Download Discovering Geometry

Discovering Geometry
Indicators/Benchmarks
Geometry and Spatial Sense Standard
8-1. Make and test conjectures about characteristics
and properties (e.g., sides, angles, symmetry)
of two-dimensional figures and threedimensional objects.
Lesson
0.1
Topic
Symmetry
Activities/Investigation
Symmetry show and tell
Materials
Work sheet
Pretty pictures
Rulers
0.2
Line designs
Astrid, 8 point slot
2&3
Days
Compass
Straightedge
0.3
Construction
tools
Work sheet
Curves by lines
Ruler
Dot paper
2&3
Days
.5
Day
Ruler
1.5
Days
Radius
Regular hexagon
Recognize and
explain
Point
Line
Plane
Collinear
Coplanar
Line segment
Endpoints
Congruent
Midpoint
Bisects
Daisy Designs
0.99
Test
Geometry and Spatial Sense Standard
10-2. Recognize and explain the necessity for
certain terms to remain undefined, such as
point, line and plane.
10- 1. Formally define and explain key aspects of
geometric figures, including:
a. interior and exterior angles of polygons;
b. segments related to triangles (median,
altitude, midsegment);
c. points of concurrency related to triangles
(centroid, incenter, orthocenter,
circumcenter);
d. circles (radius, diameter, chord,
circumference, major arc, minor arc,
sector, segment, inscribed angle).
Patterns, Functions and Algebra Standard
8-13. Compute and interpret slope, midpoint and
distance given a set of ordered pairs.
1.1
1.1A
Basic definitions
Definition List
Midpoint
Formula
Technology
Pacing
1
Day
1
Day
1
Vocabulary
Symmetry (reflection,
bilateral, rotational)
Discovering Geometry
Indicators/Benchmarks
Geometry and Spatial Sense Standard
10-1. Formally define and explain key aspects of
geometric figures, including:
a. interior and exterior angles of polygons;
b. segments related to triangles (median,
altitude, midsegment);
c. points of concurrency related to triangles
(centroid, incenter, orthocenter,
circumcenter);
d. circles (radius, diameter, chord,
circumference, major arc, minor arc,
sector, segment, inscribed angle).
Lesson
1.2
Topic
Activities/Investigation
Poolroom math
Materials
Technology
Protractor
Pacing
2
Days
Protractor
Defining angles
1.3
Writing
definitions
1
Day
Straightedge
1.4
1.5
Days
Polygons
Quiz
.5
Day
Geometry and Spatial Sense Standard
8-4. Represent and analyze shapes using coordinate
geometry; e.g., given three vertices and the
type of quadrilateral, find the coordinates of
the fourth vertex.
10-1. Formally define and explain key aspects of
geometric figures, including:
a. interior and exterior angles of polygons;
b. segments related to triangles (median,
altitude, midsegment);
c. points of concurrency related to triangles
(centroid, incenter, orthocenter,
circumcenter);
d. circles (radius, diameter, chord,
circumference, major arc, minor arc,
sector, segment, inscribed angle).
Sketchpad
1.5
Triangles
Special
quadrilaterals
Triangles
Special quadrilaterals
2
Rulers
Protractors
2
Days
Vocabulary
Angle
Vertex
Sides
Degrees
Counterexample
Skew
Right
Acute
Obtuse
Vertical angles
Linear pair
Complementary
Supplementary
Polygon
Diagonal
Convex
Concave
Equilateral
Equiangular
Regular
Scalene
Equilateral
Isosceles
Base (angles)
Trapezoid
Kite
Parallelogram
Rhombus
Rectangle
Square
Discovering Geometry
Indicators/Benchmarks
Measurement Standard
10-5. Determine the measures of central and
inscribed angles and their associated major
and minor arcs.
Geometry and Spatial Sense Standard
10-1. Formally define and explain key aspects of
geometric figures, including:
d. Circles (radius, diameter, chord,
circumference, major arc, minor arc,
sector, segment, inscribed angle).
Geometry and Spatial Sense Standard
8-6. Draw nets for a variety of prisms, pyramids,
cylinders and cones.
11-5. Identify, sketch and classify the cross
sections of three-dimensional objects.
Lesson
Topic
Activities/Investigation
Materials
Technology
Pacing
1.6
Circles
Defining circle terms
Construction
tools
1
Day
1.8
Space geometry
Space Geometry
Dot paper
1
Day
1.98
1.99
Chapter 1
Review
Chapter 1 Test
2
Days
1
Day
3
Vocabulary
Circle
Radius
Diameter
Concentric
Arc
Semicircle
Minor arc
Space
Isometric drawing
Pyramid
Cylinder
Prism
Cone
Sphere
Hemisphere
Discovering Geometry
Indicators/Benchmarks
Geometry and Spatial Sense Standard
10-3. Make, test and establish the validity of
conjectures about geometric properties and
relationships using counterexample, inductive
and deductive reasoning, and paragraph or
two-column proof, including:
c. prove theorems involving properties of
lines, angles, triangles and quadrilaterals.
Geometry and Spatial Sense Standard
10-3. Make, test and establish the validity of
conjectures about geometric properties and
relationships using counterexample, inductive
and deductive reasoning, and paragraph or
two-column proof, including:
c. prove theorems involving properties of
lines, angles, triangles and quadrilaterals.
Patterns, Functions and Algebra Standard
10-1. Define function formally and with f(x)
notation.
8-2. Generalize patterns and sequences by
describing how to find the nth term.
Patterns, Functions and Algebra Standard
10-4. Use algebraic representations and functions
to describe and generalize geometric
properties and relationships.
Geometry and Spatial Sense Standard
10-3. Make, test and establish the validity of
conjectures about geometric properties and
relationships using counterexample, inductive
and deductive reasoning, and paragraph or
two-column proof, including:
c. prove theorems involving properties of
lines, angles, triangles and quadrilaterals.
Geometry and Spatial Sense Standard
10-3. Make, test and establish the validity of
conjectures about geometric properties and
relationships using counterexample, inductive
and deductive reasoning, and paragraph or
two-column proof, including:
c. prove theorems involving properties of
lines, angles, triangles and quadrilaterals.
8-2. Recognize the angles formed and the
relationship between the angles when two
lines intersect and when parallel lines are cut
by a transversal.
Lesson
2.1
Topic
Inductive
reasoning
2.2
Deductive
reasoning
2.3
Number patterns
Finding the rule
2
Days
2.4
Math models
Party handshakes
1
2.4A
2.5
2.6
Activities/Investigation
Shape Shifters
Materials
Technology
Pacing
1
Day
1
Day
Networks
Angle
relationships
Linear pair conjecture
Vertical angle conjecture
Parallel lines
Transversals
Which angles are
congruent?
Is the converse true?
4
Konigsburg
Worksheet
Protractors
Patty paper
Protractor
Sketchpad
Vocabulary
Inductive reasoning
Conjecture
Deductive
Function rule
Linear function
Mathematical models
Triangular number
Rectangular number
1
Day
1
Day
2
Days
Paragraph proof
Converse
Transversal
Corresponding angles
Alternate interior angles
Alternate exterior
angles
Discovering Geometry
Indicators/Benchmarks
Patterns, Functions and Algebra Standard
8-13. Compute and interpret slope, midpoint and
distance given a set of ordered pairs.
Lesson
Topic
Activities/Investigation
2.4A
2.98
2.99
Materials
Technology
Pacing
1
Day
Chapter 2
Review
Chapter 2 Test
2
Days
1
Day
5
Vocabulary
Discovering Geometry
Indicators/Benchmarks
Geometry and Spatial Sense Standard
10-3. Make, test and establish the validity of
conjectures about geometric properties and
relationships using counterexample, inductive
and deductive reasoning, and paragraph or
two-column proof, including:
d. test a conjecture using basic constructions
made with a compass and straightedge or
technology.
10-4. Construct right triangles, equilateral triangles,
parallelograms, trapezoids, rectangles,
rhombuses, squares and kites, using compass
and straightedge or dynamic geometry
software.
10-5. Construct congruent figures and similar
figures using tools, such as compass,
straightedge, and protractor or dynamic
geometry software.
Geometry and Spatial Sense Standard
10-1. Formally define and explain key aspects of
geometric figures, including:
b. segments related to triangles (median,
altitude, midsegment).
10-3. Make, test and establish the validity of
conjectures about geometric properties and
relationships using counterexample, inductive
and deductive reasoning, and paragraph or
two-column proof, including:
d. test a conjecture using basic constructions
made with a compass and straightedge or
technology.
10-4. Construct right triangles, equilateral triangles,
parallelograms, trapezoids, rectangles,
rhombuses, squares and kites, using compass
and straightedge or dynamic geometry
software.
10-5. Construct congruent figures and similar
figures using tools, such as compass,
straightedge, and protractor or dynamic
geometry software.
Patterns, Functions and Algebra Standard
10-9. Recognize and explain that the slopes of
parallel lines are equal and the slopes of
perpendicular lines are negative reciprocals.
Lesson
Topic
Activities/Investigation
Materials
Technology
Pacing
Vocabulary
3.1
Constructing a
duplicate
Copying a segment
Copying an angle
Construction
tools
1
Day
Sketch
Draw
Construct
3.2
Constructing
perpendicular
bisectors
Finding the right
bisector, right down
the middle
Patty paper
Construction
tools
1
Day
Perpendicular bisector
Coincide
Median
Midsegment
3.3
Constructing
Perpendiculars to
a line
Constructing
angle
Bisectors
3.1 – 3.4
Finding the right line
Patty paper
Perpendiculars
Angle bisecting by
folding
Angle bisecting by
compass
Constructing parallel
lines by folding
Algebra skills 3
3.4
Review
Quiz
3.5
Constructing
parallel lines
2.6A
Altitude
1
Day
1
1
Day
1
Day
Slope
1
Day
3.5A
Slopes of parallel
and
Perpendicular
lines
1
Day
6
Discovering Geometry
Indicators/Benchmarks
Geometry and Spatial Sense Standard
10-3. Make, test and establish the validity of
conjectures about geometric properties and
relationships using counterexample, inductive
and deductive reasoning, and paragraph or
two-column proof, including:
d. test a conjecture using basic constructions
made with a compass and straightedge or
technology.
10-4. Construct right triangles, equilateral triangles,
parallelograms, trapezoids, rectangles,
rhombuses, squares and kites, using compass
and straightedge or dynamic geometry
software.
10-5. Construct congruent figures and similar
figures using tools, such as compass,
straightedge, and protractor or dynamic
geometry software.
Geometry and Spatial Sense Standard
10-1. Formally define and explain key aspects of
geometric figures, including:
b. segments related to triangles (median,
altitude, midsegment);
c. points of concurrency related to triangles
(centroid, incenter, orthocenter,
circumcenter).
Geometry and Spatial Sense Standard
10-1. Formally define and explain key aspects of
geometric figures, including:
c. points of concurrency related to triangles
(centroid, incenter, orthocenter,
circumcenter).
Lesson
Topic
Activities/Investigation
Materials
3.6
Construction
problems
Perspective drawing
Construction
tools
3.7
Constructing
points of
concurrency
Concurrence
Incenter
Circumcenter
Patty paper
Construction
tools
3.8
The centroid
Are medians concurrent?
Construction
tools
Patty paper
Balancing act
3.98
3.99
Chapter 3
Review
Chapter 3 Test
Technology
Pacing
1
Day
Sketchpad
2
Days
Concurrent
Incenter
Circumcenter
Orthocenter
Circumscribed
Inscribed
1
Days
Center of gravity
2
Days
1
Day
7
Vocabulary
Perspective
Vanishing point
Horizon line
One (two) point
perspective
Discovering Geometry
Indicators/Benchmarks
Measurement Standard
8-8. Find the sum of the interior and exterior
angles of regular convex polygons with and
without measuring the angles with a protractor.
Geometry and Spatial Sense Standard
10-1. Formally define and explain key aspects of
geometric figures, including:
a. interior and exterior angles of polygons.
Geometry and Spatial Sense Standard
10-1. Formally define and explain key aspects of
geometric figures, including:
a. interior and exterior angles of polygons.
Geometry and Spatial Sense Standard
10-3. Make, test and establish the validity of
conjectures about geometric properties and
relationships using counterexample, inductive
and deductive reasoning, and paragraph or
two-column proof, including:
b. prove theorems involving triangle
similarity and congruence.
Geometry and Spatial Sense Standard
10-3. Make, test and establish the validity of
conjectures about geometric properties and
relationships using counterexample, inductive
and deductive reasoning, and paragraph or
two-column proof, including:
b. prove theorems involving triangle
similarity and congruence.
Lesson
Topic
Activities/Investigation
Materials
Technology
Sketchpad
Demo
Pacing
1
Day
Vocabulary
Auxiliary line
4.1
Triangle sum
Investigation:
Triangle sum
Straightedge
4.2
Special triangles
Investigation:
Angles (isosceles
triangle)
Ruler
Protractor
1
Day
4.3
Quiz
Triangle
inequalities
Investigation:
Shortest path worksheet
Construction
tools
1
Day
Exterior angle
Adjacent interior angles
Remote interior angles
4.4
Congruence
theorems
Investigation:
Side, Side, Side
Side, Angle, Side
Construction
tools
1
Day
Included angle
Included side
4.5
More
congruence
theorems
Investigation ASA, SAA
4.6
4.7
4.8
4.98
4.99
CPCTC
Flowchart proofs
Isosceles triangle
proofs
Construction
tools
Legs
1
Day
1
Day
1
Day
Proof Puzzles
Investigation:
Symmetry line in an
isosceles triangle
Review
Assessment
1
Day
Construction
tools
2
Days
1
Day
8
Flowchart
Flowchart proof
Biconditional
Discovering Geometry
Indicators/Benchmarks
Measurement Standard
8-8. Find the sum of the interior and exterior
angles of regular convex polygons with and
without measuring the angles with a protractor.
Geometry and Spatial Sense Standard
10-1. Formally define and explain key aspects of
geometric figures, including:
a. interior and exterior angles of polygons.
Lesson
Topic
Activities/Investigation
Materials
Pacing
1
Day
5.1
Polygon sum
Investigation:
Polygon sum
Ruler
Protractor
Calculators
5.2
Exterior angles
Investigation:
Exterior angle sum
Ruler
Protractor
Sketchpad
demo
Calculators
5.3
Kites/trapezoid
Investigation:
Kites/trapezoid
Ruler
Protractor
2
Days
5.4
Mid segments
Investigation:
Trapezoid
5.5
Parallelogram
Parallelogram
1
Day
Systems of
Linear
equations
Substitution/Elimination
worksheet
1
Day
More
Investigation:
Rhombus
Rectangle
5.5A
5.6
Parallelogram
Ruler
Protractor
Ruler
Protractor
Graph paper
1
Day
1
Day
5.7
5.98
5.99
Vocabulary
N-gon
1
Day
1
Day
Quiz
Patterns, Functions and Algebra Standard
10-6. Solve equations and inequalities having
rational expressions as coefficients and
solutions.
10-11. Solve real-world problems that can be
modeled, using systems of linear equations
and inequalities.
8-10. Solve 2 by 2 systems of linear equations
graphically and by simple substitution.
10.3 Make, test and establish the validity of conjectures about geometric properties and relationships using counterexample, inductive and
deductive reasoning, and paragraph or twocolumn proof, including:
c. prove theorems involving properties of lines,
angles, triangles and quadrilaterals.
Technology
Trapezoid
Bases
Base angles
Vertex angles
Non vertex angles
Vector
Resultant vector
Vector sum
Rectangle
Square
Dart
Quadrilateral
proofs
1
Day
Chapter 5
Review
Chapter 5 Test
2
Days
1
Day
9
Discovering Geometry
Indicators/Benchmarks
Geometry and Spatial Sense Standard
10-1. Formally define and explain key aspects of
geometric figures, including:
d. circles (radius, diameter, chord,
circumference, major arc, minor arc,
sector, segment, inscribed angle).
10-10. Solve problems involving chords, radii and
arcs within the same circle.
Measurement Standard
10-5. Determine the measures of central and
inscribed angles and their associated major
and minor arcs.
Geometry and Spatial Sense Standard
10-1. Formally define and explain key aspects of
geometric figures, including:
d. circles (radius, diameter, chord,
circumference, major arc, minor arc,
sector, segment, inscribed angle).
Number, Number Sense and Operations
Standard
10-1. Connect physical, verbal and symbolic
representations of irrational numbers; e.g.,
construct √2 as a hypotenuse or on a number
line.
Geometry and Spatial Sense Standard
10-1. Formally define and explain key aspects of
geometric figures, including:
d. circles (radius, diameter, chord,
circumference, major arc, minor arc,
sector, segment, inscribed angle).
Lesson
Topic
Activities/Investigation
Materials
6.1
Chords
Investigation:
Angles
Central angles
Center
Perpendicular
Bisector
Construction
tools
6.2
Tangents
Investigation:
Tangents
Segments
Construction
tools
6.3
Arcs
Angles
Investigation:
Inscribed angle
Same arc
Semicircle
Quadrilaterals
Parallel lines
Construction
tools
6.4
Circle proofs
6.5
Circumference
6.6
Technology
Pacing
Vocabulary
2
Days
Sketchpad
demo
1
Day
Tangent circles
Internally tangent
Externally tangent
Tangent segments
2
Days
1
Day
Investigation:
Pi
Pi problem
solving
Sketchpad
demo
Calculators
10
1
Day
1
Day
Circumference
Discovering Geometry
Indicators/Benchmarks
Geometry and Spatial Sense Standard
10-1. Formally define and explain key aspects of
geometric figures, including:
d. circles (radius, diameter, chord,
circumference, major arc, minor arc,
sector, segment, inscribed angle).
10-10. Solve problems involving chords, radii and
arcs within the same circle.
Lesson
Topic
6.7
Arc length
6.98
6.99
Chapter 6
Review
Chapter 6 Test
Activities/Investigation
Investigation:
Arc length
Materials
Construction
tools
Technology
Pacing
Calculators
1
Day
1
Day
1
Day
11
Vocabulary
Length of an arc
Discovering Geometry
Indicators/Benchmarks
Geometry and Spatial Sense Standard
10-6. Identify the reflection and rotation
symmetries of two- and three
dimensional figures.
8-5. Draw the results of translations, reflections,
rotations and dilations of objects in the
coordinate plane, and determine properties
that remain fixed; e.g., lengths of sides
remain the same under translations.
11-2. Represent translations using vectors.
10-7. Perform reflections and rotations using
compass and straightedge constructions and
dynamic geometry software.
Geometry and Spatial Sense Standard
10-8. Derive coordinate rules for translations,
reflections and rotations of geometric figures
in the coordinate plane.
8-5. Draw the results of translations, reflections,
rotations and dilations of objects in the
coordinate plane, and determine properties
that remain fixed; e.g., lengths of sides
remain the same under translations.
Geometry and Spatial Sense Standard
10-9. Show and describe the results of
combinations of translations, reflections
and rotations (compositions); e.g.,
perform compositions and specify the
result of a composition as the outcome of a
single motion, when applicable.
8-5. Draw the results of translations, reflections,
rotations and dilations of objects in the
coordinate plane, and determine properties
that remain fixed; e.g., lengths of sides
remain the same under translations.
Lesson
Topic
7.1
Transformations
Investigation:
Reflection
7.2
Isometries
Investigation:
Coordinate plane
Minimal path
Protractor
7.3
Compositions
Investigation:
Parallel lines
Intersecting lines
Protractor
7.4
Tessellations
Investigation:
Semi-regular
7.98
7.99
Activities/Investigation
Chapter 7
Review
Chapter 7 Test
12
Materials
Technology
Sketchpad
Demo
Sketchpad
demo
Sketchpad
demo
Pacing
1
Day
Vocabulary
Image
Isometry
Nonrigid transformation
Translation
Distance
Direction
Translation vector
Rotation reflection
Line of reflection
Reflectional symmetry
Line of symmetry
Rotational symmetry
Point symmetry
3
Days
Ordered pair rules
2
Days
Composition
1
Day
Tessellation
Monohedral tiling
Regular tessellation
Vertex arrangement
Semi-regular tessellation
Archimedean tilings
Discovering Geometry
Indicators/Benchmarks
Patterns, Functions and Algebra Standard
10-3. Solve equations and formulas for a specified
variable; e.g., express the base of a triangle
in terms of the area and height.
Measurement Standard
8-9. Demonstrate understanding of the concepts
of perimeter, circumference and area by
using established formulas for triangles,
quadrilaterals, and circles to determine the
surface area and volume of prisms, pyramids,
cylinders, spheres and cones. (Note: Only
volume should be calculated for spheres and
cones.)
10-1. Explain how a small error in measurement
may lead to a large error in calculated results.
10-2. Calculate relative error.
10-3. Explain the difference between absolute error
and relative error in measurement.
10-4. Give examples of how the same absolute error
can be problematic in one situation but not in
another; e.g., compare “accurate to the nearest
foot’ when measuring the height of a person
versus when measuring the height of a
mountain.
Data Analysis & Probability Standard
10.7. Model problems dealing with uncertainty
with area models (geometric probability).
Patterns, Functions and Algebra Standard
10.3. Solve equations and formulas for a specified
variable; e.g., express the base of a triangle in
terms of the area and height.
Lesson
Topic
8.1
Area of
rectangles
Parallelogram
Area of triangles
Trapezoids
Kites
8.2
8.3
Area problems
Activities/Investigation
Materials
Scissors
1
Day
Investigation:
Triangles
Trapezoids
Kites
Investigation:
Area formulas
Scissors
1
Day
Investigation:
Regular polygon
Calculators
Investigation:
Circle area
Vocabulary
Area
Base
Height
Altitude
Commutative
Associative
Calculators
1
Day
Apothem
8.5
Circle area
Pacing
Investigation:
Parallelogram
8.4
Areas of regular
polygon
Technology
Calculators
Scissors
1
Day
8.6
Sectors
1
Day
1
Day
Sector of a circle
Segment of a circle
Annulus
Where the Chips Fall
8.6A
Geometric
probability
8.7
Surface area
1
Day
Investigation:
Regular pyramid
Cone
13
2
Days
Surface area
Bases
Lateral faces
Slant height
Discovering Geometry
Indicators/Benchmarks
Measurement Standard
8-4. Derive formulas for surface area and volume
and justify them using geometric models and
common materials. For example, find:
a. the surface area of a cylinder as a function
of its height and radius.
8-5. Determine surface area for pyramids by
analyzing their parts.
8-9. Demonstrate understanding of the concepts of
perimeter, circumference and area by using
established formulas for triangles,
quadrilaterals, and circles to determine the
surface area and volume of prisms, pyramids,
cylinders, spheres and cones. (Note: Only
volume should be calculated for spheres and
cones.)
11-3. Derive a formula for the surface area of
a cone as a function of its slant height and
the circumference of its base.
Lesson
8.7
Topic
Activities/Investigation
Materials
Technology
Pacing
Chapter 8
Review
1
Day
Chapter 8 Test
1
Day
14
Vocabulary
Discovering Geometry
Indicators/Benchmarks
Geometry and Spatial Sense Standard
10-3. Make, test and establish the validity of
conjectures about geometrid properties and
relationships using counterexample, inductive
and deductive reasoning, and paragraph or
two-column proof, including:
a. prove the Pythagorean Theorem.
Number, Number Sense and Operations
Standard
8-7. Find the square root of perfect squares, and
approximate the square root of non-perfect
squares as consecutive integers between
which the root lies, e.g., √130 is between 11
and 12.
Patterns, Functions and Algebra Standard
10-8. Graph the quadratic relationship that defines
circles.
8-13. Compute and interpret slope, midpoint and
distance given a set of ordered pairs.
Lesson
Topic
Activities/Investigation
Materials
9.1
Pythagorean
Theorem
Investigation:
Pythagorean Theorem
Scissors
9.2
Converse
Pythagorean
Theorem
Special right
triangles
Investigation:
Converse
Ruler
Protractor
9.3
9.4
Story problems
9.5
Distance
9.6
Circles
Pythagorean
Theorem
9.98
9.99
Investigation:
Isosceles 30-60-90
Isosceles Right Triangle
Technology
Sketchpad
demo
Pacing
1
Day
Calculators
1
Day
Investigation
Formula
Equation of circle
1
Day
1
Day
15
Theorem
Pythagorean triples
1
Day
2
Days
Quiz (9.1-9.3)
Chapter 9
Review
Chapter 9 Test
Vocabulary
Discovering Geometry
Indicators/Benchmarks
Geometry and Spatial Sense Standard
10-1. Formally define and explain key aspects of
geometric figures, including:
a. interior and exterior angles of polygons;
b. segments related to triangles (median,
altitude, midsegment);
c. points of concurrency related to triangles
(centroid, incenter, orthocenter, and
circumcenter);
d. circles (radius, diameter, chord,
circumference, major arc, minor arc, sector,
segment, inscribed angle).
Lesson
10.1
Topic
Solids
Activities/Investigation
Definition List
Measurement Standard
8-4. Derive formulas for surface area and volume
and justify them using geometric models and
common materials. For example, find:
b. that the volume of a pyramid (or cone) is
one-third of the volume of a prism (or
cylinder) with the same base area and
height.
8-6. Solve and determine the reasonableness of
the results for problems involving rates and
derived measurements, such as velocity and
density, using formulas, models and graphs.
8-10. Use conventional formulas to find the surface
area and volume of prisms, pyramids and
cylinders and the volume of spheres and
cones to a specified level of precision.
10.2
Prism
Cylinder Volume
Investigation:
Volume
10.3
Pyramid
Cone Volume
Investigation:
Volume
Volume
problems
Quiz (10.1-10.3)
10.4
10.3A
Net Project
Platonic Solids
Materials
Technology
Pacing
1
Day
Vocabulary
Polyhedron
Faces
Edge
Vertex
Tetrahedron
Regular polyhedron
Prism
Bases
Lateral faces
Lateral edges
Right
Oblique
Altitude
Height
Pyramid
Sphere
Radius
Center
Hemisphere
Great Circle
Cylinder
Axis cone
1
Day
1
Day
1
Day
Octahedron
Dodecahedron
Icosohedron
1
Day
10.5
Displacement
Density
Quiz (10.3-10.5)
Displacement
Density
1
Day
16
Discovering Geometry
Indicators/Benchmarks
Measurement Standard
8-10. Use conventional formulas to find
the surface area and volume of prisms,
pyramids and cylinders and the volume of
spheres and cones to a specified level of
precision.
Lesson
Topic
Activities/Investigation
10.6
Sphere volume
Investigation:
Volume formula
10.7
Sphere
Surface area
Investigation:
Surface area
Chapter 10
Review
Chapter 10 Test
Quiz (10.6-10.7)
Materials
Technology
Pacing
1
Day
Calculators
1
Day
Number, Number Sense and Operations
10-2. Explain the meaning of the nth root.
10-4. Approximate the nth root of a given number
greater than zero between consecutive
integers when n is an integer; e.g., the 4th root
of 50 is between 2 and 3.
10.98
10.99
17
1
Day
1
Day
Vocabulary
nth root
Discovering Geometry
Indicators/Benchmarks
Geometry and Spatial Sense Standard
8-3. Use appropriate levels of precision when
calculating with measurements.
Geometry and Spatial Sense Standard
10-3. Make, test and establish the validity of
conjectures about geometric properties and
relationships using counterexample, inductive
and deductive reasoning, and paragraph or
two-column proof, including:
b. prove theorems involving triangle
similarity
Measurement Standard
8-7. Apply proportional reasoning to solve
problems involving indirect measurements
or rates.
Measurement Standard
9-3. Use the ratio of lengths in similar twodimensional figures or three-dimensional
objects to calculate the ratio of their areas or
volume respectively.
Lesson
Topic
11.1
Similar polygons
Investigation:
Similarity
Dilations
Rulers
Protractors
Calculators
1
Day
11.2
Similar triangles
Investigation:
Angle, angle
Side, side, side
Side, angle, side
Rulers
Protractors
1
Day
11.3
Indirect
measurement
Investigation:
Mirror, mirror
Mirrors
1
Day
11.4
Corresponding
parts of similar
triangles
Investigation:
Corresponding parts
Opposite side ratios
Rulers
Calculators
11.5
Proportions with
area/volume
Investigation:
Area
Volume
11.6
Proportional
segments
between
parallel lines
Investigation:
Parallels
Proportionality
Extended parallel
11.98
11.99
Activities/Investigation
Chapter 11
Review
Chapter 11 Test
Materials
Technology
1
Day
1
Day
1
Day
1
Day
18
Vocabulary
Similar figures
Similar polygons
Scale factor
Dilation
Indirect measurement
1
Day
Sketchpad
demo
Rulers
Protractors
Pacing
Similar solids
Discovering Geometry
Indicators/Benchmarks
Geometry and Spatial Sense Standard
9-1. Define the basic trigonometric ratios in right
triangles: sine, cosine and tangent.
Measurement Standard
9-4. Use scale drawings and right triangle
trigonometry to solve problems that include
unknown distances and angle measures.
Geometry and Spatial Sense Standard
9-2. Apply proportions and right triangle
trigonometric ratios to solve problems
involving missing lengths and angle measures
in similar figures.
Measurement Standard
9-4. Use scale drawings and right triangle
trigonometry to solve problems that include
unknown distances and angle measures.
Geometry and Spatial Sense Standard
11-4. Use trigonometric relationships to determine
lengths and angle measures; i.e., Law of
Sines and Law of cosines.
Measurement Standard
9-4. Use scale drawings and right triangle
trigonometry to solve problems that include
unknown distances and angle measures.
Lesson
12.1
Topic
Trigonometry
ratios
Activities/Investigation
Investigation:
Trigonometry tables
12.2
Problem solving
12.3
Law of sines
Quiz (12.1-12.2)
Investigation:
Area of triangle
Law of sines
12.4
Law of cosine
Investigation:
Pythagorean identity
12.5
Problem solving
with
trigonometry
12.98
12.99
Chapter 12
Review
Chapter 12 Test
Materials
Technology
Pacing
Calculators
Protractors
Rulers
1
Day
Calculators
1
Day
Calculators
Sketchpad
demo
Angle of elevation
Angle of depression
Law of sines
1
Day
Law of cosines
1
Day
19
Opposite side
Adjacent side
Tangent
Sine
Cosine
Trigonometry
Inverse tangent
1
Day
Calculators
Quiz (12.3-12.5)
Vocabulary
1
Day
1
Day
Discovering Geometry
Indicators/Benchmarks
Geometry and Spatial Sense Standard
10-2. Recognize and explain the necessity for
certain terms to remain undefined, such as
point, line and plane.
Geometry and Spatial Sense Standard
10-3. Make, test and establish the validity of
conjectures about geometric properties and
relationships using counterexample, inductive
and deductive reasoning, and paragraph or
two-column proof, including:
a. prove the Pythagorean Theorem;
b. prove theorems involving triangle
similarity and congruence;
c. prove theorems involving properties
of lines, angles, triangles and
quadrilaterals;
d. test a conjecture using basic constructions
made with a compass and straightedge or
technology.
Geometry and Spatial Sense Standard
9-3. Analyze two-dimensional figures in a
coordinate plane; e.g., use slope and distance
formulas to show that a quadrilateral is a
parallelogram.
Lesson
13.1
13.3
13.4
Topic
Premises of
geometry
Triangle proofs
Quadrilateral
proofs
13.5
Indirect proof
13.6
13.7
Circle proofs
Similarity proofs
Activities/Investigation
Postulates
Investigation:
Parallelogram
Conjectures
Investigation:
Tangent
Conjecture
Investigation:
Side, side, side
Similarity
Conjecture
Materials
Technology
Pacing
1
Day
1
Day
1
Chapter 13,
Properties of similarity
1
Day
1
Day
1
Day
1
Day
Review
Chapter 13, Test
20
Lemma
Indirect proof
1
Day
13.7A
13.98
13.99
Vocabulary
Deductive system
Premises
Postulates
Theorems
Properties of arithmetic
Properties of equality
Definition of
congruence
Postulte of geometry
Discovering Geometry
Indicators/Benchmarks
Lesson
Topic
Pacing
Activities/Investigation
Materials
Technology
Vocabulary