Variables, constants, coefficients — algebraic expressions: Foundations

This page builds every single symbol the parent note (parent topic) leaned on, starting from a thing you already trust: counting objects.

0. The thing you already know: a number is a count

Before any letter appears, we need to be crystal-clear on what a number is here. A number like $3$ is a count of identical things — three apples, three steps, three marbles. Look at the picture: three copies of one object.

Figure — Variables, constants, coefficients — algebraic expressions

1. The symbol for "add": $+$

Plain words: put two counts together into one bigger count.
The picture: slide two groups of dots next to each other and count them all.
Why the topic needs it: expressions like $3 x + 5 x$ are built out of $+$ . Before we can write $250 t + 100 p$ we must trust that $+$ means "gather into one pile."

$3 + 5 = 8$

Read left to right: a pile of 3, a pile of 5, gathered = a pile of 8.

2. The symbol for "take away": $-$

Plain words: remove some things from a count; also marks a number below zero (owing).
The picture: cross out dots from a group.
Why the topic needs it: the parent writes $a - 1$ and $- 9$ . The minus in front of a number ( $- 9$ , $- 2$ ) means "this many owed", and it travels with the number like a shadow. That is exactly why the coefficient of $- 2 x y$ is $- 2$ and not $2$ — the minus belongs to the number.

3. The symbol for "multiply": $\times$ , and its disappearing act

Plain words: repeated addition — "so many copies of."
The picture: a rectangle grid. $3 \times 4$ is 3 rows of 4 dots.
Why the topic needs it: this is the single most important idea for coefficients. $5 x$ means $5 \times x$ , i.e. five copies of the mystery amount $x$ .

This is why the parent can say "the coefficient is the numerical factor multiplied with a variable" — the multiplication is hiding in plain sight.

Plain words: split a count into equal groups.
The picture: 12 dots dealt into 4 equal rows → 3 per row.
Why the topic needs it: the parent uses $\frac{3}{4} y^{2}$ and $\frac{2}{7}$ . A fraction bar is just a stored-up division: $\frac{3}{4}$ means $3 \div 4$ , a fixed number sitting between 0 and 1. That is why $\frac{3}{4}$ counts as a constant — a division of two fixed numbers is itself fixed.

$\frac{3}{4} = 3 \div 4 = 0.75$

5. The letter: a variable

Now the star of the show.

Plain words: a letter (like $x$ , $y$ , $t$ , $w$ ) is a box whose number we haven't decided yet. It can hold different numbers on different days.
The picture: a labelled box with a "?" inside. The label ( $x$ ) is the box's name; the "?" is the unknown count inside.
Why the topic needs it: this is the whole reason algebra exists. " $x$ kg of apples" lets one expression $40 x$ stand for the cost of any amount.

6. The number bolted onto a letter: the coefficient

Plain words: the number multiplying a variable — "how many copies of the box."
The picture: the box drawn several times; the coefficient counts the copies.
Why the topic needs it: $5 x$ = five identical boxes. The $5$ counts boxes, it is not itself a box. This is the exact distinction the parent's Mistake 1 warns about.

7. The little number up high: the exponent (power)

Plain words: a small raised number saying "multiply this thing by itself that many times."
The picture: $x^{2}$ is a square of side $x$ (area = $x \times x$ ); $x^{3}$ is a cube (volume = $x \times x \times x$ ). That is literally why we say " $x$ squared" and " $x$ cubed."
Why the topic needs it: the parent's expression $5 x^{3} - 2 x y + \frac{3}{4} y^{2} - 9$ is stuffed with exponents. Without them you cannot read $x^{3}$ , and you cannot compute the degree of a term.

$x^{2} = x \times x, x^{3} = x \times x \times x .$

8. The term and the whole expression

Plain words: a term is one chunk built from numbers and letters multiplied together (e.g. $5 x^{3}$ ). An expression is several terms joined by $+$ and $-$ .
The picture: terms are LEGO bricks; the $+$ and $-$ are the studs that click them into one wall.
Why the topic needs it: "algebraic expression" is the parent's headline object. Seeing it as bricks joined by signs tells you where one term ends and the next begins — you split at every $+$ or $-$ (keeping each sign with the brick on its right).

$term 5 x^{3} term - 2 x y term + \frac{3}{4} y^{2} term - 9$

9. The constant: a number that refuses to move

Plain words: a fixed value that never changes in the problem — a plain number ( $5$ , $- 9$ , $\frac{3}{4}$ ) or a named fixed number ( $π$ , $e$ ).
The picture: a bolted-down anchor, unlike the "?" box which can change.
Why the topic needs it: the parent's Mistake 3 hinges on this — $π$ looks like a variable (it's a Greek letter) but its value is glued forever ( $\approx 3.14159$ ), so it is a constant.

Recall Variable vs constant — the one test

Ask: "Can this take different numerical values in the same problem?" Yes → variable. No (it has one fixed value) → constant. ::: Yes → variable; No → constant. That single question separates $x$ (variable) from $π$ (constant).

How these feed the topic

Read top to bottom: everything starts from counting, and the two branches (numbers-with-signs and boxes-with-copies) merge into the term, and terms joined by signs make the expression.

Equipment checklist

Cover the right side and see if you can answer each before revealing.

What does

+

mean, in one picture?

Slide two groups of dots together and count them all as one pile.

What are the two jobs of the

-

symbol?

Operation ("take away",

7 - 5

) and sign ("a negative number",

- 9

What hidden symbol sits between the

5

and the

x

5 x

A multiplication sign

\times

;

5 x

means

5 \times x

Draw the picture of

x^{2}

A square whose side length is

x

; its area is

x \times x

What is the coefficient of a lonely

x

, and of

- x

1

for

x

, and

- 1

for

- x

— the invisible coefficient.

Why is

- 9

called "the coefficient of

x^{0}

Because

x^{0} = 1

, so

- 9 = - 9 \times x^{0}

; every term hides a power of the variable.

What single test separates a variable from a constant?

"Can it take different values in the same problem?" Yes → variable, No → constant.

Why is

π

a constant even though it's a letter?

Its value is glued forever (

\approx 3.14159

); it never changes.

Where do you split an expression into terms?

At every

+

and

-

, keeping each sign attached to the brick on its right.

Connections

Parent topic — the note these foundations support
Algebraic Terms and Like Terms — once terms are clear, learn to combine them
Evaluating Algebraic Expressions — put a real number into the mystery box
Equations vs. Expressions — what happens when we add an equals sign
Linear Expressions — expressions where every exponent is $1$
Polynomials — structured stacks of these terms
Word Problems in Algebra — turning stories into these symbols