Pre-calculus Synthesis

Overview

Students consolidate Pre-calculus 10 ideas by moving fluently between algebraic forms, graphs, and real situations. They solve a short sequence of tasks that blend linear systems, sequences, polynomial operations/factoring, and an application to income.

Learning intentions

• Students will be able to interpret and create function rules connecting data, graphs, and situations.
• Students will be able to solve problems using linear functions and systems of linear equations.
• Students will be able to model and extend arithmetic sequences and solve for unknown terms.
• Students will be able to multiply and factor polynomials and use the results to solve related equations.
• Students will be able to apply primary trigonometric ratios to find an unknown length or angle in a right triangle (with reasoning).

Success criteria

• I can explain how changing a variable affects a linear function and interpret slope from context and graphs.
• I can solve a system of linear equations accurately and check the solution in context.
• I can find missing terms in an arithmetic sequence using a common-difference strategy and verify.
• I can expand and factor a polynomial expression and use it to solve an equation that results from a pre-calculus situation.
• I can use sine, cosine, or tangent to solve for an unknown side/angle and state units and assumptions.

Curriculum links

• Connecting data, graphs, and situations through functions and relations (pre-calculus).
• Linear relationships: slope, intercepts, equations of lines, and interpreting graphs.
• Arithmetic sequences: general patterns, term-to-term reasoning, and solving for unknowns.
• Polynomial operations and factoring; connecting factored forms to solutions.
• Right-triangle trigonometry using primary ratios.
• Financial literacy: gross pay and net pay; using rate and deductions to model outcomes.

Lesson structure (100 minutes)

1. 0–10 — Warm-up: function-and-slope sprint Present a short context: “A ride costs a fixed fee plus $m per minute.” Students write two forms (a description and a rule), then compute the slope and interpret it as a rate. Quick share: one student explains how slope matches the context.

2. 10–25 — Linear systems in context Provide a scenario where two mobile plans differ: “Plan A: $a + r₁t” and “Plan B: $b + r₂t” where t is minutes. Students form two linear equations, solve the system, and interpret the intersection as the time when costs are equal. Require a brief check by substitution.

3. 25–40 — Arithmetic sequence modelling Use a practical pattern (e.g., “weekly savings increase by a fixed amount”): give first term and common difference, ask for the 8th term and the term number when reaching a target amount. Students show the arithmetic-sequence method and then verify by extending the pattern.

4. 40–55 — Polynomial expansion then factoring Task 1: expand a binomial product (e.g., ((x+ p)(x+ q))). Task 2: factor a related quadratic-like expression by recognising common factors and using a two-term/three-term approach taught in factoring. Students identify the structure: “What did I look for?” They use their factored form to solve for zeros.

5. 55–70 — Trigonometry application (right triangles) Give a right-triangle measurement problem: one acute angle and one side length, ask for the remaining side using sine/cosine/tangent. Students state the ratio they used, compute, and include a short sentence about which side/angle is opposite/adjacent/hypotenuse.

6. 70–85 — Financial literacy: gross vs net pay model Provide an example pay situation: hourly wage, hours worked, and deductions (e.g., a fixed benefit deduction plus a percentage tax, or a typical simplified structure your class has used). Students write expressions for gross pay and net pay, compute both, and interpret the difference as deductions in context.

7. 85–100 — Integrated mini-task + exit check Students complete one blended worksheet problem tying everything together: for example, choose a linear plan (systems), compare sequence-based savings (arithmetic sequence), then factor to solve a resulting equation, and use trigonometry for a final geometry sub-part. End with a 3-question exit check: one system question, one sequence question, one factoring/solution reasoning question.

Resources

• Printed worksheet with 5–6 progressively integrated problems (linear systems, arithmetic sequence, polynomial operations/factoring, one trig ratio, one gross/net pay model).
• Graphing calculator or app (optional) to confirm line intersections and equation solutions.
• Reference card: slope as rise/run; sine/cosine/tangent definitions for opposite/adjacent/hypotenuse.
• Reference card: arithmetic sequence formulas and/or term-to-term method.
• Marker/whiteboard for exemplar “how to set up equations” steps.
• Small “error check” prompts: substitute back, units check, and reasonableness check.
• Student notebook space for method explanations, not just answers.
• Timer for short segments and a rubric strip for quick teacher feedback.

Assessment

• Formative: teacher circulates during each segment, checking setup steps (equations, sequence strategy, factoring method) and listening for correct reasoning.
• Formative: brief solution checks (substitution for systems; verifying the arithmetic term; checking polynomial solutions by substitution).
• Summative (mini): the integrated end task plus the 3-question exit check to confirm mastery of the targeted strands.

Differentiation

• Support: provide sentence starters for setting up equations (“Let t be…”, “Total cost equals…”), and a factoring template with common-factor prompts.
• Support: offer a worked example step at the start for the trig ratio, then remove it gradually.
• Extension: include a “choose the best plan” justification requiring interpreting slope/intersection plus an extra step (e.g., find the break-even minutes and compare net pay).
• EAL/SEN: allow diagrams for trig (label opposite/adjacent/hypotenuse) and structured tables for arithmetic sequences and gross/net pay; reduce copying by allowing answers in a provided layout.