Mental Maths Strategies That Really Work: Round, Split and Compensate

Most of us were taught one written method for each operation and told to use it every time. Mental arithmetic rewards a different habit: look at the numbers first, then bend the problem into one that is easy. Below are four such moves, each worked through with a drawing. (Younger children building the first steps of this should start with make-ten; see Better Ways to Learn Addition and Subtraction. This post is for teens and adults.)

1. Round and compensate: 98 + 47

98 is awkward and 100 is easy. So add 100, then give back what you borrowed.

• Round: 98 is 2 less than 100, so work with 100.
• Add: 100 + 47 = 147.
• Compensate: you added 2 too many, so take 2 off: 147 - 2 = 145.

Why the last step goes the way it does: if you round up a number you are adding, the answer comes out too big, so you subtract the extra. In a subtraction it depends on which number you rounded: rounding up the number you subtract from makes the answer too big, while rounding up the number you subtract makes it too small. Always ask which way you moved.

2. Split into parts: 47 x 6

Six 47s is hard to hold in your head. Six 40s and six 7s are two easy facts.

• Split 47 into 40 and 7 (tens and ones).
• Multiply each part: 40 x 6 = 240 and 7 x 6 = 42.
• Add: 240 + 42 = 282.

The picture shows why it is allowed: the whole rectangle is the two smaller rectangles side by side. The same split works for bigger numbers; you just have more parts.

3. Halve and double: 25 x 36

• Double one number, halve the other: 25 x 36 becomes 50 x 18.
• Do it again: 50 x 18 becomes 100 x 9.
• Finish: 100 x 9 = 900.

It works because the answer does not change: 36 groups of 25 hold the same total as 18 groups of 50. Use it when one number is even and the other is something like 5, 25 or 50. It is a poor choice for, say, 47 x 6, where doubling 47 does not help; picking the method to suit the numbers is the real skill.

4. Count up to find a difference: 203 - 187

Subtracting across a hundred (203 - 187) is where mistakes creep in. Treat it as a gap and hop across it instead.

• Hop to a round number: 187 to 200 is 13.
• Hop to the target: 200 to 203 is 3.
• Add the hops: 13 + 3 = 16. Check: 187 + 16 = 203.

What research says about choosing a method

Two findings, kept to what the authors reported. Jon Star and Bethany Rittle-Johnson studied grade 7 students solving linear equations; those who compared two worked solutions side by side were more likely to use the demonstrated shortcuts afterwards than those who studied the solutions one at a time (summary from Harvard Graduate School of Education; the study is Star and Rittle-Johnson, 2008, Learning and Instruction). That was equation solving, not mental arithmetic, so treat it as a hint about comparing methods, not proof for the four above.

For mental calculation itself, Joanna Threlfall (2002, Educational Studies in Mathematics) argued that "strategy choice" is a misleading description of efficient mental calculation, and described it instead as an interaction between noticing and knowledge (ERIC record). In plain terms: the knack is spotting that 98 is close to 100, or that 25 and 36 suit halving and doubling. A list of tricks helps only if you also look at the numbers.

We are not claiming these methods make you faster or smarter. The aim is smaller: fewer slips, and a way to check your answer by a second route.

A five-minute practice

• Pick three sums from a receipt, a bill or a timetable.
• For each, look first: is a number near a round one? Does it split into tens and ones?
• Solve it one way, then check it a second way (for example 98 + 47 by rounding, then by splitting 90 + 40 and 8 + 7).
• If the two answers differ, find which step slipped. That is the useful part.

Where a puzzle adventure fits

Number-pyramid locks make you add and subtract many small numbers in a row, so they are good places to try these moves. Both books below are puzzle adventure gamebooks: you do not read them in order, and each solved lock tells you which page to turn to. They are books of puzzles, not a maths course, and they do not teach these strategies step by step. Escape the Pyramid, Level 2 (Medium) has 45 locked chambers and 6 treasure rooms, with hints and full answers in the back. In Escape the Labyrinth you choose your road through 45 locked chambers and the puzzle kind changes from door to door.

Books mentioned

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