The Concept Holding Your Students Back

Module 1  Introduction To Proportional Reasoning3 Lessons

Module 2  Understanding Measurement4 Lessons

Module 3  Understanding Comparison3 Lessons

Module 4  Spatial, Counting, and Additive Thinking4 Lessons

Module 5  Diving Into Multiplicative Thinking5 Lessons

Module 6  Ratios4 Lessons

Module 7  Sparking Curiosity With Ratios4 Lessons

Module 8  Proportional Relationships4 Lessons

Module 9  Putting It All Together5 Lessons
Lesson 3 – Proportional Reasoning Is Everywhere!
In this lesson, we are going to explicitly identify some of the many interconnected proportional reasoning concepts by building a proportional reasoning concept map.
This lesson will serve as a preview of the many interconnected ideas we will be learning about throughout the duration of this minicourse.
What You’ll Learn
 How many common concepts we teach mathematics are connected to proportional reasoning.
Action Item
Be sure to view the discussion prompt below and engage in a reflection based on the prompts. Sharing your reflection by replying to the discussion prompt is a great way to solidify your new learning and ensure that it sticks instead of washing away like footprints in the sand.
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Make Math Moments Academy › Forums › Lesson 3 – Proportional Reasoning Is Everywhere! – Discussion

Lesson 3 – Proportional Reasoning Is Everywhere! – Discussion
Posted by Jon on December 6, 2019 at 5:06 amWhat does proportional reasoning mean to you and why is it important?
Reflect on how proportional reasoning connects to your context by referencing the content, curriculum, and grade level of the students you teach.
Share your reflections and any wonders you still have.
Amy Rudd replied 3 months, 2 weeks ago 27 Members · 37 Replies 
37 Replies

As I go about my chores and renos around my home, this lesson reminds me how I think and solve problems in a relative nature than additive. For example, herbicides or radiator fluid concentrates to my seed and fertiliser spreader. I think teachers prefer “The Answer” in strict concrete undebated easy to mark abstract terms. While kids love pattern blocks, measuring liquids, and folding paper. I need to consider how my practice will better prepare students for “LIFE”.
By the way, can I get a copy of all my reflections by the end of the course? ie. keep in my own reflection journal?

These are great points. I’d argue that multiplicative thinking is more convenient in everyday life as well.. but many students get hung up in additive land. Hopefully this course will help us nudge students along that journey!
As for reflections, we’ve never exported comments before BUT I’m hoping there is a way. I’ll cc @jon on this too in order to see if he knows of a way?


These topics are really helpful in understanding how proportional reasoning is being developed throughout a students math education. Much like a doctor can diagnose nutrient deficiencies, this web makes it much easier to see how deficiencies in one area are going to impact a students learning in so many other areas. Take the simple concept of unitizing and you can see how if a student struggled to develop an understanding of this concept, they are likely going to struggle with so many of the others, but especially partitioning, scaling up and down, comparing quantities, …. oh wait…. its ALL of them! This is a helpful way to understand it all. Thanks!

Great comparison between a doctor diagnosing a problem and teachers trying to identify a gap. The web has certainly opened up my eyes!


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?”

Throughout the course we will have so many opportunities to better notice and name that multiplicative thinking – even from a fractional perspective which I now love to use (but never used to really understand).


Seeing the web of how proportional reasoning is connected within all those topics makes me want to sit down with the curriculum and explicitly find curriculum concepts that need proportional reasoning to build understanding.
Currently, I am focused in on my grade 9 math course. The first two units are Rational Number and Scale Factors & Similarities. Viewing the proportional reasoning web has shown that the gaps are coming from previous years and not building their proportional reasoning. If they struggle with the first unit of grade 9, they will most likely struggle with the second unit. My brain is starting to ponder how I can make changes to address the proportional reasoning gaps and help my students have greater success with the concepts.

This is so common in later grades. Many think this is just an elementary and middle school problem, however for our struggling students, this remains a problem – often for the rest of their lives.
I might recommend building in number talks at the beginning of your lessons. Mini lessons by Cathy Fosnot are great.


Deleted User
Deleted UserOctober 5, 2020 at 12:22 amWhen I think of proportional reasoning I usually just think of fractions, percents and ratios. The idea of scaling down and up makes complete sense because we do it all the time, especially when I try to help my little kids grasp the idea of really big numbers and measurements. The idea of building spatial reasoning by decomposing shapes and numbers makes complete sense to me now with how it relates to proportional reasoning. This is what we mean when we ask our kids to be flexible number thinkers (i.e. compose and decompose numbers) and then eventually unitize numbers or groups in different ways). I love this.

Glad you’re enjoying this and recognizing how vast proportional reasoning really is. You’re so right about trying to help our students become flexible thinkers. Only problem for me was that I wasn’t flexible enough with my own thinking that I couldn’t notice and name it in my class. Hopefully this course helps you like the learning has helped me along my own journey!


Thank you for bringing to light how interconnected this topic is.
When I taught younger students and we were in the ratios section of the math book I used examples of analogies with students who felt they understood language arts better than math. (Ex. Bird is to _____ as people are to house got them to quickly fill in nest.) We had a good discussion about how to read math problems and how “translating” using the comparative language could help them keep the proportions “in order”.

It is never too early to get students thinking comparatively and to begin using ratio language! Nice work!


As I stated in previous post, I realize that proportional reasoning is used in all are curriculum. The key is to make sure that I am able to demonstrate this to the students as well. From watching the videos, the one thing I do need to do more is use visuals to explain/demonstrate so that students can see the concept in the relative manner and not with the absolute mindset.

As a math teacher, I know that math builds upon itself and I think a lot of adults know this too. However, seeing how much 1 topic effects all these other aspects is a little mind blowing and definitely makes me happy I am here to learn how I can better help my students relate to the material we learn in middle school and beyond.

I really like the visual you presented here of the interconnectivity of proportionality. I have been working to reveal this to my students this year, which is why I jumped on this course. It’s helping me add ideas to bridge gaps and bring deeper conceptual understanding to my students.

If we think about the interconnected proportional reasoning concepts, we can spend more time to dig deeper into the concepts/standards, at the same time making students’ learning a fun experience. I think mad rush to follow pacing guides can take us away from the real teaching and learning. This thought process should make us think differently about our standards and pacing guides.

Completely agree. Our district created a pacing guide for the first time this year to help with the chaos of students moving between cohorts and I think it is causing more problems than benefits including teacher stress and possibly pressure to “push through” vs explore with intention. A guide is just that – if we are pushing through without learners learning, then we aren’t helping at all.


There is a lot to unpack and discuss in the WEB diagram. I appreciated the attempts to tie it in with child development, e.g., judging spatial relationships, etc.
This idea of Proportional Reasoning being everywhere is so important to use with the teachers and students I work with. The application to real world situations is so important, and many times more important now that we are learning remotely. There are so many examples of proportional reasoning used to help us solve problems as well – e.g. election polling models, tracking COVID data in terms of contacts, projected hospital beds needed, etc., etc. This modeling relies on relational thinking. It is an important decisionmaking tool in the world.

We will definitely be coming back to the web often throughout the course, so get ready for it 🙂


I think the map you used in this module shows just how layered proportional reasoning is and why the interconnectedness of math concepts is so deep. Currently, I am teaching an 8th grade algebra class, and three grade levels of intervention. I still wonder how I can help students make the connections to prior learning and keep them open to connect what they are learning in class with me (and other teachers) this year to lessons and to concepts they will learn in the future.

The struggle is real!
Have you tried any of our problem based math units from the tasks area?
learn.makemathmoments.com/tasks


Deleted User
Deleted UserSeptember 16, 2021 at 11:56 pmI am wondering where to start with students who are middle schoolers but don’t really understand some of the lower levels of the web. How do you go back to the basics without making them feel like this is baby stuff.

The truth that proportional reasoning is everywhere is one of the things that I know, but fail to fully realize when I’m trying to lead students into discussions. It is something that I really need to remember to be explicit about including throughout the day.

I have often thought that most of the math concepts has to do with proportional reasoning in some form or another. As the Math Interventionist for my school, I am excited to gain a deeper understanding of what all it entails and spend some quality time focusing on these concepts before sending these kids off to high school.

I love that web diagram. I just had a thought to connect scaling to daily life – maps, and googlemaps in particular, which the kids know well. A map of the neighbourhood around the school could have distances that we multiply to discover the real life distances – literally scaling up to reality!

One more thought: place value itself requires proportional thinking, and so many students don’t understand this basic principle of our number system because the multiplicative thinking underlying it escapes them. With my grade 3 and 4 students I have them compare objects, to find an object 10 times bigger and another 10 times smaller (What is 10 times smaller than a fridge? Maybe a kitty litter box?). They also measured their height and then shifted over the numbers in the place value boxes (aka moved the decimal point) to find a tenth of their height, then made little cardboard figures we called “tenth of me” figures. All this to help them develop a visceral, embodied understanding of 10 times more/10 times less.

So true and important that we help our students grapple with the multiplicative relationships in place value!
One thought to consider is when looking for 10 times bigger/smaller, consider switching the language to 1 tenth the size. This will also highlight the reciprocal relationship between “10 times” and “1 tenth” (much like the doubling / halving strategy that emerges in many number talk strings).


Watching this I am thinking about how we often rush to the algorithm instead of taking the time to really cement the important concepts. We need to spend more time actually letting them physically skip count objects before we take those away. I didn’t realize how early on we start teaching proportional reasoning.
I had also not made the connection that proportional reasoning will help with my teaching of rational numbers this Spring. I am excited to learn more!!

Yes! Proportional reasoning is happening so early and giving students opportunities to explore and inquire with these ideas will certainly help as more complex topics arise.


There is a lot that goes into proportions! It touches so many aspects of math. How do I build these in the middle grades when students have missed so much of these connections! I hope this course has some ideas.

In 4th grade as we leave additive reasoning and solidify our understanding of multiplication, the partitive and quotative models of division are especially challenging.
It is amazing to be in the room when a ratio table can be used to divide.. knowing where have come from and where my students will be going is very helpful and rewarding.

I am an intervention specialist for high school math. I typically coteach in gen ed classes where there are a multiple students with identified math needs. Then I work with the gen ed teacher and we develop lessons that will benefit everyone. I don’t know what grades I will be teaching this next school year, but I have worked with seniors in a class we call College and Career Math. I wonder if I will get a lot of ideas that I could use with seniors in high school to help them approach math with more understanding. I am particularly interested learning how to improve the concepts that will help with reasoning. Part of what we do is to review fractions, decimals and percents and teach compound and simple interest. I want to make it more about understanding and seeing how these concepts relate. Also, teaching them to check for the reasonableness of their answers.

I am a certified OGAP trainer for Additive Reasoning, Multiplicative Reasoning, and Fractional Reasoning and the resource graphic included in this lesson made me think of the “big ideas” that we guide participants to discover and disseminate. I love when they realize the interwoven nature of these ideas, along with the fact that so many of them are developed in tandem. I am looking forward to exploring the proportional reasoning concepts on a deeper level and unearthing their connections in development.

Until today, I typically thought of two equivalent ratios when I thought of proportional reasoning. After seeing there are many interconnected concepts that can be represented concretely/visually, I am excited to learn more so that I can support teachers and students when trying to make connections between what was learned before students got in their current grade level.

To me proportional reasoning is the gradual development of students’ efficiency with managing more than one quantity at the same time, by determining how they are related.

This web really shows the importance of proportional reasoning. Understanding fractions are also a huge key. I have been saying how many 1/8’s are in 1 whole for 1 divided by 1/8. The students reason better that way verses the algorithm. Scaling up and scaling down is another important part of all this. I want to review my 7th grade material this summer to see how I can better present this.

I love the visual representation of the web and how it connects all of these concepts. I had no idea how interconnected all of these concepts were to each otherI saw them more as stair steps than connected. I love this new way of thinking.
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