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Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Milwaukee Public Schools
PACING GUIDE for
High School GEOMETRY
Foundation Level
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Dear Milwaukee Public School Geometry Teachers:
With the adoption of a new textbook series for the Milwaukee Public Schools in the Spring of 2008, it became necessary to design pacing guides for
these textbooks. The following is Draft One of the Pacing Guide for Discovering Geometry. Just like the headers to this document state, this is a
draft. Our district is relying on your good judgment and teaching skills to provide us with insights, suggestions, modifications, corrections, and plans
with the intent of incorporating these into Draft 2. Members of the Key Curriculum Press have told us that a first year pacing guide must be very
tentative – we will not have a clear picture of just what our students can accomplish and what our teachers can complete until the full one year
teaching and learning cycle is complete.
Therefore, we not only suggest, but highly recommend, that you use this as a guide – modify to fit the needs of your style of teaching and the needs
of your students. This means that you might have to add more time to one section of the text and then reduce time spent on another section. This
also gives you the ability to eliminate a section or sections because time is not available. However, the Discovering Series is a series that builds on
investigations – please make sure these are a definite part of your lesson plans, and teaching and learning in your classrooms. Usually there is more
than one investigation in a section – at least one should be tried. We also hope that you are using the cooperative learning skills and strategies so
important in this math curriculum.
Many of our schools are on different school calendars – some are blocked; some start earlier in August; some are year-round schools. This pacing
guide is set for 36 weeks. Time is allotted for semester exams – review and assessment. All explorations and algebra skills activities are also
included. These are places where adjustments can be made. Six chapters are covered in Semester One and seven chapters are covered in Semester
Two. We have heard that some teachers may be struggling to cover the first six chapters in one semester – especially in the blocked schools. This is
OK! The first semester is loaded with vocabulary, concepts and skills that will be needed for semester two – taking more time on the first five
chapters may prove beneficial when you approach the last chapters of the book. From Week 28 on, fewer sections are covered each week and more
time is allowed for chapter assessments. These may be the weeks during which you can catch-up. If time does run out, we suggest that perhaps
Chapter 13, Geometry as a Mathematical System, be one that you eliminate, at least for this first year.
Please remember that all of the above are suggestions – the ultimate decision of how to cover the material in the text and to follow the pacing guide
resides with the classroom teacher. We ask that all chapters are covered (with the possibility that you will not cover Chapter 13 and some sections
may be eliminated). Your reflections and feedback will periodically be requested by your MTL and/or department chair to share with our Math
Specialists, FTF Math Instructional Coach, Key Curriculum Press, and others who are part of the mathematics leadership in our district. Thank you in
advance for your participation.
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Drafted by: Sherrie K. Akinsanya, FTF Mathematics Instructional Coahc
Time
(week)
Lesson/
Activity
MPS Learning Targets
LT1: The student will communicate mathematical ideas
and reasoning using the vocabulary of mathematics in a
variety of ways, e.g. using words, numbers, symbols,
pictures, charts, tables, diagrams, graphs, and models.
Chapter 0
Geometric
Art
CHAPTER 0
LT2*: Use a variety of mathematical and
technological tools to determine measurements
0.1 Geometry directly (ruler, protractor, compass) and to check
reasonableness of answers. When students apply
in Art and
Nature
technology, e.g. computers and graphing
calculators, they will interpret results
0.2 Line
appropriately.
Designs
0.3 Circle
Designs
0.4 Op Art
0.6 Islamic
Tiles
LT16: Apply transformations and their
compositions to analyze geometric figures and
create designs.
Benchmarks Embedded in
Chapter Lessons
District
Model
CABS
CHAPTER 0
2.2* Measure lengths using a
metric/English ruler (to the
appropriate degree of tenths of cm
and sixteenths of an inch).
16.4 Determine whether a figure has
point symmetry, line symmetry or
neither.
16.5 Identify tessellations in art,
construction and nature.
16.6 Design a real-world application
of transformations and communicate
the process used in the design.
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Suggested
vocabulary
Symmetry,
axis of
symmetry,
compass,
straightedge
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Feedback
Reflection
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Time
Lesson/
Activity
Chapter 1
Introducing
Geometry
1.1 Building
Blocks of
Geometry
Algebra
Review
Midpoint
1.2 Poolroom
Math
1.3 Widgets
MPS Learning Targets
LT1: The student will communicate mathematical ideas
and reasoning using the vocabulary of mathematics in a
variety of ways, e.g. using words, numbers, symbols,
pictures, charts, tables, diagrams, graphs, and models.
Benchmarks Embedded in
Chapter Lessons
District
Suggested
Model CABS vocabulary
CHAPTER 1
CHAPTER 1
LT2*: Use a variety of mathematical and
2.3* Measure angles using a
technological tools to determine measurements
protractor (to appropriate degrees).
directly (ruler, protractor, compass) and to check
reasonableness of answers. When students apply
technology, e.g. computers and graphing
calculators, they will interpret results
appropriately.
LT8*: Use the two-dimensional coordinate plane
to analyze, generalize and represent algebraic
properties in mathematical and real-world
applications.
LT12*: Identify and apply the definitions related
to lines and angles in order to use them to solve
problems and perform basic geometric
constructions.
8.3* Use a coordinate plane and
the distance and midpoint formulas
to determine lengths and midpoints
of segments.
12.1* Identify and name points,
lines, rays, segments and planes.
12.2* Name and classify angles
and line segments.
12.3* Identify and use angle-pair
relationships in mathematical and
real-world applications.
12.4* Classify and communicate
the differences between pairs of
lines.
12.6 * Prove statements about
segments, lines, angles and angle
relationships by using properties,
postulates, and theorems.
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Point, line,
plane, collinear,
coplanar, line
segment,
endpoint,
congruent,
midpoint,
protractor,
angle, vertex,
counterexample
, right angle,
acute angle,
obtuse angle,
complementary
angles,
supplementary
angles, vertical
angles, linear
pair, dimension
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Feedback
and
Reflection
Week 2
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Time
Lesson/
Activity
Chapter 1:
continued
MPS Learning Targets
LT1: The student will communicate mathematical ideas
and reasoning using the vocabulary of mathematics in a
variety of ways, e.g. using words, numbers, symbols,
pictures, charts, tables, diagrams, graphs, and models.
LT13*: Classify and describe polygons and 3dimensional figures.
Assessment
1.4 Polygons
1.5 Triangles
1.6 Special
Quadrilaterals
Assessment
1.7 Circles
1.8 3-D
Geometry
LT14: Use and analyze geometric relationships
and properties of polygons and 3-dimensional
figures to solve mathematical and real-world
applications.
LT15: Apply the basic theorems of congruence
and similarity to solve mathematical and realworld problems.
LT18: Use perimeter, area, volume, and surface
area to solve real-world applications.
Benchmarks Embedded in
Chapter Lessons
District
Model
CABS
13.1* Classify polygons by sides and
angles.
13.2* Classify triangles by sides and
angles.
13.3* Classify quadrilaterals by their
parts.
13.5* Describe properties of
quadrilaterals.
14.7 Make a two-dimensional
representation of a 3-dimensional object.
15.1 Identify congruent and similar
polygons.
18.8 Make a model of a 3-dimensional
figure from a 2-dimensional drawing.
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Suggested
vocabulary
Polygon, right
triangle, acute
triangle,
isosceles
triangle,
scalene
triangle,
equilateral
triangle, obtuse
triangle,
quadrilateral,
circle, chord,
diameter,
radius, tangent,
arcs in circles
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Feedback
Reflection
Week 3
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Time
Lesson/
Activity
MPS Learning Targets
LT1: The student will communicate mathematical ideas
and reasoning using the vocabulary of mathematics in a
variety of ways, e.g. using words, numbers, symbols,
pictures, charts, tables, diagrams, graphs, and models.
Benchmarks Embedded in
Chapter Lessons
District
Model
CABS
Chapter1:
continued
1.9 Pictures
Chapter 1: Learning Targets Covered:
LT12 – Lines and Angles
LT 13 – Classify Polygons
Exploration: LT 14 – Analyze Polygons
Geometric LT 15 – Congruence and Similarity
Probability LT 18 – Geometric Formulas
Review
Assessment
Chapter 2
Reasoning
in
Geometry
2.1
Inductive
Reasoning
2.2 Nth
Term
CHAPTER 2
CHAPTER 2
LT10*. Evaluate information, analyze
patterns ,identify relationships, and represent
them using algebraic expressions, functions,
and equations, e.g., linear, exponential and
quadratic, recognizing that a family of
functions can model a variety of real-world
applications.
10.7* Identify a pattern in a
numerical sequence and complete the
pattern to a given term.
LT19: Use increased rigor in justifying
conclusions to mathematical and real-world
applications and in verifying the truth of
geometric theorems.
19.3 Write conjectures using
characteristics, postulates and
theorems of geometric figures.
19.4 Use and interpret Venn
diagrams.
19.6 Use valid forms of deductive
reasoning, including the laws of
detachment and syllogism, to prove
conjectures.
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Suggested
vocabulary
Venn diagram,
probability,
justification,
inductive logic,
deductive logic,
line of best fit,
conjecture,
pattern,
strategy,
randomness,
reasonableness,
representations
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Reflection
and
Feedback:
Week 4
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Time
Lesson/
Activity
MPS Learning Targets
LT1: The student will communicate mathematical ideas
and reasoning using the vocabulary of mathematics in a
variety of ways, e.g. using words, numbers, symbols,
pictures, charts, tables, diagrams, graphs, and models.
Benchmarks Embedded in
Chapter Lessons
District
Model CABS
Suggested
vocabulary
Transversal
Chapter 2:
continued
LT10*. Evaluate information, analyze patterns,
10.7* Identify a pattern in a
identify relationships, and represent them using
numerical sequence and complete
2.3 Modeling
algebraic expressions, functions, and equations,
the pattern to a given term.
e.g.,
linear,
exponential
and
quadratic,
recognizing
Assessment
that a family of functions can model a variety of
2.4
realDeductive
world applications.
Logic
12.3* Identify and use angle-pair
LT12*: Identify and apply the definitions related to relationships in mathematical and
2.5 Angle
Relationships lines and angles in order to use them to solve
real-world applications.
problems and perform basic geometric
12.5* Classify the types of angles
2.6 Angles
constructions.
formed by two lines and a
and Parallel
transversal.
Lines
14.6 Solve problems involving
real-world applications using the
LT14: Use and analyze geometric relationships and relationships between pairs of
properties of polygons and 3-dimensional figures to angles.
solve mathematical and real-world applications.
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Feedback
and
Reflection:
Week 5
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Time
Lesson/
Activity
Chapter 2:
continued
Algebra
Review:
Slope
Review
Assessment
MPS Learning Targets
LT1: The student will communicate mathematical ideas
and reasoning using the vocabulary of mathematics in a
variety of ways, e.g. using words, numbers, symbols,
pictures, charts, tables, diagrams, graphs, and models.
Benchmarks Embedded in
Chapter Lessons
District
Model
CABS
Slope,
coordinate
system, rate of
change,
bisector,
perpendicular
Learning Targets Covered: Chapter 2
LT 12 – Lines and Angles
LT 14 – Analyze Polygons
LT 19 – Proofs
CHAPTER 3
LT2*: Use a variety of mathematical and
technological tools to determine measurements
directly (ruler, protractor, compass) and to
check reasonableness of answers. When students
3.1
apply technology, e.g. computers and graphing
Duplicating
calculators, they will interpret results
Segments and
appropriately.
Angles
CHAPTER 3
2.2* Measure lengths using a
metric/English ruler (to the
appropriate degree of tenths of cm
and sixteenths of an inch).
2.3* Measure angles using a
protractor (to appropriate degrees).
3.2 Construct
Perpendicular.
Bisectors
12.7* Perform basic geometric
constructions.
Chapter 3:
Using Tools
of
Geometry
LT12*: Identify and apply the definitions
related to lines and angles in order to use them
to solve problems and perform basic geometric
constructions.
Suggested
vocabulary
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Feedback
Reflection
Week 6
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Time
Lesson/
Activity
MPS Learning Targets
LT1: The student will communicate mathematical ideas and
reasoning using the vocabulary of mathematics in a variety of
ways, e.g. using words, numbers, symbols, pictures, charts,
tables, diagrams, graphs, and models.
Benchmarks Embedded in
Chapter Lessons
District
Model CABS
Chapter 3:
continued
LT2*: Use a variety of mathematical and
2.3* Measure angles using a
technological tools to determine measurements
protractor (to appropriate
directly (ruler, protractor, compass) and to check
degrees).
3.3 Construct
reasonableness
of
answers.
When
students
apply
Perpendicular.
technology, e.g. computers and graphing calculators,
Lines
they will interpret results appropriately.
Assessment
LT12*: Identify and apply the definitions related to
12.7* Perform basic geometric
lines and angles in order to use them to solve problem constructions.
and perform basic geometric constructions.
3.4 Angle
Bisectors
3.5 Parallel
Lines
Algebra
Review
Slopes of
perpendicular
and parallel
lines
LT16: Apply transformations and their compositions
to analyze geometric figures and create designs.
16.1 Plot and label geometric
figures using a coordinate
plane.
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Suggested
vocabulary
Altitude, angle
bisector,
parallel
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Feedback
Reflection
Week 7
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Time
Lesson/
Activity
MPS Learning Targets
LT1: The student will communicate mathematical ideas and
reasoning using the vocabulary of mathematics in a variety of
ways, e.g. using words, numbers, symbols, pictures, charts,
tables, diagrams, graphs, and models.
Benchmarks Embedded in
Chapter Lessons
Chapter 3:
continued
LT12*: Identify and apply the definitions
related to lines and angles in order to use
them to solve problems and perform basic
3.6
geometric
constructions.
Constructions
12.7* Perform basic geometric
constructions.
LT18: Use perimeter, area, volume,
and surface area to solve real-world
Exploration: applications.
18. 7 Solve real-world
application problems using scale
drawings, perspective drawings,
blueprints, or computer
drawings as models of 3dimensional objects.
Assessment
Perspective
Drawing
Learning Targets Covered: Chapter 3
LT 12 – Lines and Angles
3.7
LT 16 – Transformations
Concurrency LT 18 – Geometric Formulas
District
Model
CABS
3.8 Centroid
Review
Assessment
Feedback
Reflection
Week 8
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Suggested
vocabulary
Perspective,
concurrency,
circumcener,
incenter,
centroid,
orthocenter
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Time
Lesson/
Activity
Chapter 4:
Discovering
and Proving
Triangle
Properties
MPS Learning Targets
LT1: The student will communicate mathematical ideas and
reasoning using the vocabulary of mathematics in a variety of
ways, e.g. using words, numbers, symbols, pictures, charts,
tables, diagrams, graphs, and models.
Benchmarks Embedded in
Chapter Lessons
CHAPTER 4
LT12*: Identify and apply the definitions related to
lines and angles in order to use them to solve
problems and perform basic geometric
constructions.
CHAPTER 4
12.7* Perform basic geometric
constructions.
LT14: Use and analyze geometric relationships and
properties of polygons and 3-dimensional figures to
4.2 Isosceles
solve mathematical and real-world applications.
Triangle
14.1 Determine if lengths of
14.2 three segments can form a
triangle.
14.3 Determine the range in
which the length of the third
side of a triangle must be given
the lengths of two sides.
4.1 Triangle
Sum
Algebra
Review
Solving
Equations
4.3 Triangle
Inequalities
Assessment
LT18: Use perimeter, area, volume and surface area
to solve real-world applications.
District
Model CABS
18.5 Compare perimeters and
areas of similar 2-dimensional
figures using proportions.
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Suggested
vocabulary
Equation
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Feedback
Reflection
Week 9
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Time
Lesson/
Activity
Chapter 4,
continued
4.4
Congruence
Shortcuts
MPS Learning Targets
LT1: The student will communicate mathematical ideas and
reasoning using the vocabulary of mathematics in a variety
of ways, e.g. using words, numbers, symbols, pictures, charts,
tables, diagrams, graphs, and models.
LT15: Apply the basic theorems of
congruence and similarity to solve
mathematical and real-world problems.
4.5
Congruence
Shortcuts
4.6
Congruence
Shortcuts
4.7 Flow
Charts
4.8 Extra
Triangle
Proofs
LT19: Use increased rigor in justifying
conclusions to mathematical and real-world
applications and in verifying the truth of geometric
theorems.
Benchmarks Embedded in
Chapter Lessons
District
Model
CABS
15.2 Justify congruency or
similarity of polygons by using
formal and informal proofs.
15.3 Prove two triangles are
congruent using coordinate
methods (e.g. distance and
slope formulas).
15.4 Solve real-world
applications using the
properties of congruency and
similarity.
19.7 Select and use various
types of reasoning and
methods to prove
characteristics and
properties of algebraic models
and geometric figures.
Feedback
and
Reflection:
Week 10
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Suggested
vocabulary
Inequality
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Time
Lesson/
Activity
Review
Assessment
Algebra
Review:
Systems of
Equations
MPS Learning Targets
LT1: The student will communicate mathematical ideas and
reasoning using the vocabulary of mathematics in a variety
of ways, e.g. using words, numbers, symbols, pictures, charts,
tables, diagrams, graphs, and models.
Learning Targets Covered in Chapter 4:
Benchmarks Embedded in
Chapter Lessons
District
Model
CABS
LT12 – Lines and Angles
LT 14 – Analyze Polygons
LT 15 – Congruency and Similarity
LT 18 – Geometric Formulas
LT 19 - Proofs
WKCE
Prep
Feedback
Reflection
Week 11
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Suggested
vocabulary
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Time
Lesson/
Activity
MPS Learning Targets
LT1: The student will communicate mathematical ideas and
reasoning using the vocabulary of mathematics in a variety
of ways, e.g. using words, numbers, symbols, pictures, charts,
tables, diagrams, graphs, and models.
Benchmarks Embedded in
Chapter Lessons
District
Model
CABS
WKCE
prep
NOTE: MPS High Schools choose their own dates
within a specific window for the WKCE test.
Two weeks are given in this pacing guide for
WKCE test
WKCE review. Each school will decide who
many days for review they want to take.
The remainder of the days can be used to
catch-up on the pacing guide if four
chapters have NOT been covered and/or to
start on Chapter 5.
Feedback
Reflection
Week 12
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Suggested
vocabulary
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Time
Lesson/
Activity
MPS Learning Targets
LT1: The student will communicate mathematical ideas and
reasoning using the vocabulary of mathematics in a variety
of ways, e.g. using words, numbers, symbols, pictures, charts,
tables, diagrams, graphs, and models.
Chapter 5:
CHAPTER 5
Discovering
and Proving LT12. Identify and apply the definitions related
to lines and angles in order to use them to
Polygon
Properties solve problems and perform basic geometric
constructions.
5.1 Polygon
Sum
Conjecture LT13*: Classify and describe polygons and 3dimensional figures.
5.2 Exterior
Angles of
LT14: Use and analyze geometric relationships
Polygons
and properties of polygons and 3-dimensional
figures to solve mathematical and real-world
Assessment applications.
1
Benchmarks Embedded in
Chapter Lessons
District
Model
CABS
CHAPTER 5
12.6 Prove statements about
segments, lines, angles and
angle relationships by using
properties, postulates, and
theorems.
13.1* Classify polygons by
sides and angles.
14.4 Find the measures of each
interior and exterior angle of a
regular polygon.
14.5Find the sum of the
measures of the interior and
exterior angles of a convex
polygon.
14.6 Solve problems involving
real-world applications using
the relationships between pairs
of angles..
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Suggested
vocabulary
Exterior angles
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Feedback
Reflection:
Week 13
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Time
Lesson/ Activity
MPS Learning Targets
LT1: The student will communicate mathematical ideas
and reasoning using the vocabulary of mathematics in a
variety of ways, e.g. using words, numbers, symbols,
pictures, charts, tables, diagrams, graphs, and models.
LT2. Use a variety of mathematical and
technological tools to determine measurements
directly (ruler, protractor, compass) and to
5.3 Kite and
check reasonableness of answers. When
Trapezoid
students apply technology, e.g. computers and
Properties
graphing calculators, they will interpret results
appropriately.
5.4 Properties of LT12*: Identify and apply the definitions related
Midsegments
to lines and angles in order to use them to solve
problems and perform basic geometric
5.5 Properties of constructions.
Parallelograms LT13*: Classify and describe polygons and 3dimensional figures.
ASSESSMENT
Chapter 5,
continued
5.6 Special
Parallelograms
LT19* Use increased rigor in justifying
Algebra
conclusions to mathematical and real-world
Review:
Linear equations applications and in verifying the truth of
geometric theorems.
LT9. Model and solve mathematical and realworld applications that can be represented by
linear equations and systems of equations, linear
inequalities, and quadratic equations.
Benchmarks Embedded in
Chapter Lessons
District
Model
CABS
2.2* Measure lengths using a
metric/English ruler (to the
appropriate degree of tenths of
cm and sixteenths of an inch).
2.3* Measure angles using a
protractor (to appropriate
degrees).
12.7* Perform basic geometric
constructions.
13.4 Describe properties of a
given polygon.
13.5 Describe properties of
quadrilaterals.
13.6 Describe properties of
other polygons.
19.3 Write conjectures using
characteristics, postulates and
theorems of geometric figures
9.8 Solve linear equations,
inequalities, quadratic
equations and systems of linear
equations related to real-world
applications.
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Suggested
vocabulary
Midsegment,
Intercept
Perpendicular
Parallel
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Feedback
and
Reflection:
Week 14
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Time
Lesson/ Activity
MPS Learning Targets
LT1: The student will communicate mathematical ideas
and reasoning using the vocabulary of mathematics in a
variety of ways, e.g. using words, numbers, symbols,
pictures, charts, tables, diagrams, graphs, and models.
Chapter 5:
continued
5.7 Proving
Quadrilateral
Properties
Review
Chapter
Assessment
Benchmarks Embedded in
Chapter Lessons
LT19. Use increased rigor in justifying
conclusions to mathematical and real-world
applications and in verifying the truth of
geometric theorems.
19.6 Use valid forms of
inductive and deductive
reasoning to draw
conclusions and make
conjectures.
Learning Targets Covered in Chapter 5:
LT12 Lines and Angles
LT13 – Classify Polygons
LT14 – Analyze Polygons
LT19 - Proofs
19.7 Select and use various
types of reasoning and
methods to prove
characteristics and properties
of algebraic models
and geometric figures.
District
Model
CABS
Feedback
Reflection
Week 15
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Suggested
vocabulary
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Time
Lesson/ Activity
MPS Learning Targets
LT1: The student will communicate mathematical ideas
and reasoning using the vocabulary of mathematics in a
variety of ways, e.g. using words, numbers, symbols,
pictures, charts, tables, diagrams, graphs, and models.
CHAPTER 6
Discovering and
Proving Circle
Properties
6.1 Tangent
6.2 Chord
Assessment
6.3 Arcs/Angles
Benchmarks Embedded in
Chapter Lessons
CHAPTER 6
CHAPTER 6
LT12. Identify and apply the definitions related
to lines and angles in order to use them to
solve problems and perform basic geometric
constructions.
LT17: Use properties of circles to solve
mathematical and real-world applications.
12.7 Perform basic geometric
constructions.
District
Model
CABS
17.2 Determine measures of
inscribed and central angles.
17.3 Determine arc lengths
and areas of sectors of circles.
17.4 Write and solve
equations to find the measures
of angles formed by chords,
secants, and tangents.
17.5 Calculate lengths of
chord, tangent and secant
segments.
Feedback
Reflection
Week 16
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Suggested
vocabulary
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Time
Lesson/ Activity
MPS Learning Targets
LT1: The student will communicate mathematical ideas
and reasoning using the vocabulary of mathematics in a
variety of ways, e.g. using words, numbers, symbols,
pictures, charts, tables, diagrams, graphs, and models.
Chapter 6,
continued
LT17: Use properties of circles to
solve mathematical and real-world
applications.
6.4 Proving Circle
Conjectures
6.5 Circles and
Diameter
LT19: Use increased rigor in justifying
conclusions to mathematical and real-world
applications and in verifying the truth of
geometric theorems.
Benchmarks Embedded in
Chapter Lessons
District
Model
CABS
17.1 Determine circumference
and area of a circle.
17.5 Calculate lengths of
chord, tangent and secant
segments.
17.7 Solve real-world
applications of circles.
19. 7 Use valid forms of
inductive and deductive
reasoning to draw conclusions
and make conjectures.
19.8 Select and use various
types of reasoning and methods
to prove characteristics and
properties of algebraic models
and geometric figures.
Feedback
Reflection
Week 17
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Suggested
vocabulary
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Time
Lesson/ Activity
MPS Learning Targets
LT1: The student will communicate mathematical ideas
and reasoning using the vocabulary of mathematics in a
variety of ways, e.g. using words, numbers, symbols,
pictures, charts, tables, diagrams, graphs, and models.
Chapter 6,
continued
LT17: Use properties of circles to
solve mathematical and real-world
applications.
6.6 Around the
World
Algebra Review:
Systems of
Equations
6.7 Arc length
Review
Chapter
Assessment
Benchmarks Embedded in
Chapter Lessons
District
Model
CABS
17.1 Determine circumference
and area of a circle.
17.3 Determine arc lengths
and areas of sectors of circles.
Learning Targets Covered in Chapter 6:
LT12 – Lines and Angles
LT17 – Circles
LT19 - Proofs
Feedback
Reflection
Week 18
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Suggested
vocabulary
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Time
Lesson/ Activity
MPS Learning Targets
LT1: The student will communicate mathematical ideas
and reasoning using the vocabulary of mathematics in
a variety of ways, e.g. using words, numbers, symbols,
pictures, charts, tables, diagrams, graphs, and models.
Review for Final
Exam
Final Exam
Semester One – MA 301
LT2. Use a variety of mathematical and
technological tools to determine measurements
directly (ruler, protractor, compass) and to
check reasonableness of answers. When students
apply technology, e.g. computers and graphing
calculators, they will interpret results
appropriately.
LT12*: Identify and apply the definitions
related to lines and angles in order to use
them to solve problems and perform basic
geometric constructions.
LT13*: Classify and describe polygons
and 3-dimensional figures.
LT14: Use and analyze geometric
relationships and properties of polygons
and 3-dimensional figures to solve
mathematical and real-world applications.
LT15: Apply the basic theorems of
congruence and similarity to solve
mathematical and real-world problems.
LT17: Use properties of circles to
solve mathematical and real-world
applications.
MPS Learning Targets
Supported in Semester One:
MA301
District
Model
CABS
Semester One – MA 301
LT16 Apply transformations
and their compositions to
analyze geometric figures and
create designs.
LT18 Use perimeter, area,
volume and surface area to
solve real-world applications.
LT19 Use increased rigor in
justifying conclusions to
mathematical and real-world
applications and in verifying the
truth of geometric
theorems.
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Suggested
Vocabulary
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Feedback
Reflection:
Semester 1
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Time
Lesson/
Activity
Chapter 7:
Transformations
and Tesselations
7.1
Transformations/
Symmetry
7.2 Properties of
Isometrics
7.3 Compositions of
Transformations
Assessment
MPS Learning Targets
LT1: The student will communicate mathematical ideas
and reasoning using the vocabulary of mathematics in a
variety of ways, e.g. using words, numbers, symbols,
pictures, charts, tables, diagrams, graphs, and models.
CHAPTER 7
LT16: Apply transformations and their
compositions to analyze geometric figures and
create designs.
Benchmarks Embedded in
Chapter Lessons
District
Model
CABS
CHAPTER 7
16.2 Perform transformations
using triangles in a coordinate
plane.
16.3 Identify, given an image
and pre-image, the
transformation that has taken
place as a reflection, rotation, or
translation.
16.4 Determine whether a
figure has point symmetry, line
symmetry or neither.
16.5 Design a real-world
application of transformations
and communicate the process
used in the design.
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Suggested
vocabulary
Central angles,
inscribed
angles
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Feedback
and
Reflection
Week 20
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Time
Lesson/ Activity
MPS Learning Targets
LT1: The student will communicate mathematical ideas
and reasoning using the vocabulary of mathematics in a
variety of ways, e.g. using words, numbers, symbols,
pictures, charts, tables, diagrams, graphs, and models.
Chapter 7,
continued
7.4 Tessellations of
regular polygons
7.5 Tessellations of
non-regular polygons
Assessment
7.6 Tessellations
with translations
7.7 Tessellations
with rotations
LT16: Apply transformations and their
compositions to analyze geometric figures and
create designs.
Benchmarks Embedded in
Chapter Lessons
District
Model
CABS
16.6 Identify tessellations in
art, construction and nature.
16.7 Design a real-world
application of transformations
and communicate the process
used in the design.
Exploration:
Tessellating with
Conway Criterion
Review
Chapter Assessment
Learning Target covered in Chapter 7:
LT 16: Transformations
Feedback
and
Reflection:
Week 21
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Suggested
vocabulary
Systems of
equations
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Time
Lesson/ Activity
MPS Learning Targets
LT1: The student will communicate mathematical ideas
and reasoning using the vocabulary of mathematics in a
variety of ways, e.g. using words, numbers, symbols,
pictures, charts, tables, diagrams, graphs, and models.
Chapter 8: Area
8.1 Area of
rectangles and
parallelograms
8.2 Area of kites,
trapezoids and
triangles
Assessment
CHAPTER 8
LT18: Use perimeter, area, volume and
surface area to solve real-world applications.
Benchmarks Embedded in
Chapter Lesson
District
Model
CABS
CHAPTER 8
18.1 Determine the
perimeter/circumference and
area of quadrilaterals,
triangles and other
polygons, using real-world
applications.
8.3 Area Problems
Algebra Review:
Products, factors,
quadratic equations
8.4 Area of regular
polygons
Exploration: Pick’s
formula for area
Feedback
and
Reflection:
Week 22
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Suggested
vocabulary
Area, kite
trapezoid
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Time
Lesson/ Activity
MPS Learning Targets
LT1: The student will communicate mathematical ideas
and reasoning using the vocabulary of mathematics in a
variety of ways, e.g. using words, numbers, symbols,
pictures, charts, tables, diagrams, graphs, and models.
Benchmarks Embedded in
Chapter Lessons
District
Model
CABS
Sector of a
circle, segment
of a circle,
annulus
Chapter 8,
continued
8.5 Area of circles
Assessment
LT17: Use properties of circles to solve
mathematical and real-world applications.
8.6 Any Way you
Slice it: Area of
Circle segments
Exploration:
Geometric
Probability
LT18: Use perimeter, area, volume and surface
area to solve real-world applications.
8.7 Surface Area
Chapter 8
Assessment
Suggested
vocabulary
Learning Targets covered in Chapter 8:
LT 17: Circles
LT 18: Formulas
17.1 Determine
circumference and area of a
circle.
17.3 Determine arc lengths
and areas of sectors of
circles.
18.1 Determine the
perimeter/circumference and
area of quadrilaterals,
triangles and other polygons,
using real-world applications.
Feedback
and
Reflection:
Week 23
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Time
Lesson/ Activity
MPS Learning Targets
LT1: The student will communicate mathematical ideas
and reasoning using the vocabulary of mathematics in a
variety of ways, e.g. using words, numbers, symbols,
pictures, charts, tables, diagrams, graphs, and models.
Chapter 9:
Pythagorean
Theorem
9.1 The Theorem of
Pythagoras
9.2 The Converse of
the Pythagorean
Theorem
Algebra Review:
Radical Expressions
CHAPTER 9
LT11: Explain and use the Pythagorean
Theorem and right triangle trigonometry as
models to solve a variety of right triangle
problems in real-world applications.
Benchmarks Embedded in
Chapter Lessons
District
Model
CABS
CHAPTER 9
11.1 Apply the Pythagorean
Theorem to find side lengths
of a right triangle in
mathematical and real-world
applications.
11.2 Determine if a triangle
is right, obtuse or acute by
using the Pythagorean
Theorem.
9.3 Two Special Right
Triangles
9.4 Story Problems
Feedback
and
Reflection:
Week 24
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Suggested
vocabulary
Surface area,
Pythagorean
Formula
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Time
Lesson/ Activity
MPS Learning Targets
LT1: The student will communicate mathematical ideas
and reasoning using the vocabulary of mathematics in a
variety of ways, e.g. using words, numbers, symbols,
pictures, charts, tables, diagrams, graphs, and models.
Chapter 9,
continued
LT12: Classify and describe polygons and
3-dimensional figures.
Assessment
9.5 Distance in a
Coordinate Plane
Exploration: Ladder
Climb
9.6 Circles and the
Pythagorean Theorem
LT11: Explain and use the Pythagorean
Theorem and right triangle trigonometry as
models to solve a variety of right triangle
problems in real-world applications.
LT18: Use appropriate measurements such as
perimeter, area, volume, surface area and
degrees to solve real-world applications.
Review
Chapter 9 Assessment
Benchmarks Embedded in
Chapter Lessons
District
Model CABS
12.7 Classify geometric
properties by using the
coordinate system (e.g. distance
formula, slope).
11.1 Apply the Pythagorean
Theorem to find side lengths of a
right triangle in mathematical
and real-world applications.
18.4 Solve real-world
application problems for
missing lengths in special right
triangles.
Learning Targets covered in Chapter 9:
LT 11: Pythagorean Theorem
LT 12: Classify polygons
LT 18: Formulas
Feedback
and
Reflection:
Week 25
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Suggested
vocabulary
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Time
Lesson/ Activity
MPS Learning Targets
LT1: The student will communicate mathematical ideas
and reasoning using the vocabulary of mathematics in a
variety of ways, e.g. using words, numbers, symbols,
pictures, charts, tables, diagrams, graphs, and models.
Chapter 10:
Volume
10.1 Geometry of solids
CHAPTER 10
Benchmarks Embedded in
Chapter Lessons
CHAPTER 10
LT18: Use perimeter, area, volume and surface
area to solve real-world applications.
18.2 Determine the volume
and lateral area and total
surface area of spheres,
prisms, pyramids, cylinders
and cones.
18.3 Use volume, lateral area
and surface area of 3dimensional figures to solve
real-world application
problems.
18.8 Make a model of a 3dimensional figure from a 2dimensional drawing.
LT12: Classify and describe polygons and 3dimensional figures.
12.8 Identify 3-dimensional
objects (e.g. prisms,
pyramids, cones, cylinders
and spheres).
LT13: Use and analyze geometric
relationships and properties of polygons
and 3-dimensional figures to solve
mathematical and real-world applications.
13.7 Make a twodimensional representation of
a 3-dimensional object.
Exploration: Euler’s
formula for polyhedra
10.2 Volume of Prisms
and cylinders
10.3 Volume of
pyramids and cones
Assessment
Exploration: The Five
Platonic Solids
District
Model
CABS
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Suggested
vocabulary
Volume,
polyhedron,
prism, pyramid,
cone, sphere,
lateral face,
surface area,
cylinder
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Reflection
and
Feedback:
Week 26
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Time
Lesson/ Activity
MPS Learning Targets
LT1: The student will communicate mathematical ideas
and reasoning using the vocabulary of mathematics in a
variety of ways, e.g. using words, numbers, symbols,
pictures, charts, tables, diagrams, graphs, and models.
Chapter 10,
continued
10.4 Volume
LT18: Use perimeter, area, volume and surface
area to solve real-world applications.
10.5 Displacement and
Density
LT19: Use increased rigor in justifying
conclusions to mathematical and real-world
applications and in verifying the truth of
Exploration
Orthographic Drawing geometric theorems.
Assessment
10.6 Volume of a
sphere
Benchmarks Embedded
in Chapter Lessons
District
Model
CABS
18.2 Determine the volume and
lateral area and total surface area
of spheres, prisms, pyramids,
cylinders and cones.
18.3 Use volume, lateral area and
surface area of 3-dimensional
figures to solve real-world
application problems.
18.7 Solve real-world application
problems using scale drawings,
perspective drawings, blueprints,
or computer drawings as models
of 3-dimensional objects.
19.1 Identify the converse,
inverse and contra positive of a
conditional statement.
10.7 Surface area of a
sphere
Reflection
and
Feedback
Week 27
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Suggested
vocabulary
Displacement,
density,
orthographic
drawing,
isometric
drawing
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Time
Lesson/ Activity
Chapter 10,
continued
Algebra: Solving for
any variable
MPS Learning Targets
LT1: The student will communicate mathematical ideas
and reasoning using the vocabulary of mathematics in a
variety of ways, e.g. using words, numbers, symbols,
pictures, charts, tables, diagrams, graphs, and models
LT19: Use increased rigor in justifying
conclusions to mathematical and real-world
applications and in verifying the truth of
geometric theorems.
Benchmarks Embedded
in Chapter Lessons
District
Model
CABS
19.2 Translate short verbal
arguments into symbolic form
Assessment
Exploration Sherlock
Holmes and Forms of
Reasoning
Review
Chapter 10
Assessment
Feedback
and
Reflection
Week 28
Learning Targets covered in Chapter 10:
12: Classify Polygons
13: Analyze Polygons
18: Circles
19: Proofs
19.3
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Suggested
vocabulary
Negation
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Time
Lesson/ Activity
MPS Learning Targets
LT1: The student will communicate mathematical ideas
and reasoning using the vocabulary of mathematics in a
variety of ways, e.g. using words, numbers, symbols,
pictures, charts, tables, diagrams, graphs, and models.
Chapter 11:
Similarity
Algebra Skills:
Proportion
11.1 Similar polygons
11.2 Similar triangles
Assessment
Exploration:
Constructing a Dilation
Design
CHAPTER 11
LT13: Use and analyze geometric relationships
and properties of polygons and 3-dimensional
figures to solve mathematical and real-world
applications.
LT15: Apply the basic theorems of congruence
and similarity to solve mathematical and realworld problems.
Benchmarks Embedded
in Chapter Lessons
District
Model
CABS
CHAPTER 11
13.3 Find the measures of
segments in similar polygons.
15.1 Identify congruent and
similar polygons.
15.2 Justify congruency or
similarity of polygons by using
formal and informal proofs.
15.4 Solve real-world
applications using the properties
of congruency and similarity.
LT18: Use perimeter, area, volume and surface 18.5 Compare perimeters and
area to solve real-world applications.
areas of similar 2-dimensional
figures using proportions.
18.6 Describe how a change in
one measurement affects other
measurements of the object.
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Suggested
vocabulary
Proportion,
ratio, similar,
dilation, scale
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Reflection
and
Feedback
Week 29
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Time
Lesson/ Activity
MPS Learning Targets
LT1: The student will communicate mathematical ideas
and reasoning using the vocabulary of mathematics in a
variety of ways, e.g. using words, numbers, symbols,
pictures, charts, tables, diagrams, graphs, and models.
Chapter 11,
continued
11.3 Indirect
measurement with
similar triangles
11.4 Corresponding
parts of similar
triangles
Assessment
LT15: Apply the basic theorems of congruence
and similarity to solve mathematical and realworld problems.
LT18: Use perimeter, area, volume and surface
area to solve real-world applications.
.
Benchmarks Embedded
in Chapter Lessons
District
Model
CABS
15.5 Solve real-world
applications using the properties
of congruency and similarity.
18.5 Compare perimeters and
areas of similar 2-dimensional
figures using proportions.
18.6 Describe how a change in
one measurement affects other
measurements of the object.
11.5 Proportions with
area
11.6 Proportions with
volume
Exploration: Why
Elephants Have Big
Ears
Reflection
and
Feedback
Week 30
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Suggested
vocabulary
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Time
Lesson/ Activity
MPS Learning Targets
LT1: The student will communicate mathematical ideas
and reasoning using the vocabulary of mathematics in a
variety of ways, e.g. using words, numbers, symbols,
pictures, charts, tables, diagrams, graphs, and models.
Chapter 11,
continued
11.7 Proportion and
parallel lines
Exploration: Two more
forms of Valid
Reasoning
Review
Benchmarks Embedded in District
Chapter Lessons
Model
CABS
14.3 Find the measures of
LT14: Identify and apply the definitions related segments in similar
to lines and angles in order to use them to solve polygons.
problems and perform basic geometric
constructions.
LT 19: Use reasoning and logic to draw
conclusions to mathematical and real-world
applications.
Chapter 11
Assessment
Learning Targets covered in Chapter 11:
LT13: Classify Polygons
LT14: Analyze Polygons
LT15: Congruence and Similarity
LT18: Formulas
LT19: Proofs
19.5 Diagram conditional
statements to solve logic
puzzles.
19.7 Use valid forms of
inductive and deductive
reasoning to draw
conclusions and make
conjectures.
19.8 Select and use
various types of reasoning
and methods to prove
characteristics and
properties of algebraic
models and geometric
figures.
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Suggested
vocabulary
Converse,
inverse,
contrapositive
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Feedback
and
Reflection:
Week 31
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Time
Lesson/ Activity
MPS Learning Targets
LT1: The student will communicate mathematical ideas
and reasoning using the vocabulary of mathematics in a
variety of ways, e.g. using words, numbers, symbols,
pictures, charts, tables, diagrams, graphs, and models.
Benchmarks Embedded
in Chapter Lessons
District
Model
CABS
Chapter 12:
Trigonometry
LT11: Explain and use the Pythagorean
Theorem and right triangle trigonometry as
models to solve a variety of right triangle
12.2 Solving with right problems in real-world applications.
12.1 Trigonometric
ratios
triangles
Assessment
Exploration: Indirect
measurement
.
11.3 Solve right triangle
problems using
trigonometric ratios.
11.4 Use trigonometric
ratios to find the angle of
elevation or the angle of
depression in real world
applications
12.3 Law of Sines
12.4 Law of Cosines
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Suggested
vocabulary
Trigonometry
functions,
clinometer
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Reflection
and
Feedback:
Week 32
Time
Lesson/ Activity
MPS Learning Targets
LT1: The student will communicate mathematical ideas
and reasoning using the vocabulary of mathematics in a
variety of ways, e.g. using words, numbers, symbols,
pictures, charts, tables, diagrams, graphs, and models.
Chapter 12,
continued
12.5 Problem solving
with Trigonometry
LT11: Explain and use the Pythagorean
Theorem and right triangle trigonometry as
models to solve a variety of right triangle
problems in real-world applications.
11.3 Solve right triangle problems
using trigonometric ratios.
11.4 Use trigonometric ratios to
find the angle of elevation or the
angle of depression in real world
applications.
LT16: Apply transformations and their
compositions to analyze geometric figures and
create designs.
16.6 Design a real-world
application of transformations and
communicate the process used in
the design.
LT19: Use reasoning and logic to draw
conclusions to mathematical and real-world
applications
19.5 Diagram conditional
statements to solve logic puzzles.
19.6 Use valid forms of inductive
and deductive reasoning to draw
conclusions and make conjectures.
19.7 Select and use various types
of reasoning and methods to prove
Assessment
Exploration:
Trigonometric Ratios
and the Unit Circle
Algebra Skills:
Transforming Functions
Exploration: Three
Types of Proof
Review
Benchmarks Embedded
in Chapter Lessons
Learning Targets covered in Chapter 12:
District
Model
CABS
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Suggested
vocabulary
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Chapter 12:
Assessment
LT11: Pythagorean Theorem
LT16: Transformations
LT19: Proofs
characteristics and properties of
algebraic models and geometric
figures.
Reflection
and
Feedback:
Week 33
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Time
Lesson/ Activity
MPS Learning Targets
LT1: The student will communicate mathematical ideas
and reasoning using the vocabulary of mathematics in a
variety of ways, e.g. using words, numbers, symbols,
pictures, charts, tables, diagrams, graphs, and models.
Chapter 13:
Geometry as a
Mathematical
System
13.1 The Premises of
Geometry
13.2 Planning a
Geometry Proof
CHAPTER 13
LT14: Identify and apply the definitions related
to lines and angles in order to use them to solve
problems and perform basic geometric
constructions.
LT15: Apply the basic theorems of congruence
and similarity to solve mathematical and realworld problems.
13.3 Triangle proofs
Assessment
13.4 Quadrilateral
proofs
LT19: Use reasoning and logic to draw
conclusions to mathematical and real-world
applications.
Benchmarks Embedded
in Chapter Lessons
District
Model CABS
CHAPTER 13
14.6 Prove statements about
segments, lines, angles and angle
relationships by using properties,
postulates, and theorems.
15.2 Justify congruency or
similarity of polygons by using
formal and informal proofs.
15.3 Prove two triangles are
congruent using coordinate
methods (e.g. distance and slope
formulas).
19.4 Select and use various types
of reasoning and methods to prove
characteristics and properties of
algebraic models and geometric
figures.
Feedback
and
Reflection:
Week 34
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Suggested
vocabulary
Commutative
property,
associative
property,
transitivity,
distributive
property
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Time
Lesson/ Activity
MPS Learning Targets
LT1: The student will communicate mathematical ideas
and reasoning using the vocabulary of mathematics in a
variety of ways, e.g. using words, numbers, symbols,
pictures, charts, tables, diagrams, graphs, and models.
Chapter 13,
continued
13.5 Indirect Proof
13.6 Circle Proofs
13.7 Similarity Proofs
Chapter 13:
Assessment
District
Model
CABS
15.3 Prove two triangles are
LT15: Apply the basic theorems of congruencecongruent using coordinate methods
(e.g. distance and slope
and similarity to solve mathematical and realformulas).
world problems.
LT19: Use reasoning and logic to draw
conclusions to mathematical and real-world
applications.
Algebra Skills:
Coordinate Proof
Review
Benchmarks Embedded
in Chapter Lessons
Learning Targets covered in Chapter 13:
LT14: Analyze Polygons
LT15: Congruence and Similarity
LT19: Proofs
19.3 Write conjectures using
characteristics, postulates and
theorems of geometric figures.
19.5 Select and use various
types of reasoning and methods
to prove characteristics and
properties of algebraic models
and geometric figures.
Feedback
and
Reflection:
Week 35
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Suggested
vocabulary
Milwaukee Public Schools Suggested Curriculum Pacing Guide for Discovering Geometry Draft 10/05/08
Fall 2008 – Spring 2009
Time
Lesson/
Activity
Final
Review:
Semester
Two
Final
Assessment:
Semester
Two
MPS Learning Targets: Semester Two
MPS Learning Targets Supported in
Semester 2: MA 311
Learning Targets Covered in
Semester 2: MA 311
LT1: Communication
LT2: Measurement
LT11: Right Triangles
LT15: Congruence and Similarity
LT16: Transformations
LT18: Geometric Formulas
LT19: Proofs
Learning Targets Supported in Semester
2: MA 311:
District
Model
CABS
LT12: Lines and Angles
LT13: Classify Polygons
LT14: Analyze Polygons
Final
Reflection
Semester
Two MA
311
Developed by the Milwaukee Mathematics Partnership with support by the National Science Foundation under Grant No. 0314898.
Draft 10/05/08
Suggested
vocabulary