Word Problems: Start Unknown - First Grade Mathematics

Welcome to the most intriguing type of word problem in first grade: the Start Unknown problem! In a Start Unknown story, the beginning of the story is completely hidden from you. You are told what changed, and you are told how many items there are at the very end, but you must work backward like a time traveler to figure out how many items were there at the very start. In this chapter, you will master the art of working backward to solve Start Unknown problems.

Anatomy of a Start Unknown Problem

In a Start Unknown problem, the question mark sits at the very beginning of the timeline.

The Start Unknown Timeline:
[ START AMOUNT = ? ]  --->  [ THE CHANGE HAPPENS ]  --->  [ FINAL RESULT ]
  (The Mystery Box!)             (Known!)                    (Known!)

The Mystery at the Beginning

Look at how the equation is written: [ ? ] + 5 = 12 [ ? ] - 4 = 9 You do not know where you started! You only know the change and the ending result.

Example 1: Add To (Start Unknown)

Let us examine an addition story where the starting amount is hidden.

The Story:
Some ducks were swimming in a pond.
4 more ducks flew in and joined them.
Now there are 11 ducks swimming in the pond.
How many ducks were swimming in the pond at the start?

Math Equation:
 [ ? ]  +  4  =  11
 Start   Change  Result

Working Backward:

Think like a detective: at the end, there are 11 ducks. 4 of those ducks just arrived. If you tell those 4 ducks to fly away (undoing the change), who is left? The original ducks! To undo addition (+ 4), you use SUBTRACTION (- 4): 11 - 4 = 7 ducks! Check: 7 + 4 = 11! (True!) Answer: There were 7 ducks in the pond at the start.

Example 2: Take From (Start Unknown)

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

The Story:
Liam had a bag of grapes.
He ate 5 grapes during lunch.
Now he has 8 grapes left in his bag.
How many grapes did Liam have in his bag before lunch?

Math Equation:
 [ ? ]  -  5  =  8
 Start   Change  Result

Working Backward:

At the end, Liam has 8 grapes left. But he ate 5 grapes! To find what was in the bag before lunch, you must put the 5 grapes back into the bag! To undo subtraction (- 5), you use ADDITION (+ 5): 8 + 5 = 13 grapes! Check: 13 - 5 = 8! (True!) Answer: Liam had 13 grapes in his bag before lunch.

The Secret Rule for Start Unknown Problems

To solve a Start Unknown problem, you simply DO THE OPPOSITE OPERATION: If the story added items (+), you SUBTRACT to find the start! If the story took items away (-), you ADD to find the start!

Practice Exercises

Exercise 1: Read and solve: "Some children were playing on the playground. 6 more children came over to play. Now there are 15 children on the playground. How many children were on the playground at the beginning?" Write the equation and solve it.

Exercise 2: Read and solve: "Sophia had some colored beads. She used 7 beads to make a bracelet. Now she has 9 beads left. How many beads did Sophia have before she made the bracelet?" Write the equation and solve it.

Exercise 3: Explain why you use addition to solve a Start Unknown problem where items were taken away.

Exercise 4: Write your own Start Unknown story problem that matches the equation [ ? ] - 3 = 10.

Solutions and Explanations

Solution to Exercise 1: The equation is [ ? ] + 6 = 15. To find the start, undo the addition by subtracting: 15 - 6 = 9. There were 9 children on the playground at the beginning.

Solution to Exercise 2: The equation is [ ? ] - 7 = 9. To find the start, undo the subtraction by adding: 9 + 7 = 16. Sophia had 16 beads before making the bracelet.

Solution to Exercise 3: You use addition because the starting amount had ALL the items before anything was removed. The starting amount was the whole! To find the whole, you must combine the part that was taken away with the part that remains left over.

Solution to Exercise 4: Example story: "Noah had a box of crayons. He gave 3 crayons to his friend. Now Noah has 10 crayons left in his box. How many crayons did Noah have at the start?" (Equation: ? - 3 = 10, so 10 + 3 = 13 crayons).