Posted by: groupact | August 22, 2016

Compelling Lessons

Nicole Hansen had an intriguing post this weekend about what makes a lesson compelling.  I think she summarized the need for compelling lessons nicely in a tweet linking to her post:

I like that she focused not on the practical benefit of student engagement, but rather phrased engagement as what her students deserve.  I agree completely.  My students generally work quite hard trying to balance school, activities, and teenage life.  They deserve a course that is compelling filled with lessons that are compelling.

Of course the big question is how to make that happen, and that is why Nicole examines the question of what makes a lesson compelling which she breaks down by what “type” of lesson it is.  I will just add a few thoughts both to the general question of what makes math compelling, and to her specific framework…

  • Different people may differ on what they find compelling, but one can still strive to tell a story that most would find compelling.  I have the best chance of doing that if at the least I find the story compelling.
  • It also helps to know one’s audience.  I try to know my students’ interests in general, and their interests in mathematics.  I frequently ask them (both in class and through homework surveys) what they’ve found interesting or not so much, and what they’re curious about learning more about.  This is also one of the many benefits of the noticing/wondering framework.
  • Stories are compelling.  This is captured in the idea of conflict which Nicole quoting Daniel Willingham starts her post.  It is the idea of Dan Meyer’s  “Three Acts”.  It is also the idea of Graham Fletcher’s ShadowCon session: “Becoming a Better Storyteller.”
  • Fletcher in particular notes that the story structure goes beyond just an individual lesson, and extends to the progression of ideas across a year and even beyond which is why he encouraged everyone to read the progressions.   I  have had some success in making dimension a theme/thread that runs throughout my 9th grade math course which I believe help makes the course more compelling.  (I’ll note that Salvador Daliand Madeleine L’Engle both found the concept of higher dimension compelling).
  • I agree with Nicole (and Chris) that debatable questions can be quite compelling.  One source for that can be definitions.  For example, “Is a line parallel to itself?”  Students are shocked to learn that there is no right answer…it depends on how we choose to define parallel lines.  The real debate is over what choice to make, and that requires some discussion about  what is so important about the concept of parallelism.
  • That raises a factor that can arise in questions and problems.  Ideas or solutions that are surprising/shocking can be quite compelling.  An example of such a problem is the wheat and chessboard problem.

I’m sure I’ll be thinking about this more as the year progresses.  Thanks Nicole!


Posted by: groupact | May 17, 2014

Chapter 7: Scale

This is an initial draft for the seventh chapter of a course guide for a year-long 9th grade Algebra & Geometry course.  I am requesting your assistance. For more information on the course and this project, please see this post.

Keith Devlin had a series of columns in 2008 (June, July-August, September) arguing that multiplication is not repeated addition, but rather scaling. But, unfortunately, far too many of my students come in thinking multiplication is repeated addition (or perhaps just a symbol on a calculator). It is vital that we spend some time discussing the all important concept of scale.

When to do so, though, is a tough question. I think it would be helpful to have the discussion before getting into formal definitions of dilation and similarity (chapter 8), but I also wanted to hold off on asking students to revisit their thinking in this area for as long as possible. Thus while we certainly touched upon scale when looking at unit (chapter 2) and area (chapter 6), it is only now that we’ll closely examine multiplication as scaling.

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Posted by: groupact | April 23, 2014

Chapter 6: Area and Volume

This is an initial draft for the sixth chapter of a course guide for a year-long 9th grade Algebra & Geometry course.  I am requesting your assistance. For more information on the course and this project, please see this post.

Area and volume are incredibly subtle concepts (see for example the Banach-Tarski paradox). Even if we can’t handle all of the subtleties yet, any course on geometry should spend some time discussing these central concepts and thinking about what properties we’d like them to have. This seems like a great time to have that discussion in this course. One of the central assumptions it would seem we’d want to hold for area is that it is preserved by congruences and we’ve spent the last four chapters defining what that means. Our discussion of area and volume is a reminder of why we’ve been doing that as we returns to our original motivating exploration of dimension. A key idea that should arise in our discussion is Cavalieri’s principle which builds upon the concepts of parallelism and congruence and even introduces a new non-rigid transformation “skew” which preserves distances only in one direction.

Now is also a good time to have this discussion as we are about to explore the concepts of scale and similarity (chapters 7 and 8) in which we’ll want to explore the effects on length, area, volume (and even higher dimensions). Scale will also provide an important way to think about multiplication, but it is worth first considering the area model of multiplication. In both cases we’ll focus especially on the implications of the model for the multiplication of fractions as that’s a topic where students often rely solely on procedure with limited understanding.

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Posted by: groupact | April 18, 2014

Chapter 5: Symmetry

This is an initial draft for the fifth chapter of a course guide for a year-long 9th grade Algebra & Geometry course.  I am requesting your assistance. For more information on the course and this project, please see this post.

Now that we’ve established the formal definitions of rotations, reflections, and translations, we’re ready to define a congruence as a finite composition of these rigid transformations. That forces us to define the concept of composition and it also leads us to want to develop a more convenient notation for transformations that can handle compositions well, namely parentheses notation. It would be rather upsetting if we had a figure S which was congruent to T, but T wasn’t congruent to S. That is we’d like to be assured that a congruence from S to T necessarily implies the existence of a congruence from T to S, but that we can handle if we introduce the notion of the inverse of a transformation.

We could introduce all of these concepts just motivated by the desire to define congruence, but I have found the notion of symmetry (which for now we consider to be a congruence from a figure to itself) to be more directly engaging. It also helps motivate better why we’d be interested in identifying the specific transformations that compose, instead of just being interested in whether such a sequence exits.

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Posted by: groupact | April 17, 2014

Chapter 4: Rotation and Reflection

This is an initial draft for the fourth chapter of a course guide for a year-long 9th grade Algebra & Geometry course.  I am requesting your assistance. For more information on the course and this project, please see this post.

One of the biggest challenges in a high school geometry course is to motivate the formalism. Why do we need precise definitions? Why must we prove things that seem obvious? Even things that don’t seem obvious, like the Pythagorean Theorem, why not just accept them as facts because we’ve been “taught” that they’re true? Why are we often asked to prove things that seem irrelevant? One approach is to give up on the formalism all together. Have students measure the angles of a bunch of triangles with a protractor and “observe” that they all seem to add up to 180 degrees. Another approach is to just follow Euclid (mistakes and all) and painstakingly work ones way up carefully until eventually something interesting is proven. The common core seeks to strike a balance by starting with a few assumptions and definitions that seem reasonable, but which also allow one to quickly prove key geometric facts such as those needed to define the concepts of slope and coordinates.

Even with that compromise, though, there still remains the issue of motivation. In this course I’ve been trying my best to build up some motivation for the formalism instead of starting off on Day 1 with a bunch of definitions and axioms. In chapter one we hopefully came to realize that formalism would be necessary to abstract to study something like higher dimensions where we cannot rely on intuition and that even in lower dimensions patterns don’t always generalize as we might expect them to. In chapter two we got a glimpse of a situation where the Pythagorean Theorem wouldn’t hold and in chapter three a glimpse of a situation in which a triangle’s angles didn’t sum to 180 degrees. These explorations, even if quite brief, will hopefully challenge us to be explicit about our assumptions. Finally we saw when looking at translations in chapter three that some understanding of parallelograms is going to be essential.

So that brings us to this week where we formally define the notion of rotation and reflection, and fill in the necessary details from the previous chapter concerning translation and parallelism. I expect to make great use of Geogebra this week, and perhaps a final act of motivation for formalism can be thinking about how does the computer determine precisely where to put the image of a point under these transformations.

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Posted by: groupact | April 12, 2014

Chapter 3: Direction

This is an initial draft for the third chapter of a course guide for a year-long 9th grade Algebra & Geometry course.  I am requesting your assistance. For more information on the course and this project, please see this post.

Distance is probably the most important concept in geometry, but direction is not far behind. The two concepts complement each other wonderfully. Given two distinct points, A and B, we can think about the distance between the two points as well as the direction which is directed from A to B (which always yields a second opposite from B to A). The direction from A to any point on the ray from A to B is considered to be the same direction as A to B and we assume that translation preserves both distance and direction. Thus in one-dimension we had only two directions: the direction from 0 to 1 defined the positive direction and the direction from 1 to 0 defined its opposite, the negative direction. To extend to higher dimensions we imagine a new point that is not on our number line and thus obtain a new direction. We will still want to be able to use any two points to define a translation in such a way that if the same translation takes A to C and B to D, then the distance and direction from C to D is the same as the distance and direction from A to B and the distance and direction from A to C is the same as the distance and direction from B to D. Central to this discussion will be the notion of parallel lines. This idea of translation now forces us to consider an entire plane of points…our two-dimensional space.

One way of thinking about direction is in this “vector” sense. Two points determine a distance and a direction and define a translation by that distance in that direction. Now let us consider another way that direction will show up throughout high school mathematics. Consider a circle in the plane centered at one point A and passing through another point B. Whereas the ray from A to B considers all the points in the plane that are in the same direction as A to B, but that are at any possible distance. The circle considers all the points the in the plane that are at the same distance from A as B, but that are in any possible direction. So choose a point C on this circle. This defines a direction from A to C, but we can also think about that in terms of a direction relative to the direction from A to B. We do that by defining an angle BAC which is defined as the region between the ray A to B and the ray A to C. We can “measure” that angle in several ways that connect the angle with how much of the circle it intersects.

The notion of direction will not only be key in thinking about translation, rotation, reflection, parallel lines, and angle measurements. It will also be key when we think about the slope of a line. We are thinking of a line already as being connected with a certain direction (and its opposite). So if we establish one horizontal line and positive direction on that line, we can think of a line’s direction as being determined by its “angle of elevation” ie one of the smaller angles formed with the horizontal. Once we establish the notion of similar triangles we can say that this in turn defines a fixed ratio, the slope the line. In a later course on trigonometry we give names to the functions (tangent and arctangent) defined by this correspondence.

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Posted by: groupact | April 10, 2014

Chapter 2: Distance

This is an initial draft for the second chapter of a course guide for a year-long 9th grade Algebra & Geometry course.  I am requesting your assistance. For more information on the course and this project, please see this post.

James Tanton introduces his Geometry volumes with the following:

I make a bold and nonstandard claim. Setting technical details aside, there are
essentially only three key (“non-obvious”) components to geometry:


I agree with Tanton except I would broaden each of these notions slightly. I would broaden the third concept to similarity in general, the second concept to the notion of direction, and the first concept to distance. I’ll explain my rationale for broadening the other concepts as we get to them, but the reason for extending the first concept is to think a little more about one-dimension. Some of the most common difficulties I see from students (in particular when working with fractions, combining like terms, modeling, or understanding trigonometry) seems to stem from neglecting to think about the importance of units. So I believe there is a need to revisit the concept of unit and the number line.

The challenge of doing this, as I alluded to in the last chapter, is that students may consider it to be “complicating” something they’ve “already learned.” To help combat that I’ve made two adjustments this year to what I’ve done in the past. The first is to use the desire to understand higher dimensions to motivate thinking more carefully about lower dimensions. The second is to spread out the work done on thinking about the meaning of addition on multiplication. So in this chapter we look at the concept of the number line and addition. We wait until the unit on area (chapter 6) and scale (chapter 7) to consider multiplication.

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Posted by: groupact | April 9, 2014

PARCC’S Popcorn Problem

One of the challenges in writing modeling problems is that they are inherently messy. If every aspect is cut and dried then you’re not really modeling. The problem should call for making some estimations and making some assumptions. That, however, makes writing a standardized rubric really hard. One student might make different assumptions that hadn’t been expected by the problem writer. Let’s take, for example, the PARCC sample problem called Popcorn Inventory.
The student is given a lot of data in a table (most of which is unnecessary to solve the problem, but again that is part of modeling) that can be used to determine what happened over the last 4 days as well as some assumptions in word form to help us determine what was likely to happen on the previous (or next) 3 days. The problem seems to indicate that last week is typical of what to expect this week. The student is asked to determine a reasonable estimate of how many cups of popcorn seed to purchase in order to handle the popcorn needs for the upcoming week.

The rubric seems to expect a student to use the table to determine the average weekday consumption rate of the various size popcorn containers. It then expects one to use that and the given information to extrapolate the number of containers of each that would be consumed on the weekend (Fri-Sat). Then one would use a given conversion rate to convert the consumed popcorn to an amount of popcorn seed. One would also scale the 4 given weekdays usage of popcorn seed to get how much would be used over the 5 weekdays (ie account for Sunday) and finally throw in some extra popcorn seed to bring us up from our dangerously low level of seed into the desired comfortable range.

This sample solution (provided in the rubric) assumes that on the weekend no popcorn is popped unless it goes out in a box, but the data indicates that at least during the week quite a bit of extra popcorn seed is used. How would graders handle quite different solutions with alternate (probably more reasonable) assumptions? Stay with me past the jump for my solution when I did the problem and some other alternates.

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Posted by: groupact | April 7, 2014

Chapter 1: Dimension

This is an initial draft for the first chapter of a course guide for a year-long 9th grade Algebra & Geometry course.  I am requesting your assistance. For more information on the course and this project, please see this post.

I used to start previous versions of this course with factoring of positive integers. My thought was that it made sense to start with one of the more basic “skills” in the course (arithmetic with natural numbers) and then gradually get more complicated. I have come to believe that this is the wrong approach. The first problem with starting with the basic is that it is not terribly engaging. Just as Dan Meyer bases much of his work on the philosophy that a mathematical task should tell a story starting with a visceral first act that draws one in and motivates the learning, I have come to believe that the same should hold for a mathematical course. The second problem with starting with any material that students already “know” is that they often fail to see a need to try to understand the underlying concept. They were capable of learning “that topic” once (even if they don’t now remember “details”) and they did so not by any productive struggle, but rather by being shown a procedure and practicing it. Learning it again seems to be a waste of time as “that topic” is already checked off their list, and furthermore I’m just making it unnecessarily difficult.

My hope is that starting the year by considering higher dimensions will help with both of these issues. I used this idea in the middle of the semester last year and the students were fascinated with the idea. (End-of-semester surveys generally had it as one of their favorite topics and one they wanted to learn more about). Furthermore many of the students bought into the idea that in order to explore an arena like higher dimensions, where we lacked a great deal of intuition, it would be necessary to think more carefully and try to formalize some of our intuition from lower dimensions. (This incidentally is what I think of when I think about mathematical rigor.) The concept of dimension makes for a great first act for this course. It is both a hook that draws students in setting up a conflict they want to resolve, and a thread that runs throughout the course (see part (D) below). Throughout the course we will answer some of our initial questions about higher dimension, but raise new ones. The course should end not with every question answered, but with enough resolution about our initial questions that we leave with a sense of accomplishment.
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Posted by: groupact | April 5, 2014

Algebra and Geometry Course Overview

I need your help. I am putting together a course guide for a yearlong 9th grade course on Algebra & Geometry. I have divided the course into 27 “weeks” plus an appendix. For each week my goal is to have
(A) The key ideas for that week and why we’re focusing on them at that point in the year,
(B) Related standards from the common core,
(C) Connections between the key ideas and mathematics students have done,
(D) Connections between the key ideas and mathematics students will do,
(E) Texts and videos to which students and parents could refer to help understand key ideas,
(F) Questions and tasks that can generate discussion to explore key ideas in class,
(G) Questions and tasks that can be used to help assess student understanding of key ideas, and
(H) Interesting and challenging ways to extend and apply the key ideas.

For some weeks I might also have
(I) Standard textbook problems related to the key ideas, or
(J) Misconceptions and “tricks” to avoid (a la Tina Cardone’s Nix the Tricks).

My plan is to devote a blog post for each week where I will share what I have so far, and then hope that others can share suggestions they may have. I am especially going to be in need of suggestions for (E-G) above (i.e. references, questions, and tasks—I hope that all students will have access to Geogebra in class). My goal is to publish 3-4 posts a week so that the whole course outline in as much detail as I have it now will be available in the next 2-3 months (by the end of June at the latest). In this post, I’ll try to explain a little about the background and motivation for the course. I’ll try to be brief in this post and save the details for the subsequent posts on each week and then revisit the course overview at the end.

The audience for this course is the entire 9th grade student body at our school. We do not track, and this course is designed to fit the needs of a fairly diverse student population. Currently about two-thirds of our students come in having taken an integrated algebra course in 8th grade often scoring fairly well on a standardized state high school level exam. Even those students, though, generally come in not “remembering” much of what they learned. They memorized some procedures to get through an exam and promptly forgot them. Those students that do remember procedures nearly always still lack the conceptual understanding underlying the procedures. The key goal for this course is to provide ALL students with the conceptual understanding that they can build upon going forward. One of the challenges is to convince the students of the need for that understanding and to keep them engaged when approaching some topics that might seem at first glance to be review for them and which are completely new to others in the class.

I see the big ideas for the course as the interrelated concepts of number, similarity, slope, and solution sets. The concept of dimension acts as a hook and common thread that runs throughout the course. I’ll provide details of what I mean by these concepts and what I want students to take away with them in the subsequent focused blog posts. This course is influenced by the ideas of many people, but especially the work of three particular mathematicians whose ideas I draw upon to a great extent. First I make use of Thomas Banchoff’s ideas expressed in his book Beyond the Third Dimension and the pedagogical accompaniment he wrote for that volume which appeared in On The Shoulders of Giants. I also draw upon the ideas of Hung-Hsi Wu (especially those that appear in his course notes Pre-Algebra and Introduction to School Algebra which I believe will soon be made into a book for teachers to follow his Understanding Numbers in Elementary School Mathematics). Finally I have drawn inspiration from and will refer frequently to the work done by James Tanton. I will refer to some amazing online courses he has at as well as his Thinking Mathematics! (a 10-volume set) and Geometry (a 2 volume set).
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