doing math

courses › Mental Arithmetic

Small facts

level 1 course

Single-digit addition and subtraction until the answer is instant.

Best in your head · no pen, no calculator why?

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the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

Read the lesson · about 3 minutes

The idea

Every small sum and difference comes from the same 55 addition facts, all the pairs from 0 + 0 up to 9 + 9. Subtraction is those facts read backward: 15 − 8 asks what goes with 8 to make 15.

The goal is recall, not counting. Counting on your fingers is slow, and it is easy to land one off. Lean on two anchors you already know, ten and doubles. Every other fact is one short step from one of them. After a few rounds, the answer just arrives.

Techniques

Make ten

The two numbers add up to more than 10.

  1. Take the bigger number and ask how much it needs to reach 10.
  2. Take that much away from the smaller number.
  3. The answer is 10 plus what is left.
worked example

Example: 8 + 7

  1. 8 needs 2 to reach 10.
  2. 7 − 2 = 5 left over.
  3. 10 + 5 = 15.

Answer: 15

Doubles and near doubles

The two numbers are the same, or one apart.

  1. Know the doubles cold: 6 + 6 = 12, 7 + 7 = 14, 8 + 8 = 16, 9 + 9 = 18.
  2. For numbers one apart, double the smaller one.
  3. Then add 1.
worked example

Example: 7 + 8

  1. Double the smaller one: 7 + 7 = 14.
  2. Add 1: 14 + 1 = 15.

Answer: 15

Go through ten

A subtraction or missing number where the big number is 10 or more.

  1. Count up: read 15 − 8 as "8 plus what makes 15?"
  2. Jump from the small number to 10, then from 10 to the big number. Add the two jumps.
  3. Or count back through ten: 15 − 5 = 10, then 3 more back is 7.
worked example

Example: 15 − 8

  1. 8 up to 10 is 2.
  2. 10 up to 15 is 5.
  3. 2 + 5 = 7.

Answer: 7

Tips by skill

  • TipSums to 18: Crossing 10? Top the bigger number up to 10, then add what is left of the other one.
  • TipDifferences under 20: Read it as a missing number: count up from the number taken away to the first one, through 10 if you pass it.
  • TipMissing addend: Count up from the number you have to the total, through 10 if you pass it. Add the jumps.

Watch out for

  • Adjusting a double the wrong way: 8 + 7 is one less than 8 + 8, not one more.
  • Counting back and counting the starting number too. That makes the difference one too big.
  • Adding in a missing-number problem. In 6 + ? = 13 the answer is the gap between 6 and 13, not their sum.

skills · practice stats

From rounds of this course only: box, review and test-out answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

  1. 1 day
  2. 3 days
  3. 7 days
  4. 14 days
  5. 30 days
  6. 60 days
  7. mastered · every 90 days

your rounds

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