Subtraction

What is Subtraction?

Imagine you have a big collection of 237 cubes. If you give 114 of them to a friend, how can you find out how many cubes you have left?

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Answer

To find what is left, we subtract:$$237-114$$Visually, we can see that when we take 114 blocks away from 237, we are left with 123.

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So, you have 123 cubes left. For big numbers like this, counting blocks is slow. A much faster strategy is to use column subtraction.

Definition Subtraction

Subtraction means taking an amount away from a group to find out what is left. This result is called the difference. The minus sign ($-$) tells us to subtract.
We can show "two hundred thirty-seven minus one hundred fourteen equals one hundred twenty-three" in different ways:

With Numbers: $$237 - 114 = 123$$
With Cubes: $-$ $=$
With Column Subtraction:
With a Part-Whole Model:

Subtraction In Columns

Discover

You have learned how to subtract using the regrouping ("borrowing") method. Today, we will explore another clever strategy: the compensation method.
The big idea is that the difference between two numbers stays the same if you add the same amount to both numbers. This allows us to change a tricky problem into an easier one.

Method Subtraction by Compensation

Let's calculate:$32 - 14$

Step 1: Set up the subtraction.
Step 2: Check the ones place. We need to do $2 - 4$, but we don't have enough ones.
Step 3: Compensate. Add 10 to both numbers (10 ones to the top; 1 ten to the bottom)
Step 4: Subtract the ones. Now it's easy: $$12 \text{ ones} - 4 \text{ ones} = 8 \text{ ones}$$
Step 5: Subtract the tens.$$3 \text{ tens (from 32)} - 1 \text{ ten (from 14)} - 1 \text{ ten (compensated)} = 1 \text{ ten}$$
Result: The difference is 1 ten and 8 ones. So, $$32 - 14 = 18$$

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