Intro to Math: Circles and Squares

School is back in session! One of the students I tutor apparently has no homework tonight, otherwise I don’t know when I would have had time to put this into writing – things got busy in a hurry.

I’m doing the whole #teach180 thing so if you’re desperate to see more frequent updates from the classroom you can follow me on Instagram, @multiplefactorsi. I’ll be posting every day, held accountable by a reminder on my phone. I hate tiny little red numbers on that screen, so you know I’ll be following the rules.

Anyway! The first week of math class I always do “intro to math.” We started with some talk about what topics in math we were going to do over the year, and then we spent some time on an arts integration project inspired by these tasks for small children. This would have been awesome if we had more time, or maybe they would have gotten old – who can say? A couple students finished them beautifully at home, and others have kept their unfinished projects in case of momentary boredom. I’ll attach that whole assignment.

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PDF of the assignment: day-1-arts

The second day was all about seeing things from multiple perspectives and integrating multiple viewpoints into collaborative work. We started with a number talk, something I loved when I first saw it but just never used. I took the one from YouCubed’s “Week of Inspirational Math,” week 2 day 1. I had been very nervous that our conversation would be nothing like the thought-provoking and joyful example video, but this went amazingly. Here’s our board:

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Inspired by Sarah at Math = Love, who was in turn inspired by other bloggers, we then launched Broken Circles. That was great! I love hearing students talk about math, but I also loved the no talking rule. It’s also an inspired touch that one circle completes itself. In one group, the person with the “A” pieces sat there, self-satisfied while the rest of the group struggled, and it was kind of glorious. It’s almost like the point of this task was to show that working together and paying attention is crucial.

Between that and our next collaborative task, we went over our group norms. I translated them into Spanish for an extra touch, and perhaps that will inspire me to do more group work in Spanish class? Time will tell.

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The end of Wednesday we started the Pentomino task – I blogged about Pentominoes last time I did it, SO long ago! This year I didn’t leave as much time and consequently they didn’t come up with quite as many combinations, but I do still think it was a useful exercise in visualization and pattern recognition. I’m kind of in love with my independent reflection for that task, attached. Why indeed can you not build a 6×6 square?

Independent Reflection PDF: pentomino-ir

Things are shaping up, but I already know I won’t have nearly enough time to blog as I’d like : 0

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A Conceptual intro to Imaginaries

I feel like every single year I put in my first-draft sketchy lesson plans “conceptual intro to imaginary numbers.” Then I google that phrase and find a bunch of boring-seeming articles and dumbed-down explanations, stare at them really hard, and then end up just doing what I did the year before.

Ugh.

This year I actually came up with something, finally! We had a nice long discussion about the history of number systems, guided by this powerpoint:

Number Systems

This just goes through the different number systems in the order they were devised. It starts with counting numbers, and I asked them why people originally needed this type of number.

The next set of numbers is the natural numbers, which are the same just with 0 added in. Here I just told them that adding zero was a huge deal because humans used to think of it as useless but modern mathematics doesn’t function without it. We do a ton with zero later on for asymptotes, and my [totally realistic] hope is they will be remembering this moment in suspense!

The students correctly predicted that the next set was the integers, with the edition of negative numbers. I asked them why we have negative numbers and several of them independently came up with debt. I told them that one of the first uses was for pyramids, because they had structure above and below ground – this is a spoiler for that Crest of the Peacock chapter on 0. Here we started talking about infinity – the natural numbers are already infinite, because theoretically one can count forever. With negatives, the numbers are now infinite in both directions. Whoa.

They also saw rational numbers coming. I like to tie these into Mitch Hedberg’s joke about the 2-in-one shampoo/conditioner bottles, which he didn’t get since 2 could never fit in 1. Before rational numbers, there was no way to represent 1 divided by 2, and humans’ ancient minds were as boggled as Mitch Hedberg’s. I pointed out that with rational numbers, there are now an infinite amount of numbers BETWEEN all the whole numbers – embedded infinity! My brilliant and autistic student piped up here that of course, there are multiple levels of infinity, he already fully understood that concept. This did nothing to draw in my two math anxiety girls.

For irrational numbers, a term they remembered, I tied in Pythagoras’ cult, and the original discovery that the square root of 2 is irrational. I forget where I read this story but I assume somewhere reliable. Pythagoras, and therefore all of his followers, believed that every number could be expressed as a fraction. This one time, they were all on a boat, and one of the math guys was like “Pythagoras, I reallllllly don’t think that the square root of 2 can be expressed rationally,” demonstrating using a right triangle and Pythagoras’ own Theorem. This was unacceptable, and the cult threw him overboard. But now we all believe in irrational numbers (for the most part), and most of the students remembered pi as being labeled as one.

Oh also, we’d been talking about how the natural numbers are closed under addition, but not subtraction, so we had to add the integers to account for stuff like 4 minus 7. Then the integers are closed under multiplication but not division, so we had to have rational numbers to account for stuff like 1 divided by 2. The first example of an irrational number was the square root of a non-square number. I was like “so is there anything we can’t take the square root of even still?” and they were like “uhhhhhhhhhhhhh” and I was like “OK I’ll just tell you! It’s negative numbers! i is what you get when you take the square root of something negative!” I guess I could have handled that better?

As we were working with imaginary operations like multiplication and addition, I kept finding myself drawing out some ghosts to represent these imaginary numbers:

imaginary numbers

I started doing this last year. This is such an abstract concept that they sometimes feel like all bets are off when it came to things like how addition works. The imaginary number i was so associated with negatives that they would assume all imaginary parts should be written negative. They also sometimes want 2 imaginary numbers multiplied to still have i. I tried to come up with a metaphor – ghosts are pretty transparent, but when 2 of them multiply/ overlap you can see them, they’re just negative. This doesn’t really need a gimmick, though, since they just need to know that i * i = -1, the central concept. Drawing out the ghosts helped them to see that 2i + 3i would just be 5i, since you have 2 imaginary things and then 3 more imaginary things.

For the journal this week I had them write about imaginary/ totally constructed things that serve a purpose. They came up with a huge variety of things, from Santa Claus to the value of money to “basically everything.” These [obviously] got really philosophical, which I appreciated. One student, actually the one who talked about levels of infinity, said “imaginary numbers are extremely real, they just can’t be physically counted,” which I like a lot. I wanted to push back against the idea that these constructed things are purposeless since they aren’t “real.” Like what even IS real?!

Intro to Math Week Year 3: Arts Integration Edition

Just finished my last academic day of the first academic week of the school year. As it has been for the last couple of years, this week is “intro to math week,” and this year I focused specifically on arts integration, keeping the mathematical standards of practice front and center.

On day 1 we spent some time on the syllabus, focusing on class procedures and previewing the upcoming units. Thanks to keeping this blog I have plenty of pictures of previous classes doing the work they’ll be doing, so it’s not just a line about functions or data or whatnot on a sheet of paper. Then I had them each take a Math Attitude Survey. Somewhat unsurprisingly, most of the new ones had some pretty negative associations with math. Kids also weirdly LOVE sharing bad experiences in math classrooms, so it’s a nice first day icebreaker. I took everyone’s three words and put them into a word cloud:

math word cloud 2015

I’ve always been too scared to re-administer the survey at the end of the year, but I think I should do it, or at least have them give me three words again.

The next day was all about arts integration. I had them re-examine the mathematical practices and compare them to the National Core Arts Standards, since I had personally noticed so many great links. Some of them were a little unsure at first, or had trouble understanding what individual standards meant, but they didn’t shy away from asking each other or me for clarification, and the resulting conversations were pretty cool.

art and math

Then we watched this TED talk by Daina Taimina, or at least most of it – it’s super long [for my kids’ attention spans & non-university math levels] and the important parts are at the beginning and end. I love this talk so much though. It inspires me to see her construct her own understanding, especially given her initial troubles with the subject, I enjoy seeing stereotypes smashed, and is the perfect example of arts integration. The art form makes the mathematics more concrete and comprehensible, and the mathematics provides structure and context for some interesting-looking pieces.

Typically I have some major emphasis on group norms, but I didn’t want to insert something else random into this week, so instead today we just had an extended journal reflection and then dove right into our first topics. I’ll teach them the group norms as we start doing our first group tasks. I realized earlier that I’ve never actually blogged the group norms, so I’m working on a separate post about that.

So far things are going well BUT hey it’s only the first week.

“Learn What You Missed Week” and MI stations

Greetings! I have recently returned from this year’s senior trip. As you can see from my wrist accessories, we went to the Rock and Roll Hall of Fame in Cleveland and the Franklin Institute in Philadelphia.

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We can worry about how cool my tattoo is later. Anyway, before the trip happened we were back in class for 2 days after exams. I will be honest and say that I was not extremely in favor of this decision – I did not feel like I could come up with 2 days worth of engaging lessons on “going over the test.” Ick. Instead of trying to fix every mistake they made on the test and magically make them understand what they hadn’t just one week prior, I created an initially simple-seeming activity.

For each of my classes I identified the three most-missed objectives. In case you are curious, for math 4 they were modular arithmetic, fractals, and summation notation (sigma). For math 3 they were complex number operations, data, and exponential growth and decay. Finally, for math 2 they were factoring, similar figures, and probability. Then for each of the 3 objectives I found or created (usually created) a task for each of the 9 intelligences in Gardner’s theory. Those are: interpersonal, intrapersonal, existential, naturalistic, logical/mathematical, visual/spatial, verbal/linguistic, musical, and kinesthetic.

So that’s 3 classes x 3 objectives x 9 intelligences = 81 tasks. Luckily the art objective for exponential growth also worked for fractals so actually just 80 but still! Luckily again, these went really well, and it was all worth it.

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This was the kinesthetic data task, where they had to throw the paper glob a bunch of times and record the distance, then find the mean, median, etc. I like the looks of that full page of mathematical text – that is not the work of an un-engaged math learner. This also shows the bin of algebra tiles, which I’ve never seriously used but actually really like. I experimented with them for the kinesthetic factoring task, and I think they have potential.

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I printed the tasks on colored paper and cut them out so that they could just select one and then grab it and do it. Can you slightly see that the big one has music notes on it? I think that particular music task was one of the ones that was a bit of a stretch actually – they had to look at the notes and graph the melody on a complex plane, which probably doesn’t have musical meaning. The best ever musical task was the musical data one. I drew inspiration from this & had them record data about their favorite songs’ danceability, valence, and speechiness.

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There’s a lot going on in this picture. The picture of ghosts was for a task inspired by this which is very neat. The reason why it is ghosts is probably because of the manipulatives I created to practice multiplying complex numbers. Those are in the blue cup in the front of the picture. They are pretty much algebra tiles, only instead of “x” blocks there are positive and negative ghosts to represent i. The positive ghosts are smiling and the negative ones are frowning. It was cute, initially confusing, and potentially very effective.

I’m done teaching until the fall but I think I will still have plenty of things to write about. We shall see.

Math and Jazz

I have been hearing that math and music are strongly connected for quite some time. I’m always interested in integrating art into my classroom, so I have been listening. The only problem is that I have utterly no clue about music. I definitely cannot read music, have never played an instrument (except for the recorder in elementary school, but I accidentally snapped that in half…), and feel like Fall Out Boy is a genuinely good band. Luckily, the internet was there for me.

I looked at these lesson plans for some inspiration:
NCTM illuminations  “seeing music,” love you NCTM
PBS Ken Burns JAZZ, “Math and Jazz: The Beat Goes On

We just did the straw graphs, and then all the wrap-up needed for them to be extra familiar with the graphs of sine, cosine, and tangent. They know not just what y=sin(x) looks like, but also why it looks like that. Now they’re working on what happens when the graph is y=4sin(x), or y=sin(220x), or y=sin(x)+1 (which sidenote, I do not believe has a musical meaning).

The first introduction to sine shifts was watching this video. They agreed it was pretty cool. This also helped us to discover that the larger the wave’s amplitude, the louder the sound, and the higher the pitch, the greater the frequency of waves. This was a great class because most of them know more about music than I do, so we were all teaching each other.

Then we followed the NCTM illuminations plan pretty closely. They found the frequencies of the notes in two octaves on a piano. *SPOILERS* In addition to reviewing geometric series, they discovered that the frequency of an A note in the higher octave is twice the frequency of an A note in the lower octave.

Things got exciting when we graphed a lower A, a higher A, and then a B note. I asked them to observe what how these graphs compared. The two A notes are harmonious – they would sound nice played together, and their graphs intersect in even intervals:

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An A and a B note are too close to each other. They would clash when played together, and their graphs reflect that, kind of hitting each other randomly, as a student put it “just off”:

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For an excellent example of dissonance in jazz, listen to City of Glass by Stan Kenton. It’s on Spotify!

I’m really glad I finally did this. I feel like I understand music more, and the students have something concrete to tie the very abstract concepts of sine graph shifts to.

Since November

Haven’t blogged since November? Oops. In my defense, a great deal of the time between now and then was just winter break, exam review, and then exams. I’m also basically teaching the same things as last year – in Math 4 they just finished up the Math Words Task and are moving on to the Fractals Task, and in Math 3 we just finished up our discussion on Zero and asymptotes. In terms of new things, here are four exciting moments in math:

IM 3 Infinite Series Argument
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Before just telling them the formula for an infinite sum, I wanted the kids to really understand what infinite series really mean. We all went outside and chose a little tree. I walked halfway to the tree, then half that distance, then half that distance. I regret not having them walk halfway themselves, but I was cold and wanted to go back inside quickly. Next year I will be a better teacher. I asked, “will I ever get to the tree?” Consensus? Basically.

Before our whole-class discussion I had given them the pie graph to color in and asked them to make a prediction, basically the exact same situation as the tree only with 1/3 in there instead of all 1/2’s. Most of them said “it looks like it’s going to be 3/4 because you keep adding smaller and smaller pieces.” At least two of them simply couldn’t accept that infinite pieces added to a finite area. This led to an excellent debate and demonstrated a clear need for the a/ 1-r formula.

IM 2 Probability and Combinations
SOCK DRAWER

If I’m honest, I am not a big fan of probability. It doesn’t always make sense to me, there are lots and lots of formulas, and it never really rings true to me. In an attempt to make my probability unit feel a little less contrived, I took a picture of all of my gym socks. They just felt like a probability word problem to me. I asked the kids to, among other things, tell me the probability that I reach into my sock drawer and get a matching pair, and it’s super unlikely. I also received plenty of unsolicited advice about keeping sock pairs together.

When we got to the combinations part, it was even easier to keep things real-world. I taped some stations around the tables asking about potential boxes of cupcakes or rice bowls at ShopHouse. Have you been there, BTW? It’s Vietnamese/Thai Chipotle and it’s delicious! There are 2,304 different combinations to choose from and 192 are vegan!

combination stations

It’s disappointingly difficult to read these but the right one is the ShopHouse menu, the middle one is about getting a dozen assorted cupcakes at The Sweet Lobby (they won cupcake wars!), and the left one asks them for the probability that they can unlock my phone in the 5 tries you get. The probability is 1/2000 so I didn’t feel too nervous leaving my phone out.

This definitely felt real-world. One of the kids almost refused to do the cupcake problem because a dozen seemed too expensive. Two of them saw the ShopHouse menu and were like “OK, I want brown rice, chicken, red curry…” before I asked them to read what the problem was asking. Students in other classes did the problems just because they were curious, which I obviously liked.

To get students to do practice problems like they sometimes just need to do, never underestimate the excitement that some tape and colored paper provides. The math of counting menu items from an actual menu is not different from the math of a worksheet that says “at Todd’s sandwich shop, there are 5 types of bread, 3 types of cheese, and 8 types of meat…” BUT the investment level certainly is.

On the test, I actually wrote a really similar problem to my un-exciting example just there, only the sandwich shop was called Jenny’s Sandwich Jubilee and you could potentially choose a banana bread cottage cheese pastrami sandwich! RIP Mitch Hedberg.

Speaking of Jubilees …
The word “jubilee” is really having a moment in my classroom. Did you know that the word “Jubilee” is a Biblical-times word for a period of debt-forgiveness? Thanks NPR! Evidently they had a jubilee in Iceland for mortgages, and we listened to the story to compare mortgages there with the mortgage formula we learned for US mortgages.

Then we did a search-and-rescue that spelled out JUBILEE. Meanwhile in math 2 I not only named the fictional sandwich shop Jenny’s Sandwich Jubilee, I also asked them to compare potential re-arrangements of JUBILEE, HOLIDAY, and REVELRY.

This was a lot. I should just blog more.

Exams then travel!

Last week was exam week. These are fairly traditional around here – some of the students even say it doesn’t quite feel like our school. I think spending a week taking exams is an important skill to have, so that’s totally OK.

But also, I like to include a super exciting group portion on the test. One of my classes was dealing with Bell numbers – you should glance at the Wikipedia page for a sense of the deep math behind those if that’s your thing. The formula for the nth number looks like this:

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Cool, huh?

The nth Bell number counts the number of ways to partition a set of n things into groups. For example, the Bell number for 3 is 5 because you can split these 3 things 5 different ways:

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Bell numbers can also count the number of rhyme schemes of a poem with n lines. So naturally I asked them to write 5 different 3-lined poems with different rhyme schemes. Such as:

AAB
I hope you know addition
Because it’s a good addition
To your math repertoire

One of the other classes had to use Newton’s Law of Cooling and logarithms to solve a murder mystery. An invisible dead body, separated from them by caution tape, lay in the corner of the room. They had to test its poison levels and model them according to exponential decay. Newton’s law requires the use of delicate thermometers and other equipment.

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I think the amount of fun they were having approached the amount of fun I had making those. End behavior humor : )

Another class’ exam revolved around, among other things, the tetrahemihexahedron. Here’s a picture of my cat investigating one:

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Unlike Olive’s, which was mainly the sniff test, the students’ investigation involved cutting out the net and folding it (harder than it seems!), and then re-creating that out of origami, and toothpicks and marshmallows.

I’m always impressed with how focused and collaborative the students are with these group portions. Group work probably won’t work as an all-the-time thing, but it can be super effective.

Oh! But the main reason I’m taking to the blog is that I’m leaving tomorrow morning. I’ll be co-leading a 2-ish-week senior experience travel component. It’s not math teaching, but it’s still teaching, so I’ll definitely post about it when I’m back home! See you in 2 weeks!