Math confusion-MCP math grade 6 Order of Operations Question

[Melissa H. in GA]

Hello all,

 

The teacher's edition of the 2005 edition of Modern Curriculum Press math says when doing the order of operations:

 

First, evaluate the powers.

 

Then, do the operations inside the parentheses,

 

Then, multiply and divide in order from left to right.

 

Last, add and subtract in order from left to right.

 

OK, I always was taught PEDMAS, which puts parentheses before exponents.  What do you think they mean by "evaluate the powers"?  I'm thinking maybe go ahead and set them up for multiplying as in

2^3  2X2X2  and then do them when it's time to multiply?

 

Any thoughts?

 

Thanks,

 

Melissa

[Spring Flower]

Former math teacher here. I taught order of operations many times. I also tutored in the math lab at my university and the #1 mistake I saw was order of operations. PEMDAS was my basic go-to method but it can get confusing and tricky sometimes depending on the problem. 

 

I am having a hard time, off the top of my head, thinking of an example where solving the exponents before the parentheses would make a difference. (Maybe someone else can prove me wrong.) Parentheses and exponents often go hand in hand.  I think it is safe to "evaluate the exponents" as the book says. 

 

 

I'm thinking maybe go ahead and set them up for multiplying as in

2^3  2X2X2  and then do them when it's time to multiply?

 

 

I would not recommend doing this. 

[Melissa H. in GA]

Thanks, I'll go with that then.

[kbernt]

Here's an example where the "powers before parentheses" rule would fail if they tried to apply the power to each member inside the parentheses first:

(2+3)^2 + 1

Because if they try to evaluate the powers first, THEN do the parentheses, they could get 2^2  + 3^2  + 1 = 14, when the answer should be 26: (5)^2 + 1

Parentheses (and radicals, which basically act parenthetically too) should always, always be done first. The only way to properly teach "powers first" is to know that to "properly" evaluate powers means that if the power is applied to a parenthetical expression and if that expression involves addition/subtraction, then it needs added or subtracted first THEN taken to its power. Seems kind of weird and confusing to me, because you're still doing the parentheses first in the long run. (It doesn't matter if the only operation inside the parentheses is multiplication/division.) If I were you, I'd stick with the old PEMDAS. I've run in to multiple times over the years where I just have to explain that the author of the book is only human and thus can see things differently than others do, or, at times can make flat out mistakes (or the publisher botched the author's work), and our Creator didn't make us all the same nor perfect, so just explain the right way and move on! 

[MASHomeschooler]

I learned it as 3 "chunks" of order: PP/MD/AS - so first you go through and do any powers and parentheses, as they come up, then multiplication and division, then addition and subtraction. But, for the first two, as pp said, you can't really mess it up doing either before the other. You do have to understand that (2+3)^2 is NOT 2^2 + 3^2; in order to evaluate (2+3)^2, you need to either do the parenthesis first, or expand it to 2^2 + 2*2*3 + 3^2.

[kbernt]

I learned it as 3 "chunks" of order: PP/MD/AS - so first you go through and do any powers and parentheses, as they come up, then multiplication and division, then addition and subtraction. But, for the first two, as pp said, you can't really mess it up doing either before the other. You do have to understand that (2+3)^2 is NOT 2^2 + 3^2; in order to evaluate (2+3)^2, you need to either do the parenthesis first, or expand it to 2^2 + 2*2*3 + 3^2.

 

Well said. (Better than I did!)

[Melissa H. in GA]

Thanks for the help.  I'll think of it in the 3 chunks, keeping PEDMAS in the back of my mind.

[Spring Flower]

Here's an example where the "powers before parentheses" rule would fail if they tried to apply the power to each member inside the parentheses first:

(2+3)^2 + 1

Because if they try to evaluate the powers first, THEN do the parentheses, they could get 2^2  + 3^2  + 1 = 14, when the answer should be 26: (5)^2 + 1

 

???

 

If the exponents were evaluated first it should be (2+3)(2+3) +1. Then (5)(5) + 1 =  26.  Exponents cannot be distributed. 

[kiana]

I have no idea why they'd put "evaluate the powers" first. What if there are powers outside of parentheses? In that case, they cannot be evaluated first.  -- for example: the square root of [5-(1-5)] -- in other words, [5-(1-5)]^.5

[Spring Flower]

I have no idea why they'd put "evaluate the powers" first. What if there are powers outside of parentheses? In that case, they cannot be evaluated first.  -- for example: the square root of [5-(1-5)] -- in other words, [5-(1-5)]^.5

 

This is why order of operations is so tricky. It often happens that you don't solve in the exact order P-E-M-D-A-S when doing complicated problems. Sometimes there is more than one order you can solve it to get the correct answer.

 

Simple example: 3^3 + (2+3)^2

 

Exponents first:

9 + (2+3)(2+3)

9 + (5)(5)

9 + 25 = 34

 

Parenthesis first:

3^3 + 5^2

9 + 25 = 34

[kiana]

This is why order of operations is so tricky. It often happens that you don't solve in the exact order P-E-M-D-A-S when doing complicated problems. Sometimes there is more than one order you can solve it to get the correct answer.

 

Simple example: 3^3 + (2+3)^2

 

Exponents first:

9 + (2+3)(2+3)

9 + (5)(5)

9 + 25 = 34

 

Parenthesis first:

3^3 + 5^2

9 + 25 = 34

 

Yes. But if you have a non-integer exponent, your first method is non-applicable. Teaching kids a method (always do exponents before parentheses) that does not apply in more advanced math, that they are going to have to un-learn later, is not a good idea. 

 

An advanced student is going to recognize that they yield the same answer and choose the best method to get the correct answer. For example, if I see 2+3+5+7^2, there is no reason I can't do 2+3+5 = 10, 7^2 = 49, and then do 10+49 = 59, even though that does not strictly follow the standard order of operations.

[Spring Flower]

Yes. But if you have a non-integer exponent, your first method is non-applicable. Teaching kids a method (always do exponents before parentheses) that does not apply in more advanced math, that they are going to have to un-learn later, is not a good idea. 

 

An advanced student is going to recognize that they yield the same answer and choose the best method to get the correct answer. For example, if I see 2+3+5+7^2, there is no reason I can't do 2+3+5 = 10, 7^2 = 49, and then do 10+49 = 59, even though that does not strictly follow the standard order of operations.

 

I think we are on the same page here. I guess my point is that parentheses and exponents can be done in different orders. I would never say "always" do exponents before parenthesis and I doubt the book is saying that either. I don't think PEMDAS is meant to be done in the exact order of P-E-M-D-A-S. Learning to do order of operations efficiently is the goal. 

 

As I said in my original comment, I cannot think of an example when doing exponents before parentheses results in an incorrect answer. 

 

I agree with above posters about chunking PEMDAS into three parts. 

[kiana]

I think we are on the same page here. I guess my point is that parentheses and exponents can be done in different orders. I would never say "always" do exponents before parenthesis and I doubt the book is saying that either. I don't think PEMDAS is meant to be done in the exact order of P-E-M-D-A-S. Learning to do order of operations efficiently is the goal. 

 

As I said in my original comment, I cannot think of an example when doing exponents before parentheses results in an incorrect answer. 

 

I agree with above posters about chunking PEMDAS into three parts. 

 

PEMDAS is definitely not meant to be done as multiplication before division, addition before subtraction. This is one reason that there's been a movement in some circles of math education to use something more like GEMS (grouping-exponents-multiplication-subtraction) and to emphasize that addition and subtraction are the same operation, as are multiplication and division. You'll notice that grouping still overrides exponentiation. 

 

Grouping symbols (such as parentheses) override the normal order of operations. That is why they need to be done first. A student who's been explicitly taught that exponents come before grouping symbols is going to have an issue with an expression like 4^(1.5+2), or 4^(3x0.5). I have no idea whether MCP really is teaching this, but if -- as the op says -- it is, it is wrong. 

 

Simply because, in some specific situations, they CAN be done first does not mean that grouping symbols do not have precedence. If I have 3+4+5x6, I can do the 3+4 first and get the correct answer, but multiplication still has precedence over addition. 

 

ETA: And the reason I keep posting is that OP needs to know (if the curriculum is indeed teaching this) that this is not considered correct by mathematicians. 

[Melissa H. in GA]

Wow, ladies, thank you so much for your time on this one.  I'm going to show the book to my 21 year old son who was working all week, and see if he sees something I missed. 

 

Again, thanks and I'm trying to take  all of your replies in myself, never having done higher math in school.

 

 

[kbernt]

???

 

If the exponents were evaluated first it should be (2+3)(2+3) +1. Then (5)(5) + 1 =  26.  Exponents cannot be distributed. 

 I bet you are really good at math because it just seems natural to you that exponents cannot be distributed. My whole point was that in order for kids to initially learn the rule that exponents cannot be distributed, they can do what you did above, and do the addition inside the parentheses first (2+3=5), then take the answers of "5" times each other aka the powers/exponents second. You just naturally know to do parentheses first then powers, but for a child to whom this is a new concept they may not understand the rule that exponents can't be distributed, and by teaching them to always do everything inside of parentheses first (while ignoring everything outside) works.