Distributive Property Shortcuts

Mental Math · Axiom Academy

LESSON Distributive Property Shortcuts Cut one factor at a friendly ten, multiply each piece, and put the pieces back together — one move that turns most awkward products into two easy ones. 1. Cut One Factor, Multiply Each Piece, Add A product is an area . 6 × 34 is a rectangle 6 tall and 34 wide, and its area is the answer we want. Now cut the 34 side at 30 . You get two smaller rectangles — one that is 6 × 30 and one that is 6 × 4 — and because a cut removes nothing, those two areas have to add back to the original. So 180 + 24 = 204 , and both halves of that were easy. Cut one factor into two parts and each part gets its own product One hard product becomes two easy ones and an addition The tens-and-ones cut is always available, because every two-digit number is already written as tens plus ones. 23 × 4 becomes 80 + 12 = 92 . 34 × 5 becomes 150 + 20 = 170 . You are not learning a new fact each time — you are re-using the one cut. 2. Cut Past the End and Take the Extra Back Nothing says the friendly ten has to sit inside your number. For 7 × 48 , the nearest ten to 48 is 50 , which is 2 beyond it. So draw the friendly rectangle instead: 7 × 50 = 350 . You have drawn 2 columns of 7 that nobody asked for, so measure that overshoot — 7 × 2 = 14 — and subtract it. 350 − 14 = 336 . This is not a second rule. It is the first one with the second part counted as a debt instead of a gain. 48 sits 2 below 50. That 2 is the whole size of the correction you are about to make.

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