Geometry

Units

1. 1

   Congruence and proof

   15 lessons

   1. Precise definitions: segment, angle, circle, perpendicular and parallel linesDefine these figures and line relationships precisely, then test definitions against counterexamples.
   2. Rigid motions as functions: translations, reflections, rotations and sequencesMap points through rigid motions, describe an ordered sequence, and check every vertex of its image.
   3. Symmetry: the rigid motions that carry a figure onto itselfIdentify and justify the reflections and rotations that carry a figure onto itself.
   4. Congruence defined by rigid motions and corresponding partsUse rigid motions to establish congruence, read corresponding parts from letter order, and find missing triangle measures.
   5. Triangle congruence criteria: SSS, SAS, ASA (and AAS), and why SSA and AAA failDecide whether marked parts prove triangle congruence, and justify the decision.
   6. Conditional statements and the logic of proofWrite a claim as if-then, form its converse, inverse and contrapositive, disprove it with a counterexample, and tell a deduction from a pattern.
   7. Writing a two-column proofTurn a claim into Given and Prove, plan a proof backward, and write each statement with a reason that justifies it.
   8. Proving theorems about lines and anglesProve angle relationships and both directions of the perpendicular-bisector theorem with reasons for each step.
   9. Proving theorems about triangles: angle sum, isosceles, midsegment, mediansProve and apply four triangle theorems, including the exterior-angle result.
   10. Indirect proof: assume the opposite and reach a contradictionWrite the opposite of a claim, assume it, find the clash with a given or a proven fact, and conclude the claim.
   11. The triangle inequality and the order of sides and anglesDecide whether three lengths make a triangle, find the possible third sides, and order the sides of a triangle from its angles.
   12. Interior and exterior angles of polygonsFind the interior angle sum of any convex polygon, the interior and exterior angles of a regular polygon, and a missing angle or number of sides.
   13. Proving theorems about parallelogramsProve parallelogram properties, recognize special parallelograms, and justify tests for an unmarked quadrilateral.
   14. Quadrilaterals: trapezoids, kites and how the families nestUse the properties of trapezoids and kites, place every quadrilateral in the family tree, and classify a quadrilateral from the marks it has.
   15. Formal constructions with compass and straightedgeConstruct bisectors, a parallel line and inscribed polygons; justify each using equal distances or congruent triangles.
2. 2

   Similarity and right triangles

   7 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.
   7. Trigonometry in general triangles: area formula, Law of Sines, Law of CosinesFind a triangle’s area and missing parts; choose a law and check SSA for multiple triangles.
3. 3

   Circles

   7 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. 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.
   5. 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.
   6. Inscribed and circumscribed circles; cyclic quadrilateralsConstruct both triangle circles, locate their centres, and solve opposite angles in a cyclic quadrilateral.
   7. Arc length, radian measure and sector areaFind arc length and sector area from a radius and central angle, using degrees or radians.
4. 4

   Geometry with coordinates

   6 lessons

   1. 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.
   2. The parabola from a focus and a directrixWrite and derive a vertical parabola’s equation from its focus and directrix.
   3. Proving theorems with coordinatesUse distances, slopes, and midpoints to prove claims about a figure; use letters to prove a claim for every triangle.
   4. Slopes of parallel and perpendicular linesWrite a line through a point parallel or perpendicular to a given line, and justify its slope.
   5. Partitioning a directed segment in a given ratioFind a point that divides a directed segment in a given ratio.
   6. Perimeter and area of polygons in the coordinate planeFind a polygon’s perimeter from coordinate distances and its area by enclosing and subtracting.
5. 5

   Area and volume

   4 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.
   4. Cross-sections of solids and solids of revolutionIdentify cube cross-sections and predict the solids and volumes made by rotating flat regions about specified axes.
6. 6

   Modeling with geometry

   3 lessons

   1. Modeling real objects with geometric shapesEstimate a tree trunk’s volume from tape measurements and judge whether taper matters.
   2. Density based on area and volumeCalculate and compare amounts per unit area or volume, matching units before calculating.
   3. Design problems: satisfying constraints and minimising costKeep a design constraint fixed, compare feasible dimensions, and justify the best design.
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.