Lesson Story: Track Stars

I haven't had my own classroom in a while, so when I got the chance last summer to model a lesson for some math teachers at a summer workshop, I was eager to try a task Bill Penuel turned me on to in a paper by Schwartz and Martin (2004):

Track Stars

Bill and Joe are both on the U.S. Track Team. They also both broke world records last year. Bill broke the world record for the high jump with a jump of 8 ft. Joe broke the world record for the long jump with a jump of 26 ft, 6 in. Now Bill and Joe are having an argument. Each of them think that his record is the best one. You need to help them decide. Based on the data in the table, decide if 8 ft shattered the high jump record more than 26 ft 6 in. shattered the long jump record.

Top High Jumps in 2000		Top Long Jumps in 2000
Height	Number of Jumps	Length	Number of Jumps
6'6"	1	21'6"	1
6'8"	2	22'0"	2
6'10"	3	22'6"	2
7'0"	5	23'0"	9
7'2"	6	23'5"	9
7'4"	7	24'6"	4
7'6"	4	25'0"	1
7'8"	1	25'6"	1
8'0"		26'6"

When I used this task with teachers a few years ago in our task analysis research it was rated quite highly: 5 out of 6 teachers said it rated as "Doing Mathematics" in Smith and Stein's (1998) cognitive demand framework and the task was unanimously judged as a good example of a task likely to engage students in Standard for Mathematical Practice #3, construct viable arguments and critique the reasoning of others.

Context

For the summer workshop I was working with about 20 math teachers who would be grouped by grade band (elementary, middle high) and I asked them to attempt the task using the abilities expected of students at their grade level. I admit, this makes for a somewhat artificial exercise, but I wanted to see if this task would stretch across a lot of different levels of student ability and elicit a very wide range of student strategies (even if the "students" were teachers).

One of my greatest teaching weaknesses has always been in my questioning strategies. Too often I accept quick choral responses to questions in the initiate-respond-evaluate pattern, and I don't do much to (a) push student thinking and (b) promote equitable participation, so for this lesson I used a combination of these resources:

• As a planning guide I used Smith and Stein's 5 Practices for Orchestrating Productive Mathematics Discussions.
• I used CMP's Launch, Explore, Summarize as an instructional model.
• My focus practice was facilitate meaningful discourse from Principles to Actions, and I especially wanted to sharpen my pressing and linking talk moves. The Talk Science Primer from TERC is an excellent resource on talk moves.
• I used a form of "student self-check" in the style of practical meausures to help me reflect.

There is a lot of overlap in the 5 Practices, Launch/Explore/Summarize, and the goal of facilitating meaningful discourse. That's a good thing.

The Lesson

I anticipated (the first of 5 Practices) different strategies across the three groups:

• I expected the elementary school group to focus on measuring distances and visual comparisons, and to bring up struggles around working with feet and inches and the under-developed sense of ratio.
• I expected the middle school group to calculate means and use proportional reasoning (like, "The record is 110% of the average), and perhaps use mean absolute deviation (MAD) as a measure of variability. I expected to see struggles in accounting for the multiple jumps at each distance, in calculating MAD, and debates around using mean vs. median as a measure of center.
• I expected the high school group to be similar to the middle school group, but to use standard deviation instead of MAD.

For the launch phase I avoided giving away any hints or clues about possible strategies. It was difficult to design a launch that connected to prior knowledge because of the artificial nature of teachers playing the role of students, so I took a moment to ask the teachers to think about the knowledge they'd expect students to have given the standards at their grade levels.

During the explore phase of the lesson I monitored (the second of 5 Practices) the groups for the strategies I anticipated. I wanted to use pressing questions here to push people's thinking, such as:

• "Can you tell me why you think that is correct?"
• "What do you mean by 'farther'? Is it because you added? What else might you do to measure 'farther'?"

Questions like this designed to press for student thinking were often met with teacher speculation about student thinking. As solution strategies came together, I noted them on my phone with the goal of selecting (the third of 5 Practices) two strategies per group to discuss during the whole-group summary phase of the lesson. The sequencing plan (the fourth of 5 Practices) was to discuss elementary first, then middle, then high school, with the less sophisticated strategy presented first at each level.

Here are the two posters from the elementary group:

The elementary group could quickly work through multiple strategies, so from this group I got more than just the two strategies I planned for. One set of strategies focused on how much more the record was than the next longest/highest jump, and the other set used a graphical representation of the jumps. Here are the posters from the middle school group:

One set of strategies compared the record jumps to the mean jumps, and the other set used a graphical display and interquartile range. Here are the two posters from the high school group:

There was less to differentiate these two strategies, as both groups calculated standard deviations and z-scores as a way of measuring how far above the mean was each record jump.

In the summarize phase of the lesson I focused my questioning around linking moves, such as:

• "How does your strategy compare to the first one from the elementary group?"
• (Following an explanation by Kathryn) "Tammy, do you have any questions for Kathryn?"
• "Phillip, how might your argument change if you used Dan's method?"

With questions like these, I hoped to draw connections (the fifth of 5 Practices) between ideas, such as:

• Connecting the visual centers of graphical displays with the calculated centers of the data
• Connecting MAD and SD
• Connecting the "measuring stick" idea between proportional reasoning at lower levels and the counting of MAD/SD units

Reflection

I had some hits and misses in my anticipation of the strategies I saw. The elementary teachers didn't share my expectation of focusing on measurement and comparing those measurements. Instead, they made some useful comparisons between the record and second-best jumps. I also didn't anticipate the dot plots and fitted curves in the second poster. I know it's uneasy to underestimate the capabilities of elementary students, but these kinds of graphs were not something I anticipated their teachers producing. The middle school group used proportional reasoning, as I expected, but instead of MAD they used IQR as a reference for judging the two jump records. There was one "student" who quickly worked through some MAD calculations towards the end of the work time, but it was a bit late to fit into my selection strategy. For high school, the work was less differentiated and more advanced than I anticipated. Some of this can be attributed to just labeling the group "high school" rather than "9th grade" or "AP Stats."

I was able to practice my talk moves to some degree, but this artificial scenario was less than ideal. In the explore phase of the lesson my questions were generally met with speculation about student strategies, not answers as students might give them. That was great for us all to think through the task together, but it interrupted the flow of responses you'd expect with talk moves in a more typical classroom scenario.

The discussion in the summarize phase was pretty good. Not only did we compare strategies and connect ideas in the way I anticipated, there was a welcome amount of analysis of the task itself and the different layers of ambiguity in how the data was presented. For example, we don't know if the jumps all represent different jumpers, or if the jumps represent jumps in one vs. multiple competitions. We generally agreed that some amount of ambiguity would be good when using this task in a classroom, particularly to hit the "make sense of problems" part of SMP #1.

As part of the reflection I collected data in the form of a "self-check," created in the style of "practical measures" that we've used in our research projects. In hindsight, this data doesn't focus much on my choice of teaching practice (facilitating meaningful discourse), but I like the idea of asking students for feedback that go beyond mastery of content.


Link to Google Form

The responses are a bit difficult to interpret because I'm not sure how many participants responded as teachers versus the students they were sort-of-pretending to be. The results seem mostly positive, and I agree with the very last comment: While the task had reach across many grade levels, first grade was too much of a stretch.

References

Schwartz, D. L., & Martin, T. (2004). Inventing to prepare for future learning: The hidden efficiency of encouraging original student production in statistics instruction. Cognition and Instruction, 22(2), 129–184. http://doi.org/10.1207/s1532690xci2202_1

Smith, M. S., & Stein, M. K. (1998). Reflections on practice: Selecting and creating mathematical tasks: From research to practice. Mathematics Teaching in the Middle School, 3(5), 344–350.

A First Day Statistics Activity

I have the honor of again teaching our undergraduate statistics course in the School of Education, better known here as EDUC 4716 Basic Statistical Methods. Perhaps the most interesting thing about the course is that it's not required for any education programs, minors, or certificates. Instead, the course attracts students largely from the Department of Speech, Language, and Hearing Sciences (who don't need it to graduate, but do need it to apply to grad school or, more recently, to get certified) and sociology majors. So how does this course end up in the School of Ed? Probably due to the legacy we have in quantitative methods, thanks to people like Robert Linn, Gene Glass, Lorrie Shepard, and now faculty like Derek Briggs and Greg Camilli. Somehow all of their hard work and success filters down and gives a relative stats-hack like me a chance to teach undergrads.

Many of my students are upperclassmen and have spent much of their college experience avoiding math courses. In fact, on last year's FCQ (Faculty Course Questionnaire) my students' average rating for the item "Personal interest in this subject prior to enrollment" was a 1.8 out of 6 -- a response the university tells me is at the 0th percentile across campus. I like to think of this as a great opportunity in a "nowhere to go but up" kind of way, a chance for me to change the way students think of mathematics and see themselves as mathematical beings. Then again, it's hard to make big changes in only 15 class meetings of 2.5 hours each. If I'm going to make a difference, class has to get off to a solid start.

My opening activity this year started with the preparation of four simple index cards with different distribution shapes:

Four common distributions, clockwise from top left: normal, left skewed, right skewed, and normal.

I have the benefit of a small class of 14 students. So I cut my graphs into a total of 14 pieces:

14 pieces for 14 students. Note on the bottom I've provided the hints A, B, C, and D.

When class started, I mixed up the graph pieces and handed one to each student. Then I told the class to find the other people in class who had the graph pieces that aligned with theirs. Once they had a completed graph, form a group at one of the tables and discuss which of the following they thought their group's graph might describe:

• People born each month of the year
• Student GPAs at this university
• Student heights at this university
• Starting salaries of new graduates from this university

It took my class less than 3-4 minutes to find their groups and then I gave them another 3-4 minutes to discuss what their graph shape might describe. As a class, I had each group share their ideas and then we discussed them. Not everybody agreed initially about which shape matched which description, which led into important comments about how we might think about unbiased sampling of students and imagining different scales and labels along the horizontal axes.

So in less than 15 minutes I combined group-making, statistics, and active, student-centered problem solving into one activity. This activity also gets students thinking about distribution shapes, which I sometimes worry we ignore in the rush to calculate centers and spreads. If you're wondering how to adapt this for your classroom, I offer these suggestions:

• If you have a few more students, cut more slices.
• If you have twice as many students, consider making two of each distribution shape and scaling the x-axis to match one of 8 potential descriptions. (i.e., a normal distribution scaled for heights in inches could be distinguished from one scaled for SAT scores.)
• If you want to use this for Algebra 1, you can make graphs that describe things like, "Toni walked to the bus stop at 2 mph, rode the bus at 30 mph to the bike shop, then rode a bike back home at 12 mph." Such an activity begins CPM's Algebra Connections and was the inspiration for my activity.
• If you want to use this for Algebra 2 or higher, you can use graphs of functions that students will become familiar with (parabolas, cubics, hyperbolas, etc.). I don't think it's worth fretting over vocabulary at this point -- just give students an opportunity to think about how the functions behave and what phenomena they could possibly model.

Modeling Dimensional Analysis

I generally ask myself two questions when I examine the design of a mathematical task:

1. What is the context?
2. How can we model the mathematics?

Mathematical concepts with tasks for which these two questions can be answered easily tend to be easier to learn, while teaching and learning generally becomes more difficult when one or both of those questions can't be answered. For dimensional analysis (sometimes called the unit factor method or the factor-label method), the first question is easy to answer. It doesn't take much of an imagination to design a measurement conversion task that is set in a real-world context. A model, however -- whether visual, mental, or a concrete manipulative -- is generally absent. Typical dimensional analysis problems look like this:

Q: What is 60 miles per hour in meters per second?

A: \( \frac{60 \mbox{mi}}{1 \mbox{hr}} \times \frac{5280 \mbox{ft}}{1 \mbox{mi}} \times \frac{12 \mbox{in}}{1 \mbox{ft}} \times \frac{2.54 \mbox{cm}}{1 \mbox{in}} \times \frac{1 \mbox{m}}{100 \mbox{cm}} \times \frac{1 \mbox{hr}}{60 \mbox{min}} \times \frac{1 \mbox{min}}{60 \mbox{sec}} = \frac{9656064 \mbox{m}}{360000 \mbox{sec}} = \frac{26.8224 \mbox{m}}{\mbox{sec}} \)

For those who successfully learn dimensional analysis this way, there's a certain beauty to how the units drive the problem and how the conversion factors are nothing more than cleverly written values of one, the multiplicative identity. Unfortunately, many students struggle with this method. Some are intimidated by the fractions, some can't get the labels in the right place, and some just can't get the problem started.

What we need is a model. Let's start with the most basic of unit conversion models, a ruler with both inches and centimeters:

(Yes, I'm still using the same ruler I got as a 7th grader in a regional MathCounts competition.)

With only simple visual inspection, students should be able to use a ruler to estimate conversions between inches and centimeters. This is an informal model, one students can literally get their hands on. We can assist the learning by making the models progressively more formal. Here we model a trivial conversion from one inch to centimeters with a double number line:

(Yes, you still have to know your conversion factors!)

Such a simple example looks almost too easy to be useful, but we can add number lines for more complex conversions. We can even abstract the model further and go beyond conversions of distance. Suppose we wanted to convert 3 gallons to liters. I could model that conversion with number lines this way:

(I could have used any number of transition units, but I knew 1 quart was roughly 946 milliliters.)

Filling in the question marks from top to bottom, I'll see that 3 gallons, 12 quarts, 11,352 milliliters, and 11.352 liters are all the same volume. It's easy to see they're the same because on each number line those values are the same distance from zero. Because we're only converting one kind of unit (volume), we only need one dimension.

In our initial example we were converting 60 miles per hour to meters per second. That's two kinds of units, distance and time, so our model needs two dimensions. Furthermore, it can help to think of 60 miles per hour as a line, not just a point. After all, we often travel at a speed of 60 miles per hour without actually traveling a distance of 60 miles in exactly one hour.

Can you guess where our double (or however many are necessary) number lines will go in this model? The following video will demonstrate what I would call the graphing model or two dimensional model for performing conversions.

With the work shown in the video, we haven't just done one conversion. In fact, we're prepared to write 60 miles per hour 15 different ways, not that we'll ever be asked to do that. If we needed 60 miles per hour in centimeters per minute or feet per second, all the work is done. Just choose the appropriate quantity from the vertical and divide by the appropriate quantity from the horizontal. Of course, if we're in a hurry, we won't find all those intermediate figures and instead just proceed from miles to meters and hours to seconds as quickly as possible. Will that be quicker than the traditional method shown above? Probably not, but the purpose of using a model is understanding, not speed. Once the understanding is established, students can move on to a formal method or use technology when appropriate.

Functions of Functions

Mathematical functions are usually introduced formally to students somewhere around the end of Algebra 1 or maybe in Algebra 2. If my Algebra 2 final exam had included the question, "What do you know about functions?" I probably would have said: (a) You use \( f(x) \) instead of \( y \), and (b) if a graph fails the vertical line test, it's not a function. With all due respect to my high school math teacher, this would have been a lousy answer. I might have known some peripheral information, but not the core understanding. Sadly, the mathematical importance of functions is not that difficult conceptually, yet it's crucial to the majority of the content learned in Algebra 1 and 2. Yet students still struggle with what makes a function a function.

My first experience trying to teach the definition in a non-traditional way was using the "Cola Machine" problem in CPM Algebra (Math 1). The problem, several days into Unit 11, describes the following:

The cola machine at your school offers several types of soda. Your favorite drink, Blast!, has two buttons dedicated to it, while the other drinks (Slurp, Lemon Twister, and Diet Slurp) each have one button.

1. Explain how the soda machine is a relation.
2. Describe the domain and range of this soda machine.
3. While buying a soda, Mr. Hagen pushed the button for Lemon Twister and got a can of Lemon Twister. Later he went back to the same machine but this time pushing the Lemon Twister button got him a can of Blast! Is the machine functioning consistently? Why or why not?
4. When Karen pushed the top button for Blast! She received a can of Blast! Her friend, Miguel, decided to be different and pushed the second button for Blast! He, too, received a can of Blast! Is the machine functioning consistently? Why or why not?
5. When Loufti pushed a button for Slurp, he received a can of Lemon Twister! Later, Tayeisha also pushed the Slurp button and received a can of Lemon Twister. Still later, Tayeisha noticed that everyone else who pushed the Slurp button received a Lemon Twister. Is the machine functioning consistently? Explain why or why not. (Sallee, et. al., 2002, p. 375)

Most textbooks define functions approximately the same way: a relation is a function if there exists no more than one output for each input. Without a context, however, that definition might not carry much meaning, and relying solely on the vertical line test in a graph may not be helpful enough for many students. The soda machine problem, with its emphasis on consistency, gives both teacher and student a very approachable context within which to discuss what a function is and is not.

Early in my teaching career I thought Algebra 1 basically boiled down to two big ideas: solving equations and graphing lines. If students could do those two things, I felt pretty good about them passing my class. Now I see the big ideas of Algebra 1 differently and basically aligned with Colorado's revised standards for high school mathematics. The six expectations listed for Colorado's high school algebra standard can be summarized as follows:

1. Functional representations (equations, graphs, and tables)
2. Function behavior (qualitative)
3. Function transformations (parameters and parent graphs)
4. Equivalent expressions, equations, and inequalities
5. Solving equations, inequalities, and systems of equations
6. Mathematical modeling using functions

Four of the six expectations explicitly mention functions. The last expectation, mathematical modeling using functions, represents (for many math educators) the ultimate goal for math instruction: to give students the mathematical power to describe and understand the world around them. Instead of just solving and graphing, the big idea of high school algebra is functions, with most linear function work in Algebra 1 and most non-linear function work in Algebra 2.

I think it's a mistake to delay the explanation and definition of functions until late in Algebra 1 or later. Algebra 1 and younger students can understand the cola machine problem or other, similar contexts. Suppose the class plays an "exchange" game. Student A gives the teacher three triangles in exchange for two squares. What should Student B expect to get in exchange for his three triangles? What exchange would represent a function versus a non-function? For something based more in the real-world, this conversation could be set in the context of currency exchange. Also, a helpful model for learning functions might be function machines, such as:

Function machines are helpful models for learning functions, function composition, inverse functions, and even solving equations. They help stress the input-output relationship in a way that words or equations alone might not. Students will also naturally expect a single output for each input. The real challenge for Algebra 1 or younger students is to present a variety of equations that aren't functions, or else risk having students think that every two-variable equation describes a function.

Patterns can also be used to teach functions. If students are given the sequence 2, 4, 8, …, some students are likely to predict 14 as the next term (adding 8 plus 6, the next consecutive even number), while other students might predict 16 (the fourth power of two). Because the fourth term could reasonably be two different values, we can't establish a functional relationship to describe the sequence. This could even be an example worth graphing to discuss the meaning of the vertical line test:

Our calculators also help us distinguish functions from non-functions. If you try to graph a circle on a Texas Instruments graphing calculator, you have to enter two functions: one for the top half of the circle, and one for the bottom half. Therefore, the written forms of the equation of a circle (such as \( x^2 + y^2 = 1\) or \(y= \pm \sqrt{1-x^2} \)) can't be functions. The graphing calculator is also a good tool for discussing the square root function, and why its graph must only be half of a parabola's inverse if we want it to be a function.

There two key obstacles that are likely to remain in a student's way of understanding functions. First, students will continue to focus the vast majority of the time and effort on functions, and there are too few real-world examples of useful non-functions to help distinguish the two. Non-functions are less powerful and less common, but without them we risk having students who casually accept that any equations with an input-output relationship is a function. The second obstacle is notation, and it's not an obstacle we can likely avoid. The change to function notation, such as using \(f(x) = 3x - 2\) instead of \(y = 3x - 2\), is not only difficult to explain as something other than an arbitrary change in symbols, but includes two aspects that are directly contrary to a student's prior knowledge. Now a letter (such as the \(f\) in the preceding example) is no longer a variable, but a name with no numerical value by itself. Function notation also uses parentheses to represent something other than multiplication, adding more work to the already overloaded duties mathematicians place on these two simple arcs. The power of function notation is the preservation of the input (\(x\)) with the output, but it's confusing that the output (\(f(x)\)) reuses the input variable, uses a letter that isn't a variable, and uses symbolism for multiplication of variables that's no longer multiplication. Perhaps a question like this could help students realize the power of the notation:

1. If \(y=16\), \(y=49\), and \(y=64\), what might be an equation relating an input, \(x\), to the output, \(y\)?
2. If \(f(4)=16\), \(f(7)=49\), and \(f(-8)=64\), what is \(f(x)\)?
3. Are the two questions above the same? Which one is easiest to understand? Why?

References
Sallee, T., Kysh, J., Kasimatis, E., & Hoey, B. (2002). College Preparatory Mathematics 1 (Algebra 1). (L. Dietiker, Ed.) (2nd ed., Vols. 1-2, Vol. 2). Sacramento, CA: CPM Educational Program.

Patterns of Patterns

As a young math student I knew tons of formulas and how to use them, but when it came to counting and generalizing sequences of numbers I often had to resort to brute force, or at least guess-and-check. It frustrated me that I knew there should be an easier way to generalize number patterns, but because I could usually get the right answer (patience and accuracy were on my side, thankfully), I didn't force myself to understand patterns more deeply.

For my first three years of teaching I used the original CPM series, and in their Algebra (Math 1) text they presented students with three key number sequences: the square numbers, the rectangular numbers, and the triangular numbers. These sequences appear so frequently that knowing and understanding their generalizations can be helpful when the sequences appear explicitly or when they are embedded in the foundation of another pattern.

The square number pattern, shown below, is the simplest of the three patterns, although students who are struggling to move beyond a recursive view of the pattern are likely to describe the sequence as "adding the next consecutive odd number." Because of this, the square numbers become a great example for students to see the importance of moving beyond recursive descriptions of patterns and towards expressions that yield the number of tiles for any figure.

The rectangular numbers are the next sequence in the progression of patterns. Instead of the expression (or ) as with square numbers, the rectangular numbers use for one dimension.

The triangular numbers look like a new pattern:

But in fact, the triangular numbers can be seen as half of each rectangular number. This means we can modify the rectangular number expression and represent triangular numbers using .

There are real-life representations of the triangular numbers, for sure (bowling pins, stacks of cans or boxes, etc.), but the real mathematical power behind leading students through this progression is that they see two examples of how to modify a previous pattern and generalization to get a new pattern and generalization.

Our class was presented with a problem known as the "Skeleton Tower," borrowed from http://www.wcer.wisc.edu/archive/nise/Publications/Briefs/Vol_2_No_1/. I've attempted to re-illustrate the tower in the figure below:

Suddenly our figures jump from 2-dimensional to 3-dimensional, but in doing so we've opened up the strategies students might use to generalize the pattern. If a student looks at how the construction of the tower progresses from figure to figure, he or she may think of the tower as a sum of horizontal layers. If a student were only given the table, or the sequence 1, 6, 15, 28,..., he or she would most likely see the mathematics of the sequence the same way.

On the other hand, if a student were familiar with the power of the square, rectangular, and triangular numbers, they may spatially reason the tower to be a single column of blocks of height , surrounded by four "wings," each in the pattern of the triangular numbers. Because the height of each triangular wing is , not , an adjustment to the triangular number expression must be made, giving us a new expression of to describe each of the tower's wings. All together, we can express the number of blocks in a skeleton tower of height as follows:

If a student were to rewrite this expression as , they might visualize the tower as rearranged with two opposite wings removed and each stacked on top of the remaining wings, making a rectangular shape with height and width . Remember, the triangular wings were undersized to begin with; the only reason the width is not is because the center column remains in the middle. Here is the pattern "flattened," which hints at solutions using both the overall dimensions (as just discussed) or using the rectangular numbers plus a center column:

The progression through the square, rectangular, and triangular numbers certainly makes the skeleton tower more approachable. But what if we build a tower that is more complex? What if our three fundamental number sequences are more hidden, and spatial "flattening" strategies are less apparent? Below is one such possible tower, an extension of the skeleton tower.

Unfortunately, after several hours of trying, no generalized expression for the number of blocks in this new tower was found. No amount of spatial reasoning would allow me to rearrange the blocks into an easier shape, but fortunately there were two indications that the generalization would be a cubic.

First, unlike the skeleton tower, this shape is much closer to a true pyramid, so it was difficult to imagine it losing its 3-dimensional character no matter how many blocks were rearranged. (I tried variations of the formula for the volume of a pyramid, , with no success, but that did make me hypothesize that thirds of a cube could be involved in the correct generalization.) My second clue that the generalized expression was cubic was seen in the difference between the terms of the sequence (which I had to extend to a 5th figure):

Because the third differences reach a constant term (four), the generalized expression must be a cubic. This patterning of differences is a very handy tool to have in these cases, but I must admit I don't fully understand why it works, or if it can be used as an aid in determining the expression.

(UPDATE: See update below for ways I could/should have found the generalized form.)

With no real hope of deriving the generalized expression via non-mechanical means, I gave up and used my calculator to find a cubic regression. The generalized expression for this second cube stack is . (The thirds are there, as I suspected, although that was far more luck than intuition.) Even with this expression I still can't imagine a physical restacking of cubes (or fractions of cubes) that would lead a student to this result, nor would knowledge of square, rectangular, and triangular numbers.

I find these five patterns intriguing for a number of reasons. First, we go from very basic to very difficult in a quick but logical way. The skeleton tower is a good choice because it incorporates the triangular and rectangular patterns in its solution, and it adds another dimension (literally and figuratively) that suggests new problem-solving approaches, including 3-dimensional spatial reasoning. As for my pyramid-like tower, I like that it re-establishes a disequilibrium for students (and myself). For Algebra 1 students that aren't ready for quadratic regression, being exposed to such a complex pattern might provide some motivation and could be revisited either later in the course or in Algebra 2. It might also be an example I give to students for a "design-your-own" pattern workshop, where students manipulate different permutations of the square, rectangular, and triangular patterns in 2-dimensional and 3-dimensional shapes.

UPDATE (2010.07.02): With great thanks to two friends/followers, I have two more methods of finding the generalization for my pyramidial stack of blocks.

@msmathaddict sent a tweet a link to the Dr. Math portion of the Math Forum that filled in my missing step from my pattern of differences. Now that I've seen the solution it seems obvious, but we can use a system of equations to devise the correct degree equation we're looking for.

My third differences reached a constant, so I knew the equation describing the equation would be third degree. Third degree (cubic) equations are generalized in this form:

From my original figures and table I had four known pairs: (1,1), (2,6), (3,19), and (4,44). Substituting into the generic cubic form, I can create a system of four equations with four variables:

Simplified, the system becomes:

The system might be most easily solved using matrices (because they are easily entered and manipulated in most graphing calculators):

Solving for , we get , so the general cubic becomes .

The second method for deriving the forumla for my pyramidial stack of blocks was sent to me by my friend Andrew Drenner. He opted to use a summation property to summarize the recursive nature of adding increasingly larger layers. He explains:

In a given horizontal slice of the pyramid there is cubes of height . Thus, the problem of finding the total blocks () in a pyramid tall becomes

.

Given that the summation sequence for squares is

,

can be expressed as follows: