Why do we need models to add or subtract in decimals?
Visual models are useful for setting the foundation for addition or subtraction in decimals. The models are used to introduce the concept of place value in decimals.
What are base 10 blocks?
Base 10 blocks or models as we know, are math manipulatives that are used for building a strong understanding of place values in decimals. Here is an example of base 10 blocks.
If you notice carefully, the cubes represent tens, the flats represent ones, the singular rods are for tenths, and the units represent hundredths. Let us begin from the left of the place value chart, tens. The cube as a decimal represents 1 or 1.00. The flat is a tenth of the block, \(\frac{1}{10}\), or 0.1 in decimal. The rod is a hundredth as it takes 100 of those rods to make up a single cube. We can write it as \(\frac{1}{100}\), or 0.01 in decimal. The tiny cube represents the thousandths of the cube,\(\frac{1}{1000}\), or 0.001 in decimal. Remember that each place to the left of the base 10 blocks is 10 times the size of the place to the right.
Here is a representation of a decimal number as per the base 10 blocks. We will show 24.69 in the following base 10 blocks.
If you notice the digits, there are 2 cubes that represent 2 in the tens place, 4 flats that represent the 4 in the ones place, 6 rods that represent the 6 in the tenths place, and 9 tiny cubes that represent 9 in the hundredths place.
How can we add and subtract decimals by regrouping?
Adding and subtracting decimals by regrouping is similar to how it is done for whole numbers. Regrouping is done either by carrying over or borrowing. In regrouping, each group can be grouped by 10 and be moved to the immediate left as 1 group. For example, 10 hundredths will be grouped to form 1 tenths and that group will be moved to the left, thereby increasing tenths by 1.
Solved Examples
1. Find: 3.5 + 2.4
Answer: Let us first align these decimal numbers as per their place values while keeping the decimal points intact. Now, we will use models to add these decimal numbers.
We can represent this in models like the following image.
We first add the tenths and then the ones. This will give us 5.9 as the answer.
2. Find: 3.42 – 2.24
Answer: Let us first align these decimal numbers as per their place values while maintaining the decimal points intact.
As you can see, we cannot subtract 4 hundredths from 2 hundredths. So, we will regroup 1 tenth into 10 hundredths. In the following model diagram, we will show how we regroup.
We will subtract the hundredths first, then the tenths, and then the ones.
This will give us 1.18 as the answer.
3. Use models to add 3.25 and 2.18.
Answer: Let us prepare a model to add these decimal numbers.
In addition, we will regroup 10 hundredths as 1 tenth. We will then proceed with the addition where we add the hundredths first, then the tenths, and then the ones.
This will give us 5.43 as the answer.
Decimals are used for everyday activities such as calculating money, time, distance, measurements and many more. They give an accurate understanding of numbers that sometimes whole numbers aren’t able to. The decimal numbers represent precise values.
We use “models” to understand how decimals, fractions, and whole numbers are related. The decimal point represents parts of a whole number, thus separating the tenths from the ones, and so on. The decimal model is based on the place value system and helps in understanding the different parts of a whole number.