Year 7 Maths Whole Numbers

Multiplying Numbers

Q: How do you multiply by 4 in your head?

Double twice: 26 × 4 = 52 × 2 = 104.

Q: How do you multiply by 9 quickly?

Multiply by 10, then take away the number: 17 × 9 = 170 - 17 = 153.

Q: What is the distributive law?

Splitting a number and multiplying each part: 7 × 43 = 7 × 40 + 7 × 3.

Q: How do you multiply by 5?

Multiply by 10 and halve: 34 × 5 = 170.

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Theory

Multiplying finds the total of equal groups. Split, double or adjust the numbers to multiply in your head, and use the written method for bigger numbers. This Year 7 maths page shows the mental strategies, the distributive law, the area model and the written method with its placeholder zero.

Split into tens and ones, multiply each part, then add. This is the distributive law: $7 \times 43 = 7 \times 40 + 7 \times 3 = 280 + 21 = 301$.

Double. $\times 2$ is double once, $\times 4$ is double twice, $\times 8$ is double three times ($8 = 2 \times 2 \times 2$).

Halve. $\times 5$ is $\times 10$, then halve (or halve, then $\times 10$): $5 = \dfrac{1}{2} \times 10$. Double and halve: halve one number and double the other; the answer stays the same: $14 \times 25 = 7 \times 50 = 350$.

Adjust. $\times 9$ is ten lots take away one lot: $17 \times 9 = 170 - 17 = 153$. In the same way $\times 99$ is $\times 100$ take away one lot, $\times 101$ is $\times 100$ add one lot, and $\times 102$ is $\times 100$ add two lots.

$\times 10$ puts one zero on the end of a whole number; $\times 100$ puts two zeros on the end.

Written method. Estimate first. Multiply by the ones digit, carrying any tens to the next column. Then multiply by the tens: that row starts with a placeholder zero. Add the rows and compare with the estimate.

Split $24 = 20 + 4$: $36 \times 20 = 720$ and $36 \times 4 = 144$, so $36 \times 24 = 720 + 144 = 864$.

The two rows of the written method are the two parts of the rectangle. The $0$ at the end of $720$ is the placeholder zero.

Mental strategies:

Multiplying by	Method
$2$	double once
$4$	double twice
$8$	double three times
$5$	halve, then $\times 10$
$9$	$\times 10$, then take away the number
$10$	put $0$ on the end
$100$	put $00$ on the end
$99$ / $101$	$\times 100$, then take away / add the number

          Take away one lot of the number, not 1. \(17 \times 9 = 170 - 17 = 153\), not \(170 - 1\).
        

          Estimate to catch mistakes. \(57 \times 34 \approx 60 \times 30 = 1800\). An answer of \(399\) is far too small.
        

How to use the written method (2-digit multiplier)

Estimate by rounding each number to its leading digit.
Multiply by the ones digit, carrying tens.
Write a placeholder zero, then multiply by the tens digit.
Add the two rows and compare with the estimate.

Example 1 — Doubling

Work out
(i) $43 \times 2$
(ii) $43 \times 4$
(iii) $43 \times 8$

Solution

$43 \times 2$	$=$	$86$
$43 \times 2 \times 2$	$=$	$86 \times 2 = 172$
$43 \times 2 \times 2 \times 2$	$=$	$172 \times 2 = 344$

43 \times 2 = 86, 43 \times 4 = 86 \times 2 = 172, 43 \times 8 = 172 \times 2 = 344

(i) $86$
(ii) $172$
(iii) $344$

Example 2 — Times 5, 9, 10 and 100

Work out
(i) $34 \times 5$
(ii) $26 \times 9$
(iii) $56 \times 10$
(iv) $78 \times 100$

Solution

$34 \times \dfrac{1}{2} \times 10$	$=$	$17 \times 10 = 170$
$26 \times 10 - 26$	$=$	$260 - 26 = 234$

34 \times 5 = 34 \times 1 / 2 \times 10 = 17 \times 10 = 17026 \times 9 = 26 \times 10 - 26 = 260 - 26 = 234

(i) $170$
(ii) $234$ (take away one lot of $26$, not $9$)
(iii) $560$
(iv) $7800$

Example 3 — Split to multiply

Work out $8 \times 57$ by splitting $57$ into tens and ones.

Solution

$8 \times 50 + 8 \times 7$	$=$	$400 + 56$
	$=$	$456$

8 \times 57 = 8 \times 50 + 8 \times 7 = 400 + 56 = 456

$456$

Example 4 — Mental strategies

Work out
(i) $18 \times 50$
(ii) $38 \times 99$
(iii) $27 \times 101$

Solution

$9 \times 100$	$=$	$900$
$3800 - 38$	$=$	$3762$
$2700 + 27$	$=$	$2727$

18 \times 50 = 9 \times 100 = 90038 \times 99 = 3800 - 38 = 376227 \times 101 = 2700 + 27 = 2727

(i) Halve $18$, double $50$: $900$
(ii) $38 \times 100 - 38 \times 1 = 3762$
(iii) $27 \times 100 + 27 \times 1 = 2727$

Example 5 — The written method

Use the written method to work out $58 \times 43$.

Solution

$58 \times 3$	$=$	$174$
$58 \times 40$	$=$	$2320$
$174 + 2320$	$=$	$2494$

58 \times 3 = 174, 58 \times 40 = 2320, 174 + 2320 = 2494

Estimate $60 \times 40 = 2400$. $58 \times 43 = 2494$.

Example 6 — School camp

Camp costs $\$185$ for each student, and $42$ students go.
(i) What is the total cost?
(ii) The P&C pays $\$2000$. How much do the families pay?

Solution

$185 \times 2$	$=$	$370$
$185 \times 40$	$=$	$7400$
$370 + 7400$	$=$	$7770$
$7770 - 2000$	$=$	$5770$

185 \times 42 = 370 + 7400 = 7770, 7770 - 2000 = 5770

(i) $\$7770$ (estimate $200 \times 40 = 8000$)
(ii) $\$5770$

Example 7 — Largest product

Use each of the digits $2$, $5$, $7$ and $9$ once to make two 2-digit numbers with the largest product.

Solution

$92 \times 75$	$=$	$6900$
$95 \times 72$	$=$	$6840$

92 \times 75 = 6900, 95 \times 72 = 6840

Put the biggest digits ($9$ and $7$) in the tens places, then try both ways: $92 \times 75 = 6900$ is the largest.

Common pitfalls

No placeholder zero. In $57 \times 34$, the second row is $57 \times 30 = 1710$, not $171$.

Adjusting by 1. $\times 9$ takes away one lot of the number, not $1$.

Doubling the wrong number of times. $\times 8$ is three doublings; $\times 4$ is two.

Skipping the estimate. Always compare the answer with a rough estimate.

Frequently asked questions

How do you multiply by 4 in your head?

Double twice: $26 \times 4 = 52 \times 2 = 104$.

How do you multiply by 9 quickly?

Multiply by $10$, then take away the number: $17 \times 9 = 170 - 17 = 153$.

What is the distributive law?

Splitting a number and multiplying each part: $7 \times 43 = 7 \times 40 + 7 \times 3$.

Why is there a zero in the second row?

Because you are multiplying by tens. The zero holds the ones place.

How do you multiply by 5?

Multiply by $10$ and halve: $34 \times 5 = 170$.

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Dividing Numbers

Multiplying by	Method
\(2\)	double once
\(4\)	double twice
\(8\)	double three times
\(5\)	halve, then \(\times 10\)
\(9\)	\(\times 10\), then take away the number
\(10\)	put \(0\) on the end
\(100\)	put \(00\) on the end
\(99\) / \(101\)	\(\times 100\), then take away / add the number

\(43 \times 2\)	\(=\)	\(86\)
\(43 \times 2 \times 2\)	\(=\)	\(86 \times 2 = 172\)
\(43 \times 2 \times 2 \times 2\)	\(=\)	\(172 \times 2 = 344\)

\(34 \times \dfrac{1}{2} \times 10\)	\(=\)	\(17 \times 10 = 170\)
\(26 \times 10 - 26\)	\(=\)	\(260 - 26 = 234\)

\(8 \times 50 + 8 \times 7\)	\(=\)	\(400 + 56\)
\(\)	\(=\)	\(456\)

\(9 \times 100\)	\(=\)	\(900\)
\(3800 - 38\)	\(=\)	\(3762\)
\(2700 + 27\)	\(=\)	\(2727\)

\(58 \times 3\)	\(=\)	\(174\)
\(58 \times 40\)	\(=\)	\(2320\)
\(174 + 2320\)	\(=\)	\(2494\)

\(185 \times 2\)	\(=\)	\(370\)
\(185 \times 40\)	\(=\)	\(7400\)
\(370 + 7400\)	\(=\)	\(7770\)
\(7770 - 2000\)	\(=\)	\(5770\)

\(92 \times 75\)	\(=\)	\(6900\)
\(95 \times 72\)	\(=\)	\(6840\)

Multiplying Numbers

📖 Theory

How to use the written method (2-digit multiplier)

Common pitfalls

Frequently asked questions

Theory