I recently had the opportunity to work with a group of students who were testing activities that treat geometric transformations as functions (what I call *geometric functions*). I got lots of good ideas for improving the activities not only by watching the students, but also but also from their suggestions and the recommendations of their teacher, Jessica Shupik, and of my colleagues Daniel Scher and Mike Pflueger.

After several sessions of exploring reflections, translations, and rotations from a function perspective, we turned to the remaining transformation, dilations. The dilation activity emphasizes the way a function rule relates the independent variable (a point) and a dependent variable (its dilated image).

Ms. Shupik’s school promotes a mastery approach to grading and encourages performance-based assessments. With this in mind, we came up with three dilation games for assessing her students’ understanding:

- Game 1: Given an independent variable (a point) and a function rule (a center and scale factor), find the dependent variable (the dilated point).
- Game 2: Given the independent and dependent variables and the center point, find the scale factor.
- Game 3: Given the dependent variable and the function rule (the center and scale factor), find the independent variable. This third game is not solvable in the general case, but fortunately the dilate function has an inverse.

I later added Game 4: Given both independent and dependent variables and the scale factor, find the center of dilation.

Here’s the websketch containing all four games.

(Note the labeling of the points, which is explained in the activity that precedes the assessment. The independent variable is *x,* to ease the later transition to algebraic functions, and the dependent variable is *D _{C,s}*(

As students progress through the levels of each game, the problems become harder and the hints become fewer, ramping up the challenge bit by bit. (To move from one level to the next, first press Reset to start a new game, and then drag the Level slider.)

Ms. Shupik set a level of achievement for each game, and told students to email her a screenshot when they met that level. In retrospect, it might be even more motivating to students to set a hierarchy of levels of mastery, perhaps something like this:

I was surprised while creating and testing this game to discover how much fun I had playing its higher levels. Without the security of measurements and numbers, I had to develop a feel for dilation, a sense for what a dilation by 7/4, or by –0.625, feels like.

The nice thing about this assessment, from my point of view, is that the students were learning about dilation, getting (like me) an experience-based understanding of it, even as they were trying to achieve a particular level of mastery. The assessment and learning occurred simultaneously.

While the rhombus task worked well, it did expose one of the challenges of using Sketchpad: The software features lots of menu commands and toolbox options and navigating all of the functionality can sometimes distract Sketchpad newcomers from the mathematics at hand.

Web Sketchpad differs from Sketchpad by offering a more streamlined approach to mathematical construction. It allows a teacher or curriculum developer to create and provide only those tools needed for a particular task. We can use this approach to lead students to think about a problem in new mathematical ways, just by limiting them to carefully chosen tools.

Let’s consider, as we did in our professional development classes, the challenge of constructing a rhombus. Below are four different toolsets, each of which focuses students on different mathematical properties of a rhombus. Your task is to construct a rhombus with each toolset that stays a rhombus when its vertices are dragged. These are elegant challenges. Each toolset includes a Quadrilateral tool for indicating the four vertices and interior of your rhombus, but it’s really the other tools—the Compass tool, the Parallel Line tool, the Reflect Point tool, and Perpendicular Bisector tool—that are responsible for ensuring that the quadrilateral you construct is a rhombus.

Try it yourself—use each collection of tools to construct a rhombus that stays a rhombus when you drag any of its vertices. Are some of your rhombi more general than others? How would you compare the behavior of your four rhombi when their vertices are dragged? What characteristics of a rhombus does your construction exploit?

If you need help, watch the movie at the end of this post that demonstrates four rhombus constructions, one for each toolset. And please, let us know if you come up with a different construction in addition to those in the movie.

In a recent article from the online science magazine Quanta, Pradeep Mutalik reviews a gorgeous new math book, Creating Symmetry: The Artful Mathematics of Wallpaper Patterns, by Frank A. Farris. Below is a Web Sketchpad model of a parametric equation from the book.

The Quanta article notes that the curve has fivefold symmetry and challenges readers to explain the role of the coefficients 6 and 14 in producing this symmetry. You’ll find readers’ explanations for this behavior in the comments section of the article.

In addition to investigating this particular equation, the article asks readers to explore the effects of changing the coefficients of *t* to other values . The Web Sketchpad model makes this exploration simple: just drag the red, blue, and green sliders in the upper-right corner. Which curves do you find most interesting?

In the original design of tools (which you can still try here), tapping on a tool like a segment produced a preview image of the segment, with two glowing endpoints. To position the segment in the sketch, the user dragged each endpoint to its desired location. This method seemed sensible at at the time, but we gradually questioned our choice when even the simple task of constructing a triangle began to feel burdensome.

In our new tool design, it’s still possible to drag points to where you’d like them to go, but now there is an easier way as well. Tap the Segment tool icon in the websketch below, and then tap twice in the sketch to place the two endpoints wherever you want. Use the Segment tool twice more to construct a triangle with just a few taps of your finger or mouse.

Try all three of the tools in this websketch to get a feel for how this interface works. I’ve also included a video that demonstrates how the new interface works. Let us know what you think!

]]>Students quickly discover that the puzzle isn’t quite as straightforward as it looks. The numbers 1, 4, and 16 are needed to make a sum of 21, but the 16 is also needed to make a sum of 26 (2 + 8 + 16). How can the 16 be in both circles at once? The key insight is to realize that circles can overlap each other so that a number can reside in more than one circle.

Below is a twist on the Arranging Addends puzzles that I’m presenting here for the first time. Now, rather than using addends that are powers of 2, the addends are powers of 3, and there are two of each addend. Try solving some puzzles using this new set of addends. Unlike the original collection of addends where there was just one solution to each puzzle, the powers of 3 often yield multiple solutions.

And below, to continue the theme, is yet one more version of the puzzle, this time with powers of 5.

After you’ve solved some puzzles using the powers of 5, press the arrow in the lower-right corner of the sketch to move to a second model. Notice that in addition to the circles and the addends, this model contains boxes for you to input your answers as a code. How does the code work? Well, as a hint, notice that the 25’s sit above one column of boxes, the 5’s sits above another column, and the 1’s above the remaining column of boxes.

Even if your students have never been exposed to working in different bases, they can still understand how these boxes can be used for record keeping. For example, if the green sum is 60, we can spare ourselves the trouble of saying that we need two 25’s, two 5’s, and zero 1’s and simply write “220” (which happens to be 60 in base 5).

My colleague Scott notes that these codes can help students devise a strategy for putting the addends into the circles. For example, suppose we’re given a puzzle where the green sum is 80, the blue sum is 26, and the red sum is 96. First, we write these numbers in code:

A student might then reason as follows to determine how to arrange the circles and addends:

“Let me do the 25’s first. The circles all need at least 1, so I put 1 in the overlap of all three, and then I need 2 more 25’s for the green/red overlap.”

“Now the 5’s. I don’t put any in the overlap of all three circles, but I put 1 in the overlap of the green/red. Now I just need three more 5’s in the red circle.”

“Finally the 1’s. I only need one of those, in the blue/red overlap. All done.”

Younger students might simply solve the puzzles without the code. Older students can be challenged to use the codes to record their solutions, and can also be challenged both to explain their solution strategies and to find ways to make their strategies more efficient. Some students are likely to recognize the correspondence between the codes and the solutions and invent the strategy described above: reading the solution directly from the code.

–Guest post by Mirek Majewski

In this blog post, I will show how the mosaic in the entrance to the Sultan Ahmed Mosque in Istanbul can be created using tiles in the shape of regular hexagons with the help of Sketchpad. I will then show how you can embellish the mosaic by varying the patterns, shapes, and colors on the tiles.

The mosaic contains zig-zag paths of double lines. Between them are formed regular hexagons and regular six-pointed stars. The picture below shows the pattern.It is easy to draw the outlines of large regular hexagons onto this pattern

The above illustration suggests a very natural way to design the pattern. We start with a regular hexagon, divide each side into thirds, and draw a grid connecting the trisection points. With this grid, we can easily build the Sultan Ahmed mosaic pattern. The Geometer’s Sketchpad construction is shown below.

After removing all unnecessary elements of this construction we get a nice tile.

With this tile complete, we can save it as a reusable tool for easily constructing additional copies of the tile (See the interactive Web Sketchpad model at the end of this post.)

Using a bit of imagination, you can modify and colorize the tile. Here is an example.

You can create many other patterns on the hexagonal tile using the grid from above. However, you should bear in mind that the paths on one tile should lead into the paths on adjacent tiles without any breaks and bends. This means a path passing through the edges should be straight. Of course all tiles should have the same size.

To add even more variety to our patterns, we observe that hexagonal tiles can be arranged so that there are triangular gaps between them. Thus we can create a triangular tile that will fit these gapes to match the hexagonal tiles surrounding it.

We can also construct square tiles to match the hexagonal and triangular tiles. Here are some examples.

In order to match lines on the square with lines on a hexagon, they should form 30° angles with a line perpendicular to the point on the edge of a tile.

Starting from a hexagonal tile based on the mosaic in the Sultan Ahmed Mosque, we branched out into a set of three tiles – a hexagon, a triangle and a square. Your creativity will benefit significantly from this expanded choice of tiles. Using these three tiles, you can construct many interesting designs. Here is one of them:

And here is another pattern built with triangles and hexagons only.

Both patterns are regular tilings of the plane. In other words, they cover the plane without overlapping or gaps between tiles, and the pattern has a transitive symmetry in two different directions. How many regular patterns we can create with our tiles? What about less regular patterns? Can we produce them with this small set of tiles?

Below is a Web Sketchpad model for experimenting and building patterns of your own. The model comes with several tools, including a hexagonal tile tool based on the Sultan Ahmed mosque and two supplementary tile tools. Use the Link buttons to move from page to page.

The very last page of this geometric playground contains two red points. When you create your first tile, drag the two glowing points onto the red points to attach them. Continue by creating more tiles and attaching them to the ones in your sketch. If you’d like change the scale of your mosaic pattern or rotate it, just drag either of the two red points. If you wrongly place a tile or decide that you’d like to try something different, use the arrows above the tools to undo and redo each step of your work.

]]>You can see this method below in a picture from the 17th-century manuscript *Sive de Organica Conicarum Sectionum in Plano Descriptione, Tractatus* (*A Treatise on De**vic**es for Drawing Conic Sections*) by the Dutch mathematician Frans van Schooten.

When I wrote about the pins-and-string construction, I provided a pre-built Web Sketchpad model of an ellipse based on this definition. Now it’s time for you to construct this model yourself using the tools I’ve provided in the websketch below. To get you started, I’ve included a short movie at the end of this post that walks you through the steps.

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This is a classic problem, dating back to an 1854 examination at Cambridge University.

As I considered how to solve the puzzle, I recalled a related probability challenge, Wait for a Date, that I featured in this blog last year: *Two friends arrange for a lunch date between 12:00 and 1:00. A week later, however, neither of them remembers the exact meeting time. As a result, each person arrives at a random time between 12:00 and 1:00 and waits exactly 10 minutes for the other person. When the 10 minutes have passed, each person leaves if the other person has not come. What is the probability the friends will meet?*

In both puzzles, there are two pieces of information: the two locations where the stick is broken or the times when the two friends arrive. Based on the information, the sticks either do or do not form a triangle, and the friends either do or do not cross paths.

In Wait for a Date, I represented the two arrival times as a single point in the plane, with the *x*– and *y-*coordinates indicating the two times. Doing so allowed me to run a simulation where I randomly varied the two coordinates and colored the resulting points either red or green, depending on whether the two friends met. I suspected that this geometric approach would work equally well with the broken stick puzzle.

Below is my Web Sketchpad model. The stick is represented by the interval from 0 to 1 on the *x-*axis, and points *A* and *B* represent the locations where the stick is broken. Point *B* is displayed on the *y-*axis as well, allowing me to plot a single point, *P,* whose x-coordinate is the location of the break at point *A* and whose *y*-coordinate is the location of the break at point *B*. Press *Run Simulation Once* several times. Notice that the point *P* is green when the three broken sticks can form a triangle and red otherwise. Now, press *Run Simulation Repeatedly* and watch as the screen fills with green and red points. What is the probability that the broken sticks form a triangle?

As the image below suggests, the probability of making a triangle is 1/4.

I enjoyed watching Sketchpad create this picture, and now wanted a way to understand why certain regions were red and others were green. I started with the all-red square regions in the lower-left and upper-right corners. I dragged points *A* and *B* so that point *P* sat in one of these regions and examined the situation. Visually, it was clear that breaking the stick at *A* and *B* resulted in two smaller sticks whose sum was less than 1/2. Since the remaining third stick was greater than 1/2, the triangle inequality tells us that no triangle is possible.

I then turned my attention to the the shared hypotenuses of the red and green right triangles. What was true about the stick lengths when point *P* sat on a hypotenuse? I suspected that at these locations, one stick was exactly equal to 1/2, meaning that the remaining two sticks summed to 1/2 as well. To check my conjecture, I altered my model slightly. To view the new model, press the arrow in the lower-right corner of the websketch. Now, when you drag point *A,* you’ll see that point *A’* is always exactly a distance of 1/2 away. Thus we can see all the possible stick breaks that result in one stick having a length of 1/2. Dragging point *A* back and forth between 0 and 1 reveals that my hunch was correct—point (*A, A’*) traces the two hypotenuses separating the red and green regions.

Digging deeper into the model, I picked a random spot for point *B,* left it in place, and dragged point *A* back and forth along the *x*-axis. This traced a single “slice” of successes and failures and allowed me to make intuitive sense of the situation.

The picture below shows an example of this strategy (Try it yourself on the first page of the websketch.) In the uppermost image, points *A* and *B* are to the left of 1/2 so as I described above, they form two sticks whose sum is less than 1/2. No good! In the middle image, point *A* is to the right of 1/2 and the distance between *A* and *B* is less than 1/2. In this case, *AB* is clearly less than the sum of the other two sticks and the sum of *AB* with either of the other two sticks is greater than the remaining stick. We have a triangle! In the remaining image, the distance between *A* and *B* is greater than a 1/2 so no triangle can be formed.

I’ll leave you with two variations of the broken stick puzzle:

- Suppose point
*A*is chosen first, at random, and the stick is broken at this location. Point*B*is then chosen at random from the longer of the two resulting sticks. What is the probability these three sticks form a triangle? You can run a simulation of this problem on the third page of the websketch above. - Suppose point
*A*is chosen first, at random, and the stick is broken at this location. One of the two resulting sticks is then chosen at random and split at a random location. What is the probability these three sticks form a triangle? This may sound the same as the original problem, but it isn’t! You can read more about this variant, as well as very clever alternative solution to the original puzzle, in this Martin Gardner column.

Last week’s puzzle caught my attention because it seemed tailor made for Sketchpad. I’ve described the puzzle in the Web Sketchpad model below, but you can learn about its origins from the mysteriously named Dr. W, who first brought the puzzle to the attention of the Numberplay editor.

I recommend trying the puzzle first before viewing my solutions on the second, third, and fourth pages of the Web Sketchpad model. (Use the arrows in the lower-right corner to navigate between pages.) Dr. W recommends building yourself a physical model of the triangles. Indeed the puzzle makes a fine hands-on challenge for children.

You’ll notice that nowhere in my Web Sketchpad model do I prove any results. That’s left for you!

]]>In a sense I got ahead of myself because I skipped straight to decimal approximations without focusing first on integer estimation. In the interactive Web Sketchpad model below, the red point sits at an integer—perhaps 87? To check, press *Zoom* to magnify the portion of the number line between 80 and 90. The magnified view reveals that the point is actually at 86. Pressing *New Problem* hides the second number line and moves the red point to a new location.

To progress to the next level of challenges, press the arrow in the lower-right corner of the sketch. Again, the red point sits at an integer. What is its location? 50 is a reasonable guess. Pressing *Zoom* reveals that this estimate is a bit off. The magnified view reveals the location of the point is slightly larger than 50. Perhaps 53? Zooming in one more time settles it—the location of the point is indeed 53.

The remaining two pages of the sketch ask students to estimate the location of a point between 0 and 10,000, and 0 and 100,000. As the range of possible values grows, students have the opportunity to zoom in more and more, each time refining their estimate of the point’s location.

You can download the desktop Sketchpad version of this activity, along with teacher notes, and you can find several other activities involving decimals and place value on the Dynamic Number website.

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