doing math

courses › Thinking

Mental shortcuts

level 19 course

Tricks that make hard-looking products quick.

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

Learn first (about 3 minutes)

Opens at level 19.

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Builds on: Bigger multiplication (not open yet)

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

A few products look hard but follow a pattern that makes them quick. Each shortcut is ordinary algebra in disguise, so it always works when its pattern fits.

The neatest one squares a number ending in 5. Multiply the number in front of the 5 by one more than itself, then write 25 after it. For 65², 6 × 7 = 42, so the answer is 4,225. It works because 65 × 65 = 60 × 70 + 25.

Techniques

Times 11: sum in the middle

A two-digit number times 11.

  1. Write the two digits with a gap between them.
  2. Put their sum in the gap.
  3. If the sum is 10 or more, write its last digit and carry the 1 to the left digit.
  4. It works because × 11 is × 10 plus the number once more.
worked example

Example: 86 × 11

  1. 8 + 6 = 14, so there is a carry.
  2. Write the 4 in the middle and add 1 to the 8.
  3. That gives 9, 4, 6.

Answer: 946

Around a round number

The two numbers are the same distance either side of a round number.

  1. Find the round number halfway between them, and the distance to each.
  2. Square the round number.
  3. Take away the distance squared.
worked example

Example: 47 × 53

  1. Both are 3 away from 50.
  2. 50 × 50 = 2,500, and 3 × 3 = 9.
  3. 2,500 − 9 = 2,491.

Answer: 2,491

Near 100: cross over

Both numbers are a little below 100, or both a little above.

  1. Find how far each number is from 100.
  2. Below 100, take one number's distance off the other number. Above 100, add it instead.
  3. That result is the number of hundreds.
  4. Add the two distances multiplied together.
worked example

Example: 96 × 93

  1. Short by 4 and 7.
  2. 96 − 7 = 89, so 8,900.
  3. 4 × 7 = 28.
  4. 8,900 + 28 = 8,928.

Answer: 8,928

Tips by skill

  • Tip× 11: Put the sum of the two digits between them. If the sum is 10 or more, carry the 1 to the left digit.
  • TipSquares ending in 5: Multiply the number in front of the 5 by one more than itself, then write 25 after it.
  • TipAround a round number: Find the round number halfway between them. Square it, then subtract the distance squared.
  • TipNear 100: Below 100, take one distance off the other number; above, add it. That is the hundreds. Then add the distances multiplied.

Watch out for

  • Forgetting the carry in × 11. When the digits add to 10 or more, the 1 goes onto the left digit: 74 × 11 is 814.
  • Squaring the front digit for a number ending in 5. For 65² it is 6 × 7, not 6 × 6.
  • Adding the distance squared instead of taking it away. 48 × 52 is 4 less than 2,500, not 4 more.
  • Dropping the last step near 100. For 97 × 96, the 3 × 4 = 12 still has to be added.

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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