| Learning Outcomes
(grade level expectations or standards)
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- identify, name(in multiple ways where appropriate), compare, understand, and use: point, line, plane, segment, ray, angle, collinear, coplanar, opposite rays, intersection, distance, vertex, measure, bisect, midpoint, angle bisector, counterexample, perpendicular, parallel, skew, transversal.
- make and verify conjectures about patterns using inductive reasoning.
- make and verify conclusions from conditional statements and syllogisms using deductive reasoning.
- use counterexamples to disprove statements.
- compare and contrast similarity/congruence.
- apply postulates and definitions to support logical arguments.
- use indirect methods(distance formula, segment addition postulate, segment bisector, midpoints, etc) and algebra to measure segments and solve measurement problems.
- use indirect methods (angle addition postulate, complementary/supplementary relationships, linear pairs, angle bisectors, etc.) and algebra to measure angles and solve angle problems
- classify angles as acute, right, obtuse, or straight.
- identify & contrast/compare vertical angles, linear pairs, complementary angles, and supplementary angles.
- compare and contrast (both in a plane and in three dimensions where appropriate): parallel, perpendicular, & skew lines .
- determine slopes of perpendicular or parallel lines or if told two lines are parallel or perpendicular, and given the slope of one line, find the slope of a second line.
- identify angles formed by transversals and use theorems/postulates about these angles to solve angle problems.
- use indirect methods (properties of perpendicular lines, converse of corresponding angles, converse of alternate interior angles, converse of alternate exterior angles, etc.) and algebra to determine if two lines are parallel, perpendicular or neither.
- express conjectures, conclusions, and logical arguments modeling CAPT written format.
- compare and contrast different classifications of triangles based on angle and side measures (e.g. comparison matrix).
- solve problems using different parts of a triangle (altitudes, medians, angle bisectors, perpendicular bisectors, midsegment, etc.).
- apply the angle sum, exterior angle, triangle inequality, and hinge theorems.
- solve problems using algebra and triangle angle sum theorem.
- find a missing side in a right triangle using the Pythagorean theorem or Pythagorean triples.
- determine if a triangle is obtuse, right or acute using the Pythagorean theorem.
- define and generate Pythagorean triples.
- use proportion and ratios to solve real-life problems.
- apply triangle similarity theorems (AA, SSS, ASA) & proportions to calculate measures of angles and segments.
- solve problems involving similar right triangles formed by the altitude drawn to the hypotenuse.
- use proportionality theorems to determine lengths of segments in a triangle.
- apply triangle congruency theorems to determine if two triangles are congruent.
- determine if two parts of a triangle are congruent by first determining congruency between the two triangles and then using CPCTC.
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- compare and contrast classifications of polygons (e.g. convex/concave, regular/irregular, number of sides, etc.).
- identify, name, and describe polygons (specifically with 3 to 12 sides).
- use inductive reasoning to derive an algebraic representation of the sum of the angles of polygons and then apply it.
- use algebra to solve problems involving interior and exterior angles of convex polygons.
- use transversal/parallel line properties and the Pythagorean Theorem to derive properties of parallelograms.
- use algebra and formulas to solve problems regarding parallelograms such as finding side lengths, angle measures, or variables.
- solve real-world problems using properties of parallelograms.
- classify a quadrilateral based on the geometric markings on a picture.
- compare/contrast properties of different quadrilaterals (especially within the grouping of parallelograms).
- use algebra, formulas, and indirect methods to solve problems involving the areas of squares, rectangles, circles, trapezoids, triangles, equilateral triangles, parallelograms, rhombi & irregular figures (including CAPT Pizza problems).
- use algebra to derive a formula to measure the area of a sector, and to solve problems involving sectors of a circle.
- use indirect methods, not using measuring tools, (Pythagorean Theorem, trigonometric ratios and proportions in similar figures) to find a missing measurement and then calculate the area of a regular or irregular figure.
- use algebra and properties of similar figures (dilation/scale factor) to solve area problems.
- use algebra and polygon formulas to solve real world problems involving area of irregular shapes.
- use algebra and estimation to solve real-world problems involving perimeter, circumference, and area of regular and irregular figures.
- solve CAPT problems involving perimeter, circumference, and area of regular and irregular figures, and support their answers in written form.
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- compare and contrast classifications of polyhedra (e.g. convex/concave, regular/irregular, number of sides, etc.).
- compare and contrast classifications of solids (e.g. prisms, pyramids, cones, cylinders, spheres).
- use algebra and properties of solids to solve surface and lateral area problems.
- use indirect methods (Pythagorean Theorem, trigonometric ratios and proportions in similar figures) to find a missing measurement and then calculate the surface area
- use algebra and formulas to solve problems involving the volume of solids such as right prisms, pyramids, cones, cylinders, spheres
- use algebra and volume formulas to solve real world problems involving volume of irregular shapes.
- use algebra and formulas and direct reasoning to compare and contrast the effect of change of one or more dimensions on the volume of a cylinder and rectangular prism.
- use indirect methods (Pythagorean Theorem, trigonometric ratios and proportions) to find a missing measurement and then calculate the volume of a regular or irregular solid
- solve CAPT problems involving surface area and volume of regular and irregular solids, and support their answers in written form.
- derive the ratios of the sides of a 45/45/90 and 30/60/90 triangles by at least one of the following methods: technology, manipulatives, or proof.
- solve problems using algebra and the properties of 45/45/90 and 30/60/90 triangles.
- solve basic segment and angle measurement problems using sine, cosine, and tangent on right triangles.
- use trigonometric ratios to solve real-world problems (heights of objects, distances, etc.)
- express conjectures, conclusions, and logical arguments modeling CAPT written format.
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- compare and contrast various parts of a circle (chord, diameter, radius, secant, tangent, point of tangency, major arc, minor arc, semicircle, etc.).
- solve problems using algebra & theorems relating to arc measures, arc length and central angles.
- solve problems using algebra relating to angles formed with a vertex outside, on, or inside a circle (but not at center).
- find the lengths of segments of chords, tangents, and secants.
- solve problems using the Pythagorean Theorem, trigonometric ratios and properties of tangents to find a missing measurement.
- solve problems relating to arcs in a circle.
- graph and solve problems using the equation of a circle: (x-h)2 + (y-k)2 = r2.
- complete a proof in a format of their choice.
- prove two triangles are congruent by SAS, SSS, ASA, or HL.
- prove CPCTC from a given situation.
- compare, contrast, and identify the four types of transformations.
- translate an object from a given set of directions (either in coordinate notation or vector notation).
- determine the translation of an object and express it in coordinate or vector notation.
- determine missing lengths in two objects given the dilation factor.
- Find the vertices of a reflected object on a Cartesian plane when the object has been reflected over any of the following lines: y = x, y = -x, y-axis, or the x-axis.
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