Every lesson in the Algebra 1 slide course, in full text: 106 decks, 6848 slides.
Chapter 1: Connections to AlgebraChapter 1 of Algebra 1: Concepts and Skills, built for a visual learner. Variables as labelled boxes, powers as factor counts, the order-of-operations ladder, checking solutions on a balance scale, translating English into symbols, the four-step problem solving plan, tables and graphs, and a first look at functions as machines.
1.1 Variables in AlgebraVariables as placeholders, the four operations in algebraic shorthand, the write-substitute-simplify routine for evaluating an expression, substituting into the formulas for distance and perimeter, and writing an expression of your own from a described situation.
1.2 Exponents and PowersThe base, the exponent and the power; reading and writing powers in words and in exponential form; evaluating powers including powers of a variable; how grouping symbols decide what an exponent is attached to; and the area and volume formulas that give the second and third powers their names.
1.3 Order of OperationsThe four-step order of operations, the left-to-right rule for operations of equal priority, grouping symbols including the fraction bar, whether a calculator can be trusted to apply the order itself, and evaluating a multi-operation expression that models a real situation.
1.4 Equations and InequalitiesEquations as statements with two sides, checking whether a number is a solution, solving simple equations by reading them as questions, the four inequality symbols and what the bar underneath two of them changes, and checking solutions of inequalities in real situations.
1.5 Translating Words into Mathematical SymbolsTurning English into algebra: the phrases that signal addition, subtraction, multiplication and division; the two operations whose order the words can reverse; the difference between a phrase and a sentence; and building a complete equation or inequality from a described situation.
1.6 A Problem Solving Plan Using ModelsThe five-stage plan for word problems: write a verbal model in words, assign labels with units to every quantity, translate into an algebraic model, solve and answer the original question, and check that the answer is reasonable rather than merely arithmetically correct.
1.7 Tables and GraphsOrganising data in a table and reading it in both directions, building and interpreting bar graphs and line graphs, choosing between them, and recognising how a broken or uneven vertical scale can make a truthful set of numbers tell a misleading story.
1.8 An Introduction to FunctionsFunctions as rules pairing each input with exactly one output, input-output tables, the domain and the range, deciding whether a pairing is a function, and moving between the four representations of a function: words, a rule, a table and a graph.
Chapter 2: Properties of Real NumbersChapter 2 of Algebra 1: Concepts and Skills, built for a visual learner. The real number line, absolute value as distance, adding and subtracting as jumps, the negative-factor count for multiplying and dividing, the distributive property as an area model, and combining like terms with algebra tiles.
2.1 The Real Number LineThe real number line and the vocabulary that goes with it: positive, negative and zero; integers and whole numbers; graphing a number as a point; comparing two numbers by their positions rather than their digits; and plotting decimals and fractions on a line whose scale marks need not be integers.
2.2 Absolute ValueOpposites as reflections across zero, absolute value as distance from zero and its three-case definition, solving simple absolute value equations and recognising when they have no solution, the difference between velocity and speed, and using a counterexample to disprove a general statement.
2.3 Adding Real NumbersAddition as movement on a number line, the two rules of addition decided by whether the signs agree, the five properties of addition that those rules follow from, adding a long list of signed numbers efficiently, and modelling profits and losses as positive and negative quantities.
2.4 Subtracting Real NumbersThe subtraction rule — to subtract a number, add its opposite — applied to single subtractions and to chains of them, the fact that subtraction is not commutative, evaluating a function containing a subtraction, identifying the terms of an expression written as a sum, and computing a change as a signed difference.
2.5 Multiplying Real NumbersMultiplication of signed numbers: why a positive times a negative is negative and why two negatives give a positive, the counting rule for the sign of a product, the six properties of multiplication including the properties of zero and of negative one, simplifying products containing variables, and computing a change in position as velocity times time.
2.6 The Distributive PropertyThe distributive property justified by an area model, its four versions with the factor on either side and a plus or minus inside, distributing a negative factor and preserving signs, using the property backwards for mental arithmetic, and the errors that come from failing to reach every term inside the bracket.
2.7 Combining Like TermsCoefficients including the invisible ones, identifying like terms by matching the variable part exactly, combining them by adding coefficients as the distributive property run backwards, simplifying expressions that contain grouping symbols, and knowing when an expression counts as simplified.
2.8 Dividing Real NumbersReciprocals and the inverse property of multiplication, the division rule that turns any division into a multiplication, the sign rule for quotients, simplifying complex fractions, why division by zero is undefined, and evaluating expressions and velocities that involve division.
Chapter 3: Solving Linear EquationsChapter 3 of Algebra 1: Concepts and Skills, built for a visual learner. Inverse operations on a balance scale, unwrapping multi-step equations in reverse, variables on both sides plus the no-solution and infinite-solution cases, clearing fractions and decimals, rearranging formulas, and ratio, rate and percent problems drawn as bar models.
3.1 Solving Equations Using Addition and SubtractionEquivalent equations and the balance model, inverse operations, transforming an equation by adding or subtracting the same number from both sides, isolating the variable, checking every solution in the original equation, and building one-step equations from described situations.
3.2 Solving Equations Using Multiplication and DivisionMultiplication and division as inverse operations, the multiplication and division properties of equality, dividing by a coefficient or equivalently multiplying by its reciprocal, the sign change caused by a negative coefficient, fractional coefficients, and modelling equal shares of a total.
3.3 Solving Multi-Step EquationsEquations needing more than one transformation: why the addition is undone before the multiplication, simplifying one side by combining like terms before solving, building a two-step equation from a verbal model, and checking a multi-step solution against the original equation.
3.4 Solving Equations with Variables on Both SidesEquations with variable terms on both sides: why a variable term may be added to or subtracted from both sides, choosing the side with the greater coefficient so the result stays positive, combining like terms on each side first, and modelling situations where two changing quantities become equal.
3.5 More on Linear EquationsThe four-step procedure covering every linear equation: simplify each side by distributing and combining, collect the variable terms on the side with the greater coefficient, isolate the variable, and check in the original. Includes brackets on both sides, fractional factors, and what it means when the variable terms cancel entirely.
3.6 Solving Decimal EquationsExact and approximate solutions: solving to an exact value and rounding only at the end, using the approximately-equal symbol honestly, checking a rounded answer and knowing how close to expect, clearing decimals by multiplying through, and rounding error in real situations.
3.7 FormulasSolving a formula for one of its variables: rearranging with the same inverse operations used on numerical equations, the temperature and area formulas, using a rearranged formula to compute values, checking a rearrangement by its units, and knowing when rearranging first is worth the effort.
3.8 Ratios and RatesRatios comparing like quantities and rates comparing unlike ones, unit rates and why they make comparison possible, averaging a rate by totalling both quantities, unit analysis as a way of converting units and checking a setup, and using a rate to compute a total.
3.9 PercentsPercents as ratios comparing a number to one hundred, the three notations, the percent verbal model and its algebraic form, the three kinds of percent question distinguished by which letter is unknown, converting a percent to a decimal before substituting, and identifying the base number in a real problem.
Chapter 4: Graphing Linear Equations and FunctionsChapter 4 of Algebra 1: Concepts and Skills, built for a visual learner. The coordinate plane and its quadrants, graphing from a table, horizontal and vertical lines, intercepts, slope drawn as a staircase, direct variation through the origin, slope-intercept form read straight off the equation, and the vertical line test.
4.1 The Coordinate PlaneThe coordinate plane and its parts: the two axes and the origin, ordered pairs and their two coordinates, plotting a point from the origin, reading coordinates off a graph, the four quadrants and their sign patterns, and scatter plots as pictures of paired data.
4.2 Graphing Linear EquationsSolutions of two-variable equations as ordered pairs, checking a candidate pair, the standard form that makes an equation linear, rewriting into function form to find solutions easily, building a table of values, and plotting it to obtain the straight line that is the graph of the equation.
4.3 Graphing Horizontal and Vertical LinesThe two special cases of the standard form: when A is zero the equation reduces to y equals a constant and graphs as a horizontal line, and when B is zero it reduces to x equals a constant and graphs as a vertical line. Includes writing the equation of such a line from its graph, constant functions with their domain and range, and why a vertical line is not a function.
4.4 Graphing Lines Using InterceptsThe x-intercept and y-intercept of a line, found by substituting zero for the other variable, and the quick-graph method that uses just those two points to draw a line without building a table. Includes which lines are missing an intercept, and what an intercept means in a real situation.
4.5 The Slope of a LineSlope as the ratio of vertical rise to horizontal run, the slope formula using subscripted coordinates, and the four cases: positive slope for a line rising left to right, negative for one falling, zero for a horizontal line and undefined for a vertical one. Includes why any two points on a line give the same slope.
4.6 Direct VariationTwo quantities that vary directly, meaning their ratio is a constant k called the constant of variation, so that y equals kx. Includes finding k from one pair of values, graphing a direct variation model as a line through the origin whose slope is k, and fitting an approximate model to real data.
4.7 Graphing Lines Using Slope-Intercept FormThe slope-intercept form y equals mx plus b, in which the coefficient of x is the slope and the constant is the y-intercept. Includes rewriting an equation into that form, graphing a line from its two numbers without any table, interpreting slope and intercept in a real model, and identifying parallel lines by equal slopes.
4.8 Functions and RelationsRelations as any set of ordered pairs, and functions as the relations in which every input has exactly one output. Includes the vertical line test, function notation f of x with its evaluation by substitution, and linear functions of the form f of x equals mx plus b graphed from the slope-intercept form.
Chapter 5: Writing Linear EquationsChapter 5 of Algebra 1: Concepts and Skills, built for a visual learner. Writing a line from its slope and intercept, point-slope form derived from the slope formula, equations from two points, converting to standard form, modelling real situations where the intercept is the start and the slope is the rate, and parallel and perpendicular slopes.
5.1 Slope-Intercept FormWriting the equation of a line from its slope and y-intercept, and from a graph. Includes reading a y-intercept off a graph and computing the slope from two points on it, handling axes drawn to different scales, and modelling a falling quantity with a negative slope.
5.2 Point-Slope FormThe point-slope form of a linear equation, used when the slope and any one point are known rather than the slope and the y-intercept. Includes substituting negative coordinates without losing a sign, converting to slope-intercept form, writing the equation of a parallel line through a given point, and choosing between the two forms.
5.3 Writing Linear Equations Given Two PointsWriting the equation of a line from any two points on it, by computing the slope first and then using either point. Includes the shortcut when one of the two points is the y-intercept, checking the finished equation against both given points, and modelling a descent with a negative slope.
5.4 Standard FormThe standard form Ax plus By equals C, converting into it from slope-intercept and point-slope form, clearing fractions to obtain integer coefficients, and writing a standard-form equation from a point and a slope or from two intercepts. Includes why the form is not unique and why it is the only form that covers vertical lines.
5.5 Modeling with Linear EquationsBuilding a linear model of a real situation from a rate of change and one observation, choosing a variable that counts from a stated zero, predicting with the model both algebraically and graphically, and writing a model from a verbal description of a fixed total. Includes what a prediction assumes and where a model stops being trustworthy.
5.6 Perpendicular LinesTwo non-vertical lines are perpendicular exactly when the product of their slopes is negative one, so each slope is the negative reciprocal of the other. Includes testing a pair of lines, writing the equation of a line perpendicular to a given one through a given point, the horizontal-and-vertical special case, and why a graph can check but not prove perpendicularity.
Chapter 6: Solving and Graphing Linear InequalitiesChapter 6 of Algebra 1: Concepts and Skills, built for a visual learner. Solution sets drawn on the number line, the flip rule explained as a reflection through zero, compound and and or inequalities as stacked shadings, absolute-value equations as two symmetric solutions, absolute-value inequalities as close-or-far, and shaded half-planes in two variables.
6.1 Solving Inequalities Using Addition or SubtractionGraphing one-variable inequalities on a number line with open and solid endpoints, the addition and subtraction properties of inequality, solving one-step inequalities by adding or subtracting, checking a solution set with numbers inside and outside it, and writing an inequality from a described situation.
6.2 Solving Inequalities Using Multiplication or DivisionThe multiplication and division properties of inequality, split into the case of a positive multiplier, which preserves the direction, and a negative one, which reverses it. Includes solving one-step inequalities of both kinds, graphing the results, and recognising that only the sign of the multiplier decides whether the symbol turns.
6.3 Solving Multi-Step InequalitiesInequalities requiring more than one operation: undoing an addition and a multiplication in sequence, distributing to clear brackets, collecting variable terms from both sides, and choosing which side to collect on so that no reversal is needed. Includes a profit model whose answer must be rounded in the direction the situation requires.
6.4 Solving Compound Inequalities Involving “And”Compound inequalities joined by and, whose solutions must satisfy both parts. Includes writing them as a single statement with the variable between two bounds, graphing the overlap, solving by separating the parts or by operating on all three expressions at once, and recognising when the overlap is empty.
6.5 Solving Compound Inequalities Involving “Or”Compound inequalities joined by or, whose solutions need satisfy only one of the two parts. Includes graphing the union as two rays with a gap between them, solving each part independently with the methods of earlier lessons, reversing both symbols when dividing by a negative, and a velocity model in which direction is carried by a sign.
6.6 Solving Absolute-Value EquationsAbsolute-value equations solved by splitting them into two related linear equations, one for each sign the inside expression could take. Includes isolating the absolute value first, recognising when the right side is negative and there is no solution, checking both answers, and building an equation from two given solutions using their midpoint and distance.
6.7 Solving Absolute-Value InequalitiesAbsolute-value inequalities rewritten as compound inequalities: less than becomes an and statement giving a band around the centre, and greater than becomes an or statement giving two rays. Includes reversing the symbol on the negative branch, checking one value from each region, and reading such inequalities as statements about distance and tolerance.
6.8 Graphing Linear Inequalities in Two VariablesLinear inequalities in two variables, whose solutions are ordered pairs filling a half-plane. Includes checking a pair by substitution, the three-step graphing procedure with dashed and solid boundaries, using the origin as a test point, and rewriting into slope-intercept form to read the shading directly.
Chapter 7: Systems of Linear Equations and InequalitiesChapter 7 of Algebra 1: Concepts and Skills, built for a visual learner. Systems solved by graphing, substitution and elimination, with the three outcomes shown as crossing, parallel and coincident lines; word problems with two unknowns drawn as bar models; and systems of inequalities as overlapping shaded regions.
7.1 Graphing Linear SystemsSystems of two linear equations and their solutions as ordered pairs satisfying both. Includes reading a solution off the point where two graphs intersect, the graph-and-check method with its rewriting and verification steps, why a graphical answer is only an estimate, and modelling two growing quantities to find when they become equal.
7.2 Solving Linear Systems by SubstitutionThe substitution method: solving one equation for one variable, substituting that expression into the other equation to reduce the system to one variable, back-substituting to recover the second coordinate, and checking in both originals. Includes choosing which variable to isolate and why the method gives exact answers a graph cannot.
7.3 Solving Linear Systems by Linear CombinationsThe linear-combination method: multiplying one or both equations by constants so that a variable's coefficients become opposites, adding to eliminate it, solving for the survivor, and back-substituting. Includes the case where no multiplication is needed, choosing multipliers and which variable to eliminate, and deciding between this method and substitution.
7.4 Linear Systems and Problem SolvingUsing linear systems to model real situations, particularly mixture problems with one counting equation and one value equation. Includes building the verbal model and labels, choosing the efficient solution method from the coefficients, clearing decimals, and checking the answer against the situation rather than only the algebra.
7.5 Special Types of Linear SystemsSystems whose lines do not cross exactly once: parallel lines giving no solution and coincident lines giving infinitely many. Includes recognising each case graphically from the slopes and intercepts, and algebraically from a false or always-true statement when the variables vanish.
7.6 Systems of Linear InequalitiesTwo or more linear inequalities in the same variables, whose solutions are the ordered pairs satisfying every one of them. Includes graphing each half-plane as in Lesson 6.8 and reading the overlap, handling three or more conditions, testing a point against every inequality, and recognising a bounded region and its corners.
Chapter 8: Exponents and Exponential FunctionsChapter 8 of Algebra 1: Concepts and Skills, built for a visual learner. Every exponent rule rebuilt by drawing and counting factors, zero and negative exponents forced by a halving pattern, scientific notation as a travel log for the decimal point, and exponential growth and decay curves compared against straight lines.
8.1 Multiplication Properties of ExponentsThe three multiplication properties of exponents: the product of powers rule, which adds exponents; the power of a power rule, which multiplies them; and the power of a product rule, which distributes the exponent over the factors. Includes why each rule holds, telling them apart, and the difference between doubling a length and doubling an area.
8.2 Zero and Negative ExponentsExtending exponents beyond the counting numbers. Includes deriving that a nonzero base to the zero power must be one and that a negative exponent means a reciprocal, both forced by the product of powers rule, evaluating such powers, distinguishing a negative exponent from a negative value, and applying the Lesson 8.1 rules unchanged.
8.3 Graphs of Exponential FunctionsExponential functions of the form y equals a times b to the x, their tables of values and their graphs. Includes the rising curve when the base exceeds one and the falling curve when it lies between zero and one, the role of a as the y-intercept, why the curve never meets the horizontal axis, and how such a graph differs from a line.
8.4 Division Properties of ExponentsThe two division properties of exponents: the quotient of powers rule, which subtracts exponents, and the power of a quotient rule, which distributes the exponent over numerator and denominator. Includes negative results rewritten with positive exponents, checking by cancelling factors, and comparing quantities by ratio.
8.5 Scientific NotationWriting numbers as a coefficient between one and ten times a power of ten. Includes converting in both directions by moving the decimal point, reading the exponent's sign as large or small, and multiplying, dividing and raising such numbers to powers using the exponent rules with a final adjustment back into standard form.
8.6 Exponential Growth FunctionsModelling a quantity that increases by the same percentage in each unit of time with the exponential growth model. Includes converting a percentage rate into a growth factor, distinguishing a rate from a factor, applying the model to compound interest and to population growth, and seeing why compounding outpaces adding a fixed amount.
8.7 Exponential Decay FunctionsModelling a quantity that decreases by the same percentage in each unit of time with the exponential decay model. Includes converting a decay rate into a decay factor, evaluating and graphing a depreciation model, classifying a model as growth or decay from its base, and reading why a decay curve flattens without ever reaching zero.
Chapter 9: Quadratic Equations and FunctionsChapter 9 of Algebra 1: Concepts and Skills, built for a visual learner. Square roots drawn as square sides, both roots kept on the number line, radicals simplified with factor trees, the anatomy of a parabola, solving by graphing, the quadratic formula read as a centre plus a distance, the discriminant as a root counter, and quadratic inequalities as regions.
9.1 Square RootsEvaluating and approximating square roots. Includes the definition of a square root, the positive and negative roots and the plus-or-minus notation, how many square roots a number has, perfect squares against irrational roots, evaluating radical expressions with the radical bar as a grouping symbol, and handling plus-or-minus expressions on a calculator.
9.2 Solving Quadratic Equations by Finding Square RootsSolving a quadratic equation with no linear term by isolating the squared variable and taking square roots. Includes the standard form of a quadratic equation and the leading coefficient, the three cases for how many solutions an equation has, rewriting before taking roots, and using the falling object model to answer a question about a dropped object.
9.3 Simplifying RadicalsWriting radical expressions in simplest form. Includes the three conditions for simplest form, the product property of radicals and removing perfect square factors, choosing an efficient factorisation, the quotient property, rationalising a denominator, and applying the whole procedure to a boat speed model.
9.4 Graphing Quadratic FunctionsSketching the graph of a quadratic function. Includes the standard form of a quadratic function, the parabola and which way it opens, the vertex and the axis of symmetry, the formula for the vertex's x-coordinate, building a table of values around the vertex, and reading the y-intercept straight off the equation.
9.5 Solving Quadratic Equations by GraphingUsing a graph to find or check the solutions of a quadratic equation. Includes the connection between x-intercepts and roots, rewriting an equation into standard form before graphing, estimating solutions from a sketch and confirming them algebraically, reading the number of solutions off a graph, and applying the method to a bridge cable model.
9.6 Solving Quadratic Equations by the Quadratic FormulaSolving any quadratic equation with the quadratic formula. Includes stating and applying the formula, rewriting an equation into standard form before reading its coefficients, handling irrational solutions, using the formula to find the x-intercepts of a graph, and applying the vertical motion model for a thrown object.
9.7 Using the DiscriminantUsing the discriminant to determine the number of solutions of a quadratic equation. Includes locating the discriminant inside the quadratic formula, the three cases for its sign, computing it with correct signs, predicting the number of x-intercepts of a graph, seeing how changing the constant term moves a parabola through all three cases, and answering a yes-or-no question about whether a height is ever reached.
9.8 Graphing Quadratic InequalitiesSketching the graph of a quadratic inequality in two variables. Includes the four kinds of quadratic inequality, deciding whether a point lies inside or outside a parabola, using a dashed or solid boundary curve, the test-point method for choosing which region to shade, and the above-or-below method that reads the region straight off the inequality symbol.
Chapter 10: Polynomials and FactoringChapter 10 of Algebra 1: Concepts and Skills, built for a visual learner. Polynomials added in power columns, multiplication drawn as area models so no cross-product is lost, the two special products shown as cancelling or doubling middle cells, the zero-product property, factoring as a finite factor-pair search, and factoring completely including grouping.
10.1 Adding and Subtracting PolynomialsAdding and subtracting polynomials. Includes monomials and the degree of a monomial, the vocabulary of polynomials, binomials and trinomials, writing a polynomial in standard form and naming it by degree and by number of terms, adding in vertical and horizontal formats, subtracting by adding the opposite, and modelling an area as a difference of polynomials.
10.2 Multiplying PolynomialsMultiplying polynomials. Includes using the distributive property twice on a product of binomials, the FOIL pattern and what its four letters name, multiplying longer polynomials in a vertical format, distributing horizontally so that every term meets every term, and writing an area as a product of two binomials.
10.3 Special Products of PolynomialsRecognising and using the special product patterns. Includes the sum and difference pattern, the square of a binomial in both signs, the area model that explains where the middle term comes from, the classic error of squaring term by term, and using the patterns to write the area of a region as a difference.
10.4 Solving Quadratic Equations in Factored FormSolving polynomial equations that are already written as a product equal to zero. Includes factored form and the zero-product property, setting each factor equal to zero, repeated factors giving a single solution, equations with three or more factors, sketching a parabola from its factored form, and using a factored quadratic model of an arch.
10.5 Factoring x^2 + bx + cFactoring a trinomial whose leading coefficient is one. Includes what factoring a trinomial means, the two conditions the pair of numbers must satisfy, how the signs of b and c determine the signs of the pair, a systematic search through the factor pairs of the constant, checking by multiplying back, and using factoring to solve a border problem.
10.6 Factoring ax^2 + bx + cFactoring a trinomial whose leading coefficient is not one. Includes the four numbers a factorisation must supply, testing trial factors by their Outer and Inner products, using the sign rules to narrow the search, removing a common factor before factoring, solving quadratic equations by factoring, and applying the method to a vertical motion model.
10.7 Factoring Special ProductsFactoring the special products of Lesson 10.3 in reverse. Includes the difference of two squares pattern, the two perfect square trinomial patterns, the test for recognising when a pattern applies, expressions that fit no pattern, removing a constant factor first, and solving an equation whose factored form has a repeated factor.
10.8 Factoring Cubic PolynomialsFactoring polynomials of degree three. Includes finding a greatest common factor that includes variables, prime polynomials and what it means to factor completely, factoring a four-term polynomial by grouping, the sum and difference of two cubes patterns, and reading the dimensions of a box from a factored volume.
Chapter 11: Rational Expressions and EquationsChapter 11 of Algebra 1: Concepts and Skills, built for a visual learner. Proportions as scaled bar models, direct versus inverse variation drawn side by side, cancelling factors rather than terms, multiplying and dividing rational expressions, common denominators as shared units, and rational equations with the extraneous-solution check made unmissable.
11.1 ProportionsSolving proportions. Includes the vocabulary of ratios, proportions, extremes and means, the reciprocal property, the cross product property, proportions whose cross product is quadratic, cross multiplying with polynomial expressions and excluding values that make a denominator zero, and modelling a count with a proportion.
11.2 Direct and Inverse VariationUsing direct and inverse variation. Includes both models and the constant of variation, finding the constant from a single pair of values, comparing the two models numerically over the same inputs, comparing them graphically as a line and a hyperbola, and building an inverse variation model relating a bicycle's banking angle to its turning radius.
11.3 Simplifying Rational ExpressionsSimplifying rational expressions. Includes the definition of a rational expression and simplest form, the rule for dividing out a common factor, the distinction between factors and terms, factoring the numerator and denominator before cancelling, recognising opposite factors that cancel to negative one, and dividing a polynomial by a binomial.
11.4 Multiplying and Dividing Rational ExpressionsMultiplying and dividing rational expressions. Includes both rules, factoring before multiplying so that factors cancel across the multiplication sign, treating a polynomial as a fraction with denominator one, dividing by multiplying by the reciprocal of the divisor, dividing by a polynomial, and collecting the excluded values a product or quotient carries.
11.5 Adding and Subtracting with Like DenominatorsAdding and subtracting rational expressions that share a denominator. Includes the rule for combining numerators over a common denominator, bracketing a subtracted numerator so the negative distributes correctly, simplifying the result by factoring and cancelling, sums that reduce to a constant, and the excluded values such a combination carries.
11.6 Adding and Subtracting with Unlike DenominatorsAdding and subtracting rational expressions whose denominators differ. Includes finding a least common denominator from prime factorisations, rewriting each expression over that denominator, adding and subtracting once the denominators match, handling binomial denominators whose least common denominator is their product, and modelling a journey's total time as a sum of two rational expressions.
11.7 Rational EquationsSolving equations that contain rational expressions. Includes cross multiplying when each side is a single fraction, multiplying through by the least common denominator as the general method, factoring denominators before choosing the LCD, checking every candidate against the excluded values, and setting up a work problem in which rates of work add.
Chapter 12: Radicals and More Connections to GeometryChapter 12 of Algebra 1: Concepts and Skills, built for a visual learner. Square root functions and their domains, radicals combined as like terms, radical equations with the extraneous-solution check, rational exponents split into root and power, completing the square drawn as an actual square, and Pythagoras, distance and midpoint on the coordinate grid.
12.1 Functions Involving Square RootsEvaluating and graphing functions that contain a square root. Includes the square root function and its domain and range, finding the domain by requiring the radicand to be non-negative, building a table of values and sketching the curve, the effect of a constant outside the radical on the range, and the effect of a constant inside it on the domain.
12.2 Operations with Radical ExpressionsAdding, subtracting, multiplying and dividing radical expressions. Includes combining radicals with the same radicand using the distributive property, simplifying first so that unlike radicands may turn out to match, multiplying with the product and distributive properties, the sum and difference pattern that removes a radical entirely, and rationalising a denominator that is a single radical or a sum.
12.3 Solving Radical EquationsSolving equations that contain radicals. Includes squaring both sides to remove a radical, isolating the radical before squaring, extraneous solutions and why squaring produces them, equations with no solution because a radical cannot be negative, and applying the method to a model relating pressure and flow rate.
12.4 Rational ExponentsEvaluating expressions with rational exponents. Includes cube roots and nth roots, why the nth root equals the one-over-n power, rational exponents of the form m over n and the two equivalent routes through them, the exponent properties applied to fractional exponents, and recovering a radius from a volume with a one-third power.
12.5 Completing the SquareSolving quadratic equations by completing the square. Includes adding the square of half the coefficient of x to build a perfect square trinomial, solving an equation by completing the square and taking roots, deriving the quadratic formula by the same method, reading a vertex from the completed form, and choosing an appropriate solution method for a given equation.
12.6 The Pythagorean Theorem and Its ConverseUsing the Pythagorean theorem and its converse. Includes the vocabulary of legs, hypotenuse and theorem, finding a hypotenuse from two legs, finding a leg from the hypotenuse, solving when the sides are given as expressions, using the converse to test whether a triangle is right-angled, and applying the theorem to diagonals.
12.7 The Distance FormulaFinding the distance between two points in the coordinate plane. Includes deriving the formula from the Pythagorean theorem, applying it to pairs of points, why the order of subtraction does not matter, using it with the converse of the theorem to test three points for a right angle, and measuring a real distance across a coordinate grid.
12.8 The Midpoint FormulaFinding the midpoint of a line segment in the coordinate plane. Includes the midpoint as an average of the coordinates, computing each average separately, checking a midpoint with the distance formula, recovering an endpoint from a midpoint and the other endpoint, and using midpoints to place objects on a screen.
12.9 Logical Reasoning: ProofHow mathematical statements are justified. Includes axioms and postulates accepted without proof, definitions, theorems and what proving one requires, conjectures and why examples never suffice, disproving a general statement with a single counterexample, and indirect proof by assuming the opposite and reaching a contradiction.
Slope-Intercept FormA mock teaching deck for Slope-Intercept Form in Algebra 1, used as an offline dry run.
Solving Linear EquationsAn equation is a claim that two expressions are equal, and this deck starts there. From that idea it builds one-step equations, two-step equations, and equations with the variable on both sides, using the balance model throughout. It also covers the classic trap of operating on only one side, and every worked example ends with a verification step.
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