I was looking for a derivative-based game to play in Calculus as we were just closing out our first unit on derivatives and the semester was ending. That’s when I found Derivative Clicker:
It hit the spot with my students. I explained the game and had them all start playing simultaneously, and then saw who had earned the most money in 20-25 minutes. Yes, it’s a little addictive and “brain rot” (as one student said, but, like, it was a positive review) but they had a lot of fun.
The thing about math games, though, is that the real power is not in the game itself but in the debrief. Just the lesson before this was my students’ first exposure to the idea of higher order derivatives. They asked “But what does a second derivative actually tell us about the function” and I explained, but it still felt ungrounded to them. So I thought this would help them feel the power of derivatives viscerally.
Then we filled out some tables: what if I just had a single 1st derivative (or, in other words, f'(t) = 1), how much money would I have after time? What if instead f”(t) = 1? f”'(t) = 1? This helped build up the idea of increasing rate and how the rates grew polynomially.
They also had debate question about strategy – in the game, with $500, you can buy 1 second derivative or 65 1st derivatives. Which is better? (There’s no a clear answer here – if you were to buy the second derivative and then walk away, it’ll probably be better for you by the time you get back. But if you buy the 65 1st derivatives, you’ll have enough money to buy a second derivative way before buying a second derivative will get you 65 1sts.)
When I started at my current school two years ago, one of the first conversations I had with my coteacher (Sam Shah!) was about grading, and coming to a compromise on our different grading systems. Often teachers will break down grades into categories, the weights of which can vary. But during that conversation I came up with what I thought of as the supra-categories, which I now use for all of my classes.
The first is Knowing Mathematics. In a standards based grading system, this would be the standards for content knowledge. In a traditional grading system, this would include things like tests, quizzes, projects, presentations, interviews – anything that shows what the student knows about the math itself.
The second is Doing Mathematics. In SBG, these would be process standards. In a traditional system, this might be classwork & homework, or class participation. I evaluate this typically with a portfolio of student work. (More on that below.)
The third is Being Mathematicians. I was doing Knowing/Doing before this, but this third category was how to incorporate some of what Sam had been doing with his portfolios, that I loved (and served a different purpose than mine). The assignments are about reflecting how the student fits into mathematical society – both on a small scale (in the class, reflecting on groupwork) and a larger scale (learning about other mathematicians, especially those from underrepresented populations, and about other mathematics outside of the scope of the class). This is also evaluated with a portfolio.
I first wrote about my portfolios in this post, and the general idea there still applies to my Doing Mathematics portfolio, but the structure is different.
Now I do the portfolio as an ongoing Google Slide. You can see the template for the portfolio I used for Calculus last fall here. Our school has a 7-day cycle, and once a cycle we have a double-period. So for that class, in the second half of the double I’d have a quiz about that cycle’s content (counting for Knowing Mathematics), and then afterwards they would work on their portfolio, picking one piece of classwork and one piece of homework from the cycle to reflect upon and include. This let me keep on top of the grading of the portfolio better than saving it for the end of the quarter/marking period, and also made it easier to make sure the portfolio was actually a collection of their work from the whole semester. They had to do two pieces per habit of mind, although usually I would wind up with only 16 done for the semester, not 18, to have a little wiggle room (because it’s hard to get them all). To make up for that, I include two extra slides for the work they were most proud of that quarter.
The Being Mathematicians Portfolio has a wider variety of assignments. Some will be reflections on how they work with their peers:
Some will be about mathematical debates:
Some will be learning about mathematicians (usually from underrepresented groups) or mathematics from underrepresented cultures.
Sometimes news in the math world, or other modern mathematics:
Sometimes about what it even means to do mathematics:
Students will occasionally get these as homework assignments, and we’ll usually discuss them the next class. (I’d like to be more consistent about it, as I am with the Doing Math – maybe that’s a goal for this year.) I’d also gladly take suggestions for assignments in any of these categories, or if you think there are subcategories I didn’t really hit on!
Here you can find the entire year’s worth of BM Portfolios for Geometry and Calculus. (About half of the slides were made by Sam. Wonder if you can guess which are made by whom!)
In both geometry and calculus this year the opportunity has come up to ask, “What is a mean average?” Students can usually pull together an answer that gives me a procedure (“add them up and then divide by how many there are”) but not one that really explains what that procedure shows.
With my tutoring students, I usually get to show them the method that my mother taught me when I was young, that I’ve never seen elsewhere. The procedure works by answering the question from the previous paragraph: a mean average is the value you’d have if the quantity were distributed evenly. (If we have 20 total cookies, how many to give each person so they are the same. If we have a certain budget for salaries, how to adjust them so everyone gets paid the same.)
So the method my mother taught me works that way – not adding and dividing, but redistribution. Let’s see an example.
Let’s take these five numbers I got from rolling a d100 five times. I want to average them. First, I’m gonna take a guess of around what the average is. The 10 is gonna pull it down a lot, so let’s guess the average is 70. So first I’ll redistribute the numbers from the ones larger than 70 to the ones lower than 70, like so.
Okay, now 4 of the numbers are the same, but the last one is too low. At least now when I redistribute I can take the same amount from all of them. Let’s do 5.
Pretty close! We have that 1 spare, so we’ll break that into a fraction, giving us a mean average of 65.2.
This idea of redistribution helps clarify the average value of a function or the average rate of change in calculus. Average value of a function is the y-value we’d have if we had the same total value (area) but redistributed so that all the y-values are the same (a constant).
The function in black, and its average value in purple.
The average rate of change of a function is the rate we would be going if we were going at a constant rate (aka draw a straight line, the secant).
You may be thinking that’s great, but often adding and dividing will be a cleaner and faster algorithm, and you’re not wrong. However, this method really shines when you have a question like one of these.
So let’s think. We have four tests, but we don’t know the fourth one.
We do know that we want an 89 average, which means we want all of the tests to be equal to 89. So let’s do that.
So we need a total of 10 extra points to get to that 89 average. Those points can’t come from nowhere – they have to come from the 4th test.
Therefore, that last test must be 99. Perfectly balanced, as all things should be.
This year when I was in my intro to integrals unit, I tried to look back at this blog for the second integral game I know I played (besides this one), and saw I hadn’t blogged about it. I had tweeted about it, but now I’m thinking, you know, I should, uh, archive things that I only tweeted about in a more permanent place, in guess Twitter doesn’t last much longer.
Anyway, this game is based on The Product Game, with the same structure of turns – players take turns moving a token on the bottom rows, that then determine which square in the top section, where the first player to get 4 in a row is the winner. (I usually have students play in teams of 2, but I’ll keep saying “player” go forward.)
The idea here is that the bottom rows represent the limits of a definite integral. One player plays as the Upper Limit, and the other as the Lower Limit. Once both limits are placed, the player who most recently went calculates the value of the definite integral on the accompanying graph, then covers the square in the top section with the area. (Remember that if the lower limit is greater than the upper limit, the sign is switched!)
Making the function that would give a variety of answers was a fun challenge. After coming up with a graph I thought looked good, I wound up making an excel sheet to calculate all the possible definite integrals to see how balanced it was, and adjusted.
I’ll include that excel sheet as well, as it’s useful for checking answers (as a teacher), although of course each team should be checking each other. After doing a bunch of different integrals on the same function, students often realize they can use their previous work to help them find new answers, reinforcing the cumulative nature of integrals.
I had an extra day to fill for one of my sections of calculus, thanks to the PSAT, so I set about thinking up a game I could play. I gotta tell you, as much as there is a paucity of good content-related math games out there, it’s extra so as you move up the years in high school. I can still find a good amount of algebra and geometry games online, but Algebra 2? Precalc? Calc? Fuhgeddaboutit. Well, I’m teaching both Pre-Calc and Calc this year, so I guess I’ll just have to make them myself. (I brainstormed two more during said PSAT, so those might happen soon enough.)
Starting layout for The Integral Struggle
We just covered function transformations and how they affected integrals, so this game hits on that topic. (Yeah, we’re doing integrals first.) Here’s how it works: there are 3 functions/graphs, each of which has a total area of 0 on the interval [–10,10]. One team is Team Positive, and the other is Team Negative. Teams take turns placing numbers from –9 to 9 that transform the function and therefore transform the area. Most importantly, they also place the numbers in the limits of integration, so they can just look at a specific part of the graph.
With three functions, it becomes a matter of best of 3 – if the final integral evaluates to a positive number, Team Positive gets a point, and vice versa. If it’s a tie at the end of the game, because one or more of the integrals evaluated to 0 or were undefined, then redo those specific integrals. (I discussed with the students whether it would be better to have it be the total value of all 3 integrals instead of best of 3, but I think that makes the vertical stretch too powerful.)
That’s it! Let me know what you think. Materials below.
Last year I went to a PD at Math for America that was about approaching calculus from a geometric point of view. The presenter mentioned during it that, historically, the idea of the integral was developed first, followed by the derivative, and then the limit. Yet in many calculus courses, they are taught in the exact reverse order. I decided that, should I teach calc again in the fall, I’d do integration first.
Well, school is rapidly approaching, and so I’ve been thinking about it again. I did so searching and found this intense forum discussion (oh, old Internet), which pointed me in the direction of the Apostol’s Calculus 1 textbook, which starts off with integrals. The post also had a bunch of arguments about why I shouldn’t do it. One of the notable arguments was that in order to fully teach integration (including u-substitution and integration by parts), you need differentiation. But I actually view that as a benefit, not a downside, because it forces a more spiraled approach. I can start with integrals, then go to differentiation, and then tie them together.
In general, I feel like area is a much more approachable subject than slope. My years of teaching Algebra I to 9th graders certainly seems to support that claim. But I also think it’s easier to understand the linearity of integration than the linearity of slope. “If you add together two functions, the area under the new function is the sum of the areas under the old functions” seems much more evidently true than “If you add together two functions, the slope of the tangent line for each point of the new function is equal to the sum of the slopes of the tangent lines at the same points on the old functions.”
Of course, Jonathan has already worked to restructure his calculus course, and I plan on taking a number of cues from his more spiraled sequence – but still with integrals first.
Here’s what I’m thinking:
Q1 (Intro to Integrals) – (Sam’s Abstract Functions, Area Under Stepwise Functions/Definite Integrals, Properties of Integrals, Riemann Sums, Area Under a Curve, Power Rule for Integrals, Trig Integrals, some applications)
Q2 (Intro to Derivatives) – (Average vs Instantaneous RoC, Tangent/Secant Lines, Power Rule, Trig Derivatives, some applications)
Q3 (Fundamental Theorem) – FTC, Chain Rule/u-substitution, Product Rule/Quotient Rule/Integration by Parts, Curve Sketching/Shape of a Graph)
Q4 (More Applications) – Related Rates, Optimization, Volume, etc
James Cleveland-Tran
The Roots of the Equation 