Four operations with rationals - Year 8 Mathematics

Start here: the only new thing is the sign

You already know $3 \times 4 = 12$ . The Year 8 question is: what about $(-3) \times 4$ , or $(-3) \times (-4)$ ?

The numbers are the same — you always multiply $3$ by $4$ and get $12$ . The only new skill is deciding whether the answer is positive or negative. That’s it.

What you will learn

multiply and divide integers using the sign rules,
extend the four operations to negative fractions and decimals,
apply the order of operations (BIDMAS) with signed rational numbers,
estimate and check the reasonableness of results.

1. Multiplying and dividing integers

In Year 7 you compared, ordered, added and subtracted integers. In Year 8, multiplication and division of integers become part of the core toolkit.

Worked example E Very easy: just two factors

Work out $(-3) \times 4$ .

Multiply the numbers: $3 \times 4 = 12$ .
Count negatives: one. Odd, so the answer is negative.
Answer: $-12$ .

Try another: $(-3) \times (-4)$ .

Multiply the numbers: $3 \times 4 = 12$ .
Count negatives: two. Even, so the answer is positive.
Answer: $+12$ .

Sign rules for × and ÷

Same signs give positive

(+)\times(+) = +, \qquad (-)\times(-) = +.

Different signs give negative

(+)\times(-) = -, \qquad (-)\times(+) = -.

Division follows the same pattern

\frac{+}{+} = +, \quad \frac{-}{-} = +, \quad \frac{+}{-} = -, \quad \frac{-}{+} = -.

Count the negatives

An even number of negative factors gives a positive product; an odd number gives a negative product.

Worked example 1 A mixed product

Evaluate $\;(-3) \times 4 \times (-2) \times (-5)$ .

Multiply magnitudes: $3 \times 4 \times 2 \times 5 = 120$ .
Count negatives: three. Odd, so the answer is negative.

(-3) \times 4 \times (-2) \times (-5) = -120.

2. Operations with negative rationals

The same sign rules apply to fractions and decimals.

Worked example E2 Very easy: a simple fraction sum

Evaluate $\;\dfrac{1}{2} + \dfrac{1}{4}$ .

Common denominator is $4$ : rewrite as $\dfrac{2}{4} + \dfrac{1}{4} = \dfrac{3}{4}$ .

Worked example 2 Negative fraction times fraction

Evaluate $\;\left(-\dfrac{3}{4}\right) \times \dfrac{2}{9}$ .

Magnitudes: $\dfrac{3}{4} \times \dfrac{2}{9} = \dfrac{6}{36} = \dfrac{1}{6}$ .
Sign: one negative factor; answer is negative.

\left(-\dfrac{3}{4}\right) \times \dfrac{2}{9} = -\dfrac{1}{6}.

Worked example 3 Dividing with a negative

Evaluate $\;-\dfrac{3}{8} \div \dfrac{9}{4}$ .

Keep, change, flip: $-\dfrac{3}{8} \times \dfrac{4}{9}$ .
Magnitudes: $\dfrac{12}{72} = \dfrac{1}{6}$ .
Sign: one negative; answer is negative.

-\dfrac{3}{8} \div \dfrac{9}{4} = -\dfrac{1}{6}.

Worked example 4 Signed decimals

Evaluate $\;-2.4 \times 0.5$ .

-2.4 \times 0.5 = -1.2.

(One negative $\Rightarrow$ negative; magnitudes $2.4 \times 0.5 = 1.2$ .)

3. Order of operations with signed rationals

BIDMAS still applies: Brackets, Indices, Division & Multiplication (left to right), Addition & Subtraction (left to right).

Worked example 5 Mixed operations

Evaluate $\;-\dfrac{1}{2} + \dfrac{3}{4} \times (-2)^2$ .

Indices: $(-2)^2 = 4$ .
Multiplication: $\dfrac{3}{4} \times 4 = 3$ .
Addition: $-\dfrac{1}{2} + 3 = \dfrac{5}{2}$ .

Practice: Year 8 core

Fluency

Integer × and ÷

Evaluate $(-1) \times 5$ .
Evaluate $(-2) \times (-3)$ .
Evaluate $(-6) \times 7$ .
Evaluate $(-8) \times (-5)$ .
Evaluate $9 \times (-4)$ .
Evaluate $(-3)^3$ .
Evaluate $(-2)^4$ .
Evaluate $(-42) \div 6$ .
Evaluate $(-72) \div (-8)$ .
Evaluate $(-2) \times (-3) \times (-4)$ .
Evaluate $(-1)^{100}$ .
Evaluate $(-100) \div (-25)$ .

Fluency

Negative fractions and decimals

Evaluate $-\dfrac{1}{2} + \dfrac{1}{3}$ .
Evaluate $-\dfrac{2}{5} - \dfrac{3}{10}$ .
Evaluate $-\dfrac{4}{9} \times \dfrac{3}{8}$ .
Evaluate $-\dfrac{5}{6} \div \left(-\dfrac{10}{9}\right)$ .
Evaluate $-0.25 \times 8$ .
Evaluate $-1.6 + 0.4$ .
Evaluate $-3.2 \div (-0.8)$ .
Evaluate $(-0.5)^2$ .

Reasoning

Order of operations

Evaluate $-3 + 2 \times (-4)$ .
Evaluate $(-6)^2 - 10$ .
Evaluate $-\dfrac{1}{4} \times 8 + 5$ .
Evaluate $\dfrac{-12 + 4}{-2}$ .
Evaluate $-2 \times (3 - 7)^2$ .
Evaluate $-\dfrac{3}{4} - \dfrac{1}{2} \times (-4)$ .

Reasoning

Explain and spot the mistake

Sam says $-3^2 = 9$ . Explain what is wrong and give the correct value.
Without calculating, decide whether $(-17) \times (-19) \times (-2)$ is positive or negative. Explain.
Explain in plain words why dividing a negative by a negative gives a positive.
Lee writes $\dfrac{-6 + 2}{-2} = \dfrac{-4}{-2} = 2$ . Verify whether this is right.

Problem solving

Applications

A hot-air balloon rises at $2.5$ m/s for $20$ seconds, then descends at $1.8$ m/s for $30$ seconds. What is its net change in altitude?
Temperatures in a week were $-4, -1, 3, 5, 2, -2, -3$ (°C). Find the mean temperature.
A share price drops $4\%$ one day and then rises $4\%$ the next. Is it back to the original price? Justify with a specific starting value.
A student’s score on four tests are $-3, +5, +2, -4$ (changes from the class average). What is the student’s total deviation from the average?

Challenge

Reasoning

Harder reasoning

Evaluate $\dfrac{(-2)^3 \times (-3)^2}{(-6)}$ .
Solve for $x$ : $-\dfrac{2}{3} x = 8$ .
A number $n$ satisfies $(-n)^3 = -27$ . Find $n$ .
Simplify $-\dfrac{3}{5} \left( \dfrac{10}{9} - \dfrac{2}{3} \right)$ .

Answers

Answer key

Attempt the practice first. When you're ready to check, expand the answers below.

Show the full answer key

Year 8 core - answers

Fluency

Integer × and ÷

$-5$
$6$ (negative times negative)
$-42$
$40$ (two negatives give a positive)
$-36$
$-27$ (three negative factors: odd, so negative)
$16$ (four negative factors: even, so positive)
$-7$
$9$ (two negatives cancel)
$-24$ (three negative factors: odd)
$1$ (even power of $-1$ )
$4$

Fluency

Negative fractions and decimals

$-\dfrac{1}{6}$ . Method: common denominator $6$ ; $-\dfrac{3}{6} + \dfrac{2}{6}$ .
$-\dfrac{7}{10}$ . Method: $-\dfrac{4}{10} - \dfrac{3}{10}$ .
$-\dfrac{1}{6}$ . Method: $-\dfrac{12}{72}$ .
$\dfrac{3}{4}$ . Method: $-\dfrac{5}{6} \times -\dfrac{9}{10} = \dfrac{45}{60} = \dfrac{3}{4}$ .
$-2$
$-1.2$
$4$
$0.25$

Reasoning

Order of operations

$-11$ .
$26$ .
$3$ . Method: $-\dfrac{1}{4} \times 8 = -2$ ; $-2 + 5 = 3$ .
$4$ . Method: $\dfrac{-8}{-2} = 4$ .
$-32$ . Method: $(3 - 7)^2 = 16$ ; $-2 \times 16 = -32$ .
$\dfrac{5}{4}$ . Method: $-\dfrac{1}{2} \times (-4) = 2$ ; $-\dfrac{3}{4} + 2 = \dfrac{5}{4}$ .

Reasoning

Explain and spot the mistake

$-3^2 = -(3 \times 3) = -9$ , not $9$ . The power applies only to the $3$ , so the minus sign stays in front. $(-3)^2 = 9$ ; that’s the version Sam was thinking of.
Negative. There are three negative factors; three is odd.
Dividing asks “how many of the second fit into the first”. Two negatives cancel because flipping the sign of both the “how many” and the “of what” gives the same answer.
Correct. $-6 + 2 = -4$ ; $\dfrac{-4}{-2} = 2$ .

Problem solving

Applications

Net change: $2.5 \times 20 - 1.8 \times 30 = 50 - 54 = -4$ m. The balloon is $4$ m below the starting altitude.
$0$ °C. Method: sum $= -4 - 1 + 3 + 5 + 2 - 2 - 3 = 0$ ; $0 \div 7$ .
Not back to original. Starting at $100$ : drop gives $96$ ; then rise $4\%$ gives $99.84$ . Net loss.
$0$ . Method: $-3 + 5 + 2 - 4 = 0$ .

Challenge - answers

Reasoning

Harder reasoning

$12$ . Method: top $= (-8) \times 9 = -72$ ; $-72 \div -6 = 12$ .
$x = -12$ . Method: multiply both sides by $-\dfrac{3}{2}$ .
$n = 3$ . Method: $(-n)^3 = -n^3 = -27$ , so $n^3 = 27$ , so $n = 3$ .
$-\dfrac{4}{15}$ . Method: bracket $= \dfrac{10}{9} - \dfrac{6}{9} = \dfrac{4}{9}$ ; $-\dfrac{3}{5} \times \dfrac{4}{9} = -\dfrac{12}{45} = -\dfrac{4}{15}$ .

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