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Contents

- 1 counting stamps
- 2 counting stamps
- 3 multiplication or addition
- 4 check 1
- 5 check 1
- 6 Division
- 7 counting more stamps
- 8 counting more stamps
- 9 Priority of Powers
- 10 check 2
- 11 check 2
- 12 More Operations
- 13 order of operations 1
- 14 order of operations 1
- 15 order of operations 2
- 16 order of operations 2

Now that we have a handle on writing numbers in different ways and working with exponents, let's look at how we can calculate answers. Let's start with a story. I set aside a stamp to mail a letter. Then, I made 2 piles of 3 stamps to mail larger packages. How many stamps did I use? Alright, this might be a fairly simple problem, but let's come up with a diagram to model it. This is going to be the first quiz for this section. I'm going to draw some diagrams and I want you to choose which one best represents my story. A square is going to represent a stamp, and a circle around them will represent a pile. Okay, is it diagram A, diagram B, diagram C, or diagram D? Go ahead. Make your choice.

Okay, in my story I had one single stamp, so I notice that in A, B, C, and D. I have a single stamp here. So the difference has to be with the piles. I made 2 piles of 3 stamps, so I know I need to find 2 piles in my diagram. Answer choice B has 3 piles, cause I see the 3 red circles of 2. So this one can't be right. In answer choice C, I have one pile of 6. So, that's not right either. When I look at answer choice A, I have 3 stamps in one pile and has to be correct. D is not correct. I have 4 stamps in 1 pile and 2 stamps in another pile.

So here's my question again, and this is the model that we agreed upon. Let's pair some math with this model. I would have 1 stamp plus 2 piles of 3, or 2 3. Okay, here's what's interesting about this math, I could actually calculate this math two ways. I could do 1 + 2 and get 3 3 which is 9. Or I could do 2 3 which is 6 and then add 1 which is 7. I know that the second answer must be correct. I have 7 stamps in my diagram. So I know that I need to do multiplication first. This 2 has a choice. It can either go with multiplication or it can go with addition. I know that multiplication takes priority. That's so that way, I can get the correct answer of 7. Just like in my model. Okay, so when I think about the order of operations, I know multiplication occurs before addition. Multiplication has to happen before addition, it's at a higher priority level. It turns out that multiplication is also done before subtraction. We know this because subtraction is the same as adding the opposite. For example, 3 - 1 2, subtracting a positive 1 is the same thing as adding a negative 1, or adding the opposite. So I could have 3 + (-1)2. Here the -1 could go with either the multiplication or the addition. I know that multiplication comes first, it has higher priority. So I should do 3 + -2, and I get 1. So when thinking about the order of operations we know addition and subtraction should be on the bottom, they come last. Multiplication should come before addition and subtraction. I'm going to keep track of our order of operations right here. And we'll come back to this later. Alright. Let's check your understanding of this with a quiz.

Okay. I want you to find the value of this expression. Use the order of operations that we discovered. Is your answer 34, 44, make your choice.

When performing the order of operations, you want to make sure we do multiplication first. So I understand multiplication here and here. 2 20 is 40 and 4 3 is 12, now can keep working. I am going to do addition first and then my subtraction. 1 + 40 + 5 makes 46 and then I subtract 12.

So now let's consider division. I know
division is the inverse of multiplication,
so it should also take higher priority
over addition and subtraction. So we know
in our hierarchy of the order of
operations multiplication and division
should come first. Let's consider this
problem: -3+24/4*3.*
Here are the two expressions that I could
write. This first expression has the
parentheses around the 24 and the four.
The second one has parentheses around the
four and the three. In this problem, it's
unclear whether or not we're dividing just
by four or dividing by 4*3.
We assume that this is true whenever we*
see the division symbol without
parentheses. The number immediately after
it is included or divided into the number
before it. So there's no need to solve
this problem. Here's one other word of
caution. Whenever we think about
simplifying this expression, we don't
actually want to use PEMDAS. If you've
heard of PEMDAS then you probably remember
that multiplication comes before division,
but this is not necessarily true.
Multiplication and division have the same
priority whenever we do the order of
operations. Students who learn that
multiplication come first using PEMDAS
would be incorrect. We know we should
carry out this division before we do the
multiplication. In general, we should
think about doing our simplification from
left to right, because multiplication and
division have the same priority in the
order of operations. So these parentheses
are assumed to be the case, so let's
evaluate this expression. I have -3+6*3*
and -3+18.
-3+18 makes 15. In doing our order of
operations, it's important to remember the
hierarchy. That multiplication and
division have the same priority and should
be done from left to right in the order of
operation.

Let's take this question and agree on the number of stamps that we bought using an exponent. Which of these expressions matches my story?

When thinking about an expression that
includes an exponent, let's start with one
of these squares. I have a 2x2 square. I
know I can write 2*2 as 2^2, but I don't*
have one square in my book of stamps, I
have five, so I need to multipy this by
on one book of stamps, but I have two
books of stamps, so we multiply this
expression by two. If you chose the first
one, nice work. This is a pretty tough
quiz, so just hang in there and keep
trying with these order of operation
problems.

Here is the expression that we agreed upon for the last 2 books of stamps. I'm going to show you two ways of doing this and then we're going to figure out which way is correct, based upon our diagram. For this first one I'm going to start in the parenthesis, I know I need to simplify containers first. When I look at this 2 I can choose between the exponent or I can choose multiplication, I'm going to choose the exponent first. So I'll get 2 5 4, I know 2^2 is 4. Right here I have multiplication. Remember that, whenever you have a number outside of a parenthesis, we just have multiplication in front. 5 4 is 20. and I finished simplifying my parenthesis. Now, I can do second one, I'm still going to start in parenthesis. But for this 2, I'm going to choose the multiplication over the exponent. So I'm going to do 5 times 2 first. So I have 2 10^2. I know 10^2 is 100 so I have 2 100 and I get 200. Okay, one of these answers is correct. I know that in 1 book of stamps, I have 20 stamps. So, in 2 books, I must have 40 stamps. So here are two books of stamps. And I can see that I should have method was correct. So let's go back and look. Here's where I made the choice. The two could have been paired with multiplication, or with exponents. I know that the two should be paired with the exponent. The exponent takes priority over, Over multiplication. So with our order of operations we know that we need to add in another priority. Okay, so when face with the decision between an exponent and multiplication I know that the exponent or the power should come first. So powers take the highest priority in our order of operation. When doing my order of operations I know power should come first. Multiplication and division should come second and then addition and subtraction should come third. So you might be wondering where parentheses fit on our order of operations. Well we just need to make sure we do the order of operations within each pa rentheses. We need to simplify any parentheses or brackets before we do any outside operations on them.

Okay, let's try out our order of operations with a quiz. Here's the problem and I want you to tell me its value. You can put your answer in this box here. Remember, if you don't get the question right the first time around, ask yourself, are you making the right choices based on the priority of the order of operations?

All right, when doing the order of
operations, I'm going to start within any
parentheses I have. I need to simplify
this container. So I'm going to leave the
I know I have a choice. This 2 could go
with multiplication, or with exponents.
The power, I know it should go with the
power because powers take priority. So I
have 4*2^3 is really 2 22 or 8-202.
Okay, let's keep going with simplifying*
these parentheses. I have 3 on the
outside, now with this 8 I have a choice,
it could be paired with multiplication or
subtraction. I know multiplication comes
first, so I'm going to have 32-20

Let's use our order of operations to help us solve this problem. We should solve this problem going from left to right following the hierarchy of the order of operations. Parenthesis and brackets serve as containers and they should be evaluated before we do any operations on to them. Brackets are typically used when we have parenthesis inside of them but we still evaluate them the same way. We start at the very inside and then work our way out. I'm going to simplify this -5^2. I know this 5^2 only goes with the power so I have 25 or -25. Now I simplify this parenthesis and then I add -20 to 3. I lose my brackets because I no longer have parenthesis. I just have -17 so I write that in parenthesis. I can't simplify this parenthesis anymore so now I should go back and do my order of operations. Let's look for powers. I have powers here and here, so I should simplify these numbers. I want to be very careful. I have the negative of 3 ^ 2, and the negative of 2 ^ the 2. So, I have -9, -17, and -4. Adding the opposite is the same thing as subtraction. That's why I can write -17. Finally, I can add all these negative numbers up, and I'll get my final answer. And I get -30. You can think of doing parenthesis and brackets first. We don't include them in our order of operations because, well, they're not operations, they're just containers.

Here's an order of operations problem you can solve, good luck.

Well Chris, that's a really tough order of
operations problem. But if you got the
answer of -88 great job. The first thing
we want to do when we look at a order of
operation problem is to do the inner most
parenthesis first. I can see right here
there is the inner most parenthesis of
So the first thing I'm going to do is
calculate 6-6^2.
doing these order of operation problems,
that you write down all the operations
that you have not yet performed. So, I'm
going to copy down the rest of the
problem. So for this first step I simply
evaluated what was in the innermost
parenthesis, which 6-36 is -30. Now I'm
working in the brackets, and I know that I
have to do my powers, or my exponents
first. So, -5^2 is -25, and 6^2 is 36. Now
that I have evaluated all my exponents or
powers, the next thing I want to do is my
multiplication. When I do my
multiplication its a good idea to work
from the left to the right, I'm still
working in these brackets so the 2
multiplications that I have to do are
-25*-2, which gives me 50+. I leave the*
-36, and the other multiplication that I
have to do is -1*-30, which gives me +30.*
Now that I have finished all of the
exponents, multiplication, and division
within my brackets, now I can do my
addition and my subtraction, 50-36 is 14,
and then I can add the 14 and the 30,
to do is this last operation, which is
multiplication. -2*44 gives me -88.*

Here's a second problem for the order of operations. Try this one out.

Wow Chris, that's another really hard
order of operations problem. If you got
-675, you got the answer right. Now this
was a tough problem and I even had to use
my calculator. Now let's see how I got
-675. When I'm looking at this problem,
once again I want to work from the inner
parenthesis outwards following my order of
operations. So the first thing I see is an
innermost parenthesis of 1+2.
So that's the first operation I'm going to
perform. Again, I want to make sure that I
copy down all of the other operations that
I have not yet performed. Now that I've
performed the operation in the innermost
parenthesis, I can do the next level of
parenthesis which are these brackets. So I
want to do -4(3), which is -12.
From here I see that I have another
operation in parenthesis, which is 4-10.
So I can evaluate 4-10, which is -6. Now
that I've performed all the operations,
inside parentheses, I can move on to my
exponents or my powers. So I have three
exponents, I have (-6)^2.
Remember when the negative is inside the
parentheses, it's part of the base, so
it's -6*-6 which is 36, 3^2 of course is*
all my exponents I can do my
multiplication and division. The only
multiplication I have is -5*144.*
So I can write 36+9, -5*144 is -720.
Now that I've done all my multiplication*
and division, I can do my addition and
subtraction in order from left to right.
So first thing I have to do is 36+9 which
is 45-720, 45-720 gives me the -675. Great
job.