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Square roots and nth roots, including \\sqrt[n]{a}=a^{\\frac1n}, rational exponents a^{\\frac{m}{n}}, simplifying radicals, roots of negative numbers, and rationalizing denominators using roots or conjugates.

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Logarithm algebra including the definition \\log_a b=x \\Leftrightarrow a^x=b, base 10 and natural logs, evaluating logs exactly or with technology, and using the rules \\log_a x+\\log_a y=\\log_a(xy), \\log_a x-\\log_a y=\\log_a(x/y), \\log_a(x^m)=m\\log_a x, change of base, and logarithms to solve exponential equations.

\n","hasLesson":true},{"id":170,"name":"Exponential functions","slug":"exponential-functions","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757030057079-Artboard%204.png?alt=media&token=eabb4444-acbe-474e-8671-d9c3c8b4b523","type":"image"},"skills":[493,494,797,904,901,902],"description":"

Exponential functions f(x)=a^x and logarithmic functions f(x)=\\log_a x, their domains and ranges, inverse relationship and symmetry in y=x, and graphing/making sense of exponential growth and decay models such as ca^x+k or Ab^t+c.

\n","hasLesson":true}],"description":"

Exponents, roots and rational powers, logarithm laws and change of base, and exponential and logarithmic functions and equations.

\n"},{"id":104,"name":"Function Theory","slug":"function-theory","topic":2,"subjects":["AAHL","AASL","AIHL","AISL","ecolint-y11-hl"],"isVideoReady":true,"videoId":"5-tzjROivOo","concepts":[{"id":252,"name":"Functions and their properties","slug":"functions-and-their-properties","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1770684953385-function%20machine.png?alt=media&token=c59ac711-975a-4c73-ae39-f10fda053427","type":"image"},"skills":[384,1073,1074,1075,1085],"description":"The concept of a function as a mathematical machine, the notation f\\left(x\\right)=\\ldots, and function tables / diagrams.","hasLesson":true},{"id":258,"name":"Function Graphs","slug":"function-graphs","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1770685518993-function%20graphs.png?alt=media&token=d16df97f-db93-49ec-a84f-bd64de28e3ec","type":"image"},"skills":[415,847,849,260,505,373,391,375,374,376,1035,907],"description":"

Graphing functions with a calculator, characteristics of a function, even and odd functions, x and y intercepts, horizontal and vertical asymptotes, maxima and minima

","hasLesson":true},{"id":253,"name":"Function Composition","slug":"function-composition","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1770854427967-function%20composition.png?alt=media&token=81948aeb-8bb9-4784-8de2-4a1944a38553","type":"image"},"skills":[763],"description":"

Composing functions like f\\circ g

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Function notation and representations, including evaluating, graphing and modeling functions, domain and range, composition and inverse functions, and using technology to analyze graphs, intercepts, extrema and asymptotes.

\n"},{"id":865,"name":"Rounding & Error","slug":"rounding-error","topic":1,"subjects":["AISL","AASL","AIHL","AAHL"],"isVideoReady":false,"videoId":null,"concepts":[{"id":866,"name":"Rounding and estimation","slug":"rounding-and-estimation","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757030041198-Artboard%205.png?alt=media&token=25922b18-0ebb-4291-af0d-e8584c14c0e9","type":"image"},"skills":[869,877],"description":"

Rounding, significant figures, decimal places

","hasLesson":true},{"id":868,"name":"Scientific Notation","slug":"scientific-notation","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1773515918578-Artboard%206.png?alt=media&token=69a03129-81d4-4eeb-a181-bef451d1b1cd","type":"image"},"skills":[882,885,887],"description":"

Converting numbers to and from scientific notation \\left(a\\times10^{k}\\right)

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Rounding, significant figures, and scientific notation.

"},{"id":30,"name":"Sequences & Series","slug":"sequences-series","topic":1,"subjects":["AAHL","AIHL","AISL","AASL","ecolint-y11-hl"],"isVideoReady":true,"videoId":"DI2cudLIWEk","concepts":[{"id":27,"name":"Arithmetic Sequences","slug":"arithmetic-sequences","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029981178-Artboard%207.png?alt=media&token=8a340859-acc8-458a-a5be-e962cb927b61","type":"image"},"skills":[602,603],"description":"

Introduction to the definition, properties, and general term of an arithmetic sequence

","hasLesson":true},{"id":28,"name":"Geometric Sequences","slug":"geometric-sequences","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757030011943-Artboard%208.png?alt=media&token=96b137e1-fd1c-43b5-9749-9222bc9e9393","type":"image"},"skills":[604,40],"description":"

Introduction to definition, identification, and general term of geometric sequences

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Definition of series, formula for arithmetic series

","hasLesson":true},{"id":860,"name":"Summation notation","slug":"summation-notation","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029991958-Artboard%2010.png?alt=media&token=dcfdc9b6-9e57-47bc-b6bc-9b4088d76eb5","type":"image"},"skills":[47,837,861,862,863,905],"description":"

Summation notation, properties of sums

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Finite geometric series with sum S_{n}=\\frac{u_1(1-r^n)}{1-r}, and infinite geometric series convergence when |r|<1 so that S_\\infty=\\frac{u_1}{1-r}.

\n","hasLesson":true},{"id":906,"name":"Sequence Mode (Calculator)","slug":"sequence-mode-calculator","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029904498-Artboard%2011.png?alt=media&token=542aa577-0f10-43f9-b206-99a890877c96","type":"image"},"skills":[],"description":"

Working with sequences on a calculator, computations and applications of a calculator

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Compound growth, depreciation, interest, inflation, real value

","hasLesson":true}],"description":"Arithmetic sequences u_{n}=u_1+\\left(n-1\\right)d and geometric sequences v_{n}=v_1r^{n-1}, as well as their series and the \\Sigma notation for summation."},{"id":100,"name":"Counting & Binomials","slug":"counting-binomials","topic":1,"subjects":["AAHL","AASL","ecolint-y11-hl"],"isVideoReady":true,"videoId":"w08O6aqQegU","concepts":[{"id":246,"name":"Pascal's Triangle and nCr","slug":"pascals-triangle-and-ncr","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029889677-Artboard%2013.png?alt=media&token=8abbdcf6-30e5-4df2-a898-50033c4a5db7","type":"image"},"skills":[631,630,629,1025],"description":"

Pascal's triangle and binomial coefficients, using factorials and ^{n}C_{r}=\\frac{n!}{r!(n-r)!} to find combinations and the coefficients in binomial expansions.

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Expansion of \\left(a+b\\right)^n using binomial coefficients, including the general term \\binom{n}{r}a^{n-r}b^r and finding the coefficient of a specified term.

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Factorials, combinations and Pascal's triangle, and using the binomial theorem to expand \\left(a+b\\right)^n and find specific coefficients.

\n"},{"id":102,"name":"Proof and Reasoning","slug":"proof-and-reasoning","topic":1,"subjects":["AAHL","AASL"],"isVideoReady":true,"videoId":"syZ6AKBabRA","concepts":[{"id":240,"name":"Proof by deduction","slug":"proof-by-deduction","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029849287-Artboard%2017.png?alt=media&token=312ca3d8-5599-4f64-8439-464901c0de57","type":"image"},"skills":[652,650],"description":"

Strict logical proofs by deduction, including parity arguments with even and odd integers written as n=2k and n=2k+1, and direct proofs that two expressions are equivalent by showing \\text{LHS}\\equiv\\text{RHS} for all variables.

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Simple deductive proofs to show LHS\\equiv RHS.

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Linear transformations of graphs and points: translations by a vector \\begin{pmatrix}a\\\\ b\\end{pmatrix}, vertical and horizontal stretches y=af(x) and y=f(qx), vertical and horizontal translations y=f(x)+b and y=f(x-a), and reflections in the x- and y-axes.

\n","hasLesson":true},{"id":255,"name":"Rational functions","slug":"rational-functions","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029160193-Artboard%2039.png?alt=media&token=cced4769-0d16-429a-8ae4-bd3a62d893ca","type":"image"},"skills":[808,364],"description":"

The reciprocal function f\\left(x\\right)=\\frac{1}{x}, its undefined value at x=0, self-inverse property, and the vertical and horizontal asymptotes of linear rational functions f\\left(x\\right)=\\frac{ax+b}{cx+d}.

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Translations, reflections and stretches of graphs, including reciprocal and linear rational functions with vertical and horizontal asymptotes.

\n"},{"id":356,"name":"2D & 3D Geometry","slug":"2d-geometry","topic":3,"subjects":["AAHL","AASL","AIHL","AISL","ecolint-y11-hl"],"isVideoReady":true,"videoId":"GnG2jU8VpUg","concepts":[{"id":82,"name":"Right-angled triangles","slug":"right-angled-triangles","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029099320-Artboard%2046.png?alt=media&token=803487ed-7a2d-4c58-88e5-31cba1785454","type":"image"},"skills":[87,36,56,716,83,717],"description":"

Pythagoras's theorem, SOHCAHTOA, finding side length from angles

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Area of non-right-angled triangles using A=\\frac12ab\\sin C, the sine rule \\frac{a}{\\sin A}=\\frac{b}{\\sin B}=\\frac{c}{\\sin C}, the cosine rule c^2=a^2+b^2-2ab\\cos C, and the ambiguous case when solving triangles.

\n","hasLesson":true},{"id":65,"name":"Radians, arcs and sectors","slug":"radians-arcs-and-sectors","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757028966537-Artboard%2049.png?alt=media&token=21e1f320-b306-4587-8651-9991150cbd34","type":"image"},"skills":[492,66,720,4,68,14],"description":"

Radian measure, converting between radians and degrees, and calculating arc length and sector area using l=r\\theta, A=\\frac12\\theta r^2, C=2\\pi r, and A=\\pi r^2.

\n","hasLesson":true},{"id":722,"name":"3D volume and surface area","slug":"3d-volume-and-surface-area","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757028955870-Artboard%2050.png?alt=media&token=3f2ceb70-0712-4f69-81cc-c257607a3013","type":"image"},"skills":[1080,1094,723,724,725,834,835],"description":"

Spheres, cylinders, prisms, right cones, right pyramids, and combinations of solids

","hasLesson":true},{"id":89,"name":"Angles of elevation and depression","slug":"angles-of-elevation-and-depression","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757028981407-Artboard%2048.png?alt=media&token=5b9c00d8-8c4e-40c8-afef-26030c38db55","type":"image"},"skills":[90],"description":"Angles measured between a horizontal line and some elevated or depressed object.","hasLesson":true}],"description":"

2D and 3D geometry including circle measures and sectors, right-triangle and non-right-triangle trigonometry, bearings and angles of elevation/depression, 3D coordinates and angles in space, and surface area and volume of common solids.

\n"},{"id":58,"name":"Trig equations & identities","slug":"trig-equations-identities","topic":3,"subjects":["AAHL","AASL","AIHL","ecolint-y11-hl"],"isVideoReady":true,"videoId":"M87nP0Y-Pds","concepts":[{"id":65,"name":"Radians, arcs and sectors","slug":"radians-arcs-and-sectors","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757028966537-Artboard%2049.png?alt=media&token=21e1f320-b306-4587-8651-9991150cbd34","type":"image"},"skills":[492,66,720,4,68,14],"description":"

Radian measure, converting between radians and degrees, and calculating arc length and sector area using l=r\\theta, A=\\frac12\\theta r^2, C=2\\pi r, and A=\\pi r^2.

\n","hasLesson":true},{"id":166,"name":"The Unit Circle","slug":"the-unit-circle","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757028896524-Artboard%2053.png?alt=media&token=6f0c6598-c2ab-4090-99f1-8e7ccf54e5df","type":"image"},"skills":[69,70,729,730,728,732,731,734,389,61],"description":"

The unit circle: using \\cos\\theta and \\sin\\theta as the x- and y-coordinates of a point on the circle, key exact values and quadrants, symmetry rules, periodicity, tangent as \\tan\\theta=\\frac{\\sin\\theta}{\\cos\\theta}, and \\sin^2\\theta+\\cos^2\\theta=1.

\n","hasLesson":true},{"id":735,"name":"Sinusoidal functions","slug":"sinusoidal-functions","isPaywalled":false,"isPublic":true,"thumbnail":{"type":"graph","graph":{"data":"$44","mode":"geometry"}},"skills":[736,76,14,75],"description":"

Radian measure, the sine and cosine functions with domain x\\in\\R and range (-1,1), the tangent function \\tan x=\\frac{\\sin x}{\\cos x} with its asymptotes and roots, and sinusoidal functions of the form a\\sin\\left(b\\left(x+c\\right)\\right)+d or a\\cos\\left(b\\left(x+c\\right)\\right)+d.

\n","hasLesson":true},{"id":949,"name":"Tangent and reciprocal trig functions","slug":"tangent-and-reciprocal-trig-functions","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757028852382-Artboard%2054.png?alt=media&token=c153b834-4d14-48e7-b00c-ce7ed61b764b","type":"image"},"skills":[],"description":"

Reciprocal trigonometric functions: secant \\sec x = \\frac{1}{\\cos x}, cosecant \\cosec x = \\frac{1}{\\sin x}, and cotangent \\cot x = \\frac{1}{\\tan x}, including their basic values and graph behaviour where defined.

\n","hasLesson":true},{"id":396,"name":"Trigonometric equations","slug":"trigonometric-equations","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757028815933-Artboard%2055.png?alt=media&token=283b491c-074b-4773-bc66-4c839b2c8cb5","type":"image"},"skills":[746,1021,744,745,747],"description":"

Solving trigonometric equations algebraically in a specified domain, including \\sin\\theta=a, \\cos\\theta=a, \\tan\\theta=a, equations with arguments of the form ax+b, and trigonometric quadratics using identities such as \\sin^2\\theta+\\cos^2\\theta=1.

\n","hasLesson":true},{"id":393,"name":"Pythagorean and double-angle identities","slug":"pythagorean-and-double-angle-identities","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757028794001-Artboard%2057.png?alt=media&token=6c43949e-408e-48a5-aa5d-2ee8a9e57c20","type":"image"},"skills":[61,739,740],"description":"

Trigonometric identities including \\sin^2\\theta+\\cos^2\\theta=1, \\sin2\\theta=2\\sin\\theta\\cos\\theta, and the equivalent forms of \\cos2\\theta: \\cos^2\\theta-\\sin^2\\theta, 2\\cos^2\\theta-1, and 1-2\\sin^2\\theta.

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Unit circle trigonometry, key values, quadrant/sign rules, symmetry and periodicity, double angle and Pythagorean identities, and solving basic trigonometric equations and trigonometric quadratics in a given domain.

\n"},{"id":346,"name":"Differentiation","slug":"differentiation","topic":5,"subjects":["AAHL","AASL","AIHL","AISL"],"isVideoReady":true,"videoId":"7GKpUTrLk_o","concepts":[{"id":111,"name":"Limits and Derivatives","slug":"limits-and-derivatives","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757027621748-Artboard%2086.png?alt=media&token=f862e464-90a6-482b-9b52-1be5899b056c","type":"image"},"skills":[565,566,567,570,569,571,535,534,19],"description":"

The basic idea of a limit \\lim_{x\\to a}f(x) from tables and graphs, slope as a limit, rate of change and gradient, derivatives of x^n, and linearity for sums and scalar multiples.

\n","hasLesson":true},{"id":112,"name":"Differentiation rules","slug":"differentiation-rules","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757027576694-Artboard%2088.png?alt=media&token=b370fb29-0b88-471e-867c-06e32f483988","type":"image"},"skills":[541,539,538,336,533,335,535,534],"description":"

Derivatives of x^{r},e^{x},\\ln x,\\sin x and \\cos x. The chain, product and quotient rules.

","hasLesson":true},{"id":326,"name":"Tangents and normals","slug":"tangents-and-normals","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757027557648-Artboard%2088_1.png?alt=media&token=ba008df4-2d3c-426c-bf8f-0809dadd8f51","type":"image"},"skills":[575,576],"description":"

Definitions of tangent and normal, finding tangent and normal lines to a function

","hasLesson":true},{"id":331,"name":"Applications of the First Derivative","slug":"applications-of-the-first-derivative","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757027540062-Artboard%2089.png?alt=media&token=8765b377-ec1c-498c-8770-e7f8904405d4","type":"image"},"skills":[573,16,531],"description":"

Increasing and decreasing intervals, stationary points (maxima and minima), and optimisation

","hasLesson":true},{"id":7,"name":"Second Derivatives and Applications","slug":"second-derivatives-and-applications","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757027522497-Artboard%2090.png?alt=media&token=08f7442f-e526-4cd9-af09-980e1d6aa69c","type":"image"},"skills":[348,330,574,18,532],"description":"

Second derivatives f'', using the sign of f'' to determine concavity, locate inflexion points where f''(x)=0 and changes sign, and classify stationary points with the second derivative test.

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Limits and rates of change, differentiation rules for polynomial, exponential, logarithmic and trigonometric functions, and applications to tangents, normals, stationary points, concavity and optimisation.

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Concept of an integral, areas with definite integrals, and basic anti-derivative solving skills

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Antiderivatives of polynomials, trigonometric, inverse trigonometric, exponential, and logarithmic functions

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Integration of composite functions using substitution, including integrals of the form f(ax+b), and handling definite integrals by changing the bounds or substituting back.

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Displacement and distance, velocity and acceleration, speed as |v|, average velocity and acceleration as \\Delta s/\\Delta t and \\Delta v/\\Delta t, and the relations v=\\mathrm{d}s/\\mathrm{d}t, a=\\mathrm{d}v/\\mathrm{d}t=\\mathrm{d}^2s/\\mathrm{d}t^2, \\int_{t_1}^{t_2} v(t)\\,\\mathrm{d}t for change in displacement, and \\int_{t_1}^{t_2} |v(t)|\\,\\mathrm{d}t for distance.

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Integration as reverse differentiation, including basic anti-derivatives, definite integrals, area under and between curves, substitution, and applications to displacement and distance.

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Theoretical and experimental probability, complementary events, expected number of outcomes, sample space

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Venn diagrams, intersection and union of events, conditional probability, independent events

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Tree diagrams, selection with/without replacement, dependence

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Theoretical vs experimental probability, combined events, independence vs dependence.

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Understand the basics of population, data collection, and sampling.

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Learn different ways to measure the \"center,\" or typical values, of a set of data: mean, median, and mode.

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Learn the concept of dispersion, range, IQR, outliers, and box and whisker plots.

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Learn about standard deviation and variance, which we use to measure how tightly or loosely clustered the data is around the mean.

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Summarizing and interpreting data using measures of centre and spread, frequency tables and graphs, sampling methods, quartiles, IQR, box plots, cumulative frequency, and outlier analysis.

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Scatter diagrams and lines of best fit y=ax+b, by eye or with GDC, the Pearson correlation coefficient r, and making predictions with awareness of limitations.

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Correlation and linear regression for paired data, including best fit lines, Pearson's r, predictions from regression models, and the limits of extrapolation.

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Concept of the binomial distribution, binomial PDF and CDF, expectation and variance of binomial distribution

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The normal (bell-shaped) distribution X\\sim\\mathrm{N}(\\mu,\\sigma^2), its symmetry about \\mu, z-values z=\\frac{x-\\mu}{\\sigma} and standardization to Z\\sim\\mathrm{N}(0,1), the empirical rule, and calculator-based normal and inverse normal probability calculations.

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Discrete random variables and probability distributions, including expected value and fair games, binomial models and calculator probabilities, and the normal distribution with z-values and standardization.

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