Math Fireworks

April 12, 2016

Should You Attend My Workshop on Fermi Problems at NCTM Annual?

Tyler Conference, Fermi Problems, NCTM 16, Presentations NCTM16 0 Comments

I am extremely excited to be presenting about Fermi estimation problems at NCTM's Annual Meeting in San Francisco this week! My workshop is scheduled for Saturday from 9:45 in 3006 (Moscone West).

I want to provide a bit more detail about my workshop to hopefully get more folks to attend. But, I also know that everyone's time is limited and valuable, so this may help you decide that another talk or workshop is more valuable. That's okay too! If you come to my workshop, expect to have some fun actually doing mathematics. I hope you will attend!

Here is the description I wrote last spring when I applied to NCTM:

How many bathtubs of water will you drink in your lifetime? How long would it take to count to 1 million out loud? Solve Fermi problems like these and discuss their value, their connection to the CCSSM Standards for Mathematical Practice, and strategies for implementation. Expect to work collaboratively and have fun solving problems!

That description only hints at what you will be doing. They have my workshop listed in the 6-8 grade band, but I feel that this workshop would be valuable for anyone who teaches grades 4 or higher. My outline for the presentation is as follows:

Introduce Enrico Fermi and loosely define a Fermi estimation
Have participants solve Fermi estimation problems
Discuss what value these problems (or similar types) add to our students' education
Pedagogical advice if you decide to try these out
Resources for implementing Fermi problems and for continued discussions

If you have questions about the workshop, tweet me up: @mathfireworks!

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.