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Grade 5 (age 10-11)

grade 5 Decimal Multiplication & Division

To multiply decimals, set the decimal points aside, multiply as if they were whole numbers, then give the answer as many decimal places as the two numbers have between them. Just as a rectangle 1.5 m wide and 1.2 m tall has an area of 1.5 × 1.2, a product of decimals can be drawn as an area. To divide by a decimal, move the decimal point in both numbers the same number of places to the right, so that you are dividing whole numbers. The first picture shows the two numbers as the sides of a rectangle and on a number line; the second cuts a ribbon into equal pieces.

Decimal Multiplication

📐 Decimal multiplication on a grid
①Each grid square is 0.1 wide and 0.1 tall, so its area is 0.1 × 0.1 = 0.01
②A rectangle 1.5 wide and 1.2 tall holds 15 × 12 = 180 squares, so its area is 180 hundredths, which is 1.80
③That is why two numbers with one decimal place each give a product with two decimal places. The picture always writes two places, and 1.80 is the same number as 1.8
④Make the height less than 1 and the green dot (the product) on the number line moves to the left of the blue dot (the first number), because multiplying by a number less than 1 makes the result smaller
🔢 Reading the sliders
The sliders move in steps of 0.1. One grid square is 0.1 long, so 1.5 is the length of 15 squares and 1.2 is the length of 12.
1.5
1.2

Decimal Division

✂️ How many pieces?
①Cut a 2.4 m ribbon into 0.6 m pieces and you get 4 of them: count the orange pieces and you see 2.4 ÷ 0.6 = 4
②Move the decimal point one place to the right in both numbers and you get 24 ÷ 6, which is still 4. Counting in tenths, you are splitting 24 tenths into groups of 6 tenths
③If it does not divide evenly, a green piece is left over. 2.5 ÷ 0.6 gives 4 pieces with 0.1 left, so 2.5 ÷ 0.6 = 4 R 0.1. In tenths that is 25 ÷ 6 = 4 R 1, but the 1 is one tenth, so the remainder is 0.1, not 1
④The leftover is always shorter than one piece, because if it were longer you could cut another piece
2.4
0.6

Summary

Decimal point in a product
decimal places in the product = sum of the decimal places in the two numbers
0.3 × 0.4 = 0.12 (1 place + 1 place = 2 places)
Dividing by a decimal
2.4 ÷ 0.6 = 24 ÷ 6 = 4
Moving both decimal points the same number of places keeps the quotient the same; read any remainder at the original decimal place
🎯 At a glance
①Multiplying decimals: multiply as whole numbers, then count the decimal places of both numbers
②Counting grid squares of 0.01 shows why the decimal places add up
③Dividing by a decimal: move both decimal points the same number of places and divide whole numbers
④Multiplying by a number less than 1 makes the result smaller, and dividing by a number less than 1 makes it larger
⑤A remainder is smaller than the divisor and is read at the original decimal place

Examples and Unit Test

Example 1
Work out 0.3 × 0.4.
1
Ignore the decimal points and multiply: 3 × 4 = 12.
2
Each number has one decimal place, so the answer has 1 + 1 = 2 places: 0.12.
▸ 0.12
On the grid this is 3 × 4 = 12 squares of 0.1 × 0.1 = 0.01 each, which makes 0.12.
Example 2
Work out 2.4 ÷ 0.6.
1
Move the decimal point one place to the right in both numbers: 24 ÷ 6.
2
24 ÷ 6 = 4, so 2.4 ÷ 0.6 is also 4.
▸ 4
This is the starting setting of the second picture, where 4 orange pieces fill the 2.4 exactly.
Unit Test
What is 1.5 × 1.2?
①1.7
②1.8
③18
④0.18
▸ ② 1.8
1
Multiply without the decimal points first: 15 × 12 = 180.
2
There are 1 + 1 = 2 decimal places, so the answer is 1.80, which is 1.8.
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Fraction Multiplication & Division
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Mixed Number Operations

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