Make Math Moments Academy › Forums › Full Workshop Reflections › Module 3: Teaching Through Problem Solving to Build Grit and Perseverance › Lesson 36: Tool #4 – Student Generated Solutions › Lesson 36: Discussion Prompt

Lesson 36: Discussion Prompt
Posted by Kyle Pearce on May 1, 2019 at 11:52 amWhich tool(s) from this module (concrete manipulatives, multiple representations, visuals, and student generated solutions) do you found to be most useful to make the math more accessible for your students?
Which will you try next week and how will you go about using it?
Joseph Barnas replied 1 day, 14 hours ago 33 Members · 38 Replies 
38 Replies

Fun, you showed 3 different ways of seeing the problem. My first one was different than all of them. I saw it as a nxn square on the right Plus a 1xn on the left and a 1x(n1) on the bottom. So: n^2+n+n1
This was fun!
 This reply was modified 3 years ago by Kyle Pearce.

For 2nd graders, I find that using concrete manipulatives really helps to solidify their understanding. I’ve used 1s, 10s, 100s, and 1,000s blocks to help not only with place value but with addition and subtraction with regrouping. Students who had the steps memorized made the connection. Students who had no clue had aha moments.
I also try to use multiple representations to help solidify understanding. For instance, most of my students can write the number 367 in expanded form, 300 + 60+7. However, when they are presented with a different format, they often fall apart. Providing multiple ways to ask questions and multiple ways for them to respond helps them to have a stronger grasp on numbers.

I used concrete manipulatives with students to help them sort things by color or shape. These are students who are special ed, and are probably at a kindergarten level. I used manipulatives that have Velcro on each piece to match colors and shapes with appropriate piece. I have also used other objects, such as colored shapes (like erasers) that can be sorted by color or type of shape.

I really like multiple representations mixed with studentgenerated responses (they go quite handinhand with each other), since they allow for students to dive into their own reasoning (low entry point and high ceiling). The 3 Act Tasks facilitate this kind of learning well. I’m trying one this week with cup stacking that can allow students to show their own solutions (as an intro to systems).

I find that the student generated are the most interesting for me as a teacher. For the sake of the students, I feel like using multiple respresentations is essential to make connections with the most students. Ideally, we should be using all 4 of these tools as much as possible.

I like the idea of using visuals. I teach secondary math, so many of the ideas are extremely abstract for students, and traditionally are taught that way. The task asking how many small squares are needed to go around the circle is awesome and makes pi more accessible.
As I think about it more, I guess these are tied together; manipulatives such as algebra tiles provide visual representations of abstract concepts, and the studentgenerated responses develop student understanding.

I am using concrete manipulatives this week to celebrate Pi Day. My students are “discovering” Pi using string and objects (like Jon did in the video), modelling the area of a circle using paper plates and forming into a “rectangle” and exploring surface area using mandarin orange peels. There have been lots of Aha moments. Next week I hope to use pattern blocks to help my students understand the unit circle because they always have trouble with half the circle being one pi so I hope to use an activity similar to the one shown with pattern blocks.

I reached out to some of my 6th grade teachers this week so we can collaborate and coteach a lesson. I’ve come out with amazing ideas from this module. Thank you! I’ll come back to this and share how it went.
PS. your 3.7 summary module is out of place, it showed up right after the 3.3 lesson, fyi.

Here’s a problem from a recent assessment where I used some student work to start a discussion about problem solving. The students both had similar strategies involving using subtraction to find irregular area on the coordinate plane. We can discuss efficiency (left student did a lot of extra work), connecting different representations (both students used labels pretty well), and areas where they could add strength to their solutions (confirming the answer in a different way).
I put together student work like this after assessments rather than going through the problems myself or providing answers myself, so that students have models for effective strategies. It’s a pretty effective way to show students expectations, while also showing them that they can do this, and don’t necessarily need a teacher to tell them how all the time.
 This reply was modified 8 months, 3 weeks ago by Jonathan Lind.

This week I used your sunflower model on how to solve two step equations. For the last few years I have been trying to teach my students with abstract reasoning. For this lesson I had manipulatives available to my students to come up with an expression. Most students gravitated to them almost immediately and started to use them to reason about the problem ( even though afterwards they enjoyed building other things). A physical model was so easy for them to build. From there we shared what visual models would match and then connected it to the abstract (maybe a little too soon?). It was a great start. I was a little nervous, because new things don’t always go as well as things you have done over and over. I knew by starting that this would get me to a better place as a teacher for my students, so I went for it.

I am going to use the visual of the paper and two stacks of paper and see what student generated responses I receive. I think it will provide some interesting results.
I just tried this, and only one student out of 17 did this correctly. What was interesting was that he did not think he did it correctly, and was unsure how to explain his answer. I had another student get the correct answer, but she did it by starting off estimating instead of actually doing math. She divided the 142.75 cm by 2, and then rounded that up to 72, then figured out that each pack of paper was 6 cm. While she got the right answer, and could explain her work quite well, she could have just as easily gotten a wrong answer. I encouraged her use of estimation, but also explained why it could have been wrong.
 This reply was modified 8 months ago by Terry Hill.

I don’t necessarily think that any tool is most useful for my MS students, it’s more about which suits the topic. For my G7 when first learning equations did an activity called bags and blocks (using physical blocks and bags) where the blocks represent a number/ constant and the bag a variable b/c we can’t see how many blocks are inside. Students represented equations with the manipulative and tried to find how many blocks were in each bag. The cool thing is I no longer have to teach equations with variables on both sides as a separate topic!
For my G8, last week we did speed eating hotdog Kaplinsky task to get student generated answers. We used those answers to discuss multiple representations of linear functions.

Totally agree. Being ready to use the tools as they are useful is the key!
Glad to hear you’re making some noise with problem based learning!


I have 2 questions from this module:
1. Sometimes when we use student generated solutions in a whole class discussion (our alt classes are really small), there isn’t yet enough stamina/resilience in students to carry them through seeing another person’s approach. What are your thoughts on this? We will keep encouraging students to see the benefits of multiple perspectives…
2. Would you say that “consolidation” is really about deepening students’ own thinking by connecting that thinking to a mathematical model? (double bar/line model, table of values, equation, graph, etc?) Is that a direct part of the consolidation phase?

These are great questions.
When consolidating, it can be helpful to facilitate in a way that asks students to share very specific pieces or parts of their work to avoid each group running too long. Also, every group sharing might not be necessary – especially if some approaches are very similar. You can let one share then also ask the groups “whose approaches were similar?” Then move onto another.
Also, you can consider – if you feel the thinking / focus is spent for the day, consolidating can take place at the beginning of the next day.
For what to highlight – that depends specifically on your intentionality of the lesson. The consolidation is an opportunity to drive some of those ideas home be it a big idea, strategy and/or model. From what you see during the lesson, you can decide what the next day will look like. Do we dive deeper? Did they “get it”? Did they need more practice with a model explicitly? These are all things to consider.


Thanks. Both responses are very helpful. I like the idea of sharing very specific parts. Also, thanks for the clarification re: consolidation… to drive home the point – whether model, big idea etc. Thanks!

I teach G5 so visuals and manipulatives are staple, to a lesser extent different representations and less so student generated responses. I am going to try incorporate more representations. My next topic is fraction of an amount. The context will be from Kaplinskys “”what fraction of children are in the right car seat”. I could ask them for a visual representation and see what they come up and show them others’. I can give them linking cubes to show it too, they’ll all choose those for sure!

Because I teach primarily students with IEPs, concrete manipulatives are often my goto when building basic understanding (for example, understanding that four 1/4 tiles equal one whole). I loved seeing how these manipulatives can be used to build understanding of much more complex concepts as well, and I’m excited to try that in my class this upcoming school year.
I also love the idea of using studentgenerated solutions to guide students toward understanding abstract formulas, and then “mapping” future contextless problems onto the problems with context.
Thanks for a great module!

I love these tools. I think I am going to try and make the height of the table problem into a system of equations problem.
I think I will make a video of students, or me, walking into a room and stacking the paper. I think I would use different height reams of paper. I am not sure if this would create a system, but I am going to work on it right now.

It is summer break so I don’t have a class to try this on but I would have chosen Student Generated Solutions. I would achieve this by having 9 groups of 3 work on vertical white boards on a thinking problem and then have each group share out what they did. I did a version of this in Precalculus last month and had them write a trig function for a sinusoidal graph that I gave them containing minimal information. First they did it alone, then with small group, and then with half the class. 2 people then shared out the groups solutions and how there were multiple ways to write the equations using different transformations. It was powerful to hear from their peers and work together to get a solution.

I like the manipulatives and multiple representations. These seem doable to me as I start off teaching 4 new classes. In time I hope to use visuals, but I recognize a need for me to gain more comfort with concepts in order to get students to grow into the conceptualization with me. This is true for student generated ideas as well.

I already use multiple representations, but I know I will be using studentgenerated responses more this upcoming school year! Visuals will be so helpful to use for certain units especially (commented on in the previous discussion post) and I am looking forward to finding more ways to use manipulatives. I am PUMPED about this stacking paper problem because I’m actually going to use it in school this year!

Fantastic! In the next month or so, it’ll be a full unit of study for you to use and build sense making with 🙂


As a math specialist/coach, teachers are always asking how to order student generated representations in order to bring out the mathematical concept of the lesson in a way that makes sense. My reply has been that they know where their class is at (where student thinking is at) and to go with the visual representations first. My past thinking has been for students who might not have a visual in their work, it might give them something to connect their work to. Then to choose student representations that might have visuals with equations. Now I am challenging my own thinking in that since there are so many ways to think about questions that maybe my answer ought to be: “I wonder if it is more important to find the connections in the solutions presented rather than worrying about which order to have students present?” The connections I could see in the first problem presented about finding the height of the table made me realize that all of the solutions are related – it is just up to the teacher to consolidate that learning.
So one of the activities I want to do at the beginning of next year’s professional development will be the heart representation from lesson 34. I really think that multiple representations, along with the visual representation will bring out many different teacher generated solutions for us to discuss. But most importantly it can show teachers how to connect the variety of ways they solved the same problem and how they are related.

Great plan here Julie! Most times we’ll want to find point out the connections among the solutions while ensuring that the learning goal has been explicitly discussed and made clear. I think the order still matters, however, the connections take priority.


I am out of school at the moment, but when we go back, I can even see using the stacking paper model to discuss order of operations. Have the students notice and wonder, find the height of the table and explain how. This will open up, “Why did you do that first?” “Why didn’t you subtract 5 first?” etc.
If we write out the math of one of the examples as 130.75 – 5(4.95) but have them build / tell what they did step by step, this can open that discussion. The other extensions work for this too.

I like the visual tool. This way the students have a connection to a concrete example of what the math concept/tool mean in “real life”. This might also help with the question we always get in math about “When are we every going to use this?”
Right now, I am on summer break but I am going to use some of my break to plan and make more concrete visuals to be used in my lessons. I am also going to rewrite my lesson and “flip the lesson around”

Before the end of the school year, I was using manipulatives and multiple representations often. I want to push myself for the fall to use more visuals and more student generated solutions. Once in a while I am in awe of student generated tweaks to standard algorithms, for example, but being more purposeful in planning to let students discover formulas will be a good goal for moving forward.

I have been trying to work hard on use of Manipulatives and Visual Representations over the past year. Therefore, my goal this coming year is to work on Multiple Representations and Student Generated Solutions. I am currently reading SanGiovanni and BayWilliams book on Fluency in Mathematics. The motivation for this read…I was finding my mental thought process going back to the thinking that students need Math Fact “Achievement” to be Fluent in Mathematics. Through this workshop, read and other PD I am finally seeing the benefit of Multiple Representations to gain flexibility, efficiency and finally accuracy aka achieve Math Fluency. My previous practice was to just allow a student to understand one strategy, or one representation. However, I would like to strive this year to have my students more engaged in the representations or strategies used by their peers. Hoping Peter Liljedalh’s key responses to initiate keep thinking questions with also help. : )

These are great revelations you’ve made and the resources you’re learning with are going to help you get to that same goal, just a different approach! Love it!


As I am not presently in school right now, I cannot use one with my students. However, I am excited about working these four tools into the lessons as I move forward. I loved how the paper stack and table question connected to the linear equations. This truly will give students a real world connection that is otherwise lacking on such problems. Even though many textbooks will give a story that is “real world” it generally has no connection to the actual math behind the actual equation. I loved how this connected both and will certainly be trying to incorporate all 4 tools moving forward.

We’re diving into pythagorus this week… so I’m planning to use the visuals from the “Squares to Triangles” task that Jon and Kyle created.
I also love using concrete manipulatives and have taken many from the younger classes to have at the ready with my senior high students!
Your intro to linear algebra has me inspired to follow a similar path… we have plenty of stacks of paper around here… hmm… I wonder how many?

Sounds like you’re well on your way!
Glad you’re enjoying the Squares to Triangles unit and the Stacking Paper unit! Two of our favourites!


The tool I used this week was Concrete Manipulatives. I was doing a lesson on arithmetic sequences so prior to that lesson, I used the picture Kyle posted of his daughter with the heat shirt. I gave the students plastic chips to represent the growth of hearts over the days and find an algebraic rule to represent the growth. The students were engaged, were talking with each other, and trying different strategies to find patterns. I circulated through the room and had to give very little instruction/hints. I was just there to listen and give encouragement because they were actually doing it on their own! We had a class discussion afterward. We then went on to a slide deck that only has a numerical number growth pattern. Students were quickly able to find the common difference, write an algebraic expression and find a value further down the sequence using the rule.
Arithmetic Sequences leads us into SlopeIntercept Form, so next week, I plan to use Student Generated Strategies and the Paper Stacking Task to help teach this concept.
These tools have come to me at a great time as the examples are what I am teaching right now. I also find that the tools overlap with each other. When I gave the chip manipulatives, students got to show multiple representations, they had a visual, and their answers were student generated strategies. Each of these 4 tools in isolation lend themselves to the other tool.

Because I have vertical white boards, I often have a variety of student work/examples. This week, I began a class with a problem about the temperatures where in the end, we were going to talk about dividing negative numbers. We had a couple of different strategies including a group drew a picture first, another group used repeated subtraction and then another actually divided. I talked about each group’s work in the order listed.
In the near future, I plan on using manipulatives to talk about algebraic expressions and how we represent them, what the pieces in an expression mean, and how to combine like terms.

The stacking paper lesson will be part of our linear equation unit for sure. However, I used my tangram sets with my 7th graders yesterday. We started with one whole puzzle equaling 1. So the big triangles are 1/4, the medium triangle is 1/8, and the five other pieces are 1/16. We’ve done this before so the kids understand how it all works. The test was what if the medium triangle was equal to 1. What value are the other pieces? Honestly, the kids were on task and really enjoyed it. The advantage of using concrete manipulatives is it is easy to adapt on the fly. As the students solved the initial problem I quickly changed the problem making the large triangle equal to 1. Manipulatives offer entry points for all students, that is what is so great about them.

I think concrete manipulatives is the most useful discussed in this module. I think that they are the best tool for making the math more accessible for the greatest number fo students.