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Lesson 91: Consolidating What We've Learned – Discussion
Posted by Kyle Pearce on March 7, 2020 at 12:26 pmNow that we have formally defined a proportional relationship and shared some key concepts, how has your own understanding of proportional relationships evolved since before taking this course and now?
Share your thinking below.
Luke Albrecht replied 9 months, 1 week ago 14 Members · 21 Replies 
21 Replies

My number sense has definitely grown. There have been some “aha” moments and Math is Making More Sense than it did before!

This is amazing to hear! It is so empowering when we start to make our OWN connections of how mathematics develops. Glad you’re now well along that journey also!


How much a part of everyday life proportional relationships plays. In altering recipes, in determining area coverage per gallon of paint etc.

I have enjoyed gaining the language to help my students. I wasn’t taught with much of the language and we often jumped to abstract which I was fine with because I have always thrived and understood the concepts. It is great to see where the gaps are for my students so I can help them move forward.

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.

So fantastic to hear! Allowing strategies and models to emerge is so key to ensure students conceptualize what is really going on in order to get themselves out of a tough spot when in a pinch mathematically!


My own understanding has definitely grown, especially with understanding all the definitions and the difference between things like rate, rational reasoning and scaling in tandem. Also, since I don’t spend a lot of time teaching grade 5 and beyond, building my understanding of where kids are headed has helped me better understand how I can solidify their early understanding of proportional reasoning. I still need to break it down some more, especially while looking at standards and appropriate tasks in order to develop and build their understanding at appropriate rates of understanding.

Couldn’t agree more with the importance of having a good solid understanding of where students are headed. If we don’t have that, then it is so difficult to know what to focus on as we help students develop mathematically early on.


Having this class during this semester has been very helpful. I used a lot from these videos when forming my lessons and creating good discussion – just as compare and contrast.
The biggest thing that I have gained is using the ratio and rate relationships and when to use them and why. I can tell I still need more practice on implementing these concepts more into my lesson so that students have a better understanding of them – overall these two things are key. Also understanding these two different concepts have helped me understand how students might solve a problem or were they are having difficulty.

Glad you’re seeing the value through this course. Don’t beat yourself up though – it takes quite a lot of intentional reflection to start putting these ideas into practice and to make them a natural part of your lessons. Keep at it!


I have the language that I never had before. I loved the part about the flexible thinking. I know that me students tend to see it in all different ways. To have the language to explain the different ways of thinking is invaluable.

Agreed! Takes tons of time and effort, but they will develop it over time through repeated exposure by the teacher and through multiple varied opportunities to use that language.


Besides developing the language of ratios and rates I have become much more aware of how one concept leads to another and how they are all related. I also have changed my teaching to incorporate manipulatives, double number lines and lots of drawing….

So amazing to hear! Keep us updated on how things are progressing!


I have evolved from being able to compute proportions to seeing how the computations work and how these ideas of ratio thinking and rate reasoning build a conceptional understanding of proportionality. I would love to go back to high school algebra with this understanding. I think I would have not felt like I got lucky when I did well on a test. I never understood why students would tell me they guessed and just got lucky that it was right. I think this statement is that they haven’t had enough experiences to realize that they were using ideas that make sense without confidence but they are really on the way to truly understanding.
I think another aha that I have about these concepts is that they build from such early math experiences and it isn’t about being told a fact and memorizing it. I never really considered that these ideas build from counting up to multiplication. My own fraction sense and the idea of multiplying by a reciprocal is so much stronger just because I had experience with the concept concretely and with models. I now have models that I can use with my students.

Amazing epiphanies here! Awesome work and thanks for sharing them.


I don’t think I have every deeply thought about ratios before this. I just took them for granted. Now I see that they are linked to so many aspects of the “domains” of math that really, I don’t know how we can really separate them like we try to do! This definitely changes how I look at multiplication, at problem solving and at how to represent mathematical thinking to prepare students for the more complex concepts for which they are building the foundations.

I now have a clear understanding of the difference between a ratio and a rate. I love the idea of scaling in tandem. That language makes so much more sense than trying to tell 8th graders that it “varies” in terms of the other variable. I can see clearly now that proportional thinking seeps into almost every facet of my curriculum.
Also, I teach older middle school students but I had heard of additive and multiplicative thinking. Now, though, as I work with students, I can quickly identify when students are using additive thinking as opposed to multiplicative thinking.

These are huge take aways! Your ability to notice and name where students are at in their journey will make everyone’s lives easier – both teacher and the students! Go get ‘em!


I am definitely more comfortable teaching proportional relationships now. It’s such a deep topic and so connected to many parts of mathematics that I think I just intuitively ‘got’ it, but didn’t/couldn’t quite relay ‘how’ it worked. I now have a better grasp of how to guide my students to ‘get’ it too.

I think from the very beginning of the course, the language is so key. I really like the idea of composed unit. I teach 7th grade math, 8th grade math, and algebra. Using the composed unit really pushes students toward constant of proportionality, then to the rate of change and slope language in algebra. I like your Stanley quote that with rate reasoning “you won the problem”.
I also like the distinction between quotative and partitive division. I am not sure I ever though about this distinction before. Thinking deeply again about math I have been teaching for a while has been very engaging, fun, and it has extended my own understanding. I hope it makes be better at teaching!