Building a Better Roller Coaster Part I

What does it take to motivate students to learn the complexities of polynomials?

For students in my math and programming class, it's the challenge of building a better roller coaster.

This unit was designed for my Algebra II class in an attempt to increase excitement during our study of polynomials. The accelerated course covers this topic in tedious depth and students have had a difficult time staying focused. Combining advanced concepts with a real world project has proven to be a winning combination.

I begin the unit by asking students to model various roller coaster ascents. This is the easy part since most ascents are fairly linear. Some start at ground level; others begin on an elevated platform. Students determine the slope and the y intercept. The important part is that they are modeling a real object and, for this particular class, translating the model to code. I then show this picture

and ask, how would you model that?

Students recognize the parabolic shape and know that a quadratic equation is needed. This leads to some terrific questions and an engaging discussion ensues:

How high is it?
What's the turning point?
How wide is it?
How far is it from the starting point?

We discuss various ways to represent a quadratic equation and agree to work with the vertex form. We talk through the role of each coefficient and come up with reasonable models. Everything to this point has been review. I then show this picture


and again ask the question, how would you model that?

After some discussion, we hit upon the idea of higher polynomials. And that's where we are now. We're looking at:

the shapes of functions of different degree
the effect of the sign of the leading coefficient
ways to find zeros
increasing and decreasing intervals
minima and maxima

Soon they will determine equations from data and eventually program a section of a roller coaster. More on that in Part 2. Stay tuned.

roller coaster images courtesy of Joyrides

Math Apprentice: The Companion Guide

Laura Rose, a graduate student at William and Mary, has written a comprehensive summary of Math Apprentice for Connexions, a collection of academic modules hosted by Rice University. In this summary, Ms. Rose describes how Math Apprentice can be used in a middle school classroom. She suggests the site could be the cornerstone of a semester long project about math in the real world.

I must thank Ms. Rose for doing what I had never thought to do. Her guide will surely make it easier for classroom teachers to incorporate Math Apprentice into the curriculum. It's my hope that increasing numbers of students will spend time at Math Apprentice and internalize its underlying message: math is the path to anything you want to be.

Mathematical Pursuits

Year after year, students make the steady ascent along the rocky trails of Math Mountain. Arithmetic gives way to algebra. Polygons lead to polyhedra. Functions progress from linear to quadratic to exponential. But what's at the summit? What will students do with all this knowledge?

When will we ever have to use this stuff?

Math Apprentice hopes to answer that question. Designed for students in grades 4 to 8, Math Apprentice invites students to play the role of an intern at one of eight businesses that use math. Students are given an overview of the math by an animated, virtual employee. They may then choose to freely explore math concepts or solve a specific problem.

The math in the activities is a mix of grade appropriate concepts and advanced mathematics. I think it's important for students to interact with math beyond the standards. This is often where the real joy of math can be found. Even young students can access difficult concepts if they are presented in a meaningful and engaging way.

Below are some screenshots:

At the Sweet Treat Cafe, students must analyze graphs, scale up recipes, and find the best buy.

Students learn about ratios and conversion factors at the Wheel Works Bike Shop.

At Game Pro, students use the Pythagorean Theorem to find the distance betwen the villian and the hero.




Students become computer animators at Trigon Studios. They use sine and cosine function to manipulate characters and props in a movie scene.




While interning at Doodles, students use various functions to create eye popping works of art.





At Space Logic, students match robot speeds to distance vs time graphs and program a space rover to reach its destination.




At Builders Inc, students must create room shapes whose dimensions meet the customer's specifications.




While working at Adventure Rides, students determine the height of a roller coaster hill that will give the speed that is needed.

Please share this resource and let me know what you think.

I See The Finish Line...

but I'm terrified to cross it.

I've been working almost non-stop for months on my current math project. I have one final activity to program and then it's off to my student reviewers for what I hope will be a merciful critique. Then I'll post the project online and see what happens.

I've been eager to see this project go live since its inception nearly two years ago. Each time a new milestone was reached, my heart raced with anticipation. This is by far the most ambitious project I've ever attempted. The topic is one for which I have great passion - helping kids see the connection between school math and the real world.

This project is also a significant step toward my own professional goal of creating immersive math experiences. My modus operandi as an educator has been to draw my students into a world of numbers, patterns, and relationships through challenges, investigations, and a touch of imagination. Our investigations, while based in reality, often took us to places that were just outside our reach conceptually. The math was too hard. But rather than intimidate, these experiences seemed to excite students about the possibilities. And that's what I've tried to accomplish with this project. The math in many of the activities is way beyond the project's intended age group. But if we wait until students get to advanced math before revealing the good stuff, we risk losing a large group of potentially amazing problem solvers and creative thinkers. This project aims to keep those kids onboard a little longer.

So why I am terrified? There are a number of reasons, I suppose. My work took a beating this summer and I'm still a little ebublogger-shy. Putting your work on line for all to see (perchance to bash) can leave you rather vulnerable. I never did grow a thick enough skin for this business. I also suspect that when one writes 10,000 lines of code, there's bound to be an error or ten. The thought of finding those errors and trying to correct them is a bit daunting. The greatest fear, though, comes from within. I've worked so hard on this project and have such high hopes for it that almost any success would fall short of my expectations. I'm coming to terms with the realization that, like most things, it will have its fans as well as its detractors. At least no one can fault me for trying.

Image is courtesy of brappa on Flickr.

Algebra for All

Solve for A, B, and C using the following equations:

A + B = 26
B + C = 45
C + A = 33

How would you go about it? Would you begin by combining equations? Or would you start by making substitutions? Is this a problem a 4th grader could solve? Without guessing and checking?

What if the problem looked like this?

Is it easier to solve? It's exactly the same problem, isn't it? The visual representation of the problem makes a huge difference. Now it's obvious that the two green blobs are the same size even though they exist on different scales (or different equations). And the two blue blobs are the same. Aha, the two red ones as well.

I have prealgebra and algebra students who have to ask if the A in one equation has the same value as the A in another, whether the two equations are part of a system or not. Imagine if those students had been solving math puzzles like this one throughout elementary school. Would their concept of variable be more clear?

Elementary age students at the math center think problems like this are great fun. They have no idea they are exploring linear functions or algebraic relationships. All they know is that these problems make them think. Algebraic reasoning problems give young students a chance to apply their knowledge of basic math facts within fairly complex scenarios. Problems like this one inspire young minds and satisfy their need for a greater challenge. Our students are incredibly proud when they are able to solve one of these math problems successfully.

How would a young student solve such a problem? We ask our students to compare any two scales and find what they have in common. Let's take the first two scales. Students will point out that the blue blob is common to both. Then we have them look for differences. Students notice that the scale weights differ and the partner blobs are different. We ask them what they think might be causing the weight on the second scale to be greater. It's obvious to students that the bigger red blob on scale two is causing an increase in weight.

Once they understand the effect of changing the partner from a small green blob on the first scale to a bigger red blob on the second, we can look at the quantitive aspects of the problem. We then ask what is the difference in weight between the two scales. Students will do the computation and find the difference is 19. That's the difference between scale one and scale two. What else does this number mean? What other difference does it describe? Students will relate 19 to the difference in weight of the green blob and the red blob. The red blob weighs 19 more weight units than the green blob.

We write this as:

R = G + 19

Now we have a relationship between the red blob and the green blob. This relationship tells us that if we replace a red blob with a green blob plus 19 weight units the scale reading will stay the same. So let's do it.

We head over to third scale. The equation there is:

G + R = 33

We replace the red blob. We get:

G + G + 19 = 33

What if we take 19 weight units off the scale?

What will the scale read then?

33-19 = 14

Our new equation is:

G + G = 14

At this point, students recognize this as a doubles problem and easily find the value of G to be 7. They then use this value to find the weights of the other blobs.

Modeling this problem with young students as a whole group activity is a very powerful. They excitedly share their insights and answers. We'll do several of these together before they work independently to solve similar problems. Eventually we make our way toward the original abstract problem. We replace the green blob with the letter G, then the blue blob disappears and gives way to the letter B, and finally we part ways with the red blob and bring out the letter R. The letters remain on the scale however so the context is reserved. Once the students are comfortable working with letters, we then remove the scale. Students solve systems of 3 equations by the end of 4th grade. More importantly, students learned how to think through abstract problems, a skill that will forever be of value.

Naturally, our fourth graders do not start at this level of problem solving. Early in the year, they have experience with simpler scale problems:

While the top equation looks like a guess and check problem, students can confidently find the value of the gears on the bottom scale. One they find the value is 3, they apply that value to the gears on the first scale. The problem is greatly simplified:

5xN + 6 = 26

If we multiply a number by 5 and then add 6 more to it, we get 26. what is the number?

They continue working on problems like this one and eventually move on to problems with three scales.

To make things even more interesting, we ask students to create their own scale problems. We begin with two scales which we modestly improvise with pieces of plain copy paper. We then give the students a variety of objects such as base ten blocks, colored cubes, and geometric tiles. They choose two types of objects to work with and begin creating their scale problems. They have to decide upon a value for each scale and then check it to make sure it works. Then the students switch places and try to solve the problem. It's one of their favorite activities and it wonderful to see them so actively engaged in problem solving. If you work with elementary age students, give it a try. You won't be disappointed.

Resources:

Algebraic Reasoning Scales

Systems of Equations

Resolved

A compromise has been offered and accepted in the edu-games debate. Scott has added a statement to his blog noting my request to label Math TV a learning activity rather than a game. While this compromise is not entirely satisfactory, it is enough to move the discussion forward. The main point of Scott's blog, I believe, was to initiate a dialogue about the quality of educational games. Passing judgement in the absence of critical analysis or a plan for action made it difficult to get there. It is my hope that we can now move away from simply identifying poor game design and, instead, focus on bringing about change.

For my part, I have publicly invited anyone with artistic skills or creative ideas about learning to work with me to develop a higher quality prototype. I am offering my time, experience, and programming skills. Let's combine our talents and create something that challenges the state of educational gaming, enhances the learning experiences of our students, and provides an inspiration for up and coming game designers.

Dear Edublogger

I wish I had a degree in graphic arts and could design dazzling 3D backgrounds and create characters awash with realism. I wish I had mastered Basic and C++ and Java when time was more abundant and learning difficult concepts came easily. These are skills that I could make use of today. But I chose a different path. I did not know then what my passions would be further down the road.

So I do the best I can with the limited skills I have. Six years ago, my students and I tossed around an idea for a TV station that would broadcast math shows. I took the idea in a different direction and created a character that would answer math questions, sent in by students, through mathcasts. After explaining the solution, she would invite students to try a similar problem. I spent many months developing the content. I had just discovered Flash and was learning for the first time how to draw using a computer. Eventually, I completed the project and published my work on Math Playground. My skills have improved considerably since then and those early mistakes are glaring reminders of just how inexperienced I was. I haven't updated the activity in years and I mainly keep it on the site to appease the teachers and students who continue to use it.

Imagine my surprise upon discovering this activity had been pulled out of retirement and resurrected on your widely read blog. It was used as an example of an educational game that, in your mind and in your words, sucks. As hard as I've tried to remain objective, it stings.

Why this activity was selected is a bit of a mystery; one that you have repeatedly failed to address. The activity, an odd poster child for all that is wrong with educational games, is, quite simply, NOT a game. It is not labeled as a game on my website. No one who has ever spent any time using the activity has ever confused it with being a game. It's obviously an instructional tool. As Maria Droujkova pointed out, there isn't a single game element within the activity.

One of the reasons you give for refusing to remove the image and link is your intolerance for censorship. You also say that if you make a mistake, you'll own it. An online tutorial is no more an educational game than a YouTube video, a Voicethread, or an ebook is. Yet, despite all the evidence, you will not acknowledge this error. Instead, you insist upon keeping a very flawed example to support your opinion of educational games.

My greatest concern is the false perception the casual reader will have after skimming through your blog. They will first see the title and the use of a very volatile slang word; a word that means to be repellent or disgusting. They will then see a picture of my math activity and the name of my site. And then they will move along; equating Math Playground with "games that suck." There will be no opportunity to point out the error, to change that perception. How do you justify this probable consequence?

While there is much to consider, one question I continue to revisit is why you would display another educator's work in such an overtly negative way. I may not be part of your immediate PLN but our respective networks overlap. I would never do to their work what you have done to mine. I always try to promote other people's ideas and point out what is significant about the work they do. I think it's important to encourage creativity and value contributions. I can't imagine ever calling out an individual's work in the manner you have.

The real sadness here is that I would have enjoyed a discussion about educational games, their influence and reach, and how today's online games could be redesigned for optimal learning. The wrongful inclusion of my work, the lack of respect you have shown for my opinions, and your refusal to admit a mistake, have made it impossible to continue that level of academic discussion.

While this experience has been demoralizing, I can't let it defeat me. Hundreds of thousands of students will soon descend upon Math Playground, marking the start of another school year. They enjoy the practice games, feel empowered when they create turtle art and spirographs, and love to share their math questions and ideas. And, you know what? That's all that really matters. I'm in this for them.