Download Geometric Foundations, Constructions, and Relationships

Geometry Mathematics, Quarter 1, Unit 1.1
Geometric Foundations, Constructions, and
Relationships
Overview
Number of Instructional Days:
24
(1 day = 45 minutes)
Content to Be Learned
Mathematical Practices to Be Integrated
•
Know precise definitions of geometric terms
(e.g., angle, circle, perpendicular line, parallel
line, and line segment), based on the undefined
notions of point, line, distance along a line, and
distance around a circular arc.
Construct viable arguments and critique the
reasoning of others.
•
Use definitions, theorems, postulates, and
properties to make formal arguments about
lines, angles, and triangles in a variety of ways.
Make formal geometric constructions with a
variety of tools and methods. Constructions
include:
•
Justify conclusions using definitions, theorems,
and postulates.
•
Identify flaws in arguments using logic and
reasoning based on geometric concepts.
•
•
•
o
Copying a segment/angle.
o
Bisecting a segment/angle.
o
Constructing perpendicular lines.
o
Constructing a perpendicular bisector of a
line segment.
o
Constructing a line parallel to a given line
through a point not on the line.
Use appropriate tools strategically.
Prove theorems about lines and angles.
Theorems include, but are not limited to:
•
Make formal geometric constructions with a
variety of tools including a compass,
straightedge, string, reflexive devices, paper
folding, dynamic software, etc.
•
Use technological tools to explore and deepen
understanding of concepts.
Look for and make use of structure.
o
Congruency of vertical angles.
•
o
Relationship between angles formed by
intersection of parallel lines and a
transversal.
Identify relationships in diagrams to solve
problems involving missing angles.
•
Recognize angle relationships formed by
parallel lines cut by a transversal and use them
to solve problems.
•
How do you use logic and reasoning when
proving geometric concepts?
•
What are the basic terms of geometry and how
are they used to describe figures?
Prove the triangle angle sum theorem.
Essential Questions
•
How does the use of precise definitions help
you to understand more complex geometric
ideas and theorems?
•
What are the similarities and differences
between a geometric sketch and a formal
construction?
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 1 Geometry, Quarter 1, Unit 1.1
Geometric Foundations, Constructions, and Relationships (24 days)
Written Curriculum
Common Core State Standards for Mathematical Content
Congruence
G-CO
Experiment with transformations in the plane
G-CO.1
Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment,
based on the undefined notions of point, line, distance along a line, and distance around a
circular arc.
Make geometric constructions [Formalize and explain processes]
G-CO.12 Make formal geometric constructions with a variety of tools and methods (compass and
straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).
Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing
perpendicular lines, including the perpendicular bisector of a line segment; and constructing
a line parallel to a given line through a point not on the line.
Congruence
G-CO
Prove geometric theorems [Focus on validity of underlying reasoning while using variety of ways of
writing proofs]
G-CO.9
Prove theorems about lines and angles. Theorems include: vertical angles are congruent;
when a transversal crosses parallel lines, alternate interior angles are congruent and
corresponding angles are congruent; points on a perpendicular bisector of a line segment are
exactly those equidistant from the segment’s endpoints.
G-CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle
sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints
of two sides of a triangle is parallel to the third side and half the length; the medians of a
triangle meet at a point.
Common Core Standards for Mathematical Practice
3
Construct viable arguments and critique the reasoning of others.
Mathematically proficient students understand and use stated assumptions, definitions, and previously
established results in constructing arguments. They make conjectures and build a logical progression of
statements to explore the truth of their conjectures. They are able to analyze situations by breaking them
into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them
to others, and respond to the arguments of others. They reason inductively about data, making plausible
arguments that take into account the context from which the data arose. Mathematically proficient
students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or
reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is.
Elementary students can construct arguments using concrete referents such as objects, drawings,
diagrams, and actions. Such arguments can make sense and be correct, even though they are not
generalized or made formal until later grades. Later, students learn to determine domains to which an
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 2 Geometry, Quarter 1, Unit 1.1
Geometric Foundations, Constructions, and Relationships (24 days)
argument applies. Students at all grades can listen or read the arguments of others, decide whether they
make sense, and ask useful questions to clarify or improve the arguments.
5
Use appropriate tools strategically.
Mathematically proficient students consider the available tools when solving a mathematical problem.
These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a
spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient
students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions
about when each of these tools might be helpful, recognizing both the insight to be gained and their
limitations. For example, mathematically proficient high school students analyze graphs of functions and
solutions generated using a graphing calculator. They detect possible errors by strategically using
estimation and other mathematical knowledge. When making mathematical models, they know that
technology can enable them to visualize the results of varying assumptions, explore consequences, and
compare predictions with data. Mathematically proficient students at various grade levels are able to
identify relevant external mathematical resources, such as digital content located on a website, and use
them to pose or solve problems. They are able to use technological tools to explore and deepen their
understanding of concepts.
7
Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. Young students, for
example, might notice that three and seven more is the same amount as seven and three more, or they may
sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8
equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In
the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the
significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line
for solving problems. They also can step back for an overview and shift perspective. They can see
complicated things, such as some algebraic expressions, as single objects or as being composed of several
objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that
to realize that its value cannot be more than 5 for any real numbers x and y.
Clarifying the Standards
Prior Learning
In grade 4, students drew and identified points, lines, line segments, rays, angles, and parallel and
perpendicular lines (4.G.1). Students also classified 2-D figures, based on the presence or absence of
parallel and perpendicular lines and angles, including right triangles (4.G.2).
In grade 7, students drew geometric shapes freehand as well as with tools such as rulers, protractors, and
technology (7.G.2). Students also used supplementary, complementary, vertical, and adjacent angles to
write and solve simple equations involving an unknown angle (7.G.5).
In grade 8, students used informal arguments to establish facts about angle relationships such as the angle
sum and exterior angles of triangles as well as angle relationships formed by parallel lines cut by a
transversal (8.G.5). Students explained a proof of the Pythagorean Theorem and its converse and applied
it to find unknown side lengths or the distance between two points in a coordinate system (8.G. 6, 7, 8).
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 3 Geometry, Quarter 1, Unit 1.1
Geometric Foundations, Constructions, and Relationships (24 days)
Current Learning
Students study precise definitions of angle, circle, perpendicular line, parallel line, and line segment,
based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
Students make formal geometric constructions with a variety of tools and methods using a compass and
straightedge, string, reflective devices, paper folding, dynamic geometric software, etc. (e.g., copy a
segment and an angle, bisect a segment and an angle, construct perpendicular lines, the perpendicular
bisector of a line segment, and a line parallel to a given line through a point not on the line). Students
prove theorems about lines and angles (e.g., relationships between parallel lines and a transversal, and
vertical angles) and they prove the triangle angle sum theorem.
Future Learning
Students will use the basic geometric terms and definitions throughout the year. In addition, students will
revisit and expand upon construction and proofs later this year in units on triangle congruency [unit 2.1],
right triangles [unit 2.3], polygons [unit 3.1] and circles [units 3.3, 4.1]. Students will continue to use
critical and logical thinking in algebra 2 and fourth-year math courses. The basic geometric terms and
constructions are applicable to real-life situations and professions such as civil engineering, architecture,
and construction-related work.
Additional Findings
Benchmarks for Science Literacy by the American Association for the Advancement of Science states,
“By the end of 12th grade, students should know that very complex logical arguments can be made from a
lot of small logical steps” (p. 234).
When discussing students’ exploration of the geometry of objects, this resource states, “They should also
do some mechanical drawing the old fashioned way before using the graphic capabilities of the computer”
(p. 225).
Teacher Notes
Activities utilizing patty paper, Geometer’s Sketchpad, and GeoGebra might be helpful.
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 4 Geometry, Quarter 1, Unit 1.2
Transformations
Overview
Number of instructional days:
14
(1 day = 45 minutes)
Content to be learned
Mathematical practices to be integrated
•
Represent, construct, and draw transformations
(reflections, translations, dilations, and
rotations) in the plane using a variety of tools,
such as transparencies and geometry software.
Model with mathematics.
Describe transformations as functions that take
points in the plane as inputs and give other
points as outputs.
Use appropriate tools strategically.
•
•
Compare transformations that preserve distance
and angle to those that do not (e.g., translation
versus horizontal stretch).
•
Develop definitions of rotations, reflections,
and translations in terms of angles, circles,
perpendicular lines, parallel lines, and line
segments.
•
State a sequence of transformations that will
carry a given figure onto another.
•
Verify experimentally the properties of
dilations given by a center and a scale factor:
o
The dilation of a line segment is longer or
shorter depending on the scale factor.
o
Dilation takes a line not passing through
the center of the dilation to a parallel line,
and leaves a line passing through the center
unchanged.
•
Use geometry to solve a design problem or use
a function to describe how one quantity of
interest depends on another.
•
Consider and use the available tools when they
develop an understanding of transformations.
These tools might include pencil and paper,
concrete models, a ruler, a protractor, a
calculator, a spreadsheet, a computer algebra
system, a statistical package, or dynamic
geometry software.
•
Use technology tools to explore and deepen
understanding of various transformation
concepts.
Attend to precision.
•
Communicate precisely to others.
•
Use clear definitions in discussion with others
and your own reasoning. Examine claims and
make explicit use of definitions.
•
Compare dilation to rigid motions, how are
they similar? How are they different?
•
Where would you find transformations in the
world of art?
Essential questions
•
After a transformation has taken place on the
coordinate plane, where does the image lie and
what does it look like?
•
What tools or methods would you use to
construct a figure under a reflection,
translation, rotation, and dilation?
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 5 Geometry, Quarter 1, Unit 1.2
Transformations (14 days)
Written Curriculum
Common Core State Standards for Mathematical Content
Congruence
G-CO
Experiment with transformations in the plane
G-CO.2
Represent transformations in the plane using, e.g., transparencies and geometry software;
describe transformations as functions that take points in the plane as inputs and give other
points as outputs. Compare transformations that preserve distance and angle to those that do
not (e.g., translation versus horizontal stretch).
G-CO.4
Develop definitions of rotations, reflections, and translations in terms of angles, circles,
perpendicular lines, parallel lines, and line segments.
G-CO.5
Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure
using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of
transformations that will carry a given figure onto another.
Similarity, Right Triangles, and Trigonometry
G-SRT
Understand similarity in terms of similarity transformations
G-SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor:
b.
The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
a.
A dilation takes a line not passing through the center of the dilation to a parallel line, and
leaves a line passing through the center unchanged.
Common Core Standards for Mathematical Practice
4
Model with mathematics.
Mathematically proficient students can apply the mathematics they know to solve problems arising in
everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition
equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a
school event or analyze a problem in the community. By high school, a student might use geometry to
solve a design problem or use a function to describe how one quantity of interest depends on another.
Mathematically proficient students who can apply what they know are comfortable making assumptions
and approximations to simplify a complicated situation, realizing that these may need revision later. They
are able to identify important quantities in a practical situation and map their relationships using such
tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships
mathematically to draw conclusions. They routinely interpret their mathematical results in the context of
the situation and reflect on whether the results make sense, possibly improving the model if it has not
served its purpose.
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 6 Geometry, Quarter 1, Unit 1.2
5
Transformations (14 days)
Use appropriate tools strategically.
Mathematically proficient students consider the available tools when solving a mathematical problem.
These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a
spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient
students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions
about when each of these tools might be helpful, recognizing both the insight to be gained and their
limitations. For example, mathematically proficient high school students analyze graphs of functions and
solutions generated using a graphing calculator. They detect possible errors by strategically using
estimation and other mathematical knowledge. When making mathematical models, they know that
technology can enable them to visualize the results of varying assumptions, explore consequences, and
compare predictions with data. Mathematically proficient students at various grade levels are able to
identify relevant external mathematical resources, such as digital content located on a website, and use
them to pose or solve problems. They are able to use technological tools to explore and deepen their
understanding of concepts.
6
Attend to precision.
Mathematically proficient students try to communicate precisely to others. They try to use clear
definitions in discussion with others and in their own reasoning. They state the meaning of the symbols
they choose, including using the equal sign consistently and appropriately. They are careful about
specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem.
They calculate accurately and efficiently, express numerical answers with a degree of precision
appropriate for the problem context. In the elementary grades, students give carefully formulated
explanations to each other. By the time they reach high school they have learned to examine claims and
make explicit use of definitions.
Clarifying the Standards
Prior Learning
In grade 5, students defined the coordinate system and graphed ordered pairs called coordinates. (5.G.1)
In grade 6, students drew polygons in the coordinate plane given coordinates for the vertices. (6.G.3)
In grade 7, students drew (freehand, with ruler and protractor, and with technology) geometric shapes
with given conditions. (7.G.2) Students also solved problems involving scale drawings of geometric
figures. This included computing actual lengths and areas and reproducing scale drawings at a different
scale. (7.G.1)
In grade 8, students verified experimentally the properties of rotations, reflections, and translations. They
learned that lines, segments, and angles maintain their shape and size when transformed. (8.G.1) Students
recognized that a two-dimensional figure is congruent (8.G.2)/similar (8.G.4) to another if the second can
be obtained from the first by a sequence of rotations, reflections, and translations. Furthermore, students
described the effect of dilations, translation, rotations, and reflections on two-dimensional figures using
coordinates. (8.G.3)
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 7 Geometry, Quarter 1, Unit 1.2
Transformations (14 days)
Current Learning
Students represent, construct, and draw transformations in the plane using a variety of tools such as
transparencies and geometry software.
They describe transformations as functions that take points in the plane as inputs and give other points as
outputs.
Students compare transformations that preserve distance and angle to those that do not (e.g., translation
versus horizontal stretch).
Students develop definitions of rotations, reflections, and translations in terms of angles, circles,
perpendicular lines, parallel lines, and line segments.
Students specify a sequence of transformations that will carry a given figure onto another.
Students verify experimentally the properties of dilations given by a center and a scale factor:
•
The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
•
Dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a
line passing through the center unchanged.
Future Learning
In algebra 2 and precalculus, students will use their knowledge of transformations when building
functions from existing functions. Students pursuing art-related programs and careers will continue the
study of visual transformations.
In unit 2.1, students will study rigid motion and its effect on polygons.
Additional Findings
According to A Research Companion to Principles and Standards for School Mathematics:
“Studies found that second graders did learn manual procedures for producing transformation images, but
that they did not learn to mentally perform such transformations.” (p. 162)
Even at the middle and high school level, “students’ ability to mentally perform isometries is limited …
because it demands formal operational thought” (p. 162).
Research shows that “slides appear to be the easiest motions for students, then flips and turns; however,
the direction of transformation may affect the relative difficulty of turn and flip” (p. 162).
Furthermore, “… computer environments have been found to be particularly effective in developing
conceptions of symmetry and geometric motions” (p. 162).
Principles and Standards for School Mathematics states, “In grade 9–12 all students should understand
and represent translations, reflections, rotations, and dilations of objects in the plane by using sketches,
coordinates, vectors, function notation, and matrices. They should use various representations to help
understand the effects of simple transformations and their compositions.” (p. 308)
Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin 8