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.

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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.

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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.

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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.

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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.


Solving Current Events: A Follow Up

Last week I wrote about a lesson I had developed from a video I saw about a volunteer in Lesbos using the materials of life jackets for bags. This is a reflection on how the lesson went with my class.

First of all, it was motivating, inspiring, open, and students had to persevere.

Screen Shot 2016-03-15 at 6.40.46 PMThe day after I wrote the lesson we were not going to have math since we were going on a field trip, but I had 15 minutes before we left on the trip and I was able to engage students in some wondering. We went through the first slides and noticed and wondered, not thinking about math at all. Though, my students are now mathematizing the world, so a few mathematical wonderings came through.


Then that evening they read this article and were able to answer some of their wonderings.

The math extravaganza began. We took on a math lens and thought like mathematicians. Noticing, wondering and determining what information we needed.IMG_7390 (1)

I am always so amazed with the mathematical questions students are developing now. Determining what they needed was a lot harder than in other lessons we have done. Why? It was a big scale problem with multiple questions to solve. But, through conversation they came up with some good ideas.  I then gave the students the information they asked for and they were off!


It was fascinating to watch how students approached the problem. Some just started tracing complex shapes, others grabbed rulers and drew precise lines, others drew free hand lines, but then as they saw their classmates using rulers, changed their minds. IMG_7387IMG_7380

They helped each other and after awhile all students were looking for shapes that they knew how to find the area of. I had a few students that really struggled with this and I wanted them to get to finding the area, so I provided them with my shapes.  Then they were off. It was a great way for some students to get additional reasoning with finding the area of shapes and it was through this problem that some really saw the relationships of different shapes to one another.

“I know how to find the area of triangles, so I decided to turn all these into triangles. There are triangles in every shape. ”

And then they solved….IMG_7386IMG_7388

Thinking, connecting, and then we needed a discussion. Can we find the actual area if we use the ratio 1 mm: 2 4/5m?IMG_7400We discussed how the ratio we were given was a linear ratio and we could use that for finding length, but to find area, we needed to create a new ratio of square millimeters to square meters.  And then we had some solutions…IMG_7402

We shared different strategies: converting all the lengths in millimeters to meters and then finding the area compared to finding the area in square millimeters and then converting to different units. We then found the median and mean of our results.

Then came the question: How many lifejackets are there in 40,000 square meters?

It was great to be able to review the use of ratio tables and to see the different ways students approached the problem, connecting the three different strategies. We engaged in the “Compare and Connect”  math discussion that is introduced in Intentional Talk by Elham Kazemi and Allison Hintz

And then there were the students who extended the problem…

I clearly am a terrible photographer… But did you know that 40,000 square meters is equivalent to 7 1/2 football fields, 91 1/2 NBA regulation basketball courts and about 1/2 of Buckingham palace?


And if you were to lay pencils down in 40,000 square meters, you can fit 3,007,518 pencils. hahaha

Finally came…Screen Shot 2016-03-09 at 12.22.19 PM

Hahaha, this is when I was reminded I work with 11 and 12 year olds. What would they do? “I would make a pile and build a huge slide on it.”

“I would jump in the lifejackets.”

“I would take them back to Turkey and resell them.”

“I would make a boat out of them and sell the boat.”

Clearly, we have fun seekers and entrepreneurs.  Next time, I would change the question to… What would you MAKE with 450,000 life jackets?

I then shared with them the video about how they are making bags out of the lifejackets and left students with the question. What extraneous materials do we have lying around and what useful object could we make out of it?

It was a two day lesson, but a lot of learning: using a ruler, calculating area, solving proportions, developing a plan, comparing and connecting, and persevering.



Solving Current Events: Math, Design, Global Connection

Who knew Facebook actually could be inspiring?

A friend recently posted this video


And then the wheels started to turn when I saw this picture…


How many lifejackets are there? What area is taken up by life jackets? Ooohhhh…I see rectangles, trapezoids, and triangles.  I can connect what we have been studying in geometry to what we are moving on to next: ratios and unit rates. Not only is there math, but  we can write, do some design thinking, and develop a greater understanding of the greater world. Here is the lesson. Please feel free to comment. All ideas are welcome.

Before doing the math lesson, I might have students read one of the two articles and notice and wonder about some images.



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I would then show students this image, asking what they notice and wonder. We might record their noticings and wonderings using Poll Everywhere or Pear Deck, or I might chart their ideas on chart paper. I like using technology because then everyone is able to share at once, but I also like when students share ideas orally because it tends to generate even more and creates excitement in the classroom.Screen Shot 2016-03-09 at 12.19.34 PM

We would then zoom in and add new noticings and wonderings.

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and zoom in even more to finally see what is actually there.

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Now enters math… I would start our math class telling students it is time to think like a mathematician and have them ask some mathematical questions.

From the list we generated I would hope that these two would arise. Screen Shot 2016-03-09 at 12.21.13 PM

Then I would ask what information do you need to answer those questions. We would create a list of information that is needed and then give them the following information.

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Screen Shot 2016-03-11 at 10.32.47 AMNow comes the fun…planning, persevering, and solving. After students have been working for a bit, if I notice a lot of struggling, I might stop them and ask them…Screen Shot 2016-03-09 at 12.21.50 PMScreen Shot 2016-03-09 at 12.38.27 PM

If students are still struggling, or for those students that need additional supports, I might share with them the shapes I saw and let them work form that.Screen Shot 2016-03-09 at 12.23.57 PM

After students were given time to solve for the two questions, we would share out strategies and engage in Mathematical Practice #3: Critiquing and Justifying.

I would then share with them the data that was in the article:

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At another time, I would then ask students…Screen Shot 2016-03-09 at 12.22.19 PM

We would engage in some creative design thinking and come up with some uses for 450,000 life jackets. I would then share the video that inspired this lesson and we would read. this article.

Lastly, here are a few extensions…

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Here is the Ratio and area lesson Refugee if you would like to use it.

As I write this all down, it feels long.  I wonder what I might change. What do you think?