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Unit A4: Ratios, Rates, and Linear Relationships

Connect proportional reasoning, slope, equations, graphs, and linear models. How do two quantities change together at a constant rate?

Act IV - Turning constant change into lines

The 11-lesson path

Ratios become unit rates, proportionality becomes a line through the origin, and slope generalizes constant rate. Multiple line forms reveal different information; data modeling distinguishes exact relationships from approximate trends.

Governing question

How do two quantities change together at a constant rate?

Connect proportional reasoning, slope, equations, graphs, and linear models.

Mastery design

Know when to continue

Students move among context, table, graph, and formula and explain slope and intercept with units.

Unit outcomes
  1. Use ratios, unit rates, proportions, and percent multipliers.
  2. Interpret a graph as a set of ordered-pair solutions.
  3. Calculate and interpret intercepts and slope with units.
  4. Write line equations from points, graphs, and contexts.
  5. Fit, interpret, and critique elementary linear models.
Core sequence

Lessons

Move in order when this material is new; use prerequisites and repair links when entering mid-unit.

  1. 01Ratios and equivalent ratiosCompare quantities multiplicatively and preserve the order and meaning of the comparison.
  2. 02Unit ratesReduce a comparison to change per one unit and compare rates with consistent units.
  3. 03Proportional relationshipsRecognize y=kxy=kx in tables, graphs, formulas, and contexts.
  4. 04Percent as a multiplierTreat percent change as multiplication and distinguish one-time from repeated change.
  5. 05Coordinate plane and equation solutionsInterpret graph points as ordered pairs satisfying an equation.
  6. 06InterceptsFind and interpret where one quantity is zero.
  7. 07Slope from graphs and tablesMeasure constant vertical change per horizontal change and attach units.
  8. 08Slope from two pointsUse consistent coordinate differences to calculate slope.
  9. 09Equations of linesUse slope-intercept, point-slope, and standard forms strategically.
  10. 10Parallel and perpendicular linesConnect equal and negative-reciprocal slopes with geometric relationships.
  11. 11Linear models, fit, and residualsDistinguish exact linear relationships from approximate trends and critique extrapolation.

Practice around the path

Review, mixed practice, mastery, and investigation

Every route contains a fixed set of concrete questions, with detailed solutions available after an attempt.

ReviewCumulative reviewRevisit this unit with prior-unit retrieval.PracticeMixed practiceClassify the structure before choosing a method.MasteryMastery checkUse the supplied grading and cumulative-share contract.InvestigationPerformance taskAnalyze a real cost, consumption, or public-data set and defend a linear model with a reasonable domain and limitations.Response guideMastery response guideAttempt-gated rubric support; no invented canonical answers.

Source & rights

Original storyboard, rights-separated references.

Public page content comes from the BetterGrades Algebra editorial storyboard supplied by the owner. Reference books named in provenance remain separate and are not copied into the application.