Arithmetic operations on decimal numbers share similarities with those on whole numbers. This is to be expected as the place value system for decimals in an extension of the system used for whole numbers with one key difference being the role of the decimal point. However, avoid just focusing on the decimal point and its placement without supporting your children’s understanding of how the place value of the digits works during arithmetic operations. You will find more on place value with tenths and hundredths here.
Being able to regroup and decompose decimal numbers using different units – tenths, hundredths, and thousandths – is important when adding and subtracting and this ability should come before moving onto using algorithmic methods which are typically used in 6th grade and beyond.
We can use a place value chart and counters to help with understanding the role of place value in the addition of decimals. The examples below show how this can be done. Note, there is no regrouping in the first two examples.
Note how the thirteen tenths are regrouped into one whole unit and three tenths. The examples below illustrate more regrouping of thousandths, hundredths, and tenths.
Using number lines provides an alternative visual representation of the addition of decimals as the two examples below show.
The number lines are annotated and graduated at intervals of tenths and hundredths. Encourage your children to use sketches on blank number lines as their arithmetic skills should move beyond addition or subtraction by counting single units.
Subtraction of decimals often requires the regrouping (or decomposition) of a 1 unit into 10 smaller units. Be sure your children are comfortable with this before moving on. Practice with questions like the ones below if required and/ or use Base-10 blocks to model the regrouping.
32 = (3 x 10) + (2 x 1)
32 = (2 x 10) + (12 x 1)
0.23 = (2 x 0.1) + (3 x 0.01)
0.23 = (1 x 0.1) + (13 x 0.01)
As with addition we can show the subtraction on a place value chart with counters as the examples below illustrate.
Be sure your children are aware that, as with the example above, they do not always have to regroup when carrying out an arithmetic operation.
We can use a number line to model decimal subtraction as shown below in a similar way to how we showed addition.
Your children may have developed strategies for performing arithmetic on whole numbers and these same strategies should be encouraged. For example, the subtraction shown on the above number line could be thought of as subtracting two whole units and then adding two tenths to correct the calculation as shown below.
With an understanding of decimal place value secure, your children can use the standard addition and subtraction algorithms. This understanding will help ensure that the essential step of lining up the decimal points is followed.
Be sure the decimal point is included in the answer.
You can review with your child how, when adding or subtracting whole numbers, zeros can be added to the left of a number without affecting its value.
The same logic can be applied to the tenths, hundredths, and thousandths using the base 10 blocks as a visual reference.
This will help when adding and subtracting numbers with different numbers of digits.
The example below shows a common mistake that is made when adding and subtracting with decimals.
Lining up the decimal point when doing vertical addition or subtraction makes the calculation much easier. Adding zeros to the right of the decimal numbers may also be needed. When adding with decimals and whole numbers students may also have to add a decimal point on the right of the whole number. Finally, estimating the answer and comparing it with the calculated answer will help avoid errors that do not make sense.
Try the decimals worksheet generator. It provides limitless adding decimals questions. It also provides questions on subtraction and multiplication too.