## Surprise, Surprise: Tangent Circles Produce an Ellipse

This will be the first in an occasional series of posts that offer interactive Web Sketchpad models for drawing conic sections. My interest in conic sections dates back to the mid 1990s, when I authored a conic sections book for Key Curriculum Press . That book is still available and was updated a few years ago for Sketchpad 5. You can read more about it in my prior post.

While my book is the place to go for worksheets, teacher notes, and pre-built sketches, I’ve always wanted to showcase the models on the web in an interactive form that didn’t require teachers and students to download Sketchpad. With Web Sketchpad, I can provide beautiful renditions of my conic section sketches that you can explore right here in our blog.

To inaugurate this series, I’m offering a little-known gem of a technique for drawing ellipses. The Web Sketchpad model below shows three circles. Drag point D or press Animate Point D to watch the red circle with center at point C move. As this circle moves, it remains tangent to the other two circles. Amazingly, point C traces what looks to be an ellipse!

To gather more visual evidence, drag point B to other locations so that circle c2 changes its size and location but remains inside circle c1. Then drag or animate point D again and observe the trace of point C. As another option, press the Show Locus button and then drag point B.

How can you verify that point C traces ellipses? The Show Proof Hints button offers the start of a proof that begins with this fundamental definition of an ellipse:

An ellipse is the set of points C such that the sum of the distances from C to two fixed points A and B (the foci) is constant.

The proof is simple and elegant and requires nothing more than basic geometry. While conic sections usually don’t get introduced until precalculus, proofs like this one convince me that there is much to be gained from an earlier start.

## Playing with Triangular Decompositions

Guest blogger Juan Camilo Acevedo is part of the University of Chicago’s Center for Elementary Mathematics and Science Education (CEMSE) digital team, where he develops Sketchpad-based activities for Everyday Mathematics. Currently, he teaches undergraduate language classes at the University of Chicago and is writing his doctoral dissertation on Digital Humanities. Juan holds a BA in mathematics from the University of Los Andes (Bogotá). He enjoys cross-disciplinary approaches to teaching and learning as well as the use of technology to enhance pedagogical experiences.

A great feature of Dynamic Geometry software like The Geometer’s Sketchpad is that it allows users to tinker with geometric objects. In my regular work day, it is very possible to find me lost in my thoughts playing with Sketchpad, trying to create beautiful shapes with isosceles and right triangles. The ability to drag shapes while changing their dimensions and orientation is really fun.

I don’t know if this rings true for you, but it happens to me a lot: Casual play without a specific goal in mind leads me to discover something new. I still remember the first time I played with a Rubik’s Cube, and I got a whole side of the same color. I honestly had no idea how I had done it! It was only when I tried to undo my moves and repeat the process that I understood better how the cube worked. I think the ability to play is what makes puzzles like Rubik’s Cube so powerful and educational. Play, of course, need not be limited to tangible objects. Sketchpad does a terrific job of allowing students to tinker in the virtual realm.

In the interactive Web Sketchpad model below, students are given a fixed basic shape (like a rectangle or a parallelogram) and a set of draggable “dynamic” triangles that can move, grow, and shrink. The goal is to exactly cover the fixed shape with the given triangles with no overlaps. The triangles are color-coded: orange for right, blue for isosceles, and red for equilateral. This makes it easier for students to keep track of which properties are important for each puzzle. Before you read further, try your hand at the collection of  11 challenges in this Web Sketchpad model.

Students are not required to know beforehand the underlying geometric properties of the shapes. Instead, the objective is for students to approach each challenge like they would any puzzle—it’s a game, not a problem. Just by moving the triangles, students figure out how the triangles fit the given shape. With the triangles in place, the question of why the arrangement of triangles works emerges for the student and helps to illuminate a particular geometric property of the given shapes.

An example from the fourth page of the web sketch will make this idea clear. Students are given a random ΔABC (neither right, isosceles, or equilateral) as the fixed polygon that needs to be decomposed into two right triangles. It requires some playing around to find the answer, but certainly not much. Students start with one right triangle and soon realize that two of its vertices must coincide with two vertices of  ΔABC.

From there it is a matter of resizing and rotating the triangles so they fit. The result, shown below, is that the two right angles of the right triangles are adjacent and lying on the base of ΔABC. What information does this convey? Students have just found the height of ΔABC! They also have discovered that two right angles form a line. But can a base have more than one height? Is there another way to fit the triangles together? By playing with the triangles, students will prove or refute their conjectures intuitively through hands-on play.

Now, let’s repeat this puzzle with a more complex example. Go to the eighth page of the web sketch. The student is given a right triangle, DEF, to cover and two isosceles triangles to use as pieces. Just like the previous example, the student realizes that two vertices of the isosceles triangle need to coincide with two vertices of the fixed ΔDEF.

With just a little more work, an amazing result emerges: The two isosceles triangles share an equal side in common, and hence the hypotenuse of ΔDEF is divided into two equal parts. Yay! The student has found the median of ΔDEF.

Even if it’s not explicit, the student has uncovered a theorem equivalent to the fifth postulate: a right triangle can be divided into two isosceles triangles with the equivalent sides of the same length. This is a foundational result that we do not expect the student to understand fully, but we do expect her to observe it. And it will feel very natural to the student because she built it. The difficulty of finding the right configuration will give her the idea of how strict this result is: The triangles fit into ΔDEF, but only in a very particular way. But why in such a particular way? The student might say, “Well you see, because you need this side to be equal to this and this equal to that.”

The web sketch has a total of 11 pages, all with underlying  geometric results that  students will discover through hands-on experimenting. I hope and your students you enjoy the models as much as I did building them!

## Understand the Sine Function by Dancing It

In Where Mathematics Comes From, cognitive scientists George Lakoff and Rafael Nuñez assert that our understanding of abstract mathematical concepts relies upon our sensory-motor experiences:

“For the most part, human beings conceptualize abstract concepts in concrete terms, using ideas and modes of reasoning grounded in the sensory-motor system. The mechanism by which the abstract is comprehended in terms of the concrete is called conceptual metaphor. Mathematical thought also makes use of conceptual metaphor, as when we conceptualize numbers as points on a line.” (p. 5)

If we understand numbers as points on a line, then we can conceptualize numeric variables as moveable points on a line, and functions as a particular kind of coordinated movement of two points on two lines. Just as the coordinated movement of two people often takes the form of a dance, so too we can view the coordinated movement of two points as a function dance, a dance in which the independent point’s location on its line determines the dependent point’s location on its line.

Try out the sine dance below. When you press the Go! button, three seconds later independent point x will move at a constant speed to the right along its line. Your job is to drag the bunny up and down so that the red dependent point at the tip of the bunny’s arrow is in the right place to represent sin(x). Fortunately the blue point moves up and down to show you where you should be.

Try it out now:

As with a real dance, it takes some practice to get it right. Make it faster (and harder) by moving point challenge to the right (speeding up the dance) or move it to the left for an easier dance. You can also edit the parameters to change the the nature of the dance.

What did you find challenging about doing the sine dance? Did it give you a different sense of the relative rate of change—the way the dependent variable sped up and slowed down—than you might have expected? Doing such dances can provide novel kinetic insights as you realize how fast (or slow) y moves as x travels along its domain. You get to experience the function’s rate of change through your own bodily motion.

This sketch, and other “function dance” sketches, are still being developed into finished activities. I’d really appreciate your comments and suggestions, especially if you’re willing to try them out with your students. Please email me: stek at geometricfunctions dot com.

Daniel Scher and I will be presenting this and other thought-provoking activities at the NCTM Annual Meeting next week in New Orleans. If you’re interested, please come to session #638 (Two Birds, One Stone: Transformations, Functions, and the Common Core) on Saturday at 11 am.

## Iteration in the Complex Plane

The cover story of the April 2014 issue of Mathematics Teacher is on “Iteration in the Complex Plane”, by Robin S. O’Dell. It sounds like pretty advanced mathematics, but is surprisingly accessible with The Geometer’s Sketchpad.

It’s all based on the surprising principle that you can multiply two complex numbers very easily if you express them as magnitude (sometimes called the modulus) and polar angle (sometimes called the argument). To multiply two complex numbers expressed in this form, all we need to do is to multiply the magnitudes and add the angles.

The iterations in the article are based on starting with one complex number (the seed) and multiplying repeatedly by another complex number (the multiplier). In the sketch below, you can drag S to change the seed, and you can edit the values of rM and θM to change the modulus and angle of the multiplier. (By controlling the multiplier numerically, you can make very fine adjustments to its value.) Change depth to determine how many additional times the multiplication is performed.

Begin your experimentation by changing the values as described in the sketch itself:

As a second exercise, you can produce an image similar to the first one in the article (on pp.592–593) by setting rM to 0.988, θM to 90.4°, and depth to 100. By changing these values, you can produce each of the regular polygons, rotated polygons, and star polygons described in the article.

Even more striking results ensue when you add color, using the depth of each segment to determine its color. Press the Next button to go to the second page of the sketch, and try the experiment described there.

The colors parameter determines how many colors of the spectrum are used before returning to the original color, so setting colors to 3 results in a design that alternates among three colors. When colors is not a whole number (for instance, 4.05), the fifth segment will have a very slightly different color from the first. By choosing colors that interact with the geometric nature of the complex design, you can produce some very lovely effects, effects that rival the color work shown in Figure 8 in the article.

You can download the sketch here, and even if you don’t have The Geometer’s Sketchpad you can use the Free Sketchpad Preview to open the sketch and experiment with it. (You need a licensed copy of Sketchpad to save your work.)

The beauty of the iterated designs described in this article can motivate students to experiment, to persevere, to strive to understand the structure created by these iterations, and to create stunning mathematical art.

## Simultaneous Equations in Elementary School? You Bet!

Algebra classes devote considerable time to equations in a single variable before solving multiple equations in two or more unknowns. But just because elementary-age students are not familiar with algebraic symbolism doesn’t mean they can’t solve simultaneous equations, too!

The mathematician and educator W. W. Sawyer makes a compelling argument for the early introduction of simultaneous equation in his 1964 book, Visions in Elementary Mathematics:

It is quite possible to use simultaneous equations as an introduction to algebra. Within a single lesson, pupils who previously did not know what x meant can come not merely to see what simultaneous equations are, but to have some competence in solving them. No rules need to be learnt; the work proceeds on a basis of common sense. The problems the pupils solve in such a first lesson will not be of any practical value. They will be in the nature of puzzles.

Sawyer gives an example of such a puzzle, stating that “a man has two sons. The sons are twins; they are the same height. If we add the man’s height to the height of one son, we get 10 feet. The total height of the man and the two sons is 14 feet. What are the heights of the man and his sons?”

This puzzle amounts to a simultaneous equation, but by looking at the illustration at right, it’s clear—even to a young child—that the sons are 4 feet tall and the man is 6 feet.

Inspired by Sawyer’s work, I designed my own simultaneous equations puzzle for elementary school students as part of the Dynamic Number project. An interactive Web Sketchpad version of the puzzle appears below.

Each square in the 4 x 4 grid is randomly populated with either a circle, triangle, square, or hexagon, or is left blank. Each of the four shapes is assigned a secret numerical value between 1 and 10. Students can view the sum of the shapes in any row or column of the grid simply by pressing the corresponding Show Sum button. The goal is to determine the numerical values of the shapes by reasoning about the sums.

Each time students press New Problem, a new assortment of random shapes appears in the grid and the numerical values of the shapes change as well. Some challenges are harder than others. The first puzzle that appears above is fairly straightforward since the square sits by itself in the rightmost column. But other puzzles, which contain fewer blanks, ramp up the challenge. Students might discover, for example, that a square and a circle sum to 13 while a square and two circles sum to 17. Reasoning that the two sets of shapes are the same except for the extra circle, students can deduce that the circle is equal to 4 (just as they might reason about the height of the son in Sawyer’s puzzle).

Because there are a total of eight rows and columns in the grid, students have lots of options for picking which sums will be most helpful. And if a particular grid proves too hard to solve, no big deal—students can simply press New Problem and try again.

Pressing the Make Your Own button links to a second version of the puzzle that allows students to set their own numerical values for the four shapes. Working in pairs, one student looks away while the other student enters values for the four shapes. When she’s done, she presses Hide Answers and then New Symbols, perhaps more than once, until she’s satisfied with the puzzle. Her partner then gets cracking on the solution.

You can download a desktop Sketchpad version of the simultaneous equations puzzle as well as accompanying teacher notes and worksheet at the Dynamic Number website.

And finally, just because it’s the only representation of simultaneous equations that ever made me laugh, here is Sawyer’s image to accompany m + 2s = 14 and m – s = 2. Notice how the picture makes it easy to see that the height of the sons, counted three times, is
14 –2.

## Eigenvectors of 2 x 2 Matrices: A Geometric Exploration

Shiva Gol Tabaghi obtained her PhD degree in Mathematics Education from Simon Fraser University in 2012. This guest post is based on her doctoral dissertation research. Presently, she is involved in teaching undergraduate mathematics courses at Simon Fraser University. She enjoys using dynamic geometric diagrams to influence students’ ways of thinking about mathematical concepts.

If you’ve taken linear algebra, chances are your introduction to eigenvectors and eigenvalues was not based on a geometric interpretation of these concepts. Indeed, when I reviewed a sample of linear algebra textbooks, I found a procedural algebraic approach to be dominant.

Recall that the algebraic method begins by finding eigenvalues (the roots of the characteristic equation det(A – λI) = 0) and then finding the associated eigenvectors (the non-trivial solutions for (A – λI)x = 0 given λ). If x is an eigenvector of A, then Ax = λx.

The algebraic method does not reveal the connections between linear transformations, eigenvectors, and eigenvectors. In fact, it hides the fundamental property that an eigenvector is a special vector that is transformed into its scalar multiple under a given matrix of transformation.

My findings motivated me to design a sketch that allows students to explore eigenvectors and eigenvalues from a geometric perspective. In the interactive model below, A is a 2 x 2 matrix and x is a vector. You can drag x and view its effect on Ax. You can experiment with different matrices by entering new values for the four numbers that comprise A.

Here are a few exploration ideas to try with your students:

• Drag x in a circular path and explain how x and Ax behave. Do they ever overlap?  Do they become perpendicular? How do you know when you’ve found an eigenvector of A?
• Change the entries of the matrix to match the matrix below. Drag x to find places where x overlaps Ax. Then, find a place where the vectors are collinear but point in opposite directions. Estimate the ratio of the lengths of Ax and x. Press Show Matrix Calculations and drag the point labeled λ so that its value is equal to this ratio. How can you tell by looking at the value of det(A – λI) whether you’ve found the precise eigenvalue? If necessary, adjust λ so that it is an eigenvalue. Then, if necessary, adjust x so that x represents, as precisely as possible, the associated eigenvector. How can the value of (A – λI)x help you?

• Change the entries of matrix A to create a matrix that has eigenvalues of λ = 2 and λ = 4. Find the associated eigenvectors. How many of them can you find?
• Change the entries of the matrix  to the identity matrix. Drag x to find the eigenvectors of A. Explain your findings.

My students who interacted with this interactive model derived the following benefits:

1.  They overcame their difficulties in interpreting the equality sign in Ax = λx by visualizing the collinearity of Ax with x (for specific vectors x).
2. They developed an awareness of the existence of infinitely many eigenvectors, a fact that is hidden in the algebraic procedure for finding eigenvectors of a square matrix based on finding the eigenvalues first.
3. They explored the geometric representation of eigenvectors associated with negative eigenvalues. In the process, they developed a broader interpretation of the collinearity of vectors; that is, two vectors can be collinear but point in opposite directions.

I’d be interested to hear what your students discover when they try my interactive Web Sketchpad model. You can also download the desktop Sketchpad model, Eigenvectors.gsp.

## π Day 2014

π Day has always been a special day for me, from my earliest days. In fact, I’ve never figured out whether I was so eager to celebrate my first π Day that I jumped the gun and sent my mom into labor early, or whether I just wanted be sure to experience all 24 hours of my first π Day. Whichever it was, I’ve certainly enjoyed and celebrated π Day ever since, and it’s been even more fun for me since I got involved with Sketchpad and had the opportunity to come up creative, animated decorations and diversions.

So I’m sharing here a few of my favorite circle dissections. They’re all good ways to discover that the circle’s area is given by either the half of the product of the circumference and the radius (for the dissections that yield a triangle) or by the product of half of the circumference and the radius (for the dissections that yield a rectangle). Further, increasing the number of cuts in these dissections leads to thinking about limits; the only way to turn a circle into a triangle or rectangle is to chop it into tiny, tiny sectors — but how tiny do they have to be?

I hope your students enjoy these, and that they lead to interesting discussions both about how each dissection relates the area of a circle to its radius and circumference, and about the reasoning behind the dissection: What happened to the curvature of the circle’s edge? Can you really base a logical argument on increasing the number of sectors without limit?

(Press the buttons below the sketch to look at one or another of the four models. For the explosion, reassembling doesn’t always work after an explosion, so you may have to use the Refresh button in your browser.)

## Arranging Addends Puzzles

Below is an interactive puzzle called Arranging Addends. The goal of the puzzle is to arrange the circles and the six numbers (1, 2, 4, 8, 16, and 32) so that three conditions are met simultaneously: The sum of the numbers in the green circle is 21, the sum of the numbers in the blue circle is 26, and the sum of the numbers in the red circle is 14. The numbers can be dragged into the circles, and the circles can be moved as well. Dragging the point that sits on the circumference of each circle changes the circle’s size.

Students find it straightforward to satisfy the first condition of the puzzle. Dragging 1, 4, and 16 into the green circle fills it with numbers whose sum is 21. Creating simultaneously a sum of 26 in the blue circle is more puzzling. The only way to form 26 is to add 2, 8, and 16, but the 16 already resides in the green circle. How can it perform double duty?

Most students, after pondering this dilemma for a minute or two, have an “aha!” moment: What if the green and blue circles overlap each other, with the 16 sitting in their intersection? The picture below shows how this looks.

Now, all that remains is to satisfy the final condition of the problem by making the numbers in the red circle sum to 14.  The only way to form 14 is to add 2, 4, and 8. But the way those three numbers are currently arranged makes it impossible to enclose them without including the 16 as well. Students soon realize that they need to adjust the numbers and circles to allow the 2, 4, and 8 to sit alone in the red circle while still keeping the first two conditions true (green = 21, blue = 26). The picture at right shows the final result, with all three conditions of the puzzle met simultaneously.

I’m especially pleased with the way the Arranging Addends puzzles require young learners to juggle and process multiple pieces of information. More often than not, the placement of the circles and numbers must be adjusted one or more times to ensure that the sum of each set of numbers matches the required value.

And the mathematics of the Arranging Addends puzzles runs deep. Each new challenge generated randomly by Sketchpad uses the same six numbers—1, 2, 4, 8, 16, and 32—as the elements for the three sums. These powers of two can, amazingly, form any sum from 1 to 63.

You and your students can play the Arranging Addends puzzles directly on this webpage using the interactive model at the beginning of this post. To obtain new challenges, simply press New Puzzle. To download a desktop Sketchpad version of this puzzle along with teachers notes and a student worksheet, visit the Dynamic Number website.

You might also be interested in a low-tech version of the Arranging Addends Puzzles that requires nothing more than three hula hoops and a few pieces of paper. Regardless of which version you use, these puzzles are a great way to give your students practice in addition while keeping the work varied, interesting, and challenging.

## Discovering the Angle Sum and Difference Identities

In my Advanced Methods class at Penn’s Graduate School of Education, my students are working in groups to create shared lesson plans using an inquiry approach. For a number of reasons it can be challenging for these pre-service teachers to identify appropriate topics for student inquiry, but sometimes the brainstorming they do turns into something exciting.

And so it was recently when I conferred with pre-service teacher Andrew Laskowski about planning a lesson on trig identities. His high school students were already quite familiar with the Pythagorean identities, so his group had been thinking about how to add excitement and discovery to a lesson in which students either prove or make use of the angle sum and difference formulas. As we worked on it, we came across a diagram in Wikipedia. I’ll provide the url of the Wikipedia article at the end of this post, but please don’t peek and spoil your fun.

I’ve always had trouble remembering the angle sum and difference formulas, as I often do with formulas that I’ve memorized but haven’t truly owned. Looking at the diagram, I realized that the formulas were jumping out at me. It’s an easy construction but an elegant one, and by doing it once for myself I was convinced I’d never forget it.

Andrew’s group and I spent some time working on how to present it effectively, and I hope that some of the high school students to whom this lesson will be taught will be as excited as I was about it!

We ended up with a simple diagram and challenge. It’s dynamic, of course; drag the red points to adjust the angles.

Be sure to solve it first, preferably without the hint, and be sure to see what conclusion you can draw about the purple segments, and what conclusion you can draw about the green ones. (One of the things I love is that you can solve the whole thing using just one segment length and two angles.)

When you press the “I solved it” button, several new buttons appear, and an animation shows one order in which a student might solve the triangles. The animation finishes by extracting the purple and green segments from the figure to better demonstrate the striking conclusion. (You can use the various buttons that appear to show the animation one step at a time.)

Here’s the Wikipedia article that inspired this challenge: Angle Sum and Difference Identities.

Here’s a pdf you can download if you want to use this challenge with your own students, and here’s the sketch.

## Experiments with a Color Calculator

In the 1970s, my childhood friend Tim owned an Activision console and a variety of game cartridges. Tim was the envy of our block, but no matter how much I enjoyed a rousing game of Pong, I knew that my electronic toy was even better. No, I didn’t own the rival Atari game system: I had the pleasure of playing with my father’s Casio mini-8 calculator.

As you can see at right, the Casio was nothing fancy. You could add, subtract, multiply, and divide, and that was it. I didn’t care—I loved pressing the buttons and watching the digits appear on screen.

The Casio exposed me to number patterns that I had not seen in my elementary school math classes. I remember entering 1/3 into the calculator and viewing the result of 0.3333333. How amazing to see this sequence of threes! I confess, though, that I did not consider whether the pattern continued past the rightmost 3. And if I had typed 1/7 and seen 0.1428571, I probably would have assumed this to be the precise value of the fraction.

Today’s handheld calculators show more digits than my old Casio, but they still don’t make it easy to explore patterns in the decimal expansions of fractions. Below is an interactive model of a color calculator. It’s different from a traditional calculator in several ways. It converts fractions less than one to their decimal equivalents, but it assigns each digit in the decimal expansion its own color (all 4′s are green, all 5′s are blue, etc.) By pairing each digit with a color, it becomes easier for students to spot patterns in the digits.

And whereas traditional calculators show a limited number of digits to the right of the decimal point, the color calculator shows as many digits as students would like, simply by dragging the red point. The digits are displayed in rows and columns, and students can adjust the number of rows or columns to highlight repeating patterns.

The current fraction entered into the calculator is 1/7, but you can change the numerator and denominator to create other fractions less than one.

Here are just some of the investigations students can explore with the color calculator:

• Look for color patterns in the rows, columns, and diagonals of the decimal representation. Drag the red point to change the patterns and to create new ones.
• Find fractions whose decimal representations eventually “stop” and end in all zeroes. How can you tell without checking whether a fraction has this property?
• Find fractions whose decimal representations consist of just one repeating digit. (e.g., 0.33333…) Can you find a fraction for every digit 1 through 9?
• Find  fractions whose decimal representations have really long sequences of digits that eventually repeat. Do you think there are fractions whose decimal representations never repeat?
• What do you notice about the decimal representations of the fractions 1/7, 2/7, 3/7, 4/7, 5/7, 6/7?
• What do you notice about the decimal representations of fractions with a denominator of 13? How do these patterns differ from the ones you noticed for fractions with a denominator of 7?

The color calculator concept originates from Nathalie Sinclair, a professor of mathematics education at Simon Fraser University. In her version of the color calculator, there are no numbers visible in the decimal representation of each fraction. Students focus exclusively on the patterns visible in the colors. You can read  about Sinclair’s student interviews with the color calculator in her engaging book Mathematics and Beauty.

You can also download a Sketchpad model of the color calculator, accompanying teacher notes, and a student worksheet from the Dynamic Number website.

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