Module 01
Searching for Patterns
22 days
A function is a rule with a job: every input gets exactly one output — and patterns are how we hear that rule.
Topic 1 · Quantities & Relationships (12d) — identifying functions, domain & range, and the families of graphs that describe real situations.
Topic 2 · Sequences (10d) — arithmetic and geometric sequences as the first taste of recursive and explicit rules.
You'll be able to…
- Decide whether a table, graph, or mapping represents a function.
- State the domain and range of a relationship and read them off a graph.
- Sort relationships into families — linear, exponential, quadratic — by their shape.
- Write the explicit rule for an arithmetic or geometric sequence.
Worked example · arithmetic sequence
For \(5,\ 8,\ 11,\ 14,\dots\) the common difference is \(d = 3\), so the explicit rule is
\[ a_n = 5 + 3(n - 1) = 3n + 2 \]
Then \(a_{10} = 3(10) + 2 = 32\).
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Module 02
Exploring Constant Change
32 days
A constant rate of change draws a straight line — and slope is the one number that captures it.
Topic 1 · Linear Functions (22d) — slope, intercepts, and writing \(y = mx + b\) from tables, graphs, and situations.
Topic 2 · Transforming & Comparing Linear Functions (10d) — how changes to parameters shift a line, and comparing models side by side.
You'll be able to…
- Compute slope from two points, a table, or a graph.
- Write a linear equation in slope-intercept and point-slope form.
- Interpret slope and intercept as the rate and starting value of a real situation.
- Predict how changing \(m\) or \(b\) transforms the graph — and compare two linear models.
Worked example · line through two points
Through \((2,\,1)\) and \((5,\,7)\):
\[ m = \frac{7 - 1}{5 - 2} = \frac{6}{3} = 2 \]
Using point-slope, \(y - 1 = 2(x - 2)\), which simplifies to \(y = 2x - 3\). Try these values in the Concepts in Action grapher below.
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Module 03
Modeling Linear Equations & Inequalities
27 days
Solving means finding the inputs that make a statement true — one line, or two lines meeting at a point.
Topic 1 · Linear Equations & Inequalities (10d) — solving and graphing, with inequalities like \(2x - 5 \ge 7\) modeled as constraints.
Topic 2 · Systems of Linear Equations & Inequalities (17d) — substitution, elimination, and graphing solution regions.
You'll be able to…
- Solve multi-step linear equations and justify each step.
- Solve and graph linear inequalities, flipping the sign when dividing by a negative.
- Solve a system by substitution, elimination, and graphing.
- Decide whether a system has one solution, none, or infinitely many.
Worked example · system by elimination
\[ \begin{aligned} 2x + y &= 7 \\ x - y &= 2 \end{aligned} \]
Adding the equations cancels \(y\): \(3x = 9 \Rightarrow x = 3\). Back-substitute: \(3 - y = 2 \Rightarrow y = 1\). Solution \((3,\,1)\).
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Module 04
Investigating Growth & Decay
25 days
When the rate of change is a constant multiplier instead of a constant difference, growth bends — that's exponential.
Topic 1 · Introduction to Exponential Functions (15d) — recognizing \(y = a\,b^{x}\) and the meaning of growth versus decay.
Topic 2 · Using Exponential Equations (10d) — building and applying exponential models to data.
You'll be able to…
- Tell an exponential pattern from a linear one in a table or graph.
- Read the initial value \(a\) and base \(b\) from \(y = a\,b^{x}\).
- Classify a model as growth \((b>1)\) or decay \((0
- Build an exponential model from a real scenario and use it to predict.
Worked example · growth vs. decay
A \$500 investment growing 8% per year: \(y = 500(1.08)^{x}\). After 3 years,
\[ y = 500(1.08)^{3} \approx 629.86 \]
A car worth \$20{,}000 losing 15% per year decays as \(y = 20000(0.85)^{x}\).
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Module 05 · Capstone before the EOC
Maximizing & Minimizing
44 days
A parabola has a turning point — the highest or lowest value a quadratic situation can reach — and finding it is the whole game.
Topic 1 · Introduction to Quadratic Functions (15d) — the parabola \(y = ax^2 + bx + c\), its vertex, and axis of symmetry.
Topic 2 · Polynomial Operations (10d) — adding, subtracting, multiplying, and factoring polynomial expressions.
Topic 3 · Solving Quadratic Equations (19d) — factoring, the quadratic formula, and interpreting roots.
You'll be able to…
- Identify the vertex, axis of symmetry, and direction a parabola opens.
- Add, subtract, and multiply polynomials — and factor quadratic trinomials.
- Solve quadratic equations by factoring, square roots, and the quadratic formula.
- Use the discriminant to predict whether a quadratic has two, one, or no real roots.
- Interpret roots and the vertex as answers to a real maximizing/minimizing question.
Worked example · solve by factoring
Solve \(x^2 - 5x + 6 = 0\). Factor:
\[ (x - 2)(x - 3) = 0 \;\Rightarrow\; x = 2 \text{ or } x = 3 \]
Confirm both roots on the parabola in the Concepts in Action grapher below.
Worked example · quadratic formula
For \(2x^2 + 3x - 2 = 0\),
\[ x = \frac{-3 \pm \sqrt{3^2 - 4(2)(-2)}}{2(2)} = \frac{-3 \pm 5}{4} \]
so \(x = \tfrac{1}{2}\) or \(x = -2\). The discriminant \(b^2 - 4ac = 25 > 0\) confirms two real roots.
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