- 1. What Inequality Means: Comparing Two Numbers
Insert < or > correctly between any two numbers (integers, decimals, negatives, fractions); explain a comparison using number line position, not digit size.
- 2. Two Statements, One Fact: Bidirectionality of Inequality
Rewrite any inequality with the quantities on opposite sides, choosing the correct reversed symbol; explain the connection between the two forms.
- 3. Four Symbols: Introducing ≤ and ≥
Evaluate True/False statements for all four symbols including boundary cases; match real-world vocabulary phrases to the correct inequality symbol.
- 4. Variable Inequalities: $x > n$ as a Set of Values
Answer Yes/No boundary-testing questions for comparative variable inequalities using all four symbols and both orientations.
- 5. The Number Line: Open Circle for Strict Inequalities
Draw and read number line representations for $x > n$ and $x < n$ (strict inequalities); identify the correct arrow direction and use an open circle.
- 6. Filled Circle: Inclusive Inequalities on the Number Line
Draw and read number line representations for all four inequality types; identify the correct dot type (open/filled) from the symbol (strict/inclusive) and vice versa.
- 7. Inequality Vocabulary: Translating Phrases to and from Symbols
Translate any inequality phrase into algebraic notation and vice versa; produce number line, symbolic, and verbal representations for the same inequality.
- 8. Reading Inequalities from Number Lines; Writing Both Forms
Read any number line diagram and write the corresponding inequality correctly; write both equivalent forms (variable first, and number first).
- 9. 'I Think of a Number': Narrative to Inequality
Translate narrative descriptions of an unknown number (including those involving a single operation) into a number line representation and algebraic inequality; give concrete example values.
- 10. Restrictive Inequalities: Two Boundaries, One Variable
Interpret double inequality notation; answer Yes/No boundary-testing questions for restrictive inequalities; identify and correct Sophia-style notation errors.
- 11. Number Line for Restrictive Inequalities: All Four Endpoint Combinations
Draw all four endpoint-combination number lines for restrictive inequalities; match diagrams to inequalities and write inequalities from diagrams including non-integer bounds.
- 12. Integer Solutions from Restrictive Inequalities
List all integer values satisfying a given restrictive inequality; correctly handle both inclusive and exclusive bounds, as well as non-integer boundaries.
- 13. Error Intervals: Restrictive Inequalities in Measurement
Write the error interval for any number rounded to the nearest integer, 10, 100, given number of decimal places, or significant figures.
- 14. Solving Inequalities: Single Operation (Addition and Subtraction)
Solve single-operation linear inequalities involving addition or subtraction; represent the solution set correctly on a number line.
- 15. Solving Inequalities: Multiplication and Division by Positive Numbers
Solve inequalities involving multiplication or division by a positive coefficient, including fractional forms; represent solutions on number lines.
- 16. Solving Multi-Step Linear Inequalities: Two Positive Operations
Solve two-step and bracket-containing inequalities with positive coefficients; represent solutions on number lines and include fractional solutions.
- 17. Solving Inequalities with Variables on Both Sides
Solve inequalities with variables on both sides, using the strategy of collecting $x$-terms positively; expand brackets and represent solutions on number lines.
- 18. Solving Restrictive (Double) Linear Inequalities
Solve restrictive linear inequalities by splitting them into two; represent the solution as a double inequality and on a number line; list integer solutions.
- 19. The Negative Coefficient Rule: Why the Sign Reverses
Derive the sign-reversal rule from concrete examples; apply the rule to determine the inequality for x from a known inequality for -x; complete sentence-completion exercises involving negative coefficients.
- 20. Solving Inequalities with Negative Coefficients
Solve inequalities where the coefficient of x is negative using both the "sign-reversal" rule and the "collect positively" strategy; solve restrictive double inequalities with negative coefficients.
- 21. Systems of Inequalities: Intersection on the Number Line
Find the set satisfying two simultaneous inequalities by identifying the intersection on the number line; handle all overlap scenarios including cases with no solution; solve inequalities before finding their intersection.
