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Archive for the ‘math’ Category

What does “mean” mean?

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.

Category:

algebra, Calculus, math

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Integral Limit Game

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.

Files

integral-limit-game-calculationsDownload
integral-limits-gameDownload

Category:

Calculus, games

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BYORF

One of the other games I made this year was during our rational functions unit: BYORF, which stands for Build Your Own Rational Function. (This was originally a placeholder name, but it kinda grew on me.)

BYORF is a drafting game, a la Sushi Go or 7 Wonders. You play over 2 rounds (because that fit best in our 45 minute period – 3 rounds might be better with more time?), drafting linear factor cards to build into rational functions that match certain criteria. Here’s an example of a round between two players.

In this example, the left player used only 4 of their linear factors (as you don’t need to use all 6). Then we can compare each of the 5 goal cards, which are randomized each round. L has 0 VA left of the y-axis, while R has 2, so that is 3 points to R. L has a hole at (-2, 1/3) while R has a hole at (-1, -2), so L gets 5 pts. They both have a HA at y=-1, so both score those 4. Then we have the two sign analysis cards, which score points if you have that formation somewhere in your sign analysis. R has the first one (around x=3) and both have the second one (L around x=1 and R around x=-3). So after one round, both players are tied with 11 points.

I hope that gets the idea across. The fact that students need to check each other’s work to make sure the points are being allocated correctly builds in a lot of good practice. After we played the game, I did a follow-up assignment to ask some conceptual questions (which is where the above example comes from). I’ve also attached that here.

I hope you have some fun with BYORF!

byorfDownload
byorf-rulesDownload
byorf-follow-upDownload

Category:

games, pre-calculus

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Letter Scramble

In our combinatorics unit in pre-Calculus, we tend to look at every problem as a letter rearrangement problem. This lets us move beyond permutations and combinations to model any problem involving duplicates. I wanted to build a game that had the students quickly calculate the number of arrangements for a given set of letters, so I came up with Letter Scramble.

The idea behind the game is that students have a set of goal cards with the number of arrangements they want to reach, and a hand of letter cards. On each player’s turn they can play a letter card to change the arrangement, and thus change the total possible number of arrangements. (They can also skip their turn to draw more goal cards, a la Ticket to Ride.)

I calculated all the possible answers you could get using 7 different letters and up to 8 slots, including if some of those slots are blank (and thus make disjoint groups), then determined which of those answers repeat at least once, and assigned them scores based on that.

Possible numbers of arrangements on the left, how many times they repeat in the second column, and the points I assigned them in the third.

One interesting thing about the gameplay is it promoted relational thinking. Instead of calculating each problem from scratch, you can based it on the previous answer you calculated. (So, for example, if the board read AABBCD, that would be 6!/(2!2!) = 180. But if you change that to AABBBD, one of the denominator’s 2! changes to 3!, which is the same as dividing by 3, so it’s equal to 60. No need to calculate 6!/(2!3!) directly.

Some examples of scored goals:

1! * 3!/2! = 3
5!/(2!3!) = 10
letter-scramble-rulesDownload
letter-scramble-cardsDownload

Category:

games, pre-calculus

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The Integral Struggle

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.

the-integral-struggle-rulesDownload
the-integral-struggle-game-boardsDownload
the-integral-struggle-graphsDownload
the-integral-struggle-numbersDownload

Category:

Calculus, games, math

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Slopes and Lattices Game

Okay, here’s a game I came up with off the cuff today. It kinda worked, but I guess if other people tried it and gave feedback, that’d be swell.

Players: 2 (or 2 teams), each with two colors

Board: A 10×10 grid.

The game is played in two phases. In the first phase, each team takes turns placing points on the grid, until each team has placed 5 points. The origin always is claimed as a neutral point. Every point has to be on a lattice point. (In the example below, I was blue and my student was yellow.)

In the second phase, on their turn, each player may place a new lattice point and form a line with one of their original 5 points. If that line then passes through one (or more!) of the opponent’s original 5 points, those points are stricken. If one player can strike out all of the other player’s points first, they win. (If not, then whoever strikes out the most.)

There is one caveats to round 2 – when a line is drawn, determine the slope of that line and write it below. That slope can’t be used again.

Image
A game in progress. I was blue&red, and have struck out 4 of my student’s points. They were yellow&black and have struck out two of mine. It’s their turn.

After playing the first time, it became clear that much of the game came down to placing the points. If you could place one of your points so it was collinear with two of your opponents, you can strike them both with a single line. (But this only works if there is space for a 4th, alternate color point in phase 2 to form the line.) You also want to place your points defensively, with weird slopes that don’t pass through a lot of lattice points, to keep them safe. The second player definitely has an advantage when placing points, but the first player has an advantage when drawing lines, so I’m hoping those balance out.

Thoughts?

Category:

algebra, games, geometry, math

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Vimes’ Theory of Socioeconomic Injustice

As I was planning my linear functions unit, I noticed a problem about someone choosing between two electric companies, with the standard idea of one having a larger start-up cost but a lower monthly rate. I realized these problems are very common, and they reminded me of of a quotation from my favorite author, Terry Pratchett:

The reason that the rich were so rich, Vimes reasoned, was because they managed to spend less money.

Take boots, for example. He earned thirty-eight dollars a month plus allowances. A really good pair of leather boots cost fifty dollars. But an affordable pair of boots, which were sort of OK for a season or two and then leaked like hell when the cardboard gave out, cost about ten dollars. Those were the kind of boots Vimes always bought, and wore until the soles were so thin that he could tell where he was in Ankh-Morpork on a foggy night by the feel of the cobbles.

But the thing was that good boots lasted for years and years. A man who could afford fifty dollars had a pair of boots that’d still be keeping his feet dry in ten years’ time, while the poor man who could only afford cheap boots would have spent a hundred dollars on boots in the same time and would still have wet feet.

This was the Captain Samuel Vimes ‘Boots’ theory of socioeconomic unfairness.

– Men at Arms

I wanted to share this concept with my class, so I looked for problems. One thing I realized, though, was that many of these problems involved one person choosing between two things. This makes it so there is one clearly correct answer.

But oftentimes, in the real world, people don’t have a choice. One person can afford the upfront cost to pay less in the long run, but another person can’t, and winds up paying more overall, as in the Pratchett quote above. So I decided to reframe the problems as comparisons between two people, to highlight that injustice.

The lesson started with a model problem, then I gave each group a different problem from the set below.

(I went through this page of unisex names to make all of the problems gender-neutral.)

Each table worked on a different problem (with some differentiation on which group worked on which), then they jigsawed and, in their new groups, they shared out their problems and how they solved them. Then, most importantly, they looked for similarities and differences between the problems.

We then read the Pratchett quote and discussed its meaning, and students had to agree or disagree (making a claim and warrant for each). We had a good discussion on whether it really applied to today’s world or not.

Category:

algebra, lessons, math, social justice

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The Secret of the Chormagons

(Doesn’t that title sound like it should be a YA fantasy book?)

I shared this Desmos Activity Builder I made on Twitter, but never wrote a post about it. Oops! Let’s do that now.

First, here’s the activity.

This activity is basically adapted from Sam’s Blermions post that I had helped him think about but he did all the hard work of creating questions and sequencing them. I’ve used the blermion lesson in the past and it went well, but that last part lacked the punch I was hoping for, having to work with analog points and compiling them together myself. But then I thought – wait, Desmos AB does overlays that will do this beautifully, if I can just figure out how to get the Computation Layer to do what I want. So I set to it.

The pacing helped pause things to conjecture about what chormagons are – I stopped them at slide 5 so they can make those conjectures. I then advanced the pacing to 6 for the “surprise,” and 7-8 to make further conjectures. Look at some of theirs below.

Screen Shot 2018-08-02 at 12.30.13 PM

Even showing them everyone’s, I tended to get conjectures about the shape their points make – a trapezoid, a pentagon, a heptagon, etc. (One person seemed to guess the truth, but that was uncommon.)

Then I showed them the overlay.

Screen Shot 2018-08-02 at 12.31.05 PM

I saw at least one jaw literally drop, which totally made my day. We talked about why it might be a circle, and what that means for these shapes, then I introduce the proper terminology “cyclic quadrilaterals.” (I taught this lesson as an interstitial between my Quads unit and my Circles unit.)

Speaking of proper terminology, you may notice I called them Chormagons here, as opposed to Blermions. That’s important – Blermions is very google-able, it leads right to Sam’s post! But Chormagons led them nowhere. (And they were so mad about that!)

Now Chormagons will lead right here. So this is an important tip if you use this DAB: make sure you change the name of the shape!

Category:

desmos, geometry, lessons

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One Problem, Eight Ways

I had a pretty good lesson recently that I wanted to share. It was at the end of my quadrilaterals unit, and so we were working on coordinate proofs. I love coordinate proofs because you can get so much information from just a pair of coordinates, which lends itself to lots of different ways of solving the same problem. Add to that how many different ways there are to prove something is a square, and we have the start of something good.

I gave the students the above sheet, starting off with some noticing/wondering about the graphed figure. Then I assigned each table a different method to prove that the quadrilateral is a square. Each group was off to their whiteboards to get started.

IMG_20180308_135259 (1)

It was really great to see each group discussing the problem so intently, and it reminded me how easy it is to facilitate discussion when up at the vertical whiteboards. Afterwards, the students went around in a gallery walk to compare their proofs to the other methods. They analyzed how they were similar, how they were different, and thought about which method they might prefer in the future. (Some comments included things like preferring method 2 because it only involved slopes, even though it involves more lines.)

The whole lesson went so smoothly and had tons of intra- and inter-group discussion. Need to use the structure again.

Category:

geometry, student response

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Whole Class Test

I teach an SAT Math Prep this year, which has been an interesting challenge. We basically started off with lessons on all the different content in the exam, then had a long section on tactics (which can be framed as test-taking tactics but I noticed are often just tactics for solving problems in general, which was nice). But we reached the end of those, and the (in-school) SAT is a month away. The obvious thing to do is to just keep doing practice exams, but that can get a bit boring, for both me and the students. Plus, the class that meets Tues/Thurs hasn’t had very many graded assessments this marking period, so I needed to give them something.

I had decided that grading them on correctness in a practice SAT is not appropriate. I had told them this before, and they knew their grades on their assignments were more for things like how they applied the tactic we were learning. But last class they walked in and I gave them a Part 3 exam (the non-calculator part) and told them it would be graded – but there would be a plot twist. For right now, just take it individually, except this half of the room should start from the back and go forward. Oh, and you get 5 fewer minutes than normal.

While they were working, I went around on my whiteboards and put up the numbers 1 through 20 well spread out, and an ABCD for 1-15. (I wish I had taken pictures!) This started to get them suspicious. When time was up, I told them my grading scheme: it was out of 5 pts, and they lost a point for every question they got wrong. So if you got 15 right, that’s a 0. But! They had the remaining 20 minutes of class to work together and figure out what the right answers should be. And if anyone got less than 15, the whole class lost a point – forcing them all to work together. (With limits, of course – they won’t be penalized for that kid who went to the bathroom for 15 minutes during this, for example.)

A suggestion I made to them was to go around and make votes for their answer for each question. A clear consensus might mean that that is the right answer. However! Don’t be afraid to put your answer down even if everyone else’s is different. I’ve seen questions where only one person got it right. I told them they need to convince each other of what the right answer is.

Let me tell you, I heard so many great conversations as they and I went around the room. Because it’s the SAT, no one gets them all right, so everyone is being pushed to make a convincing argument that their answer is right. Students who weren’t sure got explanations from others. It was delightful!

About halfway, I noticed a clear consensus for about 15 of the 20 questions, but the middle 5 were really quite split. So I lead the class in sharing out their reasoning for some of those questions – never saying what the right answer was, but again letting them convince each other.

It was a nice collaborative effort – I highly recommend it.

Category:

assessment, math

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