# Another way to introduce ratios: develop relative thinking

One of the most important things students need to understand to reason proportionally is the difference between comparing two quantities in relative (multiplicative) versus absolute (additive) terms.   Students often struggle with making the move to thinking multiplicatively.

How can we begin to help them make this transition?

The research suggests one way:

Make the two ways to compare quantities explicit.

What does this look like?

Here’s an example of how ratios are introduced in Chinese classrooms (Cai, 2004; Cai & Sun 2002):

Miller Middle school has 16 sixth-grade students and 12 of them said that they are basketball fans.  The remaining students are not basketball fans.  How could we describe the relationship between the students who are basketball fans and those who are not?

Once students determined that there are 12 fans and 4 non-fans, they described the relationship in the following ways:

• There are 8 more fans than non-fans.
• There are three times as many fans as non-fans.
• For every three fans, there is one non-fan.

The first description is an additive comparison, while the next two are multiplicative comparisons.  Through continued exposure to the different ways to compare quantities, students begin to move flexibly between additive and multiplicative reasoning.

Of course that’s only the beginning of developing proportional reasoning.   But being aware of the different ways to compare the situations is an important first step.  Knowing why one comparison might be more useful than another is the next step.

Cai, J. (2004). Developing algebraic thinking in the earlier grades: A case study of the Chinese elementary school curriculum. The Mathematics Educator 8(1) 107-130.

Cai, J., & Sun, W. (2002). Developing students’ proportional reasoning: A Chinese perspective. In B. H. Litwiller & G. Bright (Eds.), Making sense of fractions, ratios and proportions: National Council of Teachers of Mathematics 2002 Yearbook (pp. 195-205). Reston, VA: NCTM

# One way to introduce ratios so they make sense to students

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.

# Teaching proportions: Hold off on the cross-multiply algorithm

At a workshop last week, the following task caused a bit of confusion:

If a small gear has 8 teeth and the big gear has 12 teeth and the small gear turns 96 times, how many times will the big gear turn?

Several participants were convinced it was 144.

As we discussed the problem, it was clear that those who were getting 144 were setting up the proportion below and cross-multiplying.

Small Gear    8  =   96
Big Gear       12       x

They weren’t trying to reason about the situation. They weren’t thinking about whether it made sense that the big gear turned more times than the small gear. Instead, they identified this as a proportion problem, set up a proportion and solved for x because it was the method they were “supposed to use” to solve proportion problems.

It was a great example of why you should hold off on teaching students the cross-multiply algorithm until after they have had experience reasoning about ratios using their own strategies.

It also points to the fact that students need to develop meaning for the algorithms they use or they won’t know how or when to apply them.

Lobato and Ellis (2010) describe a student who set up a proportion and used cross-multiplication to correctly solve the following problem:

How much water would drip from a leaky faucet in 4 minutes, given that it dripped at a steady rate of 6 ounces in 8 minutes?

Although the student could provide a correct numerical answer to the problem, she could not explain why it was correct or why it made sense.

Her limited ability to reason proportionally was even more apparent when she could not make sense of the following problem:

One faucet was dripping 6 ounces in 20 minutes. A second was dripping 3 ounces in 10 minutes. Was the first one dripping slower, faster, or at the same pace as the second?

Her inability to solve the second task, as well as her inability to explain her answer in the first task, indicate that she was using a procedure that held no meaning for her.  When presented with a new task that she couldn’t solve in the same way, she didn’t know how to reason about it.

The research also shows that student have a hard time making sense of the cross-multiplication algorithm.  It doesn’t necessarily connect to their intuitive strategies.  If the algorithm is introduced before they have developed meaning for ratios and proportions, it can interfere with and even hinder their ability to develop proportional reasoning.

So how do we help ratios make sense to students so that they can begin to reason proportionally? I’ll talk more about that next time, but if you want to know know, take a look at the resources below.

Lamon, S. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework for research. In K. Lester Jr. (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (pp. 629–67). Charlotte, NC: Information Age Publishing.

Lobato, J., & Ellis, A.B. (2010). Developing Essential Understanding of Ratios, Proportions, and Proportional Reasoning for Teaching Mathematics, Grades 6–8. National Council of Teachers of Mathematics.

Teaching Ratio and Proportion in the Middle Grades:  NCTM research brief

Smith, J. (2002). The development of students’ knowledge of fractions and ratios. In B. Litwiller & G. Bright (Eds.), Making Sense of Fractions, Ratios, and Proportions (pp. 3–17). Reston, VA: National Council of Teachers of Mathematics.