Using Integers 0 to 9 (without repeating any number), what is the maximum amount of equivalent ratios that could be created?

# proportion

# Shoebacca -3Act Math

**ACT 1:**

Show them this clip and this picture:

**What questions do you have after seeing this Video and Picture?**

Ask students to write down their questions, I normally ask students to find at least 3. When I observe that most students have questions written, I ask them to share those questions with their neighbor. I then throw up a Microsoft Word document and start typing down questions students supply. Students from my classroom came up with all sorts of different questions, some we can easily answer and others that we can’t. I am looking for a key question or questions to start this lesson. If students do not ask one of these questions, I tell them that I hope I can answer most of the questions provided, but that I need them to consider one of these questions first.

*What is the shoe size? Is it like a 52?**How big of a foot would you need to wear that?**How big/tall would a person be to wear this shoe?*

Any of these type of questions will lead students down the inquiry I hope to explore with them.

**ACT 2:**

This part can take a few different paths, depending on your focus for the day (you could use this idea with Proportions, Scatter Plots and Statistics). I started this part by asking what type of mathematical operation or ideas they would need to find an answer. The most common ideas students came up with are:

- guess/ estimation
- fractions
- proportions
- equations
- graphs

These are all excellent ideas and I place them on the board as reminders to where we want to go throughout the investigation. Depending on what focus I use this for, students normally fall into 3 strategies for finding our answer.

- Proportions: Most students want to make a proportion relating themselves to Shoebacca. This is a great strategy to begin with, but since I teach 8th grade students, I want them to push themselves. This strategy would work well with our 7th grade standards.
- Statistics: One thing that comes up when we start exploring Shoebacca is the fact that we can’t accurately predict how big someone’s foot depending on their height. Students immediately think of mean (or average) and start taking down everyone’s foot size and height (this is also great review of measurement and conversion- students this past year put up their height in feet, inches and I asked for inches only, and for some classes, cm). Then the next step students wanted to take was strategy 1, they made a ratio of their class average and made a proportion. Once again, a great way to tackle this problem, but I am relentless- I want them to really explore different options of solving this problem and how those strategies relate and compare.
- Graphs: When I first designed this problem, this was naturally MY first thought. One thing all great teachers do is anticipate how students will approach a problem and then tie it into the method they are presenting. I anticipated the first two approaches and hope my students will make connections between them and this one. One of the reasons I like looking at graphs is it allows students to use software to enter data, find trends, and display it graphically. Students take measurements of their class, place it in a spreadsheet and create a graph. They can then look at lines of best fit (before you automatically create it with the program) and make a conjecture based on their data.

Some other considerations: The length of the trailer which will determine how big the shoe length is, can be varied. I normally go with what the students discover- many will do searches on trailer sizes. The most common lengths I get in class are 12′, 14′,16′, 18′ and 20′. This variation of the trailer size could be a extension of the problem.

Here are the statistics for this year: 8th grade shoe 13-14

**ACT 3:**

These are the results my students got this year:

- Proportion: varies with individual 65″/10″ = ?/240″ so ? = 1560″ or 130′
- Statistics: 66.8″/9.7″ = ?/240″ so ? = 1652.8″ or 137′ 9″
- Graph: students drew a line of best fit and got 1080 or 90′

Some great discussions resulted in the difference in these estimates for height.