I asked my 15-year-old what she learned today at school. She paused for a moment and then answered, “What did *you* learn at school today?”

It took me a while to think about what I had learned (which will make me more patient when I ask her again tomorrow), and then I remembered and shared with her:We are working with some teachers who are using the *Illustrative Mathematics 6–8 Math* curriculum. The 7th grade teachers are in Unit 1, Scale Drawings. They are working with scale drawings and maps. Today I learned to look more closely at the scale given for a map.

For example, look at the following for a moment. What’s the same? What’s different?

The last two are from IM 6-8 Math, currently available at im.openupresources.org and our IM Certified**™** partner, LearnZillion.

**What’s different about the scales on the last two?**

The standard for mathematical practice about attending to precision (MP6) says: “Mathematically proficient students try to communicate precisely to others. They state the meaning of the symbols they choose, including **using the equal sign consistently and appropriately**.”

I hadn’t noticed how frequently people record scale using an equal sign. I’m not sure that we would have noticed a difference if we had not been looking at so many questions about scale next to one another. We were trying to find some assessment items for standard 7.G.A.1, the 7th grade standard about scale, and discovered that many of those we found included a scale in the form of “1 cm = 100 miles.” I’ve looked at lots of maps, and I never noticed the incongruity of saying that 1 cm equals 100 miles. We don’t really mean that 1 cm equals 100 miles, right? Not in the same sense that we say 4 quarters equals $1, or 3+4=7. Is there any wonder that our students misuse the equal sign?

The Illustrative Mathematics curriculum is careful to say that 1 centimeter *represents* 100 miles in the scale drawing. The word *represents* may be longer than a symbol like the equal sign, but it more accurately shows the relationship. In this case, precision isn’t about efficiency, but accuracy.

**Next Step**

And so the journey continues. I am grateful for the authors of this curriculum, who help me attend to precision and for my daughter, who makes me think and share about what I’m learning. What are you learning to pay closer attention to as you teach mathematics? I look forward to hearing from you in the comments or at #LearnWithIM.

Nice observation. I have a few second thoughts:

The equal sign in map scales is related to the use of an equal sign as a notation for functions. For many reasons, it is more precise to use an arrow rather than an equal sign when defining a function. this is also true for map scales. 1 cm –> 1 mile means 1 cm of map “maps” to 1 mile of earth. But, of course we often want to use the tools of algebra with functions, e.g. graphing an equation or performing calculations, so we represent the function as an equation. You can think of maps as having a “scale” function: scale(1 mile) = 1 cm. That it is a ‘scale’ function is explicit in the header, “Scale” under which the unambiguous slang appears.

You can also justify the usage with ratios. since 1 map unit:1 earth unit is a core meaning of “map”, it is already said when we say “map” and does not need to be said again. I’ll leave it to you to complete the argument with the ratios.

The usage of scales on maps has its own history, of which I am ignorant. It is worth noting that mathematical notation is confused by its own history. We may not notice, because contexts that we understand make the meaning clear, but for students it can be confusing. Examples:

1. after all the work developing student understanding of the use of parentheses in mathematical expressions to denote the phrase structure of expressions, especially with the distributive property, we suddenly throw f(x) at them as though it were no problem.

2. after years of using “-” to denote an operation, subtraction, we use “-” to also denote the sign of a number. Explain this, -2(x – 1).

3. the use of letters in geometry to mean variously points, lines, the measure of lines is disambiguated with case and italics. But mix it in with the use of letters in algebraic expressions.

maybe we should have a strategy for helping students in such cases.

Ahh. Thank you for taking the time to help me make the connection of scale back to functions sans function notation. Being aware of mathematical notations that can be confusing for students and having strategies to minimize confusion is a must for helping our students learn mathematics. Another I think of is the use of ^(-1) for both raising to a power and naming an inverse: 4^(-1), x^(-1), f^(-1)(x), tan^(-1)(x).

Perhaps the potential confusion is compounded by the identification of scales on a single number line. So fro example on a temperature scale the number 0 (Centigrade) is placed at a point opposite the number 32 (Fahrenheit) while 100(Centigrade) is placed as a point opposite the number 212(Fahrenheit). The number line would correspond to different heights of mercury on a thermometer and the scales are created to match different units for measurement. But the identification can lead to inaccurate use of the equality without units- saying 0 = 32 and 100=212.

For these uses it may be preferable to have two distinct ( and parallel) number lines with independent scale labels and arrows between the scales to indicate the correspondence between the points on the lines and the scales. This is a good prelude to the introduction of function concepts and visualization.