Lesson 1.1: Use the Language of Algebra

New Concept

This lesson introduces the core language of algebra. You'll learn to use variables and symbols to create expressions and equations, and master the order of operations to simplify them, translating phrases into algebraic form.

What’s next

Next, you'll practice with factors and primes. Then, you'll use our interactive examples and practice cards to simplify expressions and translate algebraic phrases.

Property

A number is a multiple of $n$ if it is the product of a counting number and $n$.
If a number $m$ is a multiple of $n$, then $m$ is divisible by $n$.
If $a \cdot b = m$, and both $a$ and $b$ are integers, then $a$ and $b$ are factors of $m$.
A prime number is a counting number greater than 1 whose only factors are 1 and the number itself.
A composite number is a counting number greater than 1 that is not prime.
The prime factorization of a number is the product of prime numbers that equals the number.

Examples

• The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30, since these are the integers that multiply to give 30.
• 17 is a prime number because its only factors are 1 and 17. 25 is a composite number because it has a factor of 5 besides 1 and 25.
• The prime factorization of 42 is $2 \cdot 3 \cdot 7$. We can find this by breaking it down: $42 = 6 \cdot 7 = (2 \cdot 3) \cdot 7$.

Explanation

Think of factors as building blocks for numbers. Prime numbers are the most basic blocks. Prime factorization is breaking a number down into its unique set of prime building blocks. This is useful for simplifying fractions and other calculations.

Property

A variable is a letter that represents a number whose value may change.
A constant is a number whose value always stays the same.
An expression is a number, a variable, or a combination of numbers and variables using operation symbols.
An equation is two expressions connected by an equal sign.

Examples

• The phrase "10 less than a number $n$" translates to the algebraic expression $n - 10$.
• The sentence "The product of 4 and a number $x$ is 20" translates to the equation $4x = 20$.
• $9y - 5$ is an expression, while $9y - 5 = 22$ is an equation. The equal sign connects two expressions to make a complete mathematical statement.

Explanation

An expression is like a phrase in English, such as "five more than a number." It's an incomplete thought. An equation is a complete sentence stating two expressions are equal, like "Five more than a number is 15."

Property

Exponential Notation: $a^n$ means multiply $a$ by itself, $n$ times. The expression $a^n$ is read $a$ to the $n^{th}$ power. In $a^n$, the $a$ is the base and the $n$ is the exponent.
Order of Operations:

1. Parentheses and other Grouping Symbols
2. Exponents
3. Multiplication and Division (from left to right)
4. Addition and Subtraction (from left to right)

Examples

• The expression $4^3$ is in exponential notation and means $4 \cdot 4 \cdot 4$, which equals 64. Here, 4 is the base and 3 is the exponent.
• To simplify $20 - 3 \cdot 5$, we follow the order of operations by multiplying first: $20 - 15 = 5$.
• To simplify $2(5+1) + 3^2$, first handle parentheses: $2(6) + 3^2$. Next, exponents: $2(6) + 9$. Then multiply: $12 + 9$. Finally, add to get $21$.

Explanation

Exponents are a shortcut for repeated multiplication. To ensure everyone gets the same answer for a problem, we follow the order of operations (PEMDAS/GEMDAS). It's the universal grammar for solving math expressions, ensuring consistent results.

Property

A term is a constant or the product of a constant and one or more variables.
The coefficient of a term is the constant that multiplies the variable in a term.
Like terms are terms that are either constants or have the same variables raised to the same powers. You can simplify an expression by combining like terms.

Examples

• In the expression $5x^2 + 2y + 3x^2 - y$, the terms $5x^2$ and $3x^2$ are like terms. The terms $2y$ and $-y$ are also like terms.
• To simplify $9b + 8 + 3b - 5$, combine the like terms: $(9b + 3b) + (8 - 5)$, which simplifies to $12b + 3$.
• Translating "five times a number, decreased by two, plus three times the same number" gives the expression $5n - 2 + 3n$. Combining like terms simplifies this to $8n - 2$.

Explanation

"Like terms" are terms that represent the same kind of object. You can add apples to apples ($3x + 4x = 7x$), but you can't add apples to bananas ($3x + 4y$). Combining like terms helps simplify and organize expressions.