Back to Ganita Prakash: Class 7 Mathematics, Par...BackConcept

The Distributive Property With Integers

See with tokens why a × (b + c) = a × b + a × c works for negative numbers, and use it to find products quickly.

7 micro-lessons

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Multiply a Sum, Two Ways

5 × (4 + (−2)) equals 5 × 4 + 5 × (−2).

Two ways to work it out (Textbook p. 40)

Bracket first: $5 \times (4 + (-2)) = 5 \times 2 = 10$

Multiply each part, then add: $5 \times 4 + 5 \times (-2) = 20 + (-10) = 10$

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The Token Rectangle

4 rows of 2 positives and 3 negatives show 4 × (2 + (−3)).

A rectangle of tokens (Textbook p. 41)

For positive numbers we used a rectangle of objects to see why the distributive property works. Tokens let us do the same with negatives.

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Checking (−2) × (4 + (−3))

Both sides of the distributive property with a negative multiplier.

Problem. Check that $(-2) \times (4 + (-3)) = (-2) \times 4 + (-2) \times (-3)$ . (Textbook p. 40)

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Distributive Property Card

a × (b + c) = (a × b) + (a × c).

The distributive property (Textbook p. 41)

a \times (b + c) = (a \times b) + (a \times c)

What each part means

a

— the number multiplying the bracket

b + c

— the sum inside the bracket

When it works

Works for any integers a, b and c: positive, negative or zero.

Example

4 \times (2 + (- 3)) = 8 + (- 12) = - 4

Shortcut:

(- 547) \times 972 = (- 548 + 1) \times 972 = - 532656 + 972

Tokens for a Negative Multiplier

Describe the token picture for −4 × (2 + (−3)).

Show −4 × (2 + (−3)) with tokens

Try This, Textbook p. 41. Can you show the distributive property for $-4 \times (2 + (-3))$ with tokens?

Hint from the book: multiplying a number by $-4$ is adding the inverse of the number 4 times.

One row: the inverse of (2 + (−3))

Flip every token in the row.

The left part, 4 rows

What is −4 × 2?

The right part, 4 rows

What is −4 × (−3)?

The total

Add the two parts.

Check with the bracket first

Work out 2 + (−3) first.

Distribute and Save Time

Use the property instead of long calculation.

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$(-5) \times (18 + (-3)) = ?$ (Textbook p. 42, Q1a)

From One Product to Three

Use (−548) × 972 = −532656 to find three new products.

Find the products without long multiplication

Figure it Out, Textbook p. 44, Q13. Given $(-548) \times 972 = -532656$ , write the values of:

(a) $(-547) \times 972$ (b) $(-548) \times 971$ (c) $(-547) \times 971$

(a) (−547) × 972*

Write −547 as −548 + 1, then distribute.

(b) (−548) × 971*

Write 971 as 972 − 1.

Start from your answer to (a).

← previousMultiplying Three Or More Integers next →Pick The Pattern: Number Machines

Multiply a Sum, Two Ways

5 × (4 + (−2)) equals 5 × 4 + 5 × (−2).

Two ways to work it out (Textbook p. 40)

Bracket first: $5 \times (4 + (-2)) = 5 \times 2 = 10$

Multiply each part, then add: $5 \times 4 + 5 \times (-2) = 20 + (-10) = 10$

1 / 2

Checking (−2) × (4 + (−3))

Both sides of the distributive property with a negative multiplier.

Problem. Check that $(-2) \times (4 + (-3)) = (-2) \times 4 + (-2) \times (-3)$ . (Textbook p. 40)

1 / 4

Tokens for a Negative Multiplier

Describe the token picture for −4 × (2 + (−3)).

Show −4 × (2 + (−3)) with tokens

Try This, Textbook p. 41. Can you show the distributive property for $-4 \times (2 + (-3))$ with tokens?

Hint from the book: multiplying a number by $-4$ is adding the inverse of the number 4 times.

One row: the inverse of (2 + (−3))

Flip every token in the row.

The left part, 4 rows

What is −4 × 2?

The right part, 4 rows

What is −4 × (−3)?

The total

Add the two parts.

Check with the bracket first

Work out 2 + (−3) first.

Distribute and Save Time

Use the property instead of long calculation.

1 / 3

$(-5) \times (18 + (-3)) = ?$ (Textbook p. 42, Q1a)

From One Product to Three

Use (−548) × 972 = −532656 to find three new products.

Find the products without long multiplication

Figure it Out, Textbook p. 44, Q13. Given $(-548) \times 972 = -532656$ , write the values of:

(a) $(-547) \times 972$ (b) $(-548) \times 971$ (c) $(-547) \times 971$

(a) (−547) × 972*

Write −547 as −548 + 1, then distribute.

(b) (−548) × 971*

Write 971 as 972 − 1.

Start from your answer to (a).