Factoring: Rectangles and Double-Half Method

Factor Pairs: Double-Half Method (Grades 3-6)

A factor is a whole number that divides evenly into another number. The numbers under 100 that have the highest number of factors are 60, 72, 84, 90, and 96, which each have 12 factors. The numbers under 100 that have the second-highest number of factors are 48 and 80, which each have 10 factors. Next week, we will explore an algorithm to find the number of factors of a number without having to find them manually.

A “factor pair” is a group of two numbers that when multiplied have a product of the seed number. Below are the factor pairs for each of the seven numbers mentioned above. The older (3-6th Grade) used my method of double and half to find all such numbers. We always start with the parent rectangle of 1 and the seed number. As long as the number on the right is even, we double the number on the left and take half of the number on the right. When the number on the right is odd, we cannot use double half. So, we look at that odd number and determine its factors other than 1 and the number itself. Use the smallest factor that has not been used yet and divide that number into the seed number. If the number on the right is even, use double and half until you get another odd number on the right. When a prime number appears on the right, you are done factoring. Answer key:

FACTOR 60: FACTOR 72: FACTOR:84

1 60 1 72 1 84

2 30 2 36 2 42

4 15 4 18 4 21

3 20 8 9 3 28

6 10 3 24 6 14

12 5 6 12 12 7

FACTOR 90: FACTOR 96: FACTOR:48

1 90 1 96 1 48

2 45 2 48 2 24

3 30 4 24 4 12

6 15 8 12 8 6

5 18 16 6 16 3

10 9 32 3

FACTOR:80 After you factor these numbers, create 

1 80 the rectangles using these dimensions

2 40 on the attached pages. Indicate the 

4 20 dimensions of each rectangle and 

8 10 notice how the area of each rectangle

16 5 is identical.

I also challenged them to look at the first 9 multiples of 100 and make a conjecture about which number would have the most factors. Many of them chose 800, but a few chose 600 and 900 since it is also divisible by 3. They will find out next week. Also, I gave them the first nine square numbers greater than 100 to factor to look for patterns and of course blank pages where they should challenge themselves with larger numbers. They should choose even numbers like 240 and 4800 so they do not generate large prime numbers which are difficult to identify.

Target Area Rectangles (Grades K-2)

The K-2nd graders learned to factor by drawing rectangles on a grid to create each  target area. For example, if the target area is 10, they were taught to always start with the parent rectangle of 1 and the target number. Other than an area of one (a 1 x 1 square), all of these rectangles are longer than wide. Then try to make other rectangles of 2 by, 3 by, 4 by and so on. For the target area 10, they used 2 x 5. They were encouraged to always label each rectangle with the dimensions. The dimension is written in the middle of the segment in question. The children were challenged to record the number of rectangles possible for each target area. This will help them on the next lesson where we look at how prime numbers emerge as only having one possible rectangle. I gave them an answer key up to a target area of 25 but they should feel free to go further.