Challenge · Negative Numbers

9 marks

These problems use negative numbers in calculations, substitution, temperature contexts and BIDMAS. A worked example shows how to start; hints are at the foot.

Worked example: how to start

The temperature is $-4\,^\circ$C. It rises by $9\,^\circ$C. Work out the new temperature.

A rise means add: $-4+9$.

From $-4$, going up $9$ passes zero.

$-4+9=5\,^\circ$C.

1Work out $-7+12$.[1 mark]

Answer:

2Work out $(-3)\times(-8)$.[1 mark]

Answer:

3The temperature falls from $3\,^\circ$C to $-5\,^\circ$C. Work out the fall in temperature.[2 marks]

Answer:

4Work out the value of $2a-b$ when $a=-3$ and $b=-5$.[3 marks]

Answer:

5Work out $-2+3\times(-4)$.[2 marks]

Answer:

★Extension: Work out the value of $a^2-2a$ when $a=-4$.

Stuck? Hints (don't peek unless you need to)1. Start at $-7$ on a number line and move $12$ to the right.2. Same signs give a positive.3. The fall is the DIFFERENCE: $3-(-5)$.4. Substitute carefully, then deal with the double signs.5. BIDMAS: do the multiplication first.

Solutions & mark scheme · Negative Numbers

Total: 9 marks

Award the marks shown for each correct step; many of these have more than one valid route, so give method marks for any correct working.

1Work out $-7+12$.[1]

Model solution

$-7+12=5$.

Answer: $5$

Marks

✔1$=5$

2Work out $(-3)\times(-8)$.[1]

Model solution

Same signs $\to$ positive: $(-3)\times(-8)=24$.

Answer: $24$

Marks

✔1$=24$

3The temperature falls from $3\,^\circ$C to $-5\,^\circ$C. Work out the fall in temperature.[2]

Model solution

Fall $=3-(-5)=3+5=8\,^\circ$C.

Answer: $8\,^\circ\text{C}$

Marks

✔1Uses $3-(-5)$

✔1$=8\,^\circ$C

4Work out the value of $2a-b$ when $a=-3$ and $b=-5$.[3]

Model solution

$2a-b=2(-3)-(-5)$.

$=-6+5$.

$=-1$.

Answer: $-1$

Marks

✔1$2(-3)=-6$

✔1$-(-5)=+5$

✔1$=-1$

5Work out $-2+3\times(-4)$.[2]

Model solution

Multiply first: $3\times(-4)=-12$.

$-2+(-12)=-2-12=-14$.

Answer: $-14$

Marks

✔1$3\times(-4)=-12$

✔1$=-14$

★Extension

$a^2=(-4)^2=16$ (a square is always positive).

$-2a=-2\times(-4)=+8$.

So $a^2-2a=16+8=24$.