Download Geometry - ALC - Willmar Public Schools

Willmar Public Schools
Curriculum Map
Subject Area
Week
1
Mathematics--ALC
Content
Points, Lines, and Angles
in the Plane
Course Name
Standards Addressed
Know and apply properties of
geometric figures to solve real-world
and mathematical problems and to
logically justify results in geometry.
Construct logical arguments, based on
axioms, definitions and theorems, to
prove theorems and other results in
geometry.
2
Parallel Lines and
Transversals
Know and apply properties of
geometric figures to solve real-world
and mathematical problems and to
logically justify results in geometry.
3
Constructions
Construct logical arguments, based on
axioms, definitions and theorems, to
prove theorems and other results in
geometry.
Geometry
Skills/Benchmarks
Know and apply properties of
angles, including corresponding,
exterior, interior, vertical,
complementary and supplementary
angles, to solve problems and
logically justify results.
Use properties of polygons—
including quadrilaterals and regular
polygons—to define them, classify
them, solve problems and logically
justify results.
Date
Essential Questions
Where do I see points, lines, and angles
in the real world?
What real-life problems could I solve
with the knowledge that I have about
angle relationships?
5/6/2009
Assessments
Unit Test
Geometer’s Sketchpad Projects
Use technology tools to examine
theorems, make and test
conjectures, perform constructions
and develop mathematical
reasoning skills in multi-step
problems. The tools may include
compass and straight edge,
dynamic geometry software,
design. software or Internet applets.
Know and apply properties of
parallel and perpendicular lines,
including properties of angles
formed by a transversal, to solve
problems and logically justify
results.
Know and apply properties of
angles, including corresponding,
exterior, interior, vertical,
complementary and supplementary
angles, to solve problems and
logically justify results.
Use technology tools to examine
theorems, make and test
conjectures, perform constructions
and develop mathematical
What role has investigation using
technology played in my understanding
of parallel lines and the angles formed
by transversals?
Unit Test
Geometer’s Sketchpad Projects
Willmar Public Schools
Curriculum Map
reasoning skills in multi-step
problems. The tools may include
compass and straight edge,
dynamic geometry software,
design. software or Internet applets.
Know and apply properties of
geometric figures to solve real-world
and mathematical problems and to
logically justify results in geometry.
Know and apply properties of
parallel and perpendicular lines,
including properties of angles
formed by a transversal, to solve
problems and logically justify
results.
How can compasses, straightedges, and
geometry software be used to test
conjectures and investigate theorems?
Unit test
Geometer’s Sketchpad Projects
Constructions
Understand the roles of axioms,
definitions, undefined terms and
theorems in logical arguments.
Construct logical arguments, based on
axioms, definitions and theorems, to
prove theorems and other results in
geometry.
4
Construct logical arguments and
write proofs of theorems and other
results in geometry, including
proofs by contradiction. Express
proofs in a form that clearly
justifies the reasoning, such as twocolumn proofs, paragraph proofs,
flow charts or illustrations.
Use technology tools to examine
theorems, make and test
conjectures, perform constructions
and develop mathematical
reasoning skills in multi-step
problems. The tools may include
compass and straight edge,
dynamic geometry software, design
software or Internet applets.
Triangle Properties and
Proofs
Know and apply properties of
geometric figures to solve real-world
and mathematical problems and to
logically justify results in geometry.
Know and apply properties of
equilateral, isosceles and scalene
triangles to solve problems and
logically justify results.
Solve real-world and mathematical
geometric problems using algebraic
methods.
Use algebra to solve geometric
problems unrelated to coordinate
geometry, such as solving for an
unknown length in a figure
involving similar triangles, or using
How could the knowledge of the
properties of triangles help a person
plan a central location to build a
laundry room on a college campus?
Unit Test
Geometer’s Sketchpad Projects
Map Activity
Willmar Public Schools
Curriculum Map
the Pythagorean Theorem to obtain
a quadratic equation for a length in
a geometric figure.
Understand the roles of axioms,
definitions, undefined terms and
theorems in logical arguments.
Construct logical arguments, based on
axioms, definitions and theorems, to
prove theorems and other results in
geometry.
5
Construct logical arguments and
write proofs of theorems and other
results in geometry, including
proofs by contradiction. Express
proofs in a form that clearly
justifies the reasoning, such as twocolumn proofs, paragraph proofs,
flow charts or illustrations.
Use technology tools to examine
theorems, make and test
conjectures, perform constructions
and develop mathematical
reasoning skills in multi-step
problems. The tools may include
compass and straight edge,
dynamic geometry software, design
software or Internet applets.
Quadrilaterals and Special
Polygons
Unit Test
How can mathematical modeling and
the properties of polygons be used to
answer questions that arise in real-life?
Investigations
Projects
Know and apply properties of
geometric figures to solve real-world
and mathematical problems and to
logically justify results in geometry.
Solve real-world and mathematical
geometric problems using algebraic
methods.
6
Circle Properties and
Proofs
Calculate measurements of plane and
solid geometric figures; know that
physical measurements depend on the
choice of a unit and that they are
approximations.
Use properties of polygons—
including quadrilaterals and regular
polygons—to define them, classify
them, solve problems and logically
justify results.
Use algebra to solve geometric
problems unrelated to coordinate
geometry, such as solving for an
unknown length in a figure
involving similar triangles, or using
the Pythagorean Theorem to obtain
a quadratic equation for a length in
a geometric figure.
Compose and decompose two- and
three-dimensional figures; use
decomposition to determine the
perimeter, area, surface area and
volume of various figures.
Willmar Public Schools
Curriculum Map
Construct logical arguments, based on
axioms, definitions and theorems, to
prove theorems and other results in
geometry.
Know and apply properties of
geometric figures to solve real-world
and mathematical problems and to
logically justify results in geometry.
Solve real-world and mathematical
geometric problems using algebraic
methods.
Construct logical arguments and
write proofs of theorems and other
results in geometry, including
proofs by contradiction. Express
proofs in a form that clearly
justifies the reasoning, such as twocolumn proofs, paragraph proofs,
flow charts or illustrations.
Use technology tools to examine
theorems, make and test
conjectures, perform constructions
and develop mathematical
reasoning skills in multi-step
problems. The tools may include
compass and straight edge,
dynamic geometry software, design
software or Internet applets.
Know and apply properties of a
circle to solve problems and
logically justify results.
Unit Test
How can the properties of circles help
calculate the areas of oddly-shaped
curved figures?
Investigations
Know the equation for the graph of
a circle with radius r and center (h,
k), (x – h)2 + (y – k)2 = r2, and
justify this equation using the
Pythagorean Theorem and
properties of translations.
Use coordinate geometry to
represent and analyze line segments
and polygons, including
determining lengths, midpoints and
slopes of line segments.
7
Coordinate
Geometry
Solve real-world and mathematical
geometric problems using algebraic
methods.
Use algebra to solve geometric
problems unrelated to coordinate
geometry, such as solving for an
unknown length in a figure
involving similar triangles, or using
the Pythagorean Theorem to obtain
a quadratic equation for a length in
a geometric figure.
Three straight bike trails in a park you
are designing intersect to form a
triangle. Where would be the best
place to build a first aid station so that it
is equidistant from the three bike trails?
Unit Test
Project
Willmar Public Schools
Curriculum Map
8
9
The Pythagorean Theorem
Know and apply properties of
geometric figures to solve real-world
and mathematical problems and to
logically justify results in geometry.
Understand and apply the fact that
the effect of a scale factor k on
length, area and volume is to
multiply each by k, k2 and k3,
respectively.
Know and apply properties of
geometric figures to solve real-world
and mathematical problems and to
logically justify results in geometry.
Know and apply properties of
congruent and similar figures to
solve problems and logically justify
results.
Transformations,
Similarity, and Proportion
Trigonometry
Know and apply properties of right
triangles, including properties of
45-45-90 and 30-60-90 triangles, to
solve problems and logically justify
results.
Calculate measurements of plane and
solid geometric figures; know that
physical measurements depend on the
choice of a unit and that they are
approximations.
How can the Pythagorean theorem be
used to find the distance between two
locations?
Solve real-world and mathematical
geometric problems using algebraic
methods.
Unit Test
Map Activity
What effect does doubling the length,
width, and/or height have on a straight
line, a square, or a cube?
How can one find the height of an
object without measuring it directly?
Understand how the properties of
similar right triangles allow the
trigonometric ratios to be defined,
and determine the sine, cosine and
tangent of an acute angle in a right
triangle.
Solve real-world and mathematical
geometric problems using algebraic
methods.
10
Apply the Pythagorean Theorem
and its converse to solve problems
and logically justify results.
Unit Test
Project
Use numeric, graphic and symbolic
representations of transformations
in two dimensions, such as
reflections, translations, scale
changes and rotations about the
origin by multiples of 90˚, to solve
problems involving figures on a
coordinate grid.
Apply the trigonometric ratios sine,
cosine and tangent to solve
problems, such as determining
lengths and areas in right triangles
and in figures that can be
decomposed into right triangles.
Know how to use calculators,
tables or other technology to
What is the relationship between the
sides of similar triangles?
How can the trigonometric ratios be
used to find the missing angles and/or
sides in right triangles?
Unit Test
Willmar Public Schools
Curriculum Map
evaluate trigonometric ratios.
Use calculators, tables or other
technologies in connection with the
trigonometric ratios to find angle
measures in right triangles in
various contexts.
11
Calculate measurements of plane and
solid geometric figures; know that
physical measurements depend on the
choice of a unit and that they are
approximations.
Compose and decompose two- and
three-dimensional figures; use
decomposition to determine the
perimeter, area, surface area and
volume of various figures.
Area
Calculate measurements of plane and
solid geometric figures; know that
physical measurements depend on the
choice of a unit and that they are
approximations.
12
Volume
Determine the surface area and
volume of pyramids, cones and
spheres. Use measuring devices or
formulas as appropriate.
How can I determine how much
material I will need to cover the
surfaces in my home?
Unit Test
Project
Determine the surface area and
volume of pyramids, cones and
spheres. Use measuring devices or
formulas as appropriate.
Understand that quantities
associated with physical
measurements must be assigned
units; apply such units correctly in
expressions, equations and problem
solutions that involve
measurements; and convert
between measurement systems.
How much “stuff” can I fit inside
different closed solids?
Unit Test
Project