In these lessons, we will learn some models for decimals, how to write decimals in expanded form, how to compare and order decimals, how to estimate and round decimals.

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1. Using decimals extends the place value system to represent parts of a whole.

The use of a decimal point is a symbol that separates the tenths from the ones, or the 'part from the whole'. For example, in the number 4.2, the decimal point separates the 4 ones and the 2 tenths.

2. The base ten place value system is built on symmetry around the ones place and the decimal.

3. Decimals can represent parts of a whole, as well as mixed numbers.

4. Decimals can be interpreted and read in more than one way. For example, 4.3 may be renamed 43 tenths.

5. Decimals can be renamed as other decimals or fractions. For example, 800/1000 or 0.800 can be renamed as 80/100 or 0.80. It can also be renamed as 8/10 or 0.8

Take note that

0.5 or 0.50 or 0.500 are all equal to 1/2

0.25 or 0.250 are both equal 1/4

0.75 and 0.750 are equal to 3/4

We can use a thousands grid to model decimals.

The following shaded grid can be used to represent the number 3.146

We can also use base ten blocks to model decimals.

The following can be used to represent the number 3.231

Metre sticks can also be used to represent decimals. Measuring to the nearest millimetre is one thousandths of a metre. Centimetres are hundredths of a metre and decimetres are tenths of a metre.

Model decimals on grids and on number lines.

Using a model of a garden, students develop an understanding of tenths and hundredths places, as well as relate decimals to fractions and percents.

For example:

45.23 = 40 + 5 + 0.2 + 0.03

or

45.23 = (4 × 10) + (5 × 1) + (2 × 0.1) + (3 × 0.01).

50.302 = 50 + 0.3 + 0.002

or

50.302 = (5 × 10) + (3 × 0.1) + (2 × 0.001).

Decimals - Expanded Form

Reading/writing decimals in numeric, word, and expanded form

0.02 < 0.2 because 0 tenths is less than 2 tenths

0.021 > 0.01 because 2 hundredths is greater that 1 hundredths

We can compare decimals using a benchmark number. For example:

0.021 < 0.2 because 0.021 is less than 0.1 and 0.2 is greater than 0.1

0.8 > 0.423 because 0.8 is more than half and 0.423 is less than half.

We can compare using equivalent decimals with same number of digits. For example:

0.34 > 0.3 because 0.34 > 0.30 (34 hundredths > 30 hundredths)

8.302 < 8.32 because 8.302 < 8.320 (302 thousandths < 320 thousandths)

Ordering & Comparing Decimals.

For example, 2.9286

can be rounded to 2.929 (nearest thousandths)

can be rounded to 2.93 (nearest hundredths)

can be rounded to 2.9 (nearest tenths)

can be rounded to 3 (nearest ones)

Estimating and Rounding with Decimals

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