Decimal Subtraction
Problem String
Removal vs. Differencing â Which Strategy is Best?
ð Standards & Objectives
- Use the removal strategy to subtract
- Use the differencing strategy to subtract
- Choose the most efficient strategy
âĻ Two Powerful Strategies
There's more than one way to subtract!
Take away the amount you're subtracting by jumping backward on the number line.
Think: "I'm removing this much from my total."
Add up from the smaller number to the larger number to find the distance between them.
Think: "How far apart are these numbers?"
The best strategy depends on the numbers!
As we work through these problems, think about when each strategy works best.
ðĨ Problem 1
Solve this in your journal. Be ready to share your strategy!
How did you solve this? What jumps did you make?
ðī Removal Strategy Works Great Here!
We can remove 4 by jumping backward:
43 â â1 â 42 â â3 â 39
Or jump straight: 43 â 4 = 39
Why removal? The numbers are far apart, so taking away is easier than finding the distance.
ðĨ Problem 2
Solve this one. Is this like the last problem or different?
Notice anything about these two numbers? Are they close or far apart?
ðĩ Differencing Strategy Works Great Here!
These numbers are close together! Let's find the distance by adding up:
49 â +1 â 50 â +2 â 52
Distance: 1 + 2 = 3
Why differencing? When numbers are close, finding the distance is faster than removing!
ðŽ Turn & Talk
Discuss with your partner
Compare our first two problems:
43 â 4 vs. 52 â 49
Why might someone prefer removal for one and differencing for the other?
| Problem | Numbers Are... | Best Strategy |
|---|---|---|
| 43 â 4 | FAR apart (39 units) | ðī Removal |
| 52 â 49 | CLOSE together (3 units) | ðĩ Differencing |
ðĨ Problem 3: Decimals
Now let's try decimals! Same strategies apply.
Are these numbers close or far apart? Which strategy might be easier?
ðĩ Differencing Strategy!
19.2 and 18.9 are very close! Let's add up from 18.9 to 19.2:
18.9 â +0.1 â 19 â +0.2 â 19.2
Distance: 0.1 + 0.2 = 0.3
Think of it like money: $19.20 â $18.90 = $0.30 (thirty cents)
ðĨ Problem 4
This one's different. Look carefully at the numbers!
0.15 is a small number. Are these numbers close or far apart?
ðī Removal Strategy!
We're only removing a small amount (0.15) from 17.1:
17.1 â â0.05 â 17.05 â â0.10 â 16.95
Think of it like money: $17.10 â $0.15 = $16.95
Why removal? The numbers are far apart (17.1 vs 0.15), so we just take away!
ðŽ Turn & Talk
Let's think about our strategy choices
Look back at 19.2 â 18.9 (we used differencing)
And 17.1 â 0.15 (we used removal)
Did anyone find the distance for 17.1 â 0.15? Why might that be silly?
ðĪŠ That would be silly!
To find the distance from 0.15 to 17.1, you'd have to make tons of jumps:
0.15 â 1 â 2 â 3 â ... â 16 â 17 â 17.1
It's much faster to just remove 0.15 from 17.1!
âïļ Your Turn! Problem 5
Solve this independently. Think about which strategy is best!
Solve in your journal first. Decide: removal or differencing? Why?
ðī Removal is Faster!
We're removing a small amount (0.4 or 4 tenths) from 34.3:
34.3 â â0.3 â 34 â â0.1 â 33.9
Think money: $34.30 â $0.40 = $33.90 (taking away forty cents)
âïļ Your Turn! Problem 6
Last one! Which strategy will you choose?
Look at these numbers carefully. Are they close or far apart?
ðĩ Differencing is Easier!
31.3 and 30.8 are very close â they're both in the 30s! Find the distance:
30.8 â +0.2 â 31 â +0.3 â 31.3
Distance: 0.2 + 0.3 = 0.5
Think money: The difference between $31.30 and $30.80 is just fifty cents!
ðĄ The Big Idea
When should you use each strategy?
Numbers are FAR APART
Examples: 43 â 4, 17.1 â 0.15, 34.3 â 0.4
The number being subtracted is small compared to the first number.
Numbers are CLOSE TOGETHER
Examples: 52 â 49, 19.2 â 18.9, 31.3 â 30.8
Both numbers are almost the same â just find the small gap!
ð Summary
What did we learn today?
- Removal: Jump backward to take away (best when numbers are far apart)
- Differencing: Add up to find the distance (best when numbers are close)
- Choose wisely: Look at the numbers first to pick the most efficient strategy!
How would you explain to a friend when to use removal vs. differencing?
When you only have to take off a little bit, use removal.
When they're almost the same number, find the distance.
When both numbers are big, or both small, they're probably close together!