Lesson 1: Finding Sums and Differences with Mental Math

Property

The Associative Property of Addition states that changing the grouping of addends does not change the sum: $(a + b) + c = a + (b + c)$. The Commutative Property of Addition states that changing the order of addends does not change the sum: $a + b = b + a$. These properties allow us to break apart and regroup numbers to make addition easier.

Examples

• Make Ten Strategy:

To find $4875 + 3007$, you can break apart $3007$ into $125 + 2882$ (so that $4875 + 125$ makes a thousand).
Then, group the $125$ with the $4875$ to make $5000$.

Property

To find the difference between two numbers, you can either count up from the smaller number to the larger number, or count down from the larger number in parts.
For $a - b = c$, you can find $c$ by adding up from $b$ to $a$, or by subtracting parts of $b$ from $a$.

Examples

Counting Up to find $4825 - 4578$:

• Start at 4578. Add 22 to get 4600. Then add 200 to get 4800.
• Finally, add 25 to get 4825.
• The difference is the sum of what you added: $22 + 200 + 25 = 247$.

Counting Down to find $9034 - 5276$:

• Start at 9034. Break apart 5276 into 5000 and 276.
• First, subtract 5000 from 9034: $9034 - 5000 = 4034$.
• Then, subtract 276 from 4034: $4034 - 276 = 3758$.
• So, $9034 - 5276 = 3758$.

Property

Compensation changes the numbers in a problem to make it simpler without changing the final answer.

• Addition (Give and Take): The sum is unchanged if you add a value to one addend and subtract the same value from the other.

• Subtraction (Constant Difference): The difference is unchanged if you add the same value to both numbers.