# 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.

# A Geometry Explanation Idea

Today will likely be a two post day after too long of a layoff from this space. It’s funny, I know that this writing relaxes me and lets my brain breathe in important ways. However, I am too prone to let busy-ness in my life prevent me from taking advantage of that.

In Geometry we are examining relationships between angles and arcs. You know the drill, right? Central angles = intercepted arc measure. Inscribed angles are equal to half of the measure of their intercepted arc. Interior angles (other than those at the center) are equal to half of the sum of their intercepted arcs. Exterior angles equal half of the difference of their intercepted arcs. As I am explaining this yesterday in class I am emphasizing this half relationship over and over but I just kind of gloss over the fact that the central angle does not fit this pattern. I have gone through this explanation in the past but I do not think I was as explicit about the developing pattern with the one half scale factor in the past. So this morning I was thinking about how I can best help my students focus on this detail. I was motivated, in part, by a lovely blog post by Michael Pershan (@mpershan) which, in part, is about managing the processing load of our students. I was struck when reading by the fact that I was asking my students to juggle seemingly different rules yesterday. So I had this idea and it probably is not revolutionary in any way. But I think I want to simply say that central angles are interior angles. They happen to be interior angles that form two congruent arcs. This way I can ask my students to think about the one half scale factor for every angle/arc question that they think about.

Does anyone else out there do this? Do you agree with this idea? Do you think that there is a downside here that I am not seeing yet? I’d love to hear some thoughts in the comments or you can find me over on twitter @mrdardy

# More Calculus Fun with Series

Many many thanks to the wonderful Mike Thayer (@mthayer_nj) who sent a link to this lovely video https://www.youtube.com/watch?v=XFDM1ip5HdU in response to yesterday’s post.

We started each Calc BC class today by revisiting the rational function that caused me so many problems yesterday. Innocently enough, I decided that it would be interesting to examine 1/(1-x) first by long division where the divisor is 1 – x and we got the power series 1 + x + x^2 + x^3 + … as the quotient. Then, we used -x + 1 as the divisor and got the series -1/x – 1/x^2 – 1/x^3 – 1/x^4 … as our quotient. Class ended too soon and I was not able to answer the question of how we could consider these two very different looking series as being equal to each other since they were each results of considering the rational function 1/(1-x).

So, after sending out my call for wisdom in yesterday’s post I went to GeoGebra and discovered something lovely. Something that I was sure my students would be able to discover for themselves. I created a GGB file and I planned our day today. Can I tell you how proud I am of my students for how they handles today? Well, I won’t wait for your permission, I will just come out and tell you how thrilled I am.

I started the day by quickly revisiting yesterday’s two division results and then I called up the GGB file with only the rational function showing. They saw that I had each of the other two series expressions typed in already (out to x^5 for each) and I asked them which of the two they wanted to see first. My morning class wanted to see the series with x as the ratio first, my after lunch class wanted to see the one with the ratio of 1/x first. In each case, after unveiling the function of choice and noticing the relationship between the rational function and this new series expansion my students made the following observations:

1. The graphs only seem to match over a certain set of x values
2. If I were to add more terms, that match would improve
3. If we look at the graphs of both series then we will have a nearly complete match to the graph of the rational function

What my students realized – what I realized last night – is that the two series we can generate have completely opposite intervals of convergence. It was absolutely lovely to see geogebra help this intuition along and it was fantastic that they made this realization before seeing the second graph to confirm it.

After this breakthrough we watched the video together and all of our brains were a bit achy by the end. Amazing that Mike found/knew this link so quickly when I blogged last night.

Some other notes – A student in my after lunch class made this observation about the complementary nature of the intervals of convergence before we even looked at geogebra. I gave one last example, f(x)= x^3/(x+5) and the student who was bothered by -1 = 1 + 2 + 4 + … quickly converted x + 5 into a form of 1 – r so we could interpret the rational function as an infinite geometric series. Another student converted it to x^2/(1 + 5/x) and we, once again, had two different series’ that had complementary intervals of convergence. I have taught this course five or six years before this and never had this ‘discovery’ pop up. Now, we cannot avoid it. It provides a wonderful context for our upcoming conversations about Taylor series and it gives us the opportunity to be more aware of convergence expectations. A pretty great day!

# A Calculus Conundrum

Today we returned to school after a too long two week spring break and in Calculus BC we are beginning to engage with power series ideas. I decided to borrow and modify an exercise in our teacher resources binder. The text presents a definition of a power series and draws comparisons between them and geometric series. I decided to present four series examples and four answer choices – A) Geometric Series  B) Power Series   C) Both  D) Neither. My idea was to show the answers without the definition first and draw out a definition from those clues. That part of class worked pretty well. Then I dipped into turning a rational function (f(x) = 1/(1-x)) into a power series with a brief discussion of why we would even want to do such a thing. I showed them that this function is equal to 1 + x + x^2 + x^3 + … in three different ways. But before I could launch into this conversation I was challenged by one of my very talented students. He said that this equation could only be true if -1 < x < 1. I replied that this was a condition for convergence of an infinite geometric series but I did not want to wander into a conversation (yet) about radius of convergence so I tried to put off that conversation. To his credit he asked me to explain the equation if x had the value of 2 which would then mean that -1 = 1 + 2 + 4 + 8 + …  I did my best to congratulate this observation while still holding off a bit on talking about the radius of convergence.

So, I showed the comparison to the sum formula for an infinite geometric series and then I multiplied each side of the above equation by the denominator 1 – x showing that the series on the right telescopes into 1. Then I got myself in trouble. I did a long division to show that 1 / (1-x) expands into the infinite series. However, when I was writing notes to myself earlier int he morning I thought that my students would question why I was writing the divisor in the form 1 – x. Students are SO used to seeing expressions in a more standard form of -x + 1, so I also did the long division that way. Well, the result of the long division is pretty radically different. Instead of 1 + x + x^2 + x^3 + … the result is now -1/x – 1/(x^2) – 1/(x^3) -… I was pretty surprised by this and I thought that it was important to share this surprising difference. Well, I would have felt better about that decision if we had had about 5 more minutes to talk. Instead we were faced with the end of class and my students asked if these two drastically different expressions should be considered identical.

I conferred with another calculus teacher at my school and he was pretty surprised/intrigued by this conversation and pressed me a bit on why I wanted to open that door of rewriting the division. He also pointed out – as I should have – that the second response is not a power series since it has negative exponents. If I had pointed that out before class dismissed I would have felt better about our conversation. I need to do some thinking about how to talk about this tomorrow. Hey internet – any words of wisdom about this?

# Taxicab Geometry – A Brief Exploration

We are officially on spring break here at my school and we end the term with a week of what are called test priority days. The idea from the school’s end is that we want to protect students from having says with three (or more) major assessments as the winter term comes to a close. With a two – week break most teachers try to put a little bow on their material before taking off so as not to simply start off again on March 14 repeating a week’s worth of material. However, this leads to some awkward scheduling. My last test priority day was Monday and I met classes on Tuesday, Wednesday, and Thursday. I sent out a call for ideas on twitter (like you do, right?) and I received a handful of great suggestions. From a conversation with Henri Picciotto (@hpicciotto) and Becca Phillips (@RPhillipsMath) I decided to spend a few days with Geometry AND with Discrete Math on an intro to taxicab geometry. Henri shared a great link to one of his pages (I encourage you to download that file from Henri – the relevant ones for this discussion are labs 8.4 – 9.1) and I modified some of those ideas and created two handouts of my own (here is #1 and here is #2)

I want to take a moment here to reflect on how our two and a half days with this unit went. We worked Tuesday and Wednesday in each class and wrapped up our conversations before tackling the cool problem I wrote about here to finish our time together on Thursday. First, I want to comment on my documents and how I intend to tweak them before using them again. Then I will comment on the class action these days.

Handout #1 – First change I would make is that point B would be the point (5,4) instead of (4,5). I do not know if any student caught this, but when I imposed the map of Gainesville, FL on the situation I described, Anne is not at the point (4,5). This tweak would solely be for my comfort. I do know that students in all three periods had trouble deciding whether street location should be an x coordinate or a y coordinate. It should have been an easy decision, I think. I like the introduction of the Manhattan map as a way to discuss what a city block might mean, but I did too much talking this first day. I need to introduce the idea then get out of the way and let the students ask these questions. I also should change some of the coordinates I suggested. My students really wanted these points on the section of the grid I provided. I should probably adjust for that. Finally, I have to admit that I am pleased with the questions I asked here. I think that there is a pretty nice balance of practice, of comparing taxicab and Cartesian distances, and of asking some nice guiding questions. By the end of the day Tuesday I felt that my students were in a pretty good place.

Handout #2 – I like that I start off with the same image and the same text to reframe the conversation. I think I will take away the text here that defines a circle and make sure that this definition arises from conversation – either whole class or in small groups. I love the sense of discovery that emerges as the students begin to realize what a taxicab circle will look like. I had GeoGebra fired up on the projector and started taking ordered pair suggestions so we saw the shape emerge together. I am happy again with my questions here even though I am unsure of whether there is actually a clear formula for the number of lattice points inside the border of a Cartesian circle. We did stumble upon a formula for the lattice points inside a taxicab circle and it was pretty darned exciting to see this unfold. Since this was the last night of class work I had very little evidence that any of my students had entertained this question on their own. We had a nice enough conversation about it in class, but it would have clearly been more energetic if there had been some reflection on their own by any of my scholars.

My Geometry students seemed more engaged and interested in how the ideas unfolded in this exploration. Perhaps this is due to its clearer relationship to our ‘normal’ material for the course. My Discrete kiddos were willing to have these discussions, but they were clearly less excited/annoyed/engaged/frustrated/surprised by the discovery of the fact that circles are now squares. I felt pretty committed to the idea that we should agree on whether we wanted to limit ourselves to only considering lattice points when deciding about the nature of the taxicab circle. I had been rooting for a loosening of the idea of points here so that we would have a continuous boundary in the taxicab world as we do in our Cartesian world. Since I had so clearly framed the conversation the day before in terms of city streets and avenues almost all of my students wanted to stay with that restriction and they voted clearly to restrict to lattice points. There have been a few other places where the Geometry students were asked to agree on definitions. We agreed that a trapezoid should have only one pair of parallel sides and we agreed that kites should not have four congruent sides, they should have two pairs of congruent sides that were not congruent to the other pair. There is a clear pattern of wanting to agree to more restrictive definitions here. I have discussed this with one of my Geometry teammates and he seems a bit bothered by my willingness to allow these restrictive definitions. I understand his point about definitions later on in math, but I feel pretty committed to letting the students come to these agreements together at this level. I hope I am not undermining their future as mathematicians here. I like the placement of this material in a short, unconnected time span on our calendar. We could have this conversation at a number of times in the year and I want to keep this in my back pocket to uncover when time allows/demands a unit such as this one. I think that the fact that the students knew that this would not be part of an immediate assessment allowed them to relax a bit and just play with some of these ideas. I also think that this fed into the near complete lack of work done on finishing the questions I presented after class discussion time. I think I am willing to accept this limitation as long as the benefit of relaxation comes along with it.

I want to thank Henri and Becca for helping push me into this and I want to thank my teammate Mary who was willing to dive in and try this unit as well. My other two teammates tried some different ideas and I want to pick their brains to see how life went in their classes for these three odd days. I also want to say that I am fairly happy to have a bit of a break now and I hope to return to school on March 14 with at least a couple of weeks planned out carefully for both the Geometry and Discrete Math classes.

# A wonderful Problem

Today was our last day of school before a loooong spring break – we do not return until March 14. We were asked not to have any assessments today as some students have term finals tomorrow. So, I wanted to find a flexible problem that all 3 of my courses could wrestle with today. I teach AP Calculus BC, Discrete Math, and Geometry so this was a bit of a challenge. I found a lovely problem here :

I was so delighted by how my students engaged with this problem today. A little background first. My BC kids are on the verge of learning about power series so a series/sequence question is right up their alley right now. We have been talking convergence and divergence tests. I also had some competition problems in my back pocket because I knew this would not take them very long. My Geometry class just finished a chapter on similarity and we have spent the past two days playing with Taxicab Geometry. A blog post on that adventure is coming tonight or tomorrow night. My Discrete kids just finished their winter term where we studied patterns (numeric and visual) as well as some theory about voting and ballot strategies and they, too, have played with Taxicab Geometry this week.

I want to share a few of the insightful comments that some of the students made about this list of sequences. I prompted each class with one question first: Why does they say that these are related sequences?

In all of my classes students first focused on the rules for each arithmetic sequence and made observations about the pattern of differences moving from 2 to 4 to 8. In one of my BC classes a student instantly said ‘Each first term is 2^n and then you add 2^(n+1)’ Amazingly fast pattern recognition, but more than I hoped for right out of the gate. Most of his peers were taken aback and seemed happy to focus on smaller pieces. In each Discrete class and in my Geometry class I had students noticing the doubling pattern from one sequence to the next. Only my Calc students used recursive language technically, but all classes had students recognizing that pattern. It is interesting on reflection to see how formula driven (or is that formula comfortable) my Calculus students are compared to the other classes.

I tried to get a series of ‘what do you notice?’ comments going and the following popped up in every class;

• The first sequence is the only one with odds
• They are all arithmetic series (either by description or by use of the formal language)
• The difference in each sequence is increasing by an twice as much as the difference from the previous sequence
• The first term is a power of 2 (my Geometry kids needed prompting to remember about the 0 power)
• All the sequences other than the first have only even numbers

After gathering a series of observations about the sequences, we directed our attention to the charge of finding where the number 1000 might be hiding. Luckily no one wanted to list all the terms of a sequence until 1000 arrived or was passed by. So the following suggestions came my way;

• Subtract the first term from 1000 and divide by the common difference to see if 1000 is on the list
• Divide 1000 by 2 repeatedly until we arrive at a term that is more manageable and more clearly on one of the lists
• 1000 is 10^3 so we need to find 5^3 since 10 = 5 * 2 and we know that 2s are built up row by row

I was really pleased by the focus on 1000 being built up by factors of 5 and 2. One of the discrete classes built up to 1000 while the other kept dividing by 2 to bring it down to the 125 necessary. Once we were focused on 125 it was clear in all classes that the first sequence was the only place that 125 could live. My first class of the day is one of my AP Calculus BC classes and after realizing where the 1000 is there was no discussion of whether that 1000 could appear anywhere else. In my second class, one of my Discrete Math classes, they focused on the plural in the question and wondered whether there might be multiple landing spots for the 1000. We counted out 1 – 20 together on the lists and noticed that no number was repeated. We were pretty confident that this pattern would hold. In my second AP Calculus class – the one where a student generated a formula right away – he stepped up and showed a terrific proof that this had to be a unique solution. Writing each term as 2^n + (2^(n+1))*k where k represents some multiple of the number of differences in the sequence. By setting this equal to 1000 and factoring out a 2^n he made the argument that 1000 needed to be written as a product of a power of 2 and an odd number of the form 1 + 2k. Listing factors of 1000 it was pretty clear that only 8 * 125 satisfied the conditions of the problem.

Finally, my Geometry kiddos had the opportunity to dig into the problem and I was pretty darned pleased, I must say. It was the last period of the last day of school before a two week spring break. They are the youngest of all my students and they are the least experienced mathematically. What I saw today was real evidence that these students have been growing as problem solvers. They are more patient and persistent than they were in the fall and they are more willing to make guesses out loud than they were when we started the year together. I am so happy about the conversation we had. Other than the concern about whether 1000 exists in any of the other lists, they were able to nail all of the important pieces of this problem.

I discovered the problem at about 5:45 this morning and I could not be more pleased about the conversations I had with my students today.

Yesterday we had a two hour delay and I was looking around for an idea to engage my Geometry students at the end of the day. As I have been writing about for awhile now, we are engaged in conversations about similarity. We had some Kuta skills practice, we had some problem sets I wrote for the students for HW practice and today I wanted to have a little activity where I could introduce a question and get out of the way and listen to them debate/discuss/discover some important ideas. I looked at my own Virtual Filing Cabinet and rediscovered a great question posed by Nat Banting (@NatBanting) over on his blog called Musing Mathematically. The question I pulled was from a post last year looking at coffee cups. That particular post can be found here. Below are the two key photos that prompted the conversation yesterday.

I posed the following questions on my class handout :

1. Show that these cups are not similar.
2. If the small cup of coffee costs \$0.99, how much would you expect the large cup of coffee to cost?
3. Since they are not similar, change the height of each cup – maintaining the diameter of the top – so that the cups are similar to the small size.
4. Now, instead change the height of each cup – maintaining the diameter of the top – so that the cups are all similar to the extra large size.

Before presenting these questions/challenges I prefaced the conversation by talking about the habit of upsetting, like at a movie theater concession stand, and pointed out that larger sizes are (almost) always the better value but usually not necessary. Another important note is that I allowed them to ‘cheat’ a bit by presuming that the coffee cups are cylindrical. We have not officially touched on much in the way of volume conversations so we needed to come to an agreement, which we did quickly, on what the volume of a cylinder ought to be.

I am so thrilled with how our conversation unfolded and with the ideas that popped up during our chat. The students were quick to notice that the large and extra large cups each have the same diameter, so similarity there is thrown out the window. A student quickly nominated the ratio of height to diameter as the scale factor that was important. They were shocked by the theoretical cost for the large cup of coffee. I suggested that we ignore the pi in the calculations of volume and at first they were happy with my reason why and then they balked at the idea of just throwing it out of the calculations. This seemed to be a nearly perfect length for an exploration on a silly thirty minute class day schedule. I only hope that they remember nearly as much of the conversation as I do.