Comparing and Ordering Decimals

Review of Decimal Place Value

Decimals show values that are less than one. They use a continuation of the place values that show whole numbers. Each place value is ten times less as you move to the right. This means that the first place to the right of the decimal point (the decimal point separates the whole units from the decimal parts) is tenths, the next is hundredths, and then thousandths.

It can be helpful to visualize decimals using place value blocks. We can show one whole unit as a single cube which can be divided into ten equal parts (tenths) which can be divided into a further ten equal parts (hundredths) which can be further divided into ten equal parts (thousandths).

block showing one whole unitBlocks showing ten tenthsBlocks showing ten hundredthsBlocks showing ten thousandths

 

:

Whole Unit . Tenths Hundredths Thousandths
block showing one whole unit . Block showing one tenth Block showing one hundredth Block showing one thousandth
Example
block showing one whole unit . Blocks showing three tenths Block showing two hundredths Block showing four thousandths
1 . 3 2 4

Decimals on a Number Line

If your children have had experience using a tape measure or rule then they will already have seen decimals on a number line. The first example below shows a number line with one whole unit divided into tenths which are divided into hundredths.

number line showing tenths and hundredths

The example below shows a section of the number line magnified so that the hundredths can be seen divided into thousandths.

Number line showing hundredths and thousandths

The two examples below show four different numbers marked on number lines. The first example shows hundredths and the second example shows thousandths.

number line showing .23 and .78

number line showing .406 and .451

Comparing Decimals

Given 2 decimals, it is fairly straightforward to compare them and decide which is greater or lesser when the same number of digits are involved. e.g .7 > .3 or .67 < .92

Many students have a problem though when the number of digits is different. e.g. comparing .3 and .27

This is because they, thinking about whole numbers, relate more digits to bigger numbers. A good way to overcome this problem is to write the decimals out in columns with headings. This highlights how zeros to the right make no difference to the value of the decimal.

  . Tenths Hundredths Thousandths
.3 decimal point 3 0 0
.27 decimal point 2 7 0
.4 decimal point 4 0 0
.305 decimal point 3 0 5
.52 decimal point 5 2 0

The column layout, with the extra zeros to the right, helps compare .305 and .4 and shows clearly that .4 is greater. Zeros to the right do not change the size of decimal numbers. Notice though that sometimes the zeros are very important as they are place holders and keep other numbers in the correct place.

examples shwoing importance of the position of a zero

Zeros to the right do have their uses though. As well as helping us compare values, they can indicate how accurately something has been measured. In the example above we know that .40 has not been rounded to the nearest tenth whereas .5 might have been.

Comparing Decimals Examples

When comparing decimals, start in the tenths place. The decimal with the biggest value there is greater. If they are the same, move to the hundredths place and compare these values. If the values are still the same keep moving to the right until you find one that is greater or until you find that they are equal. If it helps, add zeros to the right so both decimals have the same number of digits.

7 Examples
.34 > .21 We only need to go as far as the tenths place to find that .34 is greater than .21
.67 < .68 The tenths are the same so we need to compare the hundredths.
.7 > .59 This can trick people on a first quick look. We just need to look at the tenths though to see that .7 is greater than .59
.3 < .34 If we add a zero to the right of the .3 we can more easily compare the hundredths and see that .3 is less than .34
.562 > .561 Here the tenths and the hundredths values are the same. We need to go to the thousandths.
.702 < .720 We need to compare hundredths. Note the important and non-important zeros.
.60 = .6 The zero to the right can be ignored.

Ordering Decimals

If students have grasped the decimal place value concept and can compare decimals as in the example above, they should find most ordering of decimals fairly straightforward. There are though still some problems that can occur. Consider the sequence below. Can you see why some students might write 4.100?

4.97, 4.98, 4.99, ?

Thinking in columns helps students see how the 9 hundredths become 10 hundredths which are exchanged for one tenth that, when added to the 9 tenths, become 10 tenths which are exchanged for 1 whole unit which adds to the 4 to give 5.00

Temporarily ignoring the decimal point can help identify the next number in the sequence although it is not the best way to promote understanding.

Try the decimal sequences worksheet to get practice with this. It includes sequences with missing numbers that are the “bridging point” between whole number parts. e.g. 5.98, 5.99, 6.01, 6.02. Practice with these types of sequences helps students to build decimal place value understanding.

Comparing and Ordering Decimals Worksheets

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