Forming and simplifying algebraic expressions: MYP 1 Mathematics Notes

What is an Algebraic Expression?

Algebra is a powerful tool that lets us describe patterns, rules, and relationships using symbols instead of words. Scientists use algebra every day , to write formulas for speed, to describe how temperature changes, or to predict the outcome of an experiment. At the heart of algebra is the algebraic expression.

Algebraic Expression: A mathematical phrase made up of variables (letters), numbers, and operations (such as +, −, ×, ÷). For example: $4 y$ , $t - 10$ , $n + 2$ , and $3 w + 5$ are all algebraic expressions. Notice there is no equals sign , an expression is not an equation.

Variable: A letter that represents an unknown or changing number. Common choices are $x$ , $y$ , $a$ , $b$ , $n$ , etc. In science, we often choose letters that remind us of what they stand for , $t$ for time, $d$ for distance, $T$ for temperature.

Term: A single number, variable, or product of numbers and variables. Terms are separated by + or − signs. In $3 a + 4$ , there are two terms: $3 a$ and $4$ .

Coefficient: The number that is multiplied by a variable in a term. In $7 x$ , the coefficient is $7$ . In $3 w + 5$ , the coefficient of $w$ is $3$ .

Constant: A number on its own in an expression , its value never changes. In $3 a + 4$ , the number $4$ is a constant.

Expression vs Equation: An expression is a mathematical phrase with no equals sign: $3 x + 5$ . An equation states that two things are equal , it always has an equals sign: $3 x + 5 = 14$ . In this unit we focus on forming and simplifying expressions.

Here are some expressions and what they mean in words:

Expression	Meaning
$x + 7$	a number plus seven
$3 m$	three times a number
$4 a - 8$	eight less than four times a number
$\frac{d}{5}$	a number divided by five
$\frac{k}{6} + 2$	two more than a number divided by six

Note

In algebra, we never write the multiplication symbol $\times$ between a number and a letter. This is because $\times$ looks too much like the letter $x$ ! Instead, we write the number directly in front of the letter: $3 w$ means $3 \times w$ .

Translating Words into Algebraic Expressions

One of the most important skills in algebra is turning an English phrase into a mathematical expression. Think of it like translating between two languages!

Here are the key "translation" words to know:

Sum / increased by / more than / added to → $+$
Difference / decreased by / less than / reduced by → $-$
Product / times / multiplied by → write the number in front of the letter (e.g. $5 n$ )
Quotient / divided by → use a fraction bar (e.g. $\frac{n}{4}$ )

Example

Translate each phrase into an algebraic expression:

Phrase	Expression
3 times a number	$3 n$
A number increased by 7	$n + 7$
A number multiplied by 5	$5 n$
12 more than a number	$n + 12$
A number reduced by 13	$n - 13$
28 divided by a number	$\frac{28}{n}$
3 more than twice a number	$2 n + 3$

Tip: Choose any letter you like for the unknown , just be consistent!

Analogy

Think of translating word phrases into algebra like translating from English to Spanish. Once you learn the vocabulary (sum → +, times → ×), the translation becomes straightforward!

Warning

"Less than" can be tricky! "6 less than a number" means $n - 6$ , NOT $6 - n$ . The phrase describes what you start with (the number) minus the amount taken away (6).

Phrase	Incorrect ❌	Correct ✅
6 less than a number	$6 - n$	$n - 6$
4 less than a number	$4 - n$	$n - 4$

A helpful trick: read the phrase backwards. "6 less than a number" , the number comes first, then subtract 6.

Translating Words into Algebraic Expressions

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