Download Distance, Midpoint, and Pythagorean Theorem

Geometry, Quarter 1, Unit 1.1
Distance, Midpoint, and Pythagorean Theorem
Overview
Number of instructional days:
8
(1 day = 45 minutes)
Content to be learned
Mathematical practices to be integrated
•
Find distance and midpoint. (2 days)
•
Identify and model points, lines, and planes.
(1 day)
Make sense of problems and persevere in solving
them.
•
Solve problems on the coordinate plane using
distance and midpoint formulas. (1 day)
•
Apply the distance and midpoint formula to solve
word problems. (1 day)
•
Solve problems using the Pythagorean Theorem
and the converse of the Pythagorean Theorem.
(1 day)
•
Derive the distance formula using the
Pythagorean Theorem. (1 day)
•
Read problems involving Pythagorean
Theorem; plan a strategy to solve and check
the problem; modify and revise, if necessary.
•
Think about simpler problems to help solve
more complex problems with distance and
midpoint.
Construct viable arguments and critique the
reasoning of others.
•
Think logically; determine if there are errors,
and explain reasoning.
•
Communicate conclusions.
•
What is the difference between how the
Pythagorean Theorem and its converse are
used?
•
Where can the Pythagorean Theorem be
applied in the real world?
Essential questions
•
What are the differences and similarities between
midpoint and distance?
•
When is the distance or midpoint formula used?
•
How is the Pythagorean Theorem related to the
distance formula?
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
C-1
Geometry, Quarter 1, Unit 1.1
Final, July 2011
Distance, Midpoint, and Pythagorean Theorem (8 days)
Written Curriculum
Grade-Span Expectations
M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric
properties, or uses theorems to solve problems involving angles, lines, polygons, circles, or right
triangle ratios (sine, cosine, tangent) within mathematics or across disciplines or contexts (e.g.,
Pythagorean Theorem, Triangle Inequality Theorem). (State)
M(G&M)–10–9 Solves problems on and off the coordinate plane involving distance and midpoint.
(State)
Clarifying the Standards
Prior Learning
In grade 4, students plotted points in the first quadrant and found horizontal and vertical distances. In fifth
grade, students plotted points in all four quadrants and reflected, translated, and rotated polygons on the
coordinate plane. In grade 7, students calculated perfect squares and related square roots. In grade 8,
students were introduced to and appled the Pythagorean Theorem.
Current Learning
Students make and defend conjectures to solve problems using the Pythagorean Theorem and the
converse of the Pythagorean Theorem. They find distance and midpoint on the coordinate plane. Students
derive the distance formula using the Pythagorean Theorem. In unit 4.3, students use the distance formula
to find the equation of a circle.
Future Learning
In precalculus and calculus, students will use the Pythagorean Theorem when studying trigonometry and
trigonometric identities.
Additional Research Findings
According to Principles and Standards for School Mathematics, students in grades 9–12 are to specify
locations and describe spatial relationships using coordinate geometry and other representational systems
(p. 308).
According to Beyond Numeracy, students at this level should be introduced to the coordinate system and
its uses in analytic geometry (pp. 10–11).
According to Principles and Standards for School Mathematics, in grades 9–12 students have to write an
algebraic justification of the Pythagorean Theorem, and students use the Pythagorean Theorem as one of
the many multiple approaches to solving geometric problems (p. 301).
According to Benchmarks for Science Literacy, in grades 6–8 students should use the Pythagorean
Theorem as a model for problem solving (p. 269).
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
C-2
Geometry, Quarter 1, Unit 1.2
Applications of Slope
Overview
Number of instructional days:
8
(1 day = 45 minutes)
Content to be learned
Mathematical Practices to be Integrated
•
Model with mathematics.
Solve problems algebraically and
graphically involving slope, parallel lines,
and perpendicular lines. (3 days)
•
•
Determine the rate of change and equation
of a line. (2 days)
Use appropriate tools strategically.
•
Solve real-world problems involving lines
on the coordinate plane. (2 days)
•
Use technology to visualize results.
•
Use tools and technology to explore and
deepen understanding of slope and be able
to detect errors.
•
What are the differences and similarities
between equations of parallel and
perpendicular lines?
Relate applications of slope to everyday
life.
Essential questions
•
What does slope tell you about a line(s),
including parallel and perpendicular lines?
•
What do the graphs of linear equations
have in common?
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
C-3
Geometry, Quarter 1, Unit 1.2
Final, July 2011
Applications of Slope (8 days)
Written Curriculum
Grade-Span Expectations
M(G&M)–10–9 Solves problems on and off the coordinate plane involving distance,
midpoint, perpendicular and parallel lines, or slope. (State)
Clarifying the Standards
Prior Learning
In grade 4, students were introduced to constant rate of change. Grade 6 students continued learning about
slope through the use of graphs. In grade 7, students solved problems involving constant rates of change,
meaning of slope, translating slope from tables to equations and vice versa (F&A 7-2). Grade 8 students
again discussed the meaning of slope and solved problems involving slope and rate of change through
tables, graphs, and equations (F&A 8-2). In grade 9, students found the equation of lines, using two
points, point and slope, etc. (F&A 10-4).
Current Learning
Students solve problems on the coordinate plane involving parallel and perpendicular lines and slope
(developmental); they also prove lines to be parallel and perpendicular by comparing slopes
(developmental). Throughout the unit, the concepts of slope and writing equations of lines are reinforced.
Also on the developmental level, students use coordinate geometry to examine the slopes of parallel and
perpendicular lines. In unit 1.4, students solve problems involving parallel and perpendicular lines (off the
coordinate plane).
Future Learning
In grades 11 and 12, students will use rate of change for polynomial, rational, and exponential functions.
Additional Research Findings
According to Benchmarks for Science Literacy, by the end of eighth grade students should know
that lines can be parallel, perpendicular, or oblique (p. 224). Also, as stated in the Principles and
Standards for School Mathematics text, by grade 8 students should understand the relationship
between symbolic expressions and graphs of lines using slope and intercepts (p. 222).
A Research Companion to Principles and Standards for School Mathematics adds that, by eighth
grade students learn that parallel lines should not intersect and are equidistant (p. 164).
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
C-4
Geometry, Quarter 1, Unit 1.3
Angles and Segments
Overview
Number of instructional days:
10
(1 day = 45 minutes)
Content to be learned
Mathematical practices to be integrated
•
Define angles, segments, and rays. (2 days)
•
Make and defend conjectures about angle and
segment addition postulates. (2 days)
Construct viable arguments and critique the
reasoning of others.
•
Make and defend conjectures about angle pairs
(adjacent, linear, vertical, supplementary, and
complementary) and angle bisectors. (5 days)
•
Use proofs to make logical arguments and
conclusions regarding angles and segments.
•
Make conjectures regarding angles and
segments and prove them, including using
counter-examples.
Attend to precision.
•
Use precise mathematical vocabulary and
symbols to communicate efficiently and
effectively.
•
How can vertical, linear, adjacent,
supplementary, and complementary angle pairs
be used to solve problems?
Essential questions
•
What are the differences and similarities
between angle pairs?
•
How do you know if an angle or segment has
been bisected?
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
C-5
Geometry, Quarter 1, Unit 1.3
Final, July 2011
Angles and Segments (10 days)
Written Curriculum
Grade Span Expectations
M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric
properties, or uses theorems to solve problems involving angles, lines, polygons, circles, or right
triangle ratios (sine, cosine, tangent) within mathematics or across disciplines or contexts (e.g.,
Pythagorean Theorem, Triangle Inequality Theorem). (State)
Clarifying the Standards
Prior Learning
Students in grade 4 classified angles relative to 90 degrees. In grade 7, vertical, adjacent, linear,
supplementary, and complementary angle properties were introduced to and reinforced.
Current Learning
At this grade level, students master and solve problems involving angle pairs (vertical, adjacent, linear,
supplementary, and complementary).
Future Learning
In future units, students will create formal proofs involving angle properties.
Additional Research Findings
•
See NCTM Principles and Standards for School Mathematics, p. 308.
•
According to A Research Companion to Principles and Standards for School Mathematics, “Angles
are the turning points in the study of geometry and special relationships” (p. 162).
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
C-6
Geometry, Quarter 1, Unit 1.4
Line and Angle Relationships
Overview
Number of instructional days:
10
(1 day = 45 minutes)
Content to be learned
Mathematical practices to be integrated
•
Construct viable arguments and critique the
reasoning of others.
•
Make and defend conjectures about parallel
and perpendicular lines off the coordinate
plane. (5 days)
Use angles formed by parallel lines and
transversals to solve real-world problems.
(4 days)
•
Think logically, determine if there are errors,
and explain their reasoning.
Model with mathematics.
•
Relate what they have learned regarding line
and angle relationships to everyday life.
Look for and make use of structure.
•
Apply prior learning about angles and lines to
new situations.
•
How do you prove that two lines cut by a
transversal are parallel or perpendicular?
Essential questions
•
What is different about the angle pairs that
form when a transversal intersects parallel lines
versus the angle pairs formed when a
transversal intersects nonparallel lines?
•
How can special angle pairs help you
determine if two lines are parallel?
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
C-7
Geometry, Quarter 1, Unit 1.4
Final, July 2011
Line and Angle Relationships (10 days)
Written Curriculum
Grade-Span Expectations
M(G&M)–10–9 Solves problems on and off the coordinate plane involving distance, midpoint,
perpendicular and parallel lines, or slope. (State)
M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric
properties, or uses theorems to solve problems involving angles, lines, polygons, circles, or right
triangle ratios (sine, cosine, tangent) within mathematics or across disciplines or contexts (e.g.,
Pythagorean Theorem, Triangle Inequality Theorem). (State)
Clarifying the Standards
Prior Learning
In grades 4–6, students were introduced to the meaning of parallel and perpendicular lines when
identifying and describing shapes. In grade 7, students learned about angles formed by two parallel and
nonparallel lines cut by a transversal.
Current Learning
Through solving real-world problems, students develop, reinforce, and master relationships of angles
formed by parallel and nonparallel lines cut by transversals off the coordinate plane. This year, students
solve problems involving parallel and perpendicular lines on the coordinate plane. Later this year, the
angle relationships between parallel and perpendicular lines cut by a transversal are applied to show that
polygons are parallelograms, rectangles, squares, etc.
Future Learning
In grade 12, students will create formal proofs of propositions involving angles and lines.
Additional Research Findings
According to Benchmarks for Science Literacy, by the end of eighth grade students should know that lines
can be parallel, perpendicular, or oblique (p. 224). Also, as shown the Principles and Standards for
School Mathematics text, in grades 9-12 students should be able to apply the angle relationships formed
by two parallel lines and a transversal to solve problems (p. 310).
Also, according to A Research Companion to Principles and Standards for School Mathematics, by
eighth grade students should know that parallel lines should not intersect and that they are equidistant
(p. 164).
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
C-8
Geometry, Quarter 1, Unit 1.5
Polygons
Overview
Number of instructional days:
4
(1 day = 45 minutes)
Content to be learned
Mathematical Practices to be Integrated
•
Use properties of polygons to classify and
solve problems involving interior and exterior
angles. (1 day)
Construct viable arguments and critique the
reasoning of others.
•
Use the exterior angle theorem to solve
problems.
(1 day)
•
Classify polygons as convex or concave.
(1 day)
•
Derive the formula for the sum of the interior
angles of a polygon. (1 day)
•
Use prior material and properties of polygons
to solve more complex problems involving
deductive and inductive reasoning.
Look for and express regularity in repeated
reasoning.
•
Look for general methods, patterns, and
repeated calculations when working with
polygons.
•
Look for patterns of polygons to find
generalizations.
•
How would you derive the formula for the sum
of the interior angles of a polygon?
Essential questions
•
What are the differences and similarities
among polygons?
•
What is the relationship between the interior
and exterior angles of a polygon?
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
C-9
Geometry, Quarter 1, Unit 1.5
Final, July 2011
Polygons (4 days)
Written Curriculum
Grade-Span Expectations
M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric
properties, or uses theorems to solve problems involving angles, lines, polygons, circles, or right
triangle ratios (sine, cosine, tangent) within mathematics or across disciplines or contexts (e.g.,
Pythagorean Theorem, Triangle Inequality Theorem). (State)
Clarifying the Standards
Prior Learning
In grade 3, students developed conceptual understanding of triangles and quadrilaterals by copying,
comparing, and drawing. In sixth grade, students classified triangles and quadrilaterals.
Current Learning
At this grade level, students solve problems involving convex polygons. Students also create formal
proofs involving polygons.
Future Learning
In grade 12, students will solve problems involving area of polygons.
Additional Research Findings
See Principles and Standards for School Mathematics, page 233.
A Research Companion to Principles and Standards for School Mathematics states that, [according to the
theory of Pierre and Dina van Hiele, “Student Progress Though Levels of Thought in Geometry (levels 04)”], at level 3 students should be at the abstract level. “Students can form abstract definitions, distinguish
between necessary and sufficient sets of conditions for a concept, and understand and sometimes even
provide logical arguments in the geometric domain” (p. 152).
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
C-10