Factor Pairs Method--Inverse Variation

Last week we learned about free fall and Newton’s equation:


Force = Mass (weight) x Acceleration (gravity)


What if force was constant at 100 Newtons? Then we see that if mass and acceleration due to gravity were whole numbers, all of the combinations would be:


Force  =   Mass  x  Acceleration   


100 = 1 x 100

100 = 2 x 50

100 = 4 x 25

100 = 5 x 20

100 = 10 x 10

100 = 100 x 1

100 = 50 x 2

100 = 25 x 4

100 = 20 x 5


What if you replaced Force with Total Dollars to be shared, replace Mass with Number of People, and replace Acceleration with Dollars each person gets? Then 100 people each get $1, 50 each get 2$, 25 each get $4, 20 each get $5, 10 each get $10 and vise versa. If an increase in one quantity causes a decrease in another quantity, or a decrease in one quantity causes an increase in another quantity, then we say that both quantities are inversely related.


Other examples of inverse proportions:

  1. Rate x Time = Distance (for a fixed distance of 100 miles, as the rate of miles per hour increases, time decreases. 

2.   Number of Workers x Effort (Time) = Results (for a fixed amount of results like building 100 lego buildings, as the number of children building legos increases, the time it takes to build them decreases)

(to build a 100 foot bridge, as the number of workers increases, the time to build decreases and vice versa)

3.   Procrastination x Grades = Time Spent (for a fixed amount of time spent, as the your level of procrastination increases such as watching more TV or playing more video games, your grade decreases, as you waste less time, your grade increases)

4. Electrical Current (amperage) x Resistance (Ohms) = Voltage (for 100 volts of power, as the current increases the resistance decreases, and as the amperage decreases, the number of Ohms of resistance increases)

5. Number of People Sharing x Amount Each Gets = Amount Shared (for an amount shared of $100, if the number of people sharing increases, the amount each gets is decreased and if the number of people sharing decreases, the amount each gets increased)

6. Number of Teachers x Number of Students in Each Class = Total Number of Students (for 100 students, if the number of teachers increases, the number of students in each class decreases and vice versa)

What are Factors? 

Integers that you can Multiply to get a Number (2 and 5 are factors of 10)

Finding Factor Pairs of a Number: (example: Factor Pairs of 16 and 36)

1. start with 1 and the number itself as your first factor pair. 1 and 16

2. double the number in the left column and take half of the 2 and   8

number in the right column and continue until the number 4 and   4

in the right column is the same number or an odd number (if that odd number is prime, you are done).

3. if the number in the right column is odd, do not double the 1 and 36

number in the right column; instead take the next smallest factor             2 and 18

of that right column number and put it in the left column and                  4 and   9

put the other factor pair in the right column (repeat steps 2 and 3) .         3 and 12

                      6 and 6


I had the children factor dozens of numbers to get a rhythm with this algorithm. 


Factor 10:  Factor 6:   Factor 8:   Factor 12   Factor 9:       Factor 14:

Factor 20:  Factor 22:   Factor 26:    Factor 28:    Factor 30:   Factor 40:

Factor 60:   Factor 72:    Factor 84:   Factor 48:   Factor 90:   Factor 96:


Factor 80:  Factor 68:  Factor 78: Factor 216:   Factor 256:   Factor 840:


The first two lines of factors can be attempted by K-2nd grade and the second two lines by 3-6th grade. I added 432 (20 factors) and 7560 (64 factors—the most factors of any four digit number.


When you get to numbers like factoring 72, you will have to use short division to cut numbers in half and by thirds. Learning this factor algorithm will make you very powerful as this algorithm is not taught in schools and is not even on the internet as I developed it when I was your children’s age.pastedGraphic.png







Factor_Pairs_K-2.pdf1.39 MB
Factor_Pairs_3-6.pdf657.68 KB