Download Unit 1 Foundations for Geometry

Geometry Mathematics Curriculum Guide – Unit 1 Foundations for Geometry
Unit 1: Foundations for Geometry
2016 – 2017
Time Frame: 10 Days
Primary Focus
Students have worked with points, lines, angles, triangles, and other geometric figures since the elementary grades. This unit revisits many ideas that may be
familiar to students, but does so in a systematic way in order to build a deductive system.
The first step in this process is to decide which terms will be assumed to be understood (undefined terms) and to give definitions of basic vocabulary words using
these undefined terms.
The next step is classical constructions in geometry involve the use of a straightedge and compass only. The straightedge allows you to draw straight lines. The
compass allows you to draw circular arcs, with all points on an arc the same distance from the point of the compass. Although neither of these tools allows you
to take measurements as a ruler would, you can still transfer lengths by opening the compass to match a given length. This allows you to copy a segment,
construct a circle with a given radius, and even construct triangles with given side lengths.
Finally, in this unit students will use the midpoint and distance formulas which are two of the key tools of coordinate geometry. Students will write coordinate
proofs later in later units and they will find that these two formulas, along with facts about the slopes of parallel and perpendicular lines, are enough to prove a
wide range of theorems.
Common Core State Standards for Mathematical Practice Unit Focus
Standards for Mathematical Practice
MP5 - Use appropriate tools strategically.
MP6 - Attend to precision.
Unit 1
How It Applies to this Topic…
Select and use appropriate tools to best model/solve problems.
Communicate using clear mathematical definitions, vocabulary, and symbols.
Construct figures efficiently and accurately and label them appropriately.
Clover Park School District 2016-2017
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Geometry Mathematics Curriculum Guide – Unit 1 Foundations for Geometry
2016 – 2017
Stage 1 Desired Results
Transfer Goals
Students will be able to independently use their learning to…
Critique a definition or a diagram with a definition.
Meaning Goals
ESSENTIAL QUESTIONS
UNDERSTANDINGS
Students will understand that…
 Geometry is a mathematical system built on accepted facts, basic terms,
and definitions.
 The basic building blocks of geometric objects are formed from the
undefined notions of point, line, distance along a line, and distance around
a circular arc.
 Formal geometric constructions can be made with a variety of tools and
methods.
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What are the undefined building blocks of geometry and how are they
used?
What tools and methods can you use to copy a segment, bisect a segment,
and construct a circle?
What tools and methods can you use to copy an angle and bisect an angle?
How can you find midpoints of segments and distances in the coordinate
plane?
Acquisition Goals
Students will know and be skilled at…
 Naming and/or using the correct symbolic notational conventions for: angle, measure of an angle, circle, perpendicular, parallel, segment, measure of a
segment, point, line, plane, endpoint, ray, plane, angle bisector, midpoint, segment bisectors, perpendicular bisectors, betweenness, congruent segments,
congruent angles, tick marks, length of segment, measure of an angle, collinear, angle addition, and segment addition.
 Accurately constructing: a) a segment congruent to a given segment b) an angle congruent to a given angle c) an angle bisector of a given angle
 Find the distance between two points in the coordinate plane.
 Find the point on a line segment that partitions the segment in a given ratio.
 Use the distance formula to compute perimeters of polygons and areas of triangles and rectangles.
 Use the distance and midpoint formulas to prove that a figure is a rectangle, parallelogram, rhombus, trapezoid, kite, or square.
Unit 1
Clover Park School District 2016-2017
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Geometry Mathematics Curriculum Guide – Unit 1 Foundations for Geometry
2016 – 2017
Stage 1 Established Goals: Common Core State Standards for Mathematics
Cluster: Standard(s)
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
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.
What students should know prior to this unit and may need to be reviewed
Definitions and formulas that may need to be reviewed prior to unit with students:
Lines and Angles: acute angle, obtuse angle, right angle, linear pair, congruent segments, congruent
angles, midpoint, bisector, perpendicular lines, parallel lines, perpendicular bisector, distance on a
number line, perimeter, circumference
Triangles: acute triangle, obtuse triangle, right triangle, isosceles triangle, scalene triangle,
equilateral triangle, altitude, Pythagorean Theorem, area of triangle
Polygons: polygon, regular polygon, trapezoid, isosceles trapezoid, properties of parallelograms,
area of parallelogram, area of trapezoid, area of rectangle, area of square
Circles: radius, arc, circumference, area of circle
Explanations, and Comments
Students may use geometric software to make geometric constructions.
Geometry is a mathematical system built on accepted facts, basic terms, and definitions. A postulate
or axiom is an accepted statement of fact.
Students should be able to perform basic processes, such as:
-name a point, line, plane, ray, segment, and angle
- make a visual representation of a point, line, plane, ray, segment, and angle
-measure a line segment using a ruler (metric/standard)
Unit 1
Clover Park School District 2016-2017
Holt Geometry Chapter 1 Lesson 1
Holt Geometry Chapter 1 Lesson 2
Holt Geometry Chapter 1 Lesson 3
Holt Geometry Chapter 1 Lesson 4
Constructions from Holt Geometry 1-2 & 1-3 and use
Georgia Analytic Geometry EOCT Study Guide Pages 47-52
(Focus on copying a segment; copying an angle; bisecting a
segment; bisecting an angle)
Supplemental Materials: Discovering Geometry Lessons
Lesson 1.1: Building Blocks of Geometry
Lesson 3.1: Duplicating Segments and Angles
Lesson 3.2: Constructing Perpendicular Bisectors
Lesson 3.3: Constructing Perpendiculars to a Line
Lesson 3.4: Constructing Angle Bisectors
Lesson 3.6: Construction Problems
Additional Resources for Constructions
Construction Resource
TI Activities:
TI-Nspire Activity: Points, Lines and Planes
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Geometry Mathematics Curriculum Guide – Unit 1 Foundations for Geometry
2016 – 2017
-find the midpoint, given two coordinates
-use segment addition postulate
-use angle addition postulate
-measure an angle using protractor (degrees)
-classify types of angles (right, acute, obtuse)
Vocabulary: point, line, plane, line segment, ray, endpoint, angle, vertex, sides, angle bisector,
congruent angles, adjacent angles, supplementary angles, vertical angles, complementary angles
Cluster: Standard(s)
Use coordinates to prove simple geometric theorems algebraically
G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the
coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).
G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
G.GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.
What students should know prior to this unit and may need to be reviewed
Represent the vertices of a figure in the coordinate plane using variables.
Use coordinates to prove or disprove a claim about a figure.
G.GPE.7 provides practice with the distance formula and its connection with the
Pythagorean Theorem.
G.GPE.6 includes the midpoint formula.
(G.GPE.4,7)
Holt Geometry Chapter 1 Lesson 5: Area formulas
Holt Geometry Chapter 1 Lesson 6
Georgia CCGPS: Geometry Properties in the Plane (Discovering the
distance formula)
EngageNY Geometry: Module 4 Lesson 9 & 10 Teacher - Student
(G.GPE.6)
Holt Geometry Chapter 1 Lesson 6
Georgia CCGPS: Learning Task: New York City
EngageNY Geometry: Module 4 Lesson 12 Teacher - Student
Calculator Activity
TI-Nspire Activity: Points, Lines and Distance
Unit 1
Clover Park School District 2016-2017
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Geometry Mathematics Curriculum Guide – Unit 1 Foundations for Geometry
2016 – 2017
Stage 2 - Evidence
Evaluative Criteria/
SBAC Assessment Level Descriptors (ALDs):
Claim 1 Clusters:
NONE
Claim 2 Clusters:
NONE
Claim 3 Clusters:
Experiment with transformations in the
plane
Sample Assessment Evidence
Concepts and Procedures
Problem Solving
Communicating Reasoning
Level 3 students should be able to use stated assumptions, definitions, and previously established results and
examples to test and support their reasoning or to identify, explain, and repair the flaw in an argument. Students
should be able to break an argument into cases to determine when the argument does or does not hold.
Level 4 students should be able to use stated assumptions, definitions, and previously established results to
support their reasoning or repair and explain the flaw in an argument. They should be able to construct a chain of
logic to justify or refute a proposition or conjecture and to determine the conditions under which an argument
does or does not apply.
Go here for Sample SBAC items
Go here for more information about the Achievement Level Descriptors for Mathematics
Common Assessment
See Sample Assessments for Geometry Units.
Unit 1
Clover Park School District 2016-2017
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Geometry Mathematics Curriculum Guide – Unit 1 Foundations for Geometry
2016 – 2017
Stage 3 – Learning Plan: Sample
Summary of Key Learning Events and Instruction that serves as a guide to a detailed lesson planning
LEARNING ACTIVITIES:
Extended Geometry:
Pre-assess on definitions and formulas related to lines and angles, triangles, polygons, and circles. 1 to 2 days should
be spent helping students revisit these terms prior to beginning this section. Use this time to set up notebook for
definitions, theorems, formulas, properties and constructions. Include Unit 1 Stuff I Should Know (SISK).
Regular Geometry:
Day 1: Holt Geometry Lesson 1-1 and 1-2 (Prioritize problems utilizing algebra, e.g. L1-2 #7,9,10,15)
Opening Activity: http://threeacts.mrmeyer.com/bestmidpoint/ Consider using this activity to help establish a
thinking classroom. Show video. Have students answer questions individually or in groups.
Exit Task: p35 1-8
NOTES:
Before starting work on Coordinate
Geometry. Use pre-assessment data to
determine what topics to scaffold for
the unit especially area and perimeter
formulas (G.GPE.4,6 & 7)
Day 2: Holt Geometry Lesson 1-3 and 1-4 (1)
Exit Task: p35 9-14
ALGEBRA REVIEW:
Solving 1-variable Equations
Holt Geometry: Chapter 1. Throughout
the unit there are opportunities to
select problems which review solving
equations. Example: Pgs. 17, 9, & 10
Day 3-4: Patty Paper and Compass/Straightedge Constructions from Holt Geometry 1-2 & 1-3 and use Georgia Analytic
Geometry EOCT Study Guide Pages 47-52 (1)
(Focus on copying a segment; copying an angle; bisecting a segment; bisecting an angle)
Exit Task Day 3: p35 15-20
Exit Task Day 4: p35 21-25
Linear Inequalities
EngageNY Geometry: Module 4, Lesson
11 defines the region in a plane with
Linear Inequalities and connects to GGPE.7
Day 5-6: Holt Geometry Lesson 1-5: Problem Solving worksheet and EngageNY Geometry: Module 4 Lesson 9 & 10
Simplifying Square Roots
Holt Geometry: Skills Bank S55. Working
with the distance formula as part of GGPE.7 creates an opportunity to review
simplifying radicals.
Day 7: Holt Geometry Lesson 1-6 and Georgia Performance Task: Geometry Properties in the Plane pg 50-57
(Discovering the distance formula)
Day 8: Holt Geometry 1-6: Challenge worksheet and Georgia Performance Task: Learning Task: New York City pg 22-36
Common Assessment
Unit 1
Clover Park School District 2016-2017
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Geometry Mathematics Curriculum Guide – Unit 1 Foundations for Geometry
2016 – 2017
Stage 3 – Learning Plan: Sample
Daily Lesson Components
Learning Target
Opening Activities (Pick Two: Fluency, Vocab, SLOT, Hmwk Review)
Activities
 Whole Group/Small Group/Guided/Collaborative/Independent
Checking for Understanding (before, during and after):
Assessments
Unit 1
Clover Park School District 2016-2017
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