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[Makeover] Ferris Wheel

What I Did

TLDR: Here’s the 101questions page.

• Start concretely. Saying “it takes 40 seconds to complete one revolution” isn’t the same as seeing a ferris wheel travel at that speed. ¶ The task also asks students to trace the path of a car on a ferris wheel, precisely, point by point, for a given domain. We’ll get to that kind of precise abstraction in a minute but for now I’d like an actual sketch. I want to know how my students see the ferris wheel’s motion.
• Give a reason to give a damn. Here, as with our last Makeover Monday, you’re asked to create a graph simply because we said so that’s why. There isn’t any sense that a graph could be useful for anything more interesting than receiving a grade.
• Raise the ceiling on the task. We’re attempting to lower the ceiling by starting more concretely, with a sketch, but we’ll also help students develop the periodic concept further than the current task does. (The “critical thinking” extension task here doesn’t develop periodics so much as call back to circumference.)
• Prove math works. The task asks students to “Predict where you will be at 3 minutes” but we don’t get the payoff. Do periodic functions actually predict where you’ll be?

Play this video.

Pass out these empty graphs [pdf]. Ask students to graph the height of the red cart above the ground as best as they can for two complete spins. We aren’t asking for exactness yet. We’re looking to see:

Do they create smooth maxima and minima or pointy, non-differentiable cusps? If they’re making cusps, they’re suggesting the cart bounces at the top of the ferris wheel.

Are all their maxima at the same height? Are all their minima at the same depth? If they’re different, they’re suggesting this metallic ferris wheel shrinks or expands over time.

Is the horizontal distance between maxima the same? If they’re different, your students are suggesting the cart isn’t moving at a constant rate, that it speeds up or slows down during the ride.

Are their minima a few feet above ground level? Or are they suggesting the red cart hits the ground at the bottom of each turn?

In every case, push your students to attend to precision. (Common Core achievement unlocked!) Give them more of these graphs or have them hit erase on the tablet and start again. Get it right. Be precise.

Now that we’re pretty clear on the structure of this new kind of function called “periodic,” let’s step our game up. Ask them to guess how many full turns the red car will take before its ride is over. Tell them their other goal is to figure out exactly how high off the ground the cart will be at the end of three minutes. (Real-world relevance achievement … still pretty much locked.)

Now we’re ready to start the task as written. Use this screenshot with information. Head to Desmos, kids. Punch in points that are more obvious to you. The minima. The maxima. The position at far left and far right. Keep that going.

Students need to see that their old models are useless to describe these points. So give them a new one. Out with y=mx+b. In with y=asinb(x+c)+d.

Have them mess with the parameters until they get a perfect fit. Then use it to find the position at 3:00.

Now show them whether or not the model actually works.

• How long should a ride last so the person ends at the bottom for an easy exit? A: Lots of different correct answers here. That’s the fun of a periodic function. Hey write out all the correct answers in order and subtract one from the next. What’s happening here?
• If the ferris wheel spun backwards, how would that change your periodic function and your calculation?
• Where else would periodic functions make sense as a model?

What You Did

Over on the blogs:

Previously. The preview post where you’ll find some other interesting ideas.

[Makeover] Ferris Wheel Preview

2013 Aug 22. And here’s Ferris Wheel.

Happy August. There are four #MakeoverMondays left and I’ll be posting their previews on the previous Thursday. What would you do with this introductory task from a unit on periodics and why?

[Makeover] Postage Rates

This is from Pearson’s Algebra 1 textbook for iPad.

What I Did

• Establish a need for the graph, in general. Why are we drawing a graph? What’s the point? Does my ability to draw a graph serve any larger purpose than getting me points on an assignment? This task doesn’t have an answer to that question.
• Establish a need for the step graph, in particular. Why are we drawing a step graph? What’s the point. Does the step graph have any advantage over other graphs? This task doesn’t have an answer to that question.

Tom Ward has me covered on the first point. Nothing’s topping this aspirational save-the-date for his 2019 marriage to Ms. Stone. What will postage cost then?

Graphs and equations of data are useful when they let us predict something external to the data we know. We don’t know the price of postage stamps in 2019 so we can extend a linear model beyond the data and find out what it might be.

Mr. Ward will need to scrounge up 54 cents per invitation.

But he still hasn’t given us a reason to care about the step graph. For that we look at internal data. We tell the kid, “Hey, your graph is messed up. If you hand that graph to someone, it says the cost of postage in 2003 was 39 cents and the cost in 2005 was 41 cents. But the cost in both those years was only 37 cents.”

If you’re going to make a graph that tells the story of the data accurately you’re going to need a different model than a straight line. Enter the step.

What You Did

Aside from Tom Ward’s superb work, over on the blogs:

• Scott Hills seizes the opportunity to show students the benefits of a well-scaled axis.
• Beth Ferguson removes step graphs from the task, which is one way to handle the problem.
• Evan Weinberg goes digital, though I’m not sure what the digital medium adds here. He also just asserts that the student’s graph “should be a step function,” which highlights the difficulty again of motivating a need for this function family.

Featured Comment

Instead of a wedding invitation, change it to a graduation invitation. Have the kids estimate how many invitations they would have to mail out. They could then calculate the cost of their invitations. You could also have them calculate the cost that their parents/grandparents paid for their graduation invitations.

[Makeover] Tire Marks

What I Did

• Reduce extraneous literacy demand. A lot of visual information has been encoded in text. Let’s get that information back in its natural medium.
• Delay the abstraction. Tables and graphs and equations will eventually be useful but let’s delay their introduction until we need them.
• Get a better image. The illustration here is a member of the “job testimonial” genre. ie. “Trooper Bob uses math, so you should too.” I’m unconvinced that message will sway classroom opinion on Algebra even a little. Instead let’s put the student into Trooper Bob’s shoes, doing Trooper Bob’s work.
• Ask a better question. Neither of the two questions here addresses any of Trooper Bob’s concerns. The first has you extend a graph for no discernible purpose. (And why extend the graph from 60 feet to 100 feet. Is that just arbitrary?) The second poses the fantastic scenario where Trooper Bob comes to the scene of a wreck already aware of how fast the car was traveling and then proceeds to do math to figure out the length of the tire marks in front of him. Which he could just measure.

So show this picture of a wreck. Ask your students to guess how fast you think that car was going when it hit the brakes. Tell them they have to figure out if it broke the law. Do they think it was speeding?

Then show them this image.

Ask your students to rank the cars from fastest to slowest. Ask them how they know. They’ve decided the variables “length of skid” and “speed” are positively related. But what kind of relationship is it? This is where a graph — a picture of a relationship — is so useful. Show them the data.

Have them graph the data. This is a little new to us. It isn’t linear. It isn’t quadratic. It isn’t exponential. Offer an explanation of the root model. It’s the inverse of a parabola. With the parabola, a little growth in the horizontal direction results in a lot of growth in the vertical direction. With the root model, a lot of growth is required in the horizontal direction before you get even a little growth in the vertical direction.

Now they can find the exact model for these data and evaluate it for 232.7 feet.

68 miles per hour in a residential zone? You won’t be needing that drivers license for a long time.

What You Did

Over on the blogs:

• Nicholas Chan encourages modeling also, where students make predictions from data.
• Eric Scholz has the same, except where I start with the accident you’re trying to solve and then get smaller data for modeling, he starts by showing students the smaller data and then ending with the accident you’re trying to solve. Is the difference substantial?
• Matthew Jones sends along this clip, which would make for interesting watching after our math work. I’m not sure what work the students would do on the video, though.
• Kate says, “bring the cop to school,” which could be great, but again what math work do the students do?

Featured Comment

Now this is the reason I follow this blog. I am a criminal justice instructor and many students come to my field as fugitives from math and science. Use of these materials is helping me develop criminal justice contextualized math resources for a class proposal.

[Makeover] Meatballs

This is from Discovering Geometry.

What I Did

Basically, I three-acted the heck out of it. Which means:

• Reduce the literacy demand. Let’s encode as much of the text as we can in a visual.
• Add perplexity. That visual will attempt to leave students hanging with the question, “What’s going to happen next?”
• Lower the floor on the task. The problem as written jumps straight to the task of calculation. We can scaffold our way to the calculation with some interesting concrete tasks.
• Add intuition. Guessing is one of those lower-floor tasks and this problem is ready for it.
• Add modeling. We’ll ask students “what information would be useful here?” before we give them that information. That’s because the first job of modeling (as it’s defined by the CCSS) is “identifying variables in the situation and selecting those that represent essential features.” The task as written does that job for students.
• Create a better answer key. Once we’ve committed to a visual representation of the task, it’ll satisfy nobody to read the answer in the back of the book. They’ll want to watch the answer.

Here’s the three-act page. Leave a response to see the entire lesson.

Show this video to students.

Ask them to write down a guess: will the sauce overflow? Ask them to guess how many meatballs it’ll take. Guess guess guess. It’s the cheapest, easiest thing I can do to get students interested in an answer and also bring them into the world of the task.

Ask them what information would be useful to know and how they would get it. Have them chat in groups about what’s important.

If they come back at you telling you they want the radius of the pot and the radius of the meatballs, push on that. Ask them how they’d get the radius. That’s tough. Is there an easier dimension to get?

Someone here may ask if the lip of the pot matters. It isn’t a perfect cylinder. Give that kid a lot of status for checking those kinds of assumptions. Tell her, “It may matter. It isn’t a perfect cylinder but modeling means asking, ‘Is it good enough?'”

Give them the information you have.

Let them struggle with it enough to realize what kind of help they’ll need. Then help them with the formula for cylinder and sphere volume. Do some worked examples.

Once they have their mathematical answer, have them recontextualize it. What are the units? If that lip matters, how many meatballs will it matter? Should you adjust your answer up or down?

Surprisingly close. The student who decided to add a couple of meatballs to her total on account of that lip is now looking really sharp.

Let’s not assume students are now fluent with these volume operations. Give them a pile of practice tasks next. Your textbook probably has a large set of them already written.

Help I Need

• Raise the ceiling on the task. My usual strategy of swapping the knowns and unknowns to create an extension task is failing me here. Watch what that looks like: “The chef adds 50 meatballs to a different pot and it overflows. Tell me about that pot and its sauce level.” I’m not proud of myself. Can you find me a better extension? I’ll give highest marks to extensions that build on the context we’ve already worked to set up (ie. don’t go running off to bowling balls and swimming pools) and that further develop the concept of volume of spheres and cylinders (ie. don’t go running off to cubes or frustums).

What You Did

Over on the blogs:

• Max Ray, Michelle Parker, and Terry Johanson are all inside my head.
• Ignacio Mancera poses a similar situation but suggests doing it live in the classroom. I don’t accept the premise that “real” always beats “digital” — there are costs and benefits to consider — but I think Ignacio and Beth have the right plan here. If you have the materials, do it their way.
• Scott McDaniel suggests changing the context from meatballs in sauce to ice cubes in an iced mocha because kids drink iced coffee but don’t make spaghetti. This introduces a pile of complications (like the non-spherical shape of the ice cubes and the non-cylindrical shape of the cup and the fact that the ice will float at the top of the cup) for unclear benefits. Time and again in this series I’ve tried to convince you that changing the context of a task does very little compared to the changes we can make to the task’s DNA. Does someone (Scott?) want to make the case that the following task is a significant improvement over the original?

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