Subtraction Strategies
Counting Backward
What is counting backward?
Counting backward is a subtraction strategy where you start from the bigger number and count down by the smaller number. Each step moves you one number less. This is the opposite of counting on for addition!
How to count backward:
- Step 1: Start with the bigger number (the number you have)
- Step 2: Count backward using the smaller number (the number you take away)
- Step 3: Say each number as you go down
- Step 4: The last number you say is the answer!
Example: 9 - 4
- Start at 9
- Count backward 4 steps: 8, 7, 6, 5
- Answer is 5
When to use counting backward:
- When subtracting a small number from a bigger number
- When numbers are within 20
- When you need a quick mental subtraction
| Subtraction | Start at | Count backward | Answer |
|---|---|---|---|
| 8 - 2 | 8 | 7, 6 | 6 |
| 10 - 3 | 10 | 9, 8, 7 | 7 |
| 15 - 4 | 15 | 14, 13, 12, 11 | 11 |
| 7 - 5 | 7 | 6, 5, 4, 3, 2 | 2 |
Use counting backward to subtract 8 - 3.
- Start at 8 (the bigger number)
- Count backward 3 steps:
- Step 1: 7
- Step 2: 6
- Step 3: 5
- The last number is 5
Answer: 5
Tom has 12 candies. He eats 4 candies. How many candies are left? Use counting backward.
- Tom starts with 12 candies
- He eats 4 candies, so subtract 4
- Start at 12, count backward 4 steps:
- 11 (1 candy eaten)
- 10 (2 candies eaten)
- 9 (3 candies eaten)
- 8 (4 candies eaten)
- Tom has 8 candies left
Answer: 8 candies
Sara counted backward from 14 to find 14 - 6. She said: 14, 13, 12, 11, 10, 9, 8 and got 8. Did she count correctly? If not, what did she do wrong?
- Start at 14
- Need to subtract 6, so count backward 6 steps
- Step 1: 13
- Step 2: 12
- Step 3: 11
- Step 4: 10
- Step 5: 9
- Step 6: 8
- Sara counted correctly! She got 8
- Check: 14 - 6 = 8 ✓
Answer: Yes, Sara counted correctly. The answer is 8.
Key Points
- Counting backward means starting from the bigger number and counting down
- You count backward by the number you are subtracting
- Each step moves one number less
- Use counting backward when the number you subtract is small
- A number line helps visualize counting backward
- The last number you say is the answer
Related Addition Facts
What are related addition facts?
Related addition facts are fact families! Every subtraction fact has a related addition fact. If you know your addition facts, you can use them to solve subtraction problems!
The fact family concept:
- Three numbers that work together
- Two addition facts
- Two subtraction facts
Example fact family: 5, 3, 8
- Addition facts: 5 + 3 = 8 and 3 + 5 = 8
- Subtraction facts: 8 - 5 = 3 and 8 - 3 = 5
How to use related facts for subtraction:
- Step 1: Look at the subtraction problem: 9 - 4 = ?
- Step 2: Think: "What plus 4 equals 9?"
- Step 3: 4 + ? = 9 → 4 + 5 = 9
- Step 4: So 9 - 4 = 5
| Subtraction | Related Addition | Answer |
|---|---|---|
| 10 - 3 = ? | 3 + ? = 10 | 7 |
| 12 - 5 = ? | 5 + ? = 12 | 7 |
| 15 - 7 = ? | 7 + ? = 15 | 8 |
| 9 - 2 = ? | 2 + ? = 9 | 7 |
Use a related addition fact to find 11 - 4.
- Think: 4 plus what equals 11?
- 4 + 7 = 11 (because 4 + 7 = 11)
- So 11 - 4 = 7
- Check: 11 - 4 = 7 ✓
Answer: 7
The fact family for 6, 8, and 14 has four facts. Write all four facts.
- Addition fact 1: 6 + 8 = 14
- Addition fact 2: 8 + 6 = 14
- Subtraction fact 1: 14 - 6 = 8
- Subtraction fact 2: 14 - 8 = 6
Answer: 6 + 8 = 14, 8 + 6 = 14, 14 - 6 = 8, 14 - 8 = 6
Riya knows that 7 + 8 = 15. How can she use this to find 15 - 7 and 15 - 8?
- If 7 + 8 = 15, then the three numbers in the fact family are 7, 8, and 15
- For subtraction, the largest number (15) minus one number equals the other number
- So 15 - 7 = 8
- And 15 - 8 = 7
- Check: 15 - 7 = 8 ✓, 15 - 8 = 7 ✓
Answer: 15 - 7 = 8 and 15 - 8 = 7
Key Points
- Related addition facts help solve subtraction problems
- Fact families have three numbers that work together
- Each fact family has 2 addition and 2 subtraction facts
- Think "what plus the smaller number equals the bigger number?"
- If you know your addition facts, subtraction becomes easy!
- The largest number in the family is the sum (for addition) or starting number (for subtraction)
Mental Subtraction, Real-Life Problems, and Missing Numbers
What is mental subtraction?
Mental subtraction means solving subtraction problems in your head without writing anything down. You use strategies like counting backward, related facts, and breaking numbers apart.
What are real-life subtraction problems?
Real-life problems use subtraction to solve everyday situations like:
- How many items are left after some are used
- How much money is left after buying something
- How many more are needed to reach a goal
What are missing number subtraction problems?
Missing number problems have a blank or question mark in the subtraction sentence. You need to find the missing number!
Types of missing number problems:
| Type | Example | How to Solve |
|---|---|---|
| Missing result | 9 - 4 = ? | Just subtract! |
| Missing first number | ? - 5 = 3 | Add: 5 + 3 = 8 |
| Missing second number | 10 - ? = 6 | Subtract: 10 - 6 = 4 |
Mental subtraction strategies:
| Strategy | When to Use | Example |
|---|---|---|
| Counting backward | Small number subtracted | 9 - 3 = 6 |
| Related facts | Numbers within 20 | 13 - 5 = 8 (since 5+8=13) |
| Break apart | Larger numbers | 15 - 7 = (15-5)-2 = 10-2=8 |
Solve mentally: 14 - 6
- Use related addition fact: 6 + ? = 14
- 6 + 8 = 14
- So 14 - 6 = 8
- Counting backward check: 14, 13, 12, 11, 10, 9, 8 ✓
Answer: 8
Real-life problem: A basket has 18 apples. 9 apples are eaten. How many apples are left?
- Start with 18 apples
- 9 are eaten, so subtract 9
- 18 - 9 = ?
- Use related fact: 9 + 9 = 18
- So 18 - 9 = 9
- Check: 18 - 9 = 9 apples left
Answer: 9 apples
Find the missing number: ? - 7 = 8
- This is a missing first number problem
- Think: What number minus 7 equals 8?
- Use related addition: 7 + 8 = 15
- Check: 15 - 7 = 8 ✓
- The missing number is 15
Answer: 15
Key Points
- Mental subtraction uses strategies without writing anything down
- Real-life problems help us use subtraction every day
- Missing number problems have a blank that needs to be found
- For missing first number: add the two numbers you see
- For missing second number: subtract the answer from the first number
- Practice all three types to become a subtraction master!