# Tag Archives: Middle School

## 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

## Number Bowling

This is a student favorite, simple to start, differentiated, with various levels of success. I’ve done this with grades 2 and up, but my 5th-6th graders have gotten the most out of it.

Most importantly students learn to be very clear in writing their expressions. This activity naturally leads to a discussion about order of operations and mathematical “grammar.” Also, in their pursuit of the strike, students often ask to be introduced to new operations.

Instructions: Begin by rolling a die three times and recording these numbers as your “1st bowl.” You may cross out (knock down) any number that you can write an equation for using those three numbers each only once. For example, if I rolled a 6, 6 and 5, I could knock down the number four by writing: 5=6-6+5, or knock down 1 by writing 1=(6÷6)^5. Notice I used all three numbers, but each only once.

The goal is to knock as many “pins” (numbers) as you can. Knocking all the pins down on your first bowl is called a “Strike.” If you can’t think of any other equations, you may bowl again and try for a “Spare.”

Notes on Implementation:

I’ve created this Number Bowling handout for students to keep track of their games. I’ve also experimented with keeping score, and I think three frames is a good length time for a game. This can get complicated though, since scoring bowling is foreign and not straightforward to many students.

This year, after a few games, I had students write down their “favorite” equations on a notecard. We used these equations during a “strategy session” where we came up with tricks to help knock down more pins. The “tricks” can all be described as using an operation to change a number, or two numbers, into another. For example, 3 can be changed into 6 by using factorial, (3!=6). Some more advanced tricks include using square root, and/or the floor and ceiling function (rounding up or down to nearest integer).

Extensions:

Last year students wondered whether a strike was possible for every combinations. We chose the brute force method of proof :). First students had to figure out how many unique outcomes were possible with 3 dice. For many I assisted them by having them look at this pdf. Obviously rolling 1,6,6 is the same as rolling 6,6,1.  Then whenever a student achieved a strike, we crossed that off the list.