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Factors and Primes

Calculation · Year 4–Year 6

A number's factors are the rectangles it makes.

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About the factors

A number of squares, and one row to try at a time. Two rows of six is twelve, so two and six are factors; five rows of two is ten with two left over, so five is not. Every rectangle you can build is a pair of factors, and when the rectangles run out so have the factors.

Finding factors is not guessing, which is what a class is doing when they are asked for the factors of 24 and start naming numbers. The rectangles are a method: try one row, then two, then three, and stop when the rows get longer than the columns because after that you are turning rectangles you already have.

A prime number is the one that only makes ONE rectangle. Thirteen squares go into a single strip and nothing else, and that strip is a better answer than “a number with exactly two factors”, which is the same fact with the picture taken out. It is also why a class will tell you that nine is prime.

A square number makes a square. Sixteen goes into four rows of four, and that is the only rectangle in the whole hunt whose two sides are the same, which is where the name comes from.

The squares left over are the point of the ones that fail. Twelve in five rows is five rows of two and two left over, and those two are drawn as the start of a sixth column that stops short: a remainder is always smaller than the number of rows, so it can never be a full column.

How to use it on your whiteboard

  1. 1Set the number of squares.
  2. 2Ask the class whether they can make a rectangle with one row, then press + on the Rows.
  3. 3Keep going. Each rectangle that works is added to the list of factor pairs.
  4. 4When the rows get as long as the columns, the hunt is over and the board says how many factors there are.
  5. 5Try 13, 17 or 23 for a number that only makes one rectangle.

Questions

Why stop when the rows get longer than the columns?
Because a rectangle turned round is the same rectangle. Four rows of three is the same pair of factors as three rows of four, so once you have passed the square root there is nothing new left to find. The board says so when you go past it.
Is 1 a prime number?
No, and the picture shows why. One square makes one rectangle, but a prime number has exactly two factors and 1 has only one factor, itself.
How high does it go?
48 squares, which is as many as a class can count on a board at a size they can see, and eight rows, which is enough to find every factor pair of every number that far.