## Weird Dice

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.

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)

The First Questions That Came to my Mind

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?

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

• 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.

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

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:

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.

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?

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

Distance to San Jose: 258 miles

Distance to Gilroy: 227 miles

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.

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?

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?

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

## Number Bracelets

“Doing mathematics should always mean finding patterns and crafting beautiful and meaningful explanations.” -Paul Lockhart, A Mathematician’s Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form.

Recently I got a chance to revisit one of my favorite activities with a fresh group of 3rd and 4th graders. I love this activity because it includes so many of the things I value. To borrow some of Dan Meyer’s Ten Design Principles for Engaging Math Tasks; perplexity is paramount, it begins with a concise description, has a low floor for entry and a high ceiling for exit, it naturally encourages a collaborative approach, it highlights the limits of students existing skills and knowledge, and students often begin making predictions and hypothesizing without any directions to do so. In summary, this task rocks!

This was the first major mathematical investigation of this year for my 3rd/4th multigrade class. I dedicated the first two block periods (2 hours each) to this investigation. Skills-wise, at its most basic level, students are practicing their single digit addition facts, as well as breaking apart the tens-place and ones-place of a sum. This serves as a perfect activity before reviewing/solidifying multi-digit addition algorithms. There are a variety of ways to extend the investigation, for example using other bases or operations. Also, once students begin to try to figure out what’s going on with all the patterns, they utilize virtually all the Mathematical Practices in a meaningful way.

The Game Intro

I introduce the number bracelets with a game. Students pair up and take a 10-sided die. First player rolls twice, writing the two number down on a piece of inch grid paper. The second player does the same. Then students add the two numbers together, but discard the tens-place, only writing the ones-place in the next number. Players continue this process until they get back to the two numbers they started with (in the same order). If students get into a loop, that means they made a mistake somewhere. What’s cool about those mistakes, is that they hint at the overall structure of the number bracelets, so the mistakes are truly valuable.

Within minutes students start to notice patterns, and making predictions. I have done this several times over the years, and each time I see new patterns myself. After about 20 minutes, I bring everyone back in for a whole class discussion. I ask them “What did you notice?” and “What do you wonder?”

Here is a list of some of the responses, compiled from two groups of 3rd/4th graders:

What do you notice?

I had 4 sequences that were the same length.

When I kept going past the first two numbers, I got into a loop, it was a pattern.

I notice that #0## comes up a lot, like 2022.

I can’t get started w/ (0,0)

Its fibonacci, starting w/random numbers.

There are patterns vertically and horizontally.

If you mess up one number, you can go in a loop, it affects everything else.

What do you wonder?

What happens if you get a zero as one of your first two numbers? How long will it be?

What happens if you get (1,2)?

I think for double numbers, 5,5 would be the shortest.

Could you make a sequence that doubled each time?

What happens when you get into an infinite loop?

What would the shortest one be, other than (0,0)? How can you be sure? How many are there to check?

Why do the zeros line up horizontally sometimes?

How is it that it repeats? How can you get out of a loop?

Why is it called number bracelets game?

What’s the longest pattern?

Are double numbers best?

Can you have a pattern that never repeats?

Number Bracelets Investigation

After a snack recess, students returned with the goal of investigating one of these questions. I asked them to choose a question to investigate, and work individually or in small groups (2-3 students). We brought out some markers, and colored pencils, so students could color code the patterns they noticed. When a student made a discovery, or had a prediction, I would ask them to write it down in their math notebook. For predictions, I asked them how they could verify this prediction?

In this first session, I tried not to steer their inquiry very much. I focused more on observing their style of inquiry, what they are noticing, how they communicate and record their findings, and how they work with their peers.

Here are some of the patterns they found.

Here’s an example from a student that had a question, and did a few experiments to answer the question. The classroom was abuzz with kids noticing patterns, making predictions, and describing their theories. If I hadn’t stopped them for lunch, they probably would have kept working through their recess. In fact, over the next few days many of them did stay in during their recess to explore further.

Week 2:

The following week we returned to this investigation to see what else we could draw out, focusing on the following question.

How many pairs of starting numbers can there be?

I first had students work in their notebooks on the question. Then had them share their findings with other students at their table. Finally we had a whole class discussion. Students came up with varying answers, including 20, 180, 100. We talked about the various methods students used, and they debated the merits of each method. I was pleasantly surprised with the amount of debate that ensued, with students changing each other’s minds.

As snack recess approached, I decided to introduce a couple classic problem solving strategies. The first strategy is to try to solve a simpler version of the problem. So what if we had fewer possible numbers to choose from, how many possible pairs would their be. We started by considering how many pairs would exist if you only had zero to work with. Then we looked at if you could use {0,1}, and then {0,1,2}. Enumerating the possibilities started to get complicated, so I introduced a second problem solving strategy: “Draw a visual model.”

The visual representation of this is a beautiful model from graph theory called a “complete bipartite graph.” This really seemed to solidify the pattern for the students.

One student then noticed something that she could hardly contain, with the classic “uu, uu, uu!” raising her hand. She shared “The first case was 1×1, then 2×2, then 3×3!”

“That’s right, so what would come next in that pattern?”

Using this model, students were able to come to an agreement that there are 100 possible starting pairs. To wrap up this discussion, we reviewed the process we went through to organize and model the information we had to answer the question.

Research Project Poster

Next, I wanted to give them a chance to explore a bit more, but with a focus on the possible lengths that a number bracelet could be. So we placed them in small groups of 2,3 students and had them focus on the following questions.

What are the different lengths a number bracelet can be?

What is the longest/shortest bracelet, and how can you be sure?

The poster needed to have examples of bracelets as well as written explanation for their reasoning. Here’s how one group began their poster:

I was hoping that students would notice that each of the different lengths contain multiple number bracelets. One group realized this for the bracelet that is 60 long (62 if you include the starting number twice).

Reflections and Extensions

Each time I’ve done this investigation, I play around a bit with how much coaching I give towards a more complete understanding of the puzzles. If time permitted, I would love to give kids as much time as they needed to fully explore the root of the patterns they find. Unfortunately, I have typically been limited to 2 or 3 block periods, which does not seem to be enough for my 3rd/4th graders to independently come up with complete explanations for the patterns they find. I still see the process as a meaningful one, despite students not fully understanding all the implications of the patterns. I think this reflects what happens often in real mathematical research and real world problem solving, when you explore a pattern to the extent you have time to, and then you move on.

As for extensions, I would ask the students how you could change the rules and see what they notice and wonder. For example what if instead of using addition modulo 10, you could try addition modulo 5, or some other modular arithmetic. Another way you could change the rules is by using an operation other than addition, and see if there are cycles.

## The Thrill of Proof in the Early Grades

What kinds of problems allow young students an opportunity to experience the, dare I say it, exhilaration of proof? Certainly there are several classics, which can give young students a sense of the beauty, elegance, and thrill of discovering a proof. The Seven Bridges of Konigsberg is a classic problem many elementary school students can figure out.

Last year, while working with a group of gifted 2nd graders, I presented an interesting problem that appeared in a New York Times NumberPlay article last April.

The puzzle asks which numbers can you draw a square made up of that many squares. The squares need not be the same size. See the article for a few examples, or read on to see the results from my students.

This group met each Friday for 30 minutes. At the end of the first session, most agreed that 2, 3 and maybe 5 were not possible. So I asked them to try to find other “squareable” numbers at home, and bring the results in for the next week. Students spent the following two Fridays investigating which numbers were squareable.

Finally by the fourth Friday we had a strong feeling that only 2, 3, and 5 were impossible. I asked them to work together to demonstrate which of the first 25 numbers were squareable. By the end of that session we had the following chart, each student participated in the creation of one or more “squareable” numbers.

In the final session, I showed them the chart and asked them “so what about the next five numbers, are they ‘squareable’?”

Students quickly started raising their hands to explain how you could make 26, 27, 28, 29 and 30 using a pair of “operations”.

Students proceeded to explain two ways of adding three squares, they dubbed these ways the “banana split” operation and the “gigantor” operation.

One student offered her “banana split” operation which cuts an existing square into four quarters (each a square).

Banana Split operation on 4, to make 7.

One student directed our attention to 23, which is basically a 20 with three squares added to make a square four times the area. We called this the “gigantor” operation to formalize it, and decided we could do this with any square we’ve made to show the “squareability” of that number plus three.

“Gigantor Operation” on 20, to make 23

Whole Class Discussion:

What follows is roughly how the conversation went:

me: “Great, now what about 30 – 35, and the next 5, etc?”

students: Yes, they are all possible!

me: “At what point can we tell that the rest of the numbers will be squareable?”

students: “What do you mean?”

me: “When is the earliest number you could stop, and build the rest using your ‘banana split’ or ‘gigantor’ operations?”

students: “Eight!”

me: “why eight?”

students: “because using the banana split operation you could build 9 from 6, 10 from 7, and 11 from 8, and so on”

me: “Awesome! lets record your proofs on paper!”

students: “huh?”

So once I felt confident that they all understood this reasoning, I asked them to write their explanations down on paper. Most struggled to put it into words, many just made diagrams with numbers. Perhaps it was the limited amount of time, or that they were just not ready or practiced in this kind of written task. Unfortunately our group didn’t meet for a few weeks after that, and the kids were ready to move on to another problem. Next time I will make sure to set aside more time to support their emerging mathematical writing skills.

While the proof that 2, 3, and 5 are impossible requires quite a bit more experience. These young students were able to come up with an explanation for why there are an infinite set of “squareable” numbers, starting at most 6. That was thrilling for me to watch, I could see the excitement in their eyes as they had the “AHA!” moment. What’s more thrilling (in mathematics) than coming up with a simple and elegant argument that applies to a infinite set of number?!

We followed it up with a “Coin Problem,” which involves similar reasoning to solve. This provides a natural opportunity for students to utilize Mathematical Practice 8 “Look for and Express Regularity in Repeated Reasoning.” This generalized problem can get quite complicated, but the basic example is accessible to early elementary. Here is what I started them with.

“If you had an unlimited number of 3 cent and 5 cent coins, which values could you make? Which values would be impossible?”

Here is the graphic organizer I gave them: Coin Problem

Try this one out for yourself, and see if you can figure out where students could employ a similar kind of reasoning as in the squareable number puzzle.

Filed under Elementary, MP.1, MP.2, MP.3, MP.7, MP.8, Proof

## Going Solar in 3 Acts

Act One: Ivanpah Solar Electric Generating Facility

Possible Questions:

What percentage of California homes could be powered by this power plant?

How many of these would we need to power a third of California homes?

How much area would that many power plants take up?

Could we fit them in the Mojave?

How much would it cost?

Act Two: What percentage of California homes could be powered by this plant?

What information do we need?

California Homes: 13,720,462 housing units

(source: Census 2010)

Extension question:

What percentage of US homes could this power?

US Housing Units: 132,312,404 (Census 2010)

I allow students to choose to use the 2010 housing data, or estimate the number of housing units as 14,000,000. The Calculations would look like one of these:

140,000/13,720,462= 1.02 %

140,000/14,000,000= 1%

This usually leads naturally into several sequel questions. Below I have listed some of my favorites with the necessary information.

Sequel: The Implications

1. How many would we need to power all California homes?

2. How much land would we need to build that many Solar Facility?

Ivanpah area: 3500 acres or 5.469 square miles

3. How much would it cost to build that many Solar Facilities?

Cost of Ivanpah: \$2.2 billion (One might expect price to go down as you scale up, so you could ask students to model the price going down in some way over time.)

4. If you were to place these side by side, what would a few possible rectangular dimensions for the facility be? What would the dimension of a square with that area be?

5. Is there enough space in the Mojave for such a facility?

I had the students look on google earth to find an area that is big enough for all those Heliostats. Here is a shot from google earth of the actual Ivanpah Facility. As a hint, I told them to start there, and look nearby.

6. Approximately what percent of California’s total electricity requirements will this supply?

Ivanpah projected to generate: 1,079,232 Megawatt-hour(MGh) per year or 1,079.232 gigawatt-hours (source: NREL)

In 2012, Total System Power for California: 302,000 gigawatt-hours (GWh)”

(Source: California Energy Commission)

Filed under 3-Act, 6.RP, 7.G, 7.RP, Middle School

## Revisiting Multiplication Tic-Tac-Toe: Common Factors and Multiples

After reading about Ultimate Tic Tac Toe, I was inspired to post a bit more about Multiplication Tic-Tac-Toe. Multiplication TTT offers similar constraints to Ultimate tic-tac-toe, but the constraints are tied to the common factors. Check out my earlier post for rules and notes about classroom use.

One main difference though, is that you can claim more than one square at a time. Below I have shared a filled out board, color coded by the amount of numbers you claim with that move. For example, if you place a token on 6 and 4 you in fact get 4 positions on the Tic-Tac-Toe board as 24 shows up 4 times. In the image below, blue represents “quadruple plays”, yellow’s represent “triple plays”, oranges represent “double plays”, and white represents “single plays.”

Here is a google doc of the Color Coded Multiplication Tic-Tac-Toe Board.

Recently I introduced the game to my 3rd and 4th graders. I mentioned you could analyze the game based on how many positions each number occupies. Then, the other day I was walking down the hallway and saw a multiplication tic-tac-toe board on the floor next to their cubbies. The cool thing was, the student had begun to color code the board herself (I have a feeling I know which student it was). This kind of investigation could naturally lead students towards questions related to common factors and multiples.

One of these days, when I have the time, I’d like to figure out the optimal strategy for the game. If anyone out there discovers it, please let me know.

## How Safe is a Tesla?

Act One: Tesla Fire

What is more likely, a fire in a conventional gasoline car or a Tesla?

Act Two: Car Fire Numbers

Conventional Gasoline Car Fires:

• 150,000 Car Fires per year
• 3 trillion miles driven by American’s per year

Tesla Fires:

• 1 Tesla Fire
• 100 million miles driven by Tesla cars

Act Three: Elon’s Response

Elon Musk Claims the following in this Blog post.

“The nationwide driving statistics make this very clear: there are 150,000 car fires per year according to the National Fire Protection Association, and Americans drive about 3 trillion miles per year according to the Department of Transportation. That equates to 1 vehicle fire for every 20 million miles driven, compared to 1 fire in over 100 million miles for Tesla. This means you are 5 times more likely to experience a fire in a conventional gasoline car than a Tesla!”

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Filed under 3-Act, 7.RP, 8.EE, Middle School