Category Archives: problem solving

Modeling Projectiles

My Functions for Modeling classes are ending on Tuesday. These are the introductory math classes at Baxter Academy. This course is paired with Modeling in Science, which has a focus on science inquiry and the physics of motion – kinematics. The final assessment is a ballistics lab, where the student groups have to measure the launch velocity of their projectile launcher and, along with some other measurements and a few guesses, identify the best launch angle and launch point to fire the projectile at a vertical target. The students are not allowed to have test shots or simulated shots. They are expected to gather the necessary data and complete all of their calculations before testing their theory with one, single shot. The target is fairly forgiving, but students are still amazed when their predictions result in a projectile going through the target.

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It’s all in how you ask the question

I teach half of an integrated math & science modeling class. On the math side, we focus on functions and a little bit of right triangle trigonometry. The science side is all about motion, one dimensional and two dimensional – hence the trigonometry. We’re now entering the final few days of the trimester, and have gotten into that 2D motion part. Did I mention that this is the introductory math/science class for 9th graders at Baxter Academy?

We started with Dan Meyer‘s Will It Hit the Hoop? concept, slightly modified. Showed Act 1 video, but captured this picture for analysis.

hoop shot-1

Interesting conversation begins. Many students are convinced that the ball will fall short of the hoop because “it is slowing down.” What makes them think that, I wondered. Maybe because up until this point, the conversation in science has been about constant velocity motion, in one dimension. Showed Act 3 of course and those who were sure the ball would go in were vindicated. But their comments still nagged at me. Maybe they just need more experiences – this was, after all, just the first day of 2D motion.

We watched part of an episode of Mythbusters, the one where they fire a bullet and drop a bullet and have them land in the same spot at the same time. It’s really a good episode. It really helps to drive home the fact that the forward motion of the bullet has nothing to do with how much time it takes to fall to the ground. It means that horizontal and vertical motion can be thought of, and modeled, independently of each other. On the science side of things they had developed the kinematics model: z(t)=\frac{1}{2}at^2+v_0t+z_0. So then we adapted that model for horizontal and vertical motion. We went back to the basketball shot. Analyzing the photo against the graph, we estimated the the initial position of the ball is at (1, 8) and the final position of the ball would be (19.75, 10). We also figured that the ball was in the air for about 1.8 seconds. From this information my students calculated the initial horizontal and initial vertical velocities to be 10.4 ft/s and 30 ft/s, respectively.

hoop shot

 

 

 

 

 

 

But Dan did not throw the ball only horizontally or only vertically. He threw it at an angle – so that it could reach the hoop, presumably. So I asked the question: “What was the launch velocity of the basketball?” and accompanied the question with this image:

hoop shot 2

 

 

 

 

Class was over, so I left them to work that out for homework.

Next day, I had them check in with each other and then asked, “How did you think about this problem?” Overwhelmingly, they agreed that the launch velocity must be the average of the horizontal and vertical velocities. This happened with both groups.  I asked them why they thought it should be the average. I asked if they thought the launch velocity should be greater than 30 ft/s, between 10.4 ft/s and 30 ft/s, or less than 10.4 ft/s. They were convinced that the launch velocity should be somewhere between 10.4 ft/s and 30 ft/s. Some thought that it should be closer to 30 ft/s since the ball is “going more up than over,” but that it would still be less than 30 ft/s.

A student in one of the classes convinced that group that it couldn’t be the average with the following reasoning: Suppose that the ball was thrown straight up. That means that the vertical velocity is 30 ft/s and the horizontal velocity is 0 ft/s. If the launch velocity is the average, then that would be 15 ft/s, but we know that the launch velocity is 30 ft/s. So it can’t be the average! So, of course I asked, “Then what could it be?” And they went with the idea that it must be the sum of the two velocities. But that would give us a launch velocity greater than 30 ft/s. We talked about this for a few minutes. They weren’t sure.

Then I showed them this picture:

hoop shot 3

 

 

 

 

Only after seeing this picture did they make any connection to a right triangle, or Pythagorean Theorem, or trigonometry. It had taken the better part of an hour to arrive at this conclusion, and then it took only 5 minutes to find the solution.

What would have happened if I had jumped directly to the right triangle representation? They would have had a quick solution, but they wouldn’t have had the opportunity to think about whether or not the launch velocity is the average of the horizontal and vertical. Maybe you think it was a waste of class time to allow my students to engage in such discussion. Maybe it was, but I don’t think so. My students had to take some time to construct meaning. They had to confront their misconception and convince themselves and each other that taking the average didn’t make sense. Sure, I could have told them, it would have been more efficient, but would that really have helped their understanding?

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Problem Solving with Algebra

That’s the name of one of the classes I’m teaching this term. We have trimesters. So each term is 12 weeks long and we have a week of “intersession” in between the terms. Except that this first term is not quite 12 weeks. The expansion area of the building wasn’t quite finished when we started school, so we had some alternative programming called “Baxter Foundations.” It included stuff like my Intro to Spreadsheets workshop. Classes started this week. And one of my classes is called Problem Solving with Algebra. I came up with that name, and I honestly don’t know exactly what it means. I have a rough idea, but it could go in a lot of different directions. Mostly, I want my students (and all the students taking this course) to think and puzzle and use algebra and solve problems.

Then I got an email that Jo Boaler has published a short paper called The Mathematics of Hope. In it she discusses the capacity of the human brain to change, rewire, and grow in a really short time based on challenging learning experiences. We’re not talking about learning experiences that are so challenging that they’re not attainable, but productive struggle. Challenging learning experiences that produce some struggle, but are achievable. The ones that make you feel really good when you solve them. You know the ones I mean.

So I decided to start this class with a bunch of patterns from Fawn Ngyuen‘s website visualpatterns.org. The kids are amazing. They jumped right in. Okay, so I taught most of them last year and they know me and what to expect from me, but seriously. Come up with some kind of formula to represent this pattern. Kinda vague, don’t you think? And I’m pushing them to come up with as many different formulas as they can, and connect those formulas to the visual representation. For example, an observation that each stage adds two cubes to the previous stage would result in a recursive formula like: C(n) = C(n-1) + 2 when C(1) = 1 (which is a recursive formula for pattern #1).

On Tuesday, different groups of students were assigned different patterns. Wednesday, each group presented what they were able to figure out. Some had really great explicit formulas, while others had really great recursive formulas. A few had both. Most were stumped at creating an explicit formula for pattern #5, pattern #7, and pattern #8.

Tuesday night, I received this email from Sam, a student:

“After staring at the problem for 2 hours, (5:45 to 7:45) and scribbling across the paper as well as two of my notebook pages, I am still unable to find a explicit equation. Then, reading the directions, I realized that the way they are worded allows the possibility of no explicit equation, as well as the fact that I only had to come up with equations as I can find. So after 2 hours, several google searches, lots of experimentation and angry muttering, I decided I have all that I can muster, and must ask you in the morning.”

I left them with the challenge to find an explicit formula related to one of these patterns. Their choice. Just put some thought into it before we meet again on Monday. Wednesday night, Sam sent me this followup email:

“After another hour at work, I found the explicit formula. I realized that the equation was quadratic, not exponential, and youtubed a how-to for quadratic formulas from tables. I kid you not, the man said the word “rectangle” and from that, I solved the problem. Then I watched the video through and took quick notes for future reference.”

Then, Dan Meyer posts this: Real work vs. Real world. Makes me think – as always. What am I asking of my students? This is real work – they are engaged and they are thinking. Sam, and the others, were not going to be defeated by a visual pattern. The fact that they are working in a “fake world” doesn’t matter.

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The Power of Interesting Questions

Today I led two groups of students through an introduction to spreadsheets as part of our Baxter Foundations workshops. Our framing question was, “How much is that Starbucks habit costing you?” Many students, of course, said $0, but we widened the question to include other vices, like Monster drinks, Red Bull, going across the street to Portland Pie every day, or down the street to Five Guys for lunch. And we broadened the question to, “What if you put your money into a retirement fund instead?” To make this real for my students, my friend Tracy admitted to her Starbucks habit and offered to be our real case study.

Before we started creating anything, I asked the students to complete this quick survey to figure out what they knew and what they didn’t. Then we looked at the results as a group. Here’s what we found:

Group 1: Mostly sophomores

Group 1: Mostly sophomores

grp2-1

Group 2: All freshmen

grp1-2

Group 1: Mostly sophomores

grp2-2

Group 2: All freshmen

grp1-3

Group 1: Mostly sophomores

grp2-3

Group 2: all freshmen

Clearly, the sophomores were bringing more to the table than the freshmen. After all, they had been instructed in spreadsheets in their engineering class last year, but they were still a bit unsure of what they knew. They thought they probably knew more than they had indicated, but didn’t know what I meant by “cell reference,” for example. And remember, I teach in Maine where 7th graders are given their own digital device. It used to be a laptop, but last year many districts changed to iPads. I would have expected the 9th graders to have had much more experience with spreadsheets, but I’m seeing that the switch to iPads is having an impact on that. Very sad.

I began by explaining the situation: Tracy spends $x each day on her Grande Soy Chai at Starbucks. If we want to figure out how much she spending, and what she could be earning instead, what information do we need? And then I had them brainstorm for a couple of minutes.

Information needed: cost of the drink, how much spent each month, and interest rate for the investment.

We made a few assumptions:

  • Tracy could find a mutual fund, or other investment, that earns an average of 7% annually
  • that she is 25 years from retiring (I don’t actually know this)
  • that the price of coffee would not change over the life of the investments (we knew this was unreasonable)
  • that Tracy would invest the same monthly amount for the life of the investment (also unlikely)

But this is also part of problem solving. Take a few minutes to watch Randall Monroe’s TED Talk and you’ll understand what I mean.

So here’s the spreadsheet that we came up with.

So what did we learn?

  • Tracy spends a lot of money on her Grande Soy Chai. But, it’s possible that the drink adds some value to her life and is worth the price.
  • Investing early and for a long time really can pay off, even if the amount invested isn’t all that much each month.
  • Learning about spreadsheets can be fun if you have an interesting question to answer.

Do I think the students in this 90-minute workshop will remember everything that we discussed? Of course not – I’ve been doing this job way too long to think that. But here’s the beauty of it all – they have their own model to reference, be it Google or Excel, they all created one and can take another look at any time. I heard from another teacher that a couple of his advisory kids started talking about making their own coffee instead. A couple of my advisory students commented on the experience at the end of the day. One said, “It was interesting to see how the numbers involved in the Starbucks added up if invested in a retirement fund. The actual application was nice.” Another said, “The spreadsheet exercise this morning was fun. I think it was the funnest way to learn how to do a spreadsheet I have ever done. So thank you.”

You’re welcome.

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What Time Will the Sun Rise?

This week I begin Exploring the MathTwitterBlogosphere. I’m looking forward to these missions and challenges because I need someone pushing me to find the time to write in this blog. It’s good for me. Like spinach.

This week’s mission: What is one of your favorite open-ended/rich problems? How do you use it in your classroom?

One of my favorite open-ended/rich problems comes at the end of a unit on trigonometric functions. After exploring, transforming, and applying trig functions to Ferris wheels, tides, pendulums, sound waves, … I assess my students’ understanding by giving them some almanac data of  sunrise and sunset times for a specific location on Earth. Their job is to analyze the data and create a trig function to model either sunrise times, sunset times, or hours of daylight – their choice.

The data looks like this

and that makes it somewhat challenging for students to even begin. They are reminded that they should have “enough” data to know if the model they develop fits well. I point out that the times are given to them in hours and minutes, but that they probably want a single unit (hours or minutes after midnight). From there, they are on their own to solve the problem. Usually, they work with a partner.

In the classes that I’ve used this task with, we’ve modified the amplitude, period, and midline of the sine and cosine functions. We haven’t introduced phase shift, yet. So, there is also a reminder about selecting a convenient “Day 0” for the function they choose to model with.

What I love about this task:

  • Students are talking math, asking each other about the number of data points they should use: “Should we just pick the same day every month? Are 12 data points enough?” or “Do we just go every 20th day?” or “What should we use for the first day?”
  • Students are problem solving. They have to convert the times into a single unit. They have to make decisions about which variable to model, when to start, which type of model to use. Then, they can collect the relevant information to modify their chosen function.
  • Students are using technology. Although they don’t have to, it’s really easiest to have the kids making scatterplots on calculators or computers and then graphing their model on top of that. Then they have a built in way to check their work – they don’t have to ask me (the teacher) if they are correct. It shows up in the picture that they create.
  • Students think that working with trig models is really hard, so they feel very proud when they are able to complete this task without any help from the teacher.
  • It’s really easy to grade. Either the model fits or it doesn’t. Kids turn in their data tables and work showing how they calculated the necessary values for their model. This precludes anyone from using the old SinReg command.
  • Even though I’ve used this task for about ten years, it’s a perfect fit with the Common Core math standards (trigonometric functions) and practices. And since I live in a SBG world, this is a very good thing.

My favorite kind of assessment is one where students have to apply what they’ve learned to a different situation. Even though we create lots of different trig models in class, sunrise, sunset, and daylight hours represent a new application. And a new challenge.

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Spreadsheets – Part 2

As a reminder, here’s the original project descriptor.

Here’s a report from one pair of students.

Mistakes made: Originally we had started out with a more expensive, brand new car. Our budget was about $100 a month to put towards a car, so it would have taken us 20 years to pay off the car, which is unreasonable. We had to downsize and settle for a less luxurious car, but one that still met almost all of our standards.

Assumptions:

      • 4 weeks is a month.
      • We have enough money saved for the down payment.
      • We take 10 years to pay off the loan
      • annual rate of 2.29% for loan

Mid-size Sedan

2. We have $100 left for a monthly car payment. Income- expenses. 3400-3300=100

3. Typically for a down-payment you would need about 11% of what the car is worth. So for our car that is worth $11,000 we have about $1,210 saved for the downpayment.

4. The requirements that our car has to meet are 4 door, 30 mpg, mid-size, seats 5.

5. We would want our car to have a working air conditioner, heat, sunroof and radio.

6. We decided to get a used Mazda6i Touring for $11,000 but with 10% off it would cost $9,900.

7. We would need to borrow $9,900 and we would get this money by taking out a loan with Bank of America.

8. (On Spreadsheet)

spreadsheet

9. Our dream car was originally a new fully equipped midsize sedan with a sunroof, but after we found that with our budget of $100 dollars a month, it would take 20 years to pay off. So we decided to get a used car. Mazda6 i Touring which still had air conditioning and heat but did not have a sunroof sadly. This car still meets most of our requirements and was much more affordable so we would be able to pay it off in 10 years.

10. Like we said in number 9, one of our problems before was a 20 year long loan. We fixed this problem by selecting a cheaper used car to buy. If the payment stays the same ($100) and the down payment is the same (11%) then the more expensive the car, the longer the duration of a loan.

https://www.google.com/search?q=Stretch+hummer+limo&rlz=1C1CHFA_enUS497US497&sugexp

11. Our dream car is a stretch hummer limo. The cost of this car brand new is $300,000 after the down payment of 11% costs $267,000. This car would be impossible for us to pay off because the interest that we would have to pay is more than we make monthly so the payment would keep increasing and we would never be able to pay it off.

12. To pay off our dream car in six years we would have to earn about $7320 a month. So subtracting expenses that leaves about $4025 a month to put into the dream car, which will pay it off in 72 months. Assuming that the interest rate and bargaining rate are the same.

What I like about this solution:

  • They stated their assumptions.
  • They made decisions.
  • They made adjustments.
  • They analyzed their results.
  • They dreamed big.
  • They used absolute addressing as part of their spreadsheet formulas.
  • They knew the difference between an annual interest rate and a monthly interest rate.
  • They understood that if their payment doesn’t even cover the monthly interest, they’ll never pay off the loan.

There were several solutions like this one. Not enough, though. Something to think about next year.

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Authentic Assessment?

My junior level math classes have begun working on a project called Buying a Car. For the past few classes we’ve been problem solving using spreadsheets. They’ve been working in teams, using Google spreadsheets to solve problems like this and this (which I adapted from our Core-Plus Mathematics text). My teaching colleague and I decided to jump into this spreadsheet mini-unit before our students had to turn in their laptops for the year. (We are a one-to-one school.)

Here are some pictures of my students hard at work.

photo7  photo6

photo3

photo2

Some things I heard as the students were working:

  • Oh, so the bank pays for the car and then you pay the bank. I get it now!
  • How much does gas cost right now?
  • Where’s the best place to get the loan from? What’s the lowest interest rate we can get?
  • Are we going to buy that truck? What’s the gas mileage on it?
  • How do we figure out the payment? What did we do before?
  • So we have to add the interest and then subtract the payment.
  • We can cut back on the money for entertainment. We can be cheap. There’s only two of us, we don’t need that much food. It’s not like we’re feeding any children.
  • How do we determine how much for a downpayment?
  • Can we afford a monthly payment of $875?

Here’s what I really like about this assessment (having never done it before):

  • There is a high degree of choice.
  • There isn’t a definitive solution.
  • Students have to make (and state) some assumptions in order to solve the problem.
  • They have to think about lots of things that go into a household budget and buying a car.
  • Students working together and helping each other to succeed.

What I’m not so sure about:

  • The quality of their results.
  • If they’ll really apply what they’ve learned during the past 4 classes learning about spreadsheets.
  • How much understanding they’ll walk away with.

It will be interesting to see what they produce as a result.

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