After writing yesterday’s post on the connections between polar and Cartesian graphs, I realized that I hadn’t said anything about how easy it is to start from scratch and create a polar graph in Sketchpad, so I decided to write this post, and include an instructional video. Here are the steps to create the graph shown on the right below.

1. Choose Graph | Plot New Function.
2. Use the Equation menu to choose r = f(θ).
3. Type “c” (for “cos”), “2″, and “th” (for “theta”).
4. Click OK.
5. If your angle units are degrees, Sketchpad may ask if you want to change to radians. (Don’t worry; the graph will be correct no matter whether you want to use radians or degrees.)

That’s it!

This short video shows how easy it is to add parameters to control the amplitude, period, and phase shift:

It’s also easy to create a family of polar functions. Once you’ve modified your graph to show f(θ) = a·sin(b·(θ – c)), here are the steps used to graph the family of functions shown below.

1. Select both the graph and parameter a.
2. Choose Construct | Family of Functions.
3. Set the domain to go from 1 to 6, and the number of samples to 11. This will create samples for these values of a: {1.0, 1.5, 2.0, …, 6.0}.
4. Click OK.

You’re done. Now experiment on your own!

The May 2013 Mathematics Teacher has an excellent article by Jonathan F. Lawes (“Graphing Polar Curves”) on the value of plotting the same function in both polar and rectangular coordinates. Doing so not only helps students understand how polar coordinates work, but also gives them a novel and revealing perspective on periodic motion and how polar coordinates reveal periodicity.

The sample activity Cartesian Graphs and Polar Graphs, shown above, comes with Sketchpad 5. After launching Sketchpad, choose Learning Center from the Help menu and navigate to “Trigonometry, Conics, and Precalculus” under Sample Activities. This activity and its accompanying sketch and worksheet provide a convenient and powerful way to engage students in exploring the relationships discussed in the article.

For each graph, the independent variable appears as a red bar that corresponds to a particular value of x (for Cartesian) or θ (for polar). The red bar has tick marks that show possible values of the dependent variable (y for Cartesian and r for polar), and a green bow-tie that shows the actual value of the dependent variable for this value of the independent variable. Here’s what it looks like for the function r = 4 cos θ when the slider for θ has been dragged to π/4:

Remarkably, the two red bars (and their green bow-ties) are identical except that the Cartesian one has been translated from θ = 0 to θ = π/4, and the polar one has been rotated from θ = 0 to θ = π/4.

We designed both the red bar and the animation to connect the two representations as clearly as we could. After showing the graphs and turning on tracing, students can animate the independent variable. The video below shows how students see both views of the function r = 4 cos 2θ as the value of θ is animated from 0 to 2π:

As the animation proceeds, students can observe how the two representations show exactly the same function, with the only difference being that the independent variable’s value undergoes translation on the left and rotation on the right.

There are many insights and discoveries students can make by dragging the independent variable and observing the graph. For instance, the seemingly counter-intuitive comparison between r = 4 cos 2θ on the left and r = 4 cos 3θ on the right becomes much clearer when students have the ability to control the value of θ by dragging. (Try it yourself first before reading the note at the end of the post.)

Other wonderful questions and realizations can come about by experimenting with a variety of functions. For instance, can you tell which of the polar graphs below represents the function r = tan θ and which represents r = 4 tan θ?

Here’s another fun challenge: Create the following polar graphs using simple trigonometric functions:

To create your own polar graphs from scratch, see my follow-up post.
This sketch implements Figures 3 through 9 from the article: Graphing Polar Curves Figures.gsp

Note: When you graph r = 4 cos 3θ by dragging θ from 0 to 2π, it’s really easy for students to see that the graph traces out six lobes, not three. The static graph shows only three, but as students drag they see how the six lobes overlap each other.

Time sure flies when you’re busy… not sure how we got to April already! Next week, we’ll be at NCSM and NCTM in Denver, presenting in workshops, demonstrating our software in the McGraw-Hill Education booth, and organizing yet another fast, fun, and inspiring Math Ignite event on Wednesday, April 17, at 3:45 pm, in Capitol Ballroom 2 at the Hyatt Regency (NCSM Session 382).

This will be my fourth year taping and editing videos of the Ignite talks at the national math conferences, and I still find these talks to be the most compelling and entertaining events at these conferences. For those of you unfamiliar with the Ignite format, the speaker gives a 5-minute talk while a 20-slide PowerPoint presentation automatically advances to the next slide every 15 seconds.

Our most recent Ignite event was the CMC-North conference in Asilomar in December. I finally finished editing all nine speakers and I want to highlight some of those talks. The complete playlist is available on our YouTube page. In my December blog post, I already featured Jennifer North Morris’ talk, so this time I want to sing the praises of former NCTM President Mike Shaughnessy, who literally sang most of his presentation.

I encourage you to view the other talks too as they were all really good. And if you’re going to be in Denver next week, please come to our next Ignite event (speakers listed at end), or attend any of these sessions in which Key folk present on Sketchpad and TinkerPlots:

• Scott Steketee, “Do the Function Dance with Sketchpad 5: Make Functions Fun and Give Function Concepts a Concrete Basis,” Tuesday, 4/16, 8:45 am, Hyatt Regency Centennial C (NCSM Session 206)

• Karen Greenhaus, “Going Uphill: Developing the Concept of Slope Using Sketchpad and TinkerPlots,” Tuesday, 4/16, 2:30 pm, Hyatt Regency Granite (NCSM Session 266)

• Scott Steketee, “Do the Function Dance with Sketchpad 5,” Thursday, 4/18, 1:00 pm, Convention Center 708/710/712 (NCTM Session 207)

• Elizabeth DeCarli, “Help Students Dig into Data, Statistics, and Probability with TinkerPlots,” Friday, 4/19, 1:00 pm, Convention Center Mile High 2B (NCTM Session 502)

Speakers at the NCSM Ignite event in Denver: Timothy Kanold, Jennifer North Morris, Karim Kai Ani, Annie Fetter, Max Ray, Jack Ashton, Robyn Silby, Lisa Carmona, Ken Gordon, Ann Hirsch, and Ellie Terry.

Of the many reasons that I chose to major in mathematics, perhaps the most compelling to me was this: Mathematics is beautiful.

Yes, mathematics could be practical, but it was the sheer beauty of its proofs that left me awestruck. How could I not be impressed, for example, by the elegance and ingenuity of the proof that the square root of 2 is irrational? I found myself charmed by the idea that even though I, a mere undergraduate student, could not have conceived of such a novel proof myself, I could still enjoy—albeit vicariously—the cleverness of others.

My interest in math led two friends who lived in my dorm to ask for my occasional help  with homework. I still remember a particular probability question that was assigned as extra credit:

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 15 minutes for the other person. When the 15 minutes have passed, each person leaves if the other person has not come. What is the probability the friends will meet?

Before reading further, think about this question and see what progress you make in solving it. Looking back 20 years, I wish I could report that I was successful in my attempts, but truth be told, I didn’t try very hard. I remembered having seen the problem in the book Fifty Challenging Problems in Probability with Solutions and pounced on the answer.

The provided solution did not disappoint. The approach to determining the chance of the two friends meeting was a perfect example of out-of-the-box thinking. The solution hinged on a very imaginative representation of the problem in which the arrival times of the two friends were plotted as (x, y) pairs on a coordinate system.

Years have passed since my college days, and while my love of mathematics hasn’t diminished, my relationship to the subject has changed. I’m no longer content to marvel at a mathematical proof the way I might sit in wonder while listening to a concert pianist.  I want to create beautiful proofs of my own, or at the very least, gain insights into the thought process that leads to the creation of such proofs.

Several years ago while co-authoring Exploring Precalculus with The Geometer’s Sketchpad, I remembered the lunch date problem and wondered whether I could model it using Sketchpad. I created a simulation that ran the lunch date scenario repeatedly, each time picking random arrival times for the two friends. As I watched the simulation, I began to feel my relationship to the problem changing. Modeling the lunch date problem with Sketchpad made the decision to represent the arrival times on a coordinate system feel less like a flash of brilliance and more like a very practical way to represent two pieces of data economically as (x, y) pairs.

In the video below, I describe step by step how I built my sketch. By watching the video and trying the construction yourself, you’ll likely learn some useful Sketchpad techniques. But I also hope that delving into the construction process will change your relationship with the mathematics, making the lunch date solution feel organic and natural.

And who knows? You might very well find other mathematical problems where a similar approach to representing data can be applied. If you do, remember what was once a stunning insight available only to the lucky few is now an established way of reasoning that is within your command.

Last week was the fourth session of my spring Advanced Secondary Math Methods class at the University of Pennsylvania. Each year I assign a semester project in which groups of three students use lesson-study techniques—on a small scale—to create, test, refine, teach, evaluate, and document specific shared instructional products, composed of a (possibly multi-day) lesson and its associated assessment. (The term, and the inspiration, comes from an NCTM Research Presession talk by Anne Morris and Jim Hiebert.)

In class we brainstormed about using an interesting and motivating problem to introduce and investigate exponential functions. I imagined the math arising from a problem connecting the keys of a piano to the frequencies of the sounds they produce.

Press a key to hear its note, and see its ratio to the previous note.

I didn’t want to flesh out a problem in advance, nor did I want us to spend a lot of time floundering to identify an appropriate problem, so I proposed two vague problem topics: a simple one (“Why do certain combinations of notes sound better than other combinations?”) and a significantly more challenging one (“How should you tune a piano to enable as many nice-sounding combinations as possible?”). To situate our brainstorming, I prepared a sequence of ten questions that started with the concept of an octave, developed the idea that harmonious sounds result from frequencies whose ratios can be represented by small integers, and eventually led to the idea of dividing the octave into evenly-spaced steps that include as many harmonious ratios as possible.

The brainstorming was extremely useful; we generated lots of questions, ideas, and implementation suggestions. It was immediately clear that this activity wouldn’t work in a silent classroom; we had to have music, by playing a song to introduce the activity, and by providing a way for students to listen to different frequencies together to find combinations that sound particularly nice. We talked about instruments, and decided the piano presents mathematical advantages (with its one-to-one correspondence between keys and frequencies) but a stringed instrument is a simpler and more easily understood mechanism. While asking students to bring musical instruments to school was a good idea, our methods class nixed it because there might not be any musicians in some classes, and even if there were musicians it didn’t seem smart to ask them to put expensive instruments at risk for the purpose of the lesson.

This discussion inspired me, the next day, to see how hard it would be to create an “instrument” just for this purpose. Here’s what I built, using a piece of scrap wood, some nails and screws, and some picture wire. You tune it with a screwdriver, and you can mark fractions of the length of the strings: 1/2, 1/3, 2/3, 1/4, 3/4, and 4/5. If I were doing this with a class, I’d have students mark the fractional distances with tape or sticky notes so I could remove them before the next class. Note, by the way, that a length of 2/3 produces a frequency of 3/2 of the fundamental — in musical terms, a fifth.

The Two-Stringed Board

Plucking the Fundamental and the Fifth (3/2)

The device is cheap and easy to make, but keeping it in tune is a bit tricky, and hearing the difference between consonant and dissonant plucked notes requires close attention, so it was a natural next step to create a sketch (available for both desktop and iPad) to make the hard parts easy. Students can choose various frequencies to play together, adjust them till they sound “nice,” and then record the frequencies and their ratio in a table. In this way they can discover various harmonies and the ratios that produce them.

Adjust the blue bars to change the string’s frequency.

Now it’s time to move to the piano, with its nice correspondence between keys and frequencies. It became clear, through our class discussion, that the starting point needed to be a piano that has only A keys, so we can play the keys from A0 through A7, having tuned A4 to 440 hertz. Here again students can make a table to see the pattern of the numbers, and they can graph it in both dynagraph and Cartesian form. This is the perfect time to ask them to describe a function that they could use to find the frequency of A10, or A15, or A4, and to figure out how to describe this function both in words and as a formula. (An important point in this class discussion is to validate both the rule expressed in words and the algebraic formula as appropriate representations of this function—as is the Sketchpad model as well.)

So this is great—-students have defined a function that links piano keys with the frequencies they produce. But we don’t have enough keys to play actual songs, because all our intervals are octaves, based on a ratio of 2:1. This is a new, more advanced problem: how can we divide up the octave intervals in a way that lets us play our favorite harmonies and melodies, the ones defined by small-integer ratios?

Unfortunately, we didn’t have time, in Methods class, even to brainstorm this follow-up problem, though it seems to me that it could be an excellent shared instructional product of its own. And for the first shared instructional product I’ve described here, we shared the brainstorming, but the draft of the worksheet and Sketchpad document are mine alone, without the benefit (yet) of any collaborators. (I haven’t finished the Teacher Notes yet either.) But I think the activity will be fun for students, I encourage you to try it out, and I’d love to have your collaboration, by telling me how it went and how it can be improved.

I was lucky enough this past week to visit some classrooms and see teachers using Sketchpad in various ways. It’s been seven years since I was in the classroom myself, so for me it was like coming home. It brought back a lot of great memories, though also some reminders of some not-so-great things that teachers have to deal with on a daily basis.

I want to share some of my observations and insights to basically sing the praises of teachers. I want to shout out to those critics of teachers and public school classrooms that until you see what a teacher faces everyday—and walk a mile in their classroom—you have no idea of the hard work, dedication, and amazing things they are doing. With little thanks and little support. Test scores don’t do justice to the true work of teachers.

I visited a high school ESL geometry class with 41 students, crammed into a small classroom with not enough desks. 41 students speaking six different languages, with one energetic teacher, who spent as much time explaining simple words like “straightedge” as she did focusing on math content. No compasses were available (in a geometry class, let’s remember), so she used various lids from containers for students to draw circles. Sketchpad was used to visually demonstrate and confirm students conjectures from what they were constructing with their lids, rulers, and paper-folding.

Then I was in a high school co-taught, special education Algebra class with 36 students, where Sketchpad was used to review concepts of slope and equations for the opening warm-up. The teacher used an interactive whiteboard and a clicker system to get feedback from students, so it was really great to see all the engagement. Later I ended up at a different high school in another Algebra  class with 35 students, where the teacher used Sketchpad on an interactive whiteboard to introduce systems of equations. A great lesson, interrupted by a fight in the hall, a student cussing because another student had written on their paper, and four boys in the back hitting and goofing off. You may ask, “why didn’t the teacher stop it?” She did, several times—but with 35 students in a class designed to hold 25 students comfortably, its physically impossible to walk around the classroom and “stay on top” of situations.

The next day I was in a 6th grade class of about 30 students—very little room to move in—but these kids all had computers at their desks and the teacher had an interactive white board. It was exciting to see all the work and creativity. The teacher had students working with Sketchpad, guiding them through the activities, and the aha moments were priceless. The end of the day found me in another middle school class with 8 alternative education students—students that everyone had given up on, but the teacher had determined he was going to get these kids to learn. According to the teacher, these were kids with no fathers at home, all on ADD medicine, and often absent to help at home. Even in the short time I was there, it was obvious that this teacher was making a difference—not only in believing these kids could learn, but more importantly modeling respect, giving them a male role model, expecting them to succeed. It was beautiful to see.

My last class was back at a high school with a repeater Algebra class. Tough kids who constantly tried to interrupt the teacher, or were sleeping at their desk, or coming in late.  It was only the 2nd day of the new semester, so you could tell they were testing the boundaries to see what their “new” teacher would let them get away with—which wasn’t much. But the fact that the 45-minute lesson was interrupted every two or three minutes by a student asking to go to the bathroom, or knocking on the door to be let in because they were late (until they got a pass they were not allowed in, but that didn’t stop the knocking), or the teacher tapping a student on the desk to put their head up—it was a wonder any teaching could happen. But it did. She used Sketchpad to help students understand slope—they played “The Slope Game,” came to the white board to move the lines to change their slopes, and explain their reasons. And, they even stayed after the bell to finish because they were so engaged.

What did I walk away with from these observations?

1. The big take-away is teachers are overworked and under-appreciated, but not a single one of these teachers had a negative attitude. They were there to do their best to help students learn.
2. Teachers are trying to use technology and engage their students. They work with what they have, but time is often against them and they are overburdened with so many district and state mandates that they often can’t do what they need or want to engage their students.
3. Class sizes are too big. 41, 36, 35 students in a room? WHAT?!!!  That’s insane. But, look around the country and you will see that is more and more the norm because of budget cuts and policy changes. So those people out there blaming teachers for not teaching, look at their class sizes. It’s hard to do all the things you are suppose to—differentiate, individualize, manage class behavior—when you can’t move in your own classroom and don’t even have enough seats for every student.
4. Inequity exists not only between school districts, but within a school district. One school has white boards and computers for students, another can’t even get compasses for their geometry teacher. Until schools can provide the same resources and efforts for all students, regardless of location, population, ethnicity, and so on, there will always be an inequity in what and how students learn.
5. Teachers are faced with so many more challenges than standardized tests, and those challenges impact everything, including results on standardized tests. We need to stop evaluating teachers solely on the basis of how their students do on standardized tests. Look at all the challenges they face—large class sizes, rude and disrespectful students, lack of resources and materials, lack of parental support, limited time, and constant “requirements” by the district and state—it’s a wonder they show up every day.

But they do. And that’s the beauty of a teacher. So, before you blame teachers for all that is wrong with education, go to a classroom and spend a day seeing what teachers really do. I guarantee that you will walk away with some disbelief and a new respect for those folks out there trying to educate our children.

As a New Yorker, I have no lack of museums to suit my every viewing whim. Matisse? Just head to the Metropolitan. Edvard Munch? The Scream is at the Museum of Modern Art. But where should I go when I’m itching for some Pythagoras or Euler? Why, the Museum of Mathematics of course!

Just two weeks ago, the Museum of Mathematics (MoMath) opened its doors. The largest museum of its kind in the United States, MoMath occupies 19,000 square feet along the north side of Madison Square Park in Manhattan.

At right is a picture of MoMath’s entrance. Unlike most museums, MoMath doesn’t have a single logo to identify itself. All of the designs in the photo were generated with Mathematica, and visitors create similar symmetric designs for themselves when they initialize their museum badge. (You can get a sense of what is involved by downloading this Sketchpad model.)

While strolling through the museum, I spotted a variety of mathematical motifs, from the door handle in the shape of π to the pattern on a bathroom wall shown in the picture below. Can you spot a hidden message in the network of segments?

There are 30 interactive exhibits to explore in the museum, and I’ll report on just a few. Formula Morph, shown below, gives visitors the chance to experiment with a variety of three-dimensional shapes and their corresponding algebraic representations. The neat twist here is the method by which you interact with the equations: To change a green-colored coefficient to a different value, you turn the corresponding green knob.

Another exhibit called Polypaint allows visitors to create wallpaper patterns by dipping an electronic paintbrush into empty color-coded paint cans and then painting onto a large touchscreen. As you apply brushstrokes to create your masterwork, the pattern is replicated across the entire screen, based on the wallpaper pattern you chose. (The iOS app, iOrnament, works similarly.)

The String Product exhibit features a giant paraboloid that cuts through the spiral staircase connecting the museum’s two floors. Visitors press any two numbers from 1 to 10 on a keypad, and the string connecting those numbers lights up on the paraboloid. By observing where the string intersects the vertical axis of the paraboloid, visitors can view the product of the two numbers. I found myself wondering about the underlying mathematics of this exhibit, so I built myself a Sketchpad model. Check it out!

The lower floor of the museum sports a Math Square that reacts to your feet as you walk across it. If you stand on the square with several friends, the Math Square maps out line segments that represent the shortest path connecting everyone. In the picture below, the Math Square displays a maze by Robert Abbott. The premise is simple: Begin at “Start” and work your way to “Finish” by walking along the path. There’s just one catch: You can only make right turns—no left turns allowed! Can you navigate through the maze?

One of my favorite exhibits in the museum is the Human Tree. By standing in front of a screen and waving your arms, you can create copy upon copy of yourself, just like a fractal.

Pictures alone aren’t enough to convey the interactive nature of the exhibits, so be sure to view George Hart’s video that leads you on a short tour of the museum. George is responsible for many of the exhibits in the museum and designed a very cool geometric puzzle sculpture sold in the museum store.

The museum has been open for less than a month, but already there are healthy crowds of adults and children each time I’ve visited. And who knows? When people think of New York museums, perhaps their train of thought will be Mondrian, Monet, Miró, Math!

Today I leave for my first proper vacation in a year and a half. Last time I took such a vacation, Key sold its high school textbooks to Kendall Hunt and transformed from a publishing company to a educational technology company. This time I just hope to survive the end of the world. Before I go, though, I want to tell you two recent stories in tribute to Karen Coe, who after more than 16 years leaves Key at the end of this week.

On December 1, we had our third annual Ignite event at the CMC-North Asilomar Math Conference. As usual, we had an impressive slate of speakers made up of teachers and leaders in the mathematics education community. I’ll be highlighting our dear friend Jennifer North Morris in a moment, but first back to Karen, who was once again our fantastic Ignite MC.

Karen first thought of adopting the Ignite format—5-minute talks with auto-advancing slides—for the mathematics education community. Over the last three years, Key has organized many Ignite events at national NCTM and NCSM conferences, as well as the CMC conferences here in California. These events have been vibrant and upbeat, often hilarious, and sometimes emotionally powerful. Organizing the Ignite events may have began as a marketing strategy, but what has emerged is truly a legacy.

If you have never attended one of our Ignite events, we have videotaped, edited, and uploaded all the talks to our YouTube page. This collection of roughly 60 talks represents an amazingly broad and diverse group of teachers and educational leaders addressing the issues that really matter in mathematics education today. Karen leaves behind a true treasure-trove of inspirational and motivational talks that are short, fast, and poignant. In fact, we’ve heard that both teacher education programs and professional development organizations use many of these talks as resources. Here is a great example—Jennifer’s fabulous presentation.

Then on December 14, we threw a huge party to honor all the people who have worked at Key over its 40-year history. It was quite a rager, including a final performance by the Key to Decibels—a band comprised of people who work (or used to work) at Key that specializes in changing the lyrics of songs to make them funny and relevant to the celebration at hand. Due to the huge role Karen has played in Key’s development, she ended up in many of the songs, but there is one we wrote just for her, “Sweet Karen Coe,” by adapting the Neil Diamond classic “Sweet Caroline.” The song continues to reverberate pleasantly in my mind.

So Sweet Karen Coe—on behalf of the mathematics education community, by the power not bestowed upon me by anyone, I hereby award you the title: Honorary Math Teacher. You’ve earned it, and we’ll miss you.

Daniel, Josephine, Karen, and Scott at Chicago NCTM

Last week I ended my fall math conference touring schedule.  It’s been a fun-filled road trip—in the span of two months I’ve been to all three NCTM regionals in Dallas, TX, Hartford, CT, and Chicago, IL, not to mention AMTNYS in Rye, NY.

One of the many things I love about these conferences is that I get to play with math and technology all day and meet lots of teachers who either a) know Sketchpad, Fathom, or TinkerPlots and just want to share what they are doing or play at the booth; b) have heard of these technologies or just came from a session on them and want to see more; or c) have never heard of these technologies (hard to believe!) and are curious when they walk by and see all the moving mathematics on our interactive white board.

My favorite question to ask teachers is, “What are you teaching next week?” because I can pretty much guarantee that I can find and demonstrate an activity, whether it be a premade model or one done on the fly, that will address whatever concept they throw at me. And then they can walk away with an idea to use in class the following week to engage their students in dynamic learning. That’s powerful!

One of the most exciting things about Sketchpad is its use for whatever math topic or grade level you are teaching. An elementary teacher needing to show factoring? Sketchpad’s got that. A middle school teacher looking to introduce quadrilateral properties? Yep—Sketchpad can do that. A  calculus teacher wanting a visual to show the antiderivative? You bet! Sketchpad can do that too.

Skeptical? Well, here are three quick video clips that demonstrate my responses to teachers in Chicago who challenged me with a math concept.

Elementary teacher: “I am working with fractions and number lines and students are really having a tough time visualizing the fractions.”

Geometry teacher: “We just finished the points of concurrency in a triangle.  Can Sketchpad show these and help me develop the Euler Line?” (I only showed her the circumcenter, in this video below. She got so excited that after that she took over and started constructing on her own!)

Calculus teacher: “I would love to be able to show the definite integral.”

In my examples I did a bit of everything—a premade model from our free users forum, Sketch Exchange, constructing from scratch, and then a premade model from within the software itself (go to the Help menu and choose either Sample Sketches or Using Sketchpad | Sample Activities).  Challenge yourself—what are you teaching next week? Can you find a Sketchpad activity to use instead of what you normally might do? Try it—your kids will LOVE it!

In the Fall as Middle School and Algebra 1 teachers look for activities to develop students’ understandings of proportional relationships, they may turn to measuring scale diagrams or using springs. One of KCP Technologies’ new online Data Games called Proximity offers a new option for teachers to help students learn about proportional relationships.

In Proximity, you shoot a ball toward a target–the closer you get, the higher you score. Try it here! To shoot the ball: click on or touch it, drag away from the target, and release.

Most computer games generate data, but they go unexamined and disappear when the game ends. But notice when you push the ball here that the data dynamically update in the graph and the portion of the table shown. Students learn that analyzing this data is the key to success in Data Games!

Students playing Proximity generally develop intuition that a proportional relationship exists between how hard the ball is pushed and how far it travels. Some students get really engaged in trying to score big just using trial and error as they figure out how hard to push. And they can achieve some modest success with this informal approach, perhaps unlocking the second level of the game. But it’s very difficult to advance to the third level, which we recommend Middle School and Algebra 1 teachers set as a goal, using only this method. We’ve designed the Data Games so that students really need to analyze the data and apply their math skills in order to succeed.

There are various strategies students might discover they can use to their advantage. Students might examine the table, shown below, and see a common ratio between push and distance, which they can then use to determine the amount of push needed for each future desired distance. Our student worksheet for Proximity also asks students to think about what the distance would be for a push of 0.

Needing to know the desired distance prompts students to use the onscreen ruler to measure the distance from ball to target. The game provides scaffolding for those players who don’t catch on to the value of measuring distance, offering a prompt if the ruler hasn’t been used for the past 60 seconds.

We chose to use a whole-number proportional constant in the first level of Proximity so that students could more easily discern the type of relationship that exists. In higher levels, decimal constants are introduced, and students generally find examining the graph to be a more useful strategy. They observe the data points fall on a line which passes through the origin. The gear menu allows students to create a movable line on the graph, which you can try below. The equation for the line is provided, and when you drag the line to fit the data, the equation updates dynamically. Students can then see that the line passes through the origin, and its slope is the proportional constant they need to determine push, given distance. (Bouncing the ball off the side walls creates points that do not all fall on the line like this, a good piece of data complexity for students to grapple with, or avoid by not bouncing.)

These beginning Algebra skills are not easy for many students, but game-based learning can motivate them to develop their understandings. Data Games are not designed to be so thrilling for kids that they’re going to throw away their Wii games at home, but they engage students at a high level during math class to improve their skills with data analysis.

Teachers can use Proximity and the other Data Games to help students master the Common Core State Standards for Mathematics involving data analysis. You can find the games and teacher resources supporting their classroom usage free on our website at play.ccssgames.com. These resources for each game include: short videos introducing the game and its data analysis tools to students; a video for teachers; a student worksheet; and teacher facilitation guidelines, including Common Core State Standards alignment. Free registration is required to access the materials.

I will also be offering a Key Curriculum Webinar on Tuesday, Dec. 4, at 4 pm PST, on using Data Games like Proximity to teach Middle School data analysis skills from the Common Core State Standards. You can register for that webinar here.