I love words. Even though I am a mathematics teacher, I often discuss the etymology of a word, or words: sometimes it is of any word used in explaining a topic, or in any conversation with the class; often it occurs with academic vocabulary; and more often, it is for one or more mathematics vocabulary words. Nearly all the time, however, my purpose is to connect the words together to help student understanding.

As we approach linear relationships, and the concept of slope, I often start discussing ** distance**,

**, and**

*time***, and how they relate to one another. Most students intuitively understand these concepts.**

*speed*When I ask if they remember how to express the relationships between these three concepts, most respond that they do not. This leads to a discovery process beginning with the word ** speed** and a discussion of common ways people describe their speed to one another, especially if riding in a car. Most often students respond with miles per hour. From here, I write

**on the board then expand it into**

*mph***. A discussion of units and units of measure typically occurs at this juncture; I am a huge fan of dimensional analysis to help students better understand their mathematics.**

*miles per hour*From ** miles per hour**, we discuss the mathematical symbol(s) used to depict the word

*typically called the fraction bar: —, or dash: ⁄ ; I prefer to use the fraction bar. From here, I write out the following representation for the units of speed.*

**per**,At this point, I proceed in one of two ways, depending upon whether I wish to make explicit connections between a series of related mathematics vocabulary first, or to emphasize an algebraic representation for one of the relationships between ** distance**,

**, and**

*time**As this post focuses on vocabulary over representations, I will discuss the latter in a later post.*

**speed**.## Explicit Vocabulary Connections

If I intend to emphasize the connections between certain mathematics vocabulary, which will become clear in a moment, I ask students for a synonym for ** speed** that they may have heard in earlier math or science classes. Few mention

**, however, at times, some do.**

*rate*Once the vocabulary word ** rate** is in play, I write out something akin to the following.

Attention then shifts back to the representation of the units for ** rate** at the end of the sentence. When I ask what name students might give to the representation for the units, students respond with

**most often, and some with the word I sought:**

*fraction***.**

*ratio*This allows me to write out the following: *rate***&** * ratio* asking if anyone notices any similarity between the two words. A few students exclaim

**! To which I say, “Correct!”, which leads to lots of giggling. Once the laughter settles down, I briefly explain the Latin root word,**

*rat***or**

*rat***, from which both**

*rata**and*

**rate***were derived.*

**ratio**With the connection between these two words established, I ask students if they know of any other mathematics vocabulary that includes either of these in whole or in part. As I have a large poster I created in the front of my classroom depicting number sets, students fairly quickly announce ** rational number**; whereupon, I show that

**is a root word of**

*ratio***This leads me to write out a rational number, such as three-fourths, as shown below.**

*rational.*This segment concludes with students observing the connections between the mathematics vocabulary: * rate*,

**ratio**, and**rational number**.
Dave

Two quick comments here

1) I love this emphasis on language. As a department, we have been having some conversations along these lines. It is SO important that we be consistent and thorough in how we use words around our students. I’d love to have all my Algebra kids have this conversation with their teacher.

2) Where do you stand on the units used for speed? I am pretty stubborn about wanting to say ‘miles per hour per hour’ rather than miles per hour squared since there is not physical unit of a squared hour. I know my physics colleague says seconds squared or hours squared.

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Re: 1) Thanks and I agree, and 2) as a teacher whose first degree is in electrical engineering and has a certificate in secondary mathematics and physics, I follow the International System of Units (SI) for all physical quantities. There are base SI units and derived SI units, the latter of which applies to acceleration. See http://physics.nist.gov/cuu/Units/index.html for further info.

At the same time, as acceleration is the change in velocity per unit of time, where speed (miles per hour) is the absolute value of velocity, you could state the units for acceleration as “miles per hour per hour” as you prefer. However, this is a complex fraction, which simplifies to “miles per hour squared”. I find this both mathematically pleasing and consistent with international standards.

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I love the use of vocabulary development to connect the mathematics to physical connections. Being a physicist, dimensional analysis is superbly important and one of the many tricks I try to show students, especially to connect to their verifying if the answer is reasonable. Additionally, I have to say I am a huge fan of writing in math classes. I have found that this is a crucial piece to having students start to connect between the words and their thinking. For example, I may have them write out their observation about what they notice before we discuss it whole class, getting them to write and give me their observations seems to produce some quality understanding.

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As a physicist, you might appreciate my response to Mr. Dardy’s inquiry above…

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There’s an awful lot of evidence that making those neurological connections between “math problems” and language is really important for being able to understand and transfer the knowledge. David Berg’s multisensory math stuff has intensive, explicit connections from the ground up. When we’re working with the pre-pre-algebra folks, we keep coming back to “parts and wholes” …

… I often connected “irrational” numbers being numbers you coudln’t make a ratio with, but also tossed in that poor ol’ Pythagoras was considered irrational as in crazy for proving to his little math society that no, some things couldn’t be expressed in integers or ratios thereof.

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Doh! I left out irrational numbers in my post…I usually speak to those at the same time to be complete, where I ask students if they know its meaning. Many know that the prefix ir- connotes “not” so that goes pretty smoothly. I also talk about the dilemma faced by the Pythagoreans with this discovery as well as the myth that Hippasus was killed for discovering them.

I might add to the post to correct this oversight, or do so with another post.

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