The Counting Up Method

Mental Math · Axiom Academy

Stand on the smaller number, hop forward to friendly stops, and add the hops — that total is the difference. Take 83 − 47 . Put yourself on 47 and walk forward until you reach 83, but only in hops that are easy to make. First hop: +3 , which lands you on the round number 50 . Second hop: +30 , landing on 80 . Last hop: +3 , and you are standing on 83 . The only thing left is to add up how far you walked: 3 + 30 + 3 = 36 . You never subtracted and you never borrowed — you kept a running hop tally and read the answer straight off it. Three forward hops from 47 to 83 A waypoint is a number that is easy to land on and easy to leave from — in practice, the round tens, then the round hundreds. Your first hop is nearly always up to the next round ten : from 47 that is +3 to 50. After that you are travelling between round numbers, which is where mental arithmetic is quickest. Here is the part people worry about, and they needn't. Between 47 and 83 you could stop at 50, 60, 70 and 80 — or just at 50 and 80. Take the four-hop route +3, +10, +20, +3 and the three-hop route +3, +30, +3 : both total 36 . The route is a matter of taste. The total is pinned by the two endpoints, and those never move. Route B — one long hop between the tens From 47, add 3 and you are on 50. This first hop is always the complement of the ones digit — the same complements you drilled in Unit 1. From 50 to 80 is a single easy 30. Splitting it into 10 + 20 is equally correct, just more to remember.

This is the written version of the interactive lesson above. See the full Mental Math course.