Category Archives: calculus

Building Functions - Thinking Ahead to Calculus

My ninth graders are working on building functions and modeling in the final unit of the year. There is plenty of good material out there for doing these tasks as a way to master the Common Core standards that describe these skills.

I had a sudden realization that a great source for these types of tasks might be my Calculus materials. Related rates, optimization, and applications of integrals in a Calculus course generally require students to write models of functions and then apply their differentiation or integration knowledge to arrive at a result. The first step in these questions usually involves writing a function, with subsequent question parts requiring Calculus methods to be applied to that function.

I dug into my resources for these topics and found that these questions might be excellent modeling tasks for the ninth grade students if I simply pull out the steps that require Calculus. Today's lesson using these adapted questions was really smooth, and felt good from a vertical planning standpoint.

I could be late to this party. My apologies if you realized this well before I did.

#Teachers Coding - Bingo Cards

When I attended a Calculus AB workshop back in 2003, one of the nice takeaways was a huge binder of materials that could be used immediately with students. I ended up scanning much of those materials and taking the digital versions with me when I moved overseas.

One of these activities was called derivative bingo. This was a set of two sheets, one with a list of expressions, and another a 5 x 5 bingo card with the derivatives of those expressions. It was perfect to use after introducing derivative rules such as the product and quotient rules to develop proficiency.

It also wasn't as fun of an activity for two reasons. The first was that I only had one bingo card provided as part of the activity. Since everyone had the same card, everyone would really obtain five in a row after doing the same set of problems. Yes, I could have made different ones at some point in the past twelve years to resolve this problem, but I never thought about it with enough advance time to do so. The other reason was that the order of the list of derivatives was carefully designed so that you only obtained five in a row after doing most of the problems provided. Good for the purposes of getting students to do more practice, but definitely an attribute that hacks the entertainment value even more.

As you might expect, I wrote a computer tool to manage this. You can visit this site and see a sample card. Reload the page, and you'll generate a new one.

This turned into a nice little competition between groups of students, and I kept a tally of how many total rows had been matched by each group as they developed. The different cards led to some great conversations between students about their results:

Screen Shot 2016-01-22 at 10.13.05 AM




I use the KaTEX rendering library to make the mathematical expressions look good. If you would like to edit the files for use with your own class, you can go to the GitHub repository here and download a zip file with all of the files. You'll find instructions there for changing the code to fit your needs. If those instructions don't make sense to you, let me know.

If you would just like a set of cards for the derivative practice activity that is ready for use with a class, that PDF is here: derivative-bingo-class-files




Intermediate Value Theorem & Elevators

I've used the elevator analogy with the intermediate value theorem before, but only after talking students through the intermediate value theorem first. This time, I took them through the following thought experiment first:

Step 1:

You enter the elevator on floor 2. You close your eyes and keep them closed until you arrive at floor 12, twenty seconds later.

Questions for discussion:

  • At approximately what time was the elevator located at floor 7? How do you know? What assumptions are you making?
  • Was there a time when the elevator was at floor 3? Floor 8? How do you know?
  • Were you ever at floor 13? How do you know? Are you really sure?

Step 2:

Another day, you again enter the elevator on floor 2. You again keep your eyes closed, but another person gets on from some floor other than floor 2. You keep your eyes closed. The other person leaves the elevator at some point. After 60 seconds, you are on floor 12, and you open your eyes.


  • Was there a time at which the elevator was at floor 7? How do you know?
  • Was there a time at which the elevator was at floor 13? How do you know?
  • What was the highest floor at which you can guarantee the elevator was located during the minute long trip? The lowest floor?

Step 3

On yet another day, you are once again entering the elevator at floor 2 to go to floor 12. You close your eyes, same story as before. Another person gets on the elevator and leaves. This time, however, you open your eyes just long enough to see that the person leaves the elevator at floor 15. As before, the entire trip takes 60 seconds.


  • Was there a time at which the elevator was at floor 7?
  • Was there a time at which the elevator was at floor 13? How do you know?
  • Make a list of all of the floors that you can guarantee that the elevator could have stopped at during the 60 second trip.
  • Can you guarantee that the elevator was never located at floor 17?

We then visited the driving principle to why we can do this thought experiment: why can we come to these conclusions without opening our eyes in the elevator? What is it about our experiences in elevators that makes this possible?

My students were primed to bring up continuity given that they worked through the concepts during the previous class. That said, there were quite a few lights that went on when I asked what it would be like to ride in a discontinuous elevator. Skipping floors, feeling the elevator move upwards and then arriving at a floor lower than where we started, or arriving at different floors just from closing or opening the doors.

Once we were comfortable with this, I threw the standard vocabulary of the intermediate value theorem:

Suppose f(x) has a maximum value M and a minimum value L over an interval [a,b]. There exists a value c in [a,b] such that L≤f(c)≤M as long as...

...and I left it there, hanging in the air until a student filled the silence with the condition of continuity over [a,b]. This was also a great time to introduce the idea of an existence theorem - it tells you that a mathematical object exists, and might give you some information on where to find it, but won't definitively tell you exactly where it is located. Fun stuff.

We then talked about other examples of functions that are or are not continuous. Students brought up crashing into a wall after moving at a non-zero velocity. I also have this group of students the following period for physics, so I brought up what the velocity versus time graph actually looks at when you zoom in to the time of impact. (I like that this wasn't a cognitive stretch for them given their experience zooming in on data on their calculators and graphs from Logger Pro.) The student that brought this up quickly argued himself back from saying that this was truly discontinuous.

This was a fun activity, and I'm glad I went through it. The concept of IVT is fairly intuitive, but we often present it in a way that doesn't emphasize why it is special. In previous years, I started with the graph of a polynomial function bouncing up and down, asked students for the max/minimum value, and then asked them to identify whether they could do this for any value in the range between the maximum and minimum. They could, but never really saw the point of why that was special. Forcing them to imagine closing their eyes, limiting the information available to them, and then seeing how far they could take that limited knowledge made a difference in how this felt on the teaching end. I've seen some pretty good responses on my assessments of this concept as well, so it seems to have done some good for the students as well. (Phew!)

2012-2013 Year In Review – Learning Standards

This is the second post reflecting on this past year and I what I did with my students.

My first post is located here. I wrote about this year being the first time I went with standards based grading. One of the most important aspects of this process was creating the learning standards that focused the work of each unit.

What did I do?

I set out to create learning standards for each unit of my courses: Geometry, Advanced Algebra (not my title - this was an Algebra 2 sans trig), Calculus, and Physics. While I wanted to be able to do this for the entire semester at the beginning of the semester, I ended up doing it unit by unit due to time constraints. The content of my courses didn't change relative to what I had done in previous years though, so it was more of a matter of deciding what themes existed in the content that could be distilled into standards. This involved some combination of concepts into one to prevent the situation of having too many. In some ways, this was a neat exercise to see that two separate concepts really weren't that different. For example, seeing absolute value equations and inequalities as the same standard led to both a presentation and an assessment process that emphasized the common application of the absolute value definition to both situations.

What worked:

  • The most powerful payoff in creating the standards came at the end of the semester. Students were used to referring to the standards and knew that they were the first place to look for what they needed to study. Students would often ask for a review sheet for the entire semester. Having the standards document available made it easy to ask the students to find problems relating to each standard. This enabled them to then make their own review sheet and ask directed questions related to the standards they did not understand.
  • The standards focus on what students should be able to do. I tried to keep this focus so that students could simultaneously recognize the connection between the content (definitions, theorems, problem types) and what I would ask them to do with that content. My courses don't involve much recall of facts and instead focus on applying concepts in a number of different situations. The standards helped me show that I valued this application.
  • Writing problems and assessing students was always in the context of the standards. I could give big picture, open-ended problems that required a bit more synthesis on the part of students than before. I could require that students write, read, and look up information needed for a problem and be creative in their presentation as they felt was appropriate. My focus was on seeing how well their work presented and demonstrated proficiency on these standards. They got experience and got feedback on their work (misspelling words in student videos was one) but my focus was on their understanding.
  • The number standards per unit was limited to 4-6 each...eventually. I quickly realized that 7 was on the edge of being too many, but had trouble cutting them down in some cases. In particular, I had trouble doing this with the differentiation unit in Calculus. To make it so that the unit wasn't any more important than the others, each standard for that unit was weighted 80%, a fact that turned out not to be very important to students.

What needs work:

  • The vocabulary of the standards needs to be more precise and clearly communicated. I tried (and didn't always succeed) to make it possible for a student to read a standard and understand what they had to be able to do. I realize now, looking back over them all, that I use certain words over and over again but have never specifically said what it means. What does it mean to 'apply' a concept? What about 'relate' a definition? These explanations don't need to be in the standards themselves, but it is important that they be somewhere and be explained in some way so students can better understand them.
  • Example problems and references for each standard would be helpful in communicating their content. I wrote about this in my last post. Students generally understood the standards, but wanted specific problems that they were sure related to a particular standard.
  • Some of the specific content needs to be adjusted. This was my first year being much more deliberate in following the Modeling Physics curriculum. I haven't, unfortunately, been able to attend a training workshop that would probably help me understand how to implement the curriculum more effectively. The unbalanced force unit was crammed in at the end of the first semester and worked through in a fairly superficial way. Not good, Weinberg.
  • Standards for non-content related skills need to be worked in to the scheme. I wanted to have some standards for year or semester long skills standards. For example, unit 5 in Geometry included a standard (not listed in my document below) on creating a presenting a multimedia proof. This was to provide students opportunities to learn to create a video in which they clearly communicate the steps and content of a geometric proof. They could create their video, submit it to me, and get feedback to make it better over time. I also would love to include some programming or computational thinking standards as well that students can work on long term. These standards need to be communicated and cultivated over a long period of time. They will otherwise be just like the others in terms of the rush at the end of the semester. I'll think about these this summer.

You can see my standards in this Google document:
2012-2013 - Learning Standards

I'd love to hear your comments on these standards or on the post - comment away please!

Volumes of Revolution - Using This Stuff.

As an activity before our spring break, the Calculus class put its knowledge of finding volumes of revolution to, well, find volumes of things. It was easy to find different containers to use for this - a sample:


We used Geogebra to place points and model the profile of the containers using polynomials. There were many rich discussions about wise placement of points and which polynomials make more sense to use. One involved the subtle differences between these two profiles and what they meant for the resulting volume through calculus methods:

Screen Shot 2013-04-08 at 4.19.33 PM

The task was to predict the volume and then use flasks and graduated cylinders to accurately measure the volume. Lowest error wins. I was happy though that by the end, nobody really cared about 'winning'. They were motivated themselves to theorize why their calculated answer was above or below, and then adjust their model to test their theories and see how their answer changes.

As usual, I have editorial reflections:

  • If I had students calculating the volume by hand by integration every time, they would have been much more reluctant to adjust their answers and figure out why the discrepancies existed. Integration within Geogebra was key to this being successful. Technology greases the rails of mathematical experimentation in a way that nothing else does.
  • There were a few many lessons that needed to happen along the way as the students worked. They figured out that the images had to be scaled to match the dimensions in Geogebra to the actual dimensions of the object. They figured out that measurements were necessary to make this work. The task demanded that the mathematical tools be developed, so I showed them what they needed to do as needed. It would have been a lot more boring and algorithmic if I had done all of the presentation work up front, and then they just followed steps.
  • There were many opportunities for reinforcing the fundamentals of the Calculus concepts through the activity. This is a tangible example of application - the actual volume is either close to the calculated volume or not - there's a great deal more meaning built up here that solidifies the abstraction of volume of revolution. There were several 'aha' moments and I saw them happen. That felt great.

Computational Thinking & Spreadsheets

I feel sorry for the way spreadsheets are used most of the time in school. They are usually used as nothing more than a waypoint on the way to a chart or graph, inevitably with one of its data sets labeled 'Series 1'. The most powerful uses of spreadsheets come from how they provide ways to organize and calculate easily.

I've observed a couple things about the problem solving process among students in both math and science.

  • Physics students see the step of writing out all of the information as an arbitrary requirement of physics teachers, not necessarily as part of the solution process. As a result, it is often one of the first steps to disappear.
  • In math, students solving non-routine problems like Three Act problems often have calculations scrawled all over the place. Even they are written in an organized way, in the event that a calculation is made incorrectly, any sets of calculations that are made must be made again. This can be infuriating to students that might be marginally interested in finding an answer in the first place.
  • Showing calculations in a hand written document is easy - doing so in a document that is to be shared electronically is more difficult. There are also different times when you want to see how the calculation was made, and other times that you want to see the results. These are often presented in different parts of a report (body vs. appendix) but in a digital document, this isn't entirely necessary.

Here's my model for how a spreadsheet can address some of these issues:
Screen Shot 2013-02-01 at 7.47.59 PM

Why I like it:

  • The student puts all of the given information at the top. This information may be important or used for subsequent calculations, or not. It minimally has all of the information used to solve a problem in one place.
  • The coloring scheme makes clear what is given and what is being being calculated.
  • The units column is a constant reminder that numbers usually have units. In my template, this column is left justified so that the units appear immediately to the right of the numerical column.
  • Many students aren't comfortable exploring a concept algebraically. By making calculations that might be useful easy to make and well organized, this sets students up for a more playful approach to figuring things out.
  • Showing work is easy in a spreadsheet - look at the formulas. Depending on your own expectations, you might ask for more or less detail in the description column.

Some caveats:

    • A hand calculation should be done by someone to confirm the numbers generated by the spreadsheet are what they should be. This could be a set of test data provided by the teacher, or part of the initial exploration of a concept. Confirming that a calculation is being done correctly is an important step of trusting (but verifying, to quote Reagan for some reason) the computer to make the calculations so that attention can be focused on figuring out what the numbers mean.
    • It does take a bit of time to teach how to enter a formula into a spreadsheet. Don't turn it into a lecture about absolute or relative addressing, or about rows and columns and which is which - this will come with practice. Show how numbers in scientific notation look, and demonstrate how to get a value placed in another cell. Get straight into making calculations happen among your students and in a way that is immediately relevant to what you are trying to do. Then change a given value, and watch the students nod when all of the values in the sheet change immediately.
    • Building off of what I just said, don't jump to a spreadsheet for a situation just to do it. The structure and order should justify itself. Big numbers, nasty numbers, lots of calculations, or lots of given information to keep track of are the minimum for establishing this from the start as a tool to help do other things, not an end in and of itself.
    • Do not NOT


      hand your students a spreadsheet that calculates everything for them. If a student wants to make a spreadsheet for a particular type of calculation, that's great. That's the student recognizing that such a tool would be useful, and making the effort to do this. If you hand them a calculator for one specific application, it perpetuates the idea among students that they have to wait for someone else that knows better than them to give them the tool to use. Students should have the ability to make their own utilities, and this is one way to do it.

Example from class yesterday:

We are exploring the way Newton's Law of Gravitation is used. I asked students to calculate the force of gravity from different planets in the solar system pulling on a 65 kilogram person on Earth, with Wolfram Alpha as the source of data. Each of them used a scientific or graphing calculator to calculate their numbers, with the numbers they used written by hand (without units) on their papers with minimal consistency. They grumbled about the sizes of the numbers. When noticeable differences arose in magnitude between different students, they checked each other until they were satisfied.

I then showed them how to take the pieces of data they found and put them in the spreadsheet in the way I described above. In red, I highlighted the calculation for the magnitude of the force for an object on Earth, and then asked a student to give me her data. This was the value she calculated! I was quickly able to confirm the values that the other students also had made.

I then had them calculate the weight of an object on Earth's surface using Newton's law of gravitation. This sent them again on a search for data on Earth's vital statistics. They were surprised to see that this value was really close to the accepted value for g = 9.8;m/s^2. I then asked them in their spreadsheet how they might figure out the acceleration due to gravity based on what they already knew. Most were able to figure out without prompting that dividing by the 65 kilogram mass got them there. I then had them use that idea and Newton's Law of Gravitation to figure out how to obtain the acceleration due to gravity at a given distance from the mass center of a planet. I then had them use the spreadsheet model on their own to calculate the acceleration due to gravity on a couple of different planets, and it went really well.

The focus from that point on was on figuring out what those numbers meant relative to Earth. Often with these types of problems, students will calculate and be done with it. These left them a bit curious about each other's answers (gravity on Jupiter compared to the Moon) and opened up the possibilities for subsequent lessons. I'll write more about how I have grown to view spreadsheets as indispensable computing tools in the classroom in the future. A pure computational tool is the lowest level on the totem pole of applications of computers for learning mathematics or science, but it's a great entry point for students to see what can be done with it.


Spreadsheet Calculation Template

Centripetal Acceleration of the Moon - a comparison we used two days ago to suggest how a 1/r^2 relationship might exist for gravity and the moon.



Differentiation Rules - Making it Interactive

I always struggle during the days spent going over differentiation rules. The mathematician in me says the students need to see where the rules come from so that they aren't just a recipe. On the other hand, I see students glazing over a bit with notation and getting lost in the midst of the overall goal: how do we find shortcuts for finding derivative functions outside of using the limit definition every time?

I have also tried going through the derivations in class and having them just watch and see the progression on their own, without copying things down. Some compulsively copied despite my repeated requests not to do so - I think it was a situation of seeing copying notes down as an alternative to really digging in to what was actually going on. It's mindless to copy down notes, a great alternative to actually going through the steps of understanding.

Last year I made videos of the derivations and asked students to watch them outside of class in a one-off attempt at flipping. That didn't work - students said they watched but 'didn't get it', so my attempt to quiz them when they arrived in class was a bust.

This is my compromise this year: for finding the derivative of a constant, a constant times a function, and the power rule, students will be guided through what has essentially my lesson plan for previous lessons. Sums of functions, products, and quotients will be given first as applications of the limit rules, but the details of getting from the start to the finish will be kept as an exercise for later.

See my handout for today here:
03 - CW - Differentiation Rules

Thank you to Patrick Honner and Dan Anderson for their comments pushing me on this.

Why SBG is blowing my mind right now.

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