagonal rectangles both have area ab. Together they explain the middle term 2ab.
For any real a and b, expand algebraically to verify the identity. The literal area picture uses nonnegative lengths.
Remember the cross term
(3x + 2)² = 9x² + 12x + 4The middle term is 2 × 3x × 2, not 6x.
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Question
Find 42² by writing 42 as 40 + 2.
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Subtract two strips, then put their overlap back
Removing two full strips removes their shared corner twice.The overlap explains the plus b²
y (units)
x (units)
- Original square · 6 by 6, area 36.
- Top strip · Remove area 12.
- Right strip · Remove area 12.
- Overlap · Area 4 was removed twice.
- 1Remaining 16 · (2, 2)
- 2Put back 4 · (5, 5)
The remaining lower-left square is 4 by 4. Its area is 36 − 12 − 12 + 4 = 16. Equal units on both axes.
Read the plotted values
| x | y |
|---|---|
| 0 | 0 |
| 6 | 0 |
| 6 | 6 |
| 0 | 6 |
| 0 | 0 |
| x | y |
|---|---|
| 0 | 4 |
| 6 | 4 |
| 6 | 6 |
| 0 | 6 |
| 0 | 4 |
| x | y |
|---|---|
| 4 | 0 |
| 6 | 0 |
| 6 | 6 |
| 4 | 6 |
| 4 | 0 |
| x | y |
|---|---|
| 4 | 4 |
| 6 | 4 |
| 6 | 6 |
| 4 | 6 |
| 4 | 4 |
- Square-of-a-difference identity
- (a − b)² = a² − 2ab + b².
Start with an a by a square. Remove the top b-wide strip and right b-wide strip. Their b by b overlap was removed twice, so add it back once.
This area argument needs a ≥ b ≥ 0. The algebraic identity itself holds for all real a and b.
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Question
Calculate 28² as (30 − 2)².
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A square difference can be rearranged into a rectangle
Remove a smaller square, then move one of the remaining strips.Same area in a new rectangle
y (units)
x (units)
- Rectangle · Length a + b = 6; width a − b = 2.
- 112 square units · (3, 1)
For a = 4 and b = 2, the area is 6 × 2 = 12, equal to 16 − 4. Equal units on both axes.
Read the plotted values
| x | y |
|---|---|
| 0 | 0 |
| 6 | 0 |
| 6 | 2 |
| 0 | 2 |
| 0 | 0 |
Rearrange exactly the same two pieces
- Blue piece
- A 2 by 4 rectangle, area 8.
- Orange piece
- A 2 by 2 square, area 4.
- Difference-of-squares identity
- a² − b² = (a + b)(a − b).
The mixed terms cancel: (a + b)(a − b) = a² − ab + ab − b².
This is useful for products equally far from a midpoint, such as 97 × 103 = (100 − 3)(100 + 3).
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Question
Find 47 × 53 using a midpoint.
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Multiply near a power of ten using two easy parts
The shortcut works because 11 = 10 + 1, 101 = 100 + 1, and 99 = 100 − 1.Separate the shifted copy from the original
- Partial product
- One multiplication result before adding the parts.
236 × 11 = 2360 + 236 = 2596. Place values align when you add; carry when a column reaches ten.
236 × 101 = 23600 + 236 = 23836. For 99, subtract rather than add: 236 × 99 = 23600 − 236.
- 1
Split 11 into 10 + 1.
- 2
Multiply 236 by each part.
- 3
Add 2360 and 236, respecting column carries.
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Question
Calculate 407 × 101.
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Distribute to every term, then collect matching powers
Terms such as x² and x are different types and cannot be combined by adding their coefficients.Sort by variable part
- Coefficient
- The numerical multiplier of a variable term.
2x(x² − 3x + 4) = 2x³ − 6x² + 8x. The 2x multiplies all three terms.
Multiplying three brackets can be done one pair at a time. For example, (a + b)³ = a³ + 3a²b + 3ab² + b³. Expand, then match variable powers.
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Question
Simplify 5x² + 2x − 3x² + 7x.
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Different counts of one picture should give equal formulas
Count a dot pattern in two ways, then use algebra to explain why they agree.Count the square with one corner missing
- Equivalent expressions
- Expressions with the same value for every allowed substitution.
A 5 by 5 dot array missing one corner has 24 dots. Count it as 5² − 1, or as a 4 by 6 rectangle after moving a row.
In general (k + 1)² − 1 = k² + 2k = k(k + 2). The equality is explained by expansion, not only by one drawing.
Go deeper: A useful sum of square identities
Adding (a + b)² and (a − b)² cancels the cross terms and gives 2(a² + b²). Subtracting these two expansions instead gives 4ab. Keep both signs and all terms to explain these patterns.
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Question
Why does (k + 1)² − 1 simplify to k(k + 2)?
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