Expressions & Equations
Understanding Expressions, Exponents, Equations, and Inequalities
Grade 6
- ✓By the end of this lesson students will be able to understand and use exponents to represent repeated multiplication.
- ✓By the end of this lesson students will be able to evaluate numerical expressions involving whole-number exponents and algebraic expressions by substituting values for variables.
- ✓By the end of this lesson students will be able to solve one-variable equations using inverse operations.
- ✓By the end of this lesson students will be able to understand and represent solutions to one-variable inequalities.
Key concepts
An exponent tells you how many times to multiply a base number by itself. The base is the number being multiplied, and the exponent (or power) is the small number written above and to the right of the base. For example, in 2³, 2 is the base and 3 is the exponent. This means 2 multiplied by itself 3 times.
A mathematical phrase that can contain numbers, variables, and operation symbols. An expression does NOT have an equals sign. A variable is a letter or symbol that represents an unknown value. To evaluate an expression, you substitute a given number for each variable and then simplify the expression using the order of operations (PEMDAS/GEMDAS).
A mathematical statement that shows two expressions are equal. An equation DOES have an equals sign. The goal when solving an equation is to find the value(s) of the variable that make the equation true. We use inverse operations to isolate the variable (get it by itself on one side of the equals sign). Whatever you do to one side of the equation, you must do to the other side.
A mathematical statement that compares two expressions using an inequality symbol. Unlike equations, which have a single solution, inequalities often have a range of solutions. The symbols used are: < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Solutions to inequalities can be represented on a number line.
Key facts to remember
- 1Exponents represent repeated multiplication (e.g., 5³ = 5 × 5 × 5).
- 2The order of operations (PEMDAS/GEMDAS) is crucial for evaluating expressions: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).
- 3An expression is a mathematical phrase without an equals sign; an equation is a statement showing two expressions are equal.
- 4Variables are symbols (usually letters) that represent unknown values.
- 5Inverse operations 'undo' each other (addition/subtraction, multiplication/division) and are used to solve equations.
- 6Inequalities compare quantities using symbols like <, >, ≤, ≥.
- 7The solution to an equation is a specific value; the solution to an inequality is often a range of values.
Worked examples
Example 1
Evaluate the expression 4² + (20 - 8) ÷ 3.
Answer
20
Remember the order of operations: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
Example 2
Solve the equation x - 15 = 37.
Answer
x = 52
To check your answer, substitute x = 52 back into the original equation: 52 - 15 = 37. Since 37 = 37, the solution is correct.
Example 3
Evaluate 3y + 8 when y = 5. Then, determine if y = 5 is a solution to the inequality 3y + 8 < 25.
Answer
When y = 5, 3y + 8 = 23. Yes, y = 5 is a solution to the inequality 3y + 8 < 25 because 23 < 25 is a true statement.
Common mistakes
- ✗Confusing bⁿ with b × n (e.g., 2³ is 2 × 2 × 2 = 8, not 2 × 3 = 6).
- ✗Incorrectly applying the order of operations, especially with multiplication/division or addition/subtraction when they appear together.
- ✗Forgetting to perform the same operation on both sides of an equation, leading to an incorrect solution.
- ✗Misinterpreting inequality symbols (e.g., thinking < means 'greater than').
- ✗Not substituting the variable's value correctly or performing the calculation inaccurately when evaluating expressions.
Exam tips
- ★Always show your work step-by-step for every problem. This helps you catch errors and allows your teacher to understand your thinking.
- ★Double-check all calculations, especially with exponents and multiple operations. A small arithmetic error can lead to a completely wrong answer.
- ★For equations, substitute your answer back into the original equation to verify that it makes the equation true.
- ★Read inequality problems carefully to understand what range of values makes the statement true. Pay close attention to the direction of the inequality symbol.
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