"Excuse me, sir" Do you mind if I take a picture of the donut display? I'm a Math teacher and I use real world images in my Math class. "No problem! Here, let me make it really pretty for you and add these chocolate ones." He was so nice! However, I wanted the display of just the original ones so that my students could quickly subitize to find out how many donuts it was. I still took the picture to show my appreciation, but I figured it was useless. After looking at the picture, I thought to myself " Well, I can still ask my kids how many donuts in all? Or, I can still ask them about the original glazed donuts and they will be able to multiply all the rows and subtract the chocolate or they can multiply the other rows and add the original glazed from the last row.
And there it was. On this October Day in 2018; I, Amber Williams, for the first time ever UNDERSTOOD Order of Operations in Real-World Context. Did I know PEMDAS, did I know how to simplify expressions using Order of Operation? Yes!. Did I know how Order of Operations were used in Real-Life? Um...can't say that I did.
I guess I did view Order of operations as a multi-step word problem. Which is a real-world context. However, I never viewed it how I viewed it at that particular moment in the donut shop while trying to get more Math Images pictures for my students. If I discovered that, of course, I had to use those images to help my students naturally build concepts of Order of Operations before they were even taught the Order of Operations. The Order of Operations is still a debate that is going on today. There will occasionally be a viral problem and people will argue over the answer, mostly not understanding if they should Divide or Multiply first. If a concept of what Order of Operations actually is then it would be easier to understand why the steps are ordered the way that they are ordered.
What I Do:
What I had already started doing with the Math Images. However, I was more strategic about showing the different equations. This is one of those things again, where students are already doing the skill. They just don't know they are doing the skill.
2. After students shared out different equations, I would write them down and all of the students would check the different equations. I would emphasize and ask questions to students that directed their thinking to seeing the different ways they could solve the problem. I would then ask them is it possible for them to switch the order of how they do each operation and still get the same answer. I let them play around with that for a while and discover on their own why the order matter.
Below, is an example of my students' equations to figure out how many lifesavers there are.
4 x 3 - 1 = 11
One student explanation is they they knew it was four in every row except one. So they multiplied 4 x 3 and subtracted 1.
4 x 2 + 3 = 11
One student explanation is that they knew two rows had four lifesavers, and the other one row had three. So they multiplied 4 x 2 and added the other 3.
3 x 3 + 2 = 11
One student explanation is that there were three columns of three and one column of two. So they multiplied 3 x 3 and added the other 2.
6 x 2 - 1 = 11
One student explanation is that they visually saw six on each side, even though one was missing. So they multiplied 6 x 6 and then subtracted one.
6 + 4 + 1 =11
One student explanation is that they noticed the colors. There were six orange lifesavers, four green lifesavers, and one red lifesaver. So they added 6 + 4 + 1.
Side Note: Did you notice all of the creativity that came through these activities? I've said a million times and i'll say it a million and one...MATH IMAGES ARE EVERYTHING!
Through my students discussing their steps in how they solved the problem, they are able to make a connection between the Order of the Operations and why they are done in a certain order. I am not about teaching kids higher grade skills, because I truly believe a deeper concept of their grade level works way better than teaching a procedure from another grade. However, if it is a heavy concept I love to have kids practice to build conceptual understanding and often times I don't tell them the skill they just did, or I tell them long after they have practiced with the concept repeatedly. My hopes that if in the 4th-grade that if I add that small component to my Math Talks then by the time they are in the grade learning the Order of Operations then they will have a clear understanding of what they are doing and why they are doing it in a certain order...and not just because they are told a rule.