Category Archives: MP.4

Weird Dice

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Last spring I did a fun and interesting 3D printing project with a group of 6th-8th grade students integrating geometry, probability, and statistics concepts. Here is an overview.

The Inspiration

Midway through the spring, our school was lucky enough to have a Makerbot Replicator 2 donated to our school. We had done a few projects over the previous two years with the original Makerbot, but I personally had never completed a build. So I decided to try it out, without necessarily thinking about how to apply it for my math classes.

The first thing I made was this, on Tinkercad, then printed on the Replicator 2.

I started with a hexagonal prism. Then I deleted 4 rectangular prisms from the hexagonal prism, as shown above. After rotating it on to its side, I printed it out as seen on the right above. Stunning, I know. But even this simple print did not come without several failures and hiccups.

3D printing requires a growth mindset

I won’t go into the details, as that is another kind of blog post, but suffice it to say that there are a lot of different things that can go wrong when trying to design and print a 3D object. Each failure brings a new understanding of the process. One way I celebrate those failures is with my Epic Fails Box. (I used to have more in there, it appears some students have been helping themselves to these trinkets)

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The First Questions That Came to my Mind

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Holding this object, made me immediately curious about the probabilities. So I labeled the two rectangular faces 1 and 10. Then I labeled the opposite faces so that they added up to 11. 

As luck had it, the next unit I was preparing for was probability. This had the added benefit of involving quite a bit of geometry in designing the die, and also with working out the surface area. 

So I decided to go all in, and do a Weird Dice Project with a group of 6th-8th students. I started by printing several copies of my die, and labeled the faces 1-10, then gave them out to groups of students to examine.

Here’s what my students noticed…

Less of a probability that is lands on 1 and 10.

Side 1 is slightly bigger then side 10.

Because side 10 is smaller than side 1, side 6 and 7 were bigger than all the other.

6,7,4,5 are bigger than all the others.

2,3,8,9 are bigger than 1 and 10.

Edges that don’t surround 1 and 10 form a square.

What they wondered…

Which side did you 3-d print it on?

Just because they are smaller does that effect the probability?

When it lands do you record what is on the top or what is on the bottom?

What is the probability of rolling 1 or 10?

Corners and edges being sharp, does that effect the probability?

What about the faces being rounded?

Is the probability the same for the numbers 1 and 10?

Does the shape of the side effect the probability?

Do the honey comb fillings effect the probability?

What are the dimensions of the die?

Is it weighted?

Experimental Probability vs. Theoretical Probability

We spent a class period working out the experimental probability (aka, rolling it a few hundred times), finding the the surface area of the sides, and creating a theoretical model of the probability based on the surface area.

Graphing the distribution for the experimental and theoretical model raised some more interesting questions. What is the purpose of a model? Does a model need to perfectly match the experimental data, or can it ever perfectly match the data? What other factors could be at play?

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Once students had sufficient experience working with the dice I made, I assigned them the task of making their own “weird dice,” and then analyzing the probability.

The Project

Students were asked to…

  • Design a die in Tinkercad, or another 3D modeling tool.
  • The die had to be “weird,” in that it cannot look like any known dice, and must have uneven/unknown probabilities.
  • Once students got approval on their design, we printed them out. Students then wrote numbers on the sides of their dice and did experimental trials.
  • They then had to use surface area as a way to model the probabilities.
  • Finally they were asked to compare the experimental probability vs. theoretical probability.

Below are some pictures of their printed dice.

I created a template, using data from my die, on google docs for them to use for their final report. Here is the link:

Weird Dice Report Template

Below are a couple examples of the final reports. Groups used a variety of methods to find the surface area. In the first example, the students had to account for the dice landing in between faces, and so they decided to find the surface area of the imaginary triangular face created by the concave sections of their die, and write a number on the four concave spaces.

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In this second example below, the students used the 3d modeling software to calculate the surface area of their irregular faces.

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In retrospect I should have had them graph both the Theoretical Probability as well as the Experimental, so that they could compare them side by side. Next iteration, I will do this.

Create a Game using Weird Dice

As a final extension, I suggested students make a game to play with their parent at the end of trimester portfolio day. 

Students used a variety of approaches, both in terms of their geometric designs, as well as calculating probability. The games students made were an interesting reflection of both their personalities, as well as the mathematical aspects of their die.

Here are a few examples:

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Low Tech Version

I fully recognize the privilege of having access to a 3D printer. One way to do this project without a 3D printer is to have the students design the dice, and then outsourcing the 3D printer. However, as a low-tech alternative you could use pattern blocks and tape to create weird dice, and do a similar investigation. In fact, I’m sure your students could come up with many other materials to use.

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Filed under 7.SP.5, 7.SP.6, 7.SP.7, Mathematical Investigations, Middle School, MP.4

When will I ever use math in life? A true story.

Here’s an answer to the classic question “when will I ever use math in the real world?” There are a lot of assumptions that have to be made, like in many problems involving math in our daily lives. And, perhaps most importantly, my solution to this math problem had a real impact on whether this would go down in my personal history as a good day or a bad day!

Prologue: The Setting

This summer I went down to LA to help watch my nephew Sebastian for 3 days. It was a lot of fun, but being the father of two boys myself I was excited to get back home to my kids in Redwood City. So on Wednesday, I packed my car ahead of time, and then went to pick up my sister after work at UCLA, in her car since she had the right size carseat for her son.

I finally got on the road around rush hour, but was happy to be heading north. Fast forward about an hour and a half later, and I was getting hungry. I stopped right before I-5 and the 99 junction, where I knew I could get the always reliable Chipotle Burrito. So I parked my car and reached for my wallet.

But my wallet was nowhere to be found! It took a few moments of retracing my steps to realize, I could last remember putting it in my sisters car when I drove to pick her up. After a few moments of mild panic, I thought through my options, and realized there were two simple choices. Go back to LA for my wallet or continue driving to Redwood City and have my sister mail me my wallet.

Obviously I didn’t want to turn back, but did I have enough gas? The following is my attempt at turning this adventure into a 3-Act Math lesson.

Act 1:  Go Back to Los Angeles for my Wallet or Continue Home?

Odometer and Gas Meter in Lebec

Above is a picture of my dashboard, I took it right after realizing I forgot my wallet. The last time I had filled up, I reset my trip odometer.

What would you do in this circumstance?

What information would you need to gather to make a decision?

Act 2: Gathering Information

Sitting in my car, I’m hungry and hot (its July in the Central Valley). I talk to my sister on the phone, asking her to double check if my wallet is in her car. She confirms, and I am faced with a dilemma.

I scrounged up all the money I could find in my car, every last coin, but all I could muster is $17.25 cents. Gas was $4.298 per gallon.

So I sat down to calculate whether it was possible to make it home with the money I had available.

Here are a few of the maps I looked at when making my decision.

Distance to Redwood City: 281 miles

 

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Distance to San Jose: 258 miles

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Distance to Gilroy: 227 miles

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Over the years I have gotten in the habit of tracking my mpg, by reseting the odometer each time I fill up my tank. So I knew that my mpg was approximately 22 miles per gallon.

I used that for my estimate, but there are a variety of methods students could use to get an MPG estimate. Here are the first two that come to mind. I wouldn’t just tell students though, as this would rob them of the opportunity to practice making reasonable assumptions, and/or finding relevant and reliable information on the internet.

  1. Google CR-V 2001 Manual Transmission MPG.
  2. The 2001 CR-V has a 15.2 Gallon Tank, and the picture shows I had driven 166 miles with about half a tank.

Summary of Information:

Available Cash: $17.25

Gas Cost: $4.298 per gallon

Approximate CR-V MPG: 22 mpg

Distance to Redwood City: 281 miles

Distance to San Jose: 258 miles

Distance to Gilroy: 227 miles

Act 3: I went for it!

My stomach yearns for the comfort of an oversized burrito, for a moment I consider just buying a burrito and heading back to LA for my wallet. But then I realize that I could use my Starbucks app (which is connected to Paypal) on my phone to buy a sandwich and some snacks, and still fill up my tank with the $17.25.

Here’s how much gas I was able to buy.

Gas Pump in Lebec, Ca

I was worried I wouldn’t quite make it, so I called up my coworker that lived in San Jose and asked if he could be my back up in case I ran out of gas before reaching home. He said he would be ok staying up until midnight, what a pal! I felt pretty sure I could make it at least to Gilroy. So I went for it!

Epilogue: How good were my assumptions?

So I got to Gilroy, before my gas light came on. I gave my coworker a call, and he told me to meet him at a gas station off the 101 in San Jose. He was able to pay for my gas, which I repaid him instantly using Paypal.

Here’s the picture of how much gas I pumped.

What do you wonder?

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Here’s some of the things I naturally wondered…

Could I have made it to Redwood City?

How far off were my estimates?

How did the assumptions I made create error in my estimates?

Did I really have exactly half a tank of gas before adding 4 gallons?

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Filed under 3-Act, 7.RP, 8.EE, Middle School, MP.1, MP.2, MP.4, MP.6, MP.7, MP.8