Our article summarizes our curriculum unit, *Connecting Geometry and Algebra Through Functions.* This unit’s Web-Sketchpad-based activities connect functions in geometry (transformations whose variables are points on the plane) with functions in algebra (whose variables are points on the number line).

In particular, students work with geometric transformations as functions that take an input point and produce an output point, and relate these functions to algebra by using them to construct the Cartesian graph of a generalized linear function. They dilate and then translate a point, restrict these points to number lines, and ultimately observe that in the algebraic equation *y* = *mx* + *b*, *m* corresponds to the scale factor for dilation and *b* corresponds to the length of the vector for translation.

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As part of that work, I just completed a websketch that nicely mixes practice with logical reasoning. Students are challenged to find a hidden treasure on the coordinate plane by guessing its location. For each guess, students are told its taxicab distance (traveling horizontally and vertically) to the treasure. Using this information as clues, students deduce where the treasure lies.

The websketch below starts by introducing students to the way distance is measured between each guess and the treasure. The second and third pages (accessible by using the arrows in the lower-right corner) present two different views of the coordinate plane, the first showing only the first quadrant and the second showing all four quadrants.

The game is geared towards elementary-age students, but it’s suitably challenging for older students, too. Personally, I find it addicting to play!

]]>Bunny Times uses carrots in a field to represent the array model of multiplication. In the first few levels, students determine the total number carrots in the field and a single bunny eats them all. Some students may count the carrots one by one, but soon, a patch of fog rolls in, obscuring many of the carrots. In later levels, bunny teams eat carrots in unison, with students able to specify the number of bunnies in each team. These game elements lead students to develop new strategies for thinking about multiplication, such as skip counting, adding on, and deconstructing problems like 7 x 8 into (5 x 8) + (2 x 8).

When we released Bunny Times in 2013 as part of the NSF-funded Dynamic Number project, you needed to either own Sketchpad or proceed through a multi-step installation process on the iPad to play it. Now, however, Web Sketchpad makes everything simple: You can play Bunny Times in your web browser by clicking the picture below.

If you tap the ‘Learn to Play’ sign post on the opening Bunny Times screen, you’ll be taken to a video that gives an overview of the game. If you’d like to read about the pedagogy behind the game and a mathematical overview of each level, download this document.

Have fun and share with us your experience of using Bunny Times with your students or children!

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I was reminded of this theorem while reading the article Using Appropriate Tools Strategically by Milan Sherman and Charity Cayton in the November 2015 issue of NCTM’s *Mathematics Teacher*. Their article addresses the pedagogical potential of Dynamic Geometry technology and focuses on the power of a point theorem as a concrete example of how the software can foster high cognitive demand tasks.

Rather than summarize Sherman and Cayton’s ideas here, I’ll describe a Dynamic Geometry application of the power of a point theorem not covered in their article:* Use the power of a point theorem to build a constant-area rectangle. A constant-area rectangle is one whose perimeter can change but whose area remains fixed.*

Since the product of *AP *and* PB *is constant as point* A *travels around the circumference of the circle, this suggests that we construct a rectangle whose dimensions are* AP *and

Does this construction show us every possible rectangle with an area of 21.3 cm^{2}? No. As point *A* spins, we don’t see rectangles where *AP’*s length is close to zero (like *AP* = 0.001 cm) and we don’t see rectangles where *AP’*s length is very large (like *AP* = 300 cm). Indeed, *AP* can be no larger than the diameter of the circle.

Might dragging point *P* very near the circumference work? Try it—you’ll notice that the constant area of the rectangle changes, and that clashes with our goal: We want to see all possible rectangles with an area of 21.3 cm^{2}. (Dragging point *P* outside the circle has interesting results; results that are beyond the scope of this post.)

Fortunately, there is another method of generating a constant-area rectangle. On page 3 of the websketch, you’ll see a right triangle *BDA*. Altitude *AC* is the geometric mean of *BC* and *CD,* meaning that *BC · CD* = *AC*^{2} = 21.3 cm^{2}. By animating point *C* along its horizontal line, distance *AC* remains constant, so we again create segments whose product is constant. But this time, as point *C* moves to the left, distance *BC* approaches 0 and distance *CD* approaches infinity.

There’s a nice connection to be made between these two methods of creating constant-area rectangles. Page 4 of the websketch again shows right triangle *BDA*, but now we see its circumcircle as well. Segments *AA’* and *BD* are chords of the circle with *BC* *· CD = AC · A’C, or * *BC · CD* = *AC*^{2} since *AC* = *A’C*. Thus the right triangle construction is really just a special case of the circle construction, with one of the two chords being a diameter of the circle.

I first wrote about constant-area rectangles in the April 1996 issue of NCTM’s *Mathematics Teacher*. Sketchpad has come a long way in twenty years, but my article’s conclusion still holds: “A chief pleasure of these investigations comes from taking theorems that may seem like nothing more than geometric curiosities and turning them into devices that perform a desired function. Specifically, the geometry that lies behind the chord theorem and the geometric-mean construction becomes the engine driving the movement of the constant-area rectangles. By setting these theorems in motion, students are able to generalize them and uncover relationships that the static counterparts in a textbook cannot reveal.”

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.

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