Exploring the beauty of numbers and ratios

I recently read A Mathematician’s Lament. The author would like to see math taught the same way as children learn to appreciate music and art. Explore the beauty by playing with it. Explore the beauty of numbers by manipulating them.

I think we need a better balance between pure and applied math. The Common Core stresses real-world application. Those scenarios answer the question, “Why do I need to know this?” but at times I think we’re missing the beauty of numbers. So when our math department met to continue our work on creating mathematical tasks I didn’t want to lose sight of that.

Our latest creation is a ratios and proportions task titled A Balanced Lunch. It’s influenced by Lure of the Labyrinth’s Managers’ Cafeteria Puzzle; however the students are required to delve deeper into equivalent ratios proportional reasoning. I’ll explain that in a bit. 

If you are not familiar with L of the L it is a fabulous puzzle-solving journey. Below is a screenshot from the Manager’s Cafeteria Puzzle. This puzzle’s goal is to determine the ratios between each of the four food items. The player has only enough “guesses” as there are number of missing food items. Twelve lights are lit to represent the twelve missing food items. I added the ratios to provide a clearer understanding of the puzzle. The puzzle has just enough information to solve.

Tray one establishes the "slop" to sushi ratio as 11:3. The middle tray shows a "slop" to turkey leg ratio as 11:7. The tray on the right shows the burger to turkey leg ratio as 9:7.
Tray one establishes the “slop” to sushi ratio as 11:3. The middle tray shows a “slop” to turkey leg ratio as 11:7. The tray on the right shows the burger to turkey leg ratio as 9:7.

The task goes beyond L of the L in a couple of ways. Besides wanting the students to use mathematical language when explaining their solution, they will explore graphing to discover that equivalent ratios form a line going through the origin. As an extension students will also create their own puzzle plus determine just the right number of values and combinations to be revealed at the start.

balanced lunch

When the students played the online game they were in pairs sharing a laptop. There was a LOT of math talk but much of it was imprecise language. For example a 1:1 ratio was stated as, “The numbers are the same”. I also heard, “This number is three times as big” when describing a 3:1 ratio. Sixth graders need to begin using more precise language so the task is designed with that in mind.

Here’s the task.

You can use any graph paper for graphing equivalent ratios but here’s one that has 10 lines per inch.

Let me know what you think.

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