Day 83 – Dividing Fractions


Math 8 – Every now and then I resort to some notes and today was one of those days. We had modeled some simple division using counters, number lines and circle graphs.  However, at some point the models start to fall apart. Or, they become unweildy. At this point we turn to some mathematical properties to help with computing a division operation with fractions.

The students seem to be ok with this process. Once again, simplifying the fractions is by far and away the most difficult part for them.

Day 75 – Birthdays

Math 8 – Friday, 2:55 pm. First week back after the winter break. 13 year olds. Solution: join forces with another class and see how many people would have the same birthday!

I think there were 68 people in this photo (I didn’t really count). With 70 people there is a 99.9% chance that there will be at least two people with the same birthday. This group had 4 pairs. Fun!

Oh, earlier in the class we also did some work with modeling multiplication of fractions. I’m not sure how much more I will have the students pursue this. Whole numbers x fractions is easy to model and gives the students a physical/visual representation of what is happening with the operation. It removes some of the abstraction. I’m convinced that this modeling loses its shine as the operations get more complex as we move towards division. In my mind, models are simplified representations that help with explanations and predictions. By the time we get to dividing fractions, I think the models no longer accomplish what a model is supposed to.WP_20170106_14_49_30_Pro.jpg

Day 73 – Restaurants and Retirement

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Math 8 – Today was a pretty good day. I started the lesson with warm-up using ratios/percents/rates. I brought in a receipt from a dim sum lunch that I had over the winter holidays. Most students did proper calculations although it certainly took a long time. I was left wondering how long it would have taken them if they were at the restaurant?

I was able to use this assessment to identify a few students that I will ask to come in for extra help. While we are finished the “percents unit” or topic, I can’t let them leave like this. All students can and should be competent in dividing a restaurant bill including tip. Even if they end up with a restaurant app on their phone, this problem is nearly identical to dozens of other situations that will require the same skills.


We then moved onto a group whiteboard problem with fractions (or ratios, depends how you look at it!). Nearly all groups first attacked the problem by finding a common denominator but then they realized that the number of married women wouldn’t match the number of married men. Hmmmm…


Day 59 – Cookie Sharing

Math 8 – Today’s class was working on sharing a cookie problem. It turned out to be a fantastic class, perhaps the best we’ve had this year. The problem had a low entry with lots of ways to solve it. We were able to walk around the room and see the different solutions.

The above solution is one of the most intuitive.  Students realize that 1/2 the people get 1/2 the cookies (21) and since they each share this equally, they each get 10.5.  This continues for the next two people.

Below is the most common solution.  Students would turn this into a rate question as students naturally understand rates with little to no instruction. They find that 3.5 cookies cost $1 and then calculate how many cookies each person can have.


The rate thing is pretty interesting. Why are students so comfortable with rates? I think it’s because rates are usually very concrete. They explain the relationship between tangible things. Cookies per unit dollar. Kilometers per unit hour. Miles per unit gallon.

Next we had the fraction solution. This one is really neat because in our discussion the students realize that they are once again dealing with percents. 7/12 is 7 parts of the whole.


Finally I would end our tour with the primary goal for the lesson: equivalent ratios.  This wasn’t the most common solution, but every class did have it.


Once you get to the equivalent ratio solution, you can see that it is perhaps the cleanest to exhibit and explain. I didn’t see this at the time, and I may come back to it and highlight this property.

The class was full of Flow, engaged students and learning.  So awesome.

Now here is my dirty secret about the day. Many of my students were on field trips so the class size was in the range of 18 kids.  While I can totally manage a class of 30, wow does the party change when the numbers drop.  Behaviour problems were almost non-existent, and the level of dialogue was amazing.

If someone tells you that class size doesn’t matter, maybe they’re right… Unless they’re referring to a class where students are engaged in collaborative work that needs to be discussed and shared, in which case they will be dead wrong.