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Mathematics1 course

Pre-Algebra

Variables, expressions, integers, exponents and one-step equations: the bridge from arithmetic to algebra.

About Pre-Algebra

Pre-algebra is the bridge between calculating with numbers and reasoning with letters. It covers negative numbers, exponents, the order of operations and one-step and two-step equations, and it introduces the habit of treating an equation as a balance that must stay level.

The key shift is from finding an answer to describing a relationship. A rule like "the cost is $3 per notebook plus $2 shipping" becomes C=3n+2C = 3n + 2, and that one line answers every question about any number of notebooks. Students who are comfortable with that translation find algebra far easier.

These skills are used whenever a formula is applied: converting temperatures with F=95C+32F = \frac{9}{5}C + 32, finding how many hours of work cover a cost, or setting up a spreadsheet column. Pre-algebra is also the main content of many placement tests, so solid fluency here often decides which math course a student starts in.

Pre-Algebra

Integers, exponents, expressions and simple equations, with the habits that prevent sign errors. (Middle school)

Integers and Signed Numbers

Integers are the whole numbers and their negatives: …,−2,−1,0,1,2,…\ldots, -2, -1, 0, 1, 2, \ldots. A number line makes their arithmetic visual: adding a positive moves right, adding a negative moves left, and subtracting a number is the same as adding its opposite.

Exponents

An exponent counts repeated multiplication: 25=2⋅2⋅2⋅2⋅2=322^5 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 32. The exponent rules all follow from counting factors. Multiplying a3⋅a4a^3 \cdot a^4 strings together seven factors of aa, so the exponents add.

One- and Two-Step Equations

An equation is a balance: whatever you do to one side you must do to the other to keep it true. A one-step equation needs a single inverse operation to isolate the variable. Addition undoes subtraction, multiplication undoes division, and vice versa.

Full Pre-Algebra course outline

How to study Pre-Algebra

  • Say each equation step out loud as an action applied to both sides: "subtract 7 from both sides", then "divide both sides by 4".
  • Track negative signs with brackets: −(x−3)=−x+3-(x - 3) = -x + 3, and (−2)2=4(-2)^2 = 4 while −22=−4-2^2 = -4.
  • Substitute your answer back into the original equation every time; it takes seconds and catches most errors.
  • Practise turning short word problems into expressions before solving anything.
Before you start
Arithmetic with fractions, decimals and percentages.
Where it leads
Algebra 1: linear equations, graphs of lines, systems and quadratics. Students who finish pre-algebra confidently usually find the first months of algebra feel like a review.

Key terms in Pre-Algebra

Variable
A letter standing for a number that can change or is not yet known, such as nn for the number of tickets sold.
Integer
A whole number that can be positive, negative or zero. Subtracting a negative is the same as adding: 5−(−3)=85 - (-3) = 8.
Ratio
A comparison of two quantities by division. A ratio of 3 to 2 can be scaled to 6 to 4 without changing the comparison.
Proportion
An equation saying two ratios are equal, like 34=x20\tfrac{3}{4} = \tfrac{x}{20}, solved by scaling or cross-multiplying to get x=15x = 15.
Like terms
Terms with the same variable part, such as 3x3x and −5x-5x, which can be combined into −2x-2x; 3x3x and 3x23x^2 cannot.
Inverse operation
The operation that undoes another: subtraction undoes addition and division undoes multiplication, which is how one-step equations are solved.

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