We know that students struggle with understanding ratios and reasoning proportionally. The cross-multiply algorithm doesn’t make sense to them.

What can we do to help ratios make sense to students?

Whenever I’m thinking of how to introduce a new concept, I like to start by thinking about what ** students already know** that can be built on so that a new concept makes sense to them.

That’s when I usually go to the research. Researchers have conducted lots of studies that focus on what students can intuitively do or what contexts have been successful in fostering particular concepts.

In the case of ratio, the research shows that recipes are an effective context for introducing ratio. Students understand how to adjust a recipe to make more or less of it without changing the taste.

What are some potential ways to build on this when introducing ratios?

- Give them a recipe that involves two quantities, such as lemons and cups of sugar and ask them how to make more lemonade that tastes the same and less lemonade that tastes the same. Have them record different values in a labeled table.
- Ask them how to make lemonade that tastes the same if you only have 1 lemon or ½ of a lemon.
- Ask them how to make lemonade that tastes the same if you only have 1 cup of sugar or ½ a cup of sugar.
- Give different recipes, varying the difficulty of the numbers used.
- Have them graph different recipes.
- Later, give two recipes and have students determine which is “more lemony”

Of course, this is just one potential route to introduce ratios. I am sure you can think of others you might use.

The development of proportional reasoning is a complex process and requires a number of conceptual steps. I only addressed the beginning steps in the suggestions above. If you want to know more, check out the work below.

Kent, L. B., Arnosky, J., & McMonagle, J. . (2002). Using representational contexts to support multiplicative reasoning. In B. Litwiller, & G. Bright (Ed.), *Making sense of fractions, ratios, and proportions* (pp. 145-152). Reston, VA: NCTM.

Noelting, G. (1980). The development of proportional reasoning and the ratio concept. *Educational Studies in Mathematics, 11*, 217 – 253.

Streefland, L. (1984). Search for the roots of ratio: Some thoughts on the long term learning process. *Educational Studies in Mathematics, 15*(4), 327 – 348.