### Key Concepts

- Finding Sums and Differences with Mental Math
- Estimate Sums and Differences

## Finding Sums and Differences with Mental Math

### Mental Math:

- Mental math is a group of skills that allow people to do the math ‘in their head’ without using a pencil, paper, or calculator.
- Mental math is useful in school and in everyday life.
- Mental math can help kids understand math concepts better and get to the answers faster.

### Solving mental math with the help of properties of addition and subtraction

**Associative Property **

*a* + (*b* + *c*) = (*a* + *b*) + *c*

**Commutative Property **

a + b = b + a

**Identity Property **

a + 0 = a

**Example 1:**

Find $1,985 + 2,595 with mental math. Use addition properties to find the sum

#### Using Associative Property:

Break apart 1,985 to get a number that makes a ten, hundred, or thousand when added to 2,595. Then use the **associative property** **of addition** to change the grouping.

1,985 + 2,595

= (1,580 + 405) + 2,595

= 1,580 + (405 + 2,595)

= 1,580 + 3,000

= 4,580

#### Using Commutative Property:

Break one addend apart and add on

You can start with either addend because of the commutative property of addition.

#### Using Identity Property:

Add 15 to 1,985. Then subtract 15 from 2,595 to compensate. Adding 15 and subtracting 15 is the same as adding zero. Adding zero does not change the sum because of the **identity property of addition**.

1,985 + 2,595

= (1,985 + 15) + (2,595 – 15)

= 2,000 + 2,580

= 4,580

**Example 2:**

Subtract 2,595 – 1,985 with mental math. Use subtraction properties to find the difference

**Count Up**

Count from 1,985 up to 2,595

Find how much you counted up

5 + 10 + 595 = 610

**Count Down**

Count down 1,985 from 2,595

**Use compensation**

Adding the same amount to both numbers in a subtraction problem does not change the difference.

(2,595 + 15) – (1,985 + 15)

= 2,610 – 2,000

= 610

**Estimate Sums and Differences**

Estimation is the process of finding an estimate or approximation

**Estimating Sums**

You could also round 5, 387 to the hundreds place,

**Step 1:** Rounding is always first decide which place to round to.

**Step 2:** Round the first addend to the thousands place and the second addend to the hundreds place.

**Step 3:** Now you can add 5, 400 and 400 mentally.

**Estimating Differences**

The actual answer is **33,209**, so our estimation is pretty accurate.

**Step 1:** Rounding is always first decide which place to round to.

**Step 2:** Round each addend to the ten thousands place.

**Step 3:** Now you can subtract 20, 000 from 50, 000 mentally.

**Example:**

12,642 books, 4,298 magazines, and 2,149 movies were checked out of the public library. About how many more books were checked out than magazines and movies combined?

Estimate: Round to the nearest thousand

Find the number of movies and magazines

4,298 –> 4,000

+2,149 –> + 2,000

= 6,000

Subtract the number of magazines and movies from the rounded number of books

13,000 – 6,000 = 7,000

About 7,000 more books were checked out

Estimate: Round to the nearest hundreds

Find the number of movies and magazines

4,298 —-> 4,300

+2,149 —-> + 2,100

= 6,400

Subtract the number of magazines and movies from the rounded number of books

12,600 – 6,400 = 6,200

**Exercise:**

**Read and answer each question:** The table shows the number of people visiting an art museum over 3 months.

January | February | March | |

Child | 28 | 34 | 56 |

Adult | 59 | ? | 55 |

Senior | 15 | 22 | ? |

Total | ? | 139 | ? |

- What is the total number of people that visited the art museum in January?
- Compared to January, how many more children go to the museum in February?
- How many adults visited the museum in February?
- 16 more seniors visited in March than the number that visited in January and February combined. How many seniors visited the museum in March?
- Which month had the highest number of visitors?
- Write an equation using
*x*and then solve the equation.

“In February, there were*x*museum pass holders admitted to the museum. 68 of the visitors did not have a museum pass.” - Ashley finds a dress that costs $116 and a hat that costs $42. How much does she need to pay if people buying two clothing items get $15 off on the second item?
- Estimate and add: 7549 + 3808 + 4261.
- Rachel shots baskets each day for a period of 2 weeks. She shot a total of 2260 baskets. Rachel shot 100 more baskets each day during the last 3 days. How many shots per day did she take during the first week?
- A school has a total of 1258 students. There are 297 primary students and 364 junior students. How many senior students are there?

**What have we learnt:**

- How to add multi-digit numbers using the standard algorithm
- How to subtract multi-digit numbers using the standard algorithm
- The different properties of addition
- Finding sums and differences with mental math
- Understand how to estimate sums and differences