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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

]]>The interactive model below shows a point sitting on a number line. What is its location? Students might reasonably propose 8.7 or 8.8. After typing their estimate into the box, they press the *Zoom* button, and watch as the interval between 8 and 9 expands, showing a magnified view divided into ten equal parts (try it!). The point still sits at the same location, but now it lies at a tick mark. What do the tick marks on the second number line represent? Students identify them as tenths and can now name the precise location of the point as 8.7. Pressing *New Problem* moves the point to a different location (agin, exact in tenths), ready for students to estimate.

To move to the next number-line model, press the arrow in the lower-right corner of the sketch. Again, the goal is to identify a point’s location. The point looks to be at.. oh, say 6.2. When students press *Zoom*, they’e in for a surprise: The point doesn’t sit at 6.2 or at any of the tenths marks. It’s a little less than 6.2. Perhaps 6.19? To check students press the second *Zoom* button and yes—the point is indeed at 6.19!

The magnification of the number line continues on the next sketch page where students make three successive predictions of the point’s location. Zooming in on the units number line reveals tenths, zooming in on tenths reveals hundredths, and zooming in on hundredths reveals thousandths.

The final page of the sketch is the most interesting of all. Students make four estimates of the point’s location, but even when they’ve burrowed down to the final number line showing ten thousandths, the point *still* doesn’t sit a tick mark. Could it be that no matter how many times they zoom in, the point will *never* lie at a tick mark? This is a great opportunity to introduce students to the notion of irrational numbers.

Finally, be sure to press the *Animate* button on the last page of the sketch. This button sets the point in motion, and students can analyze how the simultaneous movement of the point along five number lines is related. For example, why does the point move at a glacial pace along the uppermost number line but move ever faster on the number lines below it?

This zooming number line model was inspired by the work of Paul Goldenberg at Education Development Center. You can read more about this approach to place value and decimals on the *Think Math!* curriculum page or in Paul’s book, *Making Sense of Algebra*. You should also check out Paul’s 1991 article, *A Mathematical Conversation with Fourth Graders* (linked on this page under the 4th Grade heading).

You can download several desktop Sketchpad versions of the zooming number line model, along with teacher notes, at the Dynamic Number website.

]]>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.

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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!

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