My Love for Students’ Understanding of Math Grows Deeper and Deeper

I am so grateful that I get to begin each day with math! Every day starts with playing! It is incredible.

We explore and talk every morning while making connections, looking for patterns and engaging in mathematical arguments in the form of a Number Talk; a Mathematical Conversation from Intentional Talk; a Sometimes, Always, Never discussion; a True or False Reasoning activity; choral counting; or estimation activity. My biggest problem is keeping track of time.

This past week, it was time to explore ratios. I looked through our CPM Book, the 6th grade Standards, Jason Zimba’s coherence map, and didn’t feel like the Text Book was going to meet our needs exactly the way it was set up.  So, I started to think about what my goals were and what my students need to understand.

Throughout parent conferences, I kept hearing myself speak to parents about the importance of connecting mathematical concepts. Too many students were seeing math concepts as isolated and I really want them to see mathematical relationships and see all the connections.   I realized that because some hadn’t been making these connections for years, my most important job is to create opportunities and engage them in conversation so they can make these connections.

I really wanted students to understand the language of ratios, understand the different types of ratios: part to part and part to whole, and see ratios as a relationship while connecting what they knew about fractions.

I remembered a Math Assessment Project Lesson that I thought might be perfect.

  1. It allowed for multiple entry points, allowing all students to have access to it.
  2. It would allow us to make connections to their understanding of fractions, but also allow students that need to develop more understanding with fractions that opportunity as well.
  3. It would allow us to investigate what a ratio is and come up with various notations and the language of ratios without being directly told them.
  4. From this task, I would learn what my students know and where we would go next.

As I mentioned earlier, we start each day my playing with numbers. That morning we had started with an Always, Sometimes, Never for these three statements: 1/4 = 25%, 7/12>8/14, 3<4x. We had recently been comparing fractions to percents, I wanted them to make note of when comparing fractions to use a benchmark number such as 1/2, and since we have also been working with variables, I wanted them to think about what would happen if you used a fraction versus a whole number in place of the variable.  I loved that we got on the topic of the importance of the whole. “Ms. S, 1/4 = 25% only if they are both out of the same whole. 1/4 of 8 is not equal to 25% of 4.” Love!

I wanted to give them something that they might be able to connect to  and the comparison of the two fractions was perfect for that.

For the task of Fizzy Orange Drink, Screen Shot 2015-12-06 at 2.08.00 PM.pngwe first started out with some Noticing and Wondering. This I have come to believe is one of the best habits my students can develop. It creates engagement and as Jo Boaler states in Mathematical Mindsets, “the desire to understand it and to think about it.”

I then gave them the information that they had requested and we were off.Screen Shot 2015-12-06 at 2.08.27 PM

We came up with these three possibilities of most orangey to least orangey. Then students justified and explained, caught mistakes, and connected to the second problem in the warm up.”Remember in the warm up we said 7/12 was 1/12 away from 1/2 and 8/14 was 1/14 away from 1/2 and because there were the same number of parts we can compare them easily. We can do the same thing here.” Yes! The student’s noticing helped other students who naturally weren’t making the connections. We agreed on the Middle row, comparing the fractions or part to whole ratios using a common numerator or unit rate.Screen Shot 2015-12-06 at 2.20.38 PM

It was then time to stop, so for HW they were asked to respond on our classroom forum and reflect about the day’s work.Screen Shot 2015-12-06 at 2.24.18 PM.png

The next day they got into the big task of sorting and matching. But first they made some Noticings and Wonderings. They noticed that some cards were related, asked about the blank cards and decided themselves what they thought the task should be. It was awesome!Screen Shot 2015-12-06 at 2.08.50 PM

I told them, as they sort and order, to be thinking about what they think a ratio might be.IMG_1778

IMG_1781

Then it was time to Troubleshoot and Revise. So for HW, the students had to analyze two strategies that had mistakes. Then the next day in class  we held a Trouble Shoot and Revise discussion.  This was so important because we were able to surface some misconceptions and then students had an opportunity to revise their sorts.

Finally, it was time to share and compare strategies. One or two people from each group was to stay with their poster and explain their process and reasoning. The other members walked around and were to ask questions and critique their peers’ reasoning.  We then discussed mistakes and corrections, analyzed the dictionary definition of a ratio and related it to what we had just done and created a list of notations and language of ratios.

IMG_1800They then had a follow up task of mixing drinks but with 3 ingredients. My job now is to analyze their strategies and depending on how they did, we will look at my 3 favorite mistakes and Trouble Shoot and Revise or Compare and Connect Strategies.  So what comes after that? Good question.

That is what I am trying to figure out now.  In our CPM book, they are enlarging and reducing pictures and shapes. That is really a 7th grade standard, but focusing on it as scaling up or down it could be a great connection to what they were doing in 5th grade. So, I think I will have them engage in that, but with the focus of connecting scaling up or down to the work we just did. I like that it is a completely different context and think that could lead into some great conversations and big understandings.

The other thing I want to do is introduce some representations: double number lines, ratio tables, and graphs to help them see the multiplicative reasoning. I’m thinking of possibly showing them a problem solved with the three representations to see if they can connect each representation and determine how they work instead of me directly teaching. This will help place more of the cognitive load on them and require them to really engage actively.

Some are also still struggling with the notion that a fraction is division. So, pulling in some equal sharing problems with proportional reasoning is also going to be helpful. I just need to determine which I do first. Ideas?

Their minds were firing all week and they are starting to make some good sense of the math. I think we have a good start. There is a curiosity and a desire to want to make sense of it all now.

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