Welcome back, mathematicians! Justin here. I've got something very exciting to show you.
Come with me. The scientists and engineers over at the MiaPlaza Space Center have prepped our
first rocket for space launch. Here comes the countdown: Five, four, three, two, one, zero.
Uh-oh, um, it seems that the launch failed due to a miscommunication and calculations among
the space exploration team. They must not have used the order of operations.
Solving mathematical problems in the wrong order can have
some serious consequences. I think we're practicing our order of operations today!
By the end of this lesson, you'll be able to apply the order of operations
to simplify complex expressions and explain your reasoning behind the
steps performed while you're solving. Have your guided notes ready, and let's dive in!
Now,
let's begin with a review of what we've learned in our last lesson.
In our most recent lesson, you learned that operations are actions that we perform in math,
like grouping symbols, exponents, multiplication, division, subtraction, and addition.
The order of operations is the sequence in which mathematical operations should be performed.
When solving a complex problem, start with the grouping symbols like parentheses or brackets,
followed by simplifying the exponents, then tackling the multiplication and division,
and finally handling any remaining subtraction and addition. To help us remember this order,
we use "GEMS." Ready to put our knowledge to the test? Have those guided notes handy, and let's go!
In math, the term "simplify" means to change a complex expression into a more simple form. So
let's simplify the expression: 3 squared times the quantity of 2 plus 1 minus 8.
Whenever we read expressions with grouping symbols, we use terms like "quantity" or
"the result of" to emphasize that the groupings should be performed first.
Following GEMS in the order of operations, we should always start by looking for groupings,
like this one: 2 plus 1 equals 3. So we'll record 3 right underneath that section of the expression.
Be sure to bring down the rest of the unsolved pieces in the order they appear in the expression.
Now we're ready to move on to the next operation in GEMS: exponents.
Does this expression have any exponents? Yes, right here! 3 squared, or 3 to the power of 2,
is the same thing as 3 times 3, which equals 9. So we can record that right
below the spot where the exponent was in the expression and bring down everything else.
According to GEMS, multiplication and division come next. They're inverse operations or
opposites, so they are ranked equally in GEMS. Whenever we approach multiplication or division
in a problem, we work from left to right, solving whatever comes first. There isn't any division in
this example, so we just need to take care of this multiplication right here: 9 times 3 equals 27.
Bring down that last part of the expression, and we're ready for the last level of the order of
operations: subtraction and addition. Subtract 27 minus 8, and that equals 19 -- simply simplified!
Let's try another one. This time we'll simplify the expression: the quantity of 15 plus 5 divided
by the quantity of 2 plus 2 cubed. Following GEMS, we start by looking for groupings. Interesting,
this problem has two sets of groupings. Whenever we're faced with more than one case of groupings,
we work from left to right, solving whichever grouping comes first. This same concept applies to
exponents. If you ever come across a problem with more than one grouping or more than one exponent,
just work from left to right until you're ready to move on to the next step in GEMS.
So we start by adding 15 plus 5, which equals 20. Bring down the rest of the expression,
and now we're ready to tackle that second grouping: 2 plus 2 cubed.
Notice how there are two operations within this set of parentheses: addition and an exponent.
Exponents come before addition in GEMS, so we'll start by solving 2 cubed, or 2 to the power of 3.
2 cubed means 2 times 2 times 2. Two times two equals four, and times two again equals eight!
Next, we're going to add 2 plus 8, and that gives us 10. Bring down this first part of
the expression, and we're left with 20 divided by 10, which equals 2. We've reached our final
answer. You're getting pretty good at this! Ready to try a little challenge?
For our next problem, we need to simplify the expression: 7 plus 3 times the quantity of 10
minus 1, all plus 2, all within parentheses. How fascinating! There are two different kinds
of grouping symbols in this problem, and the brackets are inside the parentheses.
This is a fun one, mathematicians! Take a moment to make a prediction about how you
think we should approach this problem. Feel free to pause the video while you think.
Whenever you come across a case of groupings inside another grouping, you should always work
on the innermost set first and then work outward. In this case, we have a set of brackets within the
parentheses, so we'll first tackle what's inside the brackets. Inside the brackets,
we have 10 minus 1, which equals 9. So we'll record 9 right underneath that section of the
expression and then bring down the rest of the unsolved pieces. You think you could solve the
rest of this problem by yourself? Pause the video here, and we'll check your work when you're done!
Remaining inside the parentheses is 9 plus 2, which equals 11.
Bring down the rest, and we're left with 7 plus 3 times 11. According to GEMS, multiplication
should be performed before addition. Three times 11 equals 33. 7 plus 33 equals 40. Great job!
We've got one more practice problem, and I think you're ready to try this one all by yourself.
The final problem we have is to simplify the expression:
4 squared minus the quantity of 3 plus 3 divided by 2 plus 7.
I challenge you to think through why you're doing each step as you solve it. Imagine you're
explaining the steps of this problem to a friend or a sibling. Pause the video here and solve!
We start with the parentheses: 3 plus 3 equals 6. Bring down any unsolved parts.
Next up, exponents. We have one exponent here: 4 squared equals 16.
Bring down the rest of the problem, and we're ready to move on to multiplication and division.
6 divided by 2 equals 3. Now we're left with subtraction and addition. Subtraction and addition
are ranked equally in GEMS, so we need to work from left to right, solving whatever comes first.
16 minus 3 equals 13, and 13 plus 7 equals 20. And there's our final answer!
Great work today, mathematicians! You now know how to apply the order of operations to simplify
complex expressions. We also practiced explaining our reasoning behind each step performed. Now
that you know how to use the order of operations, let's see if we can help out the space engineers.
You are not going to believe this; I just bumped into the lead engineer. Turns out they did follow
the order of operations, and their calculations were correct. They just forgot to fuel the rocket!
I'm going to help them fuel up. Meanwhile, be sure to complete the practice questions
and the extension activities that go along with this lesson. See you right back here next time!