By contrast, Dan notes that the Smarter Balanced Assessment Consortium (SBAC) has a stronger focus on questions that ask students to “construct,” “analyze,” and “argue” as a way to test their understanding of Common Core mathematics. A Khan Academy diet of multiple-choice items and numeric answers would seem ill suited to helping students prepare for the types of mathematical reasoning expected by the Common Core.

What might explain the mismatch between the Khan Academy’s offerings and the demands of the Common Core? Dan Meyer’s answer is a practical one: Our ability to assess students’ mathematical understanding with technology is primitive. He explains, “Khan Academy asks students to solve and calculate so frequently, not because those are the mathematical actions mathematicians and math teachers value most, but because those problems are easy to assign with a computer in 2014. Khan Academy asks students to submit their work as a number or a multiple-choice response, not because those are the mathematical outputs mathematicians and math teachers value most, but because numbers and multiple-choice responses are easy for computers to grade in 2014.”

Dan’s comments about the difficulty of authoring rich, computer-gradable mathematics tasks reminded me of work that Steve Rasmussen, Scott Steketee, Nick Jackiw, and I undertook several years ago. We set out to discover whether Sketchpad could deliver assessment items that would engage students in dynamic mathematical models and have them demonstrate their understanding through their interactions with the models. Starting with this blog post and in the next two or three to follow, I’ll share some of our assessment ideas.

I’ll state upfront that our examples are not revolutionary—these are mathematical tasks that are several notches above numerical answers and multiple-choice responses, but they’re still based on traditional content. Nonetheless, they are representative of what can be done now with Web Sketchpad, and they point the way to more ambitious possibilities.

I’ll start with a question that assesses students’ basic understanding of isosceles triangles. Read the multiple-choice item below. It’s a straightforward question, and if a student knows the definition of an isosceles triangle, there isn’t any thinking required. Indeed, even if a student knows nothing about isosceles triangles, she might still pick the correct triangle, as only one triangle visibly differs from the others in having two of its angles equal.

Now, consider the assessment task below. It again centers on the fundamental definition of an isosceles triangle, but it is different from its static counterpart above. Rather than pick which of four triangles is isosceles, students are given a single triangle—one whose lengths and angles can be changed by dragging its vertices—and are asked to make it isosceles. The multiple-choice question had just one right answer; this dynamic version yields numerous solutions. *Any* isosceles triangle the student creates, whether it be one with angle measures 2°, 2°, and 176° or 50°, 50°, and 80°, counts as correct.

When compared to the passivity of identifying an isosceles triangle, the engagement required to make an isosceles triangle feels like a step up on the assessment ladder. But with better assessments comes the question of how we should grade them. The multiple-choice version is clear enough—a student’s answer is either right or wrong with no room for partial credit. But grading the dynamic version is thornier. Answering the question requires a student to use a mouse, a trackpad, or perhaps her finger to position the triangle’s vertices. Try this yourself with the Web Sketchpad model above. It can be surprisingly tricky to get the triangle’s angles to match the values you have in mind!

The motor skills needed to adjust the angles precisely led us to factor in some forgiveness to the scoring, shown below. If a student drags the triangle so that two angles are either 1 or 2 degrees apart, we award him partial credit and the benefit of the doubt, assuming that his unrealized goal was to create two congruent angles.

Are we too generous? Perhaps. But if a computer-based assessment item makes genuine use of the opportunities offered by technology (dragging, constructing, measuring, etc.) then we have to consider whether students have enough exposure to the technology prior to the test so that the assessment measures their mathematical understanding and not their proficiency with the software (Of course, if the assessment is grounded in poorly designed software, then no degree of practice may make a difference!)

Finally, imagine taking this isosceles triangle item to the next level of assessment sophistication. Rather than provide students with a triangle, we start them with a blank screen and a circle and segment tool. Using these tools, their goal is to construct a triangle that stays isosceles no matter how its vertices are dragged. This is a great example of a meaningful construction task, but it raises several questions:

- Will Common Core curricula commit to making computer-based constructions a part of their content?
- Can teachers devote enough time to such construction tasks so that students are prepared for them when assessed?
- Can we find a way for computers to grade construction tasks?

I’m curious to hear what you think!

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In the interactive Web Sketchpad model below, press *Jump Along* to watch the bunny take 2 jumps of 4 along the number line. The bunny leaves behind a trail of its path, providing a visual representation of 2 x 4 = 8.

With the bunny back at 0, it’s time to find other ways to reach 8. Enter new values for “Number of Jumps” and “Jump By.” Before pressing *Jump Along*, however, drag the red point that sits on the multi-colored segment. The color of this point controls the color of the bunny’s jumps. By making each set of jumps a different color, it’s easier to distinguish one from another, and the resulting rainbow-like pattern is an attractive artifact of students’ work.

Below are all four ways to reach 8. While students often meet the equivalence of *a* x *b* and *b* x *a* with a shrug, here we have a nice visual representation that distinguishes 2 x 4 and 1 x 8 from 4 x 2 and 8 x 1. We can also see the factors of 8 by noting where each of the four paths first lands on the number line (The red path, for example, lands first at 4, indicating that 4 is a factor of 8.)

Students can now explore other destinations on the number line. To jump to numbers larger than 12, just drag the point at 1 closer to 0 to rescale the number line.

As students explore factor rainbows, they can explore questions like: Do certain numbers create prettier factor rainbows than others? For larger target destinations, are there more ways to reach the target? Which factor rainbows have only 2 paths? Do all factor rainbows contain an even number of paths?

]]>–Guest post by Michael de Villiers

“*Mathematics is about problems, and problems must be made the focus of a student’s mathematical life. Painful and creatively frustrating as it may be, students and their teachers should at all times be engaged in the process — having ideas, not having ideas, discovering patterns, making conjectures, constructing examples and counterexamples, devising arguments, and critiquing each other’s work.*“ – Lockhart (2002)

Encouraging creative problem posing and problem solving in a mathematics classroom means that students will inevitably propose false conjectures that need to be refuted by counter-examples. The ability to refute or disprove false mathematical statements is in many ways as important as proving true ones, but is often neglected in teaching and learning.

This blog post describes a mathematical exploration that I recently undertook with a mathematical colleague, Nic Heideman. It highlights the dilation facility of Dynamic Geometry software and its role in refuting two very plausible conjectures. It* *follows on two iterative construction procedures described in De Villiers (2014) where iterated triangles converge towards an equilateral triangle.

**Investigation 1: Tangent Points of Incircles**

Start with any Δ*ABC* and its incircle and incentre *I*. Label the points where the circle touches the sides *BC, CA,* and *AB* and respectively as *A*_{1}, *B*_{1}, and *C*_{1}. Repeat the process with the new Δ*A*_{1}*B*_{1}*C*_{1} constructing the next incircle, *I*_{1}.

Then repeat the process twice more. When you’re done, connect incentre *I* to *I*_{3} with a straight line. What do you visually notice about the four incentres? Check by dragging vertices *A, B,* and *C *in the interactive Web Sketchpad model below. Can you make a conjecture? Can you prove or disprove it?** **

**Investigation 2: Excentres**

Start with any Δ*ABC *and construct its incentre *I* and excentres (The three excentres of a triangle are located at the intersection of the angle bisectors of the two exterior angles formed on each side of the triangle.) You can view this construction in the Web Sketchpad construction above by pressing the arrow in the lower-right corner of the screen.

Label the excentres formed on the sides of the sides *BC, CA,* and *AB* respectively as *A*_{1}, *B*_{1}, and *C*_{1}, and construct incentre *I*_{1 } of the new Δ*A*_{1}*B*_{1}*C*_{1}. Repeat the process with Δ*A*_{1}*B*_{1}*C*_{1}. Then repeat the process twice more. When you’re done, connect incentre *I* to *I*_{3} with a straight line. What do you visually notice about the four incentres? Check by dragging vertices *A, B,* and *C *in the Web Sketchpad model. Can you make a conjecture? Can you prove or disprove your conjecture?** **

In our first investigation, it clearly seems that all four incentres are collinear (lie on the same straight line), with *I*_{2} and *I*_{3} almost coinciding. The same seems to be true in our second investigation: Although *I*_{1} does not lie on the constructed line from *I* to *I*_{3}, the other three incentres appear to be collinear. Using Dynamic Geometry software to drag our constructions convinced us that the conjectures were valid.

Armed with compelling experimental evidence that our conjectures were true, we proceeded to attack the two conjectures trying both geometric as well as algebraic approaches. Neither approach was immediately successful, with the algebraic approach becoming especially cumbersome and messy. Scanning the literature for any mention of the results, as well as other related mathematical results we might be able to use, also proved fruitless. While we did find that Denison (2001) describes the second conjecture as unproven, he incorrectly claims that all four incentres are collinear (Our Dynamic Geometry model illustrates that *I*_{1} is not collinear with the other three incentres.)

**Refutation of Conjectures 1 and 2**

Our frustrating inability to prove Conjectures 1 and 2 gradually led us to suspect that perhaps they were false, despite the seemingly convincing experimental evidence. So we went back to the proverbial drawing board to more closely examine the conjectures, this time trying to produce counter-examples to disprove them.

Since the incentre points were grouped so closely together, we clearly needed to enlarge the figures by zooming in. This could be achieved by dragging the entire figure to make it bigger and bigger. Alternatively, and more efficiently, we could use the dilation tool of the Dynamic Geometry software to enlarge relevant portions or elements of the figure to examine them more closely.

By marking *I*_{2} as the centre of dilation, and dilating the blue line through *I* and *I*_{3} as well as the incentres *I*, *I*_{1}and *I*_{3}, by a factor of 100 to 1, we noted that the line shifted to the dashed red line, as shown in the figure below (You can also press *Show Dilated Objects *in the Web Sketchpad model and then drag the dilation slider’s scale from 1 to 100. It might be necessary to drag the vertices of Δ*ABC* to see the line clearly.)

Notice that the images of *I* and *I*_{3} still lie on the red line, whereas the image of *I*_{1} does not (Points *I´* and *I´*_{1} are off screen, but in the software one can scroll up and to the right to check where they actually lie in relation to the enlarged, red line.) More over, despite *I*_{2} appearing to lie on the constructed blue line from *I* to *I*_{3}, the line shift clearly shows that *I*_{2} is not on the line. Despite our strong, initial conviction, this showed conclusively that the incentres for Conjecture 1 were not collinear!

In the figure below showing Conjecture 2, we performed a similar dilation. Using *I*_{2} as our center of dilation, we dilated the dashed line through *I *and* I*_{3} as well as incentres *I* and *I*_{3} by a scale factor of 100. In doing so, the line shifted to the red dashed line and we can see that *I*_{2} does not lie along it (As before, you can also press *Show Dilated Objects* in the Web Sketchpad model and then drag the dilation slider’s scale from 1 to 100. You’ll need to drag the vertices of Δ*ABC* to see the line clearly.) Note again that because of the large scale factor, images *I´* and *I´*_{3} are completely off screen. Some scrolling, however, confirms that they are on the dilated red line.

Since the incentres lie so close to a straight line, it is important to emphasize that there is hardly any way we would have found these counter-examples by mere paper-and-pencil construction — unless, that is, we’d used a sheet of paper about 100 times the size of an A4 sheet, and were able to make accurate constructions using extremely large and unwieldy compasses and rulers! This episode therefore lucidly illustrates how useful computing software has become in modern day mathematical research, not only to find and formulate new conjectures, but also to enable one to disprove false statements with the production of counter-examples (compare De Villiers, 2010; Borwein, 2012).

**References**

Borwein, J.M. (2012). Exploratory Experimentation: Digitally-Assisted Discovery and Proof. In G. Hanna & M. de Villiers (Eds.), *Proof and Proving in Mathematics Education*, New ICMI Study Series 15, pp. 69-96.

Denison, B. (2001). Triangles & Quadrilaterals: A Response. *Mathematics in School*, Nov 2001, pp. 15-16.

De Villiers, M. (2014). Over and Over Again: Two Geometric Iterations with Triangles. *Learning & Teaching Mathematics*, No. 16, July 2014, pp. 40-45.

De Villiers, M. (2010). Experimentation and Proof in Mathematics. In G. Hanna, H.N. Jahnke & H. Pulte (eds.), *Explanation and Proof in Mathematics: Philosophical and Educational Perspectives* (pp. 205-221). New York: Springer.

Lockhart, P. (2002). *A Mathematician’s Lament*. In Devlin’s Angle (March 2008).

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The 17th-century Dutch mathematician Frans van Schooten developed “hands-on manipulatives” centuries before the term became popular in math education circles. Below are two images of ellipse-drawing linkages from van Schooten’s manuscript, *Sive de Organica Conicarum Sectionum in Plano Descriptione, Tractatus* (*A Treatise on De**vic**es for Drawing Conic Sections*).

Building physical models of these devices isn’t hard. A bent straw, for example, works well for the linkage on the left. As another approach, you can use Sketchpad to construct them.

The interactive Web Sketchpad models below allow you to draw ellipses using the two models above as well as two other related methods. For each model, press the *Animate* button to set it in motion. You can move between the models using the arrows in the bottom-right corner of the sketch window.

For each model, experiment with different locations of the green point that’s tracing the ellipse. How does the shape of the ellipse change based on the point’s position? You can also experiment with the length controls at the bottom left of each page.

It’s not especially difficult to prove that these four models do, in fact, draw ellipses. The *Show Proof Hints* button on the Bent Straw page offers some ideas that you can apply to the other two models as well.

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

]]>My panel was supposed to discuss his work on the UCSMP curriculum. My two co-panelists had been deeply involved in that work, but I had not, so I took the opportunity to address one aspect of the mathematics that Zal pioneered in *GATA* and that remains prominent in *UCSMP Geometry* today: the treatment of geometric transformations as functions (a treatment that I describe here as “geometric functions”).

My hope was to show how activities based on *The Geometer’s Sketchpad *not only support Zal’s insight from over 40 years ago, but validate it in ways that weren’t even well-understood at that time. I wanted to make two main points, one about cognitive science and one about mathematics:

- Treating transformations as functions is supported by the cognitive science findings regarding embodied cognition and conceptual metaphor, as described (for instance) in
*Where Mathematics Comes From*by Lakoff and Nunez. When students drag a point as the independent variable, they are experiencing variables in a physical way, and the act of varying point*x*and observing the resulting motion of point*r*(_{j}*x*) (the reflection across mirror*j*of*x*) becomes for the student a conceptual metaphor for the function that relates the two points.

- Treating transformations as functions enables students to connect geometry and algebra in a very direct and elegant way. Students can restrict such transformations to a number line, thereby turning the two-dimensional point variables of geometry into one-dimensional real numbers of algebra while simultaneously turning the transformation itself into a linear function. Having done so, they can apply a translation to the dependent variable to produce the Dynagraph representation invented by Goldenberg, Lewis, and O’Keefe, or they can apply a rotation to produce the Cartesian graph of
*y*=*mx*+*b*(where*m is*the scale factor for dilation and*b*is the vector length for translation).

Here’s a movie I made of my presentation:

And here’s a Web Sketchpad version that shows the restriction of the geometric function to a number line to turn it into a linear function, and the subsequent transformation that represents the function as a Cartesian graph.

It was an honor to participate in this symposium, and I hope I did justice to Zal’s insight from so many years ago by showing its deep connections both to cognitive science and to the unity of geometry and algebra.

Note: In advance of the symposium and dinner, Lisa Carmona (Vice President at McGraw-Hill Education, preK-5 Portfolio) put up an eloquent post on the McGraw-Hill Education blog attesting not only to Zal’s accomplishments, but to the way he inspires so many of us to advocacy as well as a commitment to students’ deep understanding of mathematics.

]]>This week, I set my sights on isosceles triangles. It’s common to encounter isosceles triangles as supporting players in geometric proofs, but how can isosceles triangles be made the stars of their own mathematical puzzles?

Above is a collection of interactive Web Sketchpad puzzles. Each puzzle begins with an isosceles triangle *ABC *and a point *R* on the coordinate grid. Your goal is to drag a vertex of the triangle to point *R* so that the new triangle is also isosceles. In the first puzzle, this is easy: Dragging either vertex *A, B,* or *C* to point *R* results in an isosceles triangle.

Press the right arrow at the top of the sketch page to move on to the second puzzle. Point *R* now sits at (3, 2). This challenge is straightforward, too: Dragging point *B* to point *R* yields an isosceles triangle.

Pressing the right arrow again reveals the third puzzle, and this one is a little different. It’s no longer possible to drag vertex *A, B,* or *C* directly to point *R* while keeping *ΔABC* isosceles. We need to add a new rule to our puzzle: *You can drag more than one vertex of ΔABC to reach point R, but each intermediate resting position of the triangle must be isosceles.*

We can solve this particular puzzle in two steps: First, drag point *B* to (3, 2). The resulting triangle is isosceles. Now drag point *A* to point *R* to solve the puzzle. Again, the triangle is isosceles.

Most of the remaining Web Sketchpad puzzles above require multiple steps to position a vertex of isosceles triangle *ABC* at point *R*. As your students solve the challenges, here are some questions and tasks to present to them:

- Name all the locations for point
*R*that allow you to solve the puzzle in just one move. Do these points, or a subset of them, share anything in common? - Can you find the fewest number of moves needed to solve each puzzle?
- What general strategies did you discover for solving the isosceles triangle puzzles? Can you describe or illustrate these strategies so that a friend could follow them?
- Can you find a way to predict how many moves it will take to solve a puzzle without carrying out each step?
- On the second-to-last page of the Web Sketchpad model, pick your own location for point
*R*and challenge a friend to solve your puzzle. - On the last page of the Web Sketchpad model, find a sequence of moves for
*ΔABC*so that one vertex of the triangle rests on point*R*and another vertex on point*P*.

Here’s how the static version of the puzzle works: Each of the ten letters in the Web Sketchpad model above has been assigned a random, secret value between 1 and 10 (It’s possible that two or more letters may share the same value.) Your job is to determine these values. To do so, you’ll use the circle. To move the circle, you can drag its interior. You can also drag the point sitting on the circumference to change the circle’s size.

Whenever two or more letters sit inside the circle, its interior turns green, and you’ll be told the sum of those letters. So, for example, when G and I are in the circle’s interior, Sketchpad reports that G + I equals 16.

To play again with new secret values, press* New Puzzle. *When your students are ready to try a different challenge, press the arrow in the bottom-right corner of the model. There are still ten letters whose values must be determined, but hey–the letters are moving (dancing) across the screen! The rules are still the same–surround two or more letters with the circle to determine their sum–but the movement of the letters makes the puzzle more challenging.

What strategies do your students use to uncover the values of all 10 letters?

When I tested this activity with a group of fourth graders, Lee solved the static version of the puzzle by dragging the circle so that it covered all ten letters. He then adjusted the circle so that it covered 9 letters, thus allowing him to determine the value of the extra letter. He continued in this fashion, scaling back the number of letters sitting in the circle from 9 to 8 to 7, and so on, all the way to just 2 letters.

When Lee moved on to the “dancing” version of the puzzle, he began with the same strategy of covering all ten letters with the circle and then scaling back one letter at a time. But because the letters did not stay put, he found it difficult, if not impossible, to remove one letter at a time from the interior of the circle. After some consideration, Lee realized he would need to adopt a new strategy: He began by encircling 2 letters, noting their value, and then adjusting the circle to include a third letter. He continued in this manner, picking other pairs of letters and adding a remaining letter to the pair. It was fascinating to watch Lee modify his original strategy to this new one, motivated by the switch from static to dynamic letters.

This activity first appeared as part of the NSF-funded Dynamic Number project. You can download the original Sketchpad model and teacher notes here.

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

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