I find it quite amazing the way in which such a random process can end up creating a beautiful pattern. In fact, quite 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.

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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!

This is a deep theorem, but one aspect of it is lovely, surprising, and entirely approachable by high-school geometry students.

Below are two maps of the United States, one a scaled copy of the other. The smaller map sits atop the larger one, with the edges of both maps parallel (The scale factor and the particular location of the smaller map arbitrary.)

Believe it or not, you can stick a pin straight through both maps so that the pin simultaneously pierces the identical geographical location on each one!

Finding this special location (the “fixed point”) and proving that it satisfies this condition is possible with nothing more than a basic knowledge of geometry. Indeed, when I posed this challenge to teachers in my NYU geometry course, they devised a variety of clever proofs. In an upcoming post I’ll share their ideas, but for now, see what you and your students can discover.

Meanwhile, let’s explore a variation of the map problem. Suppose that again we have two maps that are scaled copies, but rather than aligning their edges to be parallel, we simply place the smaller map randomly onto the other as shown below. It turns out that it is *still* possible to find a location for the pin that pierces the same spot on both maps simultaneously. Wow!

To aid us in finding this spot, let’s use Web Sketchpad. The interactive model below shows a digital artwork created by my mother, Joan Scher (If you’d prefer to work with the map, press the arrow in the lower-right corner of the sketch.) Change the value of *n* from 0 to 1. By doing so, you’ll create a rotated scaled copy of the painting placed on top of the original.

Now bump *n* from 1 to 2 and notice what happens. You’ll obtain yet another copy of the painting, created using the same shrink-and-rotate parameters, but this time applied to the scaled copy. Keep increasing the value of *n*. As you do, you’ll see a spiraling collection of scaled copies, built with the same shrink-and-rotate recipe. Try as well pressing “Repeated Copies” to watch *n* cycle from 0 to 30.

Can you convince yourself intuitively that this sequence of steadily shrinking pictures will converge to the “fixed point” shared by the original two paintings?

To experiment more, you can adjust the scale factor (the “scale by” parameter in the sketch) and the amount of rotation (the “rotate by” parameter). You can also adjust the placement of the scaled copies by dragging the two small blue points that sit near the lower-left corner of the original picture.

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As shown in the interactive Web Sketchpad model below, Danny started by constructing two concentric circles with center at point *A*. He continued by adding a radius *AD* that intersected the smaller circle at point *E*. Danny then built a right triangle with hypotenuse *DE *whose base and height are parallel to the horizontal and vertical lines that pass through point *A*.

Drag point *D* and observe the trace of point *F*. The oval you form certainly looks like an ellipse, but is it? After receiving Danny’s construction, Key Curriculum issued a challenge to teachers and students in its newsletter: Could they prove that Danny’s oval was an ellipse?

The editors at Key received hundreds of letters and a potpourri of proofs demonstrating that Danny had indeed built himself an ellipse. Can you and your students devise one or more proofs of your own?

Danny’s technique for constructing ellipses turns out to be exceptionally handy because it does not require us to know the location of the ellipse’s two foci. We only need indicate the center of the ellipse and the lengths of its major and minor axes. If you click the arrow in the bottom-right corner of the web sketch above, you’ll see a neat animation of the words “Dynamic Geometry” that was built using Danny’s method.

If you’d like to explore other methods of constructing ellipses, check out my prior blog posts, 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.

]]>*Ruth, Phyllis, and Joan each bought a different kind of fruit (orange, apple, pear) and a different vegetable (spinach, kale, carrots) at the supermarket. No one bought both an orange and carrots. Ruth didn’t buy an apple or kale.* And so on…

Now that the Common Core Standards for Mathematical Practice talk explicitly about problem solving, reasoning, and sense making, the educational benefits of logic puzzles seem more relevant than ever.

Below is an interactive Web Sketchpad model for introducing elementary students to logical reasoning. Unlike traditional logic puzzles that come pre-written, these logic puzzles can be created by your students. As an example, start by dragging two statements across the vertical divider line. Let’s assume you pick “Ann is taller than Bill” and “Bill is taller than Carlos.”

Your job is to change their heights of Ann, Bill, Carlos, and Denise (dragging the points atop their heads) so that both statements are true. Notice that the star turns from red to green when you’ve satisfied both statements simultaneously. You’re now ready to answer a question: Who is taller, Ann or Carlos?

Your diagram likely shows that Ann is taller than Carlos, but is this always true? What I really like about this model is that students can change the heights of Ann, Bill, and Carlos , but as long as Ann is taller than Bill and Bill is taller than Carlos, Ann will always be taller Carlos. I think deducing this conclusion is made easier with the interactive visual model.

By comparison, suppose that students drag the statements “Ann is shorter than Bill” and “Bill is taller than Denise” across the vertical divider. What, if anything, can be said about Ann and Denise? In this case, the students can arrange the heights of the three people so that either Ann or Denise is taller while still keeping the star green.

Here are some additional challenges:

- Can you pick statements that make it impossible to turn the star green?
- How many statements can you drag across the vertical divider line and still be able to make the star green?
- Click the arrow in the bottom-right corner of the model to try a more abstract variation of these logic problems. The names Ann, Bill, Carlos, and Denise are replaced by the letters A, B, C, and D and “shorter than” and “taller than” are replaced by the symbols < and >.

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Several months ago, I shared a Numberplay puzzle from former Key Curriculum editor Dan Bennett. Now I’d like to recap the Numberplay puzzle from last week:

A flying squirrel-frog is being chased by two cats. Here is how the chase goes:

Step 1: The cats go half the distance from where they started to where the squirrel-frog is.

Step 2: The squirrel-frog jumps into the air and glides down, landing on the opposite side of the cat farthest from it, exactly as far from that cat as when it jumped.

These steps repeat until the cats get close enough to swipe the squirrel-frog down, or else realize they will never catch the squirrelly animal.

In which starting configurations will the cats catch the flying squirrel-frog? In which configurations will the flying squirrel-frog escape?

When I read this problem, I immediately wondered whether I could model it with Sketchpad. Before long, I had a working version ready to go. Below you can try it with Web Sketchpad.

Start by dragging the cats and the squirrel-frog to any location you like. Then, press *Move Cats* and *Move Squirrel Frog* to run the simulation once. Continuing pressing the two buttons, one after the other, to see whether the cats catch the squirrel-frog.

I played with this sketch for a while, trying many different starting locations for the cats and the squirrel-frog. I made some observations, talked about them with my colleague, Scott, and then looked in the Comments section of the Numberplay blog to see how other people had approached the puzzle.

What observations and conjectures do you have?

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