My Precalculus Problem

This is Lucas’s first year in my school. He’s a senior in my precalculus class. After a few assessments his grade begins to tumble. I know little about his previous math education. I meet with him during lunch a few times to help on the content and see that he’s missing several foundational math and reasoning skills. He ends up with a D in my class.

Jennifer is bored in my class. She’s easily getting an A. I can tell that she knows most of the content because she remembers it from algebra II. At some point in the first trimester she asks me if the second trimester will be review as well. I said that a lot of this is not review and that there will less familiar content in the next trimester (which is mostly true). She ends up with an A-.


In the high school in which I teach there is honors precalculus and “regular” precalculus. I teach the regular level. This means that I get kids like Jennifer, kids like Lucas, and everyone in between. I have college bound students and students that will go into a trade. I have students that love math and some that are only there because their parents made them. I get a lot of students that have scooted by with A’s and B’s without trying and would prefer to continue not trying.

So, basically a typical high school class.

The last couple years I’ve struggled to differentiate for the diversity in this class. I’ve failed many of these students because the content either goes too deep or not deep enough. This summer I’m working on solving this problem, or at least minimizing it. The flowchart below is what I’m currently thinking, although I’m sure this will change as I continue to work on it.

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My idea is to pre-assess over algebra II skills that are needed for the unit. If students have mastered most of those skills then they take a different track then those that haven’t. I haven’t worked out a full module yet but I’m thinking I have most of the track 1 materials made and need to make most of the track 2 materials.

Most of my direct instruction is on video which means not every student has to be at the same place at the same time. I just need them to be ready for the summative assessment on a certain day.

With this model I can patch the conceptual holes for the kids that need it, and push the kids that don’t.

What problems do you see cropping up with this idea? For example, I’m worried some less motivated kids might intentionally do poorly on the pre-assessment so they have the “easier” track.

Any feedback is welcome and appreciated. Thanks for reading.

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“Opening up” Math Class

In an effort to write more I’m going to be posting shorter posts on things that are on mind regarding education and mathematics. Writing helps me process and refine my ideas and I believe it will make me a better educator.

I often think about “opening up” my math class. By “opening up” I mean developing my class in such a way that students have time to explore ideas (preferably ideas that are of interest to them, but also concepts that are in the standards).  In this setting students would be encouraged to do a number of things on a regular basis.

First, they’d be encouraged to explore wrong answers. If a student got an answer wrong they would take time to figure out why, and represent the correct solution in multiple ways (graphing, algebraically, numerically, verbally, etc.). We so often don’t have time for this and don’t value this type of exploration. I think that should change.

Second, they’d be encouraged to take ideas further on their own, in class. A good example is synthetic division vs. long division of polynomials. We always tell students that synthetic division only works in certain situations, but what about that student that wants to know why? How do we support that student? Because if that student is allowed to explore that idea he/she will likely come away with an understanding of polynomials that is far deeper than if I just told him/her the reason. (God forbid the student came up with a reason I hadn’t thought of!)

Third, students would be encouraged to work on meaningful tasks involving mathematics in small groups. These might be “real world” projects or, equally valuable, deep explorations in mathematics. The objective for the group would be not only to solve the problem(s) but to be able to communicate the solution in a meaningful (dare I say visually meaningful and appealing) way.

I do some of this on a small scale in my various classes, but I am quite often up against two major adversaries: the curriculum and time. Although I am up against this, I think that if I “opened up” my class my students would become better thinkers, communicators, and self-motivated learners. In general I think they’d become more mathematically minded and I think it is incredibly valuable to have a society of mathematically minded individuals (more on this in a future post!). I think this is why educators have to be creative, take risks, and embrace technology. That combination, for me, has been powerful in helping me to take what steps I have toward the “open” math class.

If I think of more ways in which math class could be opened up I will be sure to update. Please give me your feedback and ways in which you “open up” your class (math or otherwise)!

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Let Students Explore and Collaborate (week 1)

We had a great first week! This is my third year of teaching and this is by far the most excited I have been for the rest of the school year. Steve Kelly and I tried a few new things last year in pre calculus and calculus that we didn’t care for, so we revamped the prerequisite units in pre calculus and calculus (again). I have provided links to both documents.

In pre calculus we created a packet for the students with 5 activities. The first asked them to make a piece of art using Geogebra or Desmos and then upload that art to their blog. This had kids a little confused and some weren’t really sure how to begin. It helped that I gave them an example of art I made with Desmos. I don’t think this was a terrible situation as I think it gave activity 5 more meaning and the final works of art were much better. The intermediate activities build on one another And are designed to lead the student to an understanding of the families of functions and transformations. This should be review, but quite often the families of functions are taught in isolation and students lose the big picture. This is especially true for the understanding of how any function is moved left, or right or up-and-down, or reflected. Our objective was that by the end of the packet students would be able to create a better piece of art and understand why their art looked the way that it did.

In calculus, Steve and I sat down together and determined the concepts in mathematics that are most important to be successful in calculus. These included skills like understanding composition of functions, graphical reasoning skills, algebraic manipulation skills, domain and range analysis, and a myriad of other skills. This was successful, as students were given an opportunity to refresh their brains, get back into math mode, and collaborate.

Both of these activities, although in structure were quite different, set the stage for a year full of collaboration and communication.

(I will post some the artwork in another post.)

Calculus packet

Precalculus Packet