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?

]]>This and similar activities involving “technologically embodied geometric functions” are designed to familiarize students with important function concepts in an active and concrete way corresponding to cognitive science findings. Still, students’ conceptual understanding in the geometric realm may not transfer well unless it’s explicitly linked to their work with families of symbolically defined algebraic functions.

Making that connection was the topic of the presentation that Daniel Scher and I gave at last April’s NCTM Annual Meeting. In that session we proposed a series of activities to make the link explicit: Students move their two-dimensional geometric functions into one dimension (by restricting the variables to a line), mark the line as a number line, and connect the geometric behavior of the variables to their numeric values.

Reduce the Dimension, shown below, is the first of those activities. The task for students in this activity is to take the initial step by turning a two-dimensional function into a one-dimensional function by putting both variables on the same line.

The activity is available in two forms:

- A Sketchpad activity in which students create the functions from scratch, restrict their domains, and so forth. The worksheet is here.
- A prepared web sketch like the one above (but also incorporating the student worksheet in a scrollable window), available directly at http://geometricfunctions.org/wsp/tegf/reduce-the-dimension

I much prefer the build-from-scratch choice, because students’ understanding and retention are enhanced by constructing the mathematics themselves. (This choice requires Sketchpad 5; if you don’t already have Sketchpad, you can download the free preview.) But I decided to also create the web-based choice because of its convenience and accessibility.

This activity is new, and there are no teacher notes available yet. We’d love to have your comments on the activity, and your suggestions for improving it.

In future posts, we’ll number the line so that our variables have numeric values, we’ll turn the line into a dynagraph with separate parallel input and output axes, and we’ll conclude by turning the dynagraph axes perpendicular to each other and generating the Cartesian graph corresponding to our original geometric function.

]]>Below are eight dials, each with ten evenly spaced tick marks. Press *Start/Stop* and asking students to observe what happens. How many times does the red counter on the far-right dial move before the counter on the dial to its left moves once? How many times does the counter on the far-right dial move before the third red counter from the right moves once? Can students predict the answers to these questions for the remaining counters on dials four through eight?

Continue by asking students to describe the connection between these dials and our base 10 number system. To make the connection concrete, press *Show Numbers* to view the count as the red counters move around their dials.

Now comes the real fun: Press the arrow in the bottom-right corner to go to the second dials model. Notice that there are just three tick marks per dial instead of 10. As before, press *Start/Stop* and ask students to observe the movement of the red counters. How many times does the red counter on the far-right dial move before the counter on the dial to its left moves once? How many times does the counter on the far-right dial move before the third red counter from the right moves once? Can students predict the answers to these questions for the remaining counters on dials four through eight?

All of these questions introduce students to base 3 counting in a natural way, aided by the dynamism of the red counters moving around their respective dials.

It’s instructive (and mesmerizing) for students to watch the counters move for five minutes or more and begin to understand and be able to predict the pattern in their movements. Students can, and undoubtedly will, hasten the counters’ action by dragging the point labeled *speed*.

After the counters have moved for a while, tell students to press *Start/Stop* to freeze their current location. Ask them to determine how many times the counter on the far-right dial has moved over the course of the entire animation. Answering this question is equivalent to converting a number from base 3 back to base 10 (Students can check their answer by pressing *Show Total Moves*.)

To explore all of these questions in bases other than 10 or 3, simply change the number of tick marks per dial.

I’ll leave you with a challenge from my colleague, Scott: Can you get the number dials to show 33333333? Try it for different bases!

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In our presentation at last April’s NCTM Annual Meeting, we took this approach one step further, by proposing a sequence of student experiences that would connect the commonly-studied geometric transformations directly to linear functions in the form *f*(*x*) = *mx* + *b*, and by extension to a variety of other algebraically-expressed functions.

Though our proposed student experiences are not yet fully fleshed out, we’re eager to bring these ideas to our blog, and in this post I’m taking the first step by setting the stage: on each page of today’s dilation challenges you’ll be asked to figure out which member of the dilation family is responsible for the function behavior you see and experience on the screen.

(Click the arrow at the bottom right of the sketch to go to the next challenge.)

These challenges are taken from our Dilation Function Family activity, which is designed as an introduction to dilations from a function point of view. You can download the full activity here including the sketch, student worksheet, and extensive teacher notes. (The downloaded version requires Sketchpad 5; if you don’t already have Sketchpad, you can download the free preview here: http://info.mheducation.com/sketchpad.trial.html.)

We’re also beginning to experiment with putting some of these activities on the web, with both the interactive sketch and the student directions on the same web page. You can try out the web version of the Dilation Function Family activity here: http://geometricfunctions.org/wsp/tegf/dilation-family/. (The main difference between the downloaded activity and the web activity is that on the web, students can’t actually construct the dilation functions themselves, so they need to press buttons in the websketch instead.)

]]>Well, technology moves fast these days, and the latest odometers are more powerful than ever. While our prior odometer featured ‘+’ buttons above each digit, our newest innovation in number-tracking technology features ‘+’ and (gasp!) ‘–’ buttons.

I kid, of course, but the introduction of ‘–’ buttons really does change the scope of questions we can ask students. In the interactive Web Sketchpad model below, check what happens when you press the various ‘+’ and ‘–’ buttons.

When I use this model with students, I start by asking them how to reach 9. Students will typically press the ‘+’ button above the 0 in the units place nine times. That’s a fine method, but can they reach 9 in fewer button presses? I reset the odometer to 0 and ask them to try again.

With a little experimentation, they discover that pressing the ‘+’ in the tens place followed by the ‘–’ in the ones place lands them at 9 in just two button presses (10 – 1 = 9). Students are excited that they were able to improve their “score” by knocking 7 presses off their prior attempt.

I continue by asking students how they can reach 99, 999, and 9,999 as economically as possible. Students quickly see the pattern and realize that they can reach any of these values in just two button presses. They find it amazing that even 999,999 is just two stops away from 0.

I then give students a variety of numbers and challenge them to see who can reach each target value in the fewest button presses. I’ll typically start with numbers less than 100. It doesn’t take long for strategies to emerge: To reach 34 quickly, follow the sequence 0, 10, 20, 30, 31, 32, 33, 34. But to reach 37, it’s better to progress to 40 first and then count down: 0, 10, 20, 30, 40, 39, 38, 37. Why is that? Can this pattern be generalized to other numbers?

Numbers with 5 in their units place are interesting. Contrast the quickest routes for reaching 45 (0, 10, 20, 30, 40, 41, 42, 43, 44, 45) and 75 (0, 100, 90, 80, 79, 78, 77, 76, 75). In the case of 45, it’s best to round down to 40 and then count up to 45. For 75, it’s more efficient to round up to 80 and then count down to 75. Why is that? Can this pattern be generalized to other numbers?

As students hone their skills, they progress to numbers larger than 100. What is the fastest way to reach 172, or 836, or 2014, or 86,555? With any number between 1 and 1,000,000 as fair game, there are lots of puzzles for students to ponder!

After students have solved a variety of these odometer challenges, I ask them to think about the strategies they’ve developed and share them with their classmates. As a homework assignment, they write a strategy guide for the game, clear enough for a newcomer who has never played it. Some students might only have strategies that work for numbers up to 100. Other students’ insights might extend to much larger numbers. It doesn’t really matter—the process of teasing apart strategies, both simple and complex, is a perfect opportunity to put the Common Core Mathematical Practices into action. In particular, I’m thinking of these practices:

- Make sense of problems and persevere in solving them;
- Construct viable arguments and critique the reasoning of others;
- Look for and make use of structure.

If you have a chance to use this odometer game with students, I’d be very interested to hear about your experience!

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I built this model as a way to support students’ development of place-value concepts. Here are two ideas for how you might use it:

- Pick a target number and ask students to name different combinations of button presses that will land the odometer at the target. To reach 100, for example, we might combine 9 tens with 10 ones or 7 tens with 30 ones. For larger targets like 1,000, there are lots of possible answers.
- Press ‘Hide Odometer.’ Then, press the various ‘+’ buttons multiple times. Ask students to determine the hidden value of the odometer based on the information in the table. For example, 3 hundreds, 5 tens, and 2 ones is 352. A more challenging problem is to convert 12 hundreds, 15 tens, and 18 ones to 1,368. Students love creating these puzzles for each other because it’s fun to press the buttons, and the more they press, the harder the problem!

In my next post, I’ll share another odometer that has ‘–’ as well as ‘+’ buttons. The inclusion of ‘–’ buttons leads to some very interesting problems.

]]>Use the sketch below to find the locus from which the striker, represented by point *S*, has a 15-degree shooting angle (plus or minus 10%).

When you drag point *S* closer to or farther from the goal, what do you notice about the point? How does this help you find the locus?

After experimenting a bit, turn on tracing and try to trace the locus as accurately as you can. Then try a different angle (by clicking and changing the target-angle parameter). What shape does the locus have? How can you explain this shape?

A related challenge is to imagine the striker running down the field, toward the goal line, as in the animation below.

When the striker makes a run from midfield straight toward the goal line, where is her shooting angle the greatest? (It may help to turn off the *Run!* button and drag the striker by hand.) Once you find this spot, mark it.

To change where she makes her run, drag the dashed red line up or down. Mark the greatest-angle spot for different runs. What pattern do you think you see? How can you explain this pattern?

Just as the fun of soccer is in the playing, the fun of these mathematical soccer challenges is in exploring them, so I provide no solutions here. But if your experimentation gives you some ideas about how to find the location of the maximum shooting angle for a given run, you can download the sketch here and try to construct that point, and even the locus of that point for the striker’s different runs. (If you don’t already have Sketchpad, you can download the free preview here.)

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