Mini-Math Lessons – Proportional Reasoning: Comparing Rates and Looking At Scaling

What size frame do I need? Why is that candle burning faster than the other one? These are questions we are going to explore today. And it all has to do with rates, and proportions, along with other factors such as type of wax for the candles. Proportional reasoning comes into play in seemingly mundane things, like determine the size frame needed for a picture that you might be enlarging (or shrinking). Like yesterday, it’s about comparing and using mathematics to help understand and model real-world situations. What I love about these types of problems is that they can be approached several different ways, and each way can provide a different perspective and answer because you get more and/or different information. This is what modeling with mathematics is really all about.

Both activities today, as with all the activities this week, are adapted from Fostering Mathematical Thinking in the Middle Grades with Casio Technology, Casio 2011. I have made version of them, but if you have handheld calculators, these same activities are available in the free Math Activities under graphing calculators for middle school at the Casio Education Website. The first activity has to do with two candles, the same height to start, but burning at different rates due to different types of wax. Students will explore fractions by looking at the fraction of each candle that is burned. They will compare using tables and graphs and use proportional reasoning to determine things such as when is one exactly half of the other. The second activity has to do with wanting to frame an image, and depending on the room it is to go in, the image will be sized-up or sized-down, so how much framing is needed and how much glass is needed?  This is a perimeter and area ratio problem and there is some nice simulations that students use to collect data on side length, perimeter, and area as side length increases. From experience, I know students struggle with the understanding that if you double the dimensions (length/width), that perimeter also doubles but area quadruples (exponential). The data collection and looking at the tables and graphs

Here are the links to the two activities and the video overview that explores the activities and some of the skills/features:

  1. Proportional Reasoning – Burning Bright
  2. Proportional Reasoning – Stretch That Picture


The tool being used in these mini-math lessons is the FREE web-based math software,

Remember – if you want to save and/or modify any of these activities, create a free account.  Some useful links below: