Since the GED exam is now the official high school equivalency exam in New York State, we have been looking into the content that is tested on the math section of the exam. We were happy to see that the GED includes standards from earlier grades, which is in contrast to the previous HSE exam in New York State, which was limited to high school standards. For example, the Quantitative Problem Solving portion of the GED includes questions based on the following standards:

• Calculate and use ratios, percents and scale factors
  • Compute unit rates. Examples include but are not limited to: unit pricing, constant speed, persons per square mile, BTUs per cubic foot.
  • Use scale factors to determine the magnitude of a size change. Convert between actual drawings and scale drawings.
  • Solve multistep, arithmetic, real-world problems using ratios or proportions including those that require converting units of measure.
  • Solve two-step, arithmetic, real world problems involving percents. Examples include but are not limited to: simple interest, tax, markups and markdowns, gratuities and commissions, percent increase and decrease.

[from The GED Assessment Guide for Educators]

I am excited that these topics are again part of our responsibility as math teachers for adults. It’s hard to imagine a topic that is more important for mathematical understanding than proportional reasoning. We now have the opportunity to incorporate ratios, rates, and percentages into preparation for the high school equivalency exam.

I wanted to share a resource I have used to think about teaching ratios. The Math Learning Center has a great series called Learning to Think Mathematically, with books and activities for teaching math. In this series is a free, downloadable book called Learning to Think Mathematically with Ratios.

Why should we teach students to use ratio tables? The author, Jeff Frykholm, Ph.D., uses the following problem as an example of why ratio tables can be useful for students:

Stop for a moment and think about how you would answer this question. Scroll down after you have come up with an answer.

.

.

.

.

.

.

Some common incorrect solutions described in the book:

• 12 boxes
• 12 remainder 5 boxes
• 12.42 boxes

What is wrong with these answers?

.

.

.

.

Of course, you could certainly use division by 12 to answer the word problem above, but skipping straight to that strategy means skipping over thinking about the actual situation, with apples in boxes. If they are told to divide, many students will not think critically about the situation. With a ratio table, you would probably start more slowly: “If one box holds 12 apples, then two boxes must hold 24 apples.” Along the way, you are thinking about how many apples per box, which may help you see why 12 boxes are not enough for 149 apples.

We are trying to find enough boxes for 149 apples, so need a lot more boxes. Maybe we could continue doubling boxes and apples:

Oops. Too many apples. But maybe I could look back and see that 48 apples + 96 apples is pretty close to 149 apples.

I’m really close. I just have 5 more apples that don’t fit. Since 12 apples fit in a box, I need one more box than 12.

This is just one way to think about the situation. There are many, many ways to reason with a ratio table. The author presents a different solution strategy with a ratio table, starting with the fact that 12 apples will fit in 1 box:

If we divide 149 by 12, we get 12.416… Of course, we might realize that this means there are extra apples that don’t fit in the 12 boxes, but it seems clearer in the ratio table. In the end, we want to be able to think about proportional situations in different ways. Each way of thinking reinforces other strategies.

There is a lot of good stuff in this book. It includes an explanation of different solution strategies with ratio tables, including multiplying by 10, doubling, and halving. It also includes activity sheets for teaching multiplication, division, and fractions with ratio tables. For more ideas, I recommend listening to episode 29 (ratio tables) of the great podcast, Math is Figure-Out-Able with Pam Harris and cohost Kim Montague.

By the way, the section on ratios at the beginning of Being Counted: Probability & Statistics, Part 1 was inspired by this book.