With desktop Sketchpad, so-called “custom” tools can come to your rescue. By building an equilateral triangle just once, you can save it as a tool that can instantly create more equilateral triangles with just a few clicks of your mouse.

Despite their obvious value, custom tools lack visibility in Sketchpad’s interface. To find them, you need to burrow into Sketchpad’s custom tool menu, as shown below. Wouldn’t it be nice if using a square tool were as convenient as using the point, compass, and straightedge tools?

With Web Sketchpad, custom tools finally come out of the shadows. The web sketch below allows students to construct equilateral triangles, squares, rhombi, hexagons, and octagons simply by clicking the icons. If you click the equilateral triangle icon for example, you’ll see a preview of the triangle you’re about to create, with two of its vertices pulsating. By dragging both vertices to locations of your choice, the triangle becomes permanent. If you then click the equilateral triangle icon again, you can create a second equilateral triangle. And if you drag the pulsating vertices to coincide with two vertices from the first triangle, you can merge these points together, creating two attached triangles.

Use the above Web Sketchpad model to create tessellations. How many different ones can you make? If you make a mistake while building a tessellation, there unfortunately isn’t yet the ability to undo it. You’ll need to refresh your web browser window and start again.

If you’re interested in knowing how I built this set of polygon tools, I’ll describe briefly what I did: I began by constructing an equilateral triangle, square, rhombus, hexagon, and octagon in desktop Sketchpad. I took a screenshot of each polygon to be used as the icon for the tool. I then assembled the polygons into one Sketchpad document, followed a simple naming convention to indicate that these polygons were tools, and then fed the sketch through an exporter. Presto! The model was complete.

The opportunities for customizing sketches to contain only those tools relevant to an investigation are vast. In upcoming posts, I’ll present many more examples!

]]>But there is one element of desktop Sketchpad that has been conspicuously missing from Web Sketchpad; a fundamental ability of Dynamic Geometry software that’s essential to the mathematical and pedagogical value of Sketchpad. That element is construction.

It’s fashionable these days to think that students can learn mathematics simply by playing games in which they interact with objects already on the screen. I can think of few iPad apps in which students build their own mathematics from scratch. Indeed, our own Sketchpad Explorer app for the iPad is guilty of not allowing students to construct objects—and we’ve heard plenty of requests for adding that functionality!

When students open a new Sketchpad file on their computer, they’re presented with a blank screen. Much like the experience of starting with a blank word-processing screen, it’s up to students to decide what to create. There to help them are a variety of tools found in the toolbox (shown at right). These tools include a point tool, a compass tool, a straightedge tool, and a polygon tool.

Below is a Web Sketchpad model that displays a blank screen with a point, compass, segment, line, and ray tool ready for you to use. For those of you familiar with desktop Sketchpad, you’ll notice that these tools don’t follow the same conventions as their desktop counterparts. Notice, for example, that when you click or tap on the segment tool, you immediately see a preview of the segment in the sketch. The endpoints of the segment pulse, indicating that you can drag these points wherever you like to place the segment. Try constructing a triangle, a circle and its radius, and an isosceles triangle. You’ll see that when you drag the flashing points, you can merge them with other points or attach them to other objects, like circles and lines. If you’d like to return to a blank screen, simply refresh your browser window.

A caveat: We’re still in the early phases of developing tool functionality for Web Sketchpad, so features like the ability to undo your actions, hide objects, change colors, and add labels are not yet implemented.

You also might wonder about other functionality in desktop Sketchpad, like the ability to construct perpendicular lines, transform objects, and use “custom” tools that construct objects like squares nearly instantly with just a few clicks of your mouse. These capabilities do exist in Web Sketchpad. You’ll be reading about them (and experimenting with them) in the weeks to come here on our blog!

]]>The picture above comes 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. The ellipse construction in the illustration is quite simple. Press two pins into a corkboard, place a loop of string around the pins, pull the string tight with a pencil, and trace the pencil tip’s path as you pull the pencil around the taut string. Guaranteeing that the traced path is an ellipse is this definition of an ellipse: *An ellipse is the set of points P such that PF _{1} + PF_{2} is constant for two fixed points, F_{1} and F_{2}.*

I don’t think any introduction to ellipses is complete without students making their own physical model of the pins-and-string construction and experimenting with how the distance between F_{1} and F_{2} as well as the length of the string affects the shape of the ellipse. But I also think there is value in building a Sketchpad version of the ellipse construction. Below is a pre-built Web Sketchpad model that your students can investigate. In a future post, I’ll show how you can construct this model from scratch using Web Sketchpad.

Depending on how it is introduced, triangle area can be either dull and formulaic (“To calculate a triangle’s area, multiply its base by height and divide by 2″) or it can be an opportunity for discovery and critical thinking. As I would expect, *Everyday Mathematics *opts for the latter approach.

Below is a triangle. Put yourself in the shoes of an elementary-age student who doesn’t know the formula for triangle area but does know how to compute the area of rectangles. Can you use the draggable and resizable rectangles provided with the model to help compute the triangle’s area?

Below is an approach that a student might take, enclosing a triangle with the two rectangles. Notice that the rectangles divide the overall triangle into two right triangles. By visual inspection, it’s clear that the area of each right triangle is half the area of its associated rectangle. Thus the area of the entire triangle is ^{1}⁄_{2} (4·2) + ^{1}⁄_{2} (4·5), or 14 square units. By pressing *New Problem* and experimenting with other triangles (all of which have a base along the grid lines), students develop on their own the area formula for right triangles. Indeed, students may begin to suspect that this formula applies to all triangles, not just those with right angles

To continue their investigation, students press the arrow button in the lower-right corner of the sketch to move on to a new page of problems. These problems are of the same type as those they just solved, but with one difference: the dimensions of the two rectangles are provided. Since students have already determined that the areas of the rectangles are essential for finding the triangle areas, pedagogically it seems reasonable to now save them the effort of counting squares.

The third page of the interactive model above switches to a different type of challenge. As before, the goal is to use one or more of the rectangles to help determine the area of the triangle, but now, the triangle only occasionally has a base that sits along the grid lines.

In the example below, a student has enclosed a triangle with a single rectangle whose area is 8·6, or 48 square units. To determine the area of the blue triangle, the student realizes that she can subtract the areas of the three right triangles from the area of the rectangle. Thus the area of the blue triangle is 48 – ^{1}⁄_{2} (4·2) – ^{1}⁄_{2} (8·4) – ^{1}⁄_{2 }(4·6), or 16 square units.

I’ll leave you with one last question: How might you adapt this method to find the area of the triangle below?

]]>Throughout, I’ve tried to show that the introduction of the computer into the assessment process need not be a limiting factor in the mathematical questions we pose. Indeed, as I discussed in my prior post, assessment when done well can be an opportunity for students to learn new mathematics, and not just repeat what they already know.

But what about teachers? What can *they* learn from their students’ assessments? Below is a Web Sketchpad-based assessment question designed by my former colleague, Steve Rasmussen. Try solving it before reading further.

There are several ways to move from an equation of a parabola to its graph. You could explain your method to me in the comments section of this blog, but perhaps that isn’t necessary. There are clues lurking in your graph! Specifically, the placement of points *A, B,* and *C* have a story to tell.

Below are four identical graphs of y = *x*^{2} – 2*x* – 3, each with different locations of points *A, B,* and *C*. Imagine that each graph represents the work of a different student . Based on the locations of the points, see if you can describe how each of the students went about graphing the parabola.

OK, how did you do? It’s impossible to know for sure the thinking behind each of the four methods, but here are some reasonable guesses:

Method 1: The student set * x*^{2} – 2*x* – 3 equal to 0, factored it to find roots at –1 and 3, and dragged points *A* and *C* to (–1, 0) and (3, 0). She concluded by setting *x* = 0, finding the *y*-intercept of (0, –3), and dragging point *B* to this location.

Method 2: The student set * x*^{2} – 2*x* – 3 equal to 0, factored it to find roots at –1 and 3, and dragged points *A* and *C* to (–1, 0) and (3, 0). The student realized that the *x*-value of the vertex was midway between the two roots at *x* = 1. Substituting *x* = 1 into the equation revealed that the vertex was at (1, –4). The student dragged point *B* to this location.

Method 3: The student completed the square to determine that the vertex of the parabola sat at (1, -4). She dragged point *B* to this location. The student set *x* = 0 to determine its *y*-intercept and dragged point *C* to this location. Since the parabola is symmetric about its line of symmetry at *x* =1, the student knew, without performing any calculations, that point *A* must sit at (2, –3).

Method 4: The student picked three *x*-values at random (–2, 2, and 4) and substituted each into the equation to determine the locations of points *A, B,* and *C*.

Even though we can’t say with certainty whether our analysis is correct, I think it’s pretty amazing how much information we can gleam from the placement of points *A, B,* and *C*!

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Since some form of testing will always be with us, I choose to think positively, believing that good tests can foster and reinforce the types of mathematical thinking we’d like students to develop. In my utopian view of assessment, tests are opportunities for students to learn and for teachers to gather meaningful information about their pupils’ understanding.

Unfortunately, with computer-based testing on the rise, we’re seeing lots of assessment items that are easy for a computer to grade (e.g., multiple-choice questions) and far fewer questions that give students the chance to flex their mathematical muscles in productive ways.

In my prior posts, I presented two Web Sketchpad-based interactive assessments relating to isosceles triangles and the Pythagorean Theorem. You can be the judge, but I think both questions are small but important steps along the path to better computer-based assessments. Now I’d like to share another Web Sketchpad assessment item that I developed in collaboration with Steve Rasmussen, Scott Steketee, and Nick Jackiw.

Let’s start, as I have in my previous two posts, with a paper-and-pencil assessment question:

It doesn’t get more routine than this. Students must know the relationship between the slopes of perpendicular lines and use this information to derive the equation of the line in question. If they’re clever, they can sidestep the algebraic manipulation by noticing that only choice (b) has the correct slope.

By comparison, try the interactive Web Sketchpad item below:

The difference between the two questions is stark: Whereas the paper-and-pencil item basks in algebra and equations, the Web Sketchpad version eschews algebra entirely. This raises an interesting issue: By asking students to manually position a line rather than derive its equation, have we designed too simple a question?

I don’t think so.

From my experience teaching algebra to college students, the standard approach to lines and their equations is endlessly confusing. Slope, slope-intercept form, standard form, point-slope form…the terminology associated with linear equations produces a jumble of rules, leaving students convinced that lines are an impossible nut to crack. And to that list of half-understand terms, I would add “negative reciprocal.”

So suppose students who have never been introduced to the slope relationship of perpendicular lines work through a sequence of four questions like the one above (Press the arrow in the lower-right corner of the sketch to move between the questions.) Sketchpad can tell students whether their lines are perpendicular (press the *Check* button), but this is not the actual assessment piece of the task. Rather, students use this opportunity to gather data—numerical, visual, some combination of both—that helps them to make sense of what’s true about the slopes of *any* pair of perpendicular lines.

The real assessment comes afterward and is formative in nature. Students use their experience with Sketchpad to describe what they observed about the slopes of perpendicular lines. This write-up could take a variety of forms. Students might have discovered the negative reciprocal relationship and describe it with numbers alone. Or, they might draw and annotate a picture of slope triangles that visually demonstrates the relationship. There are lots of possibilities for what students might do (Try it with your students and let me know!)

I’m cheating a little here because students’ explanation of the slope relationship must be evaluated by a teacher, not a computer. But it’s through the experience of using the computer to experiment that students form their insights.

This makes for a nice example of how learning can take place *during* an assessment, and not just in a conventional lecture setting. With assessments like this, testing doesn’t sound so bad!

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Why the mismatch? Dan doesn’t have a definitive answer, but he offers a reasonable explanation: Developing innovative assessment items that can be scored by a computer is hard.

Hard yes, but impossible no. In my previous post, I offered an example of a Web Sketchpad assessment that I developed in collaboration with Steve Rasmussen, Scott Steketee, and Nick Jackiw. Now I’ll showcase another sample of ours to demonstrate that the introduction of a computer into the assessment process need not mean the death of good, interesting questions.

Here is a straightforward task that tests whether students know and can apply the Pythagorean Theorem:

This question is a snap for a computer to grade, but as an example of a problem that requires higher-order thinking, it’s a dud. By contrast, try the interactive Web Sketchpad assessment item below. It deviates from the traditional approach above by providing just one piece of information—the triangle’s hypotenuse—and asks you to make a triangle with that length. When you’re satisfied with your triangle, press *Show Your Score* to check your answer. You’ll receive 3 points for a correct response and 0 points otherwise.

I don’t think I need to convince you that this Web Sketchpad assessment item is a significant step up the cognitive demand ladder from its static counterpart. I especially like the opportunities for discussion it provides for the entire class after students have completed their individual assessment. For example, what values other than √17 can serve as the hypotenuse if the triangle’s two legs have integer values? Will the square root of any integer create a solvable problem? Can students make a list of 20 possible hypotenuse lengths?

Aside from its pedagogical value, this assessment item is exceedingly easy for a computer to grade. And to customize the problem so that students nationwide don’t all receive the same hypotenuse, we need only ask Web Sketchpad to randomly pick another usable value for the hypotenuse by pressing *New Problem *(This randomization would occur behind the scenes, but it’s useful to see it in action here.)

Stay tuned for two more Web Sketchpad assessment items in my upcoming posts!

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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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