Lessons · Teaching · multiplication, from an array to the algorithm
Multiplication is repeated addition, an array, and a rectangle
Three times four means three groups of four, a row of three by a column of four, and the area of a three by four rectangle. All three are the same fact, and the third is the one that keeps working when the numbers stop being whole.
Hone is a place to practise a career, one idea a day. This is one of its lessons, written out in full and free to read without an account.
In beta. This lesson was written for Hone and has not yet been checked by a state certification officer. Practice material, not professional advice. What that means.
What it is for
A pupil who only knows multiplication as repeated addition hits a wall at one half times one third, because you cannot add something half a time. The area picture survives: half of a third of a square is a real piece of a real rectangle. Teaching the picture early is what makes fractions possible later.
How to think about it
Break a big multiplication into the parts of a rectangle. 23 times 14 is a rectangle 23 wide and 14 tall, cut into 20 by 10, 20 by 4, 3 by 10 and 3 by 4. Add the four pieces. That is exactly what the column algorithm does, in the same order, with the working hidden.
Worked example
23 x 14 split as (20 + 3) x (10 + 4)Four rectangles, and none of them needs a times table beyond twelve.
20x10 = 200, 20x4 = 80, 3x10 = 30, 3x4 = 12Every piece is a fact a ten-year-old already has.
200 + 80 + 30 + 12 = 322And the column method gives 322 too, because it is this with the boxes rubbed out.
Estimate first: 23 is about 20, 14 is about 15, so expect about 300An answer of 3,220 or 32 is caught before the working is checked.
Your turn
Name the shape whose area picture of multiplication still works for fractions.
Multiplication drawn as the area of a
Solve one, graded on the server
The trap
Believing multiplication always makes a number bigger. Multiply by a number below one and it makes it smaller: 8 times 0.5 is 4. The belief survives all through primary because every example was a whole number above one, and then it breaks.