K1. geometry.Understand the different roles played by axioms, definitions and theorems in the logical structure of mathematics, especially in geometry.
Evidence of Learning
| ADP Benchmarks |
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- K1.1.
- Identify, explain the necessity of and give examples of definitions, axioms and theorems.
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- K1.2.
- State and prove key basic theorems in geometry such as the Pythagorean theorem, the sum of the angles of a triangle is 180 degrees, and the line joining the midpoints of two sides of a triangle is parallel to the third side and half its length. [Core]
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- K1.3.
- Recognize that there are geometries, other than Euclidean geometry, in which the parallel postulate is not true.
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K2. geometry.Identify and apply the definitions related to lines and angles and use them to prove theorems in (Euclidean) geometry, solve problems, and perform basic geometric constructions using a straight edge and compass.
Evidence of Learning
| ADP Benchmarks |
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- K2.1.
- Identify and apply properties of and theorems about parallel lines and use them to prove theorems such as two lines parallel to a third are parallel to each other and to perform constructions such as a line parallel to a given line through a point not on the line. [Core]
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- K2.2.
- Identify and apply properties of and theorems about perpendicular lines and use them to prove theorems such as the perpendicular bisectors of line segments are the set of all points equidistant from the two end points and to perform constructions such as the perpendicular bisector of a line segment. [Core]
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- K2.3.
- Identify and apply properties of and theorems about angles and use them to prove theorems such as two lines are parallel exactly when the alternate interior angles they make with a transversal are equal and to perform constructions such as the bisector of an angle
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K3. geometry.Know the basic theorems about congruent and similar triangles and use them to prove additional theorems and solve problems. [Core]
Evidence of Learning
| College Readiness Standards |
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- 5.3b
- Apply and justify the applicability of transformations, congruence, similarity, ratios, and proportions in problem-solving situations.
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K4. geometry.Know the definitions and basic properties of a circle and use them to prove basic theorems and solve problems.
K5. geometry.Apply the Pythagorean theorem, its converse and properties of special right triangles to solve problems.
Evidence of Learning
| College Readiness Standards |
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- 5.3e
- Use the Pythagorean Theorem (or distance formula) in 2-D and 3-D situations when appropriate to compute unknown distances.
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K6. geometry.Use rigid motions (compositions of reflections, translations and rotations) to determine whether two geometric figures are congruent and to create and analyze geometric designs.
Evidence of Learning
| College Readiness Standards |
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- 5.3b
- Apply and justify the applicability of transformations, congruence, similarity, ratios, and proportions in problem-solving situations.
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K7. geometry.Know about the similarity of figures and use the scale factor to solve problems. [Core]
Evidence of Learning
| College Readiness Standards |
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- 5.3b
- Apply and justify the applicability of transformations, congruence, similarity, ratios, and proportions in problem-solving situations.
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K8. geometry.Know that geometric measurements (length, area, perimeter, volume) depend on the choice of a unit and that measurements made on physical objects are approximations; calculate the measurements of common plane and solid geometric figures.
Evidence of Learning
| ADP Benchmarks |
College Readiness Standards |
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- K8.1.
- Understand that numerical values associated with measurements of physical quantities must be assigned units of measurement or dimensions; apply such units correctly in expressions, equations and problem solutions that involve measurements; and convert a measurement using one unit of measurement to another unit of measurement. [Core]
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- K8.2.
- Determine the perimeter of a polygon and the circumference of a circle; the area of a rectangle, a circle, a triangle and a polygon with more than four sides by decomposing it into triangles; the surface area of a prism, a pyramid, a cone and a sphere; and the volume of a rectangular box, a prism, a pyramid, a cone and a sphere. [Core]
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- 5.3a
- Inductively generate a conjecture and deductively support it.
- 5.3b
- Apply and justify the applicability of transformations,
congruence, similarity, ratios, and proportions in problemsolving
situations.
- 5.3c
- Distinguish between area and perimeter of 2-D figures, surface area, and volume of 3-D figures.
- 5.3d
- Calculate the area and perimeter of circles, triangles, quadrilaterals, and regular polygons.
- 5.3e
- Use the Pythagorean Theorem (or distance formula) in 2-D
and 3-D situations when appropriate to compute unknown
distances.
- 5.3f
- Calculate the volume and surface area of spheres, right
rectangular prisms, and right circular cylinders, right prisms, right pyramids, and right cones.
- 5.3g
- Graph ellipses and hyperbolas whose axes are parallel to
the x and y axes, and demonstrate understanding of the
relationship between their standard algebraic form and their
graphical characteristics.
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- K8.3.
- Know that the effect of a scale factor k on length, area and volume is to multiply each by k, k_ and k_, respectively.
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k9. geometry.Visualize solids and surfaces in three-dimensional space when given two-dimensional representations (e.g., nets, multiple views) and create two-dimensional representations for the surfaces of three-dimensional objects.
K10. geometry.Represent geometric objects and figures algebraically using coordinates; use algebra to solve geometric problems.[Core]
Evidence of Learning
| ADP Benchmarks |
College Readiness Standards |
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- K10.1.
- Express the intuitive concept of the ?slant? of a line in terms of the precise concept of slope, use the coordinates of two points on a line to define its slope, and use slope to express the parallelism and perpendicularity of lines.
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- 5.2b
- Include additional line segments to represent important
known or unknown distances.
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- K10.2.
- Describe a line by a linear equation.
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- K10.3.
- Find the distance between two points using their coordinates and the Pythagorean theorem.
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- 5.3e
- Use the Pythagorean Theorem (or distance formula) in 2-D
and 3-D situations when appropriate to compute unknown
distances.
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- K10.4.
- * Find an equation of a circle given its center and radius and, given an equation of a circle, find its center and radius.
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K11. geometry.Understand basic right-triangle trigonometry and apply it to solve problems.
Evidence of Learning
| ADP Benchmarks |
College Readiness Standards |
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- K11.1.
- Understand how similarity of right triangles allows the trigonometric functions sine, cosine and tangent to be defined as ratios of sides and be able to use these functions to solve problems.
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- 5.4a
- Use sine, cosine or tangent to find unknown distances and angles.
- 5.4c
- Understand the relationship of cotangent, secant, and
cosecant to basic right-triangle ratios.
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- K11.2.
- Apply the trigonometric functions sine, cosine and tangent to solve for an unknown length of a side of a right triangle, given one of the acute angles and the length of another side. [Core]
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- 5.4a
- Use sine, cosine or tangent to find unknown distances and angles.
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- K11.3.
- Use the standard formula for the area of a triangle, A = _bh, to explain the area formula, A = _absinC where a and b are the lengths of two sides of a triangle and C is the measure of the included angle formed by these two sides, and use it to find the area of a triangle when given the lengths of two of its sides and the included angle.
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K12. geometry. * Know how the trigonometric functions can be extended to periodic functions on the real line, derive basic formulas involving these functions, and use these functions and formulas to solve problems.
Evidence of Learning
| ADP Benchmarks |
College Readiness Standards |
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- K12.1.
- * Know that the trigonometric functions sine and cosine, and thus all trigonometric functions, can be extended to periodic functions on the real line by defining them as functions on the unit circle, that radian measure of an angle between 0 and 360 degrees is the arc length of the unit circle subtended by that central angle, and that by similarity, the arc length s of a circle of radius r subtended by a central angle of measure t radians is s = rt.
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- 8.6a
- Represent and interpret trig functions using the unit circle.
- 8.6b
- Demonstrate an understanding of radians and degrees
by converting between units, finding areas of sectors, and
determining arc lengths of circles.
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- K12.2.
- * Know and use the basic identities, such as sin_(x) + cos_(x) = 1 and cos (_/2-x) = sin(x) and formulas for sine and cosine, such as addition and double angle formulas.
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- 8.6f
- Know and apply the identity cos ² x+sin ² x=1 and generate
related identities; apply sum and half-angle identities.
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- K12.3.
- * Graph sine, cosine and tangent as well as their reciprocals, secant, cosecant and cotangent; identify key characteristics.
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- 8.6d
- Sketch graphs of sine, cosine, and tangent functions, without
technology; identify the domain, range, intercepts, and
asymptotes.
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- K12.4.
- * Know and use the law of cosines and the law of sines to find missing sides and angles of a triangle.
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- 5.4d
- Use Law of Cosines and Law of Sines to solve problems.
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