Official Beast Academy guinea pigs thread!

[Sahamamama]

And my husband just walked in the room and saw me posting about BA, and he says, "I noticed you have a Pre-Algebra book on the bed. Uh... are we accelerating? That's like sixth or seventh grade, Honey?"

 

Me: "Yeah. That's for the math the Idiot Mother missed out on in seventh grade."

 

He is ROFL now. :glare: He just said, "The company should put that on a T-shirt. I'd buy it for you."

[Beth in SW WA]

Our set arrived today also. Beautifully designed books. Wow. Just wow.

 

I read for an hour w/ dd7 and then she finished a few pages of the practice book. She is in love with those monsters. She proclaimed with exuberance how happy she is with this new gift. She slogged through a few cwp pages this morning. Geometry will be a nice break for her.

 

Dd8 saw the books and said, "This looks awesome! I'm gonna love it, I'm gonna love it, I'm gonna love it!!"

 

After an hour of reading 3A, dd8 said, "Mom, I could read this all day and all evening and all night. I absolutely love it. This. is. so. awesome."

Edited March 20, 2012 by Beth in SW WA

[onaclairadeluna]

Action figures!!!!!:001_wub:

[nmoira]

 

Action figures!!!!!:001_wub:

And plushies. I'd even break our "one plushie in, one plushie out" rule (OPI-OPO).

[Alte Veste Academy]

And plushies. I'd even break our "one plushie in, one plushie out" rule (OPI-OPO).

 

:lol:

[TXBeth]

Is anyone planning to scan the workbook pages and do them on the iPad? I'm considering it. Dd was upset because I won't let her do any of the practice pages until I decide.

BTW, when I got home from my sister's house today, both the iPad and Beast had arrived. Guess which one we opened first?:D

[tothenthdegree]

Ours arrived today and we love them! It is 9pm and my daughter can't stop working on her new books....she is so excited about them. She thinks Capt. Kraken is very funny.

[MamaReads]

Just got ours in the mail! I used media mail so I was shocked we got it so fast. I had to put it away before dinner so no one got messy enchilada fingers on it :)

[Rosie]

Is anyone planning to scan the workbook pages and do them on the iPad?

 

We'll be doing this. It is actually part of the reason I relented and let my dh buy the new iPad. It will enable my dd to redo the pages if she needs more practice, plus my younger two will use it eventually, too. We just need to figure out how our printer/scanner can get scans to our computer! I wish BA sold a PDF version. It would make my life simpler!

[Barb_]

Can we go back to Order of Operations for a second? The reason for Order of Operations is "groups first." Here, imagine this problem in the real world:

 

3*6+4=

 

This is 3 groups of six items (say apples) plus 4 more individual items. So three baskets of apples, each with 6 apples in it. Additionally, there are 4 more apples someone just picked and set on the table. So 20 apples.

 

Now flip it:

 

4+3*6=

 

This is 4 individual items and three groups of six items. We cannot add 4 items to 3 groups of items unless we solve for the groups first. We are not adding 4 apples to three apples, but 4 apples to three groups. All we did was glance at the 4 apples on the table first. We still need to undo the three baskets of 6 apples each before we can add the 4 individual apples sitting there loose. If we really meant to say 4 items plus three more items (say 4 apples and 3 oranges) and then multiple the whole thing by 6 (to get the total number of fruit items), that is where parentheses and other grouping symbols come in.

 

(4+3)*6=

 

Now we are saying that there are 4 apples and 3 oranges in each basket. There are 6 baskets. So we have 72 pieces of fruit at our disposal.

 

We read from left to right, so it makes sense that first we must solve the groups first, either creating them or taking them apart, using multiplication, division, brackets, exponents, etc.

Edited March 21, 2012 by Barb F. PA in AZ

[Dmmetler]

We came home from Fl last night, and had BA waiting for us. DD took 3A to bed with her last night, and has spent most of today happily reading and playing. For her, the concepts are mostly review, but she's really, really enjoying the presentation in the guides and the fact that the practice is DIFFERENT than in SM. I think she also believes she's getting away with something, since mommy is letting her just play with math for school. Unfortunately, at this rate, the BA books I'd planned for fun summer review math probably won't last past about the end of March!

 

DD wants a plushie Lizzie and a set of action figures of ALL the characters.

[Whereneverever]

We came home from Fl last night, and had BA waiting for us. DD took 3A to bed with her last night, and has spent most of today happily reading and playing. For her, the concepts are mostly review, but she's really, really enjoying the presentation in the guides and the fact that the practice is DIFFERENT than in SM. I think she also believes she's getting away with something, since mommy is letting her just play with math for school. Unfortunately, at this rate, the BA books I'd planned for fun summer review math probably won't last past about the end of March!

 

DD wants a plushie Lizzie and a set of action figures of ALL the characters.

 

Does that mean you'll be looking to sell them? :bigear::tongue_smilie:

[nov05mama]

Can we go back to Order of Operations for a second? The reason for Order of Operations is "groups first."

We read from left to right, so it makes sense that first we must solve the groups first, either creating them or taking them apart, using multiplication, division, brackets, exponents, etc.

 

PEMDAS! Brackets/Parenthesis; Exponents; Multiplication/Division; Addition/Subtraction.

[stripe]

Can we go back to Order of Operations for a second? The reason for Order of Operations is "groups first." Here, imagine this problem in the real world:

 

 

 

Exactly. The rules were established to solidify and clarify something that made sense, not some random arbitrary thing. And I daresay there is a reason why a is first. It's not random either. I don't believe in teaching "that's the way it is" and "just memorize this formula and you'll be ok."

[zoo_keeper]

PEMDAS! Brackets/Parenthesis; Exponents; Multiplication/Division; Addition/Subtraction.

 

No, no, no! It's Please Excuse My Dear Aunt Sally (parentheses, exponents, multiplication/division, addition/subtraction). :)

[Barb_]

PEMDAS! Brackets/Parenthesis; Exponents; Multiplication/Division; Addition/Subtraction.

 

No, no, no! It's Please Excuse My Dear Aunt Sally (parentheses, exponents, multiplication/division, addition/subtraction). :)

 

Well, sure. But if we understand why the Order of Operations works, mnemonics are no longer necessary.

[Barb_]

Exactly. The rules were established to solidify and clarify something that made sense, not some random arbitrary thing. And I daresay there is a reason why a is first. It's not random either. I don't believe in teaching "that's the way it is" and "just memorize this formula and you'll be ok."

 

I'm glad you chimed in. I was wondering whether you were going to say the same thing.

[nov05mama]

No, no, no! It's Please Excuse My Dear Aunt Sally (parentheses, exponents, multiplication/division, addition/subtraction). :)

 

LOL How is that different than what I said? It's exactly the same thing :P PEMDAS is Please Excuse My Dear Aunt Sally :P :lol:

[garddwr]

You people are evil--I was so proud of myself for exercising self-control and NOT buying these books the day they came out--I was going to wait until this summer.

Then I read the review threads...

We have a package on the way!

[Sneezyone]

Just a follow up on the vocab issues...

 

DD came home from school today wanting to do fun math again so after doing our regularly scheduled SM pages (she did extras of those too...for fun :001_huh:) I printed out a page of angles for her to label. She got them all, correctly applying the terms to each one. I guess it just took a good night's sleep to move the info from short-term to long-term memory. So tomorrow, since we reviewed today, we'll tackle the triangles!! :D Fun times!

[garddwr]

Can we go back to Order of Operations for a second? The reason for Order of Operations is "groups first." Here, imagine this problem in the real world:

 

3*6+4=

 

This is 3 groups of six items (say apples) plus 4 more individual items. So three baskets of apples, each with 6 apples in it. Additionally, there are 4 more apples someone just picked and set on the table. So 20 apples.

 

Now flip it:

 

4+3*6=

 

This is 4 individual items and three groups of six items. We cannot add 4 items to 3 groups of items unless we solve for the groups first. We are not adding 4 apples to three apples, but 4 apples to three groups. All we did was glance at the 4 apples on the table first. We still need to undo the three baskets of 6 apples each before we can add the 4 individual apples sitting there loose. If we really meant to say 4 items plus three more items (say 4 apples and 3 oranges) and then multiple the whole thing by 6 (to get the total number of fruit items), that is where parentheses and other grouping symbols come in.

 

(4+3)*6=

 

Now we are saying that there are 4 apples and 3 oranges in each basket. There are 6 baskets. So we have 72 pieces of fruit at our disposal.

 

We read from left to right, so it makes sense that first we must solve the groups first, either creating them or taking them apart, using multiplication, division, brackets, exponents, etc.

 

Ah, but you are using the convention as part of your assumption--assuming that multiplication must come first, not proving why. It simply depends on what question you are asking. IF the question you want answered is 3 groups of 6 items plus 4 more, then of course you want to do the multiplication first. If the question you want answered is 6 groups of 4 plus 3 items, you want to do the addition first it makes sense to do the addition first. We need rules for order of operations because without them we could not know from a written sentence 4+3*6 which question is being asked--and which we should answer. It is the order of operations RULES that tell us such a sentence means there are three groups of 6 items plus four more not 6 groups four plus three items. The convention could have been created the other way around, so that without parentheses addition would be done first, and if you wanted to do multiplication first you would need to use parentheses. In that case the sentence above would be assumed to me 6 groups of four plus three items.

 

I agree with previous posters that doing "more powerful" before "less powerful" operations gives us a logical order to follow, but there is still something arbitrary about the rule.

 

It bothers me when textbooks teach that "left to right" is part of the order of operations--because left to right really doesn't matter. We came across this in MM3A, and my dd saw through it immediately--if you have a problem like this: 3*9+18-7, and reduce it to 27+18-7, it's obviously easier to subtract the 7 and then add 18--and the order in which you do addition and subtraction will never change the answer regardless of what comes first in the left-to-right sequence. I understand that, especially in a longer problem, working it left to right can help you keep track of what has been done and what hasn't so steps don't get forgotten--but that's really not an order of operations issue, just a organization of work issue.

 

Or am I missing something here?

 

--Sarah

[nansk]

IF the question you want answered is 3 groups of 6 items plus 4 more, then of course you want to do the multiplication first. If the question you want answered is 6 groups of 4 plus 3 items, you want to do the addition first it makes sense to do the addition first. We need rules for order of operations because without them we could not know from a written sentence 4+3*6 which question is being asked--and which we should answer. It is the order of operations RULES that tell us such a sentence means there are three groups of 6 items plus four more not 6 groups four plus three items.

 

I agree 100% with you. PEMDAS is just a convention (or rule).

 

The left-to-right rule does not always work either. What if the question was 4+3*6=?

[garddwr]

I agree 100% with you. PEMDAS is just a convention (or rule).

 

The left-to-right rule does not always work either. What if the question was 4+3*6=?

 

The left-to-right rule is presented as following after the higher order to lower order operations rule--so in the example above, you do the higher order operation (multiplication) first, then you do the addition left to right. In this case you only have one addition operation, so it really doesn't matter. But it would apply to longer problems like the following:

 

20+2*7+8-5*3-2*4

 

Doing multiplication first, we get:

 

20+14+8-15-8

 

Of course you can get the correct answer working left to right, but you can also get the correct answer working in any other order--unlike the PEMDAS rules, the order in which you work doesn't affect the outcome.

 

The problem above could be re-written as 8-8+20+14-15, then worked left to right. But even without re-writing it's easy to see that the 8's cancel each other out and isn't it so much easier to do that operation first rather than just following the "rule" of working left to right. I just don't think this should be presented as a "rule" of the same sort as the multiplication-before-addition rule.

[stripe]

Whatever the case, the important thing is that there is discussion about it. I don't like to just throw something out there without giving a reason for it. Or, whatever the book doesn't say about it, my kid won't "just do it because that's the rule." I think intellectually that is a good thing.

 

There is generally a reason for things more complex than grandfathering; I mean even our grandfathers had brains. I was referring to alphabetical order when I said there is a reason the letter "a" comes first.

 

So I am frankly disappointed that Beast Academy didn't embrace this as a concept worth understanding.

 

I think BA is interesting, but I think other things are too.

[stripe]

I agree 100% with you. PEMDAS is just a convention (or rule).

 

The left-to-right rule does not always work either. What if the question was 4+3*6=?

 

Then the point is what can move in the equation and the value stay the same. Does the child understand commutivity?

 

My son worked on it until he came up with a method for grouping with his own notation based on our conversations. Only then did he understand and get problems right. Telling him PEMDAS (which suggests addition>subtraction, for example) didn't cut it. He had to play it out in his head. MEP, which we use, had lots of examples, but nothing other than "do mulitplications and divisions first!" I was underwhelmed, because I usually find MEP more intellectually robust.

 

Mathematically there are some variations in orders of operations esp in computer applications. So it's not actually set in stone.

Edited March 21, 2012 by stripe

[nikkid]

Well, sure. But if we understand why the Order of Operations works, mnemonics are no longer necessary.

 

:iagree: Exactly!

[Kuovonne]

It bothers me when textbooks teach that "left to right" is part of the order of operations--because left to right really doesn't matter. We came across this in MM3A, and my dd saw through it immediately--if you have a problem like this: 3*9+18-7, and reduce it to 27+18-7, it's obviously easier to subtract the 7 and then add 18--and the order in which you do addition and subtraction will never change the answer regardless of what comes first in the left-to-right sequence. I understand that, especially in a longer problem, working it left to right can help you keep track of what has been done and what hasn't so steps don't get forgotten--but that's really not an order of operations issue, just a organization of work issue.

 

Or am I missing something here?

 

Yes, the left-to-right order does matter. Consider:

 

3 - 2 + 1

 

Done left to right you have

(3 - 2) +1 =

1 + 1 =

2

 

Done in a different order you have

3 - (2 + 1) =

3 - 3 =

0

 

Now, most of us familiar with how to do math can mentally switch the equation around to something like 3 + 1 - 2 and get the same results, but that works because we understand the rules for how to change things around to get the same answer, it doesn't really change the order of operations.

 

It's important to be able to recognize when doing things in a different order is valid and when it isn't, hence the commutitive, associative, and distributive laws. However, that is different from order of operations.

[boscopup]

There is generally a reason for things more complex than grandfathering; I mean even our grandfathers had brains. I was referring to alphabetical order when I said there is a reason the letter "a" comes first.

 

And that reason is... ?

[zoo_keeper]

LOL How is that different than what I said? It's exactly the same thing :P PEMDAS is Please Excuse My Dear Aunt Sally :P :lol:

 

I understand PEMDAS is the acronym for the Sally expression. I was just joking that the Sally expression (for me) is an easier mnemonic device then PEMDAS :)

 

And, yes, of course memory tricks are inferior to actually understanding the order of operations, but after encountering kids in my classroom that have already had over a decade of experience the operations and still get confused (in a math-related field!), I'll take any trick out there!

[garddwr]

Yes, the left-to-right order does matter. Consider:

 

3 - 2 + 1

 

Done left to right you have

(3 - 2) +1 =

1 + 1 =

2

 

Done in a different order you have

3 - (2 + 1) =

3 - 3 =

0

 

Now, most of us familiar with how to do math can mentally switch the equation around to something like 3 + 1 - 2 and get the same results, but that works because we understand the rules for how to change things around to get the same answer, it doesn't really change the order of operations.

 

It's important to be able to recognize when doing things in a different order is valid and when it isn't, hence the commutitive, associative, and distributive laws. However, that is different from order of operations.

 

I see what you are getting at--but the order doesn't matter if we keep the positive/negative value associated with each number--that is, if we see the number as +3, -2, and +1. (-2+1)+3 is the same as the original problem. The problem with the way you grouped them above is not that 2 and 1 got added first but that it should have been -2 added to 1. This may require a more advanced understanding of numbers and values than most 3 grade texts present, but it is a more accurate one.

 

I'll have to think about this more.

[Barb_]

Ah, but you are using the convention as part of your assumption--assuming that multiplication must come first, not proving why.

 

You are correct that the convention is part of my explanation. Mathematics is a language with rules and conventions just as English is based on Phonics. English at the foundation is based on the convention that if we blend phonemes together from left to right we can convey ideas. But in Math as well as in Phonics there are underlying, logical reasons for the conventions. "This is the way it's done because people smarter than we say so," is not an acceptable explanation in Math.

 

It bothers me when textbooks teach that "left to right" is part of the order of operations--because left to right really doesn't matter.

 

In a sense, you are right. But we usually teach Order of Operations to children before they have a firm grasp on negative numbers. In the same sense there really is no such thing as subtraction, only adding a negative. But I wait until the logic stage to pull that one on my kids.

[Barb_]

The convention could have been created the other way around, so that without parentheses addition would be done first, and if you wanted to do multiplication first you would need to use parentheses.

 

In that case the parentheses would be saying, "Do this last," instead of, "Do this first." But there would still be a logical underlying reason for the rule and it would still be based on breaking down groups (or higher order operations, if you prefer). You cannot add single items to groups of items without unpacking the groups first. But maybe I'm misunderstanding your meaning. How about an example? Maybe you could write up a problem and explain it in real terms using concrete objects.

Edited March 21, 2012 by Barb F. PA in AZ

[krisperry]

FWIW, AoPS Prealgebra does a lot with the distributive and commutative properties. Pretty sure they go into detail about the why behind the order of operations. I know the prove the properties. But the proof takes more manipulation of numbers than I think most kids are ready for in 3rd grade...

[Barb_]

If the question you want answered is 6 groups of 4 plus 3 items, you want to do the addition first it makes sense to do the addition first.

 

I'm not picking on you, I promise. I'm just having trouble following. Can you elaborate on the above? How can you add first?

[garddwr]

I'm not picking on you, I promise. I'm just having trouble following. Can you elaborate on the above? How can you add first?

 

If you want to end up with six groups of seven you have to add first. That's what I meant.

[garddwr]

You are correct that the convention is part of my explanation. Mathematics is a language with rules and conventions just as English is based on Phonics. English at the foundation is based on the convention that if we blend phonemes together from left to right we can convey ideas. But in Math as well as in Phonics there are underlying, logical reasons for the conventions. "This is the way it's done because people smarter than we say so," is not an acceptable explanation in Math.

 

 

Your argument about "multiplication is first because multiplication is grouping and we group first" is essentially a circular argument: we group (multiply) first because we group first--this is the convention because this is the convention. At some point someone had to decide to make that the convention.

 

When we do real-life problems (three people each get 2 red apples and 4 green apples how many apples are there altogether?) we realize that to answer the question we add the two and four apples, then multiply that by three people--or we could multiply two red apples by three people and 4 green apples by three people and add those together. If instead we are discussing three people who each got 2 apples and one person who got 4 and want to know the number of apples, we recognizes that we multiply 3 by 2 then add 4.

 

Without a convention for order of operations, either problem could be written as 4+2*3, but unless we wrote the problem we wouldn't know exactly what was meant. There is not some fundamental rule of the universe that says an operation written 4+2*3 means multiply two by three then add four. It's a rule because we made it a rule--just like it is a rule in English grammar that Mary kicked Sam means Mary is the one who did the kicking and Sam was the one kicked--it could just as well mean the Sam did the kicking, but we all agree that a sentence like this is formatted Subject/Verb/Object not the other way around. It's not a law of the universe, it's a convention people came up with because it's awfully nice to understand each other. The conventions in other languages work differently.

Edited March 21, 2012 by thegardener
typo

[Barb_]

If you want to end up with six groups of seven you have to add first. That's what I meant.

 

Oh!! I think I understand what you meant. So by this:

 

6 groups of 4 plus 3 items,

 

You meant 6 groups of items with each group containing 4 of one item and 3 of another? Got it.

[garddwr]

Oh!! I think I understand what you meant. So by this:

 

 

 

You meant 6 groups of items with each group containing 4 of one item and 3 of another? Got it.

 

Yep.

 

By the way, I do think your earlier explanation of order of operations helps add logic to the system, and is worth discussing versus just saying "memorize this acronym". I just don't think it addresses some fundamental natural order that is being followed. Order of operations is a workable convention that helps us understand each other, and it could be set up differently (i.e. addition first) and be equally comprehensible.

[Barb_]

Your argument about "multiplication is first because multiplication is grouping and we group first" is essentially a circular argument: we group (multiply) first because we group first--this is the convention because this is the convention. At some point someone had to decide to make that the convention....It's not a law of the universe, it's a convention people came up with because it's awfully nice to understand each other.

 

Okay, back up a second. No one is arguing that Order of Operations is a law of the universe. Of course, the order is at some level simply tradition. We are discussing whether the order convention is arbitrary or logical. I'm saying the convention exists so that we can all agree and communicate which numbers represent groups and which numbers represent individuals. Scroll up and take another look.

 

This discussion began with this quote:

 

* I continue to hate how almost everyone (apparently except me) explains order of operations. The "this is the way it's done" line did not impress me

.

 

I agree. Saying "We do this in this way because this is the way it's done," is classic circular reasoning.

[garddwr]

Okay, back up a second. No one is arguing that Order of Operations is a law of the universe. Of course, the order is at some level simply tradition. We are discussing whether the order convention is arbitrary or logical. I'm saying the convention exists so that we can all agree and communicate which numbers represent groups and which numbers represent individuals. Scroll up and take another look.

 

This discussion began with this quote:

 

 

 

I agree. Saying "We do this in this way because this is the way it's done," is classic circular reasoning.

 

 

My understanding is that the convention exists so we can understand whether an expression such as 4+2*3 means there are two groups of three, plus four more, or there are 6 groups of three. The expression could mean either one, and both represent groups. The convention determines that it is 2, not 6, groups of three that is meant--unless we add in parentheses. Just as it is arbitrary convention that says "Sally kicked John" doesn't mean it was John who did the kicking.

 

I do think that placing addition before multiplication in the order of operations would make things clunkier, especially with long problems--so the order is probably not entirely arbitrary. I'll have to think about it more. How would that play out in an expression such as this:

 

3*2+7-5*4+1 for example? Would you end up with (2+7-5+1)*3*4? I may have to go play around with numbers some more :001_smile:

[nmoira]

Saying "We do this in this way because this is the way it's done," is classic circular reasoning.

But this is what convention is. Order of Operations is a convention. North being (typically) "up" is a convention. Writing words from one direction to another is a convention. Righty tighty, lefty loosen is a convention, as is driving on a particular side of the road. It might be interesting to learn why a convention is a convention, but it is not necessary.

[TXBeth]

Your argument about "multiplication is first because multiplication is grouping and we group first" is essentially a circular argument: we group (multiply) first because we group first--this is the convention because this is the convention. At some point someone had to decide to make that the convention.

 

When we do real-life problems (three people each get 2 red apples and 4 green apples how many apples are there altogether?) we realize that to answer the question we add the two and four apples, then multiply that by three people--or we could multiply two red apples by three people and 4 green apples by three people and add those together. If instead we are discussing three people who each got 2 apples and one person who got 4 and want to know the number of apples, we recognizes that we multiply 3 by 2 then add 4.

 

Without a convention for order of operations, either problem could be written as 4+2*3, but unless we wrote the problem we wouldn't know exactly what was meant. There is not some fundamental rule of the universe that says an operation written 4+2*3 means multiply two by three then add four. It's a rule because we made it a rule--just like it is a rule in English grammar that Mary kicked Sam means Mary is the one who did the kicking and Sam was the one kicked--it could just as well mean the Sam did the kicking, but we all agree that a sentence like this is formatted Subject/Verb/Object not the other way around. It's not a law of the universe, it's a convention people came up with because it's awfully nice to understand each other. The conventions in other languages work differently.

 

:iagree:

 

Okay, back up a second. No one is arguing that Order of Operations is a law of the universe. Of course, the order is at some level simply tradition. We are discussing whether the order convention is arbitrary or logical. I'm saying the convention exists so that we can all agree and communicate which numbers represent groups and which numbers represent individuals. Scroll up and take another look.

 

This discussion began with this quote:

 

 

 

I agree. Saying "We do this in this way because this is the way it's done," is classic circular reasoning.

 

Not really. Saying "We do this because this is the convention that people decided on in years past, not because of any logical reason" is perfectly acceptable IF IT IS TRUE.

For example, why do we use a horizontal line for subtraction, and why are exponents written as small superscript numbers? These are arbitrary. We could just as well make up other ways to write them. But we use these particular symbols so that everyone else in the world can understand what we were thinking when we wrote down particular expressions.

That is the same reason we use the conventional order of operations. Someone made up the rules, and we all use them so we can communicate effectively. There is no logical reason why they need to be the way they are, and to say that we do multiplication/division first to avoid larger numbers makes no sense at all. The order of operations does not determine what the expression is; it just determines how we write it.

 

For a given problem, you might need to either add first or multiply first. The order of operations rules simply tell you that if you want whoever is working the problem to add first, you need to add parentheses. If you want them to multiply first, you leave the parentheses off. It could easily be the other way around.

[Kuovonne]

I see what you are getting at--but the order doesn't matter if we keep the positive/negative value associated with each number--that is, if we see the number as +3, -2, and +1. (-2+1)+3 is the same as the original problem. The problem with the way you grouped them above is not that 2 and 1 got added first but that it should have been -2 added to 1. This may require a more advanced understanding of numbers and values than most 3 grade texts present, but it is a more accurate one.

 

I'll have to think about this more.

 

When you keep the positive/negative associated the way you're describing, you're converting the subtraction operations into adding a negative number, which is valid. The commutitive law says that you can add numbers in any order. That's why what you describe works.

 

You're taking

3 - 2 + 1

and converting it to

3 + (-2) + 1

and then applying the commutitive law.

 

 

However, when strictly talking about order of operations

(3 - 2) + 1 and 3 - (2 + 1) have the same numbers and the same operations; you just do the operations in a different order. You aren't converting a subtraction problem to adding a negative number. Hence they are two different expressions.

[stripe]

Not really. Saying "We do this because this is the convention that people decided on in years past, not because of any logical reason" is perfectly acceptable IF IT IS TRUE.

 

I choose not to instruct my kids in math by telling them that this is the way it is, and that's the end of the story. For all the talk about wanting kids to understand math, to think about math, and so forth, I think the truth has been revealed.

 

This discussion began when I said I found BA disappointing for not explaining it in a more compelling or interesting way. I stand by my assessment.

[Barb_]

But this is what convention is. Order of Operations is a convention. North being (typically) "up" is a convention. Writing words from one direction to another is a convention. Righty tighty, lefty loosen is a convention, as is driving on a particular side of the road. It might be interesting to learn why a convention is a convention, but it is not necessary.

 

Some conventions have reasons; some are arbitrary. If a child says, "But why do we do it this way?" and we say, "Because that is the way it's done," is circular reasoning. We should answer, "It is arbitrary and we've all agreed for communication's sake." Righty-tighty is as far as I know, an arbitrary convention. I had a blender once that bucked tradition and dumped smoothie in my flatware drawer. However, if there is a logical reason for the convention (we all agree on North because that is where the magnetic pole is located...for now), then we as educators have a responsibility to get to the bottom of it when explaining it. "It is what it is" is circular.

 

I'm arguing that in this case, the logic came first and the convention followed. It sounds to me as if others are arguing that Order of Operations is completely arbitrary. Is that true? If so, then we will have to agree to disagree.

[nmoira]

When you keep the positive/negative associated the way you're describing, you're converting the subtraction operations into adding a negative number, which is valid. The commutitive law says that you can add numbers in any order. That's why what you describe works.

 

You're taking

3 - 2 + 1

and converting it to

3 + (-2) + 1

and then applying the commutitive law.

 

 

However, when strictly talking about order of operations

(3 - 2) + 1 and 3 - (2 + 1) have the same numbers and the same operations; you just do the operations in a different order. You aren't converting a subtraction problem to adding a negative number. Hence they are two different expressions.

I teach this as "putting in" and "taking out" and "steps forward" and "steps back."

 

We can group things we're putting in or taking out:

 

6 - 2 + 4 - 3 = 10 - 5 = 5

We're "putting in" 6 and 4 and "taking out" 2 and 3.

I also show this on the number line as "steps forward" and "steps back," occasionally reminding that we can "step back" to the left of the number line (DD the Younger gets negative numbers).

 

We can work first with small groups of what we're putting in and taking out, as long as we don't forget any of them or mess up Order of Operations, but unless you can see an obvious reason to do this (i.e. shortcut), go from left to right.

 

12 + 7 + 5 - 7 = 12 + 5 = 17 -- (7 - 7 = 0)

12 + 7 + 5 - 3 - 4 = 12 + 5 = 17 -- (7 - 3 - 4 = 0)

12 - 7 + 5 + 3 + 4 = 12 + 5 = 17 -- (taking out seven and putting in seven gives zero)

 

MEP encourages this type of grouping where possible.

[nmoira]

Some conventions have reasons; some are arbitrary. If a child says, "But why do we do it this way?" and we say, "Because that is the way it's done," is circular reasoning. We should answer, "It is arbitrary and we've all agreed for communication's sake."

I agree and I this is what I say to my kids. But it amounts to the same thing. :001_smile:

 

Righty-tighty is as far as I know, an arbitrary convention. I had a blender once that bucked tradition and dumped smoothie in my flatware drawer. However, if there is a logical reason for the convention (we all agree on North because that is where the magnetic pole is located...for now), then we as educators have a responsibility to get to the bottom of it when explaining it. "It is what it is" is circular.

 

I'm arguing that in this case, the logic came first and the convention followed. It sounds to me as if others are arguing that Order of Operations is completely arbitrary. Is that true? If so, then we will have to agree to disagree.

It may follow a logical pattern, but the pattern chosen is arbitrary. I think we're all mostly on the same page as far as that is concerned. :tongue_smilie:

[stripe]

Per Dr Peterson at http://mathforum.org/library/drmath/view/52582.html

 

1. The basic rule (that multiplication has precedence over addition)

appears to have arisen naturally and without much disagreement as

algebraic notation was being developed in the 1600s and the need for

such conventions arose. Even though there were numerous competing

systems of symbols, forcing each author to state his conventions at

the start of a book, they seem not to have had to say much in this

area. This is probably because the distributive property implies a

natural hierarchy in which multiplication is more powerful than

addition, and makes it desirable to be able to write polynomials with

as few parentheses as possible; without our order of operations, we

would have to write

 

ax^2 + bx + c

as

(a(x^2)) + (bx) + c

 

It may also be that the concept existed before the symbolism, perhaps

just reflecting the natural structure of problems such as the quadratic.

....

4. I suspect that the concept, and especially the term "order of

operations" and the "PEMDAS/BEDMAS" mnemonics, was formalized only in

this century, or at least in the late 1800s, with the growth of the

textbook industry. I think it has been more important to text authors

than to mathematicians, who have just informally agreed without

needing to state anything officially.

 

5. There is still some development in this area, as we frequently hear

from students and teachers confused by texts that either teach or

imply that implicit multiplication (2x) takes precedence over

explicit multiplication and division (2*x, 2/x) in expressions

such as a/2b, which they would take as a/(2b), contrary to the

generally accepted rules. The idea of adding new rules like this

implies that the conventions are not yet completely stable; the

situation is not all that different from the 1600s.

 

In summary, I would say that the rules actually fall into two

categories: the natural rules (such as precedence of exponential over

multiplicative over additive operations, and the meaning of

parentheses), and the artificial rules (left-to-right evaluation,

equal precedence for multiplication and division, and so on). The

former were present from the beginning of the notation, and probably

existed already, though in a somewhat different form, in the geometric

and verbal modes of expression that preceded algebraic symbolism. The

latter, not having any absolute reason for their acceptance, have had

to be gradually agreed upon through usage, and continue to evolve.

[garddwr]

However, if there is a logical reason for the convention (we all agree on North because that is where the magnetic pole is located...for now), then we as educators have a responsibility to get to the bottom of it when explaining it. "It is what it is" is circular.

 

I'm arguing that in this case, the logic came first and the convention followed. It sounds to me as if others are arguing that Order of Operations is completely arbitrary. Is that true? If so, then we will have to agree to disagree.

 

The fact that the magnetic north pole is north does not explain why north is placed at the top of a map or globe--there is nothing "up" about magnetic north--it is an arbitrary convention.

 

I would love to discover a true logic behind order of operations, but I do agree with your particular argument as a reason that order of operations must be as it is.

 

Tell me this, have you tried to wrap your mind around the idea of a different order of operations? Let's say we reverse them. As someone else said, all it would do is change the way we write a particular expression. So if you want to show 5 groups of 3 apples plus 2 more you could write: (5*3)+2 --does that not satisfy your need to show which numbers are grouped through multiplication? 5*3+2 would then mean you have 5 groups of 5 objects.

[garddwr]

Per Dr Peterson at http://mathforum.org/library/drmath/view/52582.html

 

1. The basic rule (that multiplication has precedence over addition)

appears to have arisen naturally and without much disagreement as

algebraic notation was being developed in the 1600s and the need for

such conventions arose. Even though there were numerous competing

systems of symbols, forcing each author to state his conventions at

the start of a book, they seem not to have had to say much in this

area. This is probably because the distributive property implies a

natural hierarchy in which multiplication is more powerful than

addition, and makes it desirable to be able to write polynomials with

as few parentheses as possible; without our order of operations, we

would have to write

 

ax^2 + bx + c

as

(a(x^2)) + (bx) + c

 

It may also be that the concept existed before the symbolism, perhaps

just reflecting the natural structure of problems such as the quadratic.

....

4. I suspect that the concept, and especially the term "order of

operations" and the "PEMDAS/BEDMAS" mnemonics, was formalized only in

this century, or at least in the late 1800s, with the growth of the

textbook industry. I think it has been more important to text authors

than to mathematicians, who have just informally agreed without

needing to state anything officially.

 

5. There is still some development in this area, as we frequently hear

from students and teachers confused by texts that either teach or

imply that implicit multiplication (2x) takes precedence over

explicit multiplication and division (2*x, 2/x) in expressions

such as a/2b, which they would take as a/(2b), contrary to the

generally accepted rules. The idea of adding new rules like this

implies that the conventions are not yet completely stable; the

situation is not all that different from the 1600s.

 

In summary, I would say that the rules actually fall into two

categories: the natural rules (such as precedence of exponential over

multiplicative over additive operations, and the meaning of

parentheses), and the artificial rules (left-to-right evaluation,

equal precedence for multiplication and division, and so on). The

former were present from the beginning of the notation, and probably

existed already, though in a somewhat different form, in the geometric

and verbal modes of expression that preceded algebraic symbolism. The

latter, not having any absolute reason for their acceptance, have had

to be gradually agreed upon through usage, and continue to evolve.

 

This confirms what I have begun to glimpse as I have thought about this today--our accepted order of operations makes for simpler and more elegant notation than most alternatives. Thanks for pulling up this explanation!

 

ETA: we end up somewhere in between the "completely arbitrary" and "necessarily logical" points of view--our system is not the only workable one, but it is a bit easier to work with than alternatives might be. Sort of like our place value numeral system being easier to work with than roman numerals, although both are capable of conveying the same information.

Edited March 21, 2012 by thegardener