Written by Sarah Stecher published 2 months ago

I’m all about making math visual. It’s so important to me that a student is able to *see* why something works and use a method that makes sense to them and is connected to the underlying mathematics, instead of just relying on a rote procedure or formula.

I was recently teaching my students how to solve for missing side lengths using right triangle trigonometry. We were thinking conceptually about what the trig ratio represents, what it tells us about the comparison of sides, and how we interpret ratios given as decimals. But I found that even after students successfully set up a proportion, they struggled to solve for the missing side. I knew I didn’t want to revert to cross-multiplying because I wanted students to think conceptually about what it meant that one number was 0.45 of another. While I used some guiding questions to try to help students, we ended up still going back to how to isolate x in the equation by using inverse operations. There is nothing particularly wrong with this method, but it didn’t feel very related to the rich work we had been doing with ratios. I knew there had to be a better way.

I realized that even when students understood trigonometry conceptually, they might still struggle with proportional reasoning, which would allow them to solve for a missing side. I had been doing some learning about proportional relationships from some resources for teaching the middle school grades, and I knew there had to be a way to apply that work to high school classes. Well, I found it.

The double number line.

The double number line is a visual tool that can help students solve all kinds of proportional relationship problems. And it's not just any tool, it's like the Swiss Army Knife of tools with a million different uses. We’ll start by looking at a classic proportional relationship problem students might encounter in middle school as a way of learning about the tool and its versatility. We’ll then look at how I think it can be leveraged for high school math.

**Example: Suppose that the instructions on a jar of instant coffee say to add 4 tablespoons of coffee granules to 5 cups of water. How many cups of water are needed for 10 tablespoons of coffee?**

Note that this problem could be represented as the proportion $\frac{4}{5}=\frac{10}{x}$, what most would consider the $\text{\textquotedblleft}$hard$\text{\textquotedblright}$ version of proportion solving because the variable is in the denominator. Isolating x is tricky for students, so most teachers teach cross multiplication. $4x=50$, so $x=12.5$. This is the correct answer of course, but the value of 50 does not seem at all related to the problem. Let’s see how using the double number line would provide students with opportunities for strategic reasoning and sense making. Here's how we might represent the problem.

Now let's look at four possible methods students could use to solve the problem using the same visual tool.

#### Method 1

We could consider this the method of iterating (repeating) the composed ratio of 4 tablespoons to 5 cups. That means 8 tablespoons would be 10 cups and 12 tablespoons would be 15 cups. Even though 10 is not a multiple of 4, it is half-way between 8 and 12, so the number of cups of water needed is half-way between 10 and 15, so 12.5.

#### Method 2A

This method could be called partitioning the composed ratio, and in this case the partition creates a unit ratio. If 4 tablespoons require 5 cups of water, then 1 tablespoon requires 1.25 cups of water, so 10 tablespoons requires 12.5 cups of water.

#### Method 2B

Partitioning does not necessarily mean finding the unit rate. We could just as easily have said that 2 tablespoons of water requires 2.5 cups of water. Since 10 tablespoons of coffee is 5 times the amount of coffee, we will need 2.5(5)=12.5 cups of water.

#### Method 3

This next method I would call the multiplicative factor method. How many times bigger is 10 tablespoons than 4 tablespoons? Note here that we are reasoning multiplicatively within a single variable.
10 tablespoons of coffee is 2.5 times bigger than 4 tablespoons of coffee. Thus, the number of cups we need for 10 tablespoons must be 2.5 times bigger than the number of cups we need for 4 tablespoons; 5(2.5)=12.5.

#### Method 4

We can also reason multiplicatively between two variables. Now we are asking the question: How many times greater is the number of cups of water than the tablespoons of coffee?

Since the number of cups of water is always 1.25 times greater than the number of tablespoons of coffee, 10 tablespoons of coffee will require 10(1.25)=12.5 cups of water.

The goal here is not to teach students $\text{\textquotedblleft}$the 4 methods for solving a proportion using the double number line$\text{\textquotedblright}$ as that would proceduralize the process again. Instead, the goal is introduce a tool (the visual representation of a double number line) that will help students think through any proportional reasoning problem in a way that makes sense to them. It is very likely that students' solutions paths will align with one of the methods described above.

### Benefits of using the double number line

- It doesn’t matter whether the variable is in the numerator or denominator of a proportion.
- Students can easily assess the reasonableness of an answer and detect values that don’t make sense, based on the visual.
- A single tool offers a wide variety of solution methods, depending on what makes sense to the student and the values given in the problem.
- Using the double number line is intuitive and builds on foundational ideas of multiplication and division; students aren’t relying on a set of procedures or algorithms they don’t understand.

So at this point you might be thinking: *This is all great, but proportional relationships are not really a high school standard. Besides just to review proportions, how could I use this in the classes I teach?*

Great question. It took me a long time to figure out that so much of what we teach in high school actually is just proportional relationships.

Consider how we might use the double number line to reason about the following problem.

If $∆FIG\sim∆TEA$, find the length of segment *FI*.

A student might see that a length of 10 on $\Delta TEA$ would correspond with a length of 6 on $\Delta FIG$ by halving. It follows that a length of 15 on $\Delta TEA$ would correspond with a length of 9 on $\Delta FIG$, since 15 is half-way between 10 and 20. Students may also split the 20 into fourths and then find three of those fourths. Additionally, since 15 is three fourths of 20, we need three fourths of 12 to get 9. Or, 20 is 1 ⅔ times 12, so 15 divided by 1 ⅔ gives 9.

And finally, how might a flexibility with the double number line support students in solving proportional relationships when the ratio is not given as a fraction or as a comparison of two concrete things, but as a complicated decimal? Is there hope for trigonometry?

This is the most abstract use of the double number line as we are no longer dealing with two concrete quantities. Note that we have no labels on this double number line because the upper part of the number line represents the ratio itself, which is unit-less.

#### Possible solution methods

- How many groups of 0.237 are in 1? Divide 1 by 0.237. That’s how many groups we need of 12.
- How many times bigger is 12 than 0.237? Divide 12 by 0.237. That’s how many times bigger the hypotenuse has to be than 1.
- How many times bigger is the whole (1), than the part (0.237)? Divide 1 by 0.237. This is how many times bigger the hypotenuse is than the adjacent side, since the adjacent side represents 0.237 parts of the 1 whole hypotenuse.

I love how this tool allows for flexible problem solving and sense making in a wide variety of contexts. Give it a try with your students and let us know how it goes!