Grade 9 · printable course

Algebra I

Grade 9 · 519 pages · 70 lessons · PDF, print at home

A full-year Algebra I course for a single homeschool student working through an adaptive learning app with parent guidance. Students move from linear relationships through systems, exponential and quadratic functions, to an introduction of data and statistics, building fluency with algebraic structure and function representations (tables, graphs, equations, verbal descriptions). The course emphasizes connected understanding over procedure-matching: students must justify why methods work, not just execute them. Delivered on grade level, aligned to the CCSS high school conceptual categories most central to Algebra I.

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What is inside

Every unit and lesson in this book, in order. The same lessons are taught online at lettersanddevotion.com, so a family can move between paper and screen.

Algebra I

70 lessons · try a lesson online free

Unit 1: Linear Relationships as Rules, Graphs, and Stories10 lessons
  1. 1.Starting value and constant rate of change in a verbal linear context
  2. 2.Slope as a computed ratio of vertical change to horizontal change
  3. 3.Invariance of slope across point pairs on one line
  4. 4.Distinguishing linear structure from graph/table appearance alone
  5. 5.Slope-intercept equation construction from varied input formats
  6. 6.Function notation f(x) as input-output pairing tied to context
  7. 7.Domain and range as context-imposed restrictions on an algebraic relationship
  8. 8.Equality of two linear expressions as the point of agreement between two contexts
  9. 9.Appropriateness of a linear model and selection among representations for a decision
  10. 10.Vocabulary: slope, x-intercept, y-intercept
Unit 2: Systems of Linear Equations9 lessons
  1. 11.The point of intersection of two lines as a shared solution to both equations
  2. 12.The graphing method for solving a linear system
  3. 13.The substitution method for solving a linear system
  4. 14.The elimination method and the general algebraic justification (given A=B and C=D, then A+kC=B+kD) that it preserves the solution set
  5. 15.The relationship between slope/intercept and the number of solutions to a linear system
  6. 16.Translation of a two-constraint real-world scenario into a system of linear equations
  7. 17.Viability of an algebraic solution against the real-world constraints of the situation it models
  8. 18.The relationship between equation structure (coefficient values, isolated variables) and the relative efficiency of substitution versus elimination
  9. 19.Solving and interpreting a system in a context that does not signal which special case or method applies
Unit 3: Linear Inequalities and Absolute Value8 lessons
  1. 20.The sign-reversal rule when multiplying or dividing an inequality by a negative number
  2. 21.The geometric reason the inequality direction reverses under multiplication by a negative number
  3. 22.Compound inequalities joined by 'and' or 'or'
  4. 23.The equivalence between absolute-value-distance statements and compound inequality statements
  5. 24.Case-splitting procedure for absolute value equations, including the no-solution and one-solution conditions
  6. 25.The and/or branching rule for absolute value inequalities based on the inequality direction
  7. 26.Systems of linear inequalities and their graphed solution (feasible) region
  8. 27.Systems of inequalities combined with absolute value constraints in an unfamiliar applied context
Unit 4: Systems with Nonlinear Functions — Function Notation Formalized8 lessons
  1. 28.Function notation f(x) as input-output correspondence
  2. 29.Comparison of function properties across representations
  3. 30.The equivalence between graphical intersection and algebraic solution of f(x)=g(x)
  4. 31.Graphical solution of f(x)=g(x) for a linear-nonlinear pair
  5. 32.Number of solutions to f(x)=g(x) as a property of the specific function pair
  6. 33.Translation of a verbal scenario into a function pair and interpretation of their intersection
  7. 34.The general equivalence of graphical and numerical solution methods for f(x)=g(x)
  8. 35.The meaning of the notation f(x) versus multiplication notation
Unit 5: Introduction to Quadratic Functions8 lessons
  1. 36.Second differences as the numerical signature of quadratic growth, contrasted with constant first differences (linear) and constant ratios (exponential)
  2. 37.The vertex and axis of symmetry as features read from a graphed parabola
  3. 38.The correspondence between the parameters h and k in vertex form and the vertex's coordinates
  4. 39.The effect of each parameter (a, h, k) on the graph of a parabola relative to the parent function y = x^2
  5. 40.The algebraic reason for reflective symmetry in a quadratic function, grounded in (x-h)^2 producing equal outputs for equal-and-opposite deviations from h
  6. 41.The real-world meaning of a parabola's vertex, intercepts, and symmetry in a modeling context
  7. 42.Differences/ratios as a general discrimination procedure across the three function families studied this year (linear, exponential, quadratic)
  8. 43.The decision to model an unfamiliar symmetric phenomenon as quadratic based on structural reasoning about accumulating change, not surface features
Unit 6: Solving Quadratic Equations10 lessons
  1. 44.Greatest common factor extraction combined with trinomial factoring
  2. 45.The zero product property applied to a factored trinomial equation
  3. 46.The logical requirement of a zero side for the zero product property
  4. 47.Structural features that distinguish difference-of-squares and trinomial forms
  5. 48.The completing-the-square procedure transforming ax^2+bx+c=0 into (x-p)^2=q
  6. 49.The derivation connecting completing the square to the quadratic formula
  7. 50.The discriminant as a predictor of solution type
  8. 51.Method selection and contextual interpretation of quadratic solutions
  9. 52.Equivalence of solution methods for a single quadratic equation
  10. 53.Domain restrictions on algebraic solutions to quadratic models
Unit 7: Piecewise Functions and Absolute Value Graphs8 lessons
  1. 54.Piecewise function evaluation at boundary and non-boundary inputs
  2. 55.Piecewise linear graphs with domain-restricted pieces
  3. 56.The one-function structure of a piecewise definition despite multiple rules
  4. 57.Absolute value as a two-piece function, compared against a general piecewise example
  5. 58.Transformations of f(x) = |x| producing shifted V-graphs
  6. 59.Piecewise models of tiered real-world rate structures, including boundary-ownership justification
  7. 60.Selecting and justifying piecewise vs. smooth modeling for a novel step-rate context
  8. 61.Internal consistency (single-output) errors in a constructed piecewise model
Unit 8: Modeling with Data: Linear Regression and an Introduction to Variability9 lessons
  1. 62.Scatterplot construction and visual association strength/direction
  2. 63.Slope and intercept of a fitted regression line in context
  3. 64.Residual as signed vertical distance between actual and predicted value
  4. 65.Comparison of candidate fitted lines via total residual magnitude
  5. 66.Residual plot pattern as evidence of model appropriateness
  6. 67.Correlation coefficient versus visual/residual evidence of linear fit quality (Anscombe-style contrasting cases)
  7. 68.Confounding variables and causal versus correlational claims
  8. 69.Linear versus exponential versus quadratic growth pattern as evidence for model choice
  9. 70.Full linear-model critique integrating fit, residual analysis, and causal limitation

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