The Count-Back Strategy - First Grade Mathematics

When you need to subtract a small number like 1, 2, or 3, you do not need to pull out a big pile of counters. You can use a lightning-fast mental math strategy called counting back! Counting back is the subtraction twin of counting on. Instead of stepping forward along the number path, you take steps backward toward zero. In this chapter, you will master the count-back strategy, learn how to keep track of your backward hops on your fingers, and know when this strategy is the best choice.

How the Count-Back Strategy Works

Counting back means starting at the minuend (the whole) and stepping down in reverse order.

Solving 8 - 3 by Counting Back:

Step 1: Start with 8:   "EIGHT..."  (Tap your head!)
Hop 1 Backward:         "Seven"     (Put up 1 finger)
Hop 2 Backward:         "Six"       (Put up 2 fingers)
Hop 3 Backward:         "Five!"     (Put up 3 fingers)

Result: 8 - 3 = 5!

The Three Rules of Counting Back:

Rule 1: Say the starting whole out loud firmly: "8!" Do not count backward from 8 until you make your first hop. Rule 2: Each backward hop corresponds to the next lower number in the counting sequence: 7, 6, 5. Rule 3: Use your fingers to count how many hops you have taken, NOT the number you are saying! When 3 fingers are raised, stop immediately. The number on your lips is your difference!

Visualizing Counting Back on a Number Path

A number path makes backward movement crystal clear.

Solving 7 - 2 on a Number Path:
          -1      -1
        <-----  <-----
| 1| 2| 3| 4|[5]| 6 |[7]| 8| 9|10|
            Land     Start

Landing on the Answer

Start on box 7. Take 1 hop backward to land on 6. Take a 2nd hop backward to land on 5. You took 2 hops backward, and you landed on 5. 7 - 2 = 5!

When to Use Counting Back (and When NOT To!)

Counting back is a fantastic strategy, but only when you are subtracting a small number!

Smart Use of Counting Back:
Use Counting Back for -1, -2, or -3:
  9 - 1 = 8  (1 hop back: "9... 8")
  8 - 2 = 6  (2 hops back: "8... 7, 6")
  10 - 3 = 7 (3 hops back: "10... 9, 8, 7")

DO NOT Use Counting Back for Large Subtrahends!
Solving 9 - 7 by counting back takes 7 hops backward: "9... 8, 7, 6, 5, 4, 3, 2!"
That is exhausting and easy to mess up!
For 9 - 7, use the COUNTING UP strategy instead!

Choose the Right Tool for the Job

A master carpenter uses a hammer for nails and a saw for wood. A master mathematician uses counting back when subtracting 1, 2, or 3, and switches to counting up when subtracting numbers that are close to the whole!

Practice Exercises

Exercise 1: Solve 9 - 2 using the count-back strategy. Describe the starting number and the two numbers you say as you hop backward.

Exercise 2: Solve these mental math problems by counting back 1, 2, or 3: 6 - 1 = __ 7 - 3 = __ 10 - 2 = __ 5 - 3 = __

Exercise 3: Why is counting back a bad strategy for solving 10 - 8? What makes it difficult?

Exercise 4: Start at 6 on a number path and take 3 hops backward. What number do you land on? Write the complete subtraction equation.

Solutions and Explanations

Solution to Exercise 1: Start at 9 and say: "Nine..." Then take two backward hops: "eight, seven." The answer is 7 (9 - 2 = 7).

Solution to Exercise 2: 6 - 1 = 5 (6... 5) 7 - 3 = 4 (7... 6, 5, 4) 10 - 2 = 8 (10... 9, 8) 5 - 3 = 2 (5... 4, 3, 2)

Solution to Exercise 3: Counting back is a bad strategy for 10 - 8 because taking 8 backward hops is very slow and requires keeping track of too many fingers, making it easy to lose your place. When the two numbers are close together, counting up is much faster.

Solution to Exercise 4: Starting at 6 and hopping back 3 steps: step 1 lands on 5, step 2 lands on 4, and step 3 lands on 3. You land on 3. The subtraction equation is 6 - 3 = 3.