# From Arithmetic to Algebra

You have been doing algebra your whole life. Every time you thought "what number plus 3 equals 10?" you were solving an equation. Today, we name the pieces.

## Variable

A variable is a letter or symbol that represents an unknown number or a number that can change. When you wrote a box or a blank in arithmetic, like □ + 3 = 10, that box was a variable. In algebra, we use letters instead: n + 3 = 10.

## Expression

An expression is a mathematical phrase that contains numbers, variables, and operations but no equals sign. An expression has value, but it does not make a claim that two things are equal.

Examples of expressions: - 3x + 5 - 2(n − 4) - 7 + 8

## Equation

An equation is a mathematical sentence that shows two expressions are equal, using an equals sign.

Examples of equations: - 3x + 5 = 20 - y = 2n − 4

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Think of it this way: an expression is like a phrase, "three times a number plus five." An equation is like a complete sentence, "three times a number plus five equals twenty." A phrase is not yet a claim; a sentence is.

# The Parts of an Expression

Let's take the expression 4n − 7 + 2n apart piece by piece.

## Terms

A term is each part of an expression separated by + or − signs. In 4n − 7 + 2n, there are three terms: 4n, −7, and 2n. Every time you see a + or − sign (that is not inside parentheses), it marks the border between terms.

## Coefficient

A coefficient is the number multiplied by a variable. In the term 4n, the coefficient is 4. Think of it as "how many of that variable you have." If you see just n with no number written, the coefficient is 1 because you always have at least one of something.

## Constant

A constant is a number that stands alone with no variable attached, so its value never changes. In 4n − 7 + 2n, the constant is −7.

## Like Terms

Terms with the same variable (raised to the same power) are called like terms and can be combined. In our expression: 4n + 2n = 6n (callback to Lesson 1, addition!). The constant −7 has no like term here, so it stays as is. The simplified expression is 6n − 7.

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### Worked Example

In the expression 5x + 3y − 2, identify each part:

| Part | Answer | |---|---| | Terms | 5x, 3y, and −2 | | Coefficients | 5 (for x) and 3 (for y) | | Constants | −2 | | Variables | x and y | | Like terms | None, each term has a different variable |

# Order of Operations: The Rules of the Road

Math needs rules about what to do first, just like traffic needs rules about who goes first at an intersection. Without these rules, everyone would get a different answer from the same problem.

## PEMDAS

PEMDAS stands for the order in which operations must be performed:

1. P - Parentheses (and all other grouping symbols) 2. E - Exponents 3. M/D - Multiplication and Division (left to right) 4. A/S - Addition and Subtraction (left to right)

> Critical clarification: Multiplication and division have equal priority, so work left to right through them. The same is true for addition and subtraction. PEMDAS does NOT mean multiplication always comes before division.

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## Example 1: No Parentheses

Evaluate: 3 + 4 × 2

| Step | Work | Result | |---|---|---| | Multiply first | 4 × 2 | = 8 | | Then add | 3 + 8 | = 11 |

Common mistake: 3 + 4 = 7, then 7 × 2 = 14 ← WRONG (addition was done before multiplication)

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## Example 2: With Parentheses

Evaluate: (3 + 4) × 2

| Step | Work | Result | |---|---|---| | Parentheses first | 3 + 4 | = 7 | | Then multiply | 7 × 2 | = 14 |

Notice: the same numbers, a different answer, because the parentheses changed the order.

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## Example 3: With Exponents

Evaluate: 2 + 3² × 4

| Step | Work | Result | |---|---|---| | Exponents first | 3² = 3 × 3 | = 9 | | Then multiply | 9 × 4 | = 36 | | Then add | 2 + 36 | = 38 |

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## Exponent Review

An exponent tells you how many times to multiply the base by itself.

- In 3², the base is 3 and the exponent is 2. - 3² = 3 × 3 = 9 - 5³ = 5 × 5 × 5 = 125

(Callback to Lesson 1, multiplication!)

# Properties of Operations

Mathematicians discovered rules that are always true about how numbers behave under addition and multiplication. These rules, called properties, let you rearrange and rewrite expressions without changing their value.

## Commutative Property

The commutative property says that the order does not matter for addition and multiplication.

- a + b = b + a     (example: 3 + 5 = 5 + 3) - a × b = b × a     (example: 4 × 7 = 7 × 4)

Important: The commutative property does NOT work for subtraction or division. - 5 − 3 = 2, but 3 − 5 = −2     ← These are NOT equal!

(Callback to Lesson 1, we saw that addition can go in any order!)

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## Associative Property

The associative property says that the grouping does not matter for addition and multiplication.

- (a + b) + c = a + (b + c)     (example: (2 + 3) + 4 = 2 + (3 + 4) = 9) - (a × b) × c = a × (b × c)     (example: (2 × 3) × 4 = 2 × (3 × 4) = 24)

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## Distributive Property

The distributive property is the bridge between multiplication and addition.

- a(b + c) = ab + ac

Example: 3(4 + 2) - Using distribution: 3(4) + 3(2) = 12 + 6 = 18 - Using parentheses first: 3(6) = 18 ← same answer either way!

The distributive property lets you "distribute" the multiplication across each term inside the parentheses.

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## Identity Property

The identity property says that adding 0 or multiplying by 1 does not change a number.

- a + 0 = a     (additive identity)     example: 17 + 0 = 17 - a × 1 = a     (multiplicative identity)     example: 9 × 1 = 9

0 is the additive identity and 1 is the multiplicative identity.

# Writing and Evaluating Expressions

## Translating Words into Algebra

Algebra is a language. Every English phrase about an unknown number has an algebraic translation.

| English Phrase | Algebraic Expression | |---|---| | Five more than a number | n + 5 | | Twice a number decreased by 3 | 2n − 3 | | The product of 7 and a number | 7n | | A number divided by 4 | n ÷ 4 | | Three less than a number | n − 3 (NOT 3 − n!) |

> Watch out: "Three less than a number" means you start with the number and subtract 3 from it: n − 3. If you write 3 − n, you have "a number less than three," the opposite!

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## Evaluating an Expression

To evaluate an expression means to substitute a specific number for the variable and then compute the result using the order of operations.

### Example 1: Movie Tickets

A movie ticket costs $12. If you and *n* friends go, the total cost is **12(n + 1)**. (The "+1" counts you.) If 3 friends come with you, evaluate when n = 3: 12(3 + 1) = 12(4) = **$48**

### Example 2: Pure Algebra

Evaluate 3x + 5 when x = 4:

| Step | Work | Result | |---|---|---| | Substitute | 3(4) + 5 | | | Multiply first | 12 + 5 | | | Add | | 17 |

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## Series Closing

Over these four lessons, you have built the vocabulary and skills that are the foundation of all 6th grade math. You now speak the language of mathematics. Every concept you encounter from here on (ratios, fractions, equations, geometry) will use the words and ideas you have learned this week.