Over the Christmas holiday, the number of LEGO bricks in my house increased significantly. My son received LEGO sets as gifts from numerous grandparents, aunts and uncles. I was a LEGO fan when I was a child and now I have an excuse to play with them again as an adult. We've had lots of fun recently building sets and designing our own creations. At some point I became inspired to create a scale model of our home.
Planning and Building
I started this small project by building a test model to try out the proportions and to see what kinds of bricks I would need. The sizes of the door and window established the overall size. I continued revising the structure it until it looked right and then started collecting the bricks I needed.
Building this model reminded me of working on an OpenMiddle.com math problem. In an "open middle" problem, there is a one starting point and one solution but many different paths to get to the solution. With LEGO, there are many different ways to create, revise and improve your model. There are lots of different building techniques that will all result in a well designed scale model.
After I created my initial rough model I did some reading up on LEGO scale. It turns out that it is a fairly complex topic that lots of different people have investigated. I found the Brick Architect web site to be very helpful. For "classic minifigure" scale a ratio of 1:42 can be used. One major difficulty in discussing scale is that the proportions of a LEGO minifigure are not even close to the proportions of an actual person. A LEGO minifigure is about 4 cm tall and 1.6 cm wide. An average male human is about 175 cm tall and 40 cm wide... about half as wide as a minifigure would be at that height. Another challenge is converting units. The architectural drawings of my house are in feet, which I converted to metric (cm), then a scale factor is applied and finally the metric units are converted into LEGO bricks. I found an awesome tool that does this all for you, the LEGO Unit Converter.
I used a lot of estimation to determine how many bricks of each type I would need. LEGO bricks are not cheap so you don't want to order more than you need (Check out Jon Orr's activity involving cost, Is LEGO Gender Biased?). I purchased the bricks I needed on BrickLink.com, a large online LEGO marketplace. BrickLink provides a detailed price guide for every brick available which makes it really easy to know if you're getting a good deal or not.
I needed lots of 45 degree angle slope bricks for the roof of my house. These price stats let me know what a reasonable price is to pay for new or used bricks of this type. It is amazing to see how many bricks are sold on this site. I think that the stats from this site could make for an interesting grade 12 math research project.
The Finished Project
I recently stumbled across Richard Garfield's Balloon Balance puzzle from the 9th World Puzzle Championship in October, 2000. The puzzle is similar traditional balance puzzles such as SolveMe Mobiles but with a couple of twists. The first twist is the use of balloons to include negative numbers. The second twist is the inclusion of torque... the farther from the balance point (i.e. fulcrum), the more force is applied.
I thought that a simplified version of this puzzle might work well for students who are practicing integer multiplication. First however I had to figure out the puzzle. I have to admit that this puzzle took me quite a while to solve and make sense of. I almost gave up a couple of times but I eventually figured it out. Solving this puzzle reminded me of how it feels to be a learner and to empathize with students struggling to understand a new concept. If you'd like to see the solution to the puzzle above, click this link.
To use this with students, I simplified the puzzle a bit and then made some examples and easier challenges to start with. I modeled these from a version of this puzzle created by Joseph DeVincentis for the Boston Area Puzzle Hunt League (BAPHL) #5.
Directions: On each set of balances, place the given balloons and weights in the open circles. Each beam should be balanced around its fulcrum (the small square) by the torque rule: sum of weight times distance from fulcrum for all weights on the left should equal that on the right. The beams themselves are weightless. The fulcrum with an x is the fixed anchor point for the system.
The first example shows students how the basic puzzle works. The second example below add additional complexity to the system while retaining only 4 weights/balloons to place.
I created two challenge puzzles for students to work on once they were confident with the examples.
I haven't tried this out with students yet so I'm not sure what to expect. Feel free to check out my Google Slides with the examples, challenges and solutions. I'd appreciate feedback if you have suggestions to improve this.
Nova Scotia Mathematics Curriculum Outcomes
Grade 8 N07 - Students will be expected to demonstrate an understanding of multiplication and division of integers, concretely, pictorially, and symbolically.
"Same and Different" is a math routine that I've been exploring recently. This is a routine that gives students a structure to compare and contrast two objects or ideas. I was introduced to this routine on twitter back in August 2017 when Brian Bushart launched the website https://samedifferentimages.wordpress.com/. Fueled by support from the #MTBoS, the site quickly gained momentum.
In mid-December, I saw a tweet from Kristie Donavan. In it, she wrote "#samediffmath is one of my favorite structures for connecting new ideas with prior understanding!" She also attached a few examples that she has used. I really liked the idea of using this routine to help students construct mathematical connections. In the Nova Scotia mathematics curriculum, "Connections" is one of the seven Mathematical Processes that are intended to be infused throughout each course. Making connections between mathematical ideas helps students see mathematics as a consistent and integrated whole and not isolated units of study. Below are two images that I created to help students make connections between related representations and ideas.
Below are two more #samediffmath images that I created to connect different representations. The first was inspired by a recent tweet from Kent Haines. He was exploring a variety of ways to model a linear equation. There were lots of replies and suggestions that I hadn't considered such as Howie Hua's reply suggesting using a "splat" model.
Be sure to check out the hash tag #samediffmath on twitter or the Same or Different Images website for new examples of this routine. I also have a Google Slides document with my images on it if you'd like to use or revise my images above.