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 1 – What Is Proportional Reasoning And Why Is It Important?
LESSON 1 VIDEO:
In lesson 1 of this course, we are going to be starting with an introduction to proportional relationships by exploring proportional reasoning. Together, we will learn together to gain a better understanding of what is proportional reasoning and why it is important.
What You’ll Learn
 What proportional reasoning is exactly; and,
 Why proportional reasoning matters.
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.
Make Math Moments Academy › Forums › Lesson 1 – What Is Proportional Reasoning And Why Is It Important? – Discussion

Lesson 1 – What Is Proportional Reasoning And Why Is It Important? – Discussion
Posted by Jon on December 6, 2019 at 5:01 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.
Kathy Wilcox replied 1 week, 1 day ago 58 Members · 89 Replies 
89 Replies

I teach 5th grade and proportional reasoning is a big part of what we do, albeit very concretely. According to our standards, the shift from additive thinking to multiplicative thinking begins in 3rd grade. I once was in a PD where the facilitator explained that the shift couldn’t happen before 3rd grade, really, because children aren’t cognitively ready for it. She pointed to Piaget’s literature to support this.
By 5th grade, it’s really obvious when a student is still thinking additively and not multiplicatively and it really holds kids back from some of the work we do. Sometimes I’m not entirely sure how to go back and take them through the shift, and I need to build my repertoire of experiences for early multiplicative thinking.

I also teach Grade 5 and I would agree that this is a big part of what we do, although I didn’t know it was called proportional reasoning either, as many have mentioned. My wonder is if it is necessary for children to master additive thinking before moving on to proportional reasoning. Can they work on both at the same time?
I also see students beginning to see proportions without realizing that they are. This makes me think that perhaps it comes with exploring our world in a tangible way. Most of my students (if not all) would be able to physically demonstrate that to make hot chocolate they would need to keep adding groups of 3 scoops to each person’s mug although they may not be able to articulate the mathematical relationship without guidance or translate it into an equation.I also realize after having watched the three videos that I have already been teaching students proportional reasoning without realizing it. It feels good to think that I’ll be doing it more intentionally even starting on Monday, but especially as I move through this course. I’m excited to see what I will be able to take away for myself and then be able to pass on to my students. 🙂
 This reply was modified 1 year, 8 months ago by Merrillee Reboullet.


I also teach 5th grade and agree with Tania, that proportional reasoning is a big part of what we do (although very concretely as Tania mentioned). The move to abstract is a difficult jump for some kids. Can we just say fractions???? The idea of a fraction being a ratio is mind blowing to many students and teachers.
 This reply was modified 3 years ago by Hayley Anderson.

As a middle school teacher, another thing that blows students’ minds is the fact that fractions mean division. And teaching how to multiply with fractions, especially a fraction and a whole number is pretty hard, too.

After all the years I have been teaching Math, I never really knew what proportional reasoning meant. Of course, I taught multiplicative thinking but never heard it called proportional reasoning…. Oh so many learning gaps to fill. But that’s why I am here!!

This will be my 25th year of teacher – 18 of those years have been in First grade but I’ll be looping with my class to virtually teach 2nd grade for this new school year. What I’ve seen is that sometimes, children just seem to “get it” when you ask for things like “equal groups” or comparing sets and others really seem to struggle. We work with hands on manipulatives frequently and still there is often difficulty learning these concepts.
As I listen to the first session, I realize (or rediscover) the importance of intentional use of language to help give words to students’ thinking. It’s all a journey and I’m looking forward to learning how to help my students develop deeper mathematical thinking as a result of this course! Thanks for reminding me that it was available!

It is important as it serves as a foundational piece in math to help students understand percentage etc. due to its multiplicative nature. As it is essential I am wondering why textbook that cost so much to develop and involve educators are not addressing this.

I have had the same question! Many textbooks don’t see the many concepts connected to proportional reasoning as being connected and therefore they are presented in siloed, disconnected ways. Hopefully this course will help!

I agree! The order text books go in doesn’t seem to make the most sense to enable students to be able to draw the connections. I am looking forward to having a course that will help teach me how to ensure that the leaning is able to build on itself and not feel disjointed.


I teach Algebra for mostly Freshman in High School. This is only my second year of teaching. Watching this video made me realize how many of my students didn’t master this concept. And now I have a work to call it and can start addressing it with them. Potential issues that I am facing: I need to follow the district curriculum, which leaves little room for “extras”. The curriculum doesn’t help my students “catchup” or recover from missconceptions they have accumulated over the years. And my students have a low math stamina so I have been looking to change my approach with them. I want them to see math as a tool to better their lives and not just a class to get their HS degree Achievable?

Definitely a challenge when you’re working from a pacing guide of jam packed curriculum. However I wonder if instead of planning to address THOSE ideas, if you use those ideas as a means to get to the ideas in proportionality and number sense?


I teach math 8. Math 7 is all about proportional reasoning, and I usually try to reference that and build on it to tie it in to linear relationships which is the focus of 8th grade math. In past years I didn’t do much reviewing of proportions, assuming that they were coming out of a full year of studying proportional relationships. A few years ago we decided to focus more on those connections.
I also have made the mistake of thinking my honors students wouldn’t have those gaps like my lab classes, but I’m finding they have many of those same gaps, but they’re just better at covering it up.
I definitely recognize gaps in all levels of learning and hopefully this will help me find ways to remedy that.

So true! We often “assume” students come out from the previous class having learned it all! Continually stretching backwards while reaching forwards is key.


I teach 6th through 8th graders including an Algebra class, so I really see the development of proportional reasoning through the grade band in into HS math. It is really fundamental to their understand of slope in Algebra.

So true! I’m sure you have some students who struggle with their multiplicative thinking and maybe even skip counting with certain values, too?


I am guilty of moving too quickly to abstract representations with classes. I need to recognize where kids are developmentally and lead them more gradually.

In the past, teaching 3rd grade, I would teach my students to approach proportional situations with the “make a chart or table” problem solving strategy. It is only recently that I am truly understanding how abstract in nature tables and number lines are to early learners. Because students can learn to identify patterns in such tables and make predictable outcomes, I made the assumption that this was it.
So I am wondering: what concrete and representational models should my students experience before the tables.

So true Nicole. It also looks like you have some “look fors” as you progress through the course.

Nicole, when you mention how young students are adept to finding patterns, this really holds true for me as an educator who works with 58 year olds. They are good at seeing patterns and it is important for us to give them the language to describe what they see. Introducing them to the idea of “half” or “twice as more as” or “double” can help develop the building blocks of proportional reasoning or multiplicative thinking.

And the best part about our younger learners is that they are much more ready to explore complex concepts than we often believe. Building that fractional and multiplicative language with them is so helpful as they move down the roadmap!



I am a high school math teacher in Alberta and have taught/teaching a math course that has very heavy use of proportional reasoning. From conversions between systems of measurement to income and currency, this course builds on previous exposure to proportional thinking that I need to work on unpacking more before instruction. I need to be more cognizant of starting with concrete thought before jumping to the abstract.
I know I need to work on reflecting upon student knowledge and setting up mindful scenarios that will help unleash prior student knowledge.

We’re so glad you’re here with us Gordon! Looks like you’ve been reflecting on your practice. Good for you.


I’ve been reading about proportional reasoning and wondering WHY, if it’s so foundational to so many other branches of mathematics and actually science too …. WHY isn’t it part of a math teachers education, or part of more math PD? Thank goodness for this class!

Such a great wonder. Sadly, I don’t see too many preservice programs that have solid math foundational courses at all – let alone proportional reasoning. They definitely should, though!


Proportional reasoning means that a student is flexible with numbers. When a student can break down numbers easily and see them in different ways, it helps the student solve problems with a variety of strategies and methods. Students need to have a toolkit of strategies to solve problems that they can quickly use.

Proportional reasoning means that students can break down numbers and manipulate them easily to get the desired result. I am looking forward to this course to help me with how I present things to students. I was raised with algorithms and I find it difficult to reason concretely and sometimes abstractly.

During the past year and a half, I’ve been engaged in writing math strategies for an education firm. One of the major areas of this work has been researching and compiling effective strategies for building conceptual understanding of fractions from K5. Through researching the progressions of fractional reasoning my go to references were, Battista, Steffe, and Olive. They identified six levels of fractional reasoning or trajectories of mathematical thinking required to develop the ability to reason proportionally. Generally, third grade marks the introduction to multiplication and fractions. Later in the elementary grades, students need to begin developing strategies for understanding fractions, decimals and percents by using manipulatives, drawings and diagrams. Division is the most important concept related to fractions, decimals and percents and all these math concepts require proportional
reasoning.Through the elementary years, students begin experiencing a shift in mathematics concepts from additive to multiplicative situations. Although multiplicative concepts are initially difficult for students to
comprehend, a—
mathematics curriculum must not wait …
to advance multiplicative concepts, such as
ratio and proportion. These principles must
be introduced early when considering additive situations. (Post et al. 1993) This was taken from: Tobias, Jennifer M., Andreason, Janet B., Developing Multiplicative Thinking from Additive Reasoning. “Teaching Children Mathematics,” September, 2013, Vol.2. Issue 2. pp. 1029
Love it – what a great share!
So true about not “waiting” but rather giving students opportunities to reason multiplicatively to encourage that shift from additive to multiplicative thinking.


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.

I’ve never reflected on what proportional reasoning encompasses in relation to mathematics besides often being a standalone unit in specific grades. As a reflect on my own proportional reasoning abilities, I don’t call a time when where I struggled with the multiplicative thinking and was always comfortable with exploring numbers and relationships between them.
As I reflect through my teaching eyes, I see so many students who are afraid to use multiplicative thinking in the math. They get focused on only one way to see it rather than exploring other possibilities and being comfortable with flexibility that can come with numbers.
When I have hit a time crunch with delivering the curriculum to my students (especially my grade 8s) I have asked other teachers which of the remaining units would be most beneficial to our students, and more times that I want to admit, they have stated the standalone unit of rates, ratios, and proportional reasoning is a unit that we can skip. I’m know questioning, why they are skipping this as we know our students struggle with the concepts in the following grade that rely on proportional reasoning.
As I move forward with this course, I will be reflecting on the way I want to deliver the my math courses to my students during the next quarters this year.

Super interesting and very common challenges regarding the time crunch! I’m curious to hear what you think after we wrap up the course to see if your thinking shifts on whether skipping the ratios, rates, and percentages unit is helpful.


I teach 6th grade math, but I have taught math as either a classroom teacher or interventionist in every grade kinder – 8th. Proportional reasoning is the ability to see multiplicative relationships in the world around us. It is important because it can be found everywhere so not understanding it makes a person miss out on understanding so very much of that world. When I was a junior high interventionist, I spent a great deal of time working with students on proportions because it is so highly tested. I had a percent wheel poster and each day for 100 days, students would take turns coloring in a slice of the percent wheel and our warmup was all about the percent of the day…kind of like a number talk. It was an 8th grade class and one of my favorite moments as as teacher was when a student came back from Spring Break all excited to tell me she knew what the sale prices would be on the percent off sale racks when she and her mom went shopping. We spent every day for 100 days on the topic they were taught in 6th and 7th grade and still that’s what it took for the conceptual understanding to come to some of them. Another thing I will mention is that when I am a classroom teacher, as I am this year, I have to be careful not to fall to the pressure of test preparation and make sure I spend enough time on conceptual understanding.

Very interesting reflection. Love how you take so much time over the year to ensure that students are developing that understanding necessary. These are concepts that will last a lifetime and truly will hold students back from more abstract concepts if they don’t get a good handle on them.

Susan,
Thank you for sharing your percent of the day strategy. I teach 5th grade and have seen positive results from number of the day talks. I am excited to try percent of the day…what a great idea and quick way to incorporate proportional reasoning on a daily basis!
I agree with so much of what others have shared, especially about the importance of giving time for students to deepen their conceptual understanding as they look for patterns and connections. Flexible thinking is so important. I still have students who really struggle with this. I am looking forward to learning more about how to help my students through this course!


Proportional reasoning is very important in the 8th grade curriculum when we discuss slope. It is important for students to understand and I can’t wait to learn more ways to have students understand this concept more through this course.

So excited that YOU’RE excited! Keep on going 🙂


Proportional thinking is used all the time in our 8th grade curriculum. There is slope, scale factor, linear relationships, scientific notation, and Pythag. What I am realizing is that I need to resort to this concept more when teaching students these concepts because they might seeing better and understand the concept better.
From watching the video, I also realized that maybe I need to show situations broken up in a proportional manner more often.

I’ve taught 712th grade math. Proportional reasoning comes up in all of it. The problem I have encountered is when students have been taught to mathematize situations too early. They tend to try to throw things into a proportion or an equation, but they don’t really know why, or how it works. Usually, their instincts are much more advanced than their technical skills, but they tend to ignore them instead of leaning into them. I’m hoping to learn more about how to get students to rely on their own intuition and to get them tapping into the world of proportions that they know and understand.

I’m looking forward to developing a better understanding of proportions. Sometimes I take for granted how I, as a person who enjoys math, see the relationships with numbers. Even when we were asked to count the blocks, my instinct was to group them by 2s. I wonder how many of my 7th graders would do the same or would they just count 1 at a time. I want my kids to realize that just because they are not “math people”, that they can learn to recognize the same things that I sometimes take for granted to help them become better problem solvers.

We just assessed our students’ learning on ratios and rates last week. This connects directly to the proportional reasoning. When we teach our 6th graders Unit Rates we are teaching them that proportional reasoning needed to find different equivalent rates.
I use the word “multiplicatively” constantly as I teach my students how to use ratio tables and create equivalent ratios.
Also we do number talks in which we are looking how to group different numbers together. I didn’t realize that would be proportional reasoning too. How cool.

It’s so awesome once we are able to better notice and name proportional reasoning… it also helps us in our planning to help craft better opportunities for learning through problem based lessons as well. Have you checked out the Hot Chocolate Unit in the tasks area? Its a doozie!


It represents the opportunity for the students to compare two or mor quantities using multiplications.

Love it!
However, I’d argue that proportional reasoning is much more broad than that as there are so many skills required to “own” before multiplicative thinking comes into play.
As you head through this course, you’ll see what I mean by this! (don’t worry, this is all very new for us as well!)


Proportional reasoning to me starts with comparing two items. I think about the reallife context when I am baking or cooking anything in general. It starts with basic concept, but in teaching, it spirals through many concepts such as ratios, comparison with fractions, unit rates, scale factor, and slope. I believe if we start teaching without building the conceptual understanding in context, it can be mind boggling.

I am an instructional coach, working mostly with K5, but some with 68.
Proportional reasoning is all about relationships between quantity/magnitude. I see it come to life in realworld scenarios that are interesting to adolescents — in art, fashion, music, etc. It is such an important shift in mathematics — going from “how many” in lower elementary grades, to “how much….(Larger/smaller)” in upper elementary. I remember doing a golden ration activity with teachers in a summer course I was facilitating, and the adults really got into measuring and figuring out what relationship there was between different parts of their bodies. As I watched the group, it reminded me of those times when younger students are curious about exploring the world of mathematics. It was a joyful moment and we need more of those in our math teaching and learning.

So glad that you’re diving in and seeing the value in learning more deeply about proportional reasoning concepts!


When I first counted the black squares in the video I saw them as 2 groups of 3 and one group of 2. I thought it was so cool to see that even the example of 2 groups of 4 or 1 more than 7 was what was given as the example. I love that about math and the way we can interact with it.
Proportional reasoning to me is one way that I enjoy interacting with math. To me, it is important to use proportional reasoning to make some problems more clear for me and my students to understand.

Love it. Having multiple perspectives in math is so important for making it accessible for all!


Proportional Reasoning is about being able to describe relationships between two or more things. It’s the ability to see groups of items in various ways and to see the connections between two correlated parts (distance & time). It is number sense! There is more to number sense than proportional reasoning but that is a big part of it… making sense of the numbers.

Love it. And you’re right… proportional reasoning is the backbone for building number sense and flexibility. Enjoy the rest of the module!


Proportional reasoning is critical in 8th grade. When we talk about linear function it is important for the students to understand that the straight line being grade is happening at a constant rate and of course if we start at the origin then we have a true proportional statement. I find that students struggle with linear functions when their understanding of proportional reasoning has not been fully developed.

I am beginning to understand that proportional reasoning is the ‘deliberate’ use of mathematics to help us find solutions to problems/questions. I see its importance as relevant throughout most/all areas of study in math and I am finally noticing that how I use it intuitively is because of my experiences throughout life and that I need to figure out how to pass this on to my students.

I teach special education and 6th grade regular math. I see huge holes in proportional reasoning in my special ed students to where they truly do not understand even basic concepts such as what a fraction even means. My sixth graders are all over the board in their understanding but even some of the top students have gaps. As with the fifth grade comments the sixth grade curriculum is based on large quantities of proportional reasoning,.

Hi Anne, I have students in 9th grade who struggle with the meaning of fractions and are weak at proportional reasoning! It’s such an important topic that we need to address and build foundations for in every grade.


Proportional reasoning is two variables that are interconnected they are changing in tandem.
I teach 7th grade math in the US so almost everything we do is linked to proportions! I hope that I can learn ways to support my diverse learners struggling through proportional reasoning tasks!

Proportional Reasoning is a foundational skill in multiplicative reasoning. It is a way to compare two or more things mathematically.

Proportional reasoning is what helps me to know whether I’m on track to meet my goals when I’m working out, driving somewhere, working to meet a deadline, or any other number of situations in my life.
I try to help my students make those types of connections as well. I teach 7th grade. My standards require that I teach financial contexts such as tax, tip, markup, commission, raise, bonus, and discount. Some of these seem like a bit of a stretch since they have so little experience using anything other than a debit card in financial transactions.
I wonder what contexts are out there that would be more relevant to my 7th graders.

Great points – we often are teaching concepts that will eventually be relevant in life, but if they aren’t relevant now, that can be a tough sell. Throughout the course you’ll see some contexts that are relevant to students – some that are relevant and useful, others that are relevant and not so useful. The key is that students can understand the context snd make comparisons from there.


I teach Grade 6 Math in Massachusetts, and in my mind, I think of proportional reasoning as a kind of cause and effect of two different variables.
We obviously deal with a lot of proportional reasoning in sixth grade, but it’s been a struggle for me to help the students grasp a hold of it on a deeper level. I feel like students have come to me with a very rote understanding of the previous material and I’m sure I’m guilty of just rushing to an algorithm as well. I’m here to try and learn how to reverse those trends and try to create deeper thinking math students.

Love it. You’ve come to the right place. We have all been there and actually I think we often believe that rushing to the algorithm is the most effective and efficient way. The result might be students able to solve familiar problems, but often times many do not build the problem solving skills and conceptual understanding to work through difficult problems. Excited to learn alongside you!


As an early years teacher, I see how important it is for students to make connections when they decompose numbers and begin to recognize, just as how the video explained, you can see numbers as groups of numbers or multiples of numbers rather than just the oneness. As they progress, soon they are able to recognize doubling and halving. But I think this is best learned when the students are able to manipulate objects concretely and make the connections when they are asked to articulate a response to “How do you know?” This translates into our everyday lives where students can transfer their understanding. So proportional reasoning is being able to take the relationships of numbers (whole and parts) and apply them to either compare or predict.

Great realization! I often think about how much math we “leave on the table” with early years learners. We tend to limit their math experiences to counting and only sometimes work in some additive thinking. I’m now using small quantities to talk fractionally and multiplicatively. They are so capable of thinking of complex mathematics if we help nudge them there.


Second time I am posting this answer, apologies for duplication.
Proportional reasoning is being able to break apart a number into groups where students can see sets and make sense of skip counting instead of oneness. Then they are able to transfer this knowledge into doubling and halving. This is important when we see the students apply this knowledge into real world contexts and be able to predict or extend based on these concepts that are best developed concretely – I am obviously an early years teacher!

Proportional reasoning means that students are making connections of real world situations or tings that are linked with numbers. For example, the amount of red dye to blue dye when creating a particular shade of purple is an art real world idea that is mathematical. Students often think math doesn’t exist in the natural world but proportions are so natural that they miss the trees because they only see the forest.
I think as a sixth grade math teacher that it is important to inspire students to see the math in the world around them. It isn’t just abstract but it is generated from the world in which they live. This exploration could help students to engage in math and persist; not working persistently just to learn to work, but working with an ultimate purpose in an area of interest.

This is so true! Grade 6 is a key year for proportional reasoning as we move from additive to multiplicative thinking. Making it relevant and connected to contextual situations is so helpful for building understanding.


As someone who teaches both 7th and 8th grade, I see proportional reasoning as a key concept towards understanding linear relationships and linear growth. Our standards and curricula constantly split up so many different pieces of proportionality in 7th grade, that students don’t see the connection between them. One example is that working with unit rates, percents, and proportions is all the exact same. I started using double number lines for all of these last year with students — all of a sudden, something that had previously felt so disconnected now was completely linked. I also think that the vocabulary we see can further encourage disconnectedness — unit rate, constant of proportionality, constant rate of change — these are all essentially the same piece. Why confuse students and make them think it’s all separate?

So true! Plus, as you’ll learn later in this course, a “unit rate” isn’t even really a thing. A rate is a rate is a rate … so much more learning fun to be had here! 🙂


I teach grades 3 through 6. And I notice that many of my students are held back because they are still thinking additively. They need to be capable of understanding abstract thought to reason multiplicatively. I’m very interested to learn about the early stages of this development, and what experiences help set the stage for proportional reasoning to develop.

I currently teach 5th grade math and have taught 7th grade math & PreAlgebra. Proportional reasoning means looking for constant relationships between numbers. It is very important at the 5th grade level in understanding equivalent fractions. I have started using rate tables with my 5th graders to help them with multiplication and division. Looking for those relationships…such as 1/2 as much, 4 x as much, 10 x, 1/10 of…has helped them in their understanding of some of their math facts and working with larger numbers.

Such important work. So glad to hear you recognize this and implement strategies to help students build their understanding and flexibility.


I’ve been teaching for 19 years but am back in Math after a 4 year stint in Reading. I love the predictability in math. There are patterns everywhere and so many different ways to see the same thing. I’m excited to make more connections so that I can help my students can see math in new ways. Proportional reasoning to me is helping our students see those patterns that seem to be hiding in plain sight.

Thanks for sharing Eric! Looking forward to joining you on your journey.


Last year I taught Math 8 for the first time and I planned on spending one day reviewing proportions before we jumped into slope, thinking that students had covered that topic in 7th grade. As my students worked on proportions, however, I realized that their understanding was pretty limited, which is not surprising because our curriculum hits ratios in a standalone, quick unit at the end of the school year.
Without an understanding of proportional reasoning, I knew my students would struggle with slope, so I spent as much time as I could teaching this concept, but I didn’t have much wiggle room in my curriculum to do it justice. When I saw this course advertised, I was excited because I feel that it will help me to teach this concept more effectively with the time I have. Already I can see that it will help not only with our understanding of slope but also our whole unit on transformations will benefit as well. I am already developing a broader definition of proportional reasoning. Thank you!

I teach 8th grade special ed (PreAlgebra), my students really struggle to make connections. I feel like I am constantly filling in the gaps. I’m excited to learn with everyone!

Would love to dig deeper on this. Have you tried any of our problem based units?

Not yet, I way behind everyone else in the modules. 🙁



Proportional reasoning is everywhere and is made easily available to students early on, so they don’t have to be “retaught” the concept later.

Agreed! Constantly incorporating it into our math lessons and lives can deepen understanding so it sticks and can be leveraged routinely!


I’m a math coach K12 and I see it beginning very early on, if it’s “caught” early on. I find it harder to reteach later on, or almost unteach it.
I see it everywhere, but I also find that it needs to be pointed out, and kids need to have practice with finding it, identifying it, and pointing it out.

As a 6th grade teacher my first thought is ratio’s and rates. It makes sense that is multiplicative thinking though since it really is all about groups. Most of my kids, especially after the remote learning year, have not mad the switch from additive to multiplicative thinking, so I am wondering how to help them make that switch this year.

It is a long process. With intentionality and constantly nudging towards multiplicative thinking, students will eventually join you on that ride! Many of our problem based lessons can help with this!


I am a francophone teacher. I taught grade 8 for a few years, this year I am teaching math in grade 7. Due to the language differences, I sometimes get mixed up in my terminology, but will definitely learn on the way (that’s my disclaimer). As for proportional reasoning, I am now better understanding the term. I observed that my students had a hard time in grade 8 and for that reason I began looking into new ways of teaching proportional reasoning last year, from rates to ratios to fractions, and found you at the OAME conference in may. My students are already enjoying your tasks and I have observed positive results (and attitudes) from all students, those who seem to get it and those who are on their way to understanding. I can’t wait to get deeper into your course and see my students reap the benefits. Thank you.

I teach 7th and 8th graders mathematics. I like saying that proportional reasoning is multiplicative. We just finished scale factor and the idea of a scale factor of 1/2 was the place where students struggles. We kept making models to show that multiplying by 1/2 is the same as dividing by 2.

I have taught 7,th, 8th and 9th math for students with IEP’s. As a special educator I don’t have a license in math, but have taken math courses. However none of the courses have dived into proportions. I have always taught the math based on the curriculum and just taught in “silo’s.” I have wanted for myself to have a deeper understanding of proportions. It has been one of my least favorite concepts to teach. My mind is expanding by listening to the Making Math Moments Podcas and the ideas presented there have increased my interest in this course. I am looking forward to learning how I can teach this concept better and help my students have a deeper understanding of a topic that is a main player in middle school math.

This is my first year teaching middle school math. I see progression and the need of proportional reasoning in grade 6. I started teaching students partitive and quotative divisions and I see the benefit of being able to remind them find the rate and using rate in different types of questions are not new stuff for them. Some 5th grade teachers here mentioned about teaching the proportional reasoning at a younger age. I wonder what age is the earliest that we can teach proportional reasoning to. I mean, my son is 6 years old and he seems to understand 2 groups of 4s and 4 groups of 2s. He seems to be able to tell me 3 groups of 3s or 3 groups of 2s and so on. I wonder how much he does retain and understand and be able to apply in his math learning.

I teach 4th grade and are in the middle of solidifying our switch from additive reasoning to multiplicative reasoning.
So much of what we do, even as far back in the year as learning facts is really introducing proportional reasoning. 6×7 is really (3×7)x2 and all the 4’s facts are doubles of the 2’s facts, which are doubles themselves, and so on for the eights, double the double the doubles. The fives facts are half of the tens. And so so so many more.
As we move into 2 digit by 1 digit multiplication and beyond, we apply these understandings to avoid rote algorithm use. Ratio tables with doubling, half of ten or hundred groups, and explorations of the area model all utilize proportional reasoning

I teach 8th grade and Algebra 1. Proportional reasoning is still used in rates of change of each function we cover. I am excited to see how else proportional reasoning might strenghthen how my students think about functions.

I am a SPED coteacher in high school math. I help all of the students in the class and share in the teaching. I can see this helping with graphing lines, similar triangles, trig, percents, rates, and more. I see a general lack of reasoning in students. I hope next year to take my students and teachers to a new level of “thinking” and “understanding” with what I learn in this course.