## Problems / Exercises

I wrote about this earlier today and I want to spend a few minutes trying to organize my thoughts.

Lots of thinking to do, luckily the summer will afford me some valuable time.

## What Do Numbers Mean?

This week I wrote about experimenting with number base systems in my AP Calculus BC class. A question came into my head yesterday about repeating decimals in base ten and whether/how we could decide if that number is also repeating in different number bases. It was really hard and the calculations got pretty ugly. So, today I started class with the following idea. I wrote a repeating decimal in a different number base and then converted it to base ten. The calculations are clearly more manageable and I had a clear idea that this could link back to our conversations about infinite series. What excited me today was that my vision of the infinite series was different than that suggested by my student Megan AND it was entirely different than a suggestion by my student Elijah.

I started with the base 3 number 0.122122122… I saw this as three different infinite geometric series’ each with a ratio of 1/27 and I worked the problem this way. Megan saw this as one series made of the first three terms with a ratio of 1/27. We, of course, arrived at the same answer and I really liked the way that her techniques was only one series to calculate instead of calculating three different series’ the way I saw it. I will put a picture below here with a different example showing Megan’s technique.

The example above started with the base 5 number 3.021021021021…

Elijah had a completely different approach, one based on how we teach converting base ten repeating decimals into fractions. The picture below shows his approach.

A couple of notes here. First, Elijah is a terrific math mind and this is a really creative approach. Second, this approach models the approach that my students have already seen. Third, this tactic encourages you to actually live more thoughtfully in this different number base.

I just came away SO impressed by the thoughtfulness, the persistence, and the creativity of my students this morning.

## Some Post AP Fun

My Calc BC kiddos took their AP test last Thursday and we still have classes through next Wednesday. So, I have some time to play with. This year is the first time through for me teaching a Discrete Math elective and one of the topics I ran through with that class was the notion of different number bases along with a little history about some counting systems and the symbols used. I decided that my Calc BC students deserved the opportunity to think about this as well and for the past two days we have had fun saying things like 5 + 4 = 13 (guess the base!) and things like 5 X 2 = A. My students have appreciated me joking that they should make sure to go home and tell their parents that I said 5 + 4 = 13. What I have appreciated is seeing the combination of discomfort and curiosity which turns into a bit of joy as my students wrap their heads around this topic. It is especially in testing to me to see that the BC kids, who are really the top math scholars here, are not inherently more comfortable with this topic than my Discrete students were. There is a pretty big gap in the comfort level with mathematical ideas between these two groups of students, but this notion of fundamentally reconstructing meaning for numbers is a great equalizer. In BC today I even threw out this question – convert the base 8 number 41.37 into a decimal number. Contextualizing the ‘decimal’ portion of this number was not obvious right away, but they were easily convinced once one of their classmates offered a rationale for it. I know that this is far from an earth-shattering ideas, but I also know that this is an idea that too many students are not exposed to in their high school experience and I am kind of pleased that I get to blow their minds a bit. Tomorrow we talk about the Mayans and the Babylonians and we wrestle with their numeration systems. A fun way to wind down the year.

## Platonic Triangles

Too long ago I started a Geometry post by suggesting that I might have a two post day in me. Needless to say, it did not unfold that way and some combination of malaise, exhaustion, and the irresistible momentum of the end of the year has kept me away from this place of peace and comfort for some time now.

I want to share something from our Geometry class this year that was largely motivated by the work of Sam Shah (@samjshah) and his colleague Brendan Kinnell (@bmk2k)

At TMC Sam and Brendan shared boatloads of ideas and docs that they had created for their Geometry class and I am still in the process of digesting them. One that jumped out to me immediately was a document that they called The Platonic Book of Triangles that they were kind enough to share and to allow me to share in this space. Sam wrote about their process here and here.

What I did this year was try to de-emphasize naming the trig functions and just concentrate on the inherent similarities tying together right triangles as a lead in to discussing the inherent similarities relating all regular polygons and circles. Part out of a whole has become a mantra in my class these days. So, what I did was I went to a local copy shop and had them print out a class set of bound copies of the above referenced book of triangles. My students are referring to it as the magic book of numbers. We reference it regularly to set up proportions to solve right triangles. I had the book laid out so that each page had complementary angles on either side. So the students recognized – with a little prompting – that the side lengths on the triangle with the 38 degree angle marked matched up with the side lengths on the triangle with the 52 degree angle marked. I have been SO happy with how they have taken to this reference. In a way it reminds me of the trig tables I used to look up in the back of my book but this has a couple of major advantages. First, it is far more visual and helps the students orient themselves. Second, it does not rely on memorization of a mnemonic about the definitions of the cosine, the sine, or the tangent of an acute angle in a right triangle. I have been careful when I do use those terms to say as clearly as I can that for now they do not want to talk about these functions for anything other than acute angles in a right triangle. There is a whole world of trig excitement waiting for them after their experience in our Geometry class is a dusty memory.

From this conversation about solving triangles and using this to lead into explorations of regular polygons I wanted to make sure to introduce the idea of radian measures to my young charges. I came up with what seemed like a clever idea. It was a chilly, drippy day here in NE PA so I called up the weather bug applet on my laptop. However, what I did before class was I changed its unit of measure to celsius rather than fahrenheit. A student mentioned that it was unpleasant outside – with a little prompting – so I called up my weather bug and expressed surprise that it was only 13 degrees outside. Students quickly pointed out that this was simply a different way to measure the same thing, that there is a way to jump from one representation to the other. Aha, the hook was baited! I then launched into a pretty unexciting, standard representation tying together radians and degrees, relying on my mantra of part out of a whole over and over again. I am not fully convinced that they are buying in and there is evidence that many of my students seem to think that attaching pi to a degree measure is simply some sort of stunt. I am also seeing evidence that simplifying fractions, especially those where the numerator is already a fraction, is a serious challenge to too many of my students. However, what I am convinced of at this point is that a seed has been planted that has a better chance of blooming in precalculus than for those students who did not see the concept of a radian presented to them before. We have our unit test on Monday and I hope not to be disappointed.