Three drills

The 42 kinds of arithmetic

Daily Math asks the whole-number addition, subtraction, multiplication and division taught in Japanese elementary school. That material is sorted into 42 kinds, which come to 9,417 problems in all. This page goes through what is in each of them.

1Why 42 kinds

Called simply "addition", 2 + 5 and 57 + 35 count as the same problem. But what a child gets stuck on is usually not the size of the numbers - it is a particular operation, such as carrying.

So the problems are divided two ways: by the operation involved (is there a carry or a borrow?) and by how large the numbers get. Divided like this, every answer records where the difficulty was, and the app can then ask more of the kinds you find hard.

In other words, the record does not say "weak at addition" - it says "weak at two-digit addition with a carry".

2Addition - 10 kinds, 3,934 problems

Five steps without a carry, and five steps with one. The steps are the same in both: single digits, then under 30, under 50, under 80, and under 100.

  • Addition, no carry (1 digit) 2 + 5 55 problems
  • Addition, no carry (under 30) 12 + 15 55 problems
  • Addition, no carry (under 50) 22 + 25 275 problems
  • Addition, no carry (under 80) 32 + 35 825 problems
  • Addition, no carry (under 100) 42 + 45 825 problems
  • Addition with carry (1 digit) 7 + 5 45 problems
  • Addition with carry (under 30) 17 + 5 90 problems
  • Addition with carry (under 50) 27 + 15 279 problems
  • Addition with carry (under 80) 47 + 25 756 problems
  • Addition with carry (under 100) 57 + 35 729 problems

Carrying means making a group of ten. A child who can do 2 + 5 may still stop at 7 + 5, so the two are kept apart and can be practised separately.

Sums such as 29 + 1 - one single-digit term, and an answer ending in zero - are not asked. That exclusion is inherited from the Android app this was ported from.

3Subtraction - 10 kinds, 4,897 problems

As with addition: five steps without a borrow, and five steps with one.

  • Subtraction, no borrow (1 digit) 7 - 5 55 problems
  • Subtraction, no borrow (under 30) 27 - 15 255 problems
  • Subtraction, no borrow (under 50) 37 - 25 475 problems
  • Subtraction, no borrow (under 80) 67 - 35 1125 problems
  • Subtraction, no borrow (under 100) 87 - 45 1025 problems
  • Subtraction with borrow (under 20) 12 - 7 45 problems
  • Subtraction with borrow (under 30) 22 - 17 90 problems
  • Subtraction with borrow (under 50) 42 - 27 297 problems
  • Subtraction with borrow (under 80) 72 - 47 783 problems
  • Subtraction with borrow (under 100) 92 - 57 747 problems

One step does not line up with addition. The series with a borrow starts at "under 20" rather than at single digits - subtracting one single digit from another never requires a borrow in the first place.

By count this is the largest of the four groups, with 4,897 problems.

4Multiplication - 11 kinds, 293 problems

The times tables are kept as nine separate kinds, one per table. That way a gap such as "never gets the seven times table" shows up in the record.

  • Times table of 1 1 × 2 9 problems
  • Times table of 2 2 × 3 9 problems
  • Times table of 3 3 × 2 9 problems
  • Times table of 4 4 × 2 9 problems
  • Times table of 5 5 × 2 9 problems
  • Times table of 6 6 × 2 9 problems
  • Times table of 7 7 × 2 9 problems
  • Times table of 8 8 × 2 9 problems
  • Times table of 9 9 × 2 9 problems
  • Multiplication (1 × 2 digits) 2 × 13 106 problems
  • Multiplication (2 × 1 digits) 13 × 2 106 problems

The last two involve a two-digit number, and only products below 100 are asked. 2 × 13 and 13 × 2 are counted separately: the answer is the same, but the way a child works it out is not.

Multiplication and division have far fewer problems than addition and subtraction simply because there are only 81 times-table facts to begin with.

5Division - 11 kinds, 293 problems

Nine kinds, one per divisor, plus two-digit by one-digit and two-digit by two-digit.

  • Division by 1 2 ÷ 1 9 problems
  • Division by 2 6 ÷ 2 9 problems
  • Division by 3 6 ÷ 3 9 problems
  • Division by 4 8 ÷ 4 9 problems
  • Division by 5 10 ÷ 5 9 problems
  • Division by 6 12 ÷ 6 9 problems
  • Division by 7 14 ÷ 7 9 problems
  • Division by 8 16 ÷ 8 9 problems
  • Division by 9 18 ÷ 9 9 problems
  • Division (2 ÷ 1 digits) 26 ÷ 2 106 problems
  • Division (2 ÷ 2 digits) 26 ÷ 13 106 problems

Nothing with a remainder is asked. Every division here comes out exactly. A division with a remainder has two numbers as its answer, which does not fit a screen where you enter a single number.

6Problems written with a box

Besides plain problems such as 7 + 5, some are written as □ + 5 = 12. The box can stand on either side.

This asks for the inverse: you look at an addition and think in subtraction. Memorising the answer is not enough, so the same fact gets checked from another direction.

These are left out when the app is choosing the questions for you. They call for a different kind of thinking, so they are kept for when you pick the kinds yourself.

7Which school year

Roughly how this maps onto Japanese elementary school.

  • Year 1 - single-digit addition and subtraction, including carrying and borrowing
  • Year 2 - two-digit addition and subtraction, and the times tables
  • Year 3 - division, and multiplication involving two-digit numbers
  • Year 4 - division by a two-digit number

The app does not gate problems by school year, though. You can pick the kinds you have been taught, or let the app decide where you are weak and weight the questions towards that.

8What is not covered

No decimals or fractions, no word problems, no geometry, no unit conversion. This drill does one thing: whole-number arithmetic, practised until it comes quickly.

No answer ever reaches 100 either. The largest is 99, from 10 + 89.

You can try it right now. No sign-in needed.

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For a walk-through with screenshots, see the full introduction.

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