Integrated Math II

Units

1. 1

   Extending the number system

   4 lessons

   1. Exponent rules, radicals and rational exponents; sums and products of rational and irrational numbersSimplify expressions with exponents and radicals, then justify whether exact results are rational or irrational.
   2. Rational exponents and radicalsEvaluate rational powers, combine them using exponent laws, and simplify radicals.
   3. The number i and arithmetic with complex numbersSimplify complex-number sums, products, powers of i, and quotients into a + bi form.
   4. Complex solutions of quadratic equationsSolve quadratics with non-real roots and connect those roots to graphs and factors.
2. 2

   Quadratic functions

   9 lessons

   1. Graphing linear, quadratic, square root, cube root, absolute value, piecewise and step functionsSketch each family from a rule, labeling anchor points, key features and included endpoints.
   2. Equivalent forms that reveal properties: zeros, vertex, growth and decay ratesChoose a form to find a quadratic’s zeros, vertex, or y-intercept, and read an exponential rate per stated time unit.
   3. Transforming graphs: f(x) + k, k·f(x), f(kx), f(x + k), and even and odd functionsDescribe graph shifts and scales, find a multiplier from a point, and classify functions as even, odd, or neither.
   4. Average rate of change over an intervalFind and interpret average rates from graphs, tables, and formulas, with appropriate units.
   5. Graphing quadratic functions: vertex, axis of symmetry and interceptsSketch a parabola from its opening, axis, vertex, intercepts and mirror points, and say what its minimum or maximum value is.
   6. Vertex form and maximum/minimum problemsRead the vertex from y = a(x − h)² + k, rewrite a quadratic into that form, and use it to find the best input and the best value in a situation.
   7. Completing the square to reveal the vertex and solveRewrite a quadratic to find its vertex, then complete the square to solve an equation.
   8. Projectile motion and other quadratic modelsRead a launch model h(t) = −16t² + v₀t + h₀, then find its peak, its landing time and the times it reaches a target height, keeping only answers that fit the story.
   9. The parabola from a focus and a directrixWrite and derive a vertical parabola’s equation from its focus and directrix.
3. 3

   Quadratic expressions and equations

   12 lessons

   1. Adding, subtracting and multiplying polynomialsAdd, subtract and multiply polynomials; use special products and identify a product’s degree.
   2. Factoring out the GCF and factoring x^2 + bx + cFactor a quadratic completely by taking out the greatest common factor, then finding the pair with the right product and sum, and read the zeros from the factors.
   3. Factoring ax^2 + bx + c by groupingFactor a trinomial whose x² term has a coefficient by splitting the middle term with the product ac, grouping, and checking by multiplying back.
   4. Special products: difference of squares and perfect square trinomialsSpot a difference of squares or a perfect square trinomial, factor it without a search, and factor completely when a common factor or a repeated pattern is hiding.
   5. Solving quadratic equations by factoringSolve a quadratic equation by making one side zero, factoring, and checking every solution.
   6. Solving quadratics by inspection and square rootsSolve by isolating a squared expression, finding all real solutions, and simplifying exact roots.
   7. The quadratic formula and the discriminantSolve quadratics with the formula and predict the number of real solutions from the discriminant.
   8. Choosing a method to solve a quadraticChoose an efficient method, solve exactly when possible, and check which solutions fit the question.
   9. Using properties of exponents to rewrite exponential expressionsRewrite an exponential model for a new time unit and interpret its growth rate.
   10. Systems of a linear and a quadratic equationFind and check every point shared by a line and a parabola; use the same substitution idea with a circle.
   11. Systems with a line and a quadratic or a circleSolve a system made of a line and a parabola or a circle by substitution, give every solution as a point, and say how many real solutions a system can have.
   12. Quadratic inequalitiesSolve a quadratic inequality by finding its zeros and the sign of the parabola between them, graph the solution on a number line and write it in interval notation.
4. 4

   Similarity and right triangles

   6 lessons

   1. Dilations: centre, scale factor and what they preserveConstruct a dilated image and explain its lengths, angles and lines.
   2. Similarity by transformations and the AA criterionProve triangle similarity by AA, match corresponding vertices, and find a missing side.
   3. Proving similarity theorems: side splitter, Pythagorean theorem and geometric meanUse AA similarity to prove proportional relationships and find missing lengths.
   4. Trigonometric ratios from similar right triangles; sine and cosine of complementary anglesFind sine, cosine, and tangent from either acute angle; explain why size does not change them.
   5. Special right triangles: 45°-45°-90° and 30°-60°-90°Find missing sides and exact trigonometric ratios using the two special right-triangle patterns.
   6. Solving right triangles in applied problems: trig, inverse trig and PythagorasSketch a situation, find an unknown length or angle, and check whether the answer fits.
5. 5

   Circles

   12 lessons

   1. All circles are similarMap one circle onto another, explain why π is constant, and compare circumference and area factors.
   2. Central angles, inscribed angles and chordsFind circle angles and chord distances, and explain each result from the diagram.
   3. Tangent lines: radius perpendicular to tangent, equal tangent segments, constructionConstruct two tangents from an external point and justify their lengths and enclosed angle.
   4. Inscribed and circumscribed circles; cyclic quadrilateralsConstruct both triangle circles, locate their centres, and solve opposite angles in a cyclic quadrilateral.
   5. Arc length, radian measure and sector areaFind arc length and sector area from a radius and central angle, using degrees or radians.
   6. The equation of a circle from the Pythagorean theorem; completing the squareWrite and read circle equations, recover them from expanded form, and locate points relative to a circle.
   7. Angles formed by chords, secants and tangentsFind the angle formed by two chords, two secants, a secant and a tangent, two tangents, or a tangent and a chord from the arcs it cuts off, and the reverse.
   8. Lengths of chords, secants and tangents: products of segmentsFind an unknown length where two chords cross, two secants leave a point, or a tangent and a secant leave a point, using products of segments.
   9. Parabolas: focus, directrix and vertex formDescribe a parabola as the points equally far from a focus and a directrix, find its focus and directrix from an equation x² = 4py, and write the equation in vertex form.
   10. Circles: centre-radius equations by completing the squareWrite and read the equation (x − h)² + (y − k)² = r² of a circle, complete the square to find the centre and radius from an expanded equation, and test whether a point is inside, on or outside.
   11. Ellipses: foci, axes and equationsRead the vertices, co-vertices, foci and centre of an ellipse from its equation, use c² = a² − b², and use the definition that the distances to the two foci add to a constant.
   12. Hyperbolas: vertices, foci and asymptotesRead the centre, vertices, foci and asymptotes of a hyperbola from its equation, use c² = a² + b², and tell the four conics apart from their equations.
6. 6

   Volume and cross-sections

   3 lessons

   1. Informal arguments for circle, cylinder, pyramid, cone and sphere formulas; Cavalieri's principleJustify circle area and solid volume formulas using rearranged pieces, matching slices and a cube partition.
   2. Volume of cylinders, pyramids, cones and spheres in problemsCalculate and compare volumes, choose the correct dimensions, and combine volumes of simple solids.
   3. Surface area of prisms, cylinders, pyramids, cones and spheresFind the surface area of a solid by adding the areas of the faces that count, using the slant height for sloping sides, and say which faces a problem needs.
7. 7

   Probability

   7 lessons

   1. Sample spaces and events: unions, intersections and complementsList event outcomes, identify “or,” “and,” and “not,” and find their probabilities.
   2. Independent events and the multiplication testDecide whether two events are independent by comparing their shared probability with the product of their individual probabilities.
   3. Conditional probabilityCalculate a probability given an event, distinguish which event is given, and test independence.
   4. Two-way frequency tables: joint, marginal and conditional; testing independence with dataBuild a two-way table, find joint, marginal and conditional shares, and test independence using rates.
   5. The addition ruleCalculate P(A or B) when events overlap or cannot overlap.
   6. The general multiplication rule and tree diagramsCalculate probabilities for two draws without replacement using a tree diagram and the general multiplication rule.
   7. Permutations and combinations to compute probabilitiesCount ordered arrangements and unordered groups, then use those counts to find probabilities.