Contents
How to use this guide
Choose only calculations that fit mathematics your pupils are already studying. Ask what they noticed first, what part they worked out, and whether there is another route. Keep a written method available so the class can check the mental route against the original expression.
If pupils need support, allow counters, fingers, a ten-frame, an open number line, an area sketch, partner talk, or brief jottings. The aim is not "no marks on paper". The aim is that the thinking can still be explained clearly.
You can pair these prompts with related resources such as math games for children and multiplication practice.
Avoid using speed as the main success measure. A more useful check is whether the answer is correct, the strategy fits the numbers, and the pupil can explain why it works.
1. Find each student’s starting strategy
Begin with one accessible calculation and listen before supplying a method. Then offer a second calculation with similar structure and ask whether the pupil would begin the same way.
Useful prompts: What did you notice first? What part did you work out? Is there another route? Record the words, drawings, or jottings the pupil uses so you can choose the next example deliberately.
The teaching examples and pupil responses below are hypothetical. Adapt the materials and questions to the children you teach.
Give 38 + 27. A pupil says, "I did 30 + 20 = 50 and 8 + 7 = 15, so 50 + 15 = 65." Then give 48 + 27. If the pupil again starts by separating tens and ones, you have evidence that place-value partitioning is a natural starting strategy for that pupil.
2. Make ten first
A ten-frame can be represented as 2 rows of 5, making 10 boxes in total. Investigating ten-frames
In one NSW ten-frame example, 10 is shown as 7 and 3, and also as 3 and 7. Ten-frame filler
Prepare number pairs that make 10 before moving to longer additions. When helpful, keep the benchmark visible with counters, fingers, or a ten-frame, and ask pupils to point to the part they would use first.
Worked example with a two-row five-column ten-frame: Show 7 counters, with all 5 spaces in the top row filled and 2 spaces in the bottom row filled. Add 3 counters to the 3 empty bottom-row spaces. Say, "7 needs 3 to make 10, so 7 + 3 = 10." Then extend to 7 + 5 by moving 3 of the 5 into the frame first: 7 + 3 = 10, and 2 are still left, so 10 + 2 = 12.
Once pupils can explain pairs to 10, repeat the structure with larger benchmarks such as bridging to 20 or 100, but keep the benchmark visible during first explanations.
3. Partition by place value
Give a two-digit addition or subtraction calculation and ask pupils to name the tens and ones they notice before anyone writes. Model one example with the parts on separate lines, then pause to discuss what each line represents.
Worked example: 56 + 37. Partition into tens and ones: 50 + 30 = 80, and 6 + 7 = 13. Combine the partial sums: 80 + 13 = 93. The explanation matters: 56 is 5 tens and 6 ones, and 37 is 3 tens and 7 ones, so the calculation is being rebuilt from place-value parts.
For partner talk, one pupil can describe the partition while the other records the spoken steps. Then compare the mental route with a written calculation.
4. Count on or back strategically
Present subtraction questions with different-sized gaps and ask pupils to decide on a direction before they begin. An open number line can help them label each jump and explain why they chose that route.
Worked example: 63 – 58. Count on from 58 because the numbers are close together. Jump from 58 to 60, which is 2, then from 60 to 63, which is 3 more. The total difference is 2 + 3 = 5, so 63 – 58 = 5.
Place two possible routes side by side when useful. For the same subtraction, one pupil might count on in small jumps while another groups the jumps into a jump to the next ten and a final jump.
5. Build from doubles and near doubles
One NSW card task asks players to look for doubles, near doubles, and combinations to 10 and 20. Turn over 3 (using known facts to 20)
Display a known double beside a nearby addition and ask pupils which double they would start from and what would need to change.
Worked example: 8 + 9. Start from the known double 8 + 8 = 16. Because 9 is 1 more than 8, add 1 more: 16 + 1 = 17. So 8 + 9 = 17.
Collect more than one explanation when possible. Some pupils may start from double 9 and subtract 1. Others may start from double 8 and add 1.
6. Compensate after rounding
Choose calculations near a whole-ten benchmark. Ask pupils to state three things separately: the number they adjusted, the temporary calculation, and the correction needed to return to the original expression.
Worked example: 39 + 26. Adjust 39 up to 40. The temporary calculation is 40 + 26 = 66. Because the original addend was 1 less than 40, subtract 1 from 66. So 39 + 26 = 65.
Subtraction example: 63 – 19. Adjust 19 up to 20. Calculate 63 – 20 = 43. Because you subtracted 1 too much, add 1 back. So 63 – 19 = 44.
Do not accept the answer alone. Ask pupils to connect both the adjustment and the correction back to the original numbers.
7. Bridge through a benchmark
Write a calculation near a multiple of 10 or 100 and ask pupils to identify the benchmark before they calculate. Then ask them to split one number so the first part reaches the benchmark exactly.
Worked example: 58 + 7. Bridge through 60 by splitting 7 into 2 and 5. First, 58 + 2 = 60. Then 60 + 5 = 65. So 58 + 7 = 65. The split is chosen because 2 reaches the benchmark exactly.
Keep the original expression visible while pupils explain. That reduces the risk of losing track of the leftover part after the bridge.
8. Distribute multiplication
NSW describes the distributive property as a way to simplify multiplication problems by breaking numbers apart. Numeracy skills for primary school
Give a multiplication expression and ask pupils how they could separate one factor into parts. Record the chosen parts clearly before any calculation begins.
Worked example: 13 × 4. Split 13 into 10 and 3. Then calculate 10 × 4 = 40 and 3 × 4 = 12. Add the partial products: 40 + 12 = 52. In an area sketch, the total rectangle is split into a 10 by 4 part and a 3 by 4 part.
Ask pupils to match every part of the sketch or expansion back to the original expression, then check with a written calculation.
9. Double and halve factors
A NSW number-talk example shows a strategy where one factor is doubled and the other is halved. Doubling and halving: number talk
Prepare factor pairs and invite pupils to rewrite the expression in equivalent forms. Ask them to say why a particular reorganisation looks convenient.
Worked example: 25 × 8. Halve 8 and double 25: 25 × 8 = 50 × 4. Repeat: 50 × 4 = 100 × 2. Now 100 × 2 = 200. So 25 × 8 = 200.
Use this only when the halving step stays exact and manageable for your pupils. If the reorganisation makes the numbers harder to handle, keep it as a comparison rather than forcing it.
10. Scale from known facts
Place one secure multiplication fact beside related calculations that change the place value. Ask pupils what stays the same in the basic fact and what changes in the size of the numbers.
Worked example: If a pupil knows 6 × 4 = 24, then 60 × 4 can be read as 6 tens × 4. That is 24 tens, which is 240. So 60 × 4 = 240.
Keep the place-value language explicit. Ask pupils to say "24 tens" before they say "240" so the zero is not treated as an attachment trick.
Transfer example: 6 × 40 is also 24 tens, so 6 × 40 = 240. Comparing 60 × 4 and 6 × 40 helps pupils see that the basic fact is the same even though the place value shifts.
11. Use square-number relationships accurately
Use this strategy only when square numbers are already part of current mathematics work. Present one relationship, then ask pupils to verify it with a written expansion or an area model before treating it as a usable shortcut.
Worked example: 19 × 21 can be seen as (20 – 1)(20 + 1). Use the square relationship in the middle number: 20 squared minus 1 squared, or 400 – 1 = 399. So 19 × 21 = 399.
Check the same example by distribution: 19 × 21 = 19 × 20 + 19 = 380 + 19 = 399. The check matters because it shows the relationship matches the original expression.
This is a more advanced route than make-ten, partitioning, or near doubles. If pupils do not yet recognise nearby square numbers confidently, use a different strategy and revisit this later.
12. Revisit strategies through retrieval practice
Mathematics Hub notes that there is no fixed rule for how often students should review learning or how far apart reviews should be spaced. Interleaved practice
Set a short, low-stakes mixed practice task in which pupils solve and then name the strategy they chose. Include earlier content as well as current work, allow quiet thinking time, and discuss at least one error through a worked example.
Mini retrieval set for previously taught methods:
1. 39 + 26
2. 8 + 9
3. 13 × 4
Ask pupils to solve, name a strategy and explain one step before showing the answers.
Hypothetical worked response for item 1: “I used compensation. I changed 39 to 40 and calculated 40 + 26 = 66. I added 1 too much, so I took 1 away: 66 – 1 = 65. Therefore 39 + 26 = 65.”
Check the other answers: 8 + 9 = 17 by near doubles; 13 × 4 = 52 by distribution. If a pupil gives 66 for item 1, ask which calculation they solved and model the correction step. Use that response to choose a similar compensation question for a later retrieval opportunity, rather than treating the answer alone as evidence that the method is secure.
Use the responses to decide what to revisit next. If many pupils get the answer correct but cannot explain the route, choose simpler numbers and model the explanation again.
Quick diagnostic and transfer set
Use this as a short check only for content already taught. Item 8 is optional and should be used only if pupils already know the square-number relationship involved.
Ask pupils to solve each item, name a strategy that fits, and explain one step aloud or in a jotting.
1) 7 + 6
2) 56 + 37
3) 63 – 58
4) 39 + 26
5) 58 + 7
6) 25 × 8
7) 60 × 4
8) 19 × 21
Explained answer guide
1) 7 + 6 = 13
One route is make ten: 7 + 3 = 10, then 3 more makes 13.
Another route is near doubles: 6 + 6 = 12, then add 1.
2) 56 + 37 = 93
Partition by place value: 50 + 30 = 80 and 6 + 7 = 13, then 80 + 13 = 93.
3) 63 – 58 = 5
Count on from 58 to 63: 58 to 60 is 2, and 60 to 63 is 3, so the total difference is 5.
4) 39 + 26 = 65
Compensate: 40 + 26 = 66, then subtract 1 because 39 is 1 less than 40.
5) 58 + 7 = 65
Bridge through 60: split 7 into 2 and 5, then 58 + 2 = 60 and 60 + 5 = 65.
6) 25 × 8 = 200
Double and halve: 25 × 8 = 50 × 4 = 100 × 2 = 200.
7) 60 × 4 = 240
Scale from a known fact: 6 × 4 = 24, so 6 tens × 4 = 24 tens = 240.
8) 19 × 21 = 399
Optional square-number relationship: (20 – 1)(20 + 1) = 20 squared – 1 squared = 400 – 1 = 399.
Responses to specific misconceptions
If a pupil says 7 + 6 = 10 after making ten, ask, "How many of the 6 have you used, and how many are still left?"
If a pupil says 63 – 58 = 15 by subtracting the smaller digit from the larger digit in each place, ask them to show the distance on a number line from 58 to 63.
If a pupil says 39 + 26 = 66, ask, "Which problem did you actually solve?" Then return to the correction step.
If a pupil changes 25 × 8 into 50 × 16, remind them that doubling both factors changes the product. Check with a smaller example such as 3 × 4 and 6 × 8.
If a pupil says 60 × 4 = 24 or 640, return to the place-value statement: 6 tens × 4 = 24 tens.
If a pupil says 19 × 21 equals 19 squared or 21 squared, ask them to identify the middle number first and then verify the result with distribution.
A correct answer does not always show flexible strategy choice. If a pupil gets the answer but cannot explain why the route works, give a similar calculation and ask them to compare two methods.
