Pre-algebra foundations help students connect arithmetic to algebraic thinking. This cheat sheet covers integers, number lines, absolute value, order of operations, expressions, equations, ratios, proportions, and percent. Students need these skills because they appear in nearly every middle school math unit and prepare them for algebra.
Key Facts
- The absolute value is the distance from to on a number line, so and .
- To add integers with the same sign, add their absolute values and keep the common sign, such as .
- To add integers with different signs, subtract the smaller absolute value from the larger absolute value and keep the sign of the number with greater absolute value.
- The order of operations is parentheses, exponents, multiplication and division from left to right, then addition and subtraction from left to right.
- The distributive property is , such as .
- Like terms have the same variable parts, so but cannot be combined.
- To solve a one-step equation, use inverse operations to isolate the variable, such as becoming .
- A proportion states that two ratios are equal, and can be checked with when and .
Vocabulary
- Integer
- An integer is a whole number or its opposite, such as , , or .
- Absolute Value
- Absolute value is the distance a number is from on the number line.
- Variable
- A variable is a letter or symbol that represents an unknown or changing number.
- Expression
- An expression is a mathematical phrase with numbers, variables, and operations but no equals sign.
- Equation
- An equation is a statement that two expressions are equal, such as .
- Ratio
- A ratio compares two quantities by division, such as or .
Common Mistakes to Avoid
- Forgetting that subtraction can be rewritten as addition of the opposite is wrong because means , not .
- Combining unlike terms is wrong because terms such as and do not have the same variable part.
- Doing operations strictly from left to right is wrong when exponents or parentheses are present because order of operations gives some operations priority.
- Changing only one side of an equation is wrong because equations stay balanced only when the same operation is applied to both sides.
- Setting up ratios in different orders is wrong because must match in a proportion.
Practice Questions
- 1 Evaluate .
- 2 Simplify .
- 3 Solve .
- 4 Explain why is never negative, even when is a negative number.
Understanding Pre-Algebra Foundations
Negative numbers describe direction, change, or an amount below a chosen starting point. A temperature of negative five degrees is five degrees below zero. A bank balance of negative five dollars means five dollars are owed.
This context helps with integer rules. When two negative changes are combined, the result moves farther below zero. When a positive and negative amount are combined, they partly cancel.
Drawing a number line can make the calculation visible before rules are memorized. Absolute value ignores direction and measures size.
This matters when comparing errors, distances, or changes. An error of negative three units has an absolute value of three units, even though its signed value is negative.
Order of operations is not a list to rush through. It is a set of rules that makes every calculation have one agreed result. Multiplication next to parentheses means that every term inside must be affected by the outside factor.
Students often distribute correctly to the first term but forget the second term. Another common error is treating subtraction as if it can be rearranged freely. Addition and multiplication can change order without changing a result, but subtraction and division usually cannot.
Work from left to right when operations have equal priority. Writing one step per line prevents small sign mistakes from becoming hidden.
An expression represents a quantity, while an equation makes a claim that two quantities have the same value. A variable is a placeholder for a number, not a label for an object. In an expression, simplifying means rewriting it in a shorter equivalent form.
In an equation, solving means finding values that make the claim true. The key idea is balance. Whatever operation is done to one side of an equation must be done to the other side.
Check a solution by putting it back into the original equation. This check catches arithmetic slips and shows whether the answer truly works. Terms can only be combined when they describe matching variable parts.
Three groups of x plus two groups of x makes five groups of x. Groups of x and groups of y remain different quantities.
Ratios compare quantities using the same units or related units. A unit rate tells how much belongs to one unit, such as cost per item or kilometers per hour. Proportions are useful only when the relationship stays constant.
Recipes, map scales, similar shapes, sale prices, and speed problems often use this idea. Before cross multiplying, decide whether the pairs are in the same order. Mixing up items with prices or distance with time can produce a neat calculation with the wrong meaning.
Percent means parts out of one hundred. It can be changed into a decimal for calculations, but students should keep track of what the final number represents. A discount is taken from an original price.
Tax and tip are added to a price. Estimation helps here. Ten percent of eighty is eight, so an answer near eighty dollars for a twenty percent discount should signal a mistake.