# 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.

# 5 ways to make simplifying expressions make sense

Student mistakes are important.  It’s important to know what mistakes students make and why they make those mistakes.

For example, last time, we looked at  why simplifying expressions isn’t so simple for students and why they tend to make mistakes like 2a +5b=7ab.

When we can anticipate what mistakes students commonly make and why they make them, we can began the hard work of figuring out how to develop understanding so these mistakes don’t make sense to students.  We can start to answer questions like the one below.

How do we make simplifying expressions make sense to students?

Here are 5 ideas from the research:

1. Write out the hidden multiplication in algebraic expressions.  We know that students often mistake the missing operation in 2a as addition.  In order to avoid this, write out the expression in expanded form at first (a x 2 or 2 x a).  Give students lots of experience simplifying expressions with this expanded notation at first and then gradually move away from it.
2. Develop meaning for the equal symbol.  I’ve talked about how important this is before and suggested some ways to do this starting in early grades.
3. Don’t use the fruit analogy.  It isn’t helpful.   Two apples plus five bananas equals seven apples-and-bananas makes perfect sense to students.  Plus it creates confusion about a variable represents.
4. Avoid using the same letter for the variable as the first initial of what it might represent.  Although it appears makes sense to us to use a to represent the number of apples, students see it as apples and they start to believe that the variable is a label for the object.   Once students have a clear understanding of what a variable is, you can use the same letter, but try to avoid it at first.
5. Give students experience with substituting values for variables before and after simplifying expressions and encourage them to find counterexamples.  Through continued experience, they will strengthen their understanding of why certain terms can be simplified and others can’t.

I’d love to hear what other suggestions you might have.

Want to know more?  Read the articles below.

Booth, L. R. (1988). Children’s difficulties in beginning algebra. The ideas of algebra, K-12, 20-32.

Carpenter, T. P., & Levi, L. (2000). Developing conceptions of algebraic reasoning in the primary grades. Madison, WI: National Center for improving student learning and achievement in mathematics and science.

MacGregor, M., & Stacey, K. (1997). Students’ understanding of algebraic notation. Educational Studies in Mathematics, 33, 1-19

# Simplifying expressions isn’t so simple

“…some seemingly simple ideas are not always as simple for students as they seem to adults.” -Lesley Booth

How often have you seen students make the following type of mistake?

2a +5b=7ab

My guess is that if you teach algebra, you’ve seen this mistake more than once.

If we believe that students do things that make sense to them, we need to ask:

Why does 2a + 5b=7ab make sense to students?

Here’s what the research has to say:

1. The nature of “answers” in algebra is different. From their prior experience in arithmetic, they have developed certain expectations of what answers look like. They assume that a “single term” answer is what is needed. There is also evidence that points to the idea that students give one term answers because they have difficulty accepting a “lack of closure.”
2. Algebraic notation is confusing to students. They don’t necessarily see the invisible operation between the variable and the number as multiplication. If anything, they may assume the hidden operation is addition given their experience so far in math. For example, when we write , it means 4 + 1/2 and when we write 43, it means 4 tens + 3 ones. It makes sense to them that 2+a+5+b would be equal to 7+a+b.
3. Students sometimes view the variables as concrete objects. For example, they see a as apples instead of “the number of apples” and b as bananas instead of the “number of bananas.” Therefore, they justify 2a+5b=7ab by saying 2 apples plus 5 bananas is equal to 7 apples-and-bananas.

Here’s the more difficult question: After reading why this mistake makes sense to students, what might we do differently in our teaching?

I’ll talk about what the research suggests next time, but if you want to know now, check out the work below.

Booth, L. R. (1988). Children’s difficulties in beginning algebra. The ideas of algebra, K-12, 20-32.

Tirosh, D., Even, R., & Robinson, N. (1998). Simplifying algebraic expressions: Teacher awareness and teaching approaches. Educational Studies in Mathematics, 35(1), 51-64.