| SL.1.1 |
Operations with numbers in the form a × 10^k where 1 ≤ a < 10 and k is an integer (scientific notation). (p. 29) |
Exponents and Radicals — Estimating with scientific notation, Expressing numbers in scientific notation, Multiplying and dividing in scientific notation |
| SL.1.3 |
Geometric sequences and series: use of the formulae for the nth term and the sum of the first n terms; use of sigma notation for the sums of geometric sequences; applications. (p. 29) |
Exponential Functions — Identify exponential decay from a dot pattern, Identify exponential growth from a dot pattern |
| SL.1.4 |
Financial applications of geometric sequences and series: compound interest and annual depreciation; calculating the real value of an investment given an interest rate and an inflation rate. (p. 30) |
Exponential Functions — Graph and compare compound interest functions, Write a compound growth or decay expression |
| SL.1.5 |
Laws of exponents with integer exponents; introduction to logarithms with base 10 and e; numerical evaluation of logarithms using technology; awareness that a^x = b is equivalent to log_a b = x, that b > 0, and that log_e x = ln x. (p. 30) |
Exponents and Radicals — Applying the power rule for exponents, Applying the product rule for exponents, Applying the quotient rule for exponents, Combining exponent rules to simplify, Dividing with prime factorizations, Evaluating zero-exponent expressions, Multiplying with prime factorizations, Rewriting with negative exponents, Simplifying multi-base quotients, Simplifying negative exponents in denominators, Simplifying products with exponent zero, Simplifying with negative exponents; Logarithms — Convert between exponential and logarithmic form, Estimate a logarithm, Evaluate a logarithm exactly, Find the exponent in an exponential equation; Exponential Functions — Express decay with a negative exponent; Polynomials — Multiplying monomials; Equations and Curves — Simplifying with exponent rules, Simplifying with exponent rules |
| SL.1.7 |
Laws of exponents with rational exponents; laws of logarithms (product, quotient and power rules); change of base of a logarithm; solving exponential equations, including using logarithms. (p. 31) |
Logarithms — Change the base of a logarithm, Convert between log bases graphically, Rewrite logarithms with the power rule, Rewrite logarithms with the product and quotient rules, Solve exponential equations with logarithms, Solve logarithmic equations, Transform a log graph using identities; Exponents and Radicals — Connecting fractional exponents to radicals, Evaluating general fractional exponents, Evaluating unit fraction exponents, Multiplying square root expressions, Simplifying and combining unlike radicals; Equations and Curves — Evaluating roots and inverse compositions, Evaluating square roots of squares, Simplifying fractional exponents |
| SL.1.8 |
Sum of infinite convergent geometric sequences. (p. 32) |
Calculus — Determining convergence of series |
| SL.2.1 |
Different forms of the equation of a straight line — y = mx + c (gradient-intercept), ax + by + d = 0 (general), y − y₁ = m(x − x₁) (point-gradient); gradient and intercepts; parallel lines m₁ = m₂ and perpendicular lines m₁ × m₂ = −1. (p. 37) |
Linear Relationships — Calculating rates of change, Computing slope, Evaluating linear equations, Finding constant change, Finding perpendicular slopes, Finding starting values, Graphing from point-slope form, Graphing lines from intercepts, Graphing lines from slope, Graphing real-world relationships, Interpreting intercepts, Writing parallel line equations, Writing perpendicular line equations, Writing point-slope equations, Writing slope-intercept equations, Writing standard form equations |
| SL.2.2 |
Concept of a function, domain, range and graph; function notation such as f(x), v(t), C(n); the concept of a function as a mathematical model; the informal concept that an inverse function reverses or undoes the effect of a function, the inverse as a reflection in the line y = x, and the notation f⁻¹(x). (p. 37) |
Introduction to Functions — Applying the vertical line test, Finding function domains, Finding function ranges, Following conditional rules, Following function rules, Identifying conditional rules, Identifying function rules, Matching input-output pairs, Recognizing non-functions, Writing numeric function rules; Equations and Curves — Connecting transformations to domain and range, Determining domain and range, Finding domain and range after transformations, Finding hyperbola domain and range, Graphing the square root function |
| SL.2.3 |
The graph of a function and its equation y = f(x); creating a sketch from information given or from a context, including transferring a graph from screen to paper; using technology to graph functions including their sums and differences. (p. 38) |
Introduction to Functions — Combining functions with operations, Evaluating floor and ceiling functions, Plotting conditional functions, Plotting numeric functions; Equations and Curves — Identifying power function graphs, Matching equations to graphs, Matching equations to graphs |
| SL.2.4 |
Determine key features of graphs: maximum and minimum values, intercepts, symmetry, vertex, zeros of functions or roots of equations, and vertical and horizontal asymptotes, using graphing technology; finding the points of intersection of two curves or lines using technology. (p. 38) |
Equations and Curves — Controlling the number of intersections, Finding intersection points, Identifying symmetric points, Reading values from graphs, Solving systems and unions of curves, Using hyperbola symmetry; Polynomials — Finding the y-intercept, Identifying transformed features; Introduction to Functions — Identifying function intervals, Reading continuous function graphs, Reading discontinuous function graphs |
| SL.2.5 |
Composite functions (f ∘ g)(x) = f(g(x)); the identity function; finding the inverse function f⁻¹(x) with (f ∘ f⁻¹)(x) = (f⁻¹ ∘ f)(x) = x; the existence of an inverse for one-to-one functions. (p. 39) |
Introduction to Functions — Determining invertibility, Evaluating composite functions, Finding inverse functions, Writing composite function expressions; Equations and Curves — Evaluating squaring and square roots |
| SL.2.6 |
The quadratic function f(x) = ax² + bx + c: its graph, y-intercept (0, c) and axis of symmetry; the factored form f(x) = a(x − p)(x − q) with x-intercepts (p, 0) and (q, 0); the vertex form f(x) = a(x − h)² + k with vertex (h, k). (p. 39) |
Quadratics — Connecting intercepts to factors, Converting to vertex form, Determining signs in products, Expanding binomial products, Factoring signed quadratics fully, Factoring simple quadratics, Factoring with leading coefficients, Factoring with negative constants, Finding maximum and minimum values, Finding the axis of symmetry, Finding the vertex, Finding vertices using symmetry, Graphing by factoring, Graphing parabolas from vertex form, Reading solutions from graphs, Recognizing perfect square patterns, Writing equations from graphs, Writing vertex form equations |
| SL.2.7 |
Solution of quadratic equations and inequalities; the quadratic formula; the discriminant Δ = b² − 4ac and the nature of the roots (two distinct real roots, two equal real roots, no real roots). (p. 39) |
Quadratics — Classifying solution types, Completing squares with fractions, Determining number of solutions, Isolating squared variables, Solving by completing the square, Solving equations by factoring, Solving shifted square equations, Solving with square roots, Using the discriminant, Using the quadratic formula |
| SL.2.8 |
The reciprocal function f(x) = 1/x, x ≠ 0, its graph and self-inverse nature; rational functions of the form f(x) = (ax + b)/(cx + d) and their graphs; equations of vertical and horizontal asymptotes. (p. 40) |
Equations and Curves — Completing hyperbola equations, Finding equation parameters, Finding hyperbola asymptotes, Locating asymptotes |
| SL.2.9 |
Exponential functions and their graphs f(x) = a^x (a > 0) and f(x) = e^x; logarithmic functions and their graphs f(x) = log_a x (x > 0) and f(x) = ln x; the relationships a^x = e^(x ln a) and log_a a^x = x, and exponential and logarithmic functions as inverses of each other. (p. 40) |
Exponential Functions — Build a decay function with a fractional base, Build an exponential function from non-integer points, Build an exponential function from plotted points, Build an exponential function over negative domain, Deduce an exponential function from two points, Evaluate an exponential at negative x-values, Evaluate an exponential at non-integer x-values, Find the base of an exponential function, Set the coefficient of an exponential function; Logarithms — Check solutions of logarithmic equations, Find the domain of a logarithmic function, Graph a logarithmic function |
| SL.2.10 |
Solving equations, both graphically and analytically; use of technology to solve a variety of equations, including those where there is no appropriate analytic approach; applications of graphing skills and solving equations that relate to real-life situations. (p. 41) |
Exponential Functions — Compare exponential growth to linear growth, Compare exponential growth to quadratic growth, Compare two exponential functions |
| SL.2.11 |
Transformations of graphs: translations y = f(x) + b and y = f(x − a); reflections in both axes y = −f(x) and y = f(−x); vertical stretch with scale factor p, y = p f(x); horizontal stretch with scale factor 1/q, y = f(qx); composite transformations. (p. 41) |
Equations and Curves — Applying reflections and equivalent stretches, Applying shifts and stretches, Combining multiple transformations, Identifying transformed graphs, Shifting the square root graph, Stretching and reflecting root graphs, Transforming hyperbola graphs, Transforming the square root graph, Writing transformations from constraints; Polynomials — Applying single transformations, Combining transformations; Introduction to Functions — Combining reflections and stretches, Reflecting functions across axes, Sliding Function Graphs, Stretching Function Graphs |
| SL.3.2 |
Use of the sine, cosine and tangent ratios to find the sides and angles of right-angled triangles; the sine rule a/sin A = b/sin B = c/sin C; the cosine rule c² = a² + b² − 2ab cos C and cos C = (a² + b² − c²)/(2ab); the area of a triangle as ½ ab sin C. (p. 45) |
Trigonometric Functions — Finding special right triangle sides |
| SL.3.4 |
The circle: radian measure of angles; length of an arc; area of a sector. (p. 46) |
Trigonometric Functions — Computing arc length, Constructing angles in radians, Constructing unit circle arcs, Converting degrees and radians, Identifying angles from geometry; Polar Coordinate Plane — Computing arc lengths from angles, Computing arc lengths from fractions, Describing features in radians, Finding angles from arc lengths, Plotting points using radians |
| SL.3.5 |
Definition of cos θ and sin θ in terms of the unit circle, including the relationship between angles in different quadrants; definition of tan θ as sin θ / cos θ; exact values of the trigonometric ratios of 0, π/6, π/4, π/3, π/2 and their multiples; extension of the sine rule to the ambiguous case. (p. 46) |
Trigonometric Functions — Applying symmetry and periodicity rules, Determining tangent sign by quadrant, Evaluating sine and cosine, Evaluating tangent at an angle, Evaluating trig functions in radians, Evaluating trig values at key angles, Finding angle pairs for trig values, Reading cosine from circular motion, Reading sine from circular motion; Polar Coordinate Plane — Computing x and y from polar, Finding quadrants from angles |
| SL.3.6 |
The Pythagorean identity cos²θ + sin²θ = 1; double angle identities for sine and cosine; the relationship between trigonometric ratios. (p. 47) |
Trigonometric Functions — Using the Pythagorean identity |
| SL.3.7 |
The circular functions sin x, cos x and tan x: amplitude, their periodic nature and their graphs; composite functions of the form f(x) = a sin(b(x + c)) + d; transformations; real-life contexts. (p. 47) |
Trigonometric Functions — Computing period from rotation speed, Finding transformed function periods, Identifying amplitude and midline, Identifying periodic function features, Matching functions with transformations, Predicting periodic values, Transforming periodic functions |
| SL.3.8 |
Solving trigonometric equations in a finite interval, both graphically and analytically, including equations leading to quadratic equations in sin x, cos x or tan x. (p. 48) |
Trigonometric Functions — Factoring non-linear trig equations, Solving equations from the tangent graph, Solving equations from trig graphs, Solving equations with inverse cosine, Solving equations with inverse sine, Solving trig equations on an interval |
| SL.4.1 |
Concepts of population, sample, random sample, discrete and continuous data; reliability of data sources and bias in sampling; interpretation of outliers; sampling techniques and their effectiveness. (p. 53) |
Everyday Statistics — Using the outlier rule |
| SL.4.2 |
Presentation of data (discrete and continuous): frequency distributions (tables); histograms; cumulative frequency and cumulative frequency graphs, used to find median, quartiles, percentiles, range and interquartile range; production and understanding of box and whisker diagrams. (p. 54) |
Exploring Data Visually — Compare boxplots across datasets, Find frequencies and percentiles from a histogram, Read median, quartiles, and extremes from a boxplot, Reading histograms; Everyday Statistics — Comparing data sets with box plots, Matching data to box plots, Reading box plots; Regression — Detecting hidden groups in data |
| SL.4.3 |
Measures of central tendency (mean, median and mode), including estimation of the mean from grouped data and the modal class; measures of dispersion (interquartile range, standard deviation and variance); the effect of constant changes on the original data; quartiles of discrete data. (p. 55) |
Everyday Statistics — Combining groups to find the mean, Comparing data above and below the mean, Comparing mean, median, and mode, Finding quartiles, Finding the mean (the average) of a data set, Finding the mean of uneven data, Finding the median, Finding totals from the mean, Predicting how new data shifts the mean, Seeing how outliers affect the mean, Splitting data into quarters, Updating the mean with new data, Updating the median with new data |
| SL.4.4 |
Linear correlation of bivariate data; Pearson's product-moment correlation coefficient r; scatter diagrams and lines of best fit by eye passing through the mean point; the equation of the regression line of y on x and its use for prediction; interpretation of the parameters a and b in y = ax + b. (p. 55) |
Regression — Checking the groups inside data, Choosing the best predictor, Fitting a line to data, Interpreting correlations in context, Modeling one subgroup at a time, Predicting from a model equation, Predicting from a regression line, Reading correlation from a scatter plot, Spotting nonlinear relationships; Exploring Data Visually — Describe the relationship in a scatter plot, Describing how two things relate, Judging how strongly things relate, Reading scatter plots |
| SL.4.5 |
Concepts of trial, outcome, equally likely outcomes, relative frequency, sample space (U) and event, with P(A) = n(A)/n(U); the complementary events A and A′; expected number of occurrences. (p. 56) |
Probability and Chance — Calculating probability, Changing probability by removing options, Comparing the likelihood of outcomes, Counting possible outcomes, Finding the probability something won't happen; Probability in Data — Estimating probabilities from past data |
| SL.4.6 |
Use of Venn diagrams, tree diagrams, sample space diagrams and tables of outcomes to calculate probabilities; combined events P(A ∪ B) = P(A) + P(B) − P(A ∩ B); mutually exclusive events P(A ∩ B) = 0; conditional probability P(A|B) = P(A ∩ B)/P(B); probabilities with and without replacement; independent events P(A ∩ B) = P(A)P(B). (p. 57) |
Probability in Data — Adding probabilities of separate outcomes, Combining probabilities across cases, Combining probabilities with a formula, Finding conditional probabilities from data, Finding probabilities of multi-step outcomes, Judging how often a rule fails; Probability and Chance — Counting outcomes across overlapping events, Finding probability given a condition, Finding probability of two unrelated events, Finding the probability of both events, Finding the probability of either event, Using Venn diagrams |
| SL.4.7 |
Concept of discrete random variables and their probability distributions; expected value (mean) for discrete data; applications. (p. 57) |
Probability and Chance — Calculating expected value, Calculating expected value for multiple events, Comparing expected values |
| SL.4.11 |
Formal definition and use of P(A|B) = P(A ∩ B)/P(B) for conditional probabilities, and of P(A|B) = P(A) = P(A|B′) for independent events; testing for independence. (p. 59) |
Probability in Data — Finding probability of two related events, Identifying when events affect each other |
| SL.5.1 |
Introduction to the concept of a limit; the derivative interpreted as a gradient function and as a rate of change. (p. 61) |
Calculus — Finding rates of change |
| SL.5.3 |
Derivative of f(x) = ax^n is f′(x) = anx^(n−1) for n ∈ ℤ, and the derivative of functions of the form f(x) = ax^n + bx^(n−1) + … where all exponents are integers. (p. 62) |
Calculus — Finding derivatives of polynomials |
| SL.5.5 |
Introduction to integration as anti-differentiation of functions of the form f(x) = ax^n + bx^(n−1) + … where n ∈ ℤ and n ≠ −1; anti-differentiation with a boundary condition to determine the constant term; definite integrals using technology; the area of a region enclosed by a curve y = f(x) and the x-axis where f(x) > 0. (p. 63) |
Calculus — Finding antiderivatives |
| SL.5.6 |
Derivatives of x^n (n ∈ ℚ), sin x, cos x, e^x and ln x; differentiation of a sum and a multiple of these functions; the chain rule for composite functions; the product and quotient rules. (p. 63) |
Calculus — Finding trig and exponential derivatives, Using the chain and quotient rules |
| SL.5.8 |
Local maximum and minimum points; testing for maximum and minimum; optimization; points of inflexion with zero and non-zero gradients. (p. 64) |
Calculus — Finding maximums and minimums, Testing for maximums and minimums |
| SL.5.11 |
Definite integrals, including the analytical approach ∫ from a to b of g′(x)dx = g(b) − g(a); areas of a region enclosed by a curve y = f(x) and the x-axis where f(x) can be positive or negative, without the use of technology; areas between curves. (p. 65) |
Calculus — Evaluating integrals |
