Mental Addition and Subtraction Strategies That Make Sense

Partitioning, bridging and compensation are three ways to reshape a calculation without changing its value. Partitioning breaks a number into place-value parts. Bridging moves through a helpful landmark such as the next multiple of 10, 100 or 1,000. Compensation changes a number to make the calculation easier, then corrects that change. The best mental method is the one that fits the numbers, can be explained clearly and keeps the working manageable.

Year 5 pupils should not have to force every addition or subtraction into one method. The national curriculum says they should add and subtract mentally with increasingly large numbers and decide which operations and methods to use in multi-step problems.[2] That is a curriculum reference point, not a claim that every child should hold every step in their head or that every 11+ paper uses the same format.

Contents

Three mental strategies at a glance

How to choose an efficient method

Year 5 example 1: 368 + 157

Year 5 example 2: 704 − 286

More examples for choosing, not guessing

A parent practice sequence

Common mistakes

Adjustment cues

Frequently asked questions

Next step

Three mental strategies at a glance

Partitioning: split by place value

Partitioning rewrites a number as useful parts. For example:

247 = 200 + 40 + 7

So 356 + 247 can become:

356 + 200 + 40 + 7

Partitioning is dependable when the parts are easy to track. It also makes place value visible. The parts do not always have to be hundreds, tens and ones: 96 might be split as 90 + 6, but it could be 4 + 92 if reaching the next hundred is more useful.

Bridging: travel through a landmark

Bridging splits one number so the first step lands on a helpful boundary. For 476 + 38, bridge through 500:

476 needs 24 to reach 500;

split 38 into 24 + 14;

476 + 24 = 500;

500 + 14 = 514.

For subtraction, bridging can mean counting up from the smaller number to the larger one. The total distance is the difference. It is especially useful when the numbers are close or when the route passes through tidy multiples.

Compensation: change, calculate, correct

Compensation rounds one number to a nearby friendly value and then undoes the change. For 476 + 38:

476 + 40 − 2 = 516 − 2 = 514

The adjustment must preserve equality. Replacing 38 with 40 adds 2 too much, so 2 must be subtracted. In 704 − 298, subtracting 300 removes 2 too much, so 2 must be added back:

704 − 300 + 2 = 406

Compensation is often efficient near a multiple of 10, 100 or 1,000. It is not automatically best merely because a number can be rounded.

How to choose an efficient method

Look at the number structure before calculating.

Choose partitioning when hundreds, tens and ones give short, secure steps: 425 + 230 or 786 − 321.

Choose bridging when one number is close to the next landmark, or when finding a difference by counting up is shorter: 596 + 27 or 498 to 503.

Choose compensation when a number is just below or above a tidy multiple: +99, −198, +302 or −49.

Choose a different mental route if your first choice creates too many held steps. Efficiency includes accuracy and explainability, not just the fewest written symbols.

EEF guidance recommends teaching a range of mental, calculator and pencil-and-paper methods and encouraging pupils to consider when different methods are appropriate and efficient.[3] Its scope is classroom teaching in Key Stages 2 and 3. Here, that principle is applied cautiously to short home conversations; the sequence below is not a tested intervention and does not imply that mental calculation is always preferable.

A useful prompt is: “What do you notice about the numbers?” This asks for structure before speed. A child might notice that 198 is almost 200, that 368 is 32 away from 400, or that 704 and 286 can be connected through 300 and 700.

Year 5 example 1: 368 + 157

All three routes below are valid. Comparing them shows why method choice depends on the numbers.

Partitioning

Split 157 by place value:

368 + 157 = 368 + 100 + 50 + 7 = 468 + 50 + 7 = 518 + 7 = 525

This route uses familiar place-value steps. It is reliable, although it asks the child to retain several intermediate totals.

Bridging through 400

368 needs 32 to reach 400. Split 157 into 32 and 125:

368 + 157 = 368 + 32 + 125 = 400 + 125 = 525

This is mathematically neat because 400 is easy to work from. However, splitting 157 into 32 and 125 may not be the most obvious choice for every child.

Compensation

157 is close to 160:

368 + 157 = 368 + 160 − 3 = 528 − 3 = 525

Here compensation is probably the shortest route: add three extra, then remove three.

Explicit verification

Use the inverse operation:

525 − 157 = 525 − 100 − 50 − 7 = 425 − 50 − 7 = 375 − 7 = 368

The inverse returns the starting addend, so 368 + 157 = 525 is verified. A rough magnitude check agrees: 368 is about 370 and 157 is about 160, giving about 530, so 525 is reasonable.

Year 5 example 2: 704 − 286

Subtraction makes the direction of compensation particularly important.

Partitioning

Split 286 into 200, 80 and 6:

704 − 286 = 704 − 200 − 80 − 6 = 504 − 80 − 6 = 424 − 6 = 418

This route is systematic. Crossing from 504 to 424 requires secure subtraction of eight tens.

Bridging by counting up

Find the distance from 286 to 704:

286 to 300 is 14;

300 to 700 is 400;

700 to 704 is 4;

total difference: 14 + 400 + 4 = 418.

Therefore, 704 − 286 = 418. This route turns subtraction into a difference and uses clear landmarks.

Compensation

286 is 14 less than 300. Subtract 300, then add 14 back:

704 − 286 = 704 − 300 + 14 = 404 + 14 = 418

The correction is addition because subtracting 300 removed 14 more than subtracting 286 should have removed.

Explicit verification

Add the difference to the smaller number:

418 + 286 = 418 + 200 + 80 + 6 = 618 + 80 + 6 = 698 + 6 = 704

Because the sum returns the original larger number, 704 − 286 = 418 is verified. A magnitude check also fits: about 700 − 300 is about 400, and 418 is plausible.

More examples for choosing, not guessing

499 + 236

Compensation exposes the useful structure:

499 + 236 = 500 + 236 − 1 = 735

Partitioning would also work, but adding 400, 90 and 9 creates more steps. Bridging by adding 1 to reach 500 and then the remaining 235 is effectively the same structural idea expressed as a journey:

499 + 1 + 235 = 735

603 − 198

Compensate around 200:

603 − 198 = 603 − 200 + 2 = 405

Verify:

405 + 198 = 405 + 200 − 2 = 605 − 2 = 603

487 − 462

The numbers are close, so bridge by counting up:

462 to 470 is 8;

470 to 480 is 10;

480 to 487 is 7;

8 + 10 + 7 = 25.

Partitioning 462 away from 487 is possible but cumbersome. The small difference makes counting up more efficient.

A word problem still needs an operation decision

A child has 704 points and spends 286. Once subtraction has been selected, any sound mental strategy may produce the difference. Choosing the operation and choosing the calculation method are separate decisions. This article stays with the second decision: once addition or subtraction has been selected, which mental route best fits the numbers?

A parent practice sequence

Use this as an adjustable sequence, not a fixed programme.

1. Secure the building blocks

Check number bonds to 10 and 100, complements to the next multiple of 10, and adding or subtracting whole hundreds and tens. If 68 + 32 or 300 − 70 is effortful, reduce the size before comparing methods.

2. Name one strategy at a time

Model a short example and say what stays equal: “I changed 198 to 200, so after subtracting 200 I add 2 back.” Ask the child to repeat the equality, not merely the answer.

3. Compare methods on one calculation

Use 368 + 157. Write partitioning, bridging and compensation side by side. Ask:

Which route has the fewest difficult steps for you?

Where is the helpful landmark?

What adjustment must be undone?

How could you verify the result?

EEF guidance says worked examples can help pupils focus on reasoning and strategies, and it advises teaching pupils to use and compare different approaches.[3] This supports comparison, but it does not establish one universal home-practice routine.

4. Offer a small mixed set

Try six calculations with visible cues in the numbers: +99, −201, a small difference, a clean place-value split, and two where more than one route is sensible. Ask for the strategy name and one line of explanation before the answer.

5. Remove the strategy label

Now ask the child to choose. Accept a method that is different from yours if it preserves the value and can be explained. The aim is flexible selection, not guessing the adult’s preferred route.

6. Add a check

Use an inverse calculation or a rough magnitude check. Keep this brief: verification should confirm the answer without repeating the entire calculation in the same way.

Common mistakes

Compensating in the wrong direction

For 603 − 198, subtracting 200 removes 2 too much, so add 2 back. Say the change aloud: “I subtracted two extra; I must restore two.”

Changing both numbers without preserving the value

Adjustments are allowed only when equality is maintained. In addition, adding 1 to one addend requires subtracting 1 elsewhere if the total is to stay unchanged. In subtraction, adding the same amount to both numbers preserves their difference: 603 − 198 = 605 − 200.

Calling every split “partitioning”

Splitting 38 into 24 and 14 specifically to land on 500 is bridging, not ordinary place-value partitioning. The labels matter less than recognising why the split helps, but precise language makes comparison clearer.

Choosing by habit

A child may always partition because it feels safe, even for 399 + 247. Do not ban the reliable method. First acknowledge it, then compare it with 400 + 247 − 1.

Holding too many steps mentally

Mental does not mean invisible. Jotting one interim number or drawing an empty number line can protect the reasoning. The goal is to understand and control the method, not to overload memory.

Adjustment cues

If place value is shaky: use smaller numbers and say each part: 243 = 200 + 40 + 3.

If bridging is unclear: draw an empty number line and mark only the start, landmark and finish.

If compensation direction is confused: use the sentence “I changed ___ by ___, so I must ___.”

If the child uses one method for everything: present pairs such as 428 + 371 and 428 + 399, then ask what changed in the number structure.

If explanations are secure but slow: practise recognising friendly numbers without completing the calculation: “Which number would you adjust, and by how much?”

If answers are quick but unreliable: require an inverse check for the next three questions and reduce the number size if needed.

If the child is consistently accurate: ask for two valid routes and a reason one is more efficient for that calculation.

Frequently asked questions

Is partitioning the same as column addition or subtraction?

No. Partitioning uses place-value parts within a mental route. Column methods are formal written algorithms and are outside this article’s scope.

Must a child always use the shortest method?

No. A slightly longer route may be more secure. Efficient means economical enough, accurate and explainable for that child and those numbers.

Is counting up valid for subtraction?

Yes. Subtraction can represent difference. If the gap is easy to build through landmarks, counting up is a valid way to find it.

When is compensation useful?

Look for a number near a multiple of 10, 100 or 1,000, such as 49, 198, 301 or 999. The correction should be easier than calculating with the original number.

Should every calculation be done mentally?

No. The calculation size, purpose and the child’s current understanding matter. This article compares three mental strategies; it does not argue against jottings, formal written methods or calculators in suitable contexts.

How can I ask for an explanation without making practice feel like a test?

Use a neutral prompt: “Show me the route you chose,” or “What did you notice first?” Ask about one calculation in depth rather than demanding a commentary on every answer.

Next step

Choose two calculations and ask your child to predict the most helpful route before solving. Then verify each result with an inverse or magnitude check. Try a Maths lesson to see guided Maths practice in context.

Sources

[2] https://www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study [3] https://educationendowmentfoundation.org.uk/education-evidence/guidance-reports/maths-ks-2-3