Area Model for Multiplication
Twenty-three rows of fourteen is a rectangle. Cut it where each ten ends and it falls into four patches; count those four and you have counted the lot. Pull an edge to see which patches change and which sit still.
23 × 14 = 322
- The four patches
- 200 + 80 + 30 + 12
- The biggest one
- 20 × 10 = 200
Try these
Drag the right-hand edge or the bottom edge of the rectangle.
Ready to try some? Two-digit multiplication worksheets
Where it usually goes wrong
Long multiplication is carried out as a memorized routine, so the four numbers being added at the end are four unexplained results and the zero on the second line is a superstition. They are the four patches of one rectangle, and the zero is there because that row is counting tens.
Try this
- Press 23 × 14 and read the four patches. 200, 80, 30 and 12 are exactly the four numbers the written method makes you find.
- Notice that 20 × 10 = 200 is most of the answer. That is why an estimate of 200 was never far off.
- Try 19 × 19 and watch the small patch grow. It is 1 × 9 at first and the corner is where the leftovers live.
Where this goes next
Multiplying out (x + 3)(x + 4) in algebra is this same rectangle with letters on two of its edges, and the four terms come from the same four patches.
More about numbers and proportion
- Completing the Squarecompleting the square adds (b/2)² because that is the exact area of the corner two b/2-wide strips leave open
- Regrouping in Addition and Subtractionthe little borrowed or carried number above a place is not a mark to copy — it is exactly what that place is worth by the moment the algorithm reaches it
- Comparing Decimalsa decimal with more digits after the point is not automatically bigger — 0.5 shades more of its grid than 0.45 does, however many digits each is written with