Make Math Moments Academy › Forums › Full Workshop Reflections › Module 5: Planning Your Problem Based Lessons › Lesson 56: Proceed or Pivot › Lesson 56: Question
Tagged: @kyle @jon, pivoting

Lesson 56: Question

We’d like you to reflect on two (2) planned lessons you delivered back to back in the past where you proceeded as planned instead of pivoting your plan.
We want you to reflect on this experience by first outlining the two lessons and then think about how you might pivot the next time you teach this concept.
Clearly, when you planned these lessons, you thought there would be a smooth transition from one lesson to the next, but something just didn’t work. Maybe you anticipated students would approach the concepts with greater ease or maybe the concept was much more complex than you had realized the first time through your grade standards or curriculum. Be sure to use the 5.6 PROCEED OR PIVOT REFLECTION template to help you.
Share your experience and any questions, comments, or thinking here.
Let’s start a great discussion.

I don’t have a particular lesson in mind but I do think there has been a number of times recently that I should have pivoted to a lesson on creating a visual model. The majority of my class can’t seem to think of using a visual model as a strategy to help solve problems. They been shown or have discovered a number of models that can be useful to solve different problems but they don’t think to go to these models as a strategy that could be helpful. This is one area I think I will pivot to when visual models are not a strategy that should come about from student work, especially when a visual model makes sense.

This is a common challenge. A strategy you can use is asking students to convince you… when they are just plugging into formulas for example, asking them to prove that it “works” can be helpful but does take time and consistency.


I wonder how often I do a full pivot. I like to bring back topics multiple times so that students brains have a chance to process. So, its more like sprinkling an idea over a few days, but how much and what you sprinkle if definetly a huge idea. Sometimes its good to move forward, then come back later when you can link 2 ideas together. For example, I like the popcorn video you shared, that has the chance to link linear functions to systems, but if they don’t fully grasp the linear function piece, I have 2 units inbetween those ideas in my curriculum, so I will bring back the idea of graphing or making a table in our domain/range unit.

Sounds like an interleaving / spiralling strategy which is great.
Your question about how much to do before moving on is key… I think it varies by topic and by group of students. You want enough there to make the experience a moment they can go back to through context, but not too drawn out where it just seems like “one moment” when you’ve squished 250 moments together! Hard thinking!


I thought that moving from adding simple fractions using grids and counters, then arrays (lesson 1) to adding simple fractions as well as mixed numbers using the algorithm (lesson 2) would not be a stretch but found that I needed to go back and do activities where they struggled with arrays with mixed numbers to cement the idea of the more concrete before the idea of the adding mixed number.

So much thinking to unpack in that area! Thanks for sharing your work!
Have you checked out the wooly worm race unit? Could be helpful for this work:
learn.makemathmoments.com/task/woolywormrace


This whole idea reminds me of 3 years ago where my administrator thought it would be a good idea to turn all learning goals into SMART goals. Which made a learning goal (which we had to post every day) into long phrases that students didn’t care about or understand. But the idea was if we did not achieve the measurable goal, we had to reteach. So really, it is an excellent practice, but I think putting the practice of pivoting into the learning goal went a little far. Also, the teacher complained about how this practice expected them to add to a timeframe that was already short. As you stated in the presentation, when pivoting to ensure student abilities to create concrete/visual representation will save time because it will allow students to understand that will fit into future learning goals. But I feel the fear is that every lesson will need to pivot, but that fear I think is never justified. But I digress and will go into the task at hand.
With my early experience with CPM, I felt I proceeded a lot when I should have pivoted because CPM trainers told me that the textbook would spiral back to concepts that students do not master. (I do think CPM is a great textbook, but that does put a lot of trust in a book that has no clue who my students are) I love the problem because it always gives me apparent struggles with the students when solving the Linear System of equations. The problem states,
“The Alpine Music Club is going on its annual music trip. The members of the club are yodelers, and they like to play the xylophone. This year they are taking their xylophones on a gondola to give a performance at the top of Mount Monch. The gondola conductor charges $2 for each yodeler and $1 for each xylophone. It costs $40 for the entire club, including the xylophones, to ride the gondola. We know two yodelers can share a xylophone, so the number of yodelers on the gondola is twice the number of xylophones. How many yodelers and how many xylophones are on the gondola?”
Great problem to break apart and slowly release information to the students (Especially since it is wordy, like my response). But the issue becomes that the following lesson has students solve the system using the equal values method. This is an excellent progression, but students often struggle writing the linear equations from situations like the yodelers and xylophone players.
So I feel a great pivot lesson would be a simple lesson where you are given starting values and rates of change to write two variable equations in slopeintercept form and then go back to uncovering different methods for solving linear systems. Sorry for the lengthy response.

Lots of great points and reflecting here.
In regards to the SMART goals for learning goals – love that idea if we had more planning time to commit to an endeavour like that. Maybe looking at big ideas from the SMART goal lens is a better middle ground?
One thing to consider is balance at all times. When we share whether we fan proceed or pivot, keep in mind we aren’t suggesting that students master or have a high level of expertise (yet) as that simply isn’t possible in just a few short days. So the trouble becomes trying to determine when is it appropriate to pivot or when can I proceed? Spiralling back is always key and if we are building out ideas conceptually starting with low floor (as you suggest BEFORE the word problem you shared), then it gives students an opportunity to revisit ideas.
The problem based unit that pops in my mind as I read your systems of equations scenario is the Shot Put unit:
learn.makemathmoments.com/task/shotput
It starts with a scenario as a single equation and builds towards a system. Again, giving students opportunities to build their understanding as they go.


This year teaching proportions to 7th graders I started with the chocolate milk task on Desmos and thougth it would automaitlly be engaging and informative. Students did not understand the double number line and I got discouraged and moved on. I tried the balloon task as well – hoew many balloons are needed to lift you. I did a notice and wonder with David Blaine and again thought this would fix the problem. I kept missing the sense making piece and then moving on to the next thing rather then strategically pivoting using data from student answers and instead followed my path of “intersting” activities.

Hey @david.diehl We were definitely guilty of this for many years. When you experience this again what will you do differently?


I am nearly one term into our new school year and I have had to pivot a number of times already. I have many students that do not have a conceptual understanding and I have started to play with the spiralling idea. Using visual, concrete strategies has helped my students and I am sure I will do a lot of pivoting with my students.

ORIGINAL PLANNED LESSON. STEP 1
Linear and affine functions
Solving proportional problems
representing in a chart
making a table of values.
ORIGINAL PLANNED LESSON. STEP 2
Linear and affine functions
Solving proportional problems
representing in a chart
making a table of values
finding the algebraic formula
POSSIBLE PIVOT LESSON
POSSIBILITY 1: it has happened this year. Some of them had problems putting the coordinates in a plot.
Plotting coordinates
We play how to sink the float. With coordinates, first the numbers should be integers and then decimals.
POSSIBILITY 2: They have problems to understand how they have to proceed to make the table of values and they have very few strategies to solve proportional problems.
The pivot lesson will be a lesson with a friendlier numbers and that the deduction of a unit reduction strategy is more obvious than in the normal lesson.

Recently while teaching Volume and Surface area to a group of ELL students who were hybrid (In person and remote), I noticed that they were having a hard time differentiating between shapes, and also as to why volume was three dimensional and SA was two dimensional. Even though I had created several graphic organizers I realized that the students needed more real world visual representations. This was especially true of the remote students, which created even more challenge. However, just by going back and re teaching the lesson with the inclusion of real world examples in videos, and also physical representations that myself and the in person students held on camera made a difference. Students needed to be able to make the connection between the shape, it’s dimensions, and their prior knowledge. In the future when teaching these lessons, I will be sure to provide more representations that students can handle themselves, and also connect to their own life experiences.

I did a review of area/surface area and I thought the kids “got it” (meaning I thought they already “had it” from Gr. 7). I gave them a short task and 1/3 of the class didn’t get it at all. Some mixed up perimeter and area, some just didn’t do the task at all, some figured out one side and thought they were done. My plan had been to have a “boxing match smackdown” where they compared surface areas of various sized boxes to determine the “winner” of each match. I was really flustered that I could not move on, leaving 1/3 completely baffled or defeated. In the “old days” I probably would have moved on anyways, hoping the others would ask questions or miraculously “get it” but with some nods to Damian Cooper’s writings and your own voices in the back of my head I decided to do two things: give the 2/3 who “got it” the boxing match independently (round 1) and have a small group lesson with the 1/3, using the boxing match materials to gap fill and answer questions. They did not ask many, but working collaboratively they caught on to the differences that had meant disaster on the preactivity. Pivoting to work with this small group allowed them to join in the subsequent “rounds” and we all finished together. Some of the kids in the 1/3 group actually outperformed some of the 2/3 group by the end of the unit, which was pretty satisfying.

That is so awesome to hear! What a great pivot by allowing the 2/3 to continue on and working in a small group. You should be proud of the learning your students have gleaned from this experience and note that your timely shift in practice surely had a great influence on the outcome! Bravo!


In third grade we introduce area of rectangles and composite shapes. The first few lessons go through area using square unit tiles and comparing square centimeters to square inches. After a few days of using the concrete manipulatives we move to solving the problems with just the measurements. Then we are presented with composite figures with measurements. Every year my students struggle going from the rectangles to the composite figures but because we are on a time table for completing lessons I follow the lesson sequence.
I think that I need to pivot and prove concrete examples of area with composite figures so that students can see the square units and how the composite figures are made up of a variety of rectangles. Students really struggle when they are not given all the side lengths of the rectangles in the composite figures because they don’t see the rectangles within the composite shape. Before teaching the lessons on area of composite shapes, I think I need a task where students are looking at the shapes within a composite figure.

The two lessons I taught back to back were completing the square and the quadratic formula. Students can derive the quadratic formula from completing the square of a quadratic function. I taught completing the square with Algebra tiles and I modeled the function x^2+8x and I walked through this example with the students. Then I had students practice with x^2+4x and asked them to complete the square with it. Students could imitate what I did in class but the understanding of an array would have really helped. I didn’t slow down but went straight to the abstract with the lesson.
The following day I derived the Quadratic Formula using completing the square and the students were stymied. They couldn’t participate in the lesson because they didn’t understand first how to complete the square. Usually, a student completes the square with a numerical problem and I model how to derive the Quadratic Formula algebraically.
Had I pivoted at completing the square by spending more time with the students and connected the Algebra tiles with an array, students would have felt more comfortable with how completing the square derives the Quadratic Formula.

Great realizations here. Both completing the square and deriving the quadratic formula are two pretty abstract concepts (at least how we typically taught it). Getting concrete and ensuring students are seeing the “why” and “how” behind both is so key.
Check out this task if you haven’t yet… it’s really helpful on getting “to” completing the square:
tapintoteenminds.com/3actmath/magicrectangle


The transition from teaching multiplication followed by division has led to frustration and lack of understanding for many of the students. With pivoting I can picture the using the spiral method to move between multiplication and division.

Moving too quickly from solving onestep equations to solving two and multistep equations is something I have done more than once (not proud of it!). Every time, I find that after another notgreat outcome, I promise myself that I will change things up and learn from my mistakes before teaching it again. Invariably, I do take the time to rethink and make plans to do things better/differently, but somehow end up right back where I started. Honestly, it makes me so mad that I continually let myself be pressured (both by myself and outside forces) into doing things I know are not in the best interest of my kids.
This time, I am hopeful in a different way because I feel the planning aspect that you have laid out encompasses so many more possibilities than I have ever thought through in the past. Additionally, now I can see that if I plan thoroughly enough to specifically see how following through will really save time in the end (and more importantly have a greater impact on student learning and retention) I am much more likely to not make the same choices of just moving on in the future. Feeling optimistic!

The argument can be made in support of either to proceed or to pivot.
Many a time I proceed and tend to go back and reintroduce the concept, when the chance presents. To a large extent, unit rate is present in solving equations and and also in proportional relations.
Pivoting on the other hand, has an advantage, because it provides opportunity to support the struggling students with multiple representation of the concept, while exposing the students to more concepts.

Sounds like you’ve come to the conclusion that there is never a one size fits all answer or path to take. There are pros and cons to both and we need to use our professional judgement to try and determine which path is most helpful at the time.


At the beginning of this past year, I started with a lesson on proportions/ratios and wanted it to lead into a lesson about writing an equation of a line. I had students plot 2 points using desmos graphing calculator. Then their partner tried to write an equation that connected those two points. Reflecting on that lesson, I should have pivoted to many more lessons that require proportional reasoning, but I would do a much better job of connecting to different representations. I also realized that students could easily extend to given whole numbers but did not have equal success with decimals as in your defrosting problem. Much more time should have been spent helping students develop their number sense.

When I was teaching my first year in a new grade level in a new state, my partner and I had planned for the series of lessons with the standard that says: Solve one and twostep problems using data in whole number, decimal, and fraction form in a frequency table, dot plot, and stem and leaf plot. Talk about a loaded standard!! It expects kids to not only read and interpret the three types of data representations, but also requires them to solve twostep problems with them! This would include adding, subtracting, and comparing fractions, decimals and whole numbers.
I think we were really crunched for time or something, because we had planned to simply use the consecutive pages in the student math book, which goes something like:
Day one – Read and interpret frequency tables, with all three number forms, and do the multistep problems at the bottom of the page.
Day two – Read and interpret dot plots, with all three number forms, and make SURE you do all the multistep problems at the bottom of the page.
Day three – Read and interpret stemandleaf plots, with all three number forms, and of course, do the multistep problems.
Day four – Read and interpret all three types of data representations mixed up on the page, and solve ALL problems on the both pages. Tada! We’re done with this and can move on next week!
I knew going in that this was WAY too fast, but since we were crunched for time, I planned to proceed in this ridiculous way. To make it even more embarrassing, I used the preprinted problems in the consumable math book, which had NO connection to the kids at all. Sometime in the middle of the first day, I thought, “What am I doing???!!”
I wasn’t thinking the actual words, “Should I proceed, or should I pivot?” but I did realize that this was ridiculous. So, in hindsight, I now know that what I did was to pivot.
The next day, we collected data on the simplest thing I could think of so everyone could access it. I think I had them count how many jumping jacks they could do in one minute. We collected the data, put it in a frequency table, dot plot, and stemandleaf plot. Then we used their actual data to write and solve problems. For example: How many more students did greater than 15 jumping jacks than did fewer than 15 jumping jacks? We repeated with different studentcollected data the next three days, and they learned so much more. By the end, they could transfer data from one form to another, and write pretty sophisticated questions.
It actually took only one day longer than the original fourday plan, but it was so much more “real!”

Last year, quite a few of my students struggled with identifying zero pairs on an Expression Mat, which lead to difficulties when moving to an Equation Mat. As we moved into simplifying expressions algebraically, many also struggled with identifying zero pairs to make simplifying easier.

When I taught 5th grade math, I spent one day teaching students how to convert mixed numbers and improper fraction. Due to time constraints, I went strait for abstract thinking. The next day we moved onto applying this when adding fractions (LCM and GCF were also in the mix). In hindsight, I should have pivoted or even backtracked by helping students gain conceptual understanding of how/why converting mixed numbers and improper fractions help can help us solve more complex problems.
****Pivot Lesson: How would you represent a number greater than one using colored tiles?**** By using manipulatives, the progression to abstraction would have been organic.

I tried NCTM’s Unusual Baker task with my grade 5 this year. They were all successful with identifying the fractional amount for each piece of cake, but they really struggled with the proportionality of what to charge for each piece of cake. The total cost of each cake had to be exactly $10.00. So they would tell me that a half a cake could be $8.00 and a fourth of a cake could be $1.00. I realized that we needed more work in fractions as parts of a set, like the apples/oranges task in the book StudentCentered Mathematics. We really needed concrete experiences finding fractions of whole numbers and then fractions of whole numbers that would lead to decimal fraction amounts.

In many ways, the need to pivot came when I realized the way math was taught, when I returned to the classroom, was so different from what I had taught +20 years ago or even what I had experienced as a child. I had to learn so many new representations of numbers, how to use manipulatives more effectively (though I had used them before), and how to put an understanding of numeracy first and foremost. Many of my first lessons were more “old school” in that I “taught” techniques and tricks instead of real understanding.
Subtraction of multidigit numbers is a good example. “More on the floor, go next door and get 10 more.” “More on the top, no need to stop.” Great sayings, right? Only for the kids who already got it. Those who didn’t just switched the digits in the numbers to subtract the smaller digit from the larger digit. I needed to step backwards and make sure they had a clear understanding of place value and quantity (conservation?) before I could teach subtraction using more concrete representations and pictorial models first.

Rates and Proportional Reasoning: 2 tasks, Which is juicier? Comparing juice boxes’ ratios of juice concentrate to water and then Kayak, comparing kayakers speed and time given in fractional form. Both lessons were rushed and in hindsight, failures.
My students incorrectly set up juice fractions using part to part, not part to whole ratios. I should have pivoted back to a fractions refresher, looking at equivalent fractions and deepening my students’ understanding. Then moved into the kayak task.

At the beginning of the school year we learn about fractions and decimals, and how to convert back and forth between them. Then towards the middle of the year we learn about equivalent fractions, decimals, and percentages. I always feel like the second lesson should be very easy since we are only learning one more component, but it seems to be the biggest struggle and it scaffolds into Data when they work with relative frequency. I’m usually very good about pivoting when needed, but this one usually gets the best of me.

I spent a day where students explored exponents to figure out for themselves the rules for multiplying exponents. This was a YouCubed activity where students identify, for example, that 2^3*2^5=(2*2*2)*(2*2*2*2*2)=2^(3+5)=2^8. They also do it for raising exponents to exponents, dividing exponents, and so forth. The next day, I had them trying to combine several of the operations at the same time. It went too quickly for many of them and they couldn’t see how what they had done the previous day applied to this assignment. I should have pivoted to help them understand that things can be combined as well, though I don’t know what kind of task would have helped with that.

Recognizing a pivot would be helpful is very important. The hard part is what to do in between. This might not happen right away, but do take note of it and think on it so you can provide a better progression next time around. Something to ask yourself is “what specifically were they getting hung up on? How can we smoothen that path a bit without doing the work for them?”


I can think of a few examples of this, the most prominent in my mind is when my students start to solve equations with 1 or 2 steps.to solve for X. I know I have moved on to 3 step equations knowing full well students were still floundering. I know I need to PIVOT, get visual, have math talks and figure out the stumbling blocks so we can work through the difficulties…

Right now, specific lessons evade me, but I suppose lessons that needed to be more fully explored is substitute and calculate.
Many of the students that are not good at math do not feel confident with calculation.
It must be BEDMAS that seems daunting or, should I say, has not stayed in front of all other lessons when needed to be recalled.
The need is for them to be able and do this by default, but this isn’t the case for many but we move on.
A pivot lesson when we see evidence that they are not strong in this area before we need this skill for formulas is definitely necessary.
 This reply was modified 2 months, 1 week ago by Velia Kearns.

While teaching the unit on fractions, we go from finding equivalent fractions to comparing fractions. They student often struggle with idea of what a fraction looks like based of the numerical representation. We could pivot to look at models of fractions and talk about fractions that are less than onehalf and fractions that are greater than onehalf.

In the past I’ve had one lesson where we use 2sided counters to model adding integers and then in the next lesson we add integers without them. That first lesson there is always a group that wants to jump and so I get excited, however the 2nd lesson can bomb. Instead I can pivot and do more visual representations, going into a more abstract way, but with still some visual. Students could use number lines or drawings to show their understanding.

Fractions! When teaching fractions with grade sevens they are always all over the place for understanding. I started the lessons while online, due to the way in which the year was, so I started with equivalent, then common denominators then multiplying, (this was suggested by another teacher), then adding and subtracting. I found the kids were very confused and I have to back up and reteach a lot.
Reflecting now, I would have pivoted at the start and found some students did not have a clear understanding of whole and part, then started with adding to make whole.

I pivot constantly! However, there were a couple lessons I did on volume of spheres, cylinders, and cones where I couple it with a lesson on the area of a circle so that students could. And I think I still pivoted! I do bring these two up though because I think the reason I didn’t have to pivot as much as normal is because I read a blog on teaching volume that mentioned standard connections and common misconceptions so I put in the work beforehand to anticipate student needs and sequence the concepts accordingly to facilitate connection.

Sounds like you’re learning and growing if you are constantly planning ahead and learning / shifting your thinking before heading into the lesson. Love it.


I was coteaching a lesson in a Grade 5/6 classroom and we were exploring equivalent fractions. The task was to draw an outline around the area that represents one half. I guess I did not give clear instructions and they started drawing in the white space to make halves. I was maybe too open to not intervening. I was exploring how much to say in the beginning. It was also tough since the coteacher and I did not have an established dynamic. Anyway the task did not go very well, and the discussion was weak.
So we pivoted to a task of making 2 and 1/2 as many ways possible with pattern blocks. The goal is to show equivalent fractions, but that learning goal adds on the concept “it matters what you define as your whole” Which came out naturally. That lesson was successful and in discussion you could see the student really understanding and lighting up and even articulating their understandings of equivalence and also defining the whole. So that worked much better. With very little teaching, only directing conversation between the students for the whole class.

In my third unit lesson 4 is “Comparing Proportional Relationships” and has students using graphs and tables of things like lemonade mix and water or money and envelopes. Lesson 5 is “Introduction to Linear Relationships” and one of the earlier tasks is predicting how much a stack of cups is going up. I remember that this lesson 5 tends to be quite difficult and the majority of students really had a hard time moving on to lesson 5.
I’m thinking of going back to a problem they have seen before but applying it in ways that work more for these topics. They have already looked at how much a park charges for people entering and figured out that they had a charge for each vehicle regardless of the size and a charge per person in the vehicle.
I would need to do more but I also think I need to see what the current group of students is doing. The second half of the pivot might need to be crafted for areas where this group needs more work. It could be something more concrete like blocks or a copied shape and recording how many squares it goes up when you add another brick or whatever. They could be encouraged to represent this with pictures or tables rather than just algorithms.

In the grade 8 program, students tend to run into difficulty problemsolving with percentages. In particular, this occurs when (a) they need to find the whole (unknown number problem) when the percentage for the other number is provided, (b) the change with percent increase/decrease, and (c) calculating the unknown percent increase/decrease. The activity you carried out with the tree drawings in Module 5 Lesson 1 could really help to solidify how proportions work using percentages which in turn would be beneficial to helping students understand how percentages work in situations abc above. I will definitely pivot after the first planned lesson before moving on to the second planned lesson. This should save a lot of time and grief for students. I can feel the confidence building in them already.

I struggle to have the courage to pivot especially when working to stay at pace with teachers in the same subject since we give common quarterly and final exams. There are many times when some students need additional time to gain a deeper understanding while several incorrectly placed students are already there. I struggled to find enough extensions and additional tasks for the students who grasp the concept quickly the first time. My goal this year is to focus on developing a conceptual understanding and I think the concrete and visual models will be very helpful in achieving this goal.
Last year I was able to develop students visual understanding of multiplying binomials and factoring trinomials in the form x^2+bx+c but I struggled to find a visual model for factoring trinomials in the form ax^2+bx+c when a>1. I tried having my advanced students work on it as a challenge but we had a difficult time explaining the why between the trick to find the two terms that add up to b and multiply to ac. The visual works so well with the simple version. Any ideas, thoughts, or suggestions would be greatly appreciated.