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Today, we’re introducing evaluating expressions with order of operations.
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Let’s jump right in.
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Evaluate six plus four times three.
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Six plus four times three is a numerical expression.
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And we need to try and evaluate this expression.
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We’re actually gonna see how two different students would evaluate this expression.
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We’ll start with student A who says six plus four equals ten.
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So that’s the first step.
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And ten times three equals thirty.
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Student A has concluded that this expression is equal to thirty.
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Student B thinks the problem should be solved a little differently.
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Student B multiplies four times three first.
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And then add six plus twelve for eighteen.
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It seems as though we have a big problem now.
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Because two students have evaluated the same expression and come up with different answers.
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This expression, six plus four times three, can only be equal to one of these things.
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It can’t be thirty and also eighteen.
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And this is why learning the order of operations is so crucial.
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In this case, student B was following the order of operations.
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And student A was not.
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So let’s just pause this example and take a look at the order of operations.
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And then we’ll come back to this.
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Order of operations are rules that ensure numerical expressions have only one value.
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Just like in our last example, one student thought it was thirty.
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And one student thought it was eighteen.
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But there’s only one correct answer.
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Expressions have only one value.
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These rules tell us the order that we should operate or solve expressions.
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Not following the order of operations is like driving your car the wrong way down a one-way street.
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So let’s keep our car going the right direction and take a look at what the order of operations are.
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We remember the order of operations with the acronym PEMDAS.
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Or as I like to say “Please Excuse My Dear Aunt Sally.”
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And what in the world did these letters stand for?
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Well to start, the P stands for parentheses.
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So here you see the parenthesis around 𝑥 plus three.
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During this step, you solve anything that is grouped together, either with parentheses or brackets.
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The E stands for exponents.
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During this step, you solve any part of the expression that has an exponent.
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The next step is a little bit different.
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The M and the D stand for multiply and divide.
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During this step, we will do multiplication and division from left to right.
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So that doesn’t mean you do all the multiplication first and then all the division.
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You look at the whole problem and do the multiplication and division from left to right.
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Our last step is addition and subtraction together.
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Addition and subtraction function in the same way multiplication and division do.
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That means we add and subtract from left to right.
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We don’t solve all the addition and then all the subtraction.
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We look at the problem as a whole and solve from left to right.
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So let’s take these tools and head back to our first example.
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So “Please Excuse My Dear Aunt Sally” will help us remember how we would evaluate this problem.
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If we look at the expression and we work our way down the list, we start with P, parentheses.
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Does this problem have any parentheses?
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It does not.
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So we’re done with that step.
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We’ll also check for exponents.
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Are there any exponents in this problem?
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There are not.
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Now we move on to multiplication and division, remember from left to right.
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So is there multiplication or division in this problem?
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And the answer to that is yes.
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So we must multiply four times three first.
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That’s our first step here.
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And you can see that that’s what student B did.
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And that was the mistake that student A made.
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They added before they multiplied.
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Once you multiply four times three, your last step is addition and subtraction.
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In this problem, that means that we’ll add six to the twelve.
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Following student B’s example, six plus four times three equals eighteen.
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Here’s our next example.
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Evaluate two minus three plus seven divided by seven.
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We’ll need the tools that we learned in order of operations, “Please Excuse My Dear Aunt Sally.”
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Start this problem by checking the first step in your order of operations, parentheses.
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Because there are parentheses in this problem, we need to solve what’s inside those parentheses first.
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In the first step, we subtracted three from two and found that that was negative one.
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Following the order of operations, we’ll check to see if there are any exponents in our problem.
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There are not.
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So we’ll move on to multiplying and dividing from left to right.
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There was no multiplication.
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But there was a division.
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And we need to divide seven by seven.
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After you divide seven by seven to equal one, there’s only one step remaining in our order of operations.
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And there is only one step remaining in our expression as well.
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We just add and subtract from left to right.
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And in this case, we’ll be adding negative one to one.
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The solution to this expression turns out to be zero.
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Here’s another one.
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Evaluate four cubed minus nine divided by three squared plus five.
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No matter how big the expressions get or how many operations are inside the expression, the steps stay exactly the same.
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We start with writing down our PEMDAS, “Please Excuse My Dear Aunt Sally,” to help us remember what the order of operations are.
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Then we copy down the problem and start working our way from the beginning of the order of operations to the end.
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Are there any parentheses or any other form of grouping in this problem?
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The answer is yes.
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We’re gonna solve nine divided by three first.
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We divided nine by three and then copied the rest of our problem across.
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So now we have four cubed minus three squared plus five.
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Our next step, check for exponents.
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This expression has two sets of exponents.
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So we need to solve both of those during this step.
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We have four cubed, which equals 64, and three squared, which equals nine.
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And that ends all of the exponents in the expression.
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Next we’ll check for multiplication and division from left to right.
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In this case, there is none.
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Finally, we have addition and subtraction from left to right.
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But remember that we wanna solve this from left to right.
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We don’t start with all of the addition.
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We just start with what’s furthest to the left.
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In that case, that’s 64 minus nine, which needs to come first.
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And we’ll solve addition and subtraction from left to right.
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64 minus nine equals 55.
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55 plus five equals 60.
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And this expression equals sixty.
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To keep your car going the right way, remember “Please Excuse My Dear Aunt Sally.”
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Parentheses, exponents, multiplication, division, addition, and then subtraction.