Posted by: groupact | April 5, 2014

27 Weeks

This post is just a list of weekly topics covered in a course guide I’m developing. For details and context, see the next post.

(1) Dimension
(2) Distance
(3) Direction
(4) Rotation and reflection
(5) Symmetry
(6) Area and volume
(7) Scale
(8) Dilations
(9) Similar triangles
(10) The Pythagorean theorem
(11) The Euclidean algorithm
(12) The fundamental theorem of arithmetic
(13) Archimedes and pi
(14) Place value and polynomials
(15) Rational expressions
(16) Solving an equation (the idea)
(17) Solving a linear equation (in practice)
(18) Linear equations in two variables
(19) Rate
(20) Solving quadratic equations
(21) Graph quadratics
(22) Circles
(23) Vectors
(24) Systems of equations
(25) Linear programming
(26) Combinatorics
(27) The binomial theorem

Posted by: groupact | March 5, 2014

Always-Sometimes-Never meets Rock-Paper-Scissors

As you might notice, I haven’t been so great about follow through on this blog.  Lots of ideas to blog, but I never get around to them.  But Tina Cardone requested posts about just what happened in a single class, and I told myself I can blog this.  I’m going to focus on my 6th period 10th grade math class from yesterday where I used one idea I had just gotten from some colleagues last weekend, and another idea I came up with on a plane last weekend.


10th grade at my school has aways been about functions.  We used to use a text where we started (see chapter 3) with the definition of a function, what is it, what do we mean by it’s graph, how can we do transformations to the graph, what does it mean to compose functions, and finally what it meant by the inverse of a function.  All of this was the first unit of the year, after which we proceeded to look at polynomial and rational functions, then exponential and inverse functions, and finally trigonometric functions and their inverses.  A few years ago I tried changing the order so that in the first semester we did all of that except we didn’t look at the concept of inverse, logarithms, inverse trig, or solving polynomial equations, exponential equations, trig equations, or solving triangles.  The theme of inverse and how it could be used to “solve” was done in the 2nd semester.  I did like this better, but this year we changed up the order again.  Now we start with looking at sequences, difference equations, and modeling without worrying about formal definitions of functions.  Our idea in doing so was to really get to know and connect with functions in specific contexts before worrying about the concept more abstractly toward the end of the year.  I should write much more about this approach later, but for now what’s relevant to Monday’s lesson is that I was about to introduce the inverse trig functions, but our class doesn’t yet have an abstract concept of function or the inverse of a function.  We do, however, have experience with logarithms and using them to find when an exponential growth model reaches a given point, and we have experience using square roots to help find when a quadratic growth model reaches a given point.  In terms of trigonometry, we’ve defined trig functions in terms of modeling the sun’s apparent motion across the sky (and  back around under the flat Earth until it rises the next day).  We’ve also explored how that understanding can help us distances from knowledge of angles.   My goal for Monday was to draw on these experiences to lay the ground work for defining the inverse trig functions with the details presented in an assigned reading Monday night to be followed up on the next day (today).

The Bagel

So this past weekend I was working with two colleagues to do some PCMI planning and they told me about this great idea they had picked up at a recent conference (NCTM?) in Orlando.  So I don’t know exactly whom to credit for this, but the way I understood the idea which they called “the bagel” was as follows.  A circle–I guess that’s the bagel–is drawn on paper with about four spokes coming out from it.  A question is given, perhaps  as homework, for which there is a short answer.  At a table each student writes their answer to the question in one of the regions formed by the dividing spokes.  Then the table can discuss and debate their answers and try to convince others of the correctness of one answer.  If the entire table becomes convinced of one answer, they can place it in the center.  I often throw out problems or questions where I want each student to give the question some individual thought first before then discussing it with the table.  The bagel seemed like a great way to try to make sure some more individual though was given before table discussions commenced, and for me to be able to quickly see all the individual responses and quickly gauge where there was agreement and where there was disagreement.  

On Monday I started class with four short problems (no calculators) with the instructions “Do by yourself first in your notes..NO DISCUSSION YET”.  The problems  (1) simplify (if possible) 8-\log_2 32 (2) solve (if possible) 4+2^x=13  (3) simplify (if possible) 12-\sqrt{49}   (4) solve (if possible) 4+x^2=14.  I also had a fifth problem for those who did the first four quickly (5) 2^{3\log_2 5}-\left(\sqrt{6}\right)^4-\sqrt{(-3)^2}.  While the students were working individually on these problems I placed a sheet divided into 4 rectangles labeled 1-4 and in each was a bagel described above.  I then gave the students the bagel instructions.  The idea in starting with these problems was that I wanted to students to do some refresher work with logarithms (we hadn’t really used them in over a month), I wanted them to think about the idea of an inverse function and how it can be used to solve an equation, and, most importantly, I wanted them to think about the subtlety involved in square roots where the radix refers to one particular root, but when solving we want to list all solutions unless there is context involved.

Almost all were able to do problems (1) and (2) and the student or two who wasn’t was quickly convinced.  They just needed a refresher.  Problems (3) and (4) though led to some interesting discussions.  Problem (3) some wanted the answer to be 5 or 19 and others just wanted 5.  Three tables became convinced of 5, two became convinced of 5 or 19, and 1 remained divided when I turned to a whole class discussion.  For problem (4) five tables became convinced of \pm\sqrt{10} although at most tables at least one student originally had just \sqrt{10}.  In fact, at one of the tables three of the four had that answer, but the fourth person convinced the other three of her answer.  The sixth table also became convinced of just \sqrt{10}.  All together this was about 10 minutes.

Whole Class Discussion

On the board I pointed out the two consensus answers for problem (3) pointing out that each was able to gain the confidence of an entire table.  I asked for a volunteer to try to convince other tables of his or her table’s answer, and then for a volunteer in the other direction.  In hindsight, I think instead of volunteers I should have called upon specific people who had already been convinced to change their own answer.  The discussion continued for a little with no input from me except calling on hands, and then I said, “Let’s come back to this and look at problem (4).”  Again I shared the answers and called upon volunteers.  This time one student defended the \pm\sqrt{10} but then noted that he wanted to change his answer on problem (3).  He reasoned that if \sqrt{49} could mean 7 or -7, then there would be no need for the \pm in front of the \sqrt{10}.  This argument seemed to convince most of the class, but I chimed in at this point also with the question if \sqrt{49} was 7 or -7 what would \sqrt{49}-\sqrt{49}.  Nobody in that class seemed to like that it would have to be 14 or 0 or -14.  (Interestingly, the other section of this class required some more convincing and more prodding from me.)  In this section this took about another 15 minutes.

Always-Sometimes-Never meets Rock-Paper-Scissors

In the second part of this lesson I wanted students to think about how in the context of a right triangle a given trig ratio should determine a unique angle even if we didn’t have a name for that angle.  I like the idea of always-sometimes-never questions but hadn’t yet done any this year.  I thought this would be a good place to try some, but as I was planning this lesson I considered how to collect their responses.  I don’t do poll everywhere because students are supposed to have cell phones in school (although of course they all do).  I thought about individual whiteboards (which I also haven’t used yet but should) but decided that was unnecessary.  Instead I thought about having students raise their hand with different hand signals.  It occurred to me that rock-paper-scissors provided some clearly recognizable hand signals and the idea of everyone needing to throw out their signals at once without waiting to see what someone else does was perfect.   So I explained the “game” to the students and said use rock (fist) for always, paper (palm) for sometimes, and scissors (peace sign) for never.  

After also taking a moment to define what it is meant by “knowing” a triangle we played always-sometimes-never with the following statements.  (Statements were given one at a time with discussion and debate in-between, but in this post I’ll present all of them together now). 

(1) If the area of a square is known, then the length of its sides is known.

(2) If the area of a rectangle is known, then the length of its sides is known.

(3) If a right angle and second angle are known, the triangle is known.

(4) If a right angle, a second angle, and a side are known, the triangle is known.

(5) If a right angle and one side is known, the triangle is known.

(6) If a right angle and two sides are known, the triangle is known.


The idea of the first question was to set the stage for the usefulness of giving a name to a number we know exists.  So in the discussion that followed it I pointed out that if the area of the square was 5, we agreed there was a determined side length \sqrt{5} but if we thought about it that symbol didn’t tell us how long the side was but rather gave the name to a number we knew was a little more than 2.  

This post is already too long and I need to get to sleep so I won’t go into detail on all of the discussions that followed from each statement, but I will note some interesting things that came up.  The game worked great.  Every student was engaged and we had some great discussions stem from it.   I always gave a few moments to think and then said “1-2-3 throw”.  If a student raised their hand immediately or didn’t raise a hand at all I compared it to what would happen if actually playing rock-paper-scissors.  Most statements had just two responses and a little debate between students reached agreement.  Sometimes often came up because of ambiguity in my statement and I said that was great.  A sometimes response would often lead me to refine my statement and revote.  For example, for statement (2) one said sometimes because if the rectangle were a square we’d know (in the other section the issue came up if the area of the rectangle was prime would we know the side lengths and other students pointed out we didn’t require integers).  For (6) one student pointed out that we needed to know which sides we had (that reasoning could have also applied to (4).    In all this was about 20 minutes and I finished the class with introducing the homework which was to do the reading including any exercises the encountered in the reading.  


Posted by: groupact | July 7, 2013

The Geometry of Multiplication

This is a follow up from yesterday’s post on what is a number. There I related how I was pleased with a class discussion I had with my 10th grade class on what a number is. The key I wanted us to agree on was that a number was both a geometric and algebraic object and that we could view addition and multiplication through both lenses. Geometrically numbers tell us both an amount and a direction. In my situation with my class I was especially interested in the geometric meaning of multiplication as I was introducing complex numbers, and the reason complex numbers are so awesome is that geometric meaning of multiplication. (Without that aspect, I see no particular advantage of thinking about a point in the plane as a complex number as opposed to a pair of real numbers).

With my class the 2nd day discussion–which by this point was more of me talking then I might have liked–focused on the idea that when we multiply by a number the length of the number (or its absolute value, or norm, or size, or magnitude, or modulus, or whatever else people like to call it) plays one role. That tells us how much to “stretch” the number by. The direction of the number (or its sign, or angle, or argument, or whatever else people like to call it) plays another role. It tells us how much to change the direction of the number. So -7 times 3 tells us both to stretch 3 by a factor of 7 (giving 21) and change the direction 180 degrees to -21. Likewise -7 times -3 should stretch -3 by a factor of 7 (giving -21) and change the direction 180 degrees to 21. This might not be concrete enough yet for a model of negative times a negative, but I feel it can and should be the basis for one.

The nice thing about this interpretation of multiplication is that it not only works well with negative numbers, but extends and motivates complex numbers. If we wanted to rotate a point/number by 90 degrees counterclockwise without any stretching, we should multiply by the point/number that has a length of 1 (ie it is 1 unit away from 0) and that is located in the direction that is 90 degrees counterclockwise from the positive direction. That number has been given the name of i. If we multiply by i again we still won’t stretch (since the length is 1) but we will rotate 180 degrees. So i times itself is -1. I like this definition of i in terms of where it is located in the number plane (we’ve now moved beyond the number line) as opposed to just saying that it is the square root of -1 for many reasons. First of all, it’s motivated. Second of all, it clearly exists–I can see that point on the plane, we’re just giving it a name now. Thirdly, it’s well-defined–i is not the only number that squares to give -1, but it is the only point located 1 unit away from 0 at a 90 degree counterclockwise angle to 1. The number 3i can now be thought of in several ways. Start with i and stretch it by a factor of 3. Start with 3 and rotate it 90 degrees counterclockwise. We could even think of it as 3 units in the “i direction” just as we may think of -3 as three units in the “negative” direction or 3 turned around or -1 stretched by 3.

Since last May when I had this discussion with my 10th graders I’ve come across this post by Ben Braun which has me thinking about the distinct models of multiplication we use in terms of whether we are multiplying by something with a unit, or a scalar that does not have a unit. This is something I’m going to think about some more and follow up on later, but any thoughts on it now are more than welcome.

Posted by: groupact | July 6, 2013

What Is A Number? (The Lesson)

My “What Is A Number?” prompt came in a lesson from the first day of a unit on complex numbers that I did in the final three weeks of a semester introducing complex numbers.  I’ll write more about the unit and its goals later, but I want to start by focusing on the goals of the task because I think it could be helpful in other settings as well.   My goal was for students to understand numbers as both algebraic and geometric objects.Image

I had students start by discussing the question of what is a number at their tables, and after about 5 minutes I asked for a response from each table which I recorded on the whiteboard. (I now wish I had kept a more permanent record) I recall that in each class I had several responses with the key words of either “how much,” “amount,” or “quantity.” I consider these responses as dealing with a geometric aspects of number since geometry focuses on measurement. I had at least one response in each class along the lines of “things you can add.” I consider this response as dealing with an algebraic aspect since the focus is less contextual and more on the formal structure. So far so good, but the last key idea I needed from students was an agreement that there was also a geometric aspect to NEGATIVE numbers. I did get from one table in one class, “It is an amount or lack of an amount.” I followed up on this and applauded the consideration of negative numbers, but I wasn’t convinced that lacking 5 watches was the same as having -5 watches. Christopher Danielson has a great post on thinking about integers in which (among other things) he passes on some work from Chisty Pettis and Aran Glancy from the University of Minnesota on characteristics of good integer contexts. The key aspect I teased out of the students after we considered a couple of examples was that negative numbers should imply in some contextual sense an opposite direction which is exactly what we see in the more abstract setting of the number line. Addition has a geometric meaning on the number line in terms of translation which can be in the positive or negative direction. My recollection is that I followed up on the more difficult concept of the geometric meaning of multiplication the following day, which is what I will do in this post as well. To be continued…

Posted by: groupact | July 4, 2013

What Is A Number?

At PCMI this summer, the focus of our Reflecting On Practice class is mathematical tasks. What makes a task worthwhile? How can we adapt a task to make it more worthwhile for our classroom? What should we be considering as we implement the task in our classroom?

On the first day we had to answer the question, “What is the last cool thing you did in your classroom, and WHY?” For me, that last cool thing was simply devoting a full most of a class period to a discussion on “What is a number?” (I had students start the discussion at their table, followed by sharing an idea from each table, followed by a whole class discussion). I’ll follow up some time next week with what my goal was for this task and why I thought it was so cool. I already plan to use the prompt (with perhaps another structure to the discussion) in another class next year, but first I’d love to hear others answer this question. Once we can answer this question, we can answer Paul’s question of “What is a numeric illustration of the fact that a negative number multiplied by a negative number is a positive number?” So, please, fire away in comments or tweets.

Posted by: groupact | April 14, 2013

Rationals, Irrationals, and Decimals

In my last post I promised a follow up explaining why I feel that the concept of repeating decimals should not be introduced to students until at least Calculus. I’ve been delayed, but finally I’m ready to explain my thinking. My short explanation is that I find that my students often have a great difficulty with the concept of irrational numbers which hinders their understanding of trigonometric, inverse trigonometric, exponential, and logarithmic functions, and even just graphing polynomial functions. This difficulty is one that I fully expect–the concept of irrationals is rather abstract and filled with many subtleties that cannot be addressed–but I believe my job in developing student understanding of the concept would be easier if the students had never learned about repeating decimals. “Learning” repeating decimals pushes students to think about rationals as true numbers and irrationals as just symbols.


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Posted by: groupact | March 10, 2013

Definitions and Subtleties

Ben Blum-Smith has a great post about honoring kids’ dissatisfaction. He notes that there are instances where questions or concepts spark heated discussion and kids aren’t satisfied with the mathematical resolution. His draws on two examples, “Is 1 prime?” and “Does 0.999… equal 1?”. He writes:

But in both situations, this would be to dishonor everyone’s dissatisfaction. It is so vital that we honor it. Everybody, school-aged through grown-up, is constantly walking away from math thinking “I don’t get it.” This is a useless perspective. Never let them say they don’t get it. What they should be thinking is that they don’t buy it.

And they shouldn’t! If it wasn’t already clear that I think the above “proof” that 0.999…=1 is bullsh*t, let me make it clear. I think that argument, presented as proof, is dishonest.

In both cases (that 1 is not prime, and that 0.999…=1) there are subtle issues at play that get into advanced mathematics and so kids are right to be dissatisfied with either an invalid proof or, in the case of 1 not being prime, an appeal to a definition that doesn’t explain why the definition should be written so as to exclude 1.

I agree wholeheartedly with Ben, but I would like to explore these issues further as I see the two examples that Ben gives as fundamentally different situations. In the case of whether 1 is a prime, Ben is correct that an appeal to the definition given by Wikipedia or a textbook does nothing to resolve the issue at hand. The real question is WHY do we define primes in such a way that 1 is not included. Before that can be addressed, though, it is important to have a discussion about definitions. Students do not generally view mathematical definitions as something over which WE (the mathematical world) have control. That is most students will not even realize that a CHOICE was made to exclude 1 from the definition. Once we can accept that, then it makes sense to talk about WHY the choice was made. Sometimes some of the reasons for a choice in a definition or convention might depend on an understanding of more advanced mathematics and it is okay to explain that to students, but as much as possible we should try to provide some understanding of the choice, either by appealing to secondary reasons or simplified versions of the deeper reasons. In the case of 1 not being a prime, Ben discusses (especially in the comments) how one key reason for the choice is a desire to have unique factorization, but that requires some time to discuss why we should care about unique factorization. He gives a great example of counting factors and another commenter, Sue, asks students to think about what would happen to our factor trees if we considered 1 a prime. More can be said on this subject, but the point I want to make today is that the most important thing to understand is that a CHOICE was made and that there are reasons for that choice.


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Posted by: groupact | July 23, 2011

Follow Through

So it’s the last day of PCMI and I’ve been convinced to start a blog and write about what I do in my classroom–things that work and things that don’t.  So in the near future I’ll write about using complex numbers, organizing a class on functions by concept as opposed to by function, grading policies and standards for standards based grading, and more.  I’ve got a lot of ideas for what to write, now I just need to follow through.

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