Download essentials of geometry - Killingly Public Schools

Grades 10 - 12
Draft Sept. 2002
Killingly Public Schools
ESSENTIALS OF GEOMETRY
Grades 10-12
Language of Plane Geometry
CONTENT STANDARD 10-12 EG 1: The student will use the properties of
points, lines, and planes to define and construct other basic geometric figures.
BENCHMARK
10-12EG
1.1
The student is able to understand the basic geometric
figures and their properties.
Objectives
• 1.1.1
The student . . .
-names, knows the properties of, draws and writes the symbols for:
points, lines, segments, rays, and planes.
•
-identifies collinear, non-collinear, coplanar, and non-coplanar points.
1.1.3
BENCHMARK
10-12EG
1.2
The student is able to understand angles and their various
forms.
Objectives
The student . . .
•
-identifies the two rays that form the sides of the angle and the vertex of
the angle.
1.2.1
•
1.2.2
● 1.2.3
-names an angle is a variety of ways.
-identifies right angles, perpendicularity, opposite rays, and angles.
BENCHMARK
10-12EG
1.3
Objectives
• 1.3.1
• 1.3.2
• 1.3.3
• 1.3.4
• 1.3.5
The student is able to understand parallel lines, parallel
planes, intersecting lines, and intersecting planes.
The student . . .
-identifies the intersection of two lines as a point.
-identifies the intersection of two planes as a line.
-defines parallels and identifies the symbols for parallel.
-names lines and planes that are parallel.
-identifies and names skew lines.
BENCHMARK
10-12EG
1.4
The student is able to use the metric system to measure 10length.
Objectives
• 1.4.1
The student . . .
-names a length using symbols.
•
1.4.2
-identifies the common units of length in the metric system.
•
1.4.3
-converts between the various metric units.
•
1.4.4
-identifies the best unit of metric measurement applied to real-world
objects.
BENCHMARK
10-12EG
1.5
The student is able to construct congruent segments and
segment bisectors.
Objectives
• 1.5.1
The student . . .
-identifies the symbols for congruence.
•
1.5.2
-identifies and constructs congruent segments.
•
•
1.5.3
1.5.4
-defines midpoint.
-identifies and constructs bisectors and perpendicular bisectors.
BENCHMARK
10-12EG
1.6
The student is able to use different types of reasoning
techniques to solve problems.
Objectives
• 1.6.1
The student . . .
-identifies deductive and inductive reasoning techniques.
•
1.6.2
-applies inductive and deductive reasoning to solving problems.
•
1.6.3
-solves problems by drawing a diagram or making a model.
Angles
CONTENT STANDARD 10-12 EG 2: The student will measure and classify
angles.
BENCHMARK
10-12EG
2.1
Objectives
• 2.1.1
•
•
2.1.2
2.1.3
The student is able to use amount of turn to measure angles
and use the relationships among parts of circles.
The student . . .
-defines and identifies a circle, its center and all radii, chords, and
diameters.
-calculates radius or diameter using d = 2r.
-identifies the amount of turn of a central angle.
BENCHMARK
10-12EG
2.2
The student is able to measure angles in degrees.
Objectives
• 2.2.1
The student . . .
-calculates the change between amount of turn and degree measure.
•
2.2.2
-uses a protractor to measure and draw angles.
•
2.2.3
-identifies adjacent angles.
•
2.2.4
-adds and subtracts angle measures.
BENCHMARK
10-12EG
2.3
The student is able to inderstand the relationships between
pairs of angles.
Objectives
• 2.3.1
The student . . .
-identifies, labels, and constructs congruent angles, angle bisectors,
vertical angles, complementary angles, and supplementary angles.
•
-solves word problems by working backwards.
2.3.2
Polygons and Polyhedrons
CONTENT STANDARD 10-12 EG 3: The student will recognize, categorize,
and/or name polygons and polyhedrons.
BENCHMARK
10-12EG
3.1
The student is able to recognize concave and convex
polygons.
Objectives
• 3.1.1
The student . . .
-identifies figures that are polygons, the vertices, and sides of a polygon.
•
3.1.2
-recognizes convex and concave polygons.
•
3.1.3
-identifies polygons by their number of sides and names a polygon by the
vertices.
•
3.1.4
-identifies consecutive angles, adjacent (consecutive) sides, adjacent
(consecutive) vertices, and the diagonals of a polygon.
BENCHMARK
10-12EG
3.2
The student is able to classify triangles.
Objectives
• 3.2.1
The student . . .
-identifies equilateral, equiangular, isosceles, scalene, acute, right, and
obtuse triangles by the number of congruent sides or by its angles.
•
-identifies the base, legs, vertex angle, and base angles of a triangle.
3.2.2
BENCHMARK
10-12EG
3.3
The student is able to recognize types of quadrilaterals and
other polygons.
Objectives
• 3.3.1
The student . . .
-names, defines, and draws a trapezoid, parallelogram, rhombus, rectangle
and a square.
•
-identifies regular polygons.
3.3.2
BENCHMARK
10-12EG
3.4
Objectives
• 3.4.1
• 3.4.2
• 3.4.3
• 3.4.4
•
3.4.5
The student is able to name polyhedrons and prisms.
The student . . .
-names the faces, edges, and vertices of a polyhedron.
-names the bases and lateral faces of a prism.
-identifies a prism by the bases.
-accurately draws a prism and identifies points of view.
-draws figures using one-point and two-point perspective.
Introduction to Transformations
CONTENT STANDARD 10-12 EG 4: The student will examine congruence
and its relationship to transformations and symmetry.
BENCHMARK
10-12EG
4.1
The student is able to understand congruence and
corresponding parts.
Objectives
• 4.1.1
The student . . .
-names corresponding sides and angles of congruent polygons.
•
-writes congruence statements for congruent sides, angles, and polygons.
4.1.2
BENCHMARK
10-12EG
4.2
The student is able to understand various congruencepreserving transformations.
Objectives
• 4.2.1
The student . . .
-names and draws reflection, translation, and rotation images.
•
4.2.2
-identifies the line of reflection and clockwise, counter-clockwise, and
reverse orientation.
•
4.2.3
-solves problems involving reflections.
BENCHMARK
10-12EG
4.3
Objectives
• 4.3.1
• 4.3.2
The student is able to identify different types of symmetry.
The student . . .
-identifies and draws lines of symmetry.
-identifies line, rotational, and point symmetry.
Triangles and Inequalities
CONTENT STANDARD 10-12 EG 5: The student will explore in-depth
triangles and their related segments.
BENCHMARK
10-12EG
5.1
The student understands the Isosceles Triangle Theorem
and related theorems.
Objectives
• 5.1.1
The student . . .
-identifies the line of symmetry of an isosceles triangle and recognizes that
it bisects the vertex angle and is the perpendicular bisector of the base.
•
5.1.2
-recognizes the base angles of an isosceles triangle are congruent.
•
5.1.3
-identifies the two sides opposite two congruent angles in a triangle are
congruent.
BENCHMARK
10-12EG
5.2
Objectives
• 5.2.1
• 5.2.2
• 5.2.3
•
•
5.2.4
5.2.5
The student . . .
-identifies or draws the altitudes of a triangle and the orthocenter.
-identifies or draws the medians of a triangle and labels the centroid.
-constructs the circumcenter of a triangle using the perpendicular bisectors
of the sides.
-constructs the incenter of a triangle using angle bisectors.
-demonstrates that a point on the perpendicular bisector of segment is
equidistant from the ends of the segment.
BENCHMARK
10-12EG
5.3
Objectives
• 5.3.1
•
•
•
•
5.3.2
5.3.3
5.3.4
5.3.5
The student is able to construct or name several types of
concurrent lines that occur with triangles.
The student is able to understand inequalities involving
side lengths and angle measure in triangles.
The student . . .
-orders the sides/angles of a triangle by size given the measure of the
angles/sides.
-identifies if three given lengths of segments can be the sides of a triangle.
-identifies included angles and sides.
-applies the Hinge Theorem.
-solves problems using the guess-and-check technique.
Congruent Triangles
CONTENT STANDARD 10-12 EG 6: The student will explore congruent
triangles and their properties.
BENCHMARK
10-12EG
6.1
The student understands various postulates and theorems
that prove that two triangles are congruent.
Objectives
• 6.1.1
• 6.1.2
The student . . .
-identifies the postulates (SSS, SAS, ASA) and theorems (AAS, HL).
-applies the appropriate postulate of theorem to show that two triangles are
congruent.
•
-identifies congruent triangles and sides of triangles based on the postulate
or theorem being applied.
6.1.3
BENCHMARK
10-12EG
6.2
The student understands the relationship between congruent
triangles and their corresponding sides and angles.
Objectives
• 6.2.1
The student . . .
-identifies Corresponding Parts of Congruent Triangles are Congruent
(C.P.C.T.C.)
•
-applies C.P.C.T.C. to determine which angles and sides are congruent
given two congruent triangles.
6.2.2
BENCHMARK
10-12EG
6.2
Objectives
• 6.2.1
The student is able to use the understanding of congruent
triangles to solve real-world applications.
The student . . .
-estimates distances using congruent triangles.
Parallel Lines
CONTENT STANDARD 10-12 EG 7: The student will explore the various
pairs of angles that are formed when two lines are cut by a transversal.
BENCHMARK
10-12EG
7.1
Objectives
● 7.1.1
The student is able to identify the angles formed when two
lines are cut by a transversal.
The student . . .
-identifies interior, exterior, alternate interior/exterior, corresponding,
vertical, adjacent supplementary and same side interior/exterior angles.
BENCHMARK
10-12EG 7.2
The student is able to understand angle relationships that
occur when parallel lines are cut by a transversal.
Objectives
The student . . .
•
7.2.1
-recognizes that the alternate interior, vertical, and corresponding angles
are congruent.
•
7.2.2
-recognizes that same side interior, same side exterior, and adjacent angles
are supplementary.
•
7.2.3
-identifies two lines are parallel by the angle measure relationships.
BENCHMARK
10-12EG 7.3
The student is able to find interior and exterior angle
measures of polygons.
Objectives
The student . . .
•
7.3.1
-calculates various interior and exterior angle measurements using the
Triangle Angle-Sum Theorem, vertical, adjacent supplementary, exterior
and remote interior angles.
•
7.3.2
-calculates the sum of the measures of the interior of a polygon with n
sides as (n-2) 180°.
•
7.3.3
-calculates exterior angle measure using the Exterior Angle-Sum
Theorem.
•
7.3.4
-interprets and solves problems using contour maps.
Properties of Quadrilaterals
CONTENT STANDARD 10-12 EG 8: The student will explore and apply
properties of quadrilaterals.
BENCHMARK
10-12EG 8.1
The student is able to use the properties of parallelograms,
including special parallelograms (rhombuses, rectangles,
and squares).
Objectives
The student . . .
•
8.1.1
-identifies the properties of each type of parallelogram.
•
8.1.2
-applies the properties to determine the measures of the sides and angles of
a given parallelogram.
BENCHMARK
10-12EG 8.2
The student is able to identify the parts of trapezoids and
isosceles trapezoids and apply the properties of each.
Objectives
The student . . .
•
8.2.1
-identifies the bases, legs, base angles, and median of trapezoids and
isosceles trapezoids.
•
8.2.2
-identifies the properties of trapezoids and isosceles trapezoids.
•
8.2.3
-applies the properties to determine the measures of the sides and angles of
a given trapezoid or isosceles trapezoid.
BENCHMARK
10-12EG 8.3
The student is able to understand the relationship between
uadrilaterals, parallelograms, and special parallelograms
(rhombuses, rectangles, and squares).
Objectives
The student . . .
•
8.3.1
-identifies theorems that determine if a quadrilateral is a parallelogram,
and if a parallelogram is a special parallelogram.
•
8.3.2
-applies the theorems to classify quadrilaterals as parallelograms or as
special parallelograms (rectangles, rhombuses, or squares).
Perimeter and Area
CONTENT STANDARD 10-12 EG 9: The student will focus on the concepts
of perimeter and area of geometric figures.
BENCHMARK
10-12EG 9.1
The student is able to determine the perimeter and area of
various polygons.
Objectives
• 9.1.1
The student . . .
-measures or calculates the perimeter of various convex and concave
polygons.
•
9.1.2
-determines the area of a polygon by counting squares.
•
9.1.3
-calculates the area of rectangles, parallelograms, triangles, trapezoids,
rhombuses, and regular polygons using appropriate formulas.
•
9.1.4
-solves formulas for unknown variables.
•
9.1.5
-calculates the area of a complex figure by breaking the problem down
into smaller pieces and using appropriate formulas.
BENCHMARK
10-12EG 9.2
Objectives
• 9.2.1
• 9.2.2
• 9.2.3
• 9.2.4
The student is able to determine the circumference and area
of a circle.
The student . . .
-identifies pi (π) and the commonly used approximations.
-calculates circumference using C= πd=2πr.
-calculates the area of a circle using A= πr2.
-writes solutions to circumference and area of circle problems as exact
and/or approximate solutions.
Similarity
CONTENT STANDARD 10-12 EG 10: The student will explore the
relationships between similarity, ratios, proportions, and scale factors.
BENCHMARK
10-12EG 10.1
The student is able to express ratios in simplest form.
Objectives
• 10.1.1
The student . . .
-defines and identifies ratios.
•
10.1.2
-rewrites ratios in various forms (a to b, a:b, and a/b).
•
10.1.3
-simplifies ratios.
BENCHMARK
10-12EG 10.2
The student is able to solve proportions.
Objectives
• 10.2.1
The student . . .
-defines proportion, “means,” and “extremes.”
•
-identifies the Means-Extremes Property and applies it to solve
proportions.
10.2.2
BENCHMARK
10-12EG 10.3
The student is able to recognize figures that appear to be
similar and identify corresponding parts of similar
polygons.
Objectives
The student . . .
•
10.3.1
-defines similar figures and the notation (~).
•
10.3.2
-determines if figures appear to be similar.
•
10.3.3
-uses proper notation when writing similarity statements.
BENCHMARK
10-12EG 10.4
The student understands the relationship between
similarity, proportions, and scale factors.
Objectives
• 10.4.1
The student . . .
-sets up various proportions given similar polygons.
•
10.4.2
-solves proportions to find missing lengths of similar polygons.
•
10.4.3
-verifies that figures are similar using proportions.
•
10.4.4
-defines “scale factor” and can identify the scale factor of similar
polygons.
•
10.4.5
-applies scale factors to calculate actual lengths from scale drawings.
Similar Triangles
CONTENT STANDARD 10-12 EG 11: The student will expand their
knowledge of, and the relationships between, similar triangles.
BENCHMARK
10-12EG 11.1
The student is able to show that pairs of triangles are
similar.
Objectives
The student . . .
•
11.1.1
-identifies similar triangles using the AA, SSS, and SAS Similarity
Postulates.
•
11.1.2
-writes similarity statements for triangles.
•
11.1.3
-calculates angle and side measures using proportions and similarity
statements.
BENCHMARK
10-12EG 11.2
The student understands the Triangle Proportionality
Theorem and its consequences.
Objectives
• 11.2.1
The student . . .
-identifies the corresponding, proportional segments of two triangles.
•
11.2.2
-calculates lengths of triangle segments given certain segment lengths.
•
11.2.3
-solves problems using indirect measurement.
•
11.2.4
-applies the fact that parallel lines divide transversals into proportional
segments.
Squares, Square Roots, and Right Triangles
CONTENT STANDARD 10-12 EG 12: The student will explore various
properties of squares, square roots, and right triangles.
BENCHMARK
10-12EG 12.1
The student is able to find squares and square roots.
Objectives
The student . . .
•
12.1.1
-defines “square,” “square root,” “perfect square,” and radical notation
(
).
•
12.1.2
-evaluates squares and square roots.
•
12.1.3
-estimates square roots of non-perfect squares.
•
12.1.4
-uses a calculator to find exact or approximate square root values.
•
12.1.5
-identifies the “Product Property of Square Roots” and applies it to
simplify square roots.
BENCHMARK
10-12EG 12.2
The student is able to use the Pythagorean Theorem and its
converse.
Objectives
The student . . .
•
12.2.1
-identifies the Pythagorean Theorem and its converse.
•
12.2.2
-applies the Pythagorean Theorem to solve for the missing side of a right
triangle.
•
12.2.3
-applies the converse of the Pythagorean Theorem to determine if a
triangle is right, acute, or obtuse.
•
12.2.4
-defines “Pythagorean Triples” and is able to identify them.
•
12.2.5
-applies the Pythagorean Theorem to find the shortest distance across a
room.
BENCHMARK
10-12EG 12.3
The student is able to manipulate products and quotients of
square root expressions.
Objectives
• 12.3.1
The student . . .
-applies the Product Property of Square Roots to find the product of a
square root expression and write it in simplified form.
•
12.3.2
-identifies the properties of “simplest radical form.”
•
12.3.3
-rewrites square root expressions, including expressions with a square root
in the denominator, in simplest radical form.
BENCHMARK
10-12EG 12.4
The student is able to understands the relationship between
triangles, sides, and altitudes of right triangles.
Objectives
The student . . .
•
12.4.1
-identifies 30-60-90 and 45-45-90 right triangles and their properties.
•
12.4.2
-applies the properties of 30-60-90 and 45-45-90 right triangles to
determine the lengths of the sides of the triangles.
•
12.4.3
defines “geometric mean” and identifies the properties involving right
triangles and geometric mean.
•
12.4.4
-applies the geometric mean properties to find various lengths within right
triangles.
Area and Volume
CONTENT STANDARD 10-12 EG 13: The student will explore area and
volume of various three-dimensional geometric figures.
BENCHMARK
10-12EG 13.1
The student is able to find area and volume of right prisms.
Objectives The student . . .
•
13.1.1
-identifies the bases and faces of a right prism.
•
13.1.2
-calculates lateral and total area of a right prism.
•
13.1.3
-calculates volume or a missing dimension (given the volume) of a right
prism.
•
13.1.4
-changes a measure of area or volume to another unit.
BENCHMARK
10-12EG 13.2
The student is able to find area and volume of right circular
cylinders.
Objectives The student . . .
• 13.2.1
-identifies the circular bases and curved lateral surface.
•
13.2.2
-calculates the lateral area using L.A.=Ch=Βdh=2Βrh.
•
13.2.3
-calculates total surface area.
•
13.2.4
-calculates volume using V=Βr2h.
•
13.2.5
-solves problems using area and volume of three-dimensional figures.
Trigonometry
CONTENT STANDARD 10-12 EG 14: The student will explore basic
trigonometric ratios.
BENCHMARK
10-12EG 14.1
Objectives
• 14.1.1
The student is able to find the ratios of the basic
trigonometry ratios for any acute angle of a right triangle
given the lengths of the sides of the triangle.
The student . . .
-calculates sine, cosine and tangent ratios for acute πA in a right triangle
as:
length of leg opposite ∠ A
sin A =
,
length of hypotenuse
length of leg adjacent to ∠ A
cos A =
, and
length of hypotenuse
length of leg opposite ∠ A
tan A =
length of leg adjacent to ∠ A