My goal was to design a lesson focusing on the triangle area formula, *A* = *bh*/2. In particular, I wanted to address the common student belief that a triangle has only one base and one height. Given a triangle like *ABC* below, for example, students can identify *AB* as a base, but are less likely to realize that any of the triangle’s three sides can serve as the base. And identifying heights can be problematic, too, especially in obtuse triangles where two of the heights do not sit inside the triangle.

While a textbook can display and label the three bases and heights of triangle *ABC, *this approach is hands off. I think it is far preferable to give students the tools to construct these heights themselves.

How might such tools work? I considered offering a perpendicular line tool so that students could construct a perpendicular through each of the triangle’s three vertices and then use these perpendiculars to construct the three heights. But as a first introduction to triangle area, this approach felt too complex. I opted instead for a tool that automates the steps of constructing a height once a student specifies a base of the triangle and its opposite vertex.

You can try the Construct Height tool for yourself in the interactive Web Sketchpad model below (Aside from experimentation, the best way to learn how the tool works is to watch the video at at the end of this post.) The tool not only builds heights, it also measures and labels the heights and their associated bases. Students use the tool three times, once each for the three identical triangles, picking a different base each time. They then use the calculator tool to compute the area three times, using the three sets of base and height measurements. Lo and behold, all three answers are the same!

After you’ve constructed the three bases and heights of the triangle on page 1 of the model, press the arrow in the lower-right corner to move to page 2. Now you’ll see three copies of an obtuse triangle. Again your goal is to construct its three bases and heights and compute the areas using the two tools. (Alternatively, you can drag any of the vertices on page 1 to change those triangles to obtuse.)

My Construct Height tool is a good first step in shaking common student misconceptions about base and height, but I ultimately wanted students to be able to identify the pairs of bases and heights themselves. With this in mind, I developed a follow-up activity with a new tool that removes some scaffolding.

In the second version of the triangle area activity below, the Construct Height tool doesn’t label the base and height of the triangles, nor does it display their lengths. Students must identify the base and height for themselves and use the Measure Base and Measure Height tools to find their lengths (Again, the video below demonstrates the details of how the tools work.)

Too often, curriculum developers and teachers are limited by the sets of tools that come pre-baked into mathematics software. Web Sketchpad makes it relatively painless for anyone with knowledge of desktop Sketchpad to design highly specific tools that focus students’ thinking on those aspects of a problem that we feel are important. This triangle area problem is just one example of how we can develop a progression of related sketches, each with a different set of tools, that gradually gives students more independence and responsibility as they move from one toolset to the next.

]]>

The first page of the lock puzzle shows a lock consisting of a single dial with 4 tick marks. The goal is to open the lock by having its pointer move clockwise around the dial and end pointing straight up, back where it began. In the lock’s initial state, notice that the pointer is set to move 3 times. Press *Go Slowly* or* Go Quickly — *you’ll discover that the pointer doesn’t quite make it once around the dial. If you press *Reset,* change the “Pointer Movement” value to 4, and set it in motion again, you’ll see that the pointer makes one complete revolution and opens the lock.

For students just learning about multiples, this lock makes for an engaging model. They soon discover that while 4 is the most efficient value for opening the lock, they can also enter any multiple of 4 to get the job done. Students are eager to make new locks with more ticks by entering a new value for the parameter “ticks.” And as you can well imagine, they enjoy experimenting with locks with considerably more than just 4 ticks on the dial!

To view the second lock, press the right arrow in the bottom-right corner of the sketch. This new lock ups the ante by adding a second dial. Now the goal is to determine how many times the pointers should move so that when they come to a rest, both are simultaneously pointing straight up, back in their starting positions. As with the one-dial lock, students can change the number of ticks on each dial to create new locks of their own.

And finally the third page of this model presents one further evolution of the lock—a lock with three dials.

Below are just some of the questions that you or your students might ask about the three locks. Let me know what other investigations your students suggest!

- Will more than one number open a lock? What do these numbers share in common?
- How can you use the one-dial lock to identify the number closest to 1,000 that is a multiple of 17?
- When does the minimum number of moves needed to open a lock equal the product of the ticks on its dials?
- How can we use the two-dial lock to check if two numbers have any factors in common other than one?
- How many two-dial locks can you make that take a minimum of 30 moves to open? How about 16 moves?
- Suppose you commission me to build a two-dial lock. The two dials must be different and each must have more than one tick mark. You specify how many moves for the lock to open, but I reply, “That’s impossible!” What are some possible numbers you might have proposed?
- How many three-dial locks can you make that take a minimum of 210 moves to open?
- Suppose we decide that each dial on the two-dial lock must have more than one but fewer than 10 ticks. Which such lock will take the most moves to open?
- Describe a two-dial lock that would take a really long time to open.
- Create a two-dial lock that takes approximately one minute to open once its pointers are set in motion (You decide whether the pointers move slowly or quickly.) Create a lock that will open in approximately five minutes. How about one class period or one school day? Be sure to check!

Mathematical habits of mind are similar to the Common Core’s Standards for Mathematical Practice in that they emphasize the process of *doing* mathematics rather than being a recipient of the content. As Cuoco, Goldenberg, and Mark explain: “A curriculum organized around habits of mind tries to close the gap between what the users and makers of mathematics do and what they say. Such a curriculum lets students in on the process of creating, inventing, conjecturing, and experimenting; it lets them experience what goes on behind the study door before new results are polished and presented. It is a curriculum that encourages false starts, calculations, experiments, and special cases. Students develop the habit of reducing things to lemmas for which they have no proofs, suspending work on these lemmas and on other details until they see if assuming the lemmas will help. It helps students look for logical and heuristic connections between new ideas and old ones. A habits of mind curriculum is devoted to giving students a genuine research experience.” [Habits of Mind, p. 2]

Here’s a nice problem from *Connected Geometry* that takes familiar content (triangle area) but spins it in a way to invite experimentation, persistence, collaboration, organization, and—best of all—creative problem solving:

*Find as many ways as you can to divide an arbitrary triangle into four equal-area triangles.*

I first wrote about this problem in the October 2000 issue of NCTM’s *Mathematics Teacher,* but I had no way include an interactive model with the article. Now, I’m able to provide a Web Sketchpad model where you can divide the triangle into smaller triangles using construction tools.

The web sketch comes with three tools. To bisect a segment, choose the Bisect Segment tool. You’ll see a segment with two glowing points appear, along with its midpoint. Drag the glowing points to the desired endpoints in your sketch to merge them together. Choose the Draw Segment tool to draw a segment. As with the Bisect Segment tool, you’ll see a segment with two glowing points appear. Drag the pulsing points to merge them to existing points in your sketch. And finally, the Trisect Segment tool works the same way, dividing any segment in the sketch into three equal parts. If you make a mistake, you can use the left-pointing arrow above the three tools to back up as many steps as you like.

How many ways do you think exist to solve the problem? You might be surprised! I’ve provided six pages of triangles for your students to display their answers, and that probably is not enough. To move from page to page, just use the arrows in the bottom-right corner of the sketch.

When your students are done, try this slight variation to the problem that removes the restriction of each piece being a triangle:

*Find as many ways as you can to divide an arbitrary triangle into four equal-area pieces.*

It’s entirely possible your students may think of a solution that requires a tool other than the three that I’ve provided. If so, let me know!

]]>Session 52 on Thursday, April 16, 2015: 8:00 AM-9:15 AM in 157 B/C (BCEC)

How better to explore rate of change than as independent and dependent variables dancing together? We’ll vary *x* and *y* by doing both real and computer-based dances based on geometric transformations, dynagraphs, and Cartesian graphs of various functions. Bring a laptop or iPad with Sketchpad. Leave with student-ready geometry and algebra activities.

Session 245 on Thursday, April 16, 2015: 2:00 PM-3:00 PM in Ballroom West (BCEC)

In grades 7–12, CCSSM expects students to understand transformations as functions. This profound link allows students to build a transformation, drag its input (a point), describe the output’s behavior, restrict the domain to a number line, and voilà!—end up with a linear function and its Cartesian graph. Leave with student-ready GSP activities.

We’re hoping you can come to one or both sessions, but if not, here are two reflection challenges from our presentations that you can try right now with Web Sketchpad.

For the first challenge, your task is to drag the red point so that its reflection, the blue point, reaches the target. But move carefully! Don’t allow the blue point to touch any of the obstacles in its path.

To view the second challenge, press the arrow in the lower-right corner of the sketch. Now, your goal is to drag red point so that its reflection, the green point, follows the border of the blue polygon. Notice that in the sketch, we use the term “range” to reference the path of the green point. By dragging the red point, you’re determining both the domain of the red point and the range of the green point. The terms ‘domain’ and ‘range’ allude to the connections between transformations and functions we’ll be highlighting in our presentations.

You’ll find more material from our NCTM sessions on the Geometric Functions website.

We look forward to meeting some of you in Boston!

]]>When I say that elementary-age students should encounter simultaneous equations, I don’t mean that they should be instructed in the standard algebraic procedure for solving pairs of equations like *x* + 2*y* = 9 and 3*x* – *y* = –1. Rather, my interest lies in presenting students with puzzle-like situations in which they solve for multiple unknown values by using their common sense (You can see two interactive Web Sketchpad models of this approach here and here in my prior posts.)

Take a look at the interactive Web Sketchpad model below. It displays eight rows of shapes, each containing a different combination of circles and squares. Pick a row and drag the shapes across the vertical divider line. When you do, the sum of those circles and squares will appear in the right-hand column. The goal is to determine the numerical values of the circle and square.

When I’ve observed this activity in classrooms with third, fourth, or fifth graders, students usually begin by taking a guess based on what they learn about a single row. For example, dragging the top row of shapes, the square and circle, across the vertical divider reveals that square + circle = 9. Might the square equal 8 and the circle equal 1? Students check by dragging another row of shapes across the divider. Two squares plus a circle equals 11, but 8 + 8 + 1 does not equal 11. Trying other values for the circle and square soon reveals that square = 2 and circle = 7. The puzzle is now solved, but students are not finished: They find it fun to predict the sums of the other rows before dragging each across the vertical divider. This helps to drive home the message that every circle in every row shares the same value, as does every square—something that’s not always clear to students.

With a little guidance, students become more systematic in their approach to solving the puzzles. For example, knowing that circle + square = 9 means that there are 10 possible pairs of values for the two shapes. Students list these in a table like the one below and then check to see which pair of values satisfies the sums of the other combinations of shapes.

One aspect of this puzzle that I especially like is that students can choose how far across the vertical divider to drag each row of shapes. Below are two different placements for the first two rows of shapes.

The placement of the shapes on the right invariably encourages a student to make a key discovery: *The shapes in both rows are nearly the same. Both contain a square and a circle, but the second row contains one extra square. That must mean that the square is equal to 11 – 9 = 2. Since the square equals 2, the circle must equal 7.*

Using common sense, the student has just done—and explained—some important algebra! A problem like this can support a stimulating number talk (algebra talk?) that encourages many students to participate by describing their own numeric and proto-algebraic thinking.

Here are some of the interesting discussion questions that naturally arise when students work with this puzzle:

- Is it always possible to determine the values of the circle and square regardless of which two rows we pick?
- Which pairs of rows make it easy to solve for the circle and square? Which pairs of rows make it harder?
- Might there be more than one answer to a puzzle?
- Could a puzzle exist that has no answer?

Use the arrow buttons in the lower-right corner of the web sketch to navigate from page to page. The second page looks just like the first, but now, the possible values of the circle and square range from 0 to 20 rather than from 0 to 8. The third page of the sketch allows students to create problems for each other, deciding for themselves what values to assign to the circle and square (Should the circle equal 3,452? Should it equal –28? Students create far harder problems for each other than we might create for them!) The fourth page of the sketch adds a third symbol to the mix—a triangle—and now students must solve for three unknowns. The final page of the sketch allows student to create their own problems with three unknowns to share with each other.

If you have a chance to use this activity with your students, I’d love to hear what discoveries they make!

]]>Friday was an amazing astronomical day, with the spring equinox arriving at 6:45 pm on the US East Coast (3:45 pm Pacific Time), exactly 13 hours after the maximum of a complete solar eclipse. We didn’t get to see the eclipse here in Philadelphia (or pretty much anywhere in the Western Hemisphere), and for us in Philly the eclipse marked the start of what we hope was our last big snowstorm of the season. Though it would have been nice if the storm had ended by the time of the equinox, that was not to be, so for us the astronomical start of spring was marked by winter’s fierce refusal to depart.

At such a time, when at least in this part of the country we’re all wondering when (or if) winter will ever be over, it was a delight for me to receive an email from Anna containing her “Hello Spring” sketch, and not only, or even mainly, because of its reminder that spring really will come, even if in its own time. More exciting for me was to see Anna’s expression of creativity, the joy she communicates through her sketch. Most students—and most teachers—use Sketchpad in a much more utilitarian fashion, as a way to explore a specific mathematical question or to produce a specific geometric or algebraic construction. That’s certainly worthwhile, and it would be hard to overemphasize either the importance of studying and learning mathematics or the advantages of doing so while using a dynamic, visual tool. But mathematics also plays an often unrealized role in art, in music, in dance, and so forth. For that reason it’s particularly exciting for me to see a student express herself through color, geometric shape, and animation as Anna has done here.

So cue the music, let the sun shine, and enjoy Anna’s animated movie.

In Anna’s original GSP construction, her “Cue the Music” button was a link button to a YouTube page of Vivaldi’s Four Seasons. Like most things in every sketch, this button worked properly when exported to Web Sketchpad (WSP), and the music and the animation work very well together. In fact, there are a couple of places in the first movement where I can actually hear Anna’s bees buzzing, thanks to the violins.

In WSP, the original Cue the Music link button did the same thing as in native Sketchpad 5: it switched you away from this blog page when you press the music button. I didn’t like that; I wanted to continue watching Anna’s animated sketch, without interruption. And with WSP, there’s fortunately an alternative. Because WSP is implemented in JavaScript and HTML5, I was able to add some custom JavaScript in the sketch’s web page that detects when the user presses the “Cue the Music” button and takes appropriate action. In this case, the appropriate action is to call on HTML5 to play the Spring movements of Four Seasons (downloaded from freemusicarchive.org). When the first movement ends, I programmed the JavaScript to automatically play the second movement, and then the third. I dressed the code up a bit by allowing you to click “Cue the Music” repeatedly to switch from one movement to the next.

So I want to express my thanks to Anna for brightening for me these last days of winter—and I also want to thank her teacher, Chris Taranta, for encouraging her creativity and for putting her in touch with me. It’s been a pleasure!

]]>This post is by guest blogger Adrienne Barrett, who’s a senior mathematics and education dual major at Rowan University. She is currently student teaching and upon graduation in May, she hopes to find a full-time position teaching high-school mathematics. She’s always loved math, and studying it in college has given her a greater understanding of just how intricate and fascinating the subject is. She says, “I’ll forever consider myself a student of math because I believe that there are always new methods, representations and concepts to be explored.”

While searching for topics for my senior seminar project last semester, I came across a theorem known as Morley’s Trisector Theorem, discovered in 1900 by mathematician Frank Morley. It states that given any triangle the points of intersection of adjacent angle trisectors are the vertices of an equilateral triangle. The fact that performing a seemingly arbitrary action, such as trisecting angles of a triangle, produces a nontrivial shape—an equilateral triangle—fascinated me. I was hooked and I had to know more about this theorem.

By trisecting each of the interior angles of a triangle, you can find their points of intersection: the vertices of the First Morley Triangle. Below is a web sketch that will allow you to create and experiment with Morley triangles. (If you get confused, watch the short video beneath the sketch.) Click the top tool (Morley1) to create a triangle with one interior angle already trisected. Drag each glowing vertex in turn, moving the triangle wherever you desire. Then drag the slider at the bottom of the sketch to rotate the trisection rays into position. To trisect a second angle, click the top tool again and drag the glowing points onto the existing vertices in a way that trisects a different vertex. Use the tool once more to trisect the third vertex. Then drag the slider again (just because it’s fun to animate the trisection), leaving the angles fully trisected.

With the angles fully trisected, use the Point tool to construct the trisector intersections closest to the sides of the triangle. (Click the tool, drag the glowing point to an intersection point, and release it when both trisectors light up. Repeat for each of the other two intersections.) These are the vertices of the First Morley Triangle.

Choose the Measured Triangle tool and drag its vertices onto the intersection points. You have now created the First Morley Triangle. What do you observe about its angle measurements? I encourage you to drag the vertices of the original triangle, as well as the slider, while you observe the behavior of ∆DEF.

If at any time you want to undo one or more tools you’ve used, just press the left arrow above the tool icons.

Pleased as I was with this pretty picture, I had more in mind. Through my research I discovered that this is not the only result Morley’s Theorem provides. The First Morley Triangle is determined by the intersections of the trisectors of the interior angles of a triangle. But what about the reflex angles?

Morley’s Theorem also addresses this. Press the arrow button at the lower right of the sketch to go to page 2, which you can use to create the Second Morley Triangle. Using the second tool (Morley2), create a triangle with three trisected reflex angles. Once you have trisected all three angles, adjust the vertices, making the triangle small enough that you can see the more distant trisector intersections. (These intersections are located on the dashed lines that extend the trisectors, and are NOT the intersections closest to the sides of the triangle.)

Now use the Point tool to construct the trisector intersections, and use the Measured Triangle tool to create the Second Morley Triangle.

Take this one step further on page 3. We know that rotating a side of an angle about its vertex by 2π leaves it right back where it started—but the resulting angle’s trisectors would be in different places. That’s what the Morley3 tool does; it trisects the reflex angle, with an extra 2π added in. Use the Morley3 tool to trisect the reflex angles (with the extra 2π thrown in), and then (as before) use the Point tool to construct the intersections and the Measured Triangle tool to measure the angles of the triangle formed by the intersections. (This time the intersections are again close to the sides of the original triangle.)

While this tool adds 2π to the reflex angles being trisected, it is also possible to obtain the Third Morley Triangle by adding an extra 2π when trisecting the interior angles of the original triangle.

It is worth noting that up to this point, you have explored three equilateral triangles that are the result of Morley’s Trisector Theorem. The equilateral triangles that were created were done so using trisectors exclusively of one type: interior, reflex, or interior plus 2π/reflex plus 2π. While doing my research for my project and playing around in Sketchpad, I was impressed with these results. What fascinated me was what happens when you use combinations of different types of trisectors on the same base triangle. The result is that more equilateral triangles are formed! Naturally I spent an inordinate amount of time in Sketchpad creating my own triangle with all of its Morley triangles. There are 24 more triangles to be found that are equilateral and I encourage you to hunt for a few. Happy exploring!

]]>I’ve often been somewhat uncertain as to how valuable typical traditional Sketchpad users find custom tools, for two reasons:

(a) They’re hard to find; you have to click the Custom tool icon even to see them.

(b) It may not be clear what to do when; you have to open the Script View, or else watch the status bar at the bottom of the Sketchpad window.

Web Sketchpad’s construction interface is designed to address both issues:

(a) There’s no problem finding tools: their detailed icons are in plain view on the left side of the screen. This post’s sketch contains three tool icons. The first shows a bounded piece of a function graph, with one point determining the left bound and another the right bound. The second shows the graph of a function, and the third shows the region bounded by two functions, one defining the top edge of the region and another defining the bottom edge.

(b) It’s clear what to do: as soon as you click the icon, the entire construction is displayed, and the points or other objects you can match to sketch objects pulse until you’ve done something with them, either matched them to an existing object or dragged them to a new location.

So here’s a Pi Day challenge for students of Algebra 2, Precalculus, or Calculus: use the piecewise function tools below to create “Pieces of Pi:” the shape of π defined by piecewise functions, similar to the example on the right.

You’ll find tips in this video that illustrate how to use these tools to create the shape of the letter N.

If you want to experiment on your own with piecewise functions in Web Sketchpad, use this link to go to a Web Sketch that includes the tools above and three additional tools: Left and Right piecewise function tools (for a left-unbounded piece and a right-unbounded piece) and a New Parameter tool (to create parameters you can use in your function definitions).

]]>I recognize that the decision to incorporate specific tools encourages some lines of student thinking while discouraging others, but that doesn’t worry me much, as I can imagine using this image in developing the idea that projectile motion is parabolic, using it again a bit later to see if we can detect the effects of friction on the recorded motion, and revisiting it yet again in calculus to approximate the instantaneous rate of change in the ball’s position. To preserve some open-endedness, the first page’s text makes an effort to draw students’ attention away from the tools and toward the image. It’s not hard to elicit ten or a dozen things that students notice, and a similar number of things they wonder. By allowing no more than a single noticing or wondering from each student, and not giving away your hand by approving or disapproving of any of them, you can quickly have contributions from more than half the class just in launching this lesson.

In case the images on the tool icons haven’t already given away my agenda in this use of the picture, I’ll make it explicit now: I envision using this as a function-fitting activity in conjunction with studying the standard and factored forms of the quadratic function. I might use it in advance, asking students to note how each of the parameters changes the shape of the graph, and explaining that we’ll investigate these function forms in detail in the next few lessons. Or I might use it as review, giving students an incentive to revisit what they know about these two forms.

The picture (by leaving the ball’s path literally in the air) and the student wonderings provide both a real-world context and useful motivation for seeing where the ball goes after the images end. To find out, students will need to fit a function to the picture.

Press the button at the lower right corner of the picture to go to page 2, start with the topmost tool, and see how well you can fit a graph defined in that way to the ball’s flight. (When you click a Web Sketchpad tool, the complete construction appears, with the tool’s given objects glowing. To use the tool, drag each glowing object to specify how it connects with your existing sketch. You can drag it on top of an existing object to connect it to that object, or you can drag it to empty space to make it independent of the already-existing objects.) If at any time you want to undo one or more tools you’ve used, just press the left arrow above the tool icons.

Pages 3 and 4 are identical to page 2, allowing you to try one tool on each page and compare your results by flipping pages. I recommend to students using only one tool on each page, as otherwise the pages can become cluttered and confusing.

As you work to fit each of the three function forms to the ball’s trajectory, you may notice that you can change the position and scale of the coordinate system, which is itself an important lesson for students: they are permitted to choose a coordinate system that makes their work easier.

One last thought: In an introduction to the standard and factored forms of the quadratic function, the third tool serves as a teaser, giving students a taste of some fascinating future mathematics. But later on this same sketch would be useful to introduce students to a unit on fitting polynomial functions to data.

Another last thought: If you would like to use this with your class and want to have a fourth tool that uses the vertex form, let me know via the comments.

]]>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 glowing. 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 glowing 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, simply press the left arrow above the tool icons to undo one or more of the tools you’ve used.

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!

]]>