5th Grade Math
Adding and Subtracting with Decimals
Unit 2
Understanding the Place Value System
Reading, Writing, and Comparing Decimals
We can read and write decimals in expanded form like shown below:
347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 x (1/100) + 2 (1/1000)
This may be a review from 4th grade, but if you would like a refresher, watch the video on the right!
We will learn to understand the size of decimal numbers and compare them to common numbers such as 0, 0.5 (0.50 and 0.500), and 1.
Example:
Let's compare 0.25 and 0.17!
25 hundredths is more than 17 hundredths. In fact, it is 8 hundredths more. We can write this comparison as 0.25 > 0.17 and recognize that 0.17 < 0.25 is another way to express this comparison.
Let's look at another problem. Let's compare 0.207 to 0.26!
Both numbers have 2 tenths, so we need to compare the hundredths. The second number has 6 hundredths and the first number has no hundredths so the second number must be larger.
Another way to think about this is we know that 0.207 is 207 thousandths (and we can write it as 207/1000). 0.26 is 26 hundredths (and we can write it as 26/100) but we can also think of it as 260 thousandths (260/1000). So, 260 thousandths is more than 207 thousandths. So the answer is 0.26 > 0.207
Practice comparing decimals using these symbols >, =, and < with the games below!
Up for some fun? Play this game to order three decimals in increasing or decreasing order of size!
Rounding Decimals
We can round decimals to a certain accuracy or number of decimal places. This is used to make calculation easier to do and results easier to understand, when exact values are not too important.
First, you'll need to remember your place values (shown on the right)
To round a number to the nearest tenth , look at the next place value to the right (the hundredths). If it's 4 or less, just remove all the digits to the right. If it's 5 or greater, add 1 to the digit in the tenths place, and then remove all the digits to the right.
(In the example above, the hundredths digit is a 4 , so you would get 51.0 .)
To round a number to the nearest hundredth , look at the next place value to the right (the thousandths this time). Same deal: If it's 4 or less, just remove all the digits to the right. If it's 5 or greater, add 1 to the digit in the hundredths place, and then remove all the digits to the right.
(In the example above, the thousandths digit is an 8 , so you would get 51.05.)
In general, to round to a certain place value, look at the digit to the right of that place value to decide.
Here's another example!
Round 14.235 to the nearest tenth.
We are told the answer must be in tenths thus, it is either 14.2 or 14.3. We can see that 14.235 is closer to 14.2 (14.20) than to 14.3 (14.30) since the 3 in the hundredths place tells us to round down.
To check our work, we can use benchmark numbers. Benchmarks are convenient numbers for comparing and rounding numbers. 0, 0.5, 1, 1.5 are examples of benchmark numbers. Try using benchmark numbers in the practice problem below
Which benchmark number is the best estimate of the shaded amount in the model to the left? Explain your thinking.
There is more than 1 correct answer. You may have said that a good benchmark number would be 50 since the block to the left is made up of 100 units. Therefore, we can easily tell whether half of the block is shaded or not. There are 62 squares shaded, and we can tell that the shaded part covers more than half of the entire block (so more than 50). Therefore, the benchmark number 50 can tell us that we are correct in saying that more than half of the entire block is shaded.
Click on the game to practice rounding decimals to the nearest hundredth :)!
Performing Operations with Decimals
Adding Decimals
Before we solve any equations or expressions with decimals, we should estimate answers. An example is given below:
Let's estimate the sum of 3.6 + 1.7
We estimate the sum to be larger than 5 because 3.6 is more than 3½ and 1.7 is more than 1½ (and 3½ + 1½ = 5).
Another important concept is that when we are adding in a vertical format, it is important that we write the numbers so that the decimal points are in a line on top of another (shown in the image at the right). That way, all the numbers are in the same place value, which means when we add decimals, we add tenths to tenths and hundredths to hundredths.
Below is an example of adding decimals using models! There are two student responses, and both are correct.
A recipe for a cake requires 1.25 cups of milk, 0.40 cups of oil, and 0.75 cups of water. How much liquid is in the mixing bowl?
Follow along with this video to see a decimal addition problem being worked out!
Ready to put what you've learned into practice? Play this game to use models to gain understanding of decimal addition!
Follow along with this video to see a decimal subtraction problem being worked out!
Subtracting Decimals
We will use what we just learned about adding decimals to learn how to subtract decimals!
Let's practice estimating the answers to equations and expressions with decimal subtraction! An example is given below:
Let's estimate the answer to 5.4 – 0.8
We estimate the answer to be a little more than 4.4 because a number less than 1 is being subtracted.
Just like adding decimals, when we are subtracting decimals in a vertical format, it is important that we write the numbers so that the decimal points are in a line on top of another!
Below is an example of subtracting decimals using models!
What is 4 - 0.3?
This is because 0.3 is less than one, so we know that the answer must be greater than 3 (since 4 - 1 = 3). Therefore, we focus on the fourth box and split into tenths (since 0.3 is in the tenths place), and we must shade in 3 tenths boxes since 0.3 is the same as three tenths, and shading the boxes means that we are taking that amount away from the total.
You try! What is 9.7 - 3.8?
It is 5.9, good job! You could have solved this by lining up the numbers in a vertical way (with 9.7 on top of 3.8) or you could have used a model like the one shown in the example problem above.
