## Thursday, September 27, 2018

### Units Coordination and Proportional Reasoning

This morning I was in a 7th grade math classroom.  The class was reviewing for a test and working their way through problems at different stations.  I sat down to talk to a group that was working on this problem:

When shopping at Target, Sam found pizzas priced at 4 for \$14.00.  He was having a party with 16 friends to watch the MLB playoffs.

• If he wants to make sure to have half a pizza for each person, how many pizzas should he buy?
• How much money does each person need to contribute to cover the cost of the pizza?

As we discussed the strategies students were using to solve this problem, I started to notice levels of units coordination embedded in the problem and how the strategies students chose were related to how many levels they were able to coordinate.  (If you have not yet read my blog posts explaining units coordination you can find them here.)

Here is a visual I created to show the levels of units involved in this problem.
There actually are 4 levels of units as well if you take each pizza in the lower left visual and slice them in half.

My discussion with the first group was so enlightening that I continued to discuss this problem with other groups as they rotated through the stations.  I found that for many students they could easily determine that we need 8 pizzas.  One pizza feeds 2 people, so 8 pizzas feed 16 people.  Most students relied on proportional reasoning, thinking about it as doubling.  They focused on whole numbers and multiplying rather than fractions and dividing.

The second question is what caught my attention.  Most of the strategies students used helped them find the price per pizza.  Once they had that, they divided by 2.  There was even one student who worked from 4 pizzas for \$14, divided by 2 to know 2 pizzas is \$7, and divided by 2 again to find 1 pizza is \$3.50.  I realized that many of these students were working through one or two levels of units at a time instead of being able to jump from \$28 for 8 directly to \$1.75 for 1/2.

I think this is an important distinction to make when discussing units coordination.  All of the students were able to get the correct answer.  All of their strategies relied on their mathematical reasoning and number sense.  From the teacher perspective, however, it is important to notice the sophistication and efficiency of their strategy.  We need to strive to understand where students are in their ability to coordinate units so that we can continue to move them forward with this type of thinking.  Otherwise as the math continues to get more complex and students begin to struggle, we might not understand the root cause of the confusion.

The other interesting thing I discovered in my discussions is that while most students did not divide by 16 to find the price per 1/2 piece, they checked their answer by multiplying by 16.  This was a great reminder to me that units coordination is reversible.  Perhaps if students had been given the price per piece they could have more easily worked their way to the cost for a group.  (I'm a little tempted create some questions and draw a visual for the levels of units I could work into a problem like that, but I am short on time.  Perhaps another day.)

Here are some questions I want to reflect on.  I hope you do too.
• Does units coordination play a role in our next unit?
• What questions can we plan that will allow us to see how students are working with and coordinating units?
• Do the questions I have planned ask my students to coordinate units both forwards and backwards?
• What supports can we put in place within the lesson for students who struggle to coordinate units in order to move them forward in their thinking?