I have never introduced students to order of operations.  I have always received students after their initial introduction.  I always feel like receiving students after their rote memorisation of BEDMAS (or PEDMAS if you call 'em parentheses, or are very apologetic about your kin), means that when I try to let them explore they just default to what they memorised, but they cannot recall that after the lesson.  It makes me feel stuck between a rock and a hard place, but I still wonder what the initial introduction can look like.

Initiate a Debate

One introduction that I have just thought of this morning, I'd like to try.  It's heavy on symbols, which I am afraid of, but I think it is basic enough that it might spawn some conversation and debate.  I just want to share it and see what you all think. I like it because I feel it introduces students to some manner of proof, and following logical principles, and I think it would be very interesting to see students' misconceptions about the order of operations.  Here goes:

Premise 1
Ask the class what is 5+12?  What is 12+5?
Can we agree that 5+12=12+5?

Premise 2 
What is 3x4? Can we agree that 3x4 is another way of writing the number 12?

Conclusion
I am going to write something on the board.  Tell me if you Totally agree, somewhat agree, somewhat disagree, or totally disagree...

5+3x4=3x4+5

Have students get in groups and say, "give me a knock down reason why you think you are right."

The students that I imagine

The agree-er- This student believes it, and would focus on the fact that 3x4 is another way of writing 12.  This student would be good to play devil's advocate with.

The choose your own adventure - This student says, "If you decide to multiply first it can work, if you work left to right it doesn't."  This is pretty easy to counter, ask them to try both ways on just one side of the equation. You get that 5+3x4≠5+3x4, "That seems ridiculous doesn't it? So which one should we choose?"

The reader - We read a book left to right, so we read math sentences left to right as well. This student gets that we need to have a common way to solve all math problems, but does not quite understand the separate operations.  (This is the student that I need help with!)

Uncharted Waters

This is where I have never been and wonder how worthwhile this exchange would be.  Would students logically piece it together? Would most students be an agree-er, a choose your own adventure, or a reader? How would I encourage each of these groups to prove themselves and come to a consensus?  What questions would you ask your class in this discussion or is this just a bad introduction?

What I Hope Students Can Learn?

I have been thinking that I would much rather introduce Order of Operations, as terms rather than BEDMAS.  This would prepare them for algebra, and would provide a more conceptual understanding of the process.  I do not know what the progression would look like, but I assume that beginning with multiplication and addition would be the natural place to start.  Where would you go next? Subtraction?  Maybe you are thinking, "Timon you are CRAZY, this is HOW you should do order operations." Please tell me about it; I want to know.
 
 
I have very strong opinions about manipulatives.  I like them.  I really do, as long as they are natural, demonstrate a clear pattern, or give students an alternative way to work out a problem.  To me this seems obvious. Why would you have students work with manipulatives if they do not clearly illustrate and support the objective of the lesson?  Unfortunately in BC our curriculum is trying to shoehorn manipulatives into the curriculum.  I get their effort to make things more concrete, but I feel as if manipulatives should be a supplement, not a requirement.  Regardless of this opinion, I still have to find ways to make these manipulatives accessible, clear, and engaging, and this is my effort to do so.

The Battle of the Integers (Army Men kick inergers chips arses)

Ok, yes I am a loser.  I just need to get that out of the way, but hey it works!  How do I do this in class?  Here goes.

Start the video: The opening sequence is enough time to hand out bags of army men.  I hand them out one ziploc bag per pair, and worksheet/notes per student.
integer_addition.docx
File Size: 104 kb
File Type: docx
Download File


Then I press pause when "Integer addition" shows up on the screen.  This is the easy part.  We talk about what happens in war.  In integer war they always kill each other off in pairs (zero pairs to be specific).  Kids catch on to this pretty quick.  So we can begin with our first example.  This is our basic addition example.  Have students start with three positive army men, then add three more.  Easy peasy!  I play the video for a few seconds to let those three positive guys come in.  Now we have six, so ask the students what if the 4 red guys came to the battle scene?  The students recreate the battle scene, write down there answer, and then view it on the video!  (This involves pausing and playing, and you really have to know where to stop it).

Then students work out their own examples, they can use the army men, and they plot their actions on a number line.  I have tried to make the work sheet one that slowly removes steps (I think I saw Jason Buell do this, and I love it).  By the end students get a pretty good grasp of integer addition. It is just practice (I pick integer addition war, or a  pre-alg with pizazz that I snagged from Dan Meyer).

Where battle starts getting tough... (AKA Subtraction, and its difficulties...)

The next day we start with some small activity (probably war), and then we jump into subtraction.  I remember when I was a kid, and learning about subtracting a negative number I felt a little like this...
Picture
Subtracting negative integers is not intuitive (to me at least) in the slightest.  In fact it was not until my modern algebra class in third year university, that I actually understood why a negative times a negative is a positive.  It is thus not a surprise that this was the hardest part for me to teach, but after a few tries I finally got it.

I start by asking the students to make a battle situation that will result in a zero, but uses ten men (in total).  Students work through it, and with a little bit of help and guiding questions, they get 5 red and 5 blue.  I then introduce them to the idea of retreat.  Subtraction is like retreating, so I tell them,"from this battle, retreat (or subtract) +3 guys.  What do we have left?"

Students show me and tell me, "Negative 3!  You have -3 left!"  I get them to write this down (I am working on a worksheet still, but for now blank paper will do).  So I decided to pull out my infinite cloner from my Smart notebook, and get them to show me on the board.
battle_of_the_integers.notebook
File Size: 100 kb
File Type: notebook
Download File

We reset to zero and build it up, make bigger battles, and try questions like 0-(+5), 0-(+10), it doesn't take long to get that subtracting a positive, is like adding a negative.  Then I have them reset to zero, and this time, with out removing any men, they have to make the battle scene (I haven't thought of a better name than battle scene) equal to +2.  In this case they should have 7 positive guys, and 5 negative guys. I ask them the same thing, "now remove 3 positive guys, (+2) - (+3)," students get really used to recognizing how it helps.  The basic steps are as follows (you can do it for any number).

  1. Reset to zero
  2. For the first number of the equation, add those army men (start with zero, go to positive, then go to negative).
  3. Subtract the number (in battle terms, tell which side should retreat, start with positive, work to negative).
  4. Fight the battle, see what remains.
  5. Repeat

Go through the combinations, and keep asking students, "Do you notice patterns?" Follow along on the smartboard.  Here is an example of what it looks like on a Smartboard.

Move on Soldier!

It took me at least three full lessons of falling flat on my face (having kids more confused than when we started), but this way finally worked.  Students could see the patterns and the rules clearly.  Kids who could memorize the rules did, those who didn't could make the battle with the army men.  I was happy, but now I have to do multiplication.  I have some ideas, but nothing that I feel inspired by, whereas I am proud of this.  Any tips?  I'd love to hear them...

A completely pointless epilogue

Just for fun, I created this, and put it up in my class using block posters
Picture