Cambridge Primary Checkpoint · Mathematics 0096 · Stage 6 · Number

Counting On and Back and Negative Numbers

Count on and back in whole-number, fraction and decimal steps past zero, and add, subtract and estimate with negative numbers.

British & IGCSE › Checkpoint 6 › Checkpoint 6 Maths — Complete Course

In the curriculum

Count on and count back in equal steps — whole numbers, fractions and decimals — and keep going past zero into the negative numbers (6Nc.01)
Estimate, add and subtract integers (positive and negative whole numbers) (6Ni.01)

Negative numbers are everywhere: the temperature on a winter night in Saint Catherine in Sinai can fall below 0 °C, a lift in a shopping mall goes down to floor −1 and −2 for the car park, and the shore of the Dead Sea is about 430 metres below sea level. On this page you will learn to count past zero, to place negative numbers on a number line, and to add and subtract with them confidently.

Key words

Word / phrase	Meaning / use
integer	عدد صحيح — a whole number: positive, negative or zero (…, −3, −2, −1, 0, 1, 2, 3, …)
negative number	عدد سالب — a number less than zero, written with a minus sign: −5
positive number	عدد موجب — a number greater than zero: 5 (we do not need to write +5)
zero	صفر — neither positive nor negative
number line	خط الأعداد — a line with numbers in order; numbers get bigger as you go right
step / constant step	الخطوة الثابتة — the same amount added or subtracted each time
count on / count back	العد للأمام / للخلف — add the step each time / subtract the step each time
difference	الفرق — how far apart two numbers are on the number line
estimate	تقدير — a quick, rounded answer to check that your exact answer makes sense

1. The number line goes both ways

Zero is not the end of the number line. Numbers carry on to the left of zero: −1, −2, −3 and so on. We say −3 as negative three (not "minus three" when it is a number on its own).

A number line from −10 to 10. Every number is bigger than all the numbers to its left.

The further right a number is, the bigger it is. So 2 > −5 and −1 > −8.
The further left a number is, the smaller it is. So −9 < −4, even though 9 is bigger than 4.
−7 and 7 are the same distance from zero (7 steps), but on opposite sides.
Temperatures: −9 °C is colder than −4 °C because −9 is further below zero.

Ordering integers: put them on a mental number line. Example: order −3, 5, −8, 0, −1 from smallest to largest → −8, −3, −1, 0, 5.

2. Counting on and counting back past zero

A sequence that goes up or down by the same amount each time has a constant step. To continue it, keep adding (counting on) or keep subtracting (counting back) the same step — and do not stop at zero.

Find the step: subtract one term from the next one (for example 4 − 7 = −3, so the step is "back 3").
Check the step is the same between every pair of terms.
Keep applying the step. When you cross zero, split the jump: from 1 counting back 3 → back 1 to 0, then back 2 more to −2.
Check your last answer by counting the other way.

Counting back in steps of 3 from 7: 7, 4, 1, −2, −5. The jump from 1 to −2 crosses zero.

3. Counting in fractions and decimals

The step can be a fraction or a decimal. The method is exactly the same. It often helps to think in smaller units: a step of 0.3 is "3 tenths"; a step of ¾ is "3 quarters".

Start	Step	Sequence	Tip
1.2	back 0.4	1.2, 0.8, 0.4, 0, −0.4, −0.8	think: 12 tenths, 8 tenths, 4 tenths, 0, −4 tenths…
−1.5	on 0.5	−1.5, −1, −0.5, 0, 0.5, 1	two steps of 0.5 make 1 whole
1	back ¼	1, ¾, ½, ¼, 0, −¼, −½	simplify: 2/4 = ½
−2	on ¾	−2, −1¼, −½, ¼, 1	see the number line below

Counting on in steps of ¾ from −2. The small ticks are quarters.

4. Adding and subtracting integers on a number line

When we add or subtract a positive number, the rule is simple:

Adding a positive number → move right (the answer gets bigger).
Subtracting a positive number → move left (the answer gets smaller).
Always start at the first number in the calculation.

Example: −3 + 5. Start at −3 and jump 5 to the right. Use zero as a stepping stone: −3 to 0 is 3, then 2 more → 2.

−3 + 5 = 2: start at −3, move 5 to the right.

Example: 2 − 6. Start at 2 and jump 6 to the left: back 2 to reach 0, then back 4 more → −4.

2 − 6 = −4: split the jump at zero (2 then 4).

Example: −4 − 9. You start below zero and move further left, so the answer is more negative: 4 + 9 = 13, so −4 − 9 = −13.

Calculation	Think	Answer
−7 + 10	−7 to 0 is 7, then 3 more	3
−8 + 5	move right 5, but you do not reach 0	−3
4 − 11	4 to 0 is 4, then 7 more	−7
−5 − 8	further left: 5 + 8	−13
−12 + 12	back to zero	0

Going further: adding a negative number

Adding a negative number moves you left, the same as subtracting: 5 + (−2) = 5 − 2 = 3. You will meet this much more in Stage 7.

5. The difference between two integers

The difference is the distance between two numbers on the number line. It is never negative. Difference questions often use temperatures.

One negative, one positive: add the two distances from zero. From −4 to 6: 4 + 6 = 10.
Both negative: subtract the distances. From −9 to −2: 9 − 2 = 7.
Both positive: subtract as usual. From 3 to 11: 8.

The difference between −4 and 6 is 10 — cross zero in two parts.

الفكرة الأساسية

الأعداد السالبة على يسار الصفر، وكل ما نروح يمين العدد يكبر.
−9 أصغر من −2 (زي الحرارة: −9 أبرد).
الجمع = نتحرك يمين، الطرح = نتحرك شمال، ولما نعدّي الصفر نقسم القفزة على مرحلتين.
الفرق بين عدد سالب وعدد موجب = نجمع المسافتين من الصفر: من −4 إلى 6 = 4 + 6 = 10.

6. Estimating with integers

To estimate, round each number to a friendly number (nearest 10 or nearest 100), then calculate. The estimate tells you the size and the sign (positive or negative) of the answer, so you can spot mistakes.

Calculation	Estimate	Exact answer
−48 + 103	−50 + 100 = 50	55
−297 + 512	−300 + 500 = 200	215
61 − 189	60 − 190 = −130	−128
−405 − 196	−400 − 200 = −600	−601

Worked examples

Worked example 1 — Here is a sequence that counts back in equal steps: 10, 6, 2, … Write the next two numbers. [1 mark]

Step: 6 − 10 = −4, and 2 − 6 = −4. The sequence counts back in 4s.
2 − 4: back 2 to 0, then back 2 more → −2.
−2 − 4 = −6.
Answer: −2, −6

Common mistake: writing 2, 0 or stopping at zero. The sequence carries on past zero.

Worked example 2 — Write the two missing numbers in this sequence: 1.5, 0.9, 0.3, ☐, ☐ [1 mark]

Step: 0.9 − 1.5 = −0.6 (15 tenths → 9 tenths is back 6 tenths).
0.3 − 0.6: back 0.3 to zero, then back 0.3 more → −0.3.
−0.3 − 0.6 = −0.9.
Answer: −0.3, −0.9

Worked example 3 — Laila counts on in steps of ¾, starting at −2. Write the first five numbers she says. [1 mark]

−2 + ¾ = −1¼ (move right ¾ but you do not reach −1).
−1¼ + ¾ = −½.
−½ + ¾ = ¼ (½ to reach 0, then ¼ more).
¼ + ¾ = 1.
Answer: −2, −1¼, −½, ¼, 1

Worked example 4 — The table shows the lowest temperature in two cities on the same night.

City	Lowest temperature
Cairo	9 °C
Moscow	−7 °C

(a) How many degrees colder was Moscow than Cairo? [1 mark] (b) The next night the temperature in Moscow fell by another 5 degrees. What was the new temperature? [1 mark]

(a) One temperature is positive and one is negative, so add the distances from zero: 9 + 7 = 16. Answer: 16 degrees
(b) "Fell" means subtract: −7 − 5 = −12. Answer: −12 °C

Common mistake: in (a) doing 9 − 7 = 2. That is the difference between 9 and positive 7.

Worked example 5 — Work out: (a) −8 + 13 (b) 6 − 15 (c) −4 − 9 [3 marks]

(a) −8 to 0 is 8, then 5 more → 5
(b) 6 back to 0 is 6, then 9 more → −9
(c) both moves are to the left: 4 + 9 = 13 → −13

Worked example 6 — Estimate the answer to −297 + 512. Then work out the exact answer. [2 marks]

Round: −297 ≈ −300 and 512 ≈ 500.
Estimate: −300 + 500 = 200.
Exact: −297 to 0 is 297. We still need to move 512 − 297 = 215 more. So the answer is 215.
Check: 215 is close to 200 ✓ Answer: estimate 200, exact 215

Common mistakes

Thinking −8 is bigger than −3 because 8 > 3. On the number line −8 is further left, so −8 < −3.
Stopping at zero when counting back, or jumping from 1 straight to −1 (forgetting 0).
Getting the wrong sign: 4 − 11 is −7, not 7. Ask: did I end up left of zero?
Difference across zero: −5 to 8 is 13 (5 + 8), not 3.
Counting the ticks instead of the jumps on a number line. Count the spaces between ticks.
Decimal steps: after 0.2, counting back 0.3 gives −0.1, not −0.3 or −1.

How Checkpoint asks this

"Write the missing numbers in the sequence" — with boxes, often crossing zero, sometimes with decimals or fractions.
Read a thermometer or a scale, then "the temperature rises/falls by … degrees".
"Complete the table" of temperatures with a difference column.
Ring the coldest temperature, or put integers in order from smallest to largest.
Missing-number calculations such as −6 + ☐ = 4 — use the number line to count the jump.
"Estimate" — show the rounded numbers you used. A sketch of a number line counts as working.
Technique: always check the sign of your answer with a quick number line sketch in the margin.

خلي بالك

كلمة rises / warmer = نجمع (يمين). كلمة falls / colder = نطرح (شمال).
في المتتابعات: ما توقفش عند الصفر — كمّل في السالب: 2، −2، −6 …

Practice

Remember

Numbers get bigger to the right on the number line: −2 > −7.
To continue a sequence, find the step and keep going — past zero into negative numbers. Fractions and decimals work the same way.
Adding a positive number → move right; subtracting → move left. Split the jump at zero.
Difference across zero = add the two distances from zero.
Estimate first (round to 10s or 100s) and check the sign of your answer.

All pages in this course

Start here

Checkpoint 6 Maths: The Test and How to Prepare

Number: integers and calculation

Number: place value, decimals, fractions, percentages and ratio

Geometry and measure

Statistics and probability

Practice tests