Word Problems: Change Unknown - First Grade Mathematics

Have you ever started playing a game with some toys, and by the end of the day you had more toys, but you were not quite sure how many were added? That is a Change Unknown problem! In a Change Unknown story problem, you know where the story started, and you know where the story ended, but the middle part—the change—is the secret mystery. Solving Change Unknown problems will make you feel like a true mathematical detective searching for the missing step. In this chapter, you will master identifying and solving Change Unknown problems.

Anatomy of a Change Unknown Problem

In this type of story, the question mark sits right in the middle of the timeline.

The Change Unknown Timeline:
[ START AMOUNT ]  --->  [ WHAT CHANGED? = ? ]  --->  [ FINAL RESULT ]
   (Known!)               (The Mystery Box!)            (Known!)

The Missing Middle

Notice how different this is from a Result Unknown problem: Result Unknown: 8 + 5 = ? Change Unknown: 8 + [ ? ] = 13 You know the start (8) and the finish (13); you need to find what happened in between!

Example 1: Add To (Change Unknown)

Let us examine an addition story where the change is unknown.

The Story:
Kaitlyn had 7 stickers in her sticker album.
Her mother gave her some more stickers.
Now Kaitlyn has 12 stickers in all.
How many stickers did her mother give her?

Math Equation:
  7  +  [ ? ]  =  12
Start   Change   Result

Solving the Mystery:

You can solve this in two ways: Way 1 (Counting On): Start at 7 and count on until you hit 12: "8, 9, 10, 11, 12." You counted 5 hops! Way 2 (Subtraction): Take the final total and subtract the starting amount: 12 - 7 = 5! Answer: Her mother gave her 5 stickers!

Example 2: Take From (Change Unknown)

Now let us look at a subtraction story where the change is unknown.

The Story:
There were 14 apples hanging on an apple tree.
Some apples fell to the ground during a windstorm.
Now there are only 8 apples left on the tree.
How many apples fell to the ground?

Math Equation:
 14  -  [ ? ]  =  8
Start   Change   Result

Solving the Subtraction Mystery:

To find how many fell, find the difference between where you started (14) and where you ended (8): 14 - 8 = 6 apples! Check: 14 - 6 = 8! Answer: 6 apples fell to the ground.

Practice Exercises

Exercise 1: Read and solve: "Benjamin had 6 toy trucks. His uncle gave him some more trucks. Now Benjamin has 14 trucks in total. How many trucks did his uncle give him?" Write the missing addend equation and solve it.

Exercise 2: Read and solve: "A bakery had 15 blueberry muffins on a display tray. Customers bought some muffins in the morning. Now there are 9 muffins left on the tray. How many muffins did customers buy?" Write the subtraction equation and solve it.

Exercise 3: How can you tell the difference between a Result Unknown story and a Change Unknown story?

Exercise 4: Write your own Change Unknown story problem that matches the equation 8 + [ ? ] = 15.

Solutions and Explanations

Solution to Exercise 1: The equation is 6 + [ ? ] = 14. To solve, subtract 14 - 6 = 8 (or count on from 6 to 14: 8 hops). Benjamin's uncle gave him 8 toy trucks.

Solution to Exercise 2: The equation is 15 - [ ? ] = 9. To solve, subtract 15 - 9 = 6. Customers bought 6 muffins.

Solution to Exercise 3: In a Result Unknown story, the mystery is what happens at the end ("How many does he have now?"). In a Change Unknown story, you already know the ending, but the mystery is what happened in the middle ("How many did he get?" or "How many went away?").

Solution to Exercise 4: Example story: "Ava had 8 seashells in her collection bucket. She walked along the shore and picked up some more seashells. When she got home, she had 15 seashells in all. How many seashells did Ava find on her walk?" (Equation: 8 + ? = 15).