Numbers and operations · Ages 12–13
Directed numbers
From “Positive and negative numbers” to “Using directed numbers”, step by step, each skill with its methods, worked examples and answers.
Skills and methods in this topic, easy to hard
- Positive and negative numbersRight of 0 is positive, left is negative; 0 is neither
Method−3 is 3 steps left of 0; −(−3) = 3
Another way: Think of a thermometer−3 is 3 degrees below zero
- Comparing directed numbersFurther right is larger; a negative further from 0 is smaller
Method−7 < −2 < 0 < 3
Another way: Think of temperatures−7°C is colder than −2°C, so −7 < −2
- Adding directed numbersSame signs: add. Different: subtract, take the further one's sign
Method(−7) + 3 = −4
Another way: Walk the number lineFrom −7, 3 steps right, to −4
- Subtracting directed numbersTaking away a number = adding its opposite
Method5 − (−3) = 5 + 3 = 8
Another way: Look at the distanceFrom −3 to 5: 8 steps right
- Multiplying directed numbersSame signs positive, different negative; count the minus signs
Method(−3) × (−4) = 12
Another way: Repeated adding3 × (−2) = (−2) + (−2) + (−2) = −6
- Dividing directed numbersSame signs positive, different negative; never divide by 0
Method(−12) ÷ 3 = −4
Another way: Think backwards with times(−4) × 3 = −12
- The four operations mixedBrackets, powers, times and divide, plus and minus; the sign first at each step
Method(−2) × [3 − (−1)] + 5 = −8 + 5 = −3
Another way: Split the signs−5 + 8 − 3 + 2 = (8 + 2) − (5 + 3) = 2
- Using directed numbersChoose the positive direction, write the sum, answer in words
Method−3°C, falling 4°C: (−3) + (−4) = −7
Another way: Draw an upright number lineFrom −3, 4 steps down, to −7
Worked examples
From different skills in this topic. Have a go first, then open each one for the answer and the working.
Example 1Positive and negative numbers →
−(−5) =
Show the answer and working
Answer: 5
The minus outside the bracket means the opposite: the opposite of −5 is 5, flipped across 0 on the number line.
Example 2Multiplying directed numbers →
(−4) × (−7) =
Show the answer and working
Answer: 28
Same signs multiply to a positive: 4 × 7 = 28.
Example 3Using directed numbers →
It was −10°C at dawn and 17°C at noon. By how much did the temperature rise?
Show the answer and working
Answer: 27°C
17 − (−10) = 17 + 10 = 27. Up to 0°C first, then on to 17°C.
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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