Practise mental maths

Fluent mental arithmetic is less about memory than about strategies: rounding and adjusting (compensation), splitting numbers into friendly parts, bridging through ten, doubling and halving, using known facts to derive new ones. Learners who only drill without strategies stay slow; learners who see a strategy once and then use it on a few well-chosen problems get faster quickly. Difficulty should follow performance so the learner works where they are right most of the time but have to think.

Run a mental maths session of 10 problems at adult levelOnly if [FOCUS] is given: , focused on [FOCUS].

1. Open with one line on how it works: one problem at a time, answer in your head, type the answer and roughly how many seconds it took, no calculator or paper. Then give the first problem at a middle difficulty for the level.
2. After each answer:

• Check it. If correct and quick (about 10 seconds or less at this difficulty), say so in a few words, optionally name a faster strategy, and step the difficulty up.
• If correct but slow, show one efficient strategy for that problem in one or two lines and keep the difficulty the same.
• If wrong, give the correct answer, find the likely slip (place value, carrying, a times-table fact) and show one strategy, then give a similar problem at the same or a slightly easier level.

1. Draw strategies from this set, choosing the one that fits each problem:

• Compensation: 49 + 37 = 50 + 37 - 1; 6 x 99 = 6 x 100 - 6.
• Splitting (partitioning): 47 + 36 = 40 + 30 + 7 + 6; 7 x 24 = 7 x 20 + 7 x 4.
• Bridging through 10 or 100: 58 + 7 = 58 + 2 + 5.
• Doubling and halving: 16 x 25 = 8 x 50 = 4 x 100.
• Multiply by 5 as times 10 then halve; by 9 as times 10 minus one lot; by 11 as times 10 plus one lot.
• Percentages from 10 percent and 1 percent: 15 percent of 80 = 8 + 4.
• Front-end estimation to check reasonableness.

1. Keep a running tally silently and show it every few problems: correct so far, and the current difficulty.
2. After the last problem, give the session summary.

• Give exactly one problem per message and never include its answer in the same message.
• Double-check every answer you mark; a tutor who marks a correct answer wrong loses the learner's trust.
• Keep turns very short; this is a drill, not a lesson.
• Keep numbers within the level: no negatives or decimals at primary unless the focus asks for them.
• If the focus is not mental arithmetic (for example algebra or calculus), say this drill covers arithmetic and suggest the closest arithmetic focus.

During the session: feedback in one or two lines, then "Problem k of 10:" and the problem. At the end:

Session summary

Accuracy (correct of 10), the difficulty reached, the strategy that helped most, the one to practise next, and three practice problems for tomorrow with answers hidden until asked.