Algebra I

This is the unit everything else depends on. Your child learns that a linear relationship — a starting amount plus a steady rate of change — can be shown as a table, a graph, an equation, or a story about a phone plan or a tank of water, and that these are four views of the same thing, not four different skills. They also meet slope, intercepts, domain/range, and function notation for the first time.

1. Starting value and constant rate of change in a verbal linear context+

   Given a real-world description of a linear situation (e.g., a text-messaging plan), the student identifies the starting amount and the constant amount of change per unit and states which is which.

2. Slope as a computed ratio of vertical change to horizontal change+

   The student computes slope from two points or a table using (change in y)/(change in x) and correctly labels the units of the result in context.

3. Invariance of slope across point pairs on one line+

   The student explains why the slope computed between any two points on the same line is identical, using the idea of equal-interval accumulation of change rather than recomputation of the formula.

4. Distinguishing linear structure from graph/table appearance alone+

   Given a table, a graph, and a verbal description of three DIFFERENT linear situations, the student sorts them by which representation was generated FIRST and justifies the sort using surface cues that turn out to be unreliable, then revises the sort using structural cues (constant difference, common ratio check).

5. Slope-intercept equation construction from varied input formats+

   The student writes a linear equation in slope-intercept form from a context, a table, or two points, choosing the correct method for the given starting information.

6. Function notation f(x) as input-output pairing tied to context+

   The student evaluates and interprets f(x) notation, stating what f(3) = 11 means in terms of the original context's two quantities.

7. Domain and range as context-imposed restrictions on an algebraic relationship+

   Given a context with a natural restriction (e.g., number of weeks cannot be negative, or a tank empties and cannot go below zero), the student determines a realistic domain and range and explains what part of the algebraic line is NOT part of the model.

8. Equality of two linear expressions as the point of agreement between two contexts+

   The student solves a one-variable linear equation that arises from setting two linear expressions equal (e.g., two phone plans costing the same), and interprets the solution as the point where two stories agree, without being told in advance that this is what the algebra represents.

9. Appropriateness of a linear model and selection among representations for a decision+

   Given an entirely new kind of everyday situation not resembling any unit example (e.g., a recipe scaling with a fixed one-time setup cost analog), the student decides, without being told, whether a linear model is appropriate and produces all four representations, then argues in writing which representation someone with a specific decision to make should trust.

10. Vocabulary: slope, x-intercept, y-intercept+

    The student recalls and correctly states the definitions of slope, x-intercept, and y-intercept when shown a labeled graph.

Now there are two conditions that both have to be true at the same time — two phone plans, two recipes, two moving cars — and 'solving' means finding where they agree, or discovering they never do. Your child learns graphing, substitution, and elimination as three ways of asking the same question, and learns to pick the efficient method by looking at the equations rather than always defaulting to one.

1. The point of intersection of two lines as a shared solution to both equations+

   Given a system of two linear equations graphed on the same coordinate plane, identify the point of intersection and state what it means for both equations to be true at that point simultaneously.

2. The graphing method for solving a linear system+

   Solve a system of two linear equations by graphing, including cases where the intersection has non-integer coordinates requiring estimation.

3. The substitution method for solving a linear system+

   Solve a system of two linear equations using substitution when one equation is already solved for a variable or can be solved for a variable in one step.

4. 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+

   Solve a system of two linear equations using elimination, including cases requiring multiplying one or both equations by a constant before adding, and, for a NEW pair of equations not previously demonstrated, explain in the student's own words why replacing an equation with the sum of itself and a multiple of the other equation does not change the solution set.

5. The relationship between slope/intercept and the number of solutions to a linear system+

   Given the graphs or equations of two lines, classify the system as having one solution, no solution, or infinitely many solutions, and explain the classification in terms of slope and intercept.

6. Translation of a two-constraint real-world scenario into a system of linear equations+

   Given a novel real-world scenario with two independent constraints (e.g., cost, distance, mixture), write a system of two linear equations that models the situation, choosing variable definitions and units without being told what the variables represent.

7. Viability of an algebraic solution against the real-world constraints of the situation it models+

   Given a system's algebraic solution in a modeling context, determine whether the solution is viable given real-world constraints (e.g., non-negative time, whole-number people) and justify the determination.

8. The relationship between equation structure (coefficient values, isolated variables) and the relative efficiency of substitution versus elimination+

   Given two systems of linear equations with different coefficient structures, compare substitution and elimination and construct a written argument for which method is more efficient for each, citing specific features of the equations.

9. Solving and interpreting a system in a context that does not signal which special case or method applies+

   Given an unfamiliar system embedded in a non-routine context (e.g., a scenario with a disguised no-solution or infinite-solution structure, or requiring a unit conversion before writing equations), select and apply an appropriate solution method and interpret the result.

This unit swaps 'equals' for 'greater than or less than,' and introduces absolute value as a short way to describe distance from a target — like a thermostat that has to stay within 5 degrees of 70. Your child solves and graphs inequalities, learns why multiplying by a negative flips the inequality sign, splits absolute value problems into cases, and graphs systems of inequalities using an actual test point rather than guessing.

1. The sign-reversal rule when multiplying or dividing an inequality by a negative number+

   Solve a one-variable linear inequality involving a negative coefficient and correctly reverse the inequality symbol, then represent the solution on a number line.

2. The geometric reason the inequality direction reverses under multiplication by a negative number+

   Explain why multiplying a true numeric inequality by a negative number reverses its direction, using a number line as evidence rather than restating the memorized rule.

3. Compound inequalities joined by 'and' or 'or'+

   Translate a verbal description containing 'and'/'or' logic (e.g., 'the temperature is above 60 and below 80') into a compound inequality and graph its solution region on a number line.

4. The equivalence between absolute-value-distance statements and compound inequality statements+

   Given a real-world tolerance context stated only in words, produce both an absolute value inequality and an equivalent compound inequality, and justify their equivalence using a shared number line.

5. Case-splitting procedure for absolute value equations, including the no-solution and one-solution conditions+

   Solve an absolute value equation of the form |ax + b| = c by splitting into two cases, correctly identifying when zero or one solution (rather than two) results.

6. The and/or branching rule for absolute value inequalities based on the inequality direction+

   Solve an absolute value inequality (|ax + b| < c or |ax + b| > c) and correctly determine whether the solution is a compound 'and' region or a compound 'or' region.

7. Systems of linear inequalities and their graphed solution (feasible) region+

   Given a two-inequality system with no algebraic labels indicating method, determine whether a given point satisfies the system and identify the boundary lines that define the feasible region on a graph.

8. Systems of inequalities combined with absolute value constraints in an unfamiliar applied context+

   Given a novel real-world constraint scenario with three or more conditions (some inequalities, at least one absolute value), construct the full system, graph it, and identify whether a specific proposed solution point is feasible, with no indication in the problem of which techniques from the unit apply.

This unit makes f(x) notation official (it was informal in Unit 1) and asks the Unit 2 question again — where do two things agree? — but now one of them is curved instead of straight. Your child compares functions given in different formats (table vs. graph vs. equation), solves f(x) = g(x) by graphing and by matching input-output pairs in a table, and gets a first look at non-constant rate of change without the full quadratic machinery yet.

1. Function notation f(x) as input-output correspondence+

   Given f(x) = 2x - 3 or a similar linear rule, evaluate f(a) for a numeric input and state what f(a) represents in a labeled context (e.g., cost after a hours).

2. Comparison of function properties across representations+

   Given two functions in different representations (one as a table, one as a graph, one as an equation), compare a stated property such as which has the greater output at x=2 or which increases faster over an interval.

3. The equivalence between graphical intersection and algebraic solution of f(x)=g(x)+

   Explain why the x-coordinate of an intersection point of y=f(x) and y=g(x) is a solution to f(x)=g(x), using a labeled graph as evidence.

4. Graphical solution of f(x)=g(x) for a linear-nonlinear pair+

   Solve f(x)=g(x) for a linear function f and a given quadratic or absolute-value function g by graphing both and reading the intersection point(s).

5. Number of solutions to f(x)=g(x) as a property of the specific function pair+

   Determine numerically, from a table of paired values, all x-values where f(x)=g(x), including cases with zero, one, or two solutions.

6. Translation of a verbal scenario into a function pair and interpretation of their intersection+

   Given a real-world scenario described verbally (e.g., a rising cost line vs. a spiking usage curve), construct the linear and nonlinear function rules from the description and identify what the intersection means in context.

7. The general equivalence of graphical and numerical solution methods for f(x)=g(x)+

   Given a new, previously unseen pair of functions (e.g., one absolute value and one linear, in unfamiliar contexts), justify using both a graph and a table why the algebraic and graphical solutions to f(x)=g(x) must be the same value.

8. The meaning of the notation f(x) versus multiplication notation+

   Recall the definition of function notation f(x) as 'the output of function f when the input is x' when asked to distinguish it from multiplication (f times x).

This is the parabola unit — but before any equation-solving. Your child learns that a quadratic's rate of change is itself changing at a steady pace (a idea Unit 5's exponential work set up as a contrast), finds this pattern numerically through 'second differences,' and learns to read vertex, axis of symmetry, and intercepts from a graph or from vertex form. Transformations reuse the shift vocabulary from Unit 1 rather than treating parabola-shifting as new.

1. Second differences as the numerical signature of quadratic growth, contrasted with constant first differences (linear) and constant ratios (exponential)+

   Given a table of (x, y) pairs with equal x-intervals, compute first and second differences and use the pattern (constant, constant-second, or constant-ratio) to classify the table as linear, quadratic, or exponential.

2. The vertex and axis of symmetry as features read from a graphed parabola+

   Identify the vertex, axis of symmetry, and y-intercept of a parabola directly from its graph.

3. The correspondence between the parameters h and k in vertex form and the vertex's coordinates+

   Rewrite a quadratic function given in vertex form, y = a(x-h)^2 + k, to state the vertex and axis of symmetry without graphing.

4. The effect of each parameter (a, h, k) on the graph of a parabola relative to the parent function y = x^2+

   Predict the direction, width, and vertical/horizontal position change of a parabola when a, h, or k in y = a(x-h)^2 + k is altered, before graphing to check.

5. 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+

   Explain why a parabola is symmetric about a vertical line through its vertex, using the structure of the squared term in vertex form.

6. The real-world meaning of a parabola's vertex, intercepts, and symmetry in a modeling context+

   Given a real-world context modeled by a quadratic (e.g., projectile height over time), interpret the vertex, an intercept, and the axis of symmetry in terms of the situation, and state what would change about the situation if the symmetry were broken.

7. Differences/ratios as a general discrimination procedure across the three function families studied this year (linear, exponential, quadratic)+

   Given a table of data from an entirely unfamiliar context (not previously classified as linear, exponential, or quadratic in this course), determine which function family best fits using differences and ratios, and justify the choice by naming what pattern would have appeared under each of the other two families.

8. The decision to model an unfamiliar symmetric phenomenon as quadratic based on structural reasoning about accumulating change, not surface features+

   Given only a verbal description of a symmetric real-world phenomenon never framed mathematically before (e.g., an arch bridge's load distribution, or a business's cost curve with a single minimum), construct a rough quadratic model (vertex location and opening direction) and justify why a quadratic, rather than linear or exponential, structure is appropriate.

Now your child actually solves ax^2+bx+c=0 — by factoring, by completing the square, and with the quadratic formula, in that order, and learns the discriminant as a way to predict how many solutions there are before solving. The real point of the unit is judgment: given a specific equation, which method is actually the efficient one, and why.

1. Greatest common factor extraction combined with trinomial factoring+

   Given a trinomial with a GCF, factor out the GCF and then factor the remaining trinomial completely.

2. The zero product property applied to a factored trinomial equation+

   Solve a quadratic equation by factoring a trinomial (a=1) and applying the zero product property.

3. The logical requirement of a zero side for the zero product property+

   Explain why the zero product property requires one side of the equation to equal zero before factoring is useful for solving.

4. Structural features that distinguish difference-of-squares and trinomial forms+

   Classify a quadratic expression as a difference of squares, a factorable trinomial, or neither, based on its term structure.

5. The completing-the-square procedure transforming ax^2+bx+c=0 into (x-p)^2=q+

   Solve a quadratic equation by completing the square, including cases where the leading coefficient is not 1.

6. The derivation connecting completing the square to the quadratic formula+

   Derive the quadratic formula from the completed-square form of ax^2+bx+c=0.

7. The discriminant as a predictor of solution type+

   Use the discriminant (b^2-4ac) to predict the number and type of real or complex solutions without solving the equation.

8. Method selection and contextual interpretation of quadratic solutions+

   Given the unfamiliar real-world context and equation specified for the Day 18 far-transfer task (a garden-path border area problem, ax^2+bx+c=0 with a non-1 leading coefficient not previously paired with an area context in this unit), select and justify the most efficient solving method and interpret the solution(s) in terms of the situation, including when a negative or non-real solution should be rejected.

9. Equivalence of solution methods for a single quadratic equation+

   Compare two different-looking correct solving methods applied to the same equation and explain in writing why both produce equivalent solutions despite different intermediate steps.

10. Domain restrictions on algebraic solutions to quadratic models+

    Given the novel garden-border word problem in Lesson 14 (item text below), whose solution requires recognizing that a quadratic model's algebraic solution set contains an extraneous or non-physical value, identify and justify which root(s) to discard.

A single relationship can need a different rule on different parts of its domain — shipping costs that jump at weight thresholds, parking that charges by the partial hour with a daily cap. Your child learns to evaluate and graph these piecewise functions, discovers that absolute value is just the simplest two-piece case, and builds models for real tiered-rate situations.

1. Piecewise function evaluation at boundary and non-boundary inputs+

   Evaluate a given piecewise function at specified input values, including at least one boundary value, correctly assigning each input to its defining piece.

2. Piecewise linear graphs with domain-restricted pieces+

   Graph a piecewise linear function over its restricted domains, correctly marking open or closed circles at each boundary.

3. The one-function structure of a piecewise definition despite multiple rules+

   Explain why a piecewise function with correctly chosen domain restrictions (no overlaps, no gaps) still represents exactly one function rather than several.

4. Absolute value as a two-piece function, compared against a general piecewise example+

   Compare the two-case definition of absolute value to a piecewise function already graphed, and generalize that |x| is a specific instance of the piecewise structure.

5. Transformations of f(x) = |x| producing shifted V-graphs+

   Graph transformations of the absolute value parent function (vertical/horizontal shifts) by identifying the new vertex location.

6. Piecewise models of tiered real-world rate structures, including boundary-ownership justification+

   Given a real multi-tier rate table (e.g., shipping cost by weight), construct a piecewise function that models it and justify which piece owns each boundary value based on the context's actual billing convention.

7. Selecting and justifying piecewise vs. smooth modeling for a novel step-rate context+

   Given a completely unfamiliar step-rate or bracket-style context not resembling shipping or tax examples used in class (e.g., a parking garage charging by partial hour with a daily cap), determine without being told the general strategy whether a piecewise model is more defensible than a single smooth equation, and construct it.

8. Internal consistency (single-output) errors in a constructed piecewise model+

   Critique a peer-style or app-generated piecewise model of a rate context by identifying a boundary where the model assigns two different outputs to the same input, and explain what change would fix it.

This closing unit turns the year's tools on real, messy data. Fitting a line to a scatterplot is a judgment call, not a single right computation, and the leftover gap between the line and each point — the residual — is itself useful information about where the model is weak. Your child builds scatterplots, fits lines, reads residual plots, and tackles correlation versus causation by first trying to poke holes in a causal claim themselves before getting the formal vocabulary.

1. Scatterplot construction and visual association strength/direction+

   Given a bivariate dataset presented as a table, construct a scatterplot with appropriately scaled axes and describe the direction and strength of the association in words.

2. Slope and intercept of a fitted regression line in context+

   Fit a linear model to a scatterplot using the app's least-squares tool and state the slope and y-intercept of the fitted line in the units and context of the original quantities.

3. Residual as signed vertical distance between actual and predicted value+

   Compute the residual for a given data point from a fitted line and explain what the sign of the residual indicates about the point's position relative to the model.

4. Comparison of candidate fitted lines via total residual magnitude+

   Given two candidate lines fit to the same scatterplot, compare them by computing total residual magnitude and determine which is the better model, justifying the choice by referring to the comparison rather than visual impression alone.

5. Residual plot pattern as evidence of model appropriateness+

   Given a residual plot, determine whether a linear model is appropriate for the data by identifying whether the plot shows random scatter or a systematic (curved) pattern.

6. Correlation coefficient versus visual/residual evidence of linear fit quality (Anscombe-style contrasting cases)+

   Given several scatterplots with similar correlation coefficients but visibly different shapes (including one with an outlier and one with a curved trend), rank them by how trustworthy a linear model would be for each and justify the ranking using shape and residual pattern rather than the r value alone.

7. Confounding variables and causal versus correlational claims+

   Given a real-world pair of strongly correlated variables never discussed in instruction, propose at least one plausible confounding variable or reverse-causal explanation and use it to argue that the correlation does not establish causation.

8. Linear versus exponential versus quadratic growth pattern as evidence for model choice+

   Given a dataset with a visibly nonlinear trend, decide whether a linear model, an exponential model (Unit 5), or a quadratic model (Unit 6) best describes the relationship, and justify the choice using the pattern of change across equal intervals.

9. Full linear-model critique integrating fit, residual analysis, and causal limitation+

   Independently collect or use a given real bivariate dataset to produce a complete written critique that fits a linear model, analyzes residuals, states the correlation strength, and explains at least one limitation of the model including the correlation-causation distinction.