Category Archives: Middle School

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

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 about ½, or all?

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?

Homes powered by Ivanpah: 140,000

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)

Act Three: The Percentage of California Homes powered by Ivanpah

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.

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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)

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

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

Fun with Collatz Conjecture

Collatz Formula

You may have heard of Collatz Conjecture, it’s simple enough to explain to a 2nd grader, yet has stumped mathematicians for the last 80 years. Paul Erdos famously referred to it when he said “Mathematics is not yet ready for such problems.” I like exposing students to unsolved problems in mathematics, because it gives them a real sense for what mathematicians do (also their’s no pressure to solve it).

I have explored it with a variety of ages, last year I had a fruitful experience with my 5th-6th grade class and thought others would enjoy the investigation. Below you will find two ways to introduce the conjecture, and a method for reversing the recursive formula to “grow the Collatz tree.” This investigation also hits a few of the Expressions and Equations standards, such as 6.EE.A.1-2, and 6.EE.B.5.

Collatz Conjecture:

1. Take any natural number n.

2. If its even divide by two, if its odd multiply by three and add one.

Repeat step 2 indefinitely. The conjecture states that you will always reach 1 eventually.

For example, say you start with 5 your sequence would go 5-16-8-4-2-1.

Introduction:

The Game Intro: One way I’ve introduced it in the past is to have the students play a game where they roll a 10 sided die for a starting number. Then the person who gets to one in the most number of steps wins. Students quickly realize they can create a diagram, or tree network, which tells them right away who will win. Sometimes we extend it, by playing the game with 20-sided die and trying to solve that version as well. Here’s what the tree network might look like for the game with a ten sided die.

Collatz Tree for a game with ten sided dice

Collatz Tree for a game with ten-sided dice

The Human Tree Intro: Another way to introduce it that is more kinesthetic, is to give each student a number on a sticky note, name tag, or index card. Then say if your number is even, look for the person that is half your number and if your number is odd find the person with one more than triple your number. Once they find their person they can connect with them by holding their hand or putting their hand on their shoulder. I usually make a stack starting with one, going up the “Collatz Tree” until I have enough numbers for the amount of students and teachers in the group. I also try to make it so there are several branchings. When everyone is “connected,” the group should work to untangle themselves so that the overall structure can be seen. I wish I had a picture of the process, and outcome, but alas I couldn’t find one.

Growing the Collatz Tree (doubling)

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Once students get familiar with the structure of the Collatz tree we talk a bit about what it would mean for the conjecture to be true. Can we prove that a certain set of numbers will always go to one. Quickly students discover that as soon as you hit a power of 2, you are dividing by two all the way down. I like calling this the “tower of powers.”

We use the powers of 2 as the “trunk” of our Collatz tree, and the first example of a way to grow the Collatz Tree. You can reverse the recursive formula and double any number to “grow a branch” of the tree. For example, 5 could have come from 10, which could have come from 20, which could have come from 40, etc.

Branching Rule (n-1)/3

Then I ask the students to notice when there are “branchings” in the tree, in other words when are there two numbers that will lead to the same number such as 5 and 32, both lead to 16. Here students have to think about when (x-1)/3 will have an integer solution. This happens when a number is one more than a multiple of three. So we reviewed the divisibility rule for 3, and tried it out on a few examples.

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Students worked in groups to apply these two methods to grow the Collatz tree. There are lots of opportunities to differentiate the process, as students noticed patterns in branchings and some wrote algebraic expressions to describe the branches. For example, the 3-6-12… branch above can be described as 3*2^n.

Here is an example of what one group made:

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Extensions and Connections

1. If you extend the recursive formula into the complex plane you get the fractal below. For a thorough explanation check out the blog post by Nathaniel Johnston.

collatz fractal

2. Dan Finkel over at mathforlove.com has a variation called “The Dr Squares Puzzle” where they come up with a similar recursive process, with a few loops.

3. Here’s another recursive process involving numbers and their written form (It also has a tree structure):

Step one: choose a natural number N

Step two: write the number in words, count the number of letters in the word and write that number.

Step 3: Repeat Step two.

For example: One-3-Three-5-Five-4-Four… stays at four indefinitely. You can also try other languages. 🙂

For English it has a tree like structure.

For English it has a tree like structure.

Finally, a post on Collatz Conjecture wouldn’t be complete without this from XKCD:

collatz_conjecture xkcd

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Filed under 6.EE, Elementary, Graph Theory, Mathematical Investigations, Middle School, Unsolved Problem

Brewster’s Millions Beta

I’m going to try this today, as a sequel to A Billion Nickels. I’m considering it in Beta, and hoping my awesome students can help concretize and enrich the task.

Act 1: $30 million in 30 Days

Possible Questions:

How much does he need to spend per day?

How do you spend money without building Assets?

Does he do it?

Can it be done?

What would 30 million 1985 dollars be be adjusted for inflation?

Act 2: Take the Wimp Clause, or go for the $300 Million?

Students work in groups to figure out how they could spend $30 million in 30 days without accumulating assets. Using the internet and recording their plan on google spreadsheet.

Act 3: What does he do, does he win the $300 million?

“He Mailed It!”

Sequel:

Changemaker Twist

Since our school aims to educate “change makers”, I wanted to put a twist on this plot line and ask students to think about how they could use $30 million to help people.

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Filed under 3-Act, Middle School

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.

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Filed under Games, Middle School