π Day has always been a special day for me, from my earliest days. In fact, I’ve never figured out whether I was so eager to celebrate my first π Day that I jumped the gun and sent my mom into labor early, or whether I just wanted be sure to experience all 24 hours of my first π Day. Whichever it was, I’ve certainly enjoyed and celebrated π Day ever since, and it’s been even more fun for me since I got involved with Sketchpad and had the opportunity to come up creative, animated decorations and diversions.
So I’m sharing here a few of my favorite circle dissections. They’re all good ways to discover that the circle’s area is given by either the half of the product of the circumference and the radius (for the dissections that yield a triangle) or by the product of half of the circumference and the radius (for the dissections that yield a rectangle). Further, increasing the number of cuts in these dissections leads to thinking about limits; the only way to turn a circle into a triangle or rectangle is to chop it into tiny, tiny sectors — but how tiny do they have to be?
I hope your students enjoy these, and that they lead to interesting discussions both about how each dissection relates the area of a circle to its radius and circumference, and about the reasoning behind the dissection: What happened to the curvature of the circle’s edge? Can you really base a logical argument on increasing the number of sectors without limit?
(Press the buttons below the sketch to look at one or another of the four models. For the explosion, reassembling doesn’t always work after an explosion, so you may have to use the Refresh button in your browser.)
Below is an interactive puzzle called Arranging Addends. The goal of the puzzle is to arrange the circles and the six numbers (1, 2, 4, 8, 16, and 32) so that three conditions are met simultaneously: The sum of the numbers in the green circle is 21, the sum of the numbers in the blue circle is 26, and the sum of the numbers in the red circle is 14. The numbers can be dragged into the circles, and the circles can be moved as well. Dragging the point that sits on the circumference of each circle changes the circle’s size.
Students find it straightforward to satisfy the first condition of the puzzle. Dragging 1, 4, and 16 into the green circle fills it with numbers whose sum is 21. Creating simultaneously a sum of 26 in the blue circle is more puzzling. The only way to form 26 is to add 2, 8, and 16, but the 16 already resides in the green circle. How can it perform double duty?
Most students, after pondering this dilemma for a minute or two, have an “aha!” moment: What if the green and blue circles overlap each other, with the 16 sitting in their intersection? The picture below shows how this looks.
Now, all that remains is to satisfy the final condition of the problem by making the numbers in the red circle sum to 14. The only way to form 14 is to add 2, 4, and 8. But the way those three numbers are currently arranged makes it impossible to enclose them without including the 16 as well. Students soon realize that they need to adjust the numbers and circles to allow the 2, 4, and 8 to sit alone in the red circle while still keeping the first two conditions true (green = 21, blue = 26). The picture at right shows the final result, with all three conditions of the puzzle met simultaneously.
I’m especially pleased with the way the Arranging Addends puzzles require young learners to juggle and process multiple pieces of information. More often than not, the placement of the circles and numbers must be adjusted one or more times to ensure that the sum of each set of numbers matches the required value.
And the mathematics of the Arranging Addends puzzles runs deep. Each new challenge generated randomly by Sketchpad uses the same six numbers—1, 2, 4, 8, 16, and 32—as the elements for the three sums. These powers of two can, amazingly, form any sum from 1 to 63.
You and your students can play the Arranging Addends puzzles directly on this webpage using the interactive model at the beginning of this post. To obtain new challenges, simply press New Puzzle. To download a desktop Sketchpad version of this puzzle along with teachers notes and a student worksheet, visit the Dynamic Number website.
You might also be interested in a low-tech version of the Arranging Addends Puzzles that requires nothing more than three hula hoops and a few pieces of paper. Regardless of which version you use, these puzzles are a great way to give your students practice in addition while keeping the work varied, interesting, and challenging.
In my Advanced Methods class at Penn’s Graduate School of Education, my students are working in groups to create shared lesson plans using an inquiry approach. For a number of reasons it can be challenging for these pre-service teachers to identify appropriate topics for student inquiry, but sometimes the brainstorming they do turns into something exciting.
And so it was recently when I conferred with pre-service teacher Andrew Laskowski about planning a lesson on trig identities. His high school students were already quite familiar with the Pythagorean identities, so his group had been thinking about how to add excitement and discovery to a lesson in which students either prove or make use of the angle sum and difference formulas. As we worked on it, we came across a diagram in Wikipedia. I’ll provide the url of the Wikipedia article at the end of this post, but please don’t peek and spoil your fun.
I’ve always had trouble remembering the angle sum and difference formulas, as I often do with formulas that I’ve memorized but haven’t truly owned. Looking at the diagram, I realized that the formulas were jumping out at me. It’s an easy construction but an elegant one, and by doing it once for myself I was convinced I’d never forget it.
Andrew’s group and I spent some time working on how to present it effectively, and I hope that some of the high school students to whom this lesson will be taught will be as excited as I was about it!
We ended up with a simple diagram and challenge. It’s dynamic, of course; drag the red points to adjust the angles.
Be sure to solve it first, preferably without the hint, and be sure to see what conclusion you can draw about the purple segments, and what conclusion you can draw about the green ones. (One of the things I love is that you can solve the whole thing using just one segment length and two angles.)
When you press the “I solved it” button, several new buttons appear, and an animation shows one order in which a student might solve the triangles. The animation finishes by extracting the purple and green segments from the figure to better demonstrate the striking conclusion. (You can use the various buttons that appear to show the animation one step at a time.)
In the 1970s, my childhood friend Tim owned an Activision console and a variety of game cartridges. Tim was the envy of our block, but no matter how much I enjoyed a rousing game of Pong, I knew that my electronic toy was even better. No, I didn’t own the rival Atari game system: I had the pleasure of playing with my father’s Casio mini-8 calculator.
As you can see at right, the Casio was nothing fancy. You could add, subtract, multiply, and divide, and that was it. I didn’t care—I loved pressing the buttons and watching the digits appear on screen.
The Casio exposed me to number patterns that I had not seen in my elementary school math classes. I remember entering 1/3 into the calculator and viewing the result of 0.3333333. How amazing to see this sequence of threes! I confess, though, that I did not consider whether the pattern continued past the rightmost 3. And if I had typed 1/7 and seen 0.1428571, I probably would have assumed this to be the precise value of the fraction.
Today’s handheld calculators show more digits than my old Casio, but they still don’t make it easy to explore patterns in the decimal expansions of fractions. Below is an interactive model of a color calculator. It’s different from a traditional calculator in several ways. It converts fractions less than one to their decimal equivalents, but it assigns each digit in the decimal expansion its own color (all 4′s are green, all 5′s are blue, etc.) By pairing each digit with a color, it becomes easier for students to spot patterns in the digits.
And whereas traditional calculators show a limited number of digits to the right of the decimal point, the color calculator shows as many digits as students would like, simply by dragging the red point. The digits are displayed in rows and columns, and students can adjust the number of rows or columns to highlight repeating patterns.
The current fraction entered into the calculator is 1/7, but you can change the numerator and denominator to create other fractions less than one.
Here are just some of the investigations students can explore with the color calculator:
Look for color patterns in the rows, columns, and diagonals of the decimal representation. Drag the red point to change the patterns and to create new ones.
Find fractions whose decimal representations eventually “stop” and end in all zeroes. How can you tell without checking whether a fraction has this property?
Find fractions whose decimal representations consist of just one repeating digit. (e.g., 0.33333…) Can you find a fraction for every digit 1 through 9?
Find fractions whose decimal representations have really long sequences of digits that eventually repeat. Do you think there are fractions whose decimal representations never repeat?
What do you notice about the decimal representations of the fractions 1/7, 2/7, 3/7, 4/7, 5/7, 6/7?
What do you notice about the decimal representations of fractions with a denominator of 13? How do these patterns differ from the ones you noticed for fractions with a denominator of 7?
The color calculator concept originates from Nathalie Sinclair, a professor of mathematics education at Simon Fraser University. In her version of the color calculator, there are no numbers visible in the decimal representation of each fraction. Students focus exclusively on the patterns visible in the colors. You can read about Sinclair’s student interviews with the color calculator in her engaging book Mathematics and Beauty.
You can also download a Sketchpad model of the color calculator, accompanying teacher notes, and a student worksheet from the Dynamic Number website.
Take a look at the interactive model below. Most of the numbers in the array are shaded orange, but several are blue. What is special about these blue values? They are the factors of 32, the largest number in the array.
Try dragging the red point to change the dimensions of the array. You’ll see that the pattern of blue and orange changes, with the blue-shaded numbers indicating the factors of the largest number in the array.
If the goal of studying factors is to factor actual numbers, then this model is horrible—it does the work for you! But mastering factoring, while important, is not nearly as interesting as exploring the mathematical relationships between numbers and their factors. And that’s the benefit of this Sketchpad model: It makes the factoring itself easy and allows students to focus on the numerical and visual patterns of the blue and orange-shaded numbers.
Here is a partial list of questions and observations that students might make while exploring the interactive model above:
The number 1 is always shaded blue.
The number in the bottom-right corner of the array is always shaded blue.
Our array has 11 rows, and I see that the number 11 is shaded blue. In general, if we have n rows in our grid, then the number n will be shaded blue.
What arrays have just two of their numbers shaded blue?
There are only two ways to display the factors of prime numbers in the array—either as a single row of circles or a single column of circles.
If the array has at least 2 columns and 2 rows, then the number it represents isn’t prime.
If the number of rows and columns in the array are equal and prime, the array will contain exactly three numbers shaded blue.
We found a way to create arrays with exactly four numbers shaded blue. Drag the red point to form a single row of numbers. Make sure the largest number in the row is prime. Then drag the red point straight up to create a prime number of columns. That does the trick.
When our array contains an even number of rows, the rows in the upper half of the array are filled entirely with orange circles. Only the number we’re factoring is shaded blue. Why is that?
We can pair every number that is shaded blue with a partner. For example, the factors of 18 are 1, 2, 3, 6, 9, and 18. Let’s pair 1 and 18, 2 and 9, and 3 and 6 together. In each pair, the product of the numbers is 18.
What happens when we pair the factors of 25? Its factors are 1, 5, and 25. We can pair 1 and 25 together, but can we pair 5 with itself?
In most of our arrays, there are an even number of circles shaded blue. But in some cases, the number of blue circles is odd. Is there a way to predict whether there will be an even or odd number of blue circles?
We dragged the red point so that the numbers from 1-20 all appeared in a single row. We wanted to find other ways to display those 20 circles in the array. The numbers in blue—1, 2, 4, 5, 10, and 20—gave us a big hint. Since 2 is a factor of 20, we can make a 2 x 10 array. Similarly, we make a 4 x 5, a 5 x 4, and a 10 x 2 array.
If the number of columns is even, the number 2 is always shaded blue. If the number of columns is odd, the number 2 alternates between orange and blue as I drag the red point up to add more rows.
We created a game. We scrolled our sketch window so that you can only see the bottom row of circles. Your challenge is to make an educated guess about the total number of circles in the array.
You can download a Sketchpad model of the factor array and accompanying teacher notes at the Dynamic Number website.
What other questions and discoveries about factors can be made with the Sketchpad factor array? Share your ideas with us!
Take a look at the two groups of shapes below. Both groups contain an equilateral triangle and a square. Now imagine that you showed students each group and asked them to identify the shapes. Do you think students would do equally well in naming the shapes in group A and group B?
As you probably suspect, identifying the square and equilateral triangle in group B is harder. Research shows that students do a better job at naming geometric objects when they are oriented in an “upright” position with their bases horizontal to the page or the screen.
Part of the blame lies with textbooks. Students are more likely to see squares and equilateral triangles drawn upright than tilted at an angle. How can we help students expand their visual definitions of an equilateral triangle and a square so that they are better able to recognize them?
In the interactive Sketchpad game below, the goal is to spot an equilateral triangle and a square. There are eight points scattered across the screen: Three of the points form the vertices of an equilateral triangle, four of the points form the vertices of a square, and the remaining point is extraneous, included solely as a distraction.
Drag the triangle and the quadrilateral onto the points so that the triangle is equilateral and the quadrilateral is square. Press Show Answer to check your solution and press New Puzzle to generate a new, random challenge.
As an adult, I find the game surprisingly fun to play and even, at times, challenging. But the real audience for this game is students. By identifying equilateral triangles and squares in a variety of orientations, will students become more adept at recognizing the shapes? Try this game with your students and let me know how they fare!
Last year, I wrote a blog post about the following probability question:
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?
This isn’t an easy question, but it is open to a beautiful geometric interpretation that makes it much easier to understand and solve.
In my original post, I provided a downloadable Sketchpad file that you could use to investigate the geometric solution. But now with new Sketchpad technology, you can do the exploration directly in your browser. Check out the model below.
Here is some information about using the model that you’ll find helpful:
Point A represents one friend; point B represents the other friend. The locations of the points along the two axes represent the times when the friends arrive.
Point P represents the two arrival times as a single point. Notice that in its original location, point P is green. This means that A and B arrive within 10 minutes of each other and successfully meet up.
Press the button Run the Simulation Once. Points A and B will move to new, random locations. Point P will remain green if the friends’ arrival times are within 10 minutes of each other, but point P will turn red if the friends do not arrive in time to meet. Press the button several timesto make sure you understand when point P is green and when it is red.
Notice that point P leaves a trace behind of all its prior locations. By running the probability simulation multiple times, you can see whether there’s a pattern to be discerned in the placement of the green and red points. Press Run the Simulation Repeatedly, grab a coffee, sit back, and watch the screen fill with color.
How can you use the pattern of green and red points to determine the likelihood that the two friends will meet?
To clear the traces of the points, press Start Again. To explore the question for different waiting times, drag point T around the circle. For example, when T is at 30, you can examine the likelihood that the two friends will meet if each is willing to wait 30 minutes for the other to arrive.
For those of you who are quite familiar with Sketchpad, I should note that the technology on display here is not JavaSketchpad, although I did build the model with Sketchpad. Would this new technology be something you’d be interested in using to create your own interactive web pages? Let us know in the comment section!
One of my favorite exhibits at the museum is the Human Tree. When you stand in front of the Human Tree screen and wave, your arms are replaced by images of your body. This structure repeats, with more and more copies of your body linked together in a tree-like form.
If you’d like to experiment with a similar effect, check out the interactive model of a Pythagorean tree below. In its starting configuration, the tree appears to be simply a geometric illustration of the Pythagorean Theorem with a flower in each square. Drag the red point labeled n to change its value from 1 to 2. Notice that the model has sprouted two more right triangles with accompanying squares. You can drag any of the red points to experiment with the model or press the Animate button to watch the model move.
Now, make the value of n larger. As you do, you’ll see this Pythagorean tree grow more branches. It’s pretty cool!
Here are some questions for you to consider: How many new squares are added to the tree when n increases by 1? How many squares are there in total? How much new area is added when n increases by 1? How much area is there in total?
If you’d like to use Sketchpad to build a Pythagorean tree of your own, check out this online tutorial.
Yesterday, I led a webinar that demonstrated how Sketchpad can be a powerful tool for exploring Common Core algebra topics. My examples included solving for unknowns with a pan balance, exploring the slopes of lines, maximizing the area of a fixed-perimeter rectangle, and graphing trigonometric functions. I touched only briefly on each example during the webinar, and so here I’d like to return to the algebra of lines.
I’ve taught algebra and pre-calculus courses to college students, and without fail, they stumble when asked to write the equations of lines. In fact, I’ve seen students deal more successfully with the properties of trigonometric functions—their amplitude, period, and phase shift—than simple lines. What’s going on here?
I think that the problem stems in part from the terminology and algebraic machinery that accompanies students’ introduction to lines. First, students learn the formula for the slope of a line. Then, they are introduced to the point-slope form of a line and the slope-intercept form. Knowing when to use each form and doing the necessary algebra soon makes lines feel incredibly difficult. Does it really need to be so complex?
I think the answer is no, and one way to convince you is to share the following problem: Given a point A at (3, 4), how many lines can you name in under a minute that pass through it? Now wait, you might say, setting up those formulas takes some work! A minute is not much time.
But put aside what you know about formulas and procedures and think logically about what the problem is asking. In the interactive model below, you can change the three numerical boxed values. Give it a try and let us know your technique. I’ve seen calculus teachers think hard about this question before having an aha! insight, so don’t worry if you need to ponder for a while.
A few days ago I led a webinar on the Common Core and Sketchpad for Sketchpad beginners, and I showed four Sketchpad activities aligned with both the Content Standards and the Standards for Mathematical Practice. I mixed it up a bit by showing two activities in which students manipulate prepared sketches, and two activities in which students start with an empty Sketchpad window and create their own construction to manipulate and to make conjectures.
The first activity was Quadrilateral Pretenders. Here’s one of its pages—a prepared sketch that students can use to develop and refine their own definition of a parallelogram:
How can you tell the difference between the quadrilaterals that are always parallelograms and those that are only pretending?
In the rest of this post I’d like to reflect on some things that I didn’t discuss explicitly during the webinar. (You may have some comments of your own; please post them! Webinar attendees will note two changes in the sketch: I changed the question on the screen to “How can you…” and I used “Pretend” as the name of the button that makes the quadrilaterals all pretend to be parallelograms again.)
Cognitive Demand and Student Ownership
The questions we ask students, and the information we provide, are very important. The question “How can you tell the difference…” is not a yes/no question, and it’s not even asking students to categorize the figures they see. As a “how can you” question, it’s asking them to describe their thinking, and it’s implicitly asking them to do something. We may have to encourage them a bit, and we might even have to ask a student questions like “What do you think you can do to these figures?” or even “Can you read me the directions out loud?” to get them started. (Or we can offer a hint, as I did with the button above—to be used only if the student really needs it.)
Standards for Mathematical Practice
By giving students ownership of the task and the responsibility of figuring out how to proceed, we’re using this activity to address at least six of the CCSS Standards for Mathematical Practice:
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
These standards remind us of the importance of reasoning and sense-making, of all the factors that go into our students’ ability to give meaning to the mathematics they experience.
Discovery by Dragging
Because students create meaning best through actual experiences, dragging is a powerful way to explore mathematical constructions.
For instance, from the starting position of quadrilateral WXYV, drag the vertices in turn: first W, then X, then Y, and finally V. What do you notice? What do you wonder? Similarly, try each of the four vertices of quadrilateral CDEF. Why do you think some of these vertices behave differently from others? How might you learn more about the mathematics behind these behaviors? These questions emerge out of curiosity about why mathematical objects behave the way they do, and they can lead to investigations that may be even more valuable than the ones envisioned by the original writer of this activity.
Reflecting on the Activity, and on the Webinar
With the pressures we experience as teachers today, we sometimes get carried away by our lesson plans, our objectives, and the timeline we’re expected to follow. I know I did in this webinar: I wanted to be sure I covered four activities in different styles and on various topics. And we have to remind ourselves of the importance of slowing down, of giving students time to develop perseverance, of prodding them to formulate a definition with greater precision, of encouraging them to discover for themselves the structure that relates the different categories of quadrilaterals. In the end it’s the Standards for Mathematical Practice that we need to emphasize; it’s the quest for reasoning and sense-making that we have to instill and nurture through our teaching.
I’m not entirely certain how I’d change my webinar presentation based on these reflections; it’s a tough format, with over 100 attendees and a remote rather than direct connection. But I’d have done better if I’d presented only three activities, and spent more time exploring how best to use the parallelogram activity to address various Standards for Mathematical Practice. This blog post is my effort to fill in that missing piece.
Thanks to Rick Gaston for suggesting the title theme of this post, to Daniel Scher for encouraging me to write it, and to the Math Forum (from whom I shamelessly stole my “What do you notice? What do you wonder?” questions).