Make Math Moments Academy › Forums › Full Workshop Reflections › Module 4: Teaching Through Problem Solving to Build Understanding › Lesson 46: How can you be more Prince? › Lesson 46: Question

Lesson 46: Question
Posted by Jon on May 1, 2019 at 11:56 amHow will you be more Prince this week in your math lessons?
Paola Tayvah replied 2 months, 3 weeks ago 36 Members · 45 Replies 
45 Replies

I will use real/concrete items as we begin our unit on multiplication. Then, I will have students start working with base 10 blocks and manipulating them to demonstrate a stronger understanding of factors and products. Eventually, I hope that students will be able to just use numbers and steps to find answers. I love it when students are able to point out math connections when they see things. Examples: an egg carton with a dozen eggs, a dozen donuts in a box, our rows of desks in the classroom, the stars on the American flag.

I will find a picture of something, like the donut box and donuts, to illustrate a factoring problem. I will use the methods to fade the concrete idea of the problem to an abstract, symbolic representation. I will try to ignite some curiosity instead of just teaching the regular way. I think I will use the American Flag or another flag.
 This reply was modified 1 year, 1 month ago by Dawn Oliver.

Great Dawn! I’m curious about your idea with the flag?

I will use the concrete model with a purpose. Manipulatives are a must, and I love my math stations. I will also be in the look out for the gaps that my students have.

I am teaching 7th and 8th graders to solve equations and I have been showing them prompts from Bedtime from Math. Many of the prompts have triggered conversations about arrays. I think it would be helpful to backtrack and tie arrays to solving equations. The donut model would be great. Where can I find those pictures?

With our Geometry unit kicking off, I plan on using more concrete manipulatives like 3D shapes, filling them with liquid and demonstrating, and also perhaps showing more images like the layered donut box to invite students to make those concrete/visual connections before abstract formulas (and show how those are derived through the first two stages).

I really like the idea of the concreteness fading model and I think that it really helps students with automaticity and flexible thinking. When I teach multiplication of polynomials, I typically do so with an area model but I will now try it with a more concrete model first like the donuts to be more “Prince” with my lesson. I think that anytime we have a model to refer back to in order to jog students memories it will help build the bridge between concrete and abstract.

I am already thinking about how I can use this model to work towards distributive property in algebra. As a grade 9 math teacher in Ontario I am struggling to make lessons with a low enough floor and still challenge students who have a good grasp of the underlying skills already. I feel like no matter what approach I take I am failing a group of students by either not challenging them enough or going over their heads and in both scenarios I lose their attention.

I remember listening to the “Be More Prince” podcast episode a while back! Yes, everyone should be more Prince <3
In my efforts to get more teachers to be more Prince, I will be pushing teachers to really see the value in concrete and visual representations. I feel that, even though teachers might have the concrete representation, some don’t trust that students will use manipulatives properly or they fear they’ll lose student’s focus on the lesson. My goal is to get teachers to trust the process and let students explore while using the manipulatives in the classroom.

Good on you! Also we need to be confident in the actual model and strategy we are hoping students will leverage with the manipulatives which can be difficult to facilitate as well. Takes lots of time and effort on our part!


I Have a bunch of 3D figures in my room and we’re just getting into our geometry unit. I am excited about planning how to incorporate these as well as some miscellaneous 3D shapes that are sitting around my house.

I have a perfect topic for this. I taught applications of systems of equations recently and did not give my students anything concrete when we started them. It did not go well. I am teaching that lesson to a different class now, and this session motivated me to create manipulatives so students can solve some of these problems in a more tangible way.

Introducing radian measure next week in Geometry, and thinking this could be an opportunity to call back to some earlier understandings of pi. We can get out the string and see how many radii it takes to get around a circle. We can talk about why we use degrees, and why it doesn’t make great mathematical sense. That’s all I’m coming up with right now…

Great start here Jonathan! Great ideas always have great starts! Let us know how this flushes out.


I am going to use more photos or items the kids and touch & feel to solve more problems.

For volume and SA we’ll use 3D shapes, cm cubes, and graduated cylinders and beakers to find volume before and let that lead into constructing formulas.

I am being more prince this week by using the shoveling the driveway task to help my students learn how to multiply fractions. We are starting with the concrete from the videos or problems, then the students are modeling the problem through drawing it out, and then finally we move to actually using numerical fractions to determine the answer. It has really been great so far, and it has really helped with student engagement.

I’m getting ready to have a summer school month with my own children and their cousins. They are all in middle school (Grade 68). My goal is to give them a month of concrete manipulatives and conceptual understanding of multiplication and ratios. I really liked the Prince example that you can’t just go straight to the symbols for students to understand. That hit home!

Fantastic goal for you and your kiddos! Have you checked out the problem based units for ideas ? I use the contexts all the time with my own kids.

I can only access the Act 1 video for the Doughnut Delight problem. Is there another video that uses the giant box?



I like the concept of fading…when they finally tire of all the extra steps and can make the connection to math facts…this would be a good day. I teach low 8th grade students who, I am sure leave my room befuddled every day because of all the crazy symbols and language used in there. Hopefully next year I can make things more concrete for them.

I work a lot on solving word problems with my students and I will start doing equation writing from those problems with the donut example and having them draw out concrete examples. I need to have them model the word problem with a visual, work with it using a finite number, and then go to the variable version. I also liked how you wrote out what each variable was before writing an equation, that is something I do but need to reemphasize in my teaching.

I love this idea of teaching a concrete version of a task to drive a need for substitution or rather a use for an algebraic representation. Usually I start with an abstract procedure for how to evaluate an expression for different values using order of operations. There is no curiosity in that. I am looking forward to replacing this with concrete representations like this donut task.

Awesome to hear. Context goes such a long way by helping all learners enter tasks and stay grounded throughout the solving of the task.


in the upcoming school year, I will utilize problems that incorporate real objects like donuts or energy drinks. [Aside: has anyone else noticed that their students engage more completely if the problem is about food?] Then I will move toward visual representations such as the use of manipulatives or sketches. I will introduce my students to abstract representation only after they have looked at a problem concretely and visually. The concreteness fading model makes a lot of sense to me if only because many of my students have an issue with abstraction. Unfortunately, traditional ways of teaching math assume that all students are ready to think abstractly. My goal in the upcoming school year is to make the math real and the concreteness fading model will help me do that.

I was pretty excited to see Prince as I am from Midwest, just as Prince! I also am excited about manipulatives and using them all the way through Algebra. I had to watch the last video a few times to make sure I was processing the information. I was soooooo happy to see that the manipulatives are natural in quadratic equations. I will be teaching pull out math classes in our high school and 8th grade Lesson 1 starts with exponents. I truly do not know that my students will know how to explain or compute exponents versus multiplication. I have have algebra kids who are not able to identify the raised exponent and know what to do when they see it.
I do not have a plan yet, but I know I want to start with manipulatives before I even get to lesson 1. Do you have any suggestions for lessons?

How comfortable are students in using manipulatives or visual models for multiplication ?

I feel like my high school students would feel like I was treating them as very young children if I pulled out base 10 blocks or any type of counters. I feel like maybe teaching them how to sketch a module would be a better fit.



I don’t think I’ve ever seen base ten blocks used in a multiplication model like this, even though I’m quite familiar with area models and frequently use them with my 7th graders when they get stuck on the standard algorithm (because they’ve just been asked to memorize it with little understanding). When I do my intervention lessons, I will use baseten blocks to represent multiplication and let students move toward an area model, eventually developing their own algorithms.
I usually introduce integer operations with twocolor counting chips, but sometimes I push students away from them too quickly. I love seeing students (on their own) decide to draw a picture and then come up with their own rules (opposite signs –> subtract, etc.). I will facilitate opportunities for them to do this with most concepts.

@christinepomatto I also used to push to fast to eliminate the need. Now I don’t even bring up the idea of eliminating them. Students choose to move to abstract models when ready.


I am also taking this during the summer and making notes for next school year. I really like the applications with the doughnut problem. The abstract algebra was so cool to see with the graphics and how to apply to factoring. I keep trying to remember not to rush to abstract and allow for the time.

Such an important skill to develop that takes a lot of time and intentional planning. We are so trained to rush and “get through it all” that we often can’t help ourselves but rush to the algorithm. Write it down and constantly come back to it!


I’m thinking about how to use this for multiplying and dividing with negative numbers. If we use the owing money model, it doesn’t work for neg x neg. IF I use colored items with different colors for neg and positive, it sort of works, but feels very contrived and less intuitive immediately. I’m hoping you have some suggestions.
For the concepts relating to adding and subtracting including negative numbers, Maybe a picture of people in line to purchase a movie ticket. Negative numbers are those without tickets yet. Positive numbers are those who have goin into the theater or passed the counter (0 point). I usually use money or + and – markers, but this may be more interesting.

I’ve never thought about starting with visual models like this! I suppose my struggle will be applying visual models to things that don’t necessarily come naturally to me. I plan to use the idea of a balance with solving multistep algebraic equations with physical blocks and brown paper sacks, then moving into maybe algebra tiles on a mat or drawing out the visual of algebra tiles on whiteboards before moving into actual “x” representation. I’m wondering the how I could slice things like geometric transformations on a coordinate plane like this? Or graphing linear equations?

I think the first item on my agenda would be to make sure the problems in the curriculum we are using needs to have a context students can relate to and that everyone has access to. If not, change the context so students can relate. Then figure out a way to create the concrete pictures in a way that can easily change into a visual model students can work with. I have always been a proponent of using manipulatives but I never thought to use pictures first to set up the representation of a problem and then to have students intuitively try to solve what they SEE. Very interesting and intriguing to see if this will work. And of course as a math specialist/coach, I am wondering if I could lead my teachers much in the way Kyle did in this lesson, to see if I can intrigue my teachers enough to try something like this on their own.

I’m finishing up my summer, so I can’t necessarily be “more Prince” this week, but I can definitely think about how I will continue to use this in my stats classes.
The Prince example resonated with me because I did not get any tingles or feelings. I get what my students must feel when we throw a phat at them.
I love the slow reveal of one process and allowing students to create their own algorithms and apply them to future work.

I think it is important to be more Prince by thinking about each concept in the concretevisualabstract progression. I think as students get older, teachers do less and less concrete and start with visual or abstract in teaching. The students who are struggling with the new concepts may end up receiving support from manipulatives. However, all students will benefit from starting concrete to develop much deeper understanding of concepts. I will be more Prince next year by always asking how I can start a concept with manipulatives first.

I think a lot of times I merge the concrete and visual. Not realizing that they are actually separate in the model. Therefore, to be more Prince I am going to have to add more concerts into my lessons aka use more random objects, manipulatives or real world contexts.

When I do start the year in a week or so I will work on using manipulatives to show proportional relationships and then move to drawings before having the students see the ratios in fraction form. Since we move through progressions in our text with recognizing proportional relationships in tables and graphs, I think it will help to grasp the true understanding of what a proportion is in a more concrete way first.

I will use pictures of fast food items to represent large orders and then how to “make the orders easier” by combining similar items. If those pictures are not available, we could also use manipulatives from the elementary schools such as the small bears, cubes, etc. We can then represent that idea similarly with geometric shapes on the mathigon polypad and then represent those shapes, foods, items with letters (variables). The goal being to combine like terms.
 This reply was modified 4 months, 1 week ago by Victoria Murphy.

Since I am teaching Slope Intercept Form, I will be using the Paper Stacking on a table to be more Princelike. Before jumping to y=mx+b, the visual of the table is the yintercept with the slope being the height of each stack of copier paper. This way when we talk about other linear functions in slope intercept form, I can connect the symbolic representation back to the visual of the table and paper.

I love the Prince analogy!
I am currently using a few resources from the Making Math Moments website to introduce linear equations to my 8th graders. I started day 1 with two pizza orders from a local pizza place, one with two toppings and one with two toppings. From this, the students derived the cost per topping (rate of change/slope) in addition to the cost of the cheese pizza (initial value). Much of this was done as a whole class and the students quickly figured out the cost per topping was $1. On day 2 they worked in small groups and did a similar problem called the “Domino Effect”. The students ascertained the cost per topping and cheese pizza cost for all three pizza sizes. Now we are using the “Bird Migration” lesson series. I am loving it and the engagement from most students is high level. Slowly I will be introducing the terminology. Yes, we are going from concrete to abstract!

I will continue to push more manipulatives and real concrete familiar to student things to show math relationships and concepts before trying to use the same to explain concepts that I have already given.

Some of those with severe dyscalculia need to break the array down to 2s, 5s,10s, and 1s.
They demonstrate a grasp of the algebraic thinking, even without fluency with their math facts.
Many get the distributive property before their neurotypical classmates who have relied solely on the algorithm for multidigit multiplication. Surprising their peers who consider them “bad” at math, some of those with dyscalculia can just write the grid and do the rest in their heads.