Key Concepts:

• The properties of addition and subtraction
• Commutative property of addition
• Associative property of addition
• Equivalent expressions
• Additive identity property
• Some common procedures for adding and subtracting decimals
• Methods or strategies to find sums and differences mentally

Introduction:

Liam is having a party. He needs to buy the party snacks and drinks from different stores. Liam spends $21.60 at the first store and $19.40 at the second store. He is wondering if he can afford to go to a third store for more snacks. He does not have a calculator but wants to quickly figure out how much he has spent so far. How can mental math help calculate how much Liam has already spent on snacks?

In this kind of situation, we can use mental math instead of going through the procedure of adding decimals regularly.

How can we use mental math to add and subtract decimals?

Let us consider the situation we discussed earlier.

Using the prior knowledge of pairs of numbers which makes ten or any whole number we can add these numbers easily.

For example, we know that 6+4 = 10 or 0.6 + 0.4 = 1.0

 Using this we can add,

                                           $21.60+$19.40

                                                   0.60 + 0.40 = 1 (Decimal part)

                                            $21 + $19 = $40 (Whole number part)

                    $40   + $1 = $41     

              ⸫ $21.60+$19.40 = $41     

That means Liam has already spent $41.  

Properties of addition

To add mentally, we need to write equivalent expressions.

What are equivalent expressions?

We can use properties of operations to write equivalent expressions that can be simplified using mental math.

The two properties of numbers are:

1. Commutative property of addition:
2. When adding, changing the order of the numbers does not change the sum.

For example

2.8 + 3 .0 = 3.0 +2.8

             5.8 = 5.8

• The commutative property let us add two decimals in any order.       

1. Associative property of addition:
2. When adding more than two numbers, the grouping of the numbers does not change the sum.

For example

5.3+ (7.7+9.4) = (5.3+7.7) +9.4

 5.3 + 17.1= 13+ 9.4

             22.4 = 22.4

• The associative property lets us change the grouping of addends.

Examples:

Use mental math to find the sum and explain the properties used.

1. Alex purchased three items, each costing $18.25, $ 11.36, and $20.75. What is the total amount he spent?

The total cost of items = $ 18.25+$11.36+ $20.75

Use the commutative property to change the order of the addends.

• $18.25 +( $20.75 +$11.36)

Now use the associative property to change the groups.

• ($18.25+ $ 20.75) + $ 11.36

Notice that 18.25 and 20.75 are compatible numbers.

            = $50.36

           ⸫ Total cost = $50.36

1. Joseph traveled a distance of 33.5 miles in the first hour, 25.8 miles in the second hour and 32.5 miles in the third hour. What is the total distance traveled by Joseph in three hours?

Total distance traveled = 33.5 miles + 25.8 miles+ 32.5 miles

                                             = (33.5 + 25.8 + 32.5) miles

By using the commutative property, change the order of the last two addends.

                                             = {33.5 + (32.5+ 25.8)} miles

By using the associative property, regroup the addends.

                                   = {(33.5 + 32.5) + 25.8} miles

Notice that 33.5 and 32.5 are compatible numbers.

(33.5 + 32.5) = 33+32+1 = 66

                           = {66+ 25.8} miles 

                           = 81.8 miles

⸫ Joseph traveled 81.8 miles in three hours.

Compensation

This is one of the most important strategies of mental math when adding multi–digits. This involves adjusting one of the addends to make the calculation easy. This is also an alternative to the left-to-right addition or the breaking up of the second number strategy.

How to Perform the Compensation Strategy?

Let us solve one question using whole numbers.

Add 64 + 39

Since 39 is close to 40

 Add 1 to 39 and subtract 1 either from the total or from 64.

Method 1:

      64 + (39+1) = 64 +40 = 104

          ⸫ 64+39 = (104 -1) = 103

Method 2:

    (64-1) + (39+1) = 63 +40 = 103

Example2:

Let us suppose we want to add 132 + 63.

        We know that 63 is close to 60.

So, subtract 3 from 63 and add 3 to the answer or to 135.

Method 1:

                      132+63 = 132 + (63-3) = 132 +60 = 192

But we have to compensate the 3 removed from 63.

                   ⸫ 132+63 = 192 +3 = 195

Method 2:

           132 + 63 = (132+3) + (63 -3)

                            = 135 + 60 = 195  

How to Perform the Compensation Strategy for Decimals?

•  To use compensation method for addition and subtraction of decimals, we have to round one of the addends to the nearest whole number.

For example

To add 4.6 + 1.5

Either we can round 4.6 to the nearest whole number or 1.5 to the nearest whole number.

One way:

                    4.6     +   1.5

                 (+0.4)      (- 0.4)

                         5   + 1.1 = 6.1

Another way:

                 4.6     + 1.5

            (- 0.5)       (+0.5)

                     4.1 + 2    = 6.1

Example 2:

Use compensation method to solve 9.8 – 2.6

One way:

Change 2.6 to 2.0 by subtracting 0.6 from 2.6 and adjust the same by subtracting from 9.8.

                         9.8    –    2.6

                       (- 0.6)     (-0.6)

                         9. 2 – 2.0 = 7.2

Another way:

Change 2.6 to 3 by adding 0.4 to 2.6 and adjust the same by adding 0.4 to 9.8.

                          9.8   – 2.6

                    (+0.4)    (+0.4)

10.2 – 3 .0 = 7.2

Concept map:

What we have learned:

• Using mental math to add and subtract decimals.
• Expressions that have the same value are called equivalent expressions.
• Commutative Property: When adding two numbers, changing the order of numbers does not make any change in the sum.
• If a and b are two numbers, then a + b = b + a.
• Commutative property lets us add numbers in any order.
• Associative Property: When adding more than two numbers, the grouping of numbers does not change the sum.
• If a, b and c are any three numbers, then a+(b+c) = (a+b) +c.
• Associative property lets us change the grouping of addends.
• Compensation: To use the compensation method for addition and subtraction of decimals, round one of the addends to the nearest whole number.