The big idea of algebra in one picture: a single letter n is an empty box that can hold any number, and 5n means "five times whatever n turns out to be."
Every number you have ever met has been a settled, finished thing. The number 7 is seven and will never be anything else; 100 is one hundred forever. That is exactly what makes arithmetic feel safe. But it is also a kind of cage. What if you want to talk about the price of one apple before you know whether apples cost a dollar or three? What if you want a rule that works for any day, any number of people, any shape in a pattern — all at once? For that you need a number that has not made up its mind yet. You need a letter.
This lesson is the doorway into algebra. A letter such as x or n is not a mysterious code; it is simply a blank box waiting to hold a number you will choose later. By the end you will be able to do five things: explain why we reach for a letter instead of a number, translate an English phrase into a symbol expression, write that expression the tidy way everyone agrees on, evaluate it by dropping in a specific value, and describe a whole pattern with a single line. We keep one steady habit of color throughout: a letter (variable) is blue, a number that multiplies it (a coefficient) is amber, and an exponent is purple.
7.1.1 Why use a letter instead of a number
Picture a fruit stand. One apple costs $5. How much do two apples cost? Ten dollars. Five apples? Twenty-five dollars. You are doing the same move every time: take the number of apples and multiply by 5. The number of apples keeps changing, but the recipe never does. Algebra's first gift is a way to write that unchanging recipe down once, even though one of its ingredients is still unknown.
Let the letter n stand for the number of apples. We have not decided what n is — it might be 2, it might be 40 — so think of n as a box with nothing in it yet. Then the cost of n apples is 5n dollars, which is shorthand for 5 times whatever number sits in the box. Drop a 2 into the box and the cost is 5×2 = $10; drop in a 40 and it is 5×40 = $200. One small expression, 5n, quietly contains every possible answer at the same time.
Whatever number drops into the box n, the rule 5n turns it into a cost. The letter holds the place; the number arrives later.
Letters earn their keep anywhere a quantity can vary. If you are a years old today, then in 10 years you will be a + 10 years old — true whether you are 11 or 35. If a movie ticket costs p dollars, then three tickets cost 3p dollars. The letter is a stand-in, a placeholder, a promise that says: "I am some number; tell me which one whenever you like, and the rule still works." A letter used this way has a name — we call it a variable, because the value it holds is free to vary.
Key idea
A variable is a letter that stands for a number — an empty box you can fill in later. Writing a rule with a variable lets you capture every case at once: 5n is the cost of n apples for all values of n at the same time. The expression doesn't compute a single answer — it stores a recipe for answers.
Worked example — naming the variable first
A taxi charges a flat $3 to get in, plus $2 for each mile. Write an expression for the total cost of a ride.
Decide what varies and give it a letter: let m = the number of miles. name the unknown
The mileage part is $2 for each of m miles: that is 2m. "for each" signals multiply
Add the flat fee that never changes: 2m + 3 dollars. the $3 is a constant
Now one expression covers every trip: a 4-mile ride costs 2×4 + 3 = $11, a 10-mile ride costs 2×10 + 3 = $23.
🎮 Try itThe blank that holds any number
Set n, the number of apples (1 to 12). Watch the same number drop into the box and watch the cost 5n compute itself. The letter is the blank; the value is what you put in it.
Apples n3
7.1.2 Translating words into symbols
Most algebra starts life as an English sentence. The skill you need is translation: turning a phrase, word by word, into a symbol expression. The trick is to read slowly and notice which everyday words are secretly math instructions. "More than" and "increased by" mean add. "Less than" and "decreased by" mean subtract. "Times," "twice," "double," and "of" mean multiply. "Per" and "divided into" mean divide.
A little dictionary. Read the phrase from left to right, swap each word for its symbol, and the expression appears.
One trap deserves its own warning, because nearly everyone falls into it once. The phrase "5 less than x" means you start with x and take 5 away from it, so it is x − 5 — not5 − x. Think of it in plain English: "5 less than your age" is your age minus 5, not 5 minus your age. The words "less than" and "subtracted from" flip the order you might expect, so write the starting quantity first and subtract the smaller piece after it.
Watch out — "less than" flips the order
"5 less than x" is x − 5, never 5 − x. The same flip happens with "subtracted from": "3 subtracted from y" is y − 3. Addition is friendlier — "5 more than x" and "x plus 5" are both x + 5, because adding doesn't care about order — but subtraction always does.
Worked example — a two-step phrase
Translate "double x, then add 1."
"double x" means two times x: write 2x. double = times 2
"then add 1" attaches a + 1 to what you have: 2x + 1. build left to right
Notice the order of words matched the order of operations: we doubled first, then added. "Double x, then add 1" is 2x + 1; the different phrase "add 1 to x, then double" would be 2(x + 1) instead.
🎮 Try itWords-to-symbols translator
Pick a phrase. See it rebuilt as an expression, piece by piece — then slide x to watch the expression turn into a number. Look closely at "5 less than x."
Let x =4
7.1.3 Writing expressions properly
Mathematicians are tidy on purpose. Over the centuries they settled on a small set of writing conventions so that an expression looks the same in every classroom and every country. None of these rules change what an expression means; they just make it cleaner and quicker to read. Learn the five below and your algebra will look like a textbook's.
Five tidying rules. Each one shortens the writing without changing the meaning.
Here are the conventions in words. (1) Drop the times sign between a number and a letter, or between two letters: 3×x becomes 3x, and a×b becomes ab. The two symbols sitting side by side already means "multiply." (2) Write the number first. A coefficient goes in front of its letter: write 3x, never x3 (which looks like it could mean something else entirely). (3) A coefficient of 1 disappears:1×y is just y, because one of something is simply that something. (4) Show division as a fraction:x÷2 is written as the stacked fraction x2. (5) Use an exponent for a repeat: when a letter is multiplied by itself, x×x = x2, read "x squared."
Watch out — 3x is not x³, and 2x is not x²
Putting a number in front of a letter means multiply: 3x is x + x + x. Putting a number up high as an exponent means repeated multiplication: x3 is x×x×x. They are completely different. At x = 4, 3x = 12 but x3 = 64.
🎮 Try itTidy it up
Each row shows a messy form. Press Clean it up to rewrite them all the proper way, and read why each rule applies. Press Mess it up to start over.
7.1.4 Evaluating an expression
An expression like 3x + 1 is a recipe with one blank. To evaluate it means to choose a number for the letter, drop that number into the blank, and work out the single value that comes out. The act of dropping a number in is called substitution. Once the letter is gone and only numbers remain, you finish with the ordinary order of operations you already know: do the multiplying before the adding.
Suppose x = 2 in 3x + 1. Substitute the 2 for x: the expression becomes 3×2 + 1. Multiply first to get 6 + 1, then add to get 7. The recipe 3x + 1 turned the input 2 into the output 7.
To evaluate, substitute the number for the letter (parentheses keep it clear), then follow the order of operations: multiply, then add.
Wrapping the substituted number in parentheses is a small habit that saves you from real mistakes, and it matters most when the value is negative. Suppose x = −2 in the same expression 3x + 1. Write 3(−2) + 1 — the parentheses make it obvious that 3 multiplies the whole −2. That gives −6 + 1 = −5. Without the parentheses it is far too easy to write "3 − 2 + 1" by accident and get the wrong answer. Always cradle a negative substitution in parentheses.
Worked example — evaluating with a negative
Evaluate 2x − 5 when x = −3.
Substitute −3 for x, in parentheses: 2(−3) − 5. cradle the negative
Multiply: 2×(−3) = −6, so we have −6 − 5. a positive times a negative is negative
Subtract: −6 − 5 = −11. moving further below zero
So 2x − 5 equals −11 when x = −3. (Quick check at x = 4: 2×4 − 5 = 3.)
🎮 Try itThe evaluate machine
Choose an expression, then step x from −3 up to 5. Watch the substitution line appear with parentheses, then watch it simplify step by step. It stays correct for negatives.
Expression
Let x =2
7.1.5 Describing patterns with letters
Here is where letters truly shine. Suppose you build a row of squares out of matchsticks. The first square takes 4 sticks. To add a second square you don't need 4 more — the squares share a side, so you only add 3. The third square adds another 3, and so on. The pattern starts at 4 and then grows by a steady 3 each time. Watching that steadiness is the secret to capturing the whole thing in one line.
The first square needs 4 sticks; each square after that adds only 3, because neighbors share a side.
Let's build a table to see the steadiness clearly. Let n be the shape number — that is, how many squares are in the strip.
shape n
1
2
3
4
5
sticks
4
7
10
13
16
+3 each time
—
+3
+3
+3
+3
The constant +3 tells you the rule is built around 3n. But 3×1 = 3, while shape 1 actually uses 4 sticks — one more. Check the others: 3×2 = 6, but we have 7 (one more); 3×3 = 9, but we have 10 (one more). Every shape uses exactly one more than three times the shape number. So the rule is 3n + 1. That single line is the whole infinite pattern, frozen into one expression.
Key idea — one line, every shape
A pattern that starts somewhere and grows by a constant step has a rule of the form (step)×n + (start-up). For the matchstick squares the step is 3 and the rule is 3n + 1. The power of the letter is that you never have to draw shape 100 to count it — just substitute: 3×100 + 1 = 301 sticks.
Worked example — predicting a far-off shape
How many matchsticks does shape n = 20 use, and how many does shape n = 100 use?
No drawing required. The letter let us leap straight to any shape we like.
🎮 Try itMatchstick patterns
Slide n to draw a strip of n squares. The sticks are counted as they are drawn, the table row fills in, and the rule 3n + 1 is checked against the real count. Press Predict n = 100.
Squares n3
★ The big ideas, in one breath
A letter is an empty box that can hold any number — a variable. We reach for one whenever a quantity can change, so that a single expression like 5n captures every case at once. To build an expression, translate the English word by word, remembering that "5 less than x" is x − 5. Then write it tidily: number before letter, no times sign (3x), an invisible coefficient of 1, division shown as a fraction, and repeats as exponents. To evaluate, substitute a number — in parentheses, especially for negatives — and follow the order of operations. And to describe a pattern, find the constant step and write the rule once, so shape 100 is just a substitution away.
Coming up next — 7.2
Now that you can build and read expressions, you will meet the family they belong to. In 7.2 The Polynomial Family: Monomials and Polynomials you'll learn the names for single-term expressions like 5n and many-term ones like 3n + 1, and how to measure their degree.
✎ Practice 7.1
Work each one out first, then open the answer to check your thinking.
A notebook costs $3. Write an expression for the cost of b notebooks.
Show answer
3b dollars. Three dollars for each of b notebooks means 3 times b — and we write the number first, with no times sign.
Translate "7 more than x" and "7 less than x." Why are they different?
Show answer
"7 more than x" is x + 7; "7 less than x" is x − 7 (start at x, take 7 away). Addition doesn't care about order, but subtraction does — it is not7 − x.
Rewrite each in proper form: (a) 5×y (b) x×x (c) 1×m (d) a÷3.
Show answer
(a) 5y (b) x2 (c) m (a coefficient of 1 disappears) (d) the fraction a3.
Evaluate 4x + 2 when x = 3.
Show answer
4(3) + 2 = 12 + 2 = 14. Multiply before you add.
Evaluate 3x + 1 when x = −2.
Show answer
3(−2) + 1 = −6 + 1 = −5. The parentheses make it clear that 3 multiplies the whole −2.
A gym charges a $20 joining fee plus $15 each month. Write an expression for the total cost after m months, then find the cost after 6 months.
Show answer
Total = 15m + 20 dollars. After 6 months: 15×6 + 20 = 90 + 20 = $110.
Translate "half of x, then subtract 4."
Show answer
Half of x is x2; subtracting 4 gives x2 − 4.
Evaluate x2 + 1 when x = −3.
Show answer
(−3)2 + 1 = 9 + 1 = 10. A negative squared is positive: (−3)×(−3) = 9.
In a matchstick pattern the rule for the number of sticks is 3n + 1. How many sticks are in shape 12? In shape 50?
A pattern of triangles starts at 3 sticks and grows by 2 each time: 3, 5, 7, 9, … Write a rule for shape n, then find shape 10.
Show answer
The step is 2, so the rule is built on 2n; since 2×1 = 2 but shape 1 has 3 (one more), the rule is 2n + 1. Shape 10: 2×10 + 1 = 21 sticks.
True or false:3x and x3 are the same thing. Use x = 2 to back up your answer.
Show answer
False.3x means 3 times x, so at x = 2 it is 6. x3 means x×x×x, so at x = 2 it is 8. Different rule, different value.
🎯 Check yourself
Six questions to lock it in. Tap the answer you think is right.
🎮 Quiz Six quick questions
§ For teachers and parents
This lesson opens EastMath Stage 7 and introduces variables and algebraic expressions, aligned to U.S. Common Core standards in grade 6 (6.EE.A.2 — write, read, and evaluate expressions in which letters stand for numbers; 6.EE.A.2a — write expressions from phrases; 6.EE.A.2c — evaluate by substitution, including for real-world formulas; 6.EE.B.6 — use a variable to represent a number in a context). The five widgets are built so that every displayed value is computed at runtime and is correct across the full input range, including negatives in the evaluate machine. The two misconceptions targeted most directly are the order flip in "5 less than x" (it is x − 5, never 5 − x) and the confusion between a coefficient and an exponent (3x versus x³) — both appear in the prose, a "Watch out" aside, the practice set, and the quiz. The matchstick task in 7.1.5 previews linear rules of the form an + b and sets up the pattern-to-formula reasoning students will lean on throughout algebra.
eastmath.com · Stage 7 · 7.1 Letters for Numbers · Intuition before notation
eastmath.com · 7.1 From Numbers to Letters: Using Letters to Stand for Numbers · 7.1.1 Why use a letter instead of a number