Ratios, Rates, and Proportional Reasoning

Every day I think…how do I help my students connect mathematical ideas and make sense of the math we are studying.

My class has been developing an understanding of how to use ratio tables through various tasks I have brought in to supplement CPM Core Connections 1. They have demonstrated a good understanding of this, though some students are having difficulty moving from additive thinking to thinking multiplicatively. (On a side note…this is now their favorite word…they love saying it for some reason… cracks me up.)


We all have the moments, or maybe it is just me, when our planning isn’t as complete as it needs to be. I had glanced at lesson 7.1.3, but had not looked at problem 7-26 as closely as I should have.  We had only been using tables that were proportional, starting at (0,0) and then students came across this situation.

Screen Shot 2016-03-25 at 10.18.10 AM

They kept telling me that there was a mistake with Tamika’s table.  “Ms. S the relationship  between 1 and 7 is not the same as 2 and 9. This doesn’t make sense.  ”

So, we reread the problem and thought about the context. What do you notice?

“They are knitting. They have a constant rate.”

Ok, what does it mean to have a constant rate?

“It is the same. But, Ms. S, it is not the same. ”

Can we look at the table in any other way? They nodded and I left them to work. Then I heard them…

“Well, we can compare 7 to 9 and 1 to 2. Oh maybe it is increasing by 2 and by 1….”

“Every hour she knits 2 inches.”

They were starting to make sense.  Then other groups were having similar struggles, so we discussed as a class. We continued on discussing, giving them time to think with their groups and process, and realized that Tamika had started knitting with 5 inches already completed. But, I could see many of them were still a bit confused about the differences in the two situations.

We had been doing a lot of linear visual patterns and making tables and graphs from them, in addition to graphing ratio tables and noticing that ratios which are proportional create a line through the origin (even though this really is formalized in 7th grade, the discussion has grown organically).

Last weekend, I engaged in a conversation on twitter about visual patterns with  .  Their thoughts and comments made me think a lot.

I realized that my students had not been able to connect all of these situations and see the relationships between them. My goal had been to let them develop reasoning skills, looking at relationships, and connecting visual patterns to expressions in order to be able to formalize their explorations in the coming grades, but seeing them struggle with the knitting problem, I thought, how can I help them connect the visual patterns to a situation like the knitting problem?

So, I came up with two patterns that I knew they would easily be able to generalize and then we could compare and connect. Which one shows a proportional relationship?

After students had time to process and look at the patterns, think about how they were growing and the relationships, we then started our discussion.

I stated, one of the things I would like you to think about has we look at these patterns is how do these patterns connect or relate to the knitting problem we had done a couple of days ago. Remember, we are always looking at math as connected and looking for relationships to help us better understand.

Screen Shot 2016-03-25 at 10.00.16 AM


We first looked at how the patterns were growing and then we engaged in a “Compare and Connect Talk” (see Intentional Talk, by Elham Kazemi and Allison Hintz).

They made some great connections and there were a lot of “ahhhhhs” “oohhh, I get it.”.

Then we looked at the differences.

“Look at the bottom left and the right. They both have arms that are the figure number, but the left one has a square of 4 and the right one doesn’t have anything.”

“Oh, so if we subtract 4 from each on the left, then we get the right pattern. “

Can you make a connection to that in the table? They turned and talked and analyzed.

Screen Shot 2016-03-25 at 10.58.32 AM

One student said, “Look, if you subtract 4 from the 7 and the 10 and the 13, you have the same numbers in the other table.”

I then asked them, what if these patterns represented Jelly Beans and Cupcakes?  Turn to your partners and try to come up with a situation that each pattern would connect to.  Remember the knitting problem. Do you see anything related?

They talked and some struggled, but through their discussions and a class discussion, we came up with the following situation. They were seeing the patterns connected to a context and were really starting to make sense.

Screen Shot 2016-03-25 at 10.09.47 AM

Where do we go next? Do we keep playing around and looking for connections? Do I formalize some of the language?  I’m not sure… Next year, I think I would do the comparison of these two patterns before we do the CPM Lesson, as I think the visual connection would help them understand that situation more. But, as for this year…I think we keep playing and exploring and connecting. They will be ready for formalizing these ideas in 7th grade.

We need to formalize division of fractions! Ahhh! But, all this work with ratios has really started to help students understand division of fractions more as we make that connection. Wow! There really is so much to think about when designing the best way to facilitate student thinking and sense making. It is amazing. I thought I really understood ratios, rates, and proportions.  My depth of understanding is growing everyday with my students.  There is so much depth to math. It is just brilliant.

Every day I think…how do I help my students connect mathematical ideas and make sense of the math we are studying.


4 thoughts on “Ratios, Rates, and Proportional Reasoning

  1. Pingback: Best of 2016 (Q1) – Teaching With Problems

  2. Hi Anna, thanks for the inspiring and valuable presentation on all of this at the SB Colloquium this past weekend. It’s fantastic stuff. Notice and Wonder is so simple but SO IMPACTFUL. And what you’re doing to build the students’ collaboration, critical thinking, and ideation skills is so important too. I’m going to ask the MV 6th grade teachers if they are doing any of this! Will point them to your site either way…


  3. Hello! This post was recommended for The Best of the Math Teacher Blogs 2016: a collection of people’s favorite blog posts of the year. We would like to publish an edited volume of the posts at the end of the year and use the money raised toward a scholarship for TMC. Please let us know by responding via http://goo.gl/forms/LLURZ4GOsQ whether or not you grant us permission to include your post. Thank you, Tina and Lani.


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