For parents
13 − 9 on a number line: subtraction by counting up
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Does 13 − 9 always mean taking nine steps backwards? Another question is: how much must we add to nine to reach thirteen? Here your child measures the gap between two numbers. The answer is the distance travelled, not the number at the finish.
Show the difference first
Arrange thirteen large blocks in one row and nine in another, lining up their starting ends. Match nine pairs. Four blocks in the longer row have no partner: that is the difference described by 13 − 9. The NCETM resource below explains subtraction as difference. Our paper activity then moves this comparison onto a number line.
Make a line with equal spaces
Mark the numbers from 0 to 15 at equal intervals on a strip of paper. Put a counter on 9 and mark 13 as the finish. Ask, “How much do we need to add to nine to make thirteen?” Nine is the starting point, so no steps have been taken yet. If ordering numbers to 15 is still difficult, stay with the blocks for now.
Take two jumps through ten
Move from 9 to 10: a jump of one. From 10 to 13 is a jump of three. Draw two arcs above the line and label them +1 and +3. Add their lengths: 1 + 3 = 4, so 13 − 9 = 4. Check with 9 + 4 = 13. Do not add 10 and 13 together: they name positions, not the distances travelled.
Play “reach the finish”
Make cards showing the pairs 8 and 12, 9 and 14, 7 and 11, and 12 and 15. The adult sets a start and finish; the child plans the route. When crossing ten, ten can be a useful stopping point. From 12 to 15, a single jump of three is enough. Write down each jump length and total them at the end. Swap roles and let your child check your route, without a timer or race.
Four questions with explanations
12 − 8 = 4: two steps from 8 to 10 and two more to 12 give 2 + 2 = 4. 14 − 9 = 5: one to 10 and four more. 11 − 7 = 4: three to 10 and one to 11. 15 − 12 = 3: three steps, with no stop at ten. Allow an attempt before revealing the answer, then check using addition, such as 8 + 4 = 12.
The starting point is not a step
An answer of 5 for 13 − 9 may come from counting 9, 10, 11, 12, 13 as five steps. Trace the four spaces between those marks instead. If the answer is 13, ask about the distance, not the finish. Write 9 + 1 = 10 and 10 + 3 = 13 separately. Do not join them into 9 + 1 = 10 + 3, because the two sides are unequal.
Choose a method that fits
Counting up can be convenient when the numbers are close. For 13 − 1, one step backwards may be simpler. A single method need not serve every subtraction. Finish by asking why the two jumps cover four units altogether. A parent can later choose a subtraction route in Otto Math, available on the App Store. The paper number line is an offline activity from this guide.