Christina Michaels
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I don’t think my understanding of proportional relationships has necessarily evolved, but my way of teaching proportional relationships definitely has.
Before I was simply presenting students with equivalent ratios and expecting them to recongize the way they were changing. 3/4 is equivalent to 6/8 because both quantities are doubling.
Now, though I’m giving students lots of experiences using models and manipulatives. They’re drawing and using multiple representations, and THEY are the ones figuring out that the ratios are two quantities that are linked together and whatever is happening to one quantity, the same is happening to the other quantity. I’ve emphasized multiplicative thinking vs. additive thinking very explicitly and repeatedly and so we are using those ideas when we’re discussing what’s happening when we scale a ratio in tandem.

In anticipating what my students would do with this task, a common misunderstanding would be for them to use additive thinking with the initial question: 25050= 200 ml of rice, so 50050= 450 ml of water. My question is, how could I guide students to use multiplicative thinking?
My initial reaction would be to get them to scale in tandem by asking them how much water would we need if we doubled the amount of rice? or how much rice would we need if we cut the amount of water in half, to try to get them to recognize the common factor that they are scaling by, in the hopes that they are recognizing by scaling both the water and rice by the same factor they’re using multiplicative thinking. What I think, though, is that I would have some students who would be able to do this, but still go to additive thinking as their first choice for solving the problem, and not realize the mistake in their thinking.
I think using rate reasoning could help in this situation, as I think many of my students would recognize that 250 is half of 500. After trying it out with many ratios (Is this a true statement? Does it work every time? What if we tried….?), I would bring them back to their answer of 450 ml of water and ask if this makes sense.

What resonated with me is the idea of really guiding students to explain their thinking. I am in the process of shifting from telling students they got the right answer to asking them to prove their answer.
I used this lesson with my students to introduce the multiple representations for ratio reasoning, but also to very explicitly point out the multiplicative thinking they were using and the idea of this ratio being a composed unit– the number of glasses is linked together with the number of scoops, so what’s happening to the number of scoops when the number of glasses is increasing or decreasing (and vice versa)?
One thing I am still in the process of adjusting in my teaching is the idea that my students need to hear this and see this and build this and draw this over and over and over again. One or two exposures to these ideas is not enough– a trap I often fall into, especially when the the problem is a relatively simple one.

I did this lesson with my 6th, 7th and 8th graders. The problem was accessible to all, and it was really interesting to question and push students into explaining their thinking– everyone was able to come to a correct answer, but it was more difficult for them to really show their reasoning. I used this lesson as an introduction to how we can represent ratio thinking (i.e. ratio tables, double number lines, graphs, tape diagrams) in multiple ways.
The extension was interesting as well– how many cups of hot cocoa can you make with 55 scoops. It was very gratifying to see students using their representations to solve the problem, and of course the discussions that occurred around the extra scoop– what does that extra scoop mean? what could you do with it? can you have just a part of a cup of hot cocoa or does it always have to be a full cup?

Multiplicative Comparison: Amy is 8 years old; Sally is 4 years old. Amy is twice as old as Sally. Sally is 1/2 of Amy’s age.
Composed unit: Jessica’s Purple Paint. Jessica mixes 2 cups of red paint and 3 cups of blue paint to make the perfect shade of purple paint. If she needs 20 cups of paint, how many cups of red paint and how many cups of blue paint will she need to mix to keep her perfect shade of purple?
I’ve done both of these with my students, the first one to have the discussion between multiplicative and additive thinking; the second to introduce the idea of using the ratio as a unit which can be iterated to solve a problem.
They’re both similar in the sense that they use relative thinking– you have to look at the relationship between the two quantities being compared; they are not separate amounts independent of each other.
They’re different in that the ratio of ages is like looking at a snapshot of two quantities, kind of like they’re a fixed point. The paint ratio lets you take that relationship and expand it– you can scale it up or down to find different amounts for different situations. Is she painting a wall? is she painting her whole house?

A composed unit allows for scaling to solve problems; i.e. if 3 ice cream cones cost $4.50, how much would 30 ice cream cones cost?
A multiplicative comparison is focused more on looking at the relationship between two or more quantities; i.e. Sally is twice as old as Amy.

Years ago, when I taught elementary school, I used to do quick images with my 3rd graders. I am going to dig through my files and bring them out again not only because they’re fun and my middle schoolers will love them, but what a great way for students to express their multiplicative and additive thinking! What great discussions students will have– “I saw 2 groups of….” What an opportunity to explicitly point out the multiplicative and additive thinking. And then, to top it off, strategically choosing the images and then comparing the quantities to give students to play with the language when describing the images in relation to each other.

Even though quantities are still being compared, explicitly pointing out the relationship between the two is the key idea. ” ____times as long”, differs from “____ more than”; I think students immediately go for the additive thinking; I’m really excited about teaching them the purposeful shift into multiplicative.

I love the cookie monster task. I think this will be a good starting activity to do with my middle schoolers– 1 to demonstrate/experience a 3act math task, 2 because it’s dealing with addition and subtraction, students will view it as a lowrisk task because they feel more confident with adding and subtracting, and 3 to have students use multiple models to share and explain their thinking.

I love linking addition and subtraction together– especially since many students feel stronger in addition than they do subtraction.
With any contextual problem, I think my students rush to an algorithm that may or may not be correct, because they have a tendency to not read the problems very carefully. Pairing it with a model– number line, balance or tape diagram, makes it so much easier for students to identify what they are looking for and answer the question accurately.

I think my students– struggling middle schoolers– would jump right in to an algorithm. I think adding and subtracting is something they view as a strength of theirs, and where they comfortably live. The problem (like the 1001999 example in the video) is that if students make mistakes in the algorithm, they often don’t recognize that they’ve made mistakes. I think using number lines is a powerful way to help build that number sense– how far apart are these numbers? does your answer make sense?

Middle school students come to my math intervention class with so many gaps. All of these lessons and videos are just really stressing the importance of giving my students these experiences and opportunities;
I absolutely have students who just need the experience of counting objects. Given 100s of items, what strategies do they implement to count? how do they keep their count?

My 8th grades are really struggling with squares and square roots, and cubes and cube roots. I would love to have a task for them to compare square roots and cube roots.
I think that even kids who can say the square root of 64 is greater than the square root of 49 because 64 is greater than 49, could benefit so much from the unpacking of that. How do you know? How can you prove it? That leads to students accessing the task at their level– some will draw or build models, some may jump right in with numbers, an indirect measurement.
If they were comparing a square root to a cube root, they might have that same direct comparison, but it would be really interesting for them to build/discover/discuss that when we are finding the square root and the cube root, we are looking for the single attribute of one dimension of an array or 3D model and that is what’s being compared. I’m so curious– what would they do if asked to compare the square root of 64 to the cube root of 64?
Completing this task leads nicely into the standard of being able to order and compare irrational numbers.

Direct comparison is visually comparing two quantities. Indirect comparison would be using an object to help you compare quantities.
As I was watching the video and being asked to pause and make estimates, I was thinking how my experiences influence my thinking– I knew the circumference of the glass would be much greater than the height of the glass because I’ve done similar experiments in the past.
What stands out to me is the lack of experiences that my students have (or rather, don’t have). So many things that I just take for granted that I did as a kid, i.e. counting coins from my dad’s change jar, or figuring out exactly how many tokens I could get for the arcade from how many dollars i had.
As students’ realities change, so do their experiences. It just drives home the importance of me, as their teacher, being able to recognize what primary experiences lead to developing and understanding concepts, and then how to make sure my students have access those experiences.

I love the idea of beginning with everything visually. How powerful it is for students to see that they’re dealing with units, which can be flexible, and it leads so beautifully from adding fractions to multiplying fractions, all with meaning and understanding.
Questions: how do I give my middle schoolers these experiences? how do I guide them through problems like this visually? how do I alleviate the internal hang ups from the years they’ve spent struggling with these concepts? how do i “undo” the tooquicktoabstract teaching they’ve been experiencing?

I, like many who have already commented, always thought of the unit as ONLY 1.
I am so excited with the idea of students creating their own units! I envision my kiddos counting their items using different units, and the great discussions that will ensue because students LOVE it when they can all have a different, yet correct, answer. How rich a conversation it will be too, when we talk about the fractional parts of their units!

In the classroom, I would start off simple with rectangular prisms, shoe boxes, game boxes, anything we could get our hands on. I would have students measure surface area and volume, using square inches and cubic inches (or square cm and cubic cm). I would have students make estimates first– how many 1 inch squares would cover the box? How many cubes would fit inside the box.

All of these concepts are developed through experiences. I think in middle school, we are too quick to put the manipulatives away, or assume that they played with blocks and Legos in elementary, they don’t need them any more. I teach in a high poverty school, where many of my students are lacking in childhood experiences that most of us take for granted.
I want to provide my students with experiences, then explicitly link them to mathematical thinking.

I keep coming back to this idea of EXPLICITLY teaching additive thinking vs. multiplicative thinking. It is all around us, and I think the more I point it out, not only will my students start seeing them, but it will bridge that ever present gap in students’ thinking: “Why do I have to learn this? When will I ever use this?”

A big “Ahha!” moment I’ve had is this idea of additive vs. multiplicative thinking in my students. They come to me with very shaky multiplication facts, and where as the teacher, the multiplicative relationships jump out at me, I spend many hours lamenting, “If only they knew their multiplication facts, these relationships would jump out at them too!”
What I’ve come to realize is not necessarily their multiplication facts (or lack thereof) holding them back, but rather they are thinking in absolute or additive terms. They are using the reasoning they know and are sure about, and trying to cram this new way of thinking into it– it doesn’t fit, they’re frustrated, I’m frustrated, it’s a hot mess.
What I am really excited about is explicitly teaching the idea of additive vs. multiplicative thinking. Of using tasks like the “Which shape is more purple” to have these discussions with students so they become more aware of where their thinking is and more aware of shifting their thinking to the multiplicative.

I am guilty of approaching proportional reasoning as just another topic/chapter to be covered. Granted, I try to make it as hands on and accessible as possible to my intervention students, but i didn’t realize the how important and connected proportional reasoning is to so many concepts in math.

This is a great example, and such a great way to be able to see where kids are at on the continuum!

I am in the same boat, teaching math intervention to 6th, 7th and 8th graders. One thing that jumps to mind is using models to build these situations– like the picture illustrated. The $400 amount = 4 $100 amounts. Build the model, then explicitly point out it’s 4x as much because there are 4 of them. Lots of building and drawing, lots of discussions.