For a while now, I’ve been intrigued by the ways in which the study of geometric transformations can provide students with a very effective introduction to function concepts. Daniel and I have written a couple of articles about this topic, and we created a number of activities to take advantage of what can arguably be the most visual and tactile functions that students ever have the opportunity to experience. (The activities are in the Geometric Functions collection on the dynamicnumber.org website, .)

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://go.keycurriculum.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.)

]]>In my prior post, I presented an interactive Web Sketchpad odometer that is a great tool for introducing young learners to place value.

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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Below is an interactive odometer built with Web Sketchpad. Press each of the ‘+’ keys and observe their effect on the odometer’s value. Also notice how your button presses are tracked in the table below the odometer.

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.

]]>With the World Cup in our hemisphere, and the US squad having started out with a win over Ghana, my thoughts turned to the mathematics of soccer. My friend Henri Picciotto has a nice page about the shooting angle, the angle within which a shot is on goal, so I thought of using Sketchpad to explore this idea.

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.)

]]>For the past eight months, my colleague Scott Steketee and I have collaborated with the authors of the elementary curriculum *Everyday Mathematics *to design interactive Web Sketchpad models for their next edition.

When it came time to create a Sketchpad representation of an isosceles triangle, I built the interactive triangle model below. Try dragging any of the triangle’s three vertices. You’ll see that while the angle measurements of Δ*ABC* change, angles *A* and *B* always remain equal.

This isosceles triangle model is a a classic—a wonderful example of Dynamic Geometry’s ability to provide instantaneous feedback about a geometric construction as its parts are being dragged. As such, I expected the *Everyday Mathematics* authors to green-light the isosceles triangle without any discussion.

That didn’t happen.

The authors maintained that the model was too passive for young learners. Specifically, Sketchpad attended to the angle measurements, leaving the students to take on faith that the displayed measurements were correct. If students could measure the angles themselves, they not only would receive much-needed measurement practice, but they also might feel more invested in the discovery that angles *A* and *B* were always equal.

Based on this feedback, my initial instinct was to remove the angle measurements and include a virtual protractor that could be moved from vertex to vertex to measure the three angles. But this approach felt too cumbersome. I wanted students to measure the angles of as many isosceles triangles as possible, and if they only had one protractor, this process could become wearying.

My solution was to create the rather funny-looking model at right with three mini protractors attached to Δ*ABC, *one per vertex. With this model, students can drag vertices *A, B,* and *C* to create a new triangle and then swing each of the three protractors into position to estimate the three angle measurements. The picture below shows the result of positioning all three protractors.

You can try this out for yourself and with your students by going to the second page of the interactive Web Sketchpad model.

The isosceles triangle is just one example of a sketch where my long-held opinions about Dynamic Geometry pedagogy were challenged by the *Everyday Mathematics* author team. I’m enjoying the opportunity to revisit and refine my beliefs!

Every week, *The New York Times* challenges its readers to solve a mathematical puzzle in its online Numberplay column. This week’s puzzle was proposed by none other than Dan Bennett, a former editor and author at Key Curriculum Press, and his colleague, Avery Pickford.

Here is their puzzle, as described in Numberplay:

Dan and Avery love playing ping-pong. They love playing ping-pong so much that they devised a new rule to make games last longer. Scoring and play is normal, except that the score is “reduced” whenever possible. In other words, the scores are divided by the greatest common factor. So if Dan is ahead 7-4 and wins a point, instead of going to 8-4 the score becomes 2-1. Like in normal ping-pong, games go to 21. Note: If Avery is leading 20-7 and scores a point, he does not win. The score would go to 3-1.

There are many questions to ask about this game. To get the ball rolling we’ll focus on just one: What are all possible final scores?

Students can explore this puzzle without technology, but I couldn’t resist building a Sketchpad model to accompany the puzzle. You can download the sketch using this link. This is a great puzzle for implementing the Common Core Standards for Mathematical Practice!

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*In this guest post, Nate Burchell describes a sketch he uses with his students to explore parametric functions. In this process students work entirely in a graphical world, manipulating graphs directly rather than by way of equations. (Nate teaches in Seoul, Korea, where I enjoyed his family’s hospitality when I attended ICME in 2012. His blog has many wonderful Sketchpad resources.)*

— Guest post by Nate Burchell

I have been playing around with an idea in Sketchpad which I have been calling a “tactile function.” It is a function defined by independent points, points that the user can move to make the function respond. Drag the light-blue points in the graph below to see what this feels like. (Drag *t* on the *x*-axis to change the independent variable.)

You define a tactile function not by writing an equation, but by grabbing the graph and shaping it. Functions are usually considered to be symbolic ideas that have graphical representations. What if instead we designed the graph directly and let the equation be what it may?

(This process reminds me of the Doing-Undoing habit of mind that Mark Driscoll describes in Fostering Algebraic Thinking. If we change our perspective to flip an idea around, we might encounter good math and interesting connections.)

Math teachers define functions by their graphs every time they scribble a curve on the board. If I want to show an increasing function whose graph changes concavity twice, I just draw a curve with those characteristics. It would be a lot more trouble, and less enlightening for students, to figure out an equation that has the desired features.

An example I draw with chalk or marker has limitations: It’s strictly graphical, I can’t get accurate numeric information from it, and I can’t easily change its shape. But Sketchpad allows me to construct a function that’s mathematically well-defined and easy to modify.

Recently my calculus students were studying parametric curves. The graphs in this topic are difficult for students to produce and comprehend, and I realized that we could explore this topic through tactile functions.

In this sketch, two functions of *t* determine *x* and *y* coordinates of a point, located at (*x*(*t*), *y*(*t*)). The path of that point defines the parametric curve on the left. Drag *t* and observe how the values of *x*(*t*) and *y*(*t*) control the location of the red point on the left as it traces out the parametric curve defined by the blue and green graphs.

Press the link button (the triangle in the upper right corner) to see examples of the parametric curves you can make by adjusting functions *x*(*t*) and *y*(*t*).

To make your own design, redefine the *x*(*t*) and *y*(*t*) functions by dragging the 8 points that determine each of them. Try to make a pretzel, a circle, a star, or your favorite letter. On the right is a sailboat one of my students made.

In this video I give some hints:

You can download my original sketch, for use with The Geometer’s Sketchpad V5, using this link.

]]>Last week, I read about a Kickstarter campaign for the math game Primo. The game assigns colors to the prime numbers 2, 3, 5, and 7 (2 = orange, 3 = green, 5 = blue, 7 = purple) and then represents composite numbers by displaying their prime factors in color-coded form. It’s a clever idea, and it reminded me of the potential for turning the normally dry subject of factors and multiples into an opportunity for playing educational games and puzzles.

In several of my earlier blog posts, I’ve written about Sketchpad activities that present factors and multiples in puzzle form (see, for example, When Factors Put on Their Dancing Shoes and When Factoring Gets Personal). Now I’d like to introduce you to another puzzle of mine called Open the Safe that also focuses on factors and multiples.

Below is an interactive safe built with Web Sketchpad. Each of the 24 squares of the safe starts with its light off. Pressing any of the 12 buttons on the safe turns the lights of certain squares on, coloring them yellow.

The best way understand how the light patterns are generated is to play with the safe. Pressing the button on square 5, for example, turns squares 5, 10, 15, and 20 from blue to yellow. Similar experimenting suggests that pressing the button on square *n* changes the colors of squares that are multiples of *n*.

Pressing two buttons in a row produces interesting results. For example, pressing the 12 button turns the 12 and 24 from blue to yellow. By then pressing the 8 button, the 8 and 16 squares turn from blue to yellow, but the 24 square turns from yellow back to blue.

After a few minutes of play, students turn their focus to the puzzles in the table below. In each puzzle, the goal is to unlock the safe by pressing buttons so that certain squares are “on” (lit in yellow) while other squares are “off” (shaded blue). Consider puzzle A: How can you turn square 12 yellow and square 24 blue? Note that the puzzle says nothing about the other squares—these squares can be either yellow or blue.

There are several solutions to the puzzle. You can, for example, press button 4 followed by button 8 or press button 12 followed by button 8.

The puzzles, of course, give students practice in recalling factors and multiples of numbers between 1 and 24. But the puzzles are really aimed at giving students an opportunity to engage deeply with these concepts, developing reasoning skills and insights into the mathematics that occur naturally while solving the challenges.

There are 15 puzzles in the table above, but don’t stop there: Ask your students to create Open the Safe puzzles for each other. Share their work on our Facebook page!

]]>Did you know that The Geometer’s Sketchpad is a great tool for creating funhouse mirror pictures? Sure, Sketchpad can reflect, rotate, translate, or dilate a picture, but these operations are rather tame: They transform images uniformly, producing pictures that are easily recognizable versions of the original. By contrast, Sketchpad’s “custom transform” feature allows you to apply non-linear transformations to pictures, resulting in images that distort the original in surprising, amusing, and instructive ways.

Last year, my colleague Scott Steketee wrote a blog post demonstrating the result of applying a sine function transformation to a picture of the Golden Gate Bridge. This transformation leaves the bridge in an a decidedly unsafe state for motorists and pedestrians!

I recalled Scott’s sine transformation while reading the just-released book, *Math Bytes*, by Tim Chartier.

Chartier devotes a section of *Math Bytes* to the visual effects of applying two polar coordinate transformations. Each transformation begins with a point (*r, θ*) and then either squares or takes the square root of *r* to obtain the transformed point:

When I read about these two polar transformations, I knew that I had to try them with Sketchpad. I began with the first transformation, taking a random point, calculating its

(*r, θ*) value, and then plotting (*r*^{2}, *θ*). With just these two points, I could teach Sketchpad (using its “custom transform” feature) how to apply this squaring rule to any point—or pixel—in the plane. Having done so, I selected a picture, choose my squaring transformation, and instantly moved each pixel in the picture to its new location.

You can download my Sketchpad file to see what I’ve done. In the sketch, I give an example of applying each of the two polar transformations to a picture, and explain how you can easily swap the picture with any image of your choosing. (Warning: If you experiment with photos of your friends and relatives, they likely will not find the results flattering. It pays to distort your own face as well.)

To whet your appetite for experimenting, I’ve used the sketch to apply the polar transformations to six famous individuals and displayed the results below. How many can you recognize? Post your answers on our Facebook page.

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Welcome back to my ongoing series in which I feature interactive Web Sketchpad models that draw conic sections. Today’s installment, like the previous one, focuses on ellipses, and dates back to the 17th-century Dutch mathematician, Frans van Schooten.

Below is an image from van Schooten’s manuscript, *Sive de Organica Conicarum Sectionum in Plano Descriptione, Tractatus* (*A Treatise on De**vic**es for Drawing Conic Sections*). The obvious level of care lavished on the illustration is consistent with all the drawings in van Schooten’s work.

Van Schooten’s linkage consists of three moveable rods hinged together. The model is built so that *HI* = *FG* and *IF* = *GH*.

The interactive Web Sketchpad model below allows you to experiment with van Schooten’s linkage. Drag point *F* and observe the trace of point *E*. Press *Animate Point F* to see the linkage move on its own and press *Show/Hide Locus* to see the curve traced by point *F* all at once. You can change the lengths of the rods by dragging the red points at the bottom left.

It certainly appears that point *E* traces an ellipse, but can you prove it? The proof is not hard and is a great opportunity for students to apply what they know about triangle congruence. Press *Proof Hints* for step-by-step guidance.

You’ll find a desktop Sketchpad model of the congruent triangle construction as well as accompanying teacher notes and a student worksheet in my book, *Exploring Conic Sections with The Geometer’s Sketchpad*.

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