Math is written in a language. It has an alphabet, nouns, verbs and grammar. Once you can read them, a page full of symbols turns into sentences you can say out loud.
There are six sessions, about 40 minutes each. Each one has a decoder table, translation practice in both directions, and one line to read aloud. The answers stay hidden until you open them, so try first and then check.
Some symbols here are Greek letters. When one shows up, this unit tells you what job it does and moves on.
Every symbol from the six sessions on two printed pages.
Session 1 · Alphabet · about 40 minutes
Writing numbers down
You can write the number of days in a year as 365, as CCCLXV, or as 365 scratches on a wall. All three name the same amount. Only one of them lets you add two numbers without a headache. The way we write numbers is a choice people made, and some choices work better than others. A number system is an agreement.
Decoder
||||| |||
Say it
eight
What it does
Tally marks: one stroke per thing, grouped in fives so they are easier to count. Anyone can read them, but big numbers take forever to write.
Where it comes from
Tallies are still used to keep score because they need no training to read.
XIV
Say it
fourteen
What it does
Roman numerals. X is ten, V is five, I is one. A smaller symbol written before a bigger one is taken away, so IV is four and XIV is fourteen.
Where it comes from
Roman numerals have no zero and no place value: X always stands for ten, never a hundred, wherever it sits.[15]
365
Say it
three hundred sixty-five
What it does
Place value. Each place is worth ten times the place to its right, so this is 3 hundreds, 6 tens and 5 ones.
Where it comes from
The base-ten place-value system grew up in India. Fibonacci's book Liber Abaci (1202) brought it to European readers.[11][13]
0
Say it
zero
What it does
A placeholder for an empty place. In 305, the 0 says there are no tens. Without it, 305 and 35 would look the same.
Where it comes from
Indian mathematicians were treating zero as a number in its own right by around 650 AD.[12]
(1, 25) in base 60
Say it
one sixty and twenty-five
What it does
Base 60: each place is worth sixty times the place to its right. This is one group of sixty plus 25 ones, which is 85.
Where it comes from
Babylonian scribes wrote numbers with place value in base 60.[10]
1011 subscript 2
Say it
one-zero-one-one in base two
What it does
Base 2 (binary): each place is worth twice the place to its right. This is one 8, no 4, one 2 and one 1, which is 11. The small 2 names the base.
Where it comes from
Gottfried Leibniz published an explanation of base-2 arithmetic in 1703.[14]
Translate: math to English
4,072Show answer
Four thousand seventy-two. That is 4 thousands, 0 hundreds, 7 tens and 2 ones.
XXIXShow answer
Twenty-nine. X + X is twenty, and IX is one before ten, so nine.
110 subscript 2Show answer
Six. One 4, one 2 and no 1.
(2, 10) in base 60Show answer
One hundred thirty. Two sixties are 120, plus 10.
Translate: English to math
Write nineteen in Roman numerals.Show answer
XIX
Write five in base 2.Show answer
101 subscript 2 (one 4, no 2, one 1)
Write three hundred six with digits.Show answer
306
Write 75 the Babylonian way, as (sixties, ones).Show answer
(1, 15)
Read it aloud
1011 subscript 2, 11 and XI
One-zero-one-one in base two, eleven, and X-I in Roman numerals are three ways to write the same number.
Related standards (reference only): VA SOL 3.NS.1 · VA SOL 4.NS.1
Session 2 · Nouns · about 40 minutes
Names for things
In "Sam has some marbles," the word "some" stands for a number nobody has told you yet. Math does the same thing with a letter. A letter in math is a name for a number. If the number can change, the letter is a variable. If the number never changes, the letter is a constant.
Decoder
x
Say it
ex
What it does
A variable: a name for a number you do not know yet, or one that can change. In x + 2, x could be any number.
Where it comes from
Using the late letters x, y and z for unknowns comes from René Descartes' La Géométrie (1637).[4]
a, b, c
Say it
ay, bee, see
What it does
Letters from the start of the alphabet often stand for numbers that are fixed in a problem but not written in yet, like the a and b in ax + b.
Where it comes from
Letters for known amounts are older than Descartes. Cardano was writing a and b for known numbers in 1570.[4]
n
Say it
en
What it does
Usually a counting number: how many. When you see n, expect a whole number like 1, 2 or 3.
Where it comes from
n is the first letter of "number," which makes it easy to remember.
t
Say it
tee
What it does
Usually time. In a formula about something moving, t is how long it has been moving.
Where it comes from
t is the first letter of "time." Many letters are picked this way: r for radius, h for height.
π
Say it
pie
What it does
A constant: the distance around any circle divided by the distance across it. It is about 3.14159 and it never changes.
Where it comes from
William Jones first used π for this number in 1706. Leonhard Euler made it popular from 1748.[5]
θ
Say it
theta
What it does
The usual name for an angle. When θ appears, someone is measuring how far something turns.
Where it comes from
θ is a Greek letter. Here its only job is to name an angle.
Translate: math to English
3nShow answer
Three times a number n. You can also say "three n."
x + 5Show answer
A number x, plus five.
2πr, where r is the radius of a circleShow answer
Two times pi times the radius. It is the distance around the circle.
a + bShow answer
A number a plus a number b.
Translate: English to math
A number plus seven.Show answer
x + 7. Any letter works, so n + 7 is just as right.
Five times a number.Show answer
5n, which means 5 times n.
Pi times four.Show answer
4π. The number goes in front by habit.
A number, minus two.Show answer
x − 2
Read it aloud
n + 1011 subscript 2
A number n, plus one-zero-one-one in base two. That is n plus eleven.
Related standards (reference only): VA SOL 5.PFA.2 · VA SOL A.EO.1 · CCSS MP.2
Session 3 · Verbs · about 40 minutes
Action words
A sentence needs a verb, or nothing happens. In math the verbs are the operation signs and the comparison signs. Plus, minus, times and divide do something to numbers. Equals, less than and greater than say how two things compare. Each of these little marks was invented by a person, and most of them are younger than you might think.
Decoder
+ and −
Say it
plus, minus
What it does
Add, take away. A minus sign in front of a number, like −5, can also mean negative five.
Where it comes from
Both signs first appeared in print in a 1489 arithmetic book by Johannes Widmann.[2]
×
Say it
times
What it does
Multiply. 3 × 4 means three groups of four, which is 12.
Where it comes from
The × sign first appears in 1618, in an appendix believed to be William Oughtred's work.[2]
÷
Say it
divided by
What it does
Divide, or share into equal groups. 12 ÷ 3 = 4 means 12 shared three ways gives 4 each.
Where it comes from
Johann Rahn first used ÷ for division in his book Teutsche Algebra (1659).[2]
=
Say it
equals, or is the same as
What it does
Both sides name the same number. It does not mean "the answer comes next." 7 = 3 + 4 is just as true as 3 + 4 = 7.
Where it comes from
Robert Recorde invented = in his book The Whetstone of Witte (1557).[1]
≠
Say it
is not equal to
What it does
The two sides name different numbers. 2 + 2 ≠ 5.
Where it comes from
Leonhard Euler used it in a letter to another mathematician, and he may have invented it.[3]
< and >
Say it
is less than, is greater than
What it does
The open side faces the bigger number. 3 < 5 and 5 > 3 say the same thing.
Where it comes from
They were introduced by the editor of Thomas Harriot's book Artis Analyticae Praxis (1631), not by Harriot himself.[3]
≤ and ≥
Say it
is less than or equal to, is greater than or equal to
What it does
Smaller or the same, bigger or the same. "Age ≥ 13" means 13 counts.
Where it comes from
John Wallis wrote a version with the bar on top in 1670. Pierre Bouguer put two bars underneath in 1734.[3]
Translate: math to English
12 ÷ 4 = 3Show answer
Twelve divided by four equals three.
7 ≠ 8Show answer
Seven is not equal to eight.
x > 10Show answer
A number x is greater than ten.
n ≤ 30Show answer
A number n is less than or equal to thirty.
Translate: English to math
Nine minus four is five.Show answer
9 − 4 = 5
A number is less than two.Show answer
x < 2
You must be at least 48 inches tall.Show answer
h ≥ 48, where h is your height in inches.
Three times a number equals twenty-one.Show answer
3 × n = 21, or 3n = 21. The number is 7.
Read it aloud
1011 subscript 2 + n ≥ 2π
One-zero-one-one in base two, plus a number n, is greater than or equal to two pi.
Related standards (reference only): VA SOL 6.PFA.4 · CCSS MP.2
Session 4 · Grammar · about 40 minutes
Sentence rules
"Let's eat, Grandma" and "Let's eat Grandma" use the same words. The comma changes everything. Math has the same problem: 2 + 3 × 4 could mean 20 or 14, depending on which part you do first. So everyone agrees on rules for the order. The rules are not laws of nature. They are grammar, an agreement that makes everyone read the same line the same way.
Decoder
3 + x
Say it
three plus x
What it does
An expression. It is like a phrase in English: it names a number but makes no claim, so it cannot be true or false.
Where it comes from
"A big red bus" is a phrase too. It names something without saying anything about it.
3 + x = 10
Say it
three plus x equals ten
What it does
An equation. It is a full sentence that claims two things are equal. The claim is true or false depending on x. Here it is true only when x is 7.
Where it comes from
An equation's verb is the = sign from session 3.
3 + x < 10
Say it
three plus x is less than ten
What it does
An inequality. It is a sentence that compares, and it can be true for many values at once. Here any x less than 7 works.
Where it comes from
Its verb is a comparison sign from session 3.
( )
Say it
the quantity
What it does
Parentheses are punctuation. Do what is inside them first. (2 + 3) × 4 is 5 × 4, which is 20. A number touching a parenthesis means multiply, so 2(x + 1) means 2 × (x + 1).
Where it comes from
Parentheses for grouping show up in rare cases in the 1500s. Niccolò Tartaglia used them in 1556.[8]
2 + 3 × 4
Say it
two plus three times four
What it does
With no parentheses, multiply and divide before you add and subtract. Work left to right within each step. So this is 2 + 12, which is 14, not 20.
Where it comes from
No single person invented this rule. It grew out of habit and was written down as a rule much later.[16]
(8 + 4)/3
Say it
the quantity eight plus four, divided by three
What it does
A fraction bar or slash with a group on top divides the whole group. Add first, then divide: 12 ÷ 3 = 4.
Where it comes from
A fraction bar does the same job as parentheses: it holds a group together.
Translate: math to English
(5 + 1) × 2Show answer
The quantity five plus one, times two. It is worth 12.
5 + 1 × 2Show answer
Five plus one times two. Multiply first, so it is worth 7.
2x + 1 = 9Show answer
Two x plus one equals nine. It is an equation, and it is true when x is 4.
10 − 2 × 3Show answer
Ten minus two times three. It is an expression worth 4.
Translate: English to math
The quantity four plus six, times three.Show answer
(4 + 6) × 3, which is 30.
Seven more than twice a number is fifteen.Show answer
2n + 7 = 15. The number is 4.
Twice the quantity of a number plus three.Show answer
2(x + 3)
Half of the quantity ten minus four.Show answer
(10 − 4) ÷ 2, which is 3.
Read it aloud
(1011 subscript 2 − n) × 2 ≠ 3
The quantity one-zero-one-one in base two minus n, times two, is not equal to three.
Related standards (reference only): VA SOL 5.CE.4 · VA SOL 7.PFA.2 · CCSS MP.7
Session 5 · Compression · about 40 minutes
Shorthand
Texting has shortcuts: "brb" means "be right back." Math has shortcuts too, and they pack a whole sentence into a few marks. They look scary because they are short, not because they are hard. Once you can unpack one into plain English, you can read it. Packing English back into symbols is how you write math fast.
Decoder
x superscript 3
Say it
x cubed, or x to the third
What it does
An exponent: x × x × x. The small raised number counts how many copies of x get multiplied. In the order of operations, powers come before multiplying.
Where it comes from
Nicolas Chuquet was already writing exponents in 1484. René Descartes is credited with the form we use today.[2]
√
Say it
the square root of
What it does
The number that, multiplied by itself, gives what is under the sign. √49 = 7, because 7 × 7 = 49.
Where it comes from
The √ sign first appeared in print in 1525, in Christoff Rudolff's book Die Coss.[2]
|x|
Say it
the absolute value of x
What it does
How far x is from zero, ignoring direction. |−6| = 6 and |6| = 6.
Where it comes from
Karl Weierstrass used it in a manuscript dated 1841. It reached print later.[6]
f(x)
Say it
f of x
What it does
A function: a rule named f, fed the input x. If f(x) = x + 2, then f(5) = 7. These parentheses do not mean multiply.
Every member of the first set is also in the second. {1, 2} ⊂ {1, 2, 3}. Some books write ⊆ for this and keep ⊂ for when the second set has extra members.
Where it comes from
Ernst Schröder introduced signs like this in 1890.[7]
Translate: math to English
5 superscript 2Show answer
Five squared. That is five times five, which is 25.
√81 = 9Show answer
The square root of eighty-one equals nine.
|−3| = 3Show answer
The absolute value of negative three is three.
f(2), if f(x) = 3xShow answer
Say "f of two." Put 2 into the rule f. Since f(x) = 3x, f(2) = 6.
7 ∈ {2, 4, 7}Show answer
Seven is an element of the set two, four, seven.
Translate: English to math
Two to the fifth power.Show answer
2 superscript 5, which is 32.
The sum of i, for i from 1 to 3.Show answer
Σ subscript i=1 superscript 3 i, which is 1 + 2 + 3 = 6.
The set a, b is a subset of the set a, b, c.Show answer
{a, b} ⊂ {a, b, c}
How far negative eight is from zero.Show answer
|−8|, which is 8.
G of x equals x squared minus one.Show answer
g(x) = x superscript 2 − 1
Read it aloud
f(n) = (n − 1010 subscript 2) superscript 2 ≥ 0
The function f of n equals the quantity n minus one-zero-one-zero in base two, squared, and that is greater than or equal to zero.
Related standards (reference only): VA SOL 6.NS.3 · VA SOL 7.NS.3 · CCSS MP.7
Session 6 · Argument · about 40 minutes
Making a case
Saying "I'm right" convinces nobody. A good argument is a chain: each step follows from the one before, and anyone can check every link. A proof is that chain written in math. The symbols in this session are the words that hold the chain together: if, then, therefore, for every and there is.
Decoder
if P, then Q
Say it
if P, then Q
What it does
Whenever P is true, Q is true too. It says nothing about what happens when P is false.
Where it comes from
P and Q are placeholders for whole sentences, the way x stands in for a number.
⇒
Say it
implies
What it does
Short for "if ... then." x = 3 ⇒ x superscript 2 = 9.
Where it comes from
David Hilbert used a single arrow for this in 1922. The Bourbaki group used the double arrow in 1954.[7]
⇔
Say it
if and only if
What it does
The arrow works both ways: each side guarantees the other. x + 1 = 4 ⇔ x = 3.
Where it comes from
The Bourbaki group used it in the same 1954 book.[7]
∴
Say it
therefore
What it does
What comes next follows from what came before.
Where it comes from
Its upside-down twin, ∵, means "because."
∀
Say it
for all, or for every
What it does
The claim holds for every value, with no exceptions. ∀n, n + 0 = n.
Where it comes from
Gerhard Gentzen introduced the upside-down A, for "all," in 1935.[7]
∃
Say it
there exists, or there is
What it does
At least one value makes the claim true. ∃x, x + 2 = 5.
Where it comes from
Giuseppe Peano used a backwards E in 1897. Bertrand Russell was the first to use it the way we do today.[7]
Read a proof like a paragraph
Two even numbers always add up to an even number.
a = 2m and b = 2nSuppose a and b are even. Then a is 2 times some whole number m, and b is 2 times some whole number n.
a + b = 2m + 2nAdd them.
2m + 2n = 2(m + n)Take out the 2 that both parts share.
∴ a + b is evenSo a + b is 2 times a whole number, and that is what even means.
The same proof as a paragraph
Suppose a and b are even, so each one is 2 times some whole number. When you add them, you get 2m + 2n, and both parts share a 2, so the sum is 2 times (m + n). Anything that is 2 times a whole number is even. Therefore the sum of two even numbers is even, every time.
Translate: math to English
x = 2 ⇒ 3x = 6Show answer
If x equals two, then three x equals six.
∀n, n × 1 = nShow answer
For every number n, n times one equals n.
∃x, x + 4 = 10Show answer
There is a number x where x plus four equals ten. It is 6.
a = b and b = c ∴ a = cShow answer
Say "a equals b and b equals c, therefore a equals c."
Translate: English to math
If a number is greater than ten, then it is greater than five.Show answer
x > 10 ⇒ x > 5
For every number x, x plus zero is x.Show answer
∀x, x + 0 = x
There is a number whose square is 49.Show answer
∃n, n superscript 2 = 49
x equals four if and only if x plus one equals five.Show answer
x = 4 ⇔ x + 1 = 5
Read it aloud
∀n ∈ {1, 10 subscript 2, 11 subscript 2}, n superscript 2 ≥ n
For every n in the set one, one-zero in base two, and one-one in base two, n squared is greater than or equal to n.
Related standards (reference only): VA SOL G.RLT.1
Capstone: your phrasebook page
A phrasebook helps a traveller read signs in a new language. Make one page of your own for math.
Pick any math book or worksheet, at any level, even one that looks too hard.
Copy five lines of real notation from it.
Under each line, write it out in plain English.
Read each line aloud to someone, using your English version.
Circle any symbol you still could not decode, and look it up.
For teachers
Use it as a short unit before algebra, as a reset for anyone who gets stuck on notation, or as independent reading. It needs no login and no account. The standards listed for each session are related codes for reference. They are not alignment claims.