The 17th-century Dutch mathematician Frans van Schooten developed “hands-on manipulatives” centuries before the term became popular in math education circles. Below are two images of ellipse-drawing linkages from van Schooten’s manuscript, *Sive de Organica Conicarum Sectionum in Plano Descriptione, Tractatus* (*A Treatise on De**vic**es for Drawing Conic Sections*).

Building physical models of these devices isn’t hard. A bent straw, for example, works well for the linkage on the left. As another approach, you can use Sketchpad to construct them.

The interactive Web Sketchpad models below allow you to draw ellipses using the two models above as well as two other related methods. For each model, press the *Animate* button to set it in motion. You can move between the models using the arrows in the bottom-right corner of the sketch window.

For each model, experiment with different locations of the green point that’s tracing the ellipse. How does the shape of the ellipse change based on the point’s position? You can also experiment with the length controls at the bottom left of each page.

It’s not especially difficult to prove that these four models do, in fact, draw ellipses. The *Show Proof Hints* button on the Bent Straw page offers some ideas that you can apply to the other two models as well.

If you’d like to explore other methods of constructing ellipses, check out my prior blog posts, Danny’s Ellipse, The Congruent Triangle Construction, and The Tangent Circles Construction. You’ll find these and many more conic section constructions in my book, Exploring Conic Sections with The Geometer’s Sketchpad.

]]>My panel was supposed to discuss his work on the UCSMP curriculum. My two co-panelists had been deeply involved in that work, but I had not, so I took the opportunity to address one aspect of the mathematics that Zal pioneered in *GATA* and that remains prominent in *UCSMP Geometry* today: the treatment of geometric transformations as functions (a treatment that I describe here as “geometric functions”).

My hope was to show how activities based on *The Geometer’s Sketchpad *not only support Zal’s insight from over 40 years ago, but validate it in ways that weren’t even well-understood at that time. I wanted to make two main points, one about cognitive science and one about mathematics:

- Treating transformations as functions is supported by the cognitive science findings regarding embodied cognition and conceptual metaphor, as described (for instance) in
*Where Mathematics Comes From*by Lakoff and Nunez. When students drag a point as the independent variable, they are experiencing variables in a physical way, and the act of varying point*x*and observing the resulting motion of point*r*(_{j}*x*) (the reflection across mirror*j*of*x*) becomes for the student a conceptual metaphor for the function that relates the two points.

- Treating transformations as functions enables students to connect geometry and algebra in a very direct and elegant way. Students can restrict such transformations to a number line, thereby turning the two-dimensional point variables of geometry into one-dimensional real numbers of algebra while simultaneously turning the transformation itself into a linear function. Having done so, they can apply a translation to the dependent variable to produce the Dynagraph representation invented by Goldenberg, Lewis, and O’Keefe, or they can apply a rotation to produce the Cartesian graph of
*y*=*mx*+*b*(where*m is*the scale factor for dilation and*b*is the vector length for translation).

Here’s a movie I made of my presentation:

And here’s a Web Sketchpad version that shows the restriction of the geometric function to a number line to turn it into a linear function, and the subsequent transformation that represents the function as a Cartesian graph.

It was an honor to participate in this symposium, and I hope I did justice to Zal’s insight from so many years ago by showing its deep connections both to cognitive science and to the unity of geometry and algebra.

Note: In advance of the symposium and dinner, Lisa Carmona (Vice President at McGraw-Hill Education, preK-5 Portfolio) put up an eloquent post on the McGraw-Hill Education blog attesting not only to Zal’s accomplishments, but to the way he inspires so many of us to advocacy as well as a commitment to students’ deep understanding of mathematics.

]]>This week, I set my sights on isosceles triangles. It’s common to encounter isosceles triangles as supporting players in geometric proofs, but how can isosceles triangles be made the stars of their own mathematical puzzles?

Above is a collection of interactive Web Sketchpad puzzles. Each puzzle begins with an isosceles triangle *ABC *and a point *R* on the coordinate grid. Your goal is to drag a vertex of the triangle to point *R* so that the new triangle is also isosceles. In the first puzzle, this is easy: Dragging either vertex *A, B,* or *C* to point *R* results in an isosceles triangle.

Press the right arrow at the top of the sketch page to move on to the second puzzle. Point *R* now sits at (3, 2). This challenge is straightforward, too: Dragging point *B* to point *R* yields an isosceles triangle.

Pressing the right arrow again reveals the third puzzle, and this one is a little different. It’s no longer possible to drag vertex *A, B,* or *C* directly to point *R* while keeping *ΔABC* isosceles. We need to add a new rule to our puzzle: *You can drag more than one vertex of ΔABC to reach point R, but each intermediate resting position of the triangle must be isosceles.*

We can solve this particular puzzle in two steps: First, drag point *B* to (3, 2). The resulting triangle is isosceles. Now drag point *A* to point *R* to solve the puzzle. Again, the triangle is isosceles.

Most of the remaining Web Sketchpad puzzles above require multiple steps to position a vertex of isosceles triangle *ABC* at point *R*. As your students solve the challenges, here are some questions and tasks to present to them:

- Name all the locations for point
*R*that allow you to solve the puzzle in just one move. Do these points, or a subset of them, share anything in common? - Can you find the fewest number of moves needed to solve each puzzle?
- What general strategies did you discover for solving the isosceles triangle puzzles? Can you describe or illustrate these strategies so that a friend could follow them?
- Can you find a way to predict how many moves it will take to solve a puzzle without carrying out each step?
- On the second-to-last page of the Web Sketchpad model, pick your own location for point
*R*and challenge a friend to solve your puzzle. - On the last page of the Web Sketchpad model, find a sequence of moves for
*ΔABC*so that one vertex of the triangle rests on point*R*and another vertex on point*P*.

Here’s how the static version of the puzzle works: Each of the ten letters in the Web Sketchpad model above has been assigned a random, secret value between 1 and 10 (It’s possible that two or more letters may share the same value.) Your job is to determine these values. To do so, you’ll use the circle. To move the circle, you can drag its interior. You can also drag the point sitting on the circumference to change the circle’s size.

Whenever two or more letters sit inside the circle, its interior turns green, and you’ll be told the sum of those letters. So, for example, when G and I are in the circle’s interior, Sketchpad reports that G + I equals 16.

To play again with new secret values, press* New Puzzle. *When your students are ready to try a different challenge, press the arrow in the bottom-right corner of the model. There are still ten letters whose values must be determined, but hey–the letters are moving (dancing) across the screen! The rules are still the same–surround two or more letters with the circle to determine their sum–but the movement of the letters makes the puzzle more challenging.

What strategies do your students use to uncover the values of all 10 letters?

When I tested this activity with a group of fourth graders, Lee solved the static version of the puzzle by dragging the circle so that it covered all ten letters. He then adjusted the circle so that it covered 9 letters, thus allowing him to determine the value of the extra letter. He continued in this fashion, scaling back the number of letters sitting in the circle from 9 to 8 to 7, and so on, all the way to just 2 letters.

When Lee moved on to the “dancing” version of the puzzle, he began with the same strategy of covering all ten letters with the circle and then scaling back one letter at a time. But because the letters did not stay put, he found it difficult, if not impossible, to remove one letter at a time from the interior of the circle. After some consideration, Lee realized he would need to adopt a new strategy: He began by encircling 2 letters, noting their value, and then adjusting the circle to include a third letter. He continued in this manner, picking other pairs of letters and adding a remaining letter to the pair. It was fascinating to watch Lee modify his original strategy to this new one, motivated by the switch from static to dynamic letters.

This activity first appeared as part of the NSF-funded Dynamic Number project. You can download the original Sketchpad model and teacher notes here.

]]>I find it quite amazing the way in which such a random process can end up creating a beautiful pattern. In fact, a wide variety of patterns can be generated this way, depending on the scale factor and the set of target points you’re using. The pattern Dan described involves a set of five target points, which are (no surprise!) the vertices of a regular pentagon.

Dan used the programming language called Processing to play his Chaos Game, but there’s no reason we can’t play the same game in Sketchpad. So here’s a sketch that describes how to play the Sketchpad Chaos Game in order to make a pentaflake. (Though you can use the sketch below to view the completed construction, you have to use regular Sketchpad to do the construction yourself.)

Page 1 describes the initial construction, up to the iteration. Page 2 describes how to do the iteration, and shows the result. Because this iteration only adds one point with each step of the iteration, it requires a much greater depth to see the pattern.

Page 3 asks another question: What if you dilate toward all five vertices simultaneously?

Finally, all three methods (the one from my last post and the two in this one) generate lots of questions. What happens if you move the initial seed (point *Q*) outside the circle? For a particular method, how many objects does the iteration produce at each depth? What happens if you change the scale factor, and why does this scale factor (approximately 0.38) produce the result it does? What if you tried the same construction with a square, or a triangle? The Chaos Game pentaflakes are missing the middle pentagon that is part of the construction from my earlier post. Why is that, and is there any way to make it appear in the Chaos Game patterns?

I hope you’ll be inspired to experiment with your own pentaflake, but if you like you can download mine. (And even if you don’t have Sketchpad, you can get the free trial version and use it to do your own construction or to play with mine.)

]]>Here’s how the puzzle works: Sketchpad has randomly chosen a “mystery” number between 1 and 25, and students must use logical reasoning and their knowledge of multiples to determine it. Students pick and press any of the “Multiple of?” buttons in the table below. Pressing “Multiple of 3?” for example, indicates that the mystery number is a multiple of 3. The goal is to press as few buttons as necessary to determine the mystery number. Sometimes, however, the information in the table may not be enough. In this case, **only** as a last resort, students press the “Show Sum of Digits” button to learn the sum of the digits of the mystery number (For example, 17’s digits sum to 8).

I’ve field tested this puzzle with elementary students, and they become very engaged in the mathematical thinking involved in solving the challenges. The students worked in pairs to solve the puzzles and then came back together as a group to discuss and share strategies. Here are some of the questions that arose:

- Strategically, what is the best button to press first?
- If you learn that the mystery number is a multiple of 3, do you know whether the number is a multiple of 6?
- If you learn that the mystery number is a multiple of 6, do you know whether the number is a multiple of 3?
- If you learn that the mystery number is a multiple of 2 and 5, what else do you now know?
- When is it necessary to press “Show Sum of Digits”?
- Is it always possible to find the mystery number?

And, as with any good problem, the students also proposed some ways to extend it:

- Suppose the list of possible mystery numbers is expanded to include every number between 1 to 30. Will it always be possible to determine its value?
- Can the puzzle be adapted to work when the mystery number is between 1 to 40?

If you have the opportunity to try this activity with your students, let me know what questions they explore!

]]>I couldn’t let Niels down, so off I went to MathWorld to bring myself up to flake speed. Whew, there were a lot of formulas there! Fortunately there was also a clear picture: dissect a regular pentagon to make six smaller pentagons, one at each vertex and one in the space that’s left in the interior. I used Sketchpad to build a pentagon and played around with a variety of constructions, and then had a breakthrough when I realized that the edges of the inner pentagon lie on the diagonals of the original large pentagon. Eureka — inscribe a star in the pentagon and you’ve located the five inner vertices of the pentaflake! With a bit more thought, I realized that those five inner vertices are sufficient: they determine all 6 of the small pentagons, so that’s all I needed to create the fractal iteration.

So here’s the sketch I sent back to Niels, in three pages. Page 1 describes the dissection, with buttons to illustrate each of the steps. Page 2 gives detailed instructions on doing the iteration, which requires some care to create the six maps that define the dissection. (Note that you can’t *create* the iteration right here on this web page; you must do so in Sketchpad itself.) And page 3 shows the resulting pentaflake in all its glory. (Drag points *A* or *B *right here on the web page, or edit the desired-depth parameter.)

With the help of the hints on pages 1 and 2, Niels was able to construct his pentaflake, and I encourage you to do the same. Notice on page 2 that I used a depth parameter so I could easily change the depth of iteration that’s displayed. (I limited how great a depth you can use — this construction grows quickly. By depth 5 you’re nearing 10,000 pentagons, and by depth 8 you have well over a million, which will slow your computer down just a tad.)

It’s better, of course, to construct your own pentaflake, but if you like you can download mine. (And even if you don’t have Sketchpad, you can get the free trial version and use it to do your own construction or to play with mine.)

]]>This year’s winner for best illusion was created by Christopher D. Blair, Gideon P. Caplovitz, and Ryan E.B. Mruczek from the University of Nevada, Reno. It’s called the Dynamic Ebbinghaus Illusion. When I viewed the illusion on YouTube, I was very impressed. And since the illusion featured circles that moved and changed size, it seemed very likely that I could replicate it using Sketchpad.

Of course, there was really no need to build the illusion from scratch, but once you’re hooked on mathematical constructions, it’s hard to resist a new challenge.

Below, you can see what I made. Press *Animate* to set the circles in motion. As the circles move, the six outer circles grow and shrink. But what about the central circle? It appears to grow and shrink as well, but amazingly, that is an illusion—the central circle does not change size at all! To check, although I’m sure you still won’t be convinced, press *Hide Outer Circles* while the circles are in motion.

]]>

Although the project has ended, I still keep in touch with a few of the teachers, and they continue to use Sketchpad when they have an opportunity. Several months ago, Brooke Precil and Matt Silverman contacted me about modifying one of our pan balance models that provides an intuitive introduction to solving algebraic equations without any *x’*s or *y’*s. A few days later, Matt used the updated model with his fifth graders and reported that the lesson was a success. I’m eager to share the balance model with you here in Web Sketchpad form.

The first balance problem shows that a circle and square weigh the same as a circle and triangle. Because a circle sits on both sides of the balance, it makes sense to students that they can drag the circles off each side and still maintain the equality (Dragging any shape off the balance and to the left of the vertical divider returns it to the unused collection of shapes.)

To view more balance problems, press the arrow in the bottom-right corner of the model. Once your students have solved the first four problems, the real fun begins: They can now create balance problems of their own. The final page in this collection of balance problems begins with all the shapes sitting to the left of the divider. To create a problem, a student drags shapes onto the balance, chooses the numerical value of the shapes, and then hides one or more values. She then tells a classmate which shape is the unknown to determine.

Invariably, students create more challenging problems than we would give them ourselves. Don’t be surprised if students try to use as many of the 36 circles, squares, and triangles as possible!

]]>Nathan Dummitt teaches mathematics and statistics at Columbia Preparatory School in New York, NY. He teaches all four years and is interested in sharing low-threshold, high-ceiling activities with his students.

— Guest post by Nathan Dummitt

I teach Geometry at a high school in New York City, and I like to start the school year by giving an open-ended problem on the first day of class. I pick problems that both promote student discussion and spur the creation of a student work product that can decorate our classroom walls and be referenced later in the year.

I spend most summers in Japan and am always on the lookout for interesting problems in Japanese textbooks and math contests. In recent years, a consistent source of excellent problems has been the Japanese Arithmetic Olympics, which is held annually for fifth and sixth-grade students. My activity for the first day of school this year is based on a problem that appeared on the 2007 Olympics:

How many different ways can you dissect a 5 × 5 square into four pieces that can be rearranged to make a 3 × 3 square and a 4 × 4 square?

I’ve made a sketch that allows students to experiment with different colorings and arrangements before they put marker to paper. An interactive Web Sketchpad version appears below. The first page of the sketch displays one solution to the problem while the second page provides students with a work environment to develop their own solutions.

I give my students the grids towards the end of the first day of class, and they then work in small groups for about 10 minutes to investigate various dissections. Whatever they don’t finish is assigned as homework for the first night. When they bring in their grids the following day, we compare them and then decorate the walls of our classroom with their work. Below, you can see photos of what that they produced.

Contrary to what I’ve seen in many textbooks, I think it is vitally important for students to be exposed to area at the beginning of the Geometry curriculum. The Olympics problem not only offers a good chance for students to discuss area, but also gain a preliminary and informal introduction to the Pythagorean Theorem, which is lurking in there as well. We can then revisit their colored grids later in the year when we study the Pythagorean Theorem more formally.

If any of you try this activity with your students, I would love to hear how many different dissections they find. Our present total is 13, but there may be more out there!