Math Fireworks

February 6, 2016

Snap Hotel and Isometric Renderings

Tyler 5th, 6th, 7th, Content, Estimated Grade Level, Geometry, Geometry, Measurement Project, Snap Hotel, Unifix Cubes 0 Comments

I am few days into my Snap Hotel Project and it is time for an update. This post is going to be a quick smattering of latte-enhanced thoughts. In an attempt to seek your forgiveness, I am going to throw in plenty of pictures.

What We Have Done So Far

Step 1 (45 minutes) - I passed out the rules and regulations document to the students and gave them about 45 minutes to play around and wrap their head around the rules and the goals. They worked individually and had no idea they would eventually be in groups. Most students missed a few rules, but that was okay at this point. They focused on developing hotels that would have lots of valuable rooms. The single cube towers became a popular strategy, but most kids got confused by the logistics of block and land costs.

Step 2 (30 minutes) - I asked each student to construct a hotel of 25 to 30 cubes and calculate its profit or loss. This was eye-opening for lots of the students. With only 25-30 cubes it is hard to make a profit, so a number of students rapidly refined their designs. Many ended up owing me money. At this point, it was helpful to review the block and land pricing with the whole class. As students worked, I looked over shoulders to make sure they were calculating their profit correctly.

Step 3 (30 minutes) - Many of my students have mentioned that they really struggle to visualize and draw 3D shapes. One of my requirements for this project is to create a rendering of the hotel. So, I spent about 30 minutes introducing my students to drawing cubes and shapes on isometric graph paper. I showed them the basics of how it works, then gave them 4 simple models to try to represent. Abilities varied wildly, but everyone finished. Students who finished early either worked on their hotels or made a sculpture and tried to render it.

Step 4 (??? minutes) - I informed students of their groups and let them work on designing their hotels as a team. This is where I am now, I am not sure how long this will take. I want to give them enough time to test multiple ideas and optimize, but I also know that they could work on this forever. I am going to feel it out as I go, but I expect this will take them 3 to 5 45-minute class periods.

Homework (4 nights) - Part of the requirements of this assignment is to complete a budget in google sheets. I set up a template, but they have to fill in the data and write equations. To support them in this, I made 2 screencasts introducing them to the basics of spreadsheets and then gave them some data to work with. The videos covered: cell addresses, types of cell data, writing formulas using numbers, writing general formulas using cell addresses, copying formulas so they automatically adjust, and the following functions: =sum, =count, =max, =sumif, and =countif.

The students LOVED this work. I may have created spreadsheet monsters!

Observations

This Geometry unit has been a good reminder that young students do not have a ton of experience optimizing. We need more of that next year
This year I have been having students work independently prior to pairing or grouping them and it has been really effective
Students do not get enough experience rendering. They got lots of drawing practice in art classes, but, in my opinion, not enough experience with design and rendering
My students had great questions about real estate and building laws. Got lots of questions about whether their family could build a sky scrapper right where their current home sits
Some students are going to try to make a model of their hotels in Minecraft, could be interesting

January 31, 2016

Snap Hotel

Tyler 5th, 6th, 7th, Content, Geometry, Lessons, Measurement 3D, Andrew Stadel, Fawn Nguyen, Lesson, NCTM, Project, Snap Hotel, Unifix Cubes 0 Comments

Sometime ago I stumbled on an amazing lesson plan from Fawn Nguyen where students design a hotel using unifix cubes.^[6]NCTM loved this lesson so much that they put it on their illuminations website. Also, it sounds like Andrew Stadel has been involved with this, so I want to give him his props as well. I loved the idea of the lesson, but it didn't really overlap with the content I was covering in fifth grade. Now that I am teaching sixth grade and the content I cover aligns with the project, the lesson is on!

Background Knowledge

My sixth graders are finishing up a unit on 2D and 3D measurement. Here's a quick hit list of what was covered earlier in the unit: units; iteration; area and perimeter of rectangles, triangles, parallelograms, and different polygons that can be decomposed; measuring angles with angle rulers; interior angle sums; exterior angle sums; non-polygons vs. simple polygons vs. complex polygons; concave vs. convex; regular vs. irregular; surface area; volume; nets; polyhedra; prisms; and pyramids. Outside of this unit they have covered fraction operations, but not operations with decimals, percents, or integers.

While we tackled the 2D topics in depth, I am beginning this project after only about three 45 minute class periods exploring 3D shapes and properties. The idea is that, before this project, my students have some basic understanding of what surface area and volume represent and can find those properties for a given rectangular prism. Though not all can do so easily or consistently.

Basics of the Hotels^[7]I have modified this pretty extensively from Fawn's set up. She had a percentage tax structure that my students do not have the background to make use of. The room conditions and values, however, are copied and pasted straight from her website.

The students will be designing a hotel which is constructed out of unifix cubes. The goal is to construct a hotel with greatest value possible. A hotel's value comes from the number of rooms it has and the quality of each room. Each room is represented by one unifix cube. A room's value increases when more of its sides are open and when it has a roof (no cube directly above it):

Room Conditions	Value
4 windows, 1 roof	$600
4 windows, 0 roof	$500
3 windows, 1 roof	$300
3 windows, 0 roof	$250
2 windows, 1 roof	$200
2 windows, 0 roof	$175
1 windows, 1 roof	$150
1 windows, 0 roof	$125

We all know that running a business is not all profit. So, many aspects of the design and construction process come with costs. First, the students need to buy the cubes (raw materials). The first 10 cubes they purchase cost $5 each, the next 10 are $10 each, the next 10 are $20, and the price continues to double for every 10 cubes that are purchased.^[8]This is supposed to simulate the idea of supply and demand and limit the total number of cubes they can reasonably purchase. The most profit that can come from a single cube is $600 but the 81st cube they purchase will cost $640. Additionally, students have to purchase the land the hotel rests on. Every unit of land the hotel occupies costs $400.^[9]No more than 60 units of land can be purchased. Finally, there is a tax of $500 per floor above the 6th floor.^[10]In general, I wasn't sure where the students' designs would head, so I put it some natural barriers to keep them from becoming too massive.

Finally, to simulate the nature of government bureaucracy, I added some building requirements. Each hotel must have a 12 cube rectangular lobby on the ground level, and 6 units of land must be left open as parking structure. I presented this information in the form of a government regulations handout and asked the students to parse it on their own. I am tweaking it based on their feedback, so I will share that document in a later post. Shoot me a note on twitter if you just can't wait.

Project Structure

I expect the project will take about six classes to complete. It could probably be finished in as few as four, but I want to mix in a few other things. First, I want to use the project as an opportunity to learn some basic spreadsheet skills. During the beginning of this project, their homework is going to be to write some equations in google spreadsheets.

I am having students work on their own to begin the project. I am giving them two classes to parse rules and regulations and then construct a "rough draft" hotel of 30 cubes or less. I am going to have them use this "rough draft" to practice calculating the value using a spreadsheet template I made. Then, finally, I will put them into mixed-ability groups and have them work on their final hotel and supporting documents.

Each group will turn in a final hotel design and completed budget spreadsheet which will include finding the volume and surface area of the hotel. I will also have each group member complete a separate reflection that will force some meta-cognition.

I'll share more as we go. This is going to be fun!

Update 2/6/16: I posted about how the project was going.

December 14, 2015

Deal or No Deal to Teach Probability

Tyler 5th, 6th, 7th, Code/Programming, First Day Activities, Lessons, Probability, Statistics and Probability Deal or no deal, math lesson, Python 0 Comments

Oftentimes, the class periods right before breaks are so crazy that it is hard to get much done. I have gotten in the habit of using these periods to introduce mathematical topics that might not otherwise be covered. This year, before thanksgiving, I set up a game of deal or no deal, so I could introduce my students to probability and expected value.

Deal or No Deal is a press-your-luck style game show where contestants try to win money. A contestant is shown a bunch of cases that contain between $1 and $1 million dollars. At the beginning of the game the contestant chooses a case but does not see its contents. Then they open other cases, each time revealing more cases which cannot be the one they chose. After opening each case they are offered a deal and they can take that deal or continue opening cases. Here's Howie's explanation:

I always found this show fascinating and thought it would be a fun game to play with students. I wanted the students to be playing for something tangible, so instead of imaginary money, I offered them skittles. I prepared 10 cases and blindly labeled them with letters. They represent 1, 5, 10, 25, 50, 100, 200, 400, 800, and 1000 skittles.^[12]Logistically, counting more than 100 skittles is a waste of time. So instead, I found the mass of an average skittle and then used a scale to measure out the winning numbers. As you'll see below, I didn't have to do this often. In case you are curious, each skittle weighs about 0.0388 ounces. The image below shows sets of "cases" for each of my four classes.

This game was fun. Really fun! The kids had a blast and it was really exciting. Here's how they did:

6th Grade - Group 1
Highest offer: 351 skittles
Result: 10 skittles
Percent of highest offer: 2.84 %

7th Grade - Group 2
Highest Offer: 183 skittles
Result: 25 skittles
Percent of highest offer: 13%

6th Grade - Group 3
Highest Offer: 155 skittles
Result: 5 skittles
Percent of highest offer: 31%

6th Grade - Group 4

Highest Offer: 560 skittles
Result: 400 skittles
Percent of highest offer: 71.4%

Not a single group took a deal. Instead they kept opening cases until only their case was left. At each decision point, the class had to vote, so students used varying rationals to convince their peers. Some students used the gambler's fallacy while others used probability, expected value, and game theory. A few students even argued that risking a large amount of skittles would be worth it because they could gloat if they won a lot. These discussions presented lots of great opportunities to discuss probability topics.

I knew I needed a computer program to offer the deals or my students would jump on my case -- saying that's not a fair offer. So I wrote a program in python (you can try it out here). In the end, my program was probably a little too cruel when it came to offering deals. I had the program calculate the expected value based on the cases left and offer a percentage of the EV. At first I only offered 50% of the EV, but this climbed to 90% of the EV by the final deal. The next time I try this -- and there will definitely be a next time -- I am going to have the percentage of the EV climb more quickly.

Let me know if you have ever tried Deal or No Deal with your kids. I am also happy to help with the coding if you want to try this in your class.

Others' Ideas

Sarah wrote about using Deal or No Deal with her students. I really like that she had her students thinking about not just expected value, but the probability of earning more than a specific threshold: 100,000. My students were thinking about thresholds a lot. Most groups made some version of the argument, "Let's keep opening until we all get at least 10 skittles."

Thanks to Dan Meyer, Mr. Lucchese wrote about the mathematics of Deal or No Deal. He argues that the "banker" is trying to minimize the amount paid out by extending each contestant's time playing the game since more contestants means more money paid out.