One Mathematical Cat, Please! Empowering Students to Teach Themselves Mathematics
[Third of three talks as a Featured Speaker at CAMT 2015; updated August 2022.]
Mae cyfathrebu yn allweddol i lwyddiant.
Have trouble reading that? When you don't understand a language, you don't have access to the ideas in that language, even when the ideas are simple.
Not understanding a language creates a lot of stress. When authors weed out ‘cryptic’ math language and replace it with English, a problem emerges: English isn't designed to say what mathematicians need to say.
Carol's free online book One Mathematical Cat, Please! takes the time to teach the structure of the mathematical language, so people are taught how to teach themselves mathematics while they're learning it.
60 Minutes, 60 Morsels
This is designed as a selfguided talk. Just click to open each morsel and then follow the suggested links. Feel free to email me with any questions or comments. Enjoy!

 Welcome, everyone! I'm really excited to be here.
 I'm very thankful to the conference organizers for inviting me.
 This is my third talk. They told me to make my three talks independent (not consecutive), so I'll be repeating some introductory material from my first two talks.

If you went to one or both of those first two talks,
you can be guessing what language this is,
and what it says:
Mae cyfathrebu yn allweddol i lwyddiant.

 I'll present about 60 morsels in 60 minutes.
 All my talks are online, so there's no need to take notes. If you're interested in (say) morsels 27 and 53, just jot those numbers next to this talk in your program guide.

 My waaayyy too long name is Dr. Carol JVF Burns.
 If you Google (just) Carol Burns, you won't get me! (I'm not the Australian actress.) Instead, Google ‘Carol Burns math’ or ‘math cat burns’. [Try them!]
 Here's my online vita. I have a Doctor of Arts in Mathematics, which is a doctorallevel degree designed for effective teaching.
 I've taught mathematics for about three decades at both the university and high school levels.

Raise your hand (or make some noise) if you have any association with each place I've taught (if you taught or went there yourself, or if you know someone who teaches or goes there):
 University of Massachusetts (Amherst): Bachelor of Science degree Magna Cum Laude in Civil Engineering
 University of Oklahoma (Norman): Master of Arts in Mathematics
 Idaho State University (Pocatello): Doctor of Arts in Mathematics
 Miss Hall's School (private allgirl's day and boarding school, Pittsfield, Massachusetts)
 Lenox Memorial High School (public high school, Lenox, Massachusetts): I was Chair of the Mathematics Deparment for a year, before I left to pursue mathontheweb fulltime.
 Northern Arizona University (Flagstaff): I moved closer to my daughter, who is pursuing her doctorate in neurolinguistics at the University of Arizona. Then, I met my husband Ray through dancing, and moved to Tucson.

Here's that earlier (probably cryptic to most of you) sentence:
Mae cyfathrebu yn allweddol i lwyddiant.
Cutandpaste it into Google Translate. [Do it! Click ‘detect language’.] (By the way, it's Welsh. WolframAlpha gave a 2005 estimate of 575,102 native speakers.)

Here's the point of that little exercise: If you don't understand a language, then you don't have access to the ideas in that language, even when the ideas are simple.
That was a very simple little sentence. Yet, most of you had no access to its simple idea—because it wasn't expressed in a language you understand.
Mathematics is no exception!1 These ideas form the philosophy of my online curriculum: Don't just teach people mathematics. Teach them how to teach themselves mathematics.

Languages evolve to make essential ideas easy to express. The language of mathematics makes it easy to express the kinds of thoughts that mathematicians need to express.
The language of mathematics is:
 PRECISE (able to make very fine distinctions): compare $\,5x^2\,$ and $\,(5x)^2\,$
 CONCISE (able to say things briefly): the expression $\,3x^2  5x + 7\,$ represents this sequence of operations: take a number, square it, multiply it by $\,3\,,$ subtract five times the original number, then add $\,7\,$
 POWERFUL (able to express complex thoughts with relative ease): It takes a full week of classes to fully flesh out this statement, which is the central idea in Calculus: $$ \begin{gather} \lim_{x\rightarrow c} f(x) = \ell\cr\cr \text{ is equivalent to }\cr\cr \forall\ \ \epsilon \gt 0\ \ \exists\ \ \delta \gt 0\cr \text{s.t. if}\ \ x\in\text{dom}(f)\ \ \text{and}\ \ 0 \lt x  c \lt \delta\cr \text{then}\ \ f(x)  \ell \lt \epsilon \end{gather} $$

Mathematics has its nouns and sentences just as English does.
Mathematical nouns are called expressions; these are just names given to mathematical objects of interest.
Mathematical sentences must express a complete thought: they can be (always) true, (always) false, or sometimes true/sometimes false.

I wrote a tiny book, called One Mathematical Cat, Please! that explores the language of mathematics.
The entire book is online. It's an easy read. No mathematics beyond basic arithmetic with numbers like $\,1\,,$ $\,2\,$ and $\,3\,$ is required. It's appropriate as a supplement to any math class, from about grade 7 through college level, or for selfenrichment.
BrooksCole Publishers loved this book. They said:
It is wonderfully written and crafted with a care you rarely see.
If you can find the right place to put it, this book will do a great service to the mathematical educational world.But, it's a supplement, not a text for an entire course—and supplements don't sell. The ‘Cat book’ (that's my short name for it) is incorporated in my free online Algebra I course . That is, if you go through the entire course, you'll get everything in the Cat Book (and lots more).

So, why the name ‘One Mathematical Cat, Please’?
‘Cat’ is a common English noun. And, ‘$\,x\,$’ is a common mathematics ‘noun’.
So, $\,x\,$ is to mathematics as ‘cat’ is to English!

I call my original Cat book the ‘CatonSwing’ book, because of the printversion cover. The entire book is available in pdf form, for free, at the bottom of this page.
The remainder of this talk will go through some of the key ideas in the Cat Book, in order.
Chapter 1: The Language of Mathematics

Remember: the mathematical analogue of an English noun is called an expression. Thus, an expression is just a name given to a mathematical object of interest. Whereas in English we talk about people, places, and things, in mathematics we talk about numbers, functions, and sets.
[Search in the Chapter 1 pdf for ‘ideas regarding expressions’.]
Numbers are the most common type of mathematical expression. And, numbers have lots of different names. (Different names are appropriate in different situations.) Take a look at some names for the number $\,5\,.$ They all look different, but they're just different names for the same number.
English has the same concept: synonyms are words that have the same (or nearly the same) meaning. However, this ‘same object, different name’ idea plays a much more fundamental role in mathematics than in English.

[Search in the Chapter 1 pdf for ‘ideas regarding sentences’.]
Just as English sentences have verbs, so do mathematical sentences. In the mathematical sentence ‘$\,3 + 4 = 7\,$’, the verb is ‘$\,=\,$’. If you read the sentence as ‘three plus four is equal to seven’, then it’s easy to ‘hear’ the verb. Indeed, the equal sign ‘$\,=\,$’ is one of the most popular mathematical verbs.

For a sentence, you can ask about its truth. Sentences can be:

(always) true, like:
$\,1 + 2 = 3\,$
$\,x + 1 = 1 + x\,$ 
(always) false, like:
$\,1 + 2 = 4\,$
$\,x = x + 2\,$ 
sometimes true, sometimes false (ST/SF), like:
$\,x = 1\,$
$\,x > 1\,$
The notion of truth (i.e., the property of being true or false) is of fundamental importance in mathematics!
(It doesn't make sense to ask about the truth of expressions! Is $\,x\,$ true? Is $\,x\,$ false? These are not valid questions!)

(always) true, like:

Languages have conventions. In English, for example, it is conventional to capitalize proper names (like my name, ‘Carol’). This convention makes it easy for a reader to distinguish between a common noun (like ‘carol’, a Christmas song) and a name.
Mathematics also has its conventions, which help readers distinguish between different types of mathematical expressions.
For example, constants are typically named with letters near the beginning of the alphabet, like $\,a\,,$ $\,b\,$ and $\,c\,.$ Variables are typically named with letters near the end of the alphabet, like $\,x\,$ and $\,y\,.$
When you see something like $\,y = ax^2 + bx + c\,,$ you use these conventions naturally!

We're going to do an exercise all together. To do exercises with a group, I often use a technique that I like to call ‘Choral Response’ (choral, as in a singing group—a chorus).
Here are the rules:
 I read the question aloud, followed by a few seconds of silence for thinking time.
 I start moving my hands in some weird way, to clue you in that you'll be responding soon. (Students are generally amused with their teachers waving arms in strange ways. Plus, it's good exercise!)
 At the instant my hands HIT the top of my head, everyone calls out their answer at once.
 Say ‘ONE’ at the appropriate time.
 Say ‘TWO’ at the appropriate time.

Let's use choral response to do the ‘Sentences Versus Expressions’ exercise together.
 I'll read 1–16 aloud, pausing for a choral response after each one.
 You'll say one of four things: English noun, mathematical expression, English sentence, mathematical sentence

If it's a sentence, we'll ask two additional questions:
 What is the verb?
 Is it always true, always false, or sometimes true/sometimes false?

Do problems 18.

Do problems 915.

This one is interesting. Although it has some of the syntax of an English sentence (capital letter at beginning, period at end, a verb), the words have not been used in a proper context to express any meaning. It is nonsensical.
It is common for beginning students of mathematics to write ‘nonsensical’ things analogous to this.

[Search for ‘simplify’ in the Chapter 1 pdf.]
To ‘simplify an expression’ means to get a different name for the expression, that in some way is simpler. But, what does ‘simpler’ really mean? It can mean:

FEWER SYMBOLS:
‘$\,3 + 1 + 5\,$’ and ‘$\,9\,$’ are both names for the same number, but ‘$\,9\,$’ uses fewer symbols—it is shorter. 
FEWER OPERATIONS:
‘$\,3+3+3+3+3\,$’ and ‘$\,5\cdot 3\,$’ are both names for the same number, but the latter uses fewer operations.

FEWER SYMBOLS:

‘Simpler’ can also mean:

BETTER SUITED FOR CURRENT USE:
The name $\,\frac{1 \text{ foot}}{12 \text{ inches}}\,$ is a great name for the number ‘$\,1\,$’ if we need to convert inches to feet. 
PREFERRED STYLE/FORMAT:
Both $\,\frac{2}{4}\,$ and $\,\frac{1}{2}\,$ are the same number, but people usually prefer the latter.

BETTER SUITED FOR CURRENT USE:

When presenting the sentence example ‘$\,1 + 2 = 3\,$’, I'm often asked:
If ‘$\,=\,$’ is the verb, then what is the ‘$\,+\,$’?
Here’s the answer. The symbol ‘$\,+\,$’ is a connective. A connective is used to ‘connect’ objects of a given type to get a ‘compound’ object of the same type. Here, the numbers $\,1\,$ and $\,2\,$ are ‘connected’ to give the new number $\,1 + 2\,.$
A familiar English connective for nouns is the word ‘and’: ‘cat’ is a noun, ‘dog’ is a noun, ‘cat and dog’ is a ‘compound’ noun.
Chapter 2: The Real Numbers

[Search in the Chapter 2 pdf for ‘size versus order’.]
With this audience, I don't have to worry about your comfort level with things like the real number line, whole numbers, integers, averaging, and so on!
However, there is a distinction made in this book (and my entire online curriculum) that is pickier than in the rest of the K–12 mathematics world: size versus order.
There are two different concepts frequently used to compare numbers:
 SIZE refers to a number's distance from zero . The words ‘bigger’ and ‘smaller’ are used to talk about size. Bigger means farther away from zero; smaller means closer to zero.
 ORDER refers to the natural left/right ordering on the number line. Given two numbers, either they are equal, or one lies to the right of the other. The words ‘greater than’ and ‘less than’ are used to talk about order. ‘Greater than’ means ‘lies to the right of’. ‘Less than’ means ‘lies to the left of’.
The words ‘bigger’ and ‘greater’ are not synonyms! You'll never hear me say ‘$\,5\,$ is bigger than $\,3\,$’ if my point is that $\,5\,$ lies further to the right on the number line!
The words ‘smaller’ and ‘lesser’ are not synonyms! You'll never hear me say ‘$\,3\,$ is smaller than $\,5\,$’ if my point is that $\,3\,$ lies further to the left on the number line!
Chapter 3: Mathematicians are Fond of Collections

Sets are a common and important type of expression. A set is a collection with the following property: given any object, either the object is in the collection, or isn't in the collection.
The key idea is this: to qualify as a set, one need only be certain that every object (like the number $\,2\,,$ or ‘chair’, or ‘grasshopper’) is either IN the collection, or NOT IN the collection.
It’s not necessary to know which of these two cases occurs (i.e., whether the object is IN or NOT IN the collection); it’s only necessary to know that exactly one of these two situations occurs!
This idea can be difficult for students, so I like to use the following example:

Is ‘the collection of all numbers that are divisible by $\,3\,$’ a set? That is, given any object, is it either IN or NOT IN the collection?
Here's a very big number. Is it divisible by $\,3\,$?
I know you could figure this out—but it would require tremendous patience, be a waste of time and prone to human error. (It also exceeds the input length allowed at WolframAlpha!)
However, we can with complete certainty say that either $\,3\,$ DOES go in evenly, or it DOESN'T. We don't need to know which of these two situations occurs!
The ‘collection of all numbers that are divisible by $\,3\,$’ IS a set.

Mathematics has primarily evolved to be a written language, not a spoken one. It largely conveys its meaning by being looked at. Imagine teaching a math class without a board or any visuals!
Consequently, people sometimes run across mathematical stuff that is not convenient to read aloud. This contributes to its difficulty for some students—how can they understand something they can't even verbalize?

Compare these sets. They all look pretty similar.
Let's use choral response—answer YES or NO:
 Is $\,2\,$ in the first set? (This is list notation.)
 Is $\,2\,$ a member of the second set? (This is interval notation.)
 Is $\,2\,$ an element of the third set? (I like this memory device—dust collects in corners!)
 Does the last set contain $\,2\,$?
Now, try choral response on these:
 Read the first set aloud.
 Read the third set aloud.
Likely garbled responses! We're used to looking at these to get their meaning, not verbalizing them.
[August 2022 update: Now, all my lessons have audio readthrus with text highlighting, so users can hear the math being read aloud while they look at it!]

In mathematics, the phrase
LET $\,S = \{1,2,3\}\,$
means:
Take the set $\,\{1, 2, 3\}\,$ and give it the name $\,S\,,$ so that it will be easier to refer to.
In this way, you can give a desired name to any expression. The word ‘LET’ is the key to knowing that something is being named!
Choral response: How would a mathematician give the name $\,n\,$ to (say) the number $\,\pi + 2\,$?

I have several setrelated exercises in my online Algebra I curriculum. (Remember—the entire ‘cat’ book is incorporated into Algebra I.)

Introduction to Sets
[quickly scroll through; try a few exercises] 
Interval and List Notation
[quickly scroll through; try a few exercises] 
Reading Set Notation
[quickly scroll through; try a few exercises]

Introduction to Sets
Chapter 4: Holding This, Holding That

I tried very hard to minimize the appearance of variables in the first few sections of the ‘cat’ book, because variables can be stressful for beginning students of mathematics.
However, taking away a mathematician’s ability to use variables is like asking to empty a swimming pool with a teaspoon—maybe the job can be done, but it will take very long, and be very laborious.
Variables are used to ‘hold’ objects. There's always some ‘supplier’ lurking in the background, specifying what the variable gets to hold. Here are a couple examples:
 If $\,h\,$ is used to represent the height of a rectangle, then $\,h\,$ can ‘hold’ any positive number.
 If $\,b\,$ is used to represent the number of books in a home library, then $\,b\,$ can ‘hold’ the numbers $\,0\,,$ $\,1\,,$ $\,2\,,$ and so on.

[Scroll to bottom of page 40 in the Chapter 4 pdf—the varable/universal set picture.]
The universal set for a variable is its supplier—its universe: the variable is allowed to ‘hold’ anything that lives in its universal set.
In other words, what the variable ‘holds’ is allowed to vary over the entire universal set; hence the name variable is appropriate.

[Search in the Chapter 4 pdf for ‘some common uses for variables’.]
There are three common uses for variables:
(1) To state a general principle
Example: For all real numbers $\,x\,,$ $\,y\,,$ and $\,z\,,$ $\,(x + y) + z = x + (y+z)\,.$Many students don't appreciate that this is the fact that allows them to write $\,x + y + z\,$ (no parentheses) without any ambiguity! What a nuisance it would be to always have to put parentheses in place!
Also, try to say this in English—maybe something like: When three numbers are being added, the grouping doesn’t affect the result. You can group the first two numbers, then add the third; or, you can add the first number to the sum of the second and third.
Verbose! English isn't finetuned to say things like this!

[Search for ‘example 2’ in the Chapter 4 pdf.]
(2) To represent a sequence of operations
The expression $\,2x + 1\,$ represents the sequence of operations: take a number, multiply by $\,2\,,$ then add $\,1\,.$Compare with the expression $\,2(x+1)\,,$ which represents the sequence: take a number, add $\,1\,,$ then multiply by $\,2\,.$
Think about reading each of these aloud—another example that mathematics is primarily meant to be looked at, not spoken!
 $\,2x+1\,$: two ex plus 1

$\,2(x+1)\,$: two times the quantity ex plus one;
or (verbatim) two times, open parenthesis, ex plus one, close parenthesis

(3) To represent something that is currently ‘unknown’, but that we would like to know
Very often in mathematics, you know something about a number, without knowing (at least initially) exactly what the number is.
Here's a general approach to such problems:
 A name is assigned to the thing you want to know (but don’t initially know). This is your variable!
 Write a mathematical sentence involving the variable, which is true when the variable takes on the desired value(s). Initially, the sentence may look complicated; it may not be the least bit obvious what value(s) of the variable make the sentence true.
 Find the choice(s) for the variable that make your sentence true. Mathematics provides lots of tools for doing this!

We'll apply these three steps to this example:
There’s a sale at your favorite clothes store. First, everything was discounted by $\,30\%\,.$ Now, they’re taking an additional $\,20\%\,$ off the previous sale prices.
You’ve got $\,\$100\,$ in your clothes budget. Since you’re anticipating a big crowd, you want to be prepared to grab the clothes and head for the register.
Accounting for $\,5\%\,$ sales tax, in addition to the discounts, how many dollars worth of clothes can you bring to the register? (You’ll be totaling up the original prices on the tags; before any discounts.)

[Search for ‘step 2’ in the Chapter 4 pdf.]
Let $\,p\,$ be the nondiscounted sum of the original prices. For this [CAMT] audience, writing the sentence that we want to be true is easy:
 $\,30\%\,$ discount means $\,70\%\,$ remains: $\,0.7p\,$
 an additional $\,20\%\,$ discount means $\,80\%\,$ of the prior amount remains: $\,(0.8)(0.7p)\,$
 apply the $\,5\%\,$ tax: $\,1.05(0.8)(0.7p)\,$
 we're willing to spend the entire $\,\$100\,$: $1.05(0.8)(0.7p) = 100$

An easier name for the number $\,1.05(0.8)(0.7)\,$ is $\,0.588\,.$ Numbers have lots of different names, and different names are better for different purposes! Now, the equation is: $0.588p = 100$
The sentence ‘$\,0.588p = 100\,$’ is not particularly convenient to work with, from the point of view of determining when it is true. In this form, you must think: ‘What number, when multiplied by $\,0.588\,,$ gives $\,100\,$?
So, we’ll ‘transform’ the sentence into one that’s easier to work with. As all of you well know, there are certain things that you can do to sentences, that will make them look different, but that won’t change when they’re true, or when they’re false.

Now, resist (for a moment) your equationsolving abilities, and just think:
 If two numbers are equal and you divide them both by the same (nonzero) thing, will the resulting numbers still be equal? Sure!
 If two numbers are not equal, and you divide them both by the same (nonzero) thing, will they remain nonequal? Sure!
 So, if we divide both sides of the sentence $\,0.588p = 100\,$ by the same number, then we’ll end up with a sentence that looks different, but that has the same truth values as the sentence we started with!
This final sentence is so simple that it tells you when it's true!
So—you could bring about \$170 worth of clothes to the counter! Any more, and you won’t have enough money. Any less, and you’ll get some change.

[Search for ‘uppercase letters’ in the Chapter 4 pdf.]
English has lots of conventions. For example, the capitalization of proper nouns clues the reader that ‘Carol’ (capital C) refers to a person, whereas ‘carol’ (lowercase c) refers to a Christmas song.
Mathematics has lots of conventions regarding the naming of variables, which help clue the reader to the type of objects the variable can ‘hold’.
 Numbers are usually represented by lowercase letters. Since a price is a number, we used a lowercase $\,p\,,$ not a capital $\,P\,.$
 Sets are usually represented by uppercase letters. The (blackboard bold) $\,\Bbb R\,$ used to represent the set of real numbers, and the (blackboard bold) $\,\Bbb Z\,$ used to represent the set of integers are both uppercase letters.

[Search for ‘middle of alphabet’ in the Chapter 4 pdf.]
 A variable with universal set $\,\Bbb R\,$ (or, any interval of real numbers) is most likely to be named with a lowercase letter from the end of the alphabet, particularly $\,t\,,$ $\,x\,,$ or $\,y\,.$
 A variable with universal set $\,\Bbb Z\,$ (or, any subset of the integers) is most likely to be named with a lowercase letter near the middle of the alphabet; particularly $\,i\,,$ $\,j\,,$ $\,k\,,$ $\,m\,,$ or $\,n\,.$ (MEMORY DEVICE: The integers use letters from $\,i\,$ to $\,n\,.$ )
These conventions are summarized in the diagram on page 51 of Holding This, Holding That.

Let's use Choral Response to practice Using Mathematical Conventions.

[Search for ‘handwriting’ in the Chapter 4 pdf.]
Whenever letters are used in a mathematical context (i.e., as variables), they are typeset in an italic style. This convention helps to visually distinguish letters being used in a mathematical way from letters being used in a nonmathematical way.
When handwriting mathematics, it’s particularly easy to confuse variables with other things, as the cautions on page 52 of Holding This, Holding That show. Oh—the times I've struggled to distinguish $\,x\,$ from $\,y\,$ when helping students!

For these reasons, when handwriting mathematics, you want to try and ‘duplicate’ an italic font, as illustrated in the table on the bottom of page 52 of Holding This, Holding That.
I know I've likely driven my students crazy over the years, admonishing them to please ‘wiggle their exes’ and ‘cross their zees’!
Chapter 5: I Live Two Blocks West Of You

[Search for ‘memory devices’ in the Chapter 5 pdf.]
This chapter explores the natural ordering of the real numbers. I offer some memory devices that my students have found helpful over the years:

Is $\,x \lt y\,$?
(Let the $\,\lt \,$ symbol ‘fall’; it turns into the letter ‘L’.) Is $\,x\,$ Less than $\,y\,$? Does $\,x\,$ lie to the Left of $\,y\,$ on a number line? 
Is $\,x > y\,$?
(The $\,>\,$ symbol can easily be made into a capital letter ‘R’.) Is $\,x\,$ gReateR than $\,y\,$? Does $\,x\,$ lie to the Right of $\,y\,$ on a number line?

Is $\,x \lt y\,$?

Recall that I make a careful distinction between size (distance from zero) and order (left/right positioning).
So, I admonish my students: don't read $\,x \t y\,$ as ‘$\,x\,$ is smaller than $\,y\,$’! Instead, read it as ‘$\,x\,$ is less than $\,y\,$’.

I offer memory devices for the phrases ‘at least’ and ‘at most’, which are sometimes difficult for students:
 The sentence ‘$\,x\,$ is at least $\,5\,$’ means that the least (furthest left) $\,x\,$ is allowed to be is $\,5\,$; it can be $\,5\,,$ or any number greater than $\,5\,.$ So, the phrase ‘$\,x\,$ is at least $\,5\,$’ means ‘$\,x \ge 5\,$’.
 The sentence ‘$\,x\,$ is at most $\,10\,$’ means that the most $\,x\,$ is allowed to be is $\,10\,$; it can be $\,10\,,$ or any number less than $\,10\,.$ So, the phrase ‘$\,x\,$ is at most $\,10\,$’ means ‘$\,x \le 10\,$’.

Let's use Choral Response to try a few problems involving order and size:

Bigger, Smaller, Greater, Lesser
Remember: I'll read the question aloud and give you a little think time. You'll answer when my hands touch my head. (I'll pickandchoose a bit on the questions, because some would lend themselves to pretty muddy responses!) 
Practice With the Phrases ‘At Least’ and ‘At Most’

Bigger, Smaller, Greater, Lesser
Chapter 6: Numbers Have Lots of Different Names

There are lots of number games that can make you look clairvoyant. One such game goes something like this:
 YOU (speaking to another person): Think of a number, but don't tell me what it is! Do (this and this and this) to the number. I’ll bet you ended up with (some number)—am I right?
 OTHER PERSON: You’re right! How did you do that?
How do these work? They're all a consequence of an extremely important fact: numbers have lots of different names!

One such game (which has made its way around the internet several times) is Chocolate Math.
When I taught an HTML and Web Design Course at Miss Hall's School, one of my incredible students created a fun rendition of the game, which you can see (and play) here: Chocolate Math
If you want to see the details on how this works, check out: Why Chocolate Math Works

In English, words that look different, but have (nearly) the same meaning, are called synonyms. For example, ‘anxious’ and ‘fretful’ are synonyms. But, there is no language in the world where the idea of ‘different name, same meaning’ is more prevalent than in the language of mathematics.
The ability to take a number, and get a name for that number that is useful to you, is a key to success in mathematics. There are two favorite ways to get a new name for a number (without changing where the number lives on a number line):
 by adding zero; or
 by multiplying by one.

Unit conversion is an incredibly useful application of ‘getting a new name by multiplying by $\,1\,$’.
Whenever two nonzero quantities $\,a\,$ and $\,b\,$ are equal, then $\displaystyle\,\frac ab\,$ is the number $\,1\,.$
So, since (say) one foot equals twelve inches, both
$\,\displaystyle\frac{1\text{ foot}}{12\text{ inches}}\,$ and $\,\displaystyle\frac{12\text{ inches}}{1\text{ foot}}\,$
are the number $\,1\,$ in disguise!
You can call them conversion factors if you want, but they're just particularly useful names for the number $\,1\,$ in a given context!

[Search for ‘my car is’ in the Chapter 6 pdf.]
I usually drive the speed limit (or pretty close). Consequently, I often find myself being passed by other cars on the freeway. I often wonder—just how fast are those other cars going as they whiz by?
At the time I wrote this little book, my car was about $\,14\,$ feet long. If it took about $\,2\,$ seconds for the front of a passing car to travel from my rear bumper to my front bumper, then the passing car’s additional speed was $\displaystyle\,\frac{14\text{ ft}}{2\text{ sec}}\,.$ How many miles per hour is this?

$$ \frac{14\text{ ft}}{2\text{ sec}} = \frac{14\text{ ft}}{2\text{ sec}}\cdot \frac{1\text{ mile}}{5280\text{ ft}}\cdot \frac{60\text{ sec}}{1\text{ min}}\cdot \frac{60\text{ min}}{1\text{ hr}} \approx 4.8 \text{ mph} $$
Look at all those beautiful ones!
Similarly, the car that whizzes by in about half a second is going about $\,20\,$ miles per hour faster than me!

Another way I've tortured my students for almost three decades is to make them use Complete Mathematical Sentences (CMS). Early on in my career, I should have had a custom ‘CMS’ stamp made—it would have saved me so much writing!
That previous car calculation is a CMS—it takes you from one name for an expression to a different name. And, whenever two expressions are compared with an equal sign, they must be actually equal. Contrary to popular belief, the equal sign is NOT simply an indicator that ‘I'm going on to the next step’!

[Go to page 85, ‘Simplifying An Expression—Preferred Formats, in the Chapter 6 pdf.]
When my students simplify expressions, I always want just one mathematical sentence that takes them from the original name to the desired name. I strongly prefer the formats shown here, because they lend themselves to adding reasons, in parentheses, for each step. The original expression stands out, as does the final desired name.
[Scroll down a bit to the unhappy face on page 85 of the Chapter 6 pdf.]
In particular, all my students know that I HATE DANGLING EQUAL SIGNS!!
Chapter 7: These Sentences Certainly Look Different

Whereas the ‘$\,=\,$’ sign gives a way to compare mathematical expressions (like numbers and sets), the idea of equivalence gives a way to compare mathematical sentences.
For example, the sentences ‘$\,2x3 = 0\,$’ and ‘$\displaystyle\,x = \frac 32\,$’ certainly look different. However, if you substitute any number for $\,x\,,$ then if it makes one true, it also makes the other true; if it makes one false, it also makes the other false. So, even though they look different, they're alike in a very important way—they're true at the same time, and false at the same time.
When two mathematical sentences always have the same truth values, then they can be used interchangeably—you can use whichever sentence is easiest for a given situation. In this case, of course, the sentence ‘$\displaystyle\,x = \frac 32\,$’ is easiest, because it tells you when it is true!

The mathematical verb used to compare the truth values of sentences is ‘is equivalent to’.
Here’s how the language of math tells you that the sentences ‘$\,2x − 3 = 0\,$’ and ‘$\,x = \frac{3}{2}\,$’ always have the same truth values:
For all real numbers $\,x\,,$
$\,2x−3=0\,$ is equivalent to $\displaystyle\,x = \frac{3}{2}\,.$Chapter 8: (Transforming Tool #1: The Addition Property of Equality) and Chapter 9: (Transforming Tool #2: The Multiplication Property of Equality) explore the two most popular ways (very familiar to all of you!) to transform equations into equivalent ones.

[Search for ‘alternate ways’ in the Chapter 7 pdf.]
The more important something is, the more ways that seem to evolve to say the same thing. For equivalence, there are four ways to say the same thing:
 is equivalent to
 if and only if
 iff
 $\,\iff\,$
When I was an undergraduate at the University of Massachusetts at Amherst, one of my math teachers told this story: his wife typed his math papers for him. She changed all his iff's into if's (until she knew better), thinking he just didn't know how to correctly spell the word ‘if’!

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