Using Fact Families to Solve Math Problems

Remember fact families?  I used to think their only purpose was to give kids busywork to do in the hopes that they would learn their math facts.  But it turns out that knowing how to create fact family equations comes in handy for problem solving.

I’ve been working with my kids on solving simple math equations with one unknown.  Unfortunately, the way we’ve traditionally taught kids to solve these problems means they have to remember a bunch of different rules for different scenarios.

I don’t like having to remember a bunch of different rules

Why remember one set of rules for solving addition problems, another set for subtraction, and yet more rules for multiplication or division.  Think I’m exaggerating?  Let’s look at the strategies often used to solve different kinds of problems:

Operation Unknown as first number Unknown as second number
Addition – same strategy regardless of where the unknown sits A + 9 = 17:  Subtract the smaller number from larger number:17 – 9 = A 9 + A = 17: Subtract the smaller number from the larger number:17 – 9 = A
Subtraction strategy depends on location of unknown A – 9 = 8:   Add the two given numbers together:9 + 8 = A 17 – A = 8:  Swap the places of the unknown and the smaller known value, then solve:17 – 8 = A
Multiplication – same strategy regardless of where the unknown sits A * 7 = 63:  Divide the larger number by the smaller number:63 ÷7 = A 7 * A = 63: Divide the larger number by the smaller number:63 ÷7 = A
Division – strategy depends on location of unknown A ÷ 7 = 9:  Multiply the two given numbers together:7 x 9 = A 63 ÷ A = 9:  Swap the places of the unknown and the smaller known value, then solve:63 ÷ 9 = A

I’d much rather have an all-purpose system – a system that works regardless of whether I’m solving an addition, subtraction, multiplication or division problem with an unknown variable.   The only thing I need to remember to use this system is the concept of a fact family.

Fact families to the rescue

Let’s start with a review of what a fact family is.  A fact family is a group of three numbers (aka “the family”) which combine in various ways to create 4 different equations.    They will either combine to form 2 addition and 2 subtraction equations, or 2 multiplication and 2 division equations.

As an example, let’s take the numbers 8, 9 and 17.  These numbers form a fact family of addition and subtraction facts, like this:

  • 8 + 9 = 17
  • 9 + 8 = 17
  • 17 – 9 = 8
  • 17 – 8 = 9

Now let’s form a fact family of multiplication and division facts using 7, 9 and 63:

  • 7 x 9 = 63
  • 9 x 7 = 63
  • 63 ÷ 9 = 7
  • 63 ÷ 7 = 9

Now that we know how to form the fact families, we can create an all-purpose system for solving basic math equations with an unknown.

The All-Purpose Fact Family System – Expanded Version

Here’s the approach you can use instead.  Keep in mind that this assumes all problems are written as (number) (operator) (number) = (number):

  1. Identify which operation is being used in the equation (addition, subtraction, multiplication or division)
  2. Identify the opposite operation (subtraction is the opposite of addition, division is the opposite of multiplication)
  3. Identify the three “members” of the family:  Typically, the three members will be two given numbers and an unknown variable.
  4. Decide which of the three numbers will be the largest.  *Note:  it isn’t always one of the known values!  The following guidelines can help:
    1. For a subtraction or division problem, the first number in the equation will be the largest.    In the following examples, A will be the largest number because it is farthest left:
      1. A – 9 = 7
      2. A ÷ 7 = 9
    2. For an addition or multiplication problem, the last number will be the largest.   In the following examples, A will NOT be the largest number because it is NOT farthest to the right:
      1. A + 9 = 17
      2. 9 + A = 17
      3. A x 9 = 63
      4. 9 x A = 63
  5. Write the first family equation, which also happens to be the original problem
  6. Write the second family equation using the same operation as the original problem.  So if the problem is an addition problem, write a second addition equation for the fact family.
  7. Write the third family equation using the opposite operation you identified in step 2
  8. Write the fourth family equation, again using the opposite operation but rearranging the members
  9. Find one equation from the group that you can solve without any rearranging – there may be one or two equations like this.  This equation will have the unknown alone on the right side of the “=” sign.
  10. Solve the equation you identified in step 9.

10 steps seems like an awful lot to remember.  So let’s condense the system down to three steps:

The All-Purpose Fact Family System – Condensed Version

  1.  The original problem is your first fact family equation.  Write three additional fact family equations using the information given in the original problem.
  2. Find the one equation that has the unknown isolated
  3. Solve the equation you found in step 2

Now let’s see how this works on an actual math problem.  For the first case, we’ll use the expanded version:

Using the fact family system for addition (expanded version)

Let’s try the approach on this problem:

A + 9 = 17

1. Identify which operation is being used in the equation The “+” tells us this is an addition problem
2. Identify the opposite operation The opposite of addition is subtraction, so our fact family equations will have 2 addition and 2 subtraction problems
3. Identify the three “members” of the family Going left to right, our three members are:

  • A
  • 9
  • 17
4. Decide which of the three numbers will be the largest Since this is an addition problem, the 17 will be largest, since it’s the farthest to the right (i.e. the last number).
5. Write the first family equation (the original problem) A + 9 = 17
6. Write the second family equation using the same operation as the original problem. For addition, we simply change the order of the addends to get
9 + A = 17
7. Write the third family equation using the opposite operation you identified in step 2 17 – 9 = A
8. Write the fourth family equation, again using the opposite operation 17 – A = 9
9. Find one equation you can solve without any rearranging Here are the 4 equations again:  Only the third equation has the unknown isolated:

  • A + 9 = 17
  • 9 + A = 17
  • 17 – 9 = A
  • 17 – A = 9
10. Solve the equation you identified in step 9 17 – 9 = A
So A must equal 8

Now let’s try the expanded version for a subtraction problem.

Using the fact family system for subtraction (expanded version)

Let’s try the approach on this problem:

A – 9 = 8

1. Identify which operation is being used in the equation The “-” tells us this is a subtraction problem
2. Identify the opposite operation The opposite of subtraction is addition, so our fact family equations will have 2 subtraction and 2 addition problems
3. Identify the three “members” of the family Going left to right, our three members are:

  • A
  • 9
  • 8
4. Decide which of the three numbers will be the largest Since this is a subtraction problem, the A will be largest, since it’s the farthest to the left (i.e. the first number).
5. Write the first family equation (the original problem) A – 9 = 8
6. Write the second family equation using the same operation as the original problem. We know A is the largest number, and for subtraction the largest number always comes first:A – 8 = 9
7. Write the third family equation using the opposite operation you identified in step 2 For addition, the largest number goes to the far right, so A will be on the right side of the equation:
8 + 9 = A
8. Write the fourth family equation, again using the opposite operation For our second addition fact, we rearrange the addends:
9 + 8 = A
9. Find one equation you can solve without any rearranging Here are the 4 equations again:  The third and fourth equations both have the unknown isolated.  For this example, we will use the 4thequation to solve the problem:

  • A – 9 = 8
  • A – 8 = 9
  • 8 + 9 = A
  • 9 + 8 = A
10. Solve the equation you identified in step 9 9 + 8 = A
So A must equal 17

 

Hopefully you’re getting the hang of this by now, so let’s use the condensed version for our multiplication example:

Using the fact family system for multiplication (condensed version)

Let’s use the system to solve this problem:

A x 9 = 63

1. The original problem is your first fact family equation.  Write three additional fact family equations using the information given in the original problem.
  • A x 9 = 63
  • 9 x A = 63
  • 63 ÷ 9 = A
  • 63 ÷ A = 9
2. Find the one equation that has the unknown isolated 63 ÷ 9 = A
3. Solve the equation you found in step 2 63 ÷ 9 = A
So A must be 7

 

And finally, let’s try a division problem using the condensed version.

Using the fact family system for division (condensed version)

Let’s use the system to solve this problem:

63 ÷ A = 7

1. The original problem is your first fact family equation.  Write three additional fact family equations using the information given in the original problem.
  • 63 ÷ A = 7
  • 63 ÷ 7 = A
  • A x 7 = 63
  • 7 x A = 63
2. Find the one equation that has the unknown isolated 63 ÷ 7 = A
3. Solve the equation you found in step 2 63 ÷ 7 = A
So A must be 9

So there you have it: one system to solve any of the four basic math operations for one unknown.  You could condense this system down even farther once you get comfortable with it.

Shortening the system even more

To shorten the time spent even farther, you don’t necessarily have to write all four equations of the fact family.   Once you start writing them and you find an equation where the unknown is alone on the right side of the “=” sign, there’s no need to continue.  You can simply use the equation you just found to solve the problem.

Hopefully those nifty fact families will make your family’s math work time go a little more smoothly.