Download Curriculum - Pleasantville High School

Pleasanville High School
Geometry Curriculum
Fall 2008
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Aim of the lesson
What is a logic, a statement, open and closed sentences? What is the negation? What is compound statement (conjunction,
disjunction)? What are the truth values of negation, conjunction and disjunction?
What is a conditional statement? What are the hidden conditionals? How do we form inverse, converse, and contrapositive?
What sentences are logically equivalent?
What is a biconditional statement? How do we draw conclusions?
TEST 1
Know and apply that if a line is perpendicular to each of two intersecting lines at their point of intersection, then the line is
perpendicular to the plane determined by them (Define perpendicular lines and right angles)
Know and apply that through a given point there passes one and only one plane perpendicular to a given line
Know and apply that through a given point there passes one and only one line perpendicular to a given plane
Know and apply that two lines perpendicular to the same plane are coplanar
Know and apply that two planes are perpendicular to each other if and only if one plane contains a line perpendicular
to a second plane
Know and apply that if a line is perpendicular to a plane, then any line perpendicular to the given line at its point of intersection
with the given plane is in the given plane
Know and apply that if a line is perpendicular to a plane, then every plane containing the line is perpendicular to the given plane
Know and apply that if a plane intersects two parallel planes, then the intersection is two parallel lines
Know and apply that if two planes are perpendicular to the same line, they are parallel
Review
TEST 2
Computer lab: Intro to GSP
Determine if two lines cut by a transversal are parallel, based on the measure of given pairs of angles formed by
the transversal and the lines (Define vertical angles)
GSP Project “Parallel lines cut by transversal”
Investigate, justify, and apply theorems about the sum of the measures of the interior angles of a triangle.
GSP Project “Triangle Sum”
Investigate, justify, and apply theorems about the sum of the measures of the exterior angles of a triangle. Define
complementary and supplementary angles.
GSP Project “Exterior Angles in a Triangle”
Investigate, justify, and apply theorems about geometric inequalities, using the exterior angle theorem
Investigate, justify, and apply the triangle inequality theorem
Determine either the longest side of a triangle given the three angle measures or the largest angle given the lengths of
three sides of a triangle
GSP Project “Triangle Inequalities”
TEST 3(?)
What are the line segments associated with the triangle? (Define the altitude, median, and angle bisector. Indicate on the
diagram = and right angles)
What conclusions can be drawn when given line and angle bisectors, and intersecting lines (Draw conclusions in a form of a
two column proof)
How can we use reflexive, symmetric and transitive properties?
How can we use operational postulates, partition and substitution postulates?
What are conditions for congruent triangles? (SSS, SAS, ASA, AAS and not AAA or ASS) (Do not use HL)
How can we prove triangles are congruent? (Use more complicated proofs; use postulates and properties)
What are the theorems involving the pairs of angle? (Use complimentary and supplementary angles in proofs)
How can we prove that triangles, which contain perpendicular lines or altitudes are congruent? (All right angles are =)
How do we use HL to prove that two triangles are congruent?
How do we use congruent triangles to prove line segments and angles are congruent?
Review
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TEST 4
How do we use the Isosceles Triangle Theorem and its converse to prove that two triangles are congruent?
How do we prove overlapping triangles are congruent?
How do we write double-congruency proofs?
How do we prove that triangles are congruent by using the properties of parallel lines cut by a transversal?
How do we prove that two lines are parallel?
TEST 5
What are the properties of similar triangles? (Define similar polygons, state that corresponding sides are in proportion and
corresponding angles are congruent. Define ratio, proportion, means and extremes. Define ratio of sides and ratio of areas.
GSP Project “Finding the Height of a Tree” (another day, if time…)
Investigate, justify, and apply theorems about geometric relationships, based on the properties of the line segment joining the
midpoints of two sides (or line segment // to one side) of the triangle
Investigate, justify, and apply theorems about the centroid of a triangle, dividing each median into segments whose
lengths are in ratio 2:1
GSP Project “Medians in a triangle”
How do we prove that triangles are similar? (AA)
How do we prove that lines segments are proportional?
How do we prove that the product of line segments are equal?
What are the proportions in a right triangle?
Investigate, justify, and apply theorems about mean proportionality:
 the altitude to the hypotenuse of a right triangle is the mean proportional between the two segments along
the hypotenuse
 the altitude to the hypotenuse of a right triangle divides the hypotenuse so that either leg of the right
triangle is the mean proportional between the hypotenuse and segment of the hypotenuse adjacent to that
leg
Investigate, justify, and apply the Pythagorean Theorem and its converse
GSP Project “The Pythagorean Theorem”
Review
TEST 6
Investigate, justify, and apply theorems about the sum of the measures of the interior and exterior angles of polygons
Investigate, justify, and apply theorems about each interior and exterior angle measure of regular polygons
GSP Project “Constructing Regular Polygons”
Investigate, justify, and apply theorems about parallelograms involving their angles, sides, and diagonals
GSP Project “Properties of Parallelograms”
Investigate, justify, and apply theorems about special parallelograms (rectangles, rhombuses, squares) involving
their angles, sides, and the diagonals
GSP Project “Properties of a Square”
Investigate, justify, and apply theorems about trapezoids (including isosceles trapezoids) involving their angles, sides, medians,
and diagonals
GSP Project “Properties of a Trapezoid”
Review
TEST 7
Find the slope of a perpendicular line, given the equation of a line. Review
Determine whether two lines are parallel, perpendicular, or neither, given their equations. Review
Find the equation of a line, given a point on the line and the equation of a line perpendicular to the given line.
Find the equation of a line, given a point on the line and the equation of a line parallel to the desired line. Review
Find the midpoint of a line segment, given its endpoints (include finding the endpoint given the midpoint)
Find the length of a line segment, given its endpoints
Find the equation of a line that is the perpendicular bisector of a line segment, given the endpoints of the line segment
How do we use coordinate geometry to prove that a figure is a parallelogram or a rectangle?
How do we use coordinate geometry to prove that a figure is a rhombus or a square?
How do we use coordinate geometry to prove that a figure is a trapezoid or an isosceles trapezoid?
How do we find areas of polygons. Review
Review
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TEST 8
How do we solve geometric problems requiring quadratic equations? (factoring only) Review
How do we graph quadratic equations on the set of coordinate axes? (1A, 3A, 5A optional)
How do we solve quadratic equation graphically?
How do we solve a system of quadratic-linear equations graphically?
How do we solve a system of quadratic-linear equations algebraically? Review
Graph circles of the form (x – h)² + (y – k)² = r²
Write the equation of a circle, given its center and radius or given the endpoints of a diameter
Write the equation of a circle, given its graph Note: The center is an ordered pair of integers and the radius is an integer.
Find the center and radius of a circle, given the equation of the circle in center-radius form
Solve systems of equations involving one linear equation and one quadratic equation (circle) graphically
Review
TEST 9
Construct a bisector of a given angle, using a straightedge and compass, and justify the construction
Construct the perpendicular bisector of a given segment, using a straightedge and compass, and justify the construction
Construct lines parallel (or perpendicular) to a given line through a given point, using a straightedge and compass, and justify
the construction
Construct an equilateral triangle, using a straightedge and compass, and justify the construction
Investigate and apply the concurrence of medians, altitudes, angle bisectors, and perpendicular bisectors of triangles
GSP Project “Medians in a Triangle”
GSP Project “Altitudes in a Triangle”
GSP Projects combined “Angle Bisectors in a Triangle” and “Perpendicular Bisectors in a Triangle”
What is the meaning of locus and how do we discover a probable locus?
How do we discover a locus in geometric problems?
How do we find loci, which satisfy two different sets of conditions?
How do we write the equation of a locus satisfying given conditions?
How do we find intersecting loci using the coordinate geometry?
Review
TEST 10
What are central and inscribed angles?
How do we solve problems with central and inscribed angles?
What are the theorems involving chords?
GSP Project “Chords in a Circle”
How do we solve circle problems with chords?
How do we determine the angles formed by tangents and secants?
GSP Projects combined “Tangents to a Circle” and “Tangent Segments”
How do we determine the measures of angles and arcs in complex circle diagrams?
How do we find the length of line segments associated with circles?
How do we write geometric proofs with circles?
GSP Project “The Cycloid” (if time)
Review
TEST 11
Include with the following lessons: isometry, invariant properties, proper functional notation, orientation, number of invariant
points, perpendicularity, parallelism, similarities, composition of transformation, and analytical representation, etc.
GSP Project “Introducing Transformations (translation, rotation, and reflection)”
How do we find a reflection in a line?
How do we find reflection in a point?
How do we find a rotation about a point?
How do we find a translation?
How do we find a glide reflection?
How do we find a dilation?
How do we find the composition of transformations?
How do we determine the equation after a line or point reflection?
GSP Project “Glide reflection” (involves composition)
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GSP Project “Tesselations That Use Rotation” or “Tesselations Using Only Translations”
Review
TEST 12
Review formulas to find the area of polygons
GSP Project “A Triangle Within a Triangle”
How do we find the areas of geometric figures using coordinate geometry?
How do we find the areas of geometric figures whose sides are not parallel or perpendicular to the coordinate axes?
GSP Project “A Rectangle with Maximum Area”
Know and apply that the lateral edges of a prism are congruent and parallel
Know and apply that two prisms have equal volumes if their bases have equal areas and their altitudes are equal
Know and apply that the volume of a prism is the product of the area of the base and the altitude
Define polyhedron, right prism, parallelepiped, etc.
Apply the properties of a regular pyramid, including:
 lateral edges are congruent
 lateral faces are congruent isosceles triangles
* volume of a pyramid equals one-third the product of the area of the base and the altitude
Define frustum (optional)
Apply the properties of a cylinder, including:
 bases are congruent
 volume equals the product of the area of the base and the altitude
* lateral area of a right circular cylinder equals the product of an altitude and the circumference of the
base
Apply the properties of a right circular cone, including:
 lateral area equals one-half the product of the slant height and the circumference of its base
 volume is one-third the product of the area of its base and its altitude
Apply the properties of a sphere, including:
 the intersection of a plane and a sphere is a circle,
 a great circle is the largest circle that can be drawn on a sphere
 two planes equidistant from the center of the sphere and intersecting the sphere do so in congruent
circles
 surface area and volume
GSP Project “Rick’s Theorem” (if time)
Review
TEST 13
GSP Project “Square Root Spiral” (if time)
GSP Project “The Golden Rectangle” (if time)
REGENTS REVIEW