I recently sat down to plan a lesson for year 7 students about the order of mathematical operations. Here in the UK, I believe this is most commonly known as"BODMAS". The B stands for brackets, D for division, M for multiplication, A for addition and S for subtraction.

But what does the O stand for?

I've heard several different answers to the above question, none of them satisfactory. One of my colleagues told me he taught it as "Orders". This random website I found agrees. But what the heck are orders? According to the aforementioned random website, they're "numbers involving powers or square roots". I have never heard this definition before, and after consulting the oracle (Wikipedia) I found no mention of indices or powers on the page for Order (mathematics). So why on earth would we teach students the word orders when we never call them that in lessons? We in the UK usually refer to these as "indices" although I believe the Americans prefer "exponents" (but I'll get to them later).

Another colleague told me he teaches that the O stands for "of" as in "powers of", and I'm ashamed to admit this was what I was taught in school. I think this one is faintly ridiculous. Firstly, O cannot stand for "Powers of" because "Powers of" clearly begins with a P not an O. Kids may be getting dumber every generation, but I have a feeling they will notice this. Also, why does the word "powers" even need an "of"? Can we not just call them powers? It reminds me a bit of learning French when we were always taught to write the following preposition after certain words like "decider de" or "je pense que" to help you form sentences. This was actually excellent advice for learning French, but this does not dilute my point.

A third colleague (it is amazing how many of them are willing to contribute to my inane Monday-morning conversations) said that he teaches that the O stands for "Other" as in, any other operations not mentioned. This is quite nice actually, because it includes not just powers and roots but also sines, logs, factorials, etc. Very handy.

I then went into my year 7 lesson and asked them what they thought the O stood for. Interestingly, the most common response was one I had not heard yet: "operations". This is perhaps the one that annoys me the most. BODMAS is the tool we use to remember in which order we should do operations. If O stands for "operations", then we are basically saying, do the bit in the brackets first, then do the operations. Oh wait, what order do I do the operations in? Use BODMAS. So I do the brackets and then the operations. But what order do I do those operations in? etc etc. Thank you Primary school teachers. Thanks a bunch. You have just created an infinite loop in my head. You have given my eleven year-old students an acronym to learn that is actually a recursive formula. After infinite iterations they will still not have found the value of 3 + 2 x 5.

So I bet you're dying to know what I taught them in the end, right? Well I told them about the conversations I'd had in the maths office. I also told them about the American version: PEMDAS. Seriously. That's what they call it. The MDAS is obvious enough. The P is for "parentheses" which my students had never heard of but which is quite useful to know I suppose, and the E is for "exponents" as I mentioned above. My year 7s were not happy that our friends across the Atlantic do their multiplication before their Division though. "Surely they'll get different answers from us and then spaceships won't work!!" they cried. (I must have told them about the metric/imperial satellite mix up in a previous lesson). This led to a nice discussion about how those two operations are interchangeable and you would still get the same answer (or would you? I have just thought of a topic for a future post).

Anyway, in the end, I taught them the O stands for Indices. That's right, I'm on Team BIDMAS. All you BIDMAS haters out there can hate hate hate but if we refer to powers as "indices" the rest of the time why not in this? And if you have a problem with me not including trig functions or logarithms or whatever in my acronym well you shouldn't because by the time you're learning that sort of stuff you shouldn't need a mnemonic to help you remember which order to do stuff in anyway!

Over to you: what did you learn at school, and, if you're a teacher, what do you teach now?

Emma x x x

I don't every remember being taught anything remotely like a Mnemonic for operations.

ReplyDeleteI thought everybody understood "order" to mean "power" though. We talk about the order of a polynomial and numbers "in the order of 100s" for example.

The first time I saw BODMAS being used, the O was "Powers Of" - I agree Emma, pretty ridiculous and not at all pretty!

I've also seen it missed out completely (MEP in year 7 don't give it any description).

I've heard it given as Order and "Over" (in other words, the same as division).

I prefer to describe it as Operations (as in "other operations"), but acknowledge the slightly unsatisfactory infinite loop! :-)

Of course the most important thing about BODMAS/BIDMAS isn't what the O stands for but rather how we make sure that students using it get the correct answer to 6 - 3 + 4!

But we don't use the word "order" to describe powers to year 7s, 8s etc. The first time we use the word is when talking about polynomials, so year 12. We do of course say "order of magnitude" which until this very second I made no connection between the word order and the fact that order of magnitude refers to which power of 10 is involved. Weird.

DeleteI love the blank space next to the O in the MEP book - so mysterious!

I think to solve the 6 - 3 + 4 problem what would be great is if we stopped thinking about that - as being a takeaway or a subtraction and instead always think of it as the negative sign, and hence it belongs to the 3, and so we wouldn't even think of doing 3 + 4 then subtracting the answer. However, that's a whole discussion in itself! I wrote a post on that ages ago.

Why have cotton when you can have silk?. We all know that one.

DeleteI say why choose ? Have cotton and silk...

If we want students to know what this tool means, clearly identify it in full.

In short if you want students to think of the I or the O as indices but it could include other things like trig, logs etc... then use them both!

Maybe we should revamp it altogether?

BOD (MAS) and BID (MAS) are so 2014 ..

.

...could this be changed to BOIDMAS.

B:Brackets,

O Other (trig etc... )

I: Indices

D: Division

M: Multiply

A: Addition

S: Subtraction

Yep!

Wasn't that a Galaxy advert?

DeleteBOIDMAS looks good. How about BLITODMAS - brackets, logs, indices, trig, other, division, multiplication, addition, subtraction. Actually, make it SBLITODMAS - the s is for square brackets [ ] that you use when you've already used normal brackets.

I am decided that O in BODMAS as Order but I want to know about the use of O in bodmas I mean how to use Orders in a problem....I u get understood well reply me with a suitable example.....

ReplyDeleteI vaguely remember it's what you do when you have division & multiplication consecutively in a math sentence after solving brackets. You work in the order of left to right before addition and/or subtraction.

DeleteIt can get messy otherwise like in 8÷2(2+2). Left to right after brackets gets 8÷2×4=16, whereas, right to left after solving brackets is 8÷2×4=1. The correct answer is 16. Try it on your calculator. Order matters, just go left to right. BODMAS is just the base rule for D,M,A and S.

DeleteDo you know what I say, forget stupid made up misleading acronyms and mnemonics like PEMDAS, BODMAS, BIDMAS, BEDMAS, GEMDAS, GEMS, MDAS, DMAS etc and start teaching the logic and mathematical reasoning behind the actual rule instead; which is the Order of Operations.

ReplyDeleteAcronyms are lazy memory aides that are very often poorly explained, and thus lead to a lack of understanding and an over reliance on simply remembering a bunch of letters at the expense thinking of the actual concepts and processes the letters represent.

The Order of Operations rule is not hard. In fact it is based on sound mathematical logic that students have no difficulty in understanding - if it is properly explained to them. Understanding the mechanics behind the actual rule and knowing why it is structured the way it is, not only leads to an improved understanding of the Order of Operations rule, but more importantly, if explained carefully, if leads to a greater understanding of the arithmetic operations and the relationships that exist between them.

The first thing to explain is that some operations are more powerful than others. And the more powerful an operations is, the higher priority and level of precedence it is calculated with.

Addition is at the bottom because it is the least most powerful operation. Multiplication is more powerful than addition because it has the ability to take a series of addition operations and condense them down into just the one multiplication operation.

5+5+5+5+5+5+5+5 requires 7 addition operators and 7 operations in order to get the answer of 40. Whereas multiplication can condense that down to just 5x8. Much quicker and more efficient, therefore it is considered to be more powerful than addition.

Next up the ladder with the greatest operational precedence is exponentiation. It is more powerful than multiplication because it can take a series of multiplication operations and condense them down into one exponentiation operation.

5x5x5x5x5x5x5x5 is long and less efficient than exponentiation which can get the result of all those multiplication operations using just the one exponent 5⁸.

SO far in our level of power we have:

Exponentiation

Multiplication

Addition

At the same time, students need to understand that some operations are equal to each other in power. These are called inverse operations and they are the equivalent operations of each other. Inverse operations undo or reverse each other and so they equal each other in power.

The inverse operation of addition is subtraction. It is the equal least powerful.

The inverse operation of multiplication is division. These are the next most powerful.

The inverse operations of exponents are roots/radicals and logarithms. They are all treated with the highest level of power and precedence.

* The relationships between all of these inverse operations also need to be explored. What is an inverse? They need to understand that the concept of addition cannot logically exist without the reverse concept of subtraction. If something can be added together it follows then that it must be able to be taken away again. And if something can be multiplied together then it must in turn be able to be divided back up.

They then need to be shown how subtraction can easily be rewritten as addition of the negative. And the importance of a numbers sign here also must be stressed.

10 - 5 + 2 - 1

+10 ＋ -5 ＋ +2 ＋ -1

And division can be rewritten as multiplication of the inverse or reciprocal.

10 ÷ 5 ⨯ 3 ÷ 2

10 × ¹∕₅ × 3 × ¹∕₂

If this is explained they should be able to understand why addition and subtraction are calculated together in the same step in the Order or Operations across the page from left to right. Likewise for multiplication and division.

Obviously the concept of exponentiation and how it relates to roots/radicals and logarithms needs to be introduced in the appropriate level of detail for the year level.

PART 2

ReplyDeleteThe next really important thing to discuss is the purpose and use of GROUPING SYMBOLS. I think it is poor practice to call them Parentheses P or Brackets B when there are many other types of symbols that can be used to show grouping.

Parentheses ( )

Brackets [ ]

Braces { }

Angle Brackets < >

The Absolute Value Modulus | |

The HORIZONTAL vinculum or bar ___

G for GROUPING SYMBOLS. Because these symbols only have that top priority when they are acting as grouping symbols and so when they contain a GROUP.

Grouping symbols are used to indicate priority. When they are used to hold one or more OPERATIONS INSIDE them, then they are being used to show a break or deviation from the standard calculatory order. I teach students that they are the most powerful tool in the Order of Operations hierarchy because they are essentially able to break the rules.

Students need to understand that they only have priority when they contain an OPERATION that needs to be calculated, otherwise their use is strictly notational.

They also need to be taught that there is no such thing as opening, removing, offing, breaking the brackets.

They hold OPERATIONS INSIDE of them that are to be calculated first as the top priority. 2(6) does not hold an operation inside it but only a single number and so the symbols in this instance are just parentheses and they do not have any kind of priority. They can either be left in to implicitly represent multiplication or they can be removed and replaced by an explicit multiplication sign.

Anything on the outside of a set of grouping symbols is not a part of them. There is no such thing as parenthetical priority, as some people try to say there is. Juxtaposed multiplication is occasionally instructed in certain works to be given priority over explicit multiplication and division, but it is not an expected arithmetic standard or rule.

Unary operations should also be discussed towards the end of the topic as these fall under Step 2 - the Exponentiation step. These include the unary plus + and unary minus (negation) as well as factorials and percentages and some functions.

I also look at the Commutative, Associative and Distributive rules towards the end of my Order of Operations explanation. I discuss terminology as well. The definition of a TERM is important.

That is just an overview of my teaching. At the end there is generally zero confusion and almost 100% accuracy from my students on even the most complex arithmetic and algebraic expressions.

I learnt BODMAS over seventy years ago when I was about 11 or 12 at my English state school (still the best education there ever was). O is for "of" as in 1/4 of 8 = 2. It's as simple as that - basically just a form of division, which is why O is adjacent to D.

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ReplyDelete