YEAR 7 · ALGEBRAIC EXPRESSIONS AND FORMULAS · 12 MIN

Mathematical notation and implied multiplication

Use short multiplication notation, powers and fraction bars without changing the operations.

In this lesson

Read adjacent factors as multiplication
Write products using coefficients and powers
Distinguish squaring from doubling

Before you start

Terms and coefficients
Multiplication facts

01 / LEARN

Understand the idea.

Multiplication without a sign

Algebra uses implied multiplication: $6 a$ means $6 \times a$ , and $3 p q$ means $3 \times p \times q$ . Write the numerical coefficient first, followed by the variable factors. A number beside brackets multiplies the entire bracketed quantity.

This shorthand relies on letters or brackets making the factors clear. Keep a multiplication sign between numerical factors when needed: $3 \times 4$ means twelve, while writing $34$ gives the separate number thirty-four.

Powers and division

A power records repeated multiplication. The expression $a^{2}$ means $a \times a$ ; $2 a$ means $a + a$ . In $3 a^{2}$ , square $a$ before multiplying by $3$ . The exponent applies to the factor immediately before it unless brackets extend its scope.

A fraction bar means division and groups its complete numerator and denominator. Thus $\frac{x + 3}{4}$ divides the whole sum by $4$ . It differs from $x + \frac{3}{4}$ , where only $3$ is divided by $4$ .

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Write a product

Write $4 \times b \times c$ in algebraic shorthand.

Worked solution

1
Identify the coefficient and variable factors.
$4 \times b \times c$
Four is the numerical factor; the variable factors are multiplied, not added.
2
Use multiplication shorthand.
$4 \times b \times c = 4 b c$
Placing the variable factors beside the coefficient preserves the product.

Answer

$4 b c$ .

EXAMPLE 02

Square or double?

Compare $a^{2}$ and $2 a$ when $a = 5$ .

Worked solution

1
Evaluate the square.
$a^{2} = 5^{2} = 5 \times 5 = 25$
Squaring uses the value as a factor twice.
2
Evaluate the double.
$2 a = 2 \times 5 = 10$
The coefficient two multiplies the value once.
3
Compare the results.
$25 \neq 10$
Squaring and doubling are different operations and give different values for this input.

Answer

$a^{2} = 25$ and $2 a = 10$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

Write $7 \times t$ in algebraic shorthand.

Show answer & reasoning

$7 t$

Place the coefficient $7$ before $t$ : $7 \times t = 7 t$ . Leaving out the multiplication sign does not change the operation.

Keep practising

6 worksheet questions · 8 marks

Open worksheet Worked solutions

Read the questions online

Write $8 \times y$ in shorthand.
1 mark
Write $a \times b$ in shorthand.
1 mark
Write $5 \times k \times k$ in shorthand.
2 marks
Show the multiplication sign in $3 (p + 2)$ .
1 mark
Evaluate $4 b$ when $b = 6$ .
1 mark
For $x = 3$ , find $x^{2}$ and $2 x$ . Are they equal?
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Write $8 \times y$ in shorthand.
1 mark
Put the coefficient before the letter.
$8 y$
Write $a \times b$ in shorthand.
1 mark
The variable factors may be written beside one another.
$a b$
Write $5 \times k \times k$ in shorthand.
2 marks
The repeated factor gives $k^{2}$ .
Keep 5 as the coefficient.
$5 k^{2}$
Show the multiplication sign in $3 (p + 2)$ .
1 mark
The 3 multiplies the whole bracket.
$3 \times (p + 2)$
Evaluate $4 b$ when $b = 6$ .
1 mark
$4 b = 4 \times 6 = 24$ .
24
For $x = 3$ , find $x^{2}$ and $2 x$ . Are they equal?
2 marks
$x^{2} = 3 \times 3 = 9$ .
$2 x = 2 \times 3 = 6$ .
9 and 6; they are not equal.

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Curriculum reference & lesson scope

Australian Curriculum v9.0 / SA Curriculum Mathematics

Year 7 variables, expression notation, substitution, simple equivalence and formula tables. Digital methods are explained with supplied data; the topic test works by hand.

Official curriculum source ↗

In this lesson

Understand the idea.

Multiplication without a sign

Powers and division

See the method in action.

Write a product

Identify the coefficient and variable factors.

Use multiplication shorthand.

Square or double?

Evaluate the square.

Evaluate the double.

Compare the results.

Try it yourself.

Check your understanding

Keep practising