5th Grade Math
Order of Operation and Whole Numbers
Unit 1
Order of Operations and Whole Numbers
This image shows what each letter in PEMDAS stands for!
What is 'Order of Operations'?
This may be a review for you from 3rd and 4th grade, but this topic is important, so let's go over it together again!
"Operations" mean things like add, subtract, multiply, divide, squaring, etc. If it isn't a number it is probably an operation.
But, when you see something like ...
9 + (3 × 72 + 4)
... what part should you calculate first?
Start at the left and go to the right?
Or go from right to left?
Please Excuse My Dear Aunt Sally is an easy way to remember PEMDAS! Say it out loud a few times to remember it!
Let's try the example from above together!
9 + (3 × 72 + 4)
Do things in Parentheses First
4 × (5 + 3) = 4 × 8 = 32 :)
4 × (5 + 3) = 20 + 3 = 23 (wrong) :(
This is because 5 + 3 is 8, and we must solve whatever is in parentheses first.
Next is Exponents!
5 × 22 = 5 × 4 = 20 :)
5 × 22 = 102 = 100 (wrong) :(
This is because 22 equals 4, and we must solve 22 before multiplying by 5.
Then Multiply or Divide before you Add or Subtract
2 + 5 × 3 = 2 + 15 = 17 :)
2 + 5 × 3 = 7 × 3 = 21 (wrong) :(
This is because 5 x 3 equals 15, and we must solve the multiplication problem before adding 2.
Finally, if Operations are multiply and divide or add and subtract, just go left to right!
In PEMDAS, multiplication comes before division and addition comes before subtraction, but that it just how it is written. Look at the first image on the left, it says multiply OR divide and add OR subtract. That means you can perform division before multiplication if division comes first in the equation, and you can perform subtraction before addition if subtraction comes first in the equation. If this is confusing, no worries! We have shown an example below.
30 ÷ 5 × 3 = 6 × 3 = 18 :)
30 ÷ 5 × 3 = 30 ÷ 15 = 2 (wrong) :(
This is because the only operations are multiplication and division, so we can just solve this problem from left to right. Therefore, we have to solve 30 ÷ 5 first before we multiply by 3.
Still Confused? Watch these videos for more help with Order of Operations and PEMDAS!
Using Parentheses, Brackets, and Braces
You have probably seen parentheses being used in problems before, like the examples that you saw above in 'What is Order of Operations?' You may not have heard of bracket or braces before, but don't worry, they are just like parentheses!
This is a bracket: [ ] it is sometimes called a 'square bracket'
These are braces: { } they are sometimes called 'curly bracket's
They are used just like parentheses, so (26 + 18) ÷ 4 is the same thing as [26 + 18] ÷ 4 and they are both the same as {26 + 18} ÷ 4! The answer to all three of these is 11.
Next, let's look at comparing expressions that have parentheses, brackets, or braces to learn more about them!
Adding parentheses, bracket, or braces into equations can change their answer.
Example:
Let's compare 3 x 2 + 5 and 3 x (2 + 5).
For 3 x 2 + 5, using order of operations (PEMDAS), we know that we must multiply 3 x 2 before we add 5, so the answer would be 6 + 5, which is 11.
For 3 x (2 + 5), using order of operations (PEMDAS), we know that we must solve whatever operation is in the parentheses before multiplying by 3, so the answer would be 3 x 7, which is 21.
You see that adding parentheses into equation can change the answer completely, so we need to be careful when solving equation and notice whether there are parentheses, brackets, or braces
You try! Evaluate (2 + 3) x (1.5 – 0.5)
The answer is 5. We solved this problem using order of operations to solve the operations in parentheses first. 2 + 3 = 5 and 1.5 - 0.5 = 1, so then we get 5 x 1, which is just 1! Good job!
Here's another one! Evaluate (26 + 18) ÷ 4
The answer is 11. We first solve the operation in parentheses, which is 26 + 18, which is equal to 44. Then, following PEMDAS, we can divide by 4. 44 divided by 4 is 11!
Yay another practice question! Compare 15 – 6 + 7 and 15 – (6 + 7)
The first expression is equal to 16 while the second expression is equal to 2.
Let's walk through the steps if you are still confused.
For 15 – 6 + 7, we solve the expression from left to right since subtraction and addition are in equal order (although in PEMDAS the A comes before the S, they are in equal order), so first we solve 15 - 6, which is 9. Then, we add 7. 9 + 7 = 16, so the answer is 16!
For 15 – (6 + 7), we must solve the operation in the parentheses first, which is 6 + 7. 6 + 7 = 13. Then, we subtract 13 from 15 since subtraction is after parentheses. 15 - 13 = 2, so the answer is 2!
What is an Expression?
Expressions are a series of numbers and symbols (+, -, x, ÷) without an equals sign.
Example:
4(5 + 3) is an expression.
When we compute 4(5 + 3) we are evaluating the expression. The expression equals 32.
It is also important to be able to interpret expression as words, so the expression 4(5 + 3) in words is 'add three to five and then multiply by 4.'
To better understand and learn about expressions, we can look for relationships between them without calculating the answer.
Example:
7(2 x 2) is 7 times larger than the expression 2 x 2 because 7(2 x 2) means I have 7 groups of 2 x 2.
To check your answer (only do this after you have described how the expressions relate to each other!) you can calculate 7(2 x 2) as 7 x 4, which is 28. You know that 2 x 2 is 4, and 28 is 7 times greater than 4, so our description was correct!
What is an Equation?
Equations result when two expressions are set equal to each other. For example, 2 + 3 = 4 + 1.
An easy way to remember the difference is that an EQUAtion has an EQUALS sign!
Example:
4(5 + 3) = 32 is an equation.
You try! Write an expression for the steps “double five and then add 26.” hint: double means to multiply by 2 :)
The answer is (2 x 5) + 26. Great job!
Double five means to multiply 5 by 2, which is shown in (2 x 5).
After that, we can just add 26.
Up for a challenge? Describe how the expression 5(10 x 10) relates to 10 x 10.
The expression 5(10 x 10) is 5 times larger than the expression 10 x 10 since I know that I that 5(10 x 10) means that I have 5 groups of (10 x 10).
Understanding the Place Value System
Relationship Between Digits
Let's do some review from 4th grade! You learned that in whole numbers with many digits, a digit in one place represents 10 times what it represents in the place to its right and 1/10 of what it represents in the place to its left.
For example, the 2 in the number 542 is different from the value of the 2 in 324. The 2 in 542 represents 2 ones or 2, while the 2 in 324 represents 2 tens or 20. Since the 2 in 324 is one place to the left of the 2 in 542 the value of the 2 is 10 times greater.
Meanwhile, the 4 in 542 represents 4 tens or 40 and the 4 in 324 represents 4 ones or 4. Since the 4 in 324 is one place to the right of the 4 in 542 the value of the 4 in the number 324 is 1/10th of its value in the number 542.
Play with base ten blocks below!
Multiplying by negative powers of 10 is the same thing as dividing by positive powers of 10! If this is confusing, just ignore the bottom half of the image above for now :).
Patterns When Multiplying by Powers of 10s
In this section, we will go over multiplication and exponents with the number 10!
"^" is the symbol for an exponent, so 10^4 means 10 raised to the power of 4.
36 x 10 = 36 x 10^1 = 360
36 x 10 x 10 = 36 x 10^2 = 3600
36 x 10 x 10 x 10 = 36 x 10^3 = 36,000
36 x 10 x 10 x 10 x 10 = 36 x 10^4 = 360,000
When you multiply a number by 10, it is just a shift in where the decimal point is. We noticed that every time we multiplied by 10, we placed a zero to the end of the number. That makes sense because each digit’s value became 10 times larger. To make a digit 10 times larger, we have to shift it one place value to the left.
When we multiplied 36 by 10, the 30 became 300. The 6 became 60 or the 36 became 360. So, we had to place a zero at the end to have the 3 tens represent 3 one-hundreds (instead of 3 tens) and the 6 ones represents 6 tens (instead of 6 ones).
We can also divide by powers of 10!
Here's an example: 350 ÷ 10^3 = 350 ÷ 1,000 = 0.350 = 0.35
Here's another example: 52.3 ÷ 101 = 5.23 The place value of the digits in 52.3 is decreased by one place.
Performing Operations with Whole Numbers
Multiplying Whole Numbers
In 3rd and 4th grade, you learned how to multiply. We will practice more with multiplication now!
There are many ways to solve a multiplication question.
Here's a sample problem and the ways that you can solve it. You might be using one of these ways or you might even find that one way is easier for you to understand. Keep on reading to find out the problem!
There are 225 dozen cookies in the bakery. How many cookies are there?
The above explanations all result in the correct answer of 2700, so they are all correct ways to solve the question!
Models are another great way to solve a problem since they can help us visualize the numbers. For this question, you can draw an array model for 225 x 12, which is show below!
Watch this video to review how to solve a multiplication problem step-by-step!
Ready to get some practice? Play this game to multiply whole numbers!
This diagram shows the different parts of a division problem and their names!
Want a review? This helpful video goes over how to divide whole numbers!
Dividing Whole Numbers
In 4th grade, you learned to divide number with only 1-digit divisors. In 5th grade, we will build on top of that and learn how to divide with 2-digit divisors! A divisor is the number that we are dividing by, as show in the diagram on the left.
Here is a sample problem and the steps that 4 students have taken to solve the problem. Each one of these ways are correct since they result in the correct answer of 107 group with a remainder of 4 students.
There are 1,716 students participating in Field Day. They are put into teams of 16 for the competition. How many teams get created? If you have left over students, what do you do with them?
Here is another step-by-step way to solve division!
This will be our problem: 2682 ÷ 25
First we can write it out using expanded notation: 2682 ÷ 25 = (2000 + 600 + 80 + 2) ÷ 25
Then, we will use our understanding of the relationship between 100 and 25:
We first solve 2000 divided by 25. We know that 100 divided by 25 is 4, so 200 divided by 25 is 8, and 2000 divided by 25 is 80.
Next, we solve 600 divided by 25. 600 divided by 25 has to be 24. This is because 100 divided by 25 is 4, so 600 divided by 25 will be 6 times greater than 4 since 600 is 6 times greater than 100, so the answer is 6 times 4, which is 24.
Then, we solve 80 divided by 25. Since 3 x 25 is 75, we know that 80 divided by 25 is 3 with a reminder of 5 (since 80 is 5 greater than 75)
Finally, 2 divided by 25. We cannot divide 2 by 25 so 2 plus the 5 (in the previous step) leaves a remainder of 7.
80 + 24 + 3 = 107. So, the answer is 107 with a remainder of 7.
If you followed those steps, good job! If you did not completely understand, that is okay, there are many different ways to solve a division question.
We can also use models to help us solve division problems!
The image below is an area model for 9984 ÷ 64.
Ready to have fun? Click on the game to practice dividing whole numbers!