Yesterday, I blogged about a proportional reasoning math question that I did in class, and why I needed to try this activity again. I was feeling good about the changes that I made to the question, but then the students surprised me with their responses.

I decided to start things off yesterday by reading through the problem with the class.

My students were finalizing their plans for our year-end field trip, and working hard to ensure that their final choices met with our project requirements. Reading this problem made them very excited: they could save some money on their trip, and help ensure that the total cost for the students remained at $25 or less. While I wanted the groups to show me how they were going to solve this problem, I also wanted to hear some initial student thought, so I decided to record this podcast.

While I was very excited about the math possibilities for this problem, I didn’t even consider the real world implications. This is not just a make believe math problem. Students are helping to design a real field trip, so they’re thinking in terms of real life.

• I would have never considered that nine students would pay for the trip, while one student got to go for free. As a teacher, I’m used to splitting up the costs to help accommodate for deals, but many of the students saw this as more of a lottery system, and they started talking about what was “fair” and “not fair.” While I was thinking in terms of proportional reasoning, they were thinking in terms of what was the right choice for everyone.
• I didn’t even think about the fact that some students may not go on the field trip. Of course this is the case, as that always seems to happen, but this was just a make believe thinking question. I assumed that everyone would figure that all 90 students are going, but if this is real world, then their thinking is real world as well.

It was interesting when even this one group of four found out that they save the same amount of money with both types of deals.

When discussing this solution with the group, the students told me that even though they saved the same amount of money because of the cost of their tickets that $2 off would probably end up really saving them more because not everyone’s going to go on the trip, so they won’t really get nine people free. The group also mentioned to me that if their tickets cost more, then the nine free tickets would save them more money, but if they only got eight people free because of trip attendance, maybe the $2 off was still a better deal. It’s amazing how real life can make you reconsider math!

Real life also had me re-thinking math when I saw this group’s solution.

Please note that they meant “81 people pay” not “don’t pay.” They told me this during our follow-up discussion.

Since we’re not learning to multiply by decimals, I told the students that they could round, and this is exactly what they did. The problem is that both of these amounts round down. If a company is really giving us this deal, will they also allow an additional deal of paying less for each ticket from the get go? And this is where I re-think the concept of rounding when it comes to decimals and dollar amounts. If the real world math application for rounding decimals is in terms of money, and this is certainly a great example of it, will stores want people paying multiple cents less for an item if they’re rounding to the nearest dollar amount? Or if an item costs $8.35, the amount rounds to $8, but just bringing $8 to the store is not going to be enough to purchase the item. The math says to round down, but the real world application says to round up.

So as my brain continues to hurt this morning from all of this math thinking, I come to see what this focus on proportional reasoning is really all about. I think it’s about the “thinking.” I think it’s about the accountable math talk. I think it’s less about the answer, and more about the reasoning. As students worked through this proportional reasoning problem in class yesterday, I saw and heard a lot of math thinking. Was all of it correct? No, but our conversations that evolved from it allowed for some meaningful math learning — for both me and my students. And as I reflect more now, my learning continues. It’s with this in mind that I’m looking forward to Monday’s Proportional Reasoning Inservice with the hope that it gets me thinking even more, as teacher thinking leads to more student thinking and student thinking leads to increased understanding. Let the teaching/learning cycle continue! 

What are your experiences with proportional reasoning and real world math problems? What advice would you offer me as I continue to work on developing “thinking skills” in math? I’d love to hear your thoughts!

Aviva