Download Unit 5 - Geometry - CMS Secondary Math Wiki

Course Name: Math iii
Congruence: (part 1)
Unit # 2
Unit Title: Geometry
Enduring understanding (Big Idea): Students will understand that a) you can determine if 2 figures are congruent by
comparing corresponding parts b) triangles can be proven congruent without having to compare all corresponding parts c)
the angles and sides of Isosceles and Equilateral triangles have special relationships.
Essential Questions: 1) How do you identify corresponding parts of congruent triangles? 2) How do you show that 2
triangles are congruent? 3) How can you tell whether a triangle is isosceles or equilateral? 4) How do you solve problems
that involve measurements of triangles?
BY THE END OF THIS UNIT:
Students will be able to…
1) use the definition of congruence in terms of rigid
motions to show that two triangles are congruent if and
Students will know…
only if corresponding pairs of sides and corresponding
1) that two figures are congruent if a series of rigid motion
pairs of angles are congruent.
carries one onto the other.
2) explain how the criteria for triangle congruence (ASA,
2) that two triangles are congruent if all corresponding pairs
SAS, and SSS) follow from the definition of congruence
of sides are congruent and all corresponding pairs of
in terms of rigid motions.
angles are congruent.
3) Use properties of midsegments to solve problems
4) Use properties of perpendicular and angle bisectors to
solve problems
Vocabulary:
5) Use properties of medians and altitudes to solve
Throughout standards
problems
Unit Resources:
Throughout standards
Suggested Pacing: (11 days total)?
Part 1 (3 days)
- G.CO. 1, 9, 10, 11, 12
Part 2 (3 days)
- G.SRT. 2, 3, 4, 5
Part 3 ( 3 days)
- G.C. 1, 2, 3, 5
G.MG.3 (throughout entire unit)
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Mathematical Practices in Focus:
1) Make sense of problems and persevere in solving them
2) Model with mathematics
3) Attend to precision
4) Model with mathematics.
5) Look for and make use of structure.
CCSS-M Included:
G.CO. 1, 9, 10, 11, 12 (part 1)
G.SRT. 2, 3, 4, 5 (part 2)
G.C. 1, 2, 3, 5 (part 3)
G.MG. 3 (throughout)
Course Name: Math iii
CORE CONTENT
Unit # 2
Unit Title: Geometry
Cluster Title: Experiment with transformations in the plane
Standard: Standard G.CO.1 Know precise definitions of angle, circle, perpendicular lines, parallel lines, and line
segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
Concepts and Skills to Master:
Define angle, circle, perpendicular line, parallel line, and line segment
Use precise definitions to identify and model an angle, circle, perpendicular line, parallel line, and line segment
Demonstrate mathematical notation for each term.
Apply the segment addition postulate and the angle addition postulate.
Apply properties of lines and transversals. Identify special angle pairs.
SUPPORTS FOR TEACHERS
Critical Background Knowledge
Solve algebraic equations; Understanding the undefined terms: point, line, and plane; Understand distance is a nonnegative quantity.
Academic Vocabulary
Angle, circle, perpendicular line, parallel line, line segment, distance, arc
Suggested Instructional Strategies:
Resources:
Have students write their own understanding of a given term Textbook Correlation:
Give students formal and informal definitions of each term
1-2 points, lines and plane
1-3 measuring segments
and compare them
Develop precise definitions through use of examples and
1-4 measuring angles
non-examples
1-5 exploring angle pairs
Discuss the importance of having precise definitions
3-1 lines and angles
Start line segment addition with integers, move to labeled
segments then expressions.
Do the same with angle addition postulate.
Sample Assessment Tasks
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Math iii
Skill-based task
State the definition of a circle, perpendicular lines, parallel
lines, and line segment.
Unit # 2
Unit Title: Geometry
Problem Task
CORE CONTENT
Cluster Title: Prove Geometric Theorems
Standard: Standard: 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 equidistace from the segment’s endpoints.
Concepts and Skills to Master:
Identify parallel, perpendicular, and skew lines and parallel planes
Classify/determine angle relationships formed by two lines and a transversal
Prove that vertical angles are congruent.
Prove that when parallel lines are cut by a transversal, pairs of alternate interior angles are congruent, pairs of alternate
exterior angles are congruent, and pairs of same-side interior angles are supplementary.
When given parallel lines, use properties of parallel lines to calculate angle measures and solve for variables
When given angle measures, determine which lines, if any, are parallel
SUPPORTS FOR TEACHERS
Critical Background Knowledge
Solve multi-step equations; know properties of supplementary, complementary, vertical, & adjacent angles (7.G.5)
Academic Vocabulary
Vertical angles, Parallel lines & planes, Transversal, Alternate Interior/Exterior angles (AIA and AEA), Corresponding
angles (CA), Same-side Interior angles (SSI), Skew lines
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Math iii
Unit # 2
Unit Title: Geometry
Suggested Instructional Strategies:
Resources:
-Use multiple formats to write justifications: narrative
Textbook Correlation: 2-6, 3-1, 3-2, 3-3, 3-4
paragraphs, flow diagrams, two-column format, and
Online Teacher Resource Center 3-1 Game: Name It –
diagrams without words.
Claim It (Suggestion, enlarge the four diagrams on an 8 x 10
-When teaching skew lines, highlight all segments in a
and allow students more room to play with their die or
rectangular prism that are not skew; thus all other segments number cube)
are skew.
Online Teacher Resource Center 3-2 Performance Task
-Use dynamic geometry software to explore angle
Activity (Suggestion, enlarge the four diagrams on an 8 x 10
relationships
and allow students more room to play with their die or
-Connect angle relationships to the creation of tessellation number cube)
patterns
Angle Hunter Video
-Formal and informal proofs do not have to be introduced at
this point. Students can argue with justification by
completing short answer/open-ended questions such as
“John says that consecutive interior angles are congruent
when lines are parallel. How would you convince John that
that these angles are actually supplementary?” Focus on the
validity of the underlying reasoning of the justifications
-The “Build a City” project or similar activity is suggested as
a real-world application follow-up assignment to assess
student understanding.
Sample Assessment Tasks
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Math iii
Unit # 2
Unit Title: Geometry
Skill-based task
I.
Lines a and b are parallel.
Solve for x.
Solve for y.
Find the measures of angles 1-5.
Based on this information, are lines c and d parallel? In
complete sentences, explain why or why not.
Problem Task
Marie is building a sandbox in her back yard.
Only equipped with the tools to measure angles, how can
Marie determine whether both pairs of opposite sides are
parallel?
If both pairs of opposite sides are parallel, and one of the
angles measures 85 degrees, what are the measures of the
remaining angles?
II.
Suppose a || b and c || d.
1. If m∠ 6 = 50, then find m∠ 11.
2. If m∠ 2 = 70, then find m∠ 6.
3. If m∠ 7= 110, then find m∠ 10.
4. If m∠ 4= 45, then find m∠ 12.
5. Which angle could you show is congruent to ∠ 11 to
prove a || b?
6. What relationship between ∠ 6 and ∠ 11 shows c || d?
Find as many angle relationships as possible in this pattern:
CORE CONTENT
Cluster Title: Make geometric constructions
Standard: 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.
Concepts and Skills to Master:
Perform the following constructions using a variety of tools and methods: 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.
Explain why these constructions result in the desired objects.
SUPPORTS FOR TEACHERS
Critical Background Knowledge
Definitions of the following terms: circle, bisector, perpendicular, and parallel
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Math iii
Unit # 2
Unit Title: Geometry
Academic Vocabulary
Segment, Angle, Bisect, Perpendicular, Parallel, Circle, Construction, Transversal, Alternate
Interior/Exterior angles, Corresponding angles, Same-side Interior angles, Skew lines, Parallel
planes
Suggested Instructional Strategies:
Resources:
Completing constructions can be used as a follow up activity Textbook Correlation: 1-6, 3-6
or as an investigation.
Investigation Activity (from CMS Curriculum Guide)
If used as an investigation, provide students with openended follow up questions to help them draw accurate
http://math.springbranchisd.com/high/classes/algebra
conclusions. For example, after giving students
_one/Laying%20The%20Foundation/Lessons/Paralle
l%20and%20Perpendicular%20Lines%20187instructions for constructing an angle bisector, pose the
following: “Fold your patty paper along this line. What
188.pdf
can you conclude about each angle? Which of the
vocabulary terms from this section have you just
constructed?”
Have students explore how to make a variety of
constructions using different tools. Ask students to justify
how they know their method results in the desired
construction.
Discuss the underlying principles that different tools rely on
to produce desired constructions (e.g. compass:circle; miro:
reflections)
Sample Assessment Tasks
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Math iii
Skill-based task
Using patty/tracing paper, pencil, straight edge, and
compass, complete the following constructions
Parallel lines
Perpendicular lines
Perpendicular bisector
Angle bisector
Unit # 2
Unit Title: Geometry
Problem Task
Jessica is studying architecture at the University of North
Carolina at Charlotte. For homework, she must find the
center point of a regular pentagon by connecting all of the
angle bisectors. Unfortunately, Jessica has her straight
edge, but has lost her protractor. What step by step
instructions would you give to Jessica to help her complete
the assignment?
CORE CONTENT
Cluster Title: Congruence-Prove Theorems about Triangles
Standard: G-CO.10 Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles
triangles are congruent; the segment joining midpoints of 2 sides of a triangle is parallel to the third side and half the
length; the medians of a triangle meet at a point.
Concepts and Skills to Master:
Use parallel lines to prove a theorem about triangles
Find measures of angles in triangles
Prove right triangles congruent using the Hypotenuse-Leg Theorem
SUPPORTS FOR TEACHERS
Critical Background Knowledge
The sum of the angle measures of triangles is always the same i.e. 180 degrees
Understanding the parts of a right triangle
Academic Vocabulary
Exterior angle, remote interior angle, auxiliary line, hypotenuse, legs
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Math iii
Unit # 2
Suggested Instructional Strategies:
Use a number of examples with exterior angles and show
how the 2 remote angles add up to the exterior angle. Draw
the correlation between triangle interior angles totaling 180°
and an exterior angle and its linear pair inside the triangle
totaling 180°
Sample Assessment Tasks
Skill-based task
Pearson website Lesson 3.5 Enrichment
Pearson Solve it! 3.5
Pearson website Lesson 4.5 Enrichment (Swan Puzzle)
Unit Title: Geometry
Resources:
Textbook Correlation: Pearson
3.5, 4.1, 4.2, 4.3, 4.5, 4.6, 5.1, 5.4
Video resource: Introduction Congruent Triangles
Problem Task
Pearson website Activities, Games and Puzzles 3.5
CORE CONTENT
Cluster Title: Polygon Angle Sum Theorems
Standard: G-CO.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite
angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms
with congruent diagonals.
Concepts and Skills to Master:
Identify, verify, and classify properties of quadrilaterals. Define and classify special types of parallelograms
SUPPORTS FOR TEACHERS
Critical Background Knowledge
Academic Vocabulary
equiangular polygon, equilateral polygon, regular polygon, irregular polygon, n-gon, diagonal of a polygon, convex
polygon, overlapping triangles, interior angles, exterior angles, consecutive angles
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Math iii
Unit # 2
Unit Title: Geometry
Suggested Instructional Strategies:
Have students sketch regular polygons (3 sided to 8 sided
shapes). Then have students make a conjecture about the
number of overlapping triangles in each after drawing
diagonals that connect all vertices – remember: triangles
cannot overlap. See if students can discover the Polygon
Angle Sum Theorem.
Resources:
Interactive Learning: 6-1 Solve It (Dynamic Activity Online
Teacher Resources –Interactive Digital Path)
www.pearsonsuccessnet.com
Wiki Exploration: Exterior Angle Sum Theorem
http://www.geogebra.org/en/wiki/index.php/Angles
(Click on Polygon Exterior Angle Sum Theorem –
explore – click next in the top right hand corner to
Extension: Have students label all interior angles of the
continue exploration)
sketched polygons, extend the vertices, and label the
Concept Byte Exploration Activity: p.352 Exterior Angles of
measures of each exterior angle formed. Ask students
Polygons
Essential Question #2 – What conjecture can be made
Cluster Review – Use Links Below:
about the sum of the exterior angles of any convex polygon? http://freedownload.is/ppt/3-4-the-polygon-angle-sumtheorems-ppt
Click on 3.4 (also labeled 3.5 in description) The Polygon
Angle Sum Theorems
http://www.mathwarehouse.com/geometry/polygon/
-Scroll entire web page to see questions.-
Sample Assessment Tasks
Skill-based task
Problem Task
Question I: A polygon has n sides. An interior angle of the
polygon and an adjacent exterior angle form a straight angle.
a. Use an algebraic expression to represent the sum of
the measures of the n straight angles?
b. Use an algebraic expression to represent the sum of
the measures of the n interior angles?
c. Using your answers above, what is the sum of the
measures of the n exterior angles?
d. What theorem do the steps above prove?
1. For each regular polygon, state the sum of the
measures of the interior angles and give the
measure of an interior angle.
Question II: A triangle has two congruent interior angles and
an exterior angle that measures 100. Find two possible sets
of interior angle measures for the triangle?
2. For each regular polygon, state the sum of the
measures of the exterior angles and give the
measure of an exterior angle.
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Math iii
Unit # 2
Unit Title: Geometry
Similarity: (part 2)
Enduring understanding (Big Idea): Students will be able to apply their previous knowledge of transformations to
determine similarity. They will focus on proving and using the AA similarity theorem and will apply all information to solve
problems, which include finding angle measures and side lengths.
Essential Questions:
1. How can you determine whether two figures are similar using similarity transformations, angle measures, and side
lengths?
2. What is the AA Similarity theorem and why does it sufficiently determine whether two triangles are similar or not?
3. How can you prove that a line parallel to one side of a triangle divides the other two sides proportionally? How can
this information be used to solve problems?
BY THE END OF THIS UNIT:
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Math iii
Unit # 2
Students will know…
· AA Similarity Theorem
· A line parallel to one side of a triangle divides the other
two proportionally, and conversely
Vocabulary:
· Similarity, similar triangles, AA Similarity theorem,
transformation, corresponding angles, corresponding
sides (corresponding parts), proportion
Students will be able to…
· Prove the AA Similarity Theorem
· Determine whether two figures and/or triangles are
similar using similarity transformations
· Apply knowledge of similar triangles to solve problems,
which include setting up proportions, and finding angle
measures & side lengths.
Unit Resources
Learning Task:
1. 7.2 Think About a Plan; 2. 7.3 Think About a Plan
*Both located at www.pearsonsuccessnet.com *
Performance Task: Ch 7 Performance Tasks:
www.pearsonsuccessnet.com
Project:
Visit link below. Note: This project will need to be edited
and personalized before implementation.
https://vasicek.wikispaces.com/file/view/The+Similar+Triang
les+Project.pdf
Mathematical Practices in Focus:
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of
others
6. Attend to precision
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Unit Title: Geometry
Course Name: Math iii
Unit # 2
Unit Title: Geometry
CORE CONTENT
Cluster Title: Understand similarity in terms of similarity transformations
Standard: G-SRT.2 – Given two figures, use the definition of similarity in terms of similarity transformations to decide if
they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all
corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
Concepts and Skills to Master:
Determine if two figures are similar using properties of transformations
Determine if two triangles are similar, given their angle measures and side lengths
Given angle measures and side lengths, determine if two triangles are similar
SUPPORTS FOR TEACHERS
Critical Background Knowledge
8.G.4 – Understand similarity as a sequence of transformations
Knowledge of the different types of transformations (rotations, reflections, translations, and dilations).
Academic Vocabulary
Similarity, Transformation, Congruence, Corresponding Parts
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Math iii
Unit # 2
Unit Title: Geometry
Suggested Instructional Strategies:
Resources:
· Give students pairs of triangles, some of which are
similar and some of which are not. Have students verify
1. Textbook Correlation:
or disprove similarity using transformations and the
o 7.2 Similar Polygons
definition of similarity.
· Use geometry software and patty paper to explore
properties of similarity.
Sample Assessment Tasks
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Math iii
Skill-based task
Unit # 2
Unit Title: Geometry
Problem Task
1. Jan uses an overhead projector to enlarge a picture 5 in.
high and 7 in. wide. She projects the picture on a
1.
blackboard 4 ft 2 in. high and 12 ft wide. What are the
dimensions of the largest picture that can be projected 2.
on the blackboard?
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
http://illustrativemathematics.org/illustrations/603 – “Are
they similar” activity
Under what conditions do two lines intersected by two
transversals form similar triangles?
Course Name: Math iii
Unit # 2
Unit Title: Geometry
CORE CONTENT
Cluster Title: Understand similarity in terms of similarity transformations
Standard: G-SRT.3 – Use the properties of similarity transformations to establish the AA criterion for two triangles to be
similar.
Concepts and Skills to Master:
Prove using the properties of similarity transformations that if two angles of one triangle are congruent to two angles of
another triangle, the triangles are similar (AA).
SUPPORTS FOR TEACHERS
Critical Background Knowledge
· 8.G.4 – Understand similarity as a sequence of transformations
· 8.G.5 – Use informal arguments to establish facts about the angle sum…of triangles and…the angle-angle
criterion for similarity of triangles
· Knowledge of the different types of transformations (rotations, reflections, translations, and dilations).
· If two angles of a triangle are congruent to two corresponding angles of a second triangle, then the third
pair of corresponding angles must be congruent.
Academic Vocabulary
Similarity, Transformation, Angle-Angle Similarity Theorem
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Math iii
Suggested Instructional Strategies:
· Review the Triangle Angle Sum Theorem and angle
relationships with parallel lines prior to addressing this
standard.
Sample Assessment Tasks
Skill-based task
Unit # 2
Unit Title: Geometry
Resources:
1. Textbook Correlation
a. 7.3 Proving Triangles are Similar
2. Puzzle: Similarity Search
a. Located at www.pearsonsuccessnet.com
under the “Activities, Games, and Puzzle”
link for section 7.3
Problem Task
1. Determine whether the two triangles are similar. Justify 1. Given two different-sized triangle cutouts with two
corresponding angles congruent, allow the students to
your conclusion.
show that the third angle is congruent, and find a dilation
that produced the two triangles.
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Math iii
Unit # 2
Unit Title: Geometry
Teacher Created Argumentation Tasks
· Your classmate provides the following solutions to the problems below. In complete sentences, identify and explain
the error in each explanation, and tell me how you would help your classmate reach an accurate conclusion. (From
www.pearsonsuccessnet.com – Chapter 7: Find the Errors! for sections 7.2-7.3)
CORE CONTENT
Cluster Title: Prove theorems involving similarity
Standard: G-SRT.4 – Prove theorems about triangles. Theorems include: A line parallel to one side of a triangle divides
the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
Concepts and Skills to Master:
· Prove that if a line intersecting a triangle is parallel to one of the sides, then it divides the other two sides of that
triangle proportionally.
· Prove that if a line divides two sides of a triangle proportionally, then it is parallel to the third side.
SUPPORTS FOR TEACHERS
Critical Background Knowledge
· Understand Angle-Angle Similarity
· Ability to set up and solve proportions
Academic Vocabulary
Parallel Lines, Similar triangles, Angle-Angle Similarity, Proportion
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Math iii
Unit # 2
Unit Title: Geometry
Suggested Instructional Strategies:
Resources:
Help students understand that the sides are proportional for · Textbook Correlation:
any segment parallel to the base, not just the midsegment.
o 7.3 Proving Triangles are Similar
o 7.4 Similarity in Right Triangles
o 7.5 Proportions in Triangles
• 7.5 Enrichment located at www.pearsonsuccessnet.com
· Section B: Right Triangles, Altitudes, the
Pythagorean Theorem, and You (Located at
www.discoveryeducation.com)
http://player.discoveryeducation.com/index.cfm?guidAssetId=
A42740C7-7915-4A90-A1CB-DD891170FFF5
Sample Assessment Tasks
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Math iii
Skill-based task
1.
Unit # 2
Unit Title: Geometry
Problem Task
Is segment SU parallel to segment RV? Explain why or
1. Given: T is the midpoint of
, U is the midpoint of
why not.
and V is the midpoint of
.
Prove: ∆QRS ~ ∆VUT
*This problem found at www.pearsonsuccessnet.com under
the Enrichment task for section 7.3*
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
,
Course Name: Math iii
CORE CONTENT
Unit # 2
Unit Title: Geometry
Cluster Title: Prove theorems involving similarity
Standard: G-SRT.5 – Use congruence and similarity criteria for triangles to solve problems and to prove relationships in
geometric figures.
Concepts and Skills to Master:
Apply knowledge of congruent and/or similar triangles to find scale factor, angle measures, side lengths, and other
measurements such as perimeter and area.
Apply knowledge of congruent and/or similar triangles to determine the similarity of triangles based on various given
information.
SUPPORTS FOR TEACHERS
Critical Background Knowledge
· Students should be able to identify corresponding parts of triangles
· Set up and solve proportions
· Understand that similar figures have congruent corresponding angles and proportional corresponding sides
Academic Vocabulary
Similar triangles, proportions, scale factor
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Math iii
Unit # 2
Unit Title: Geometry
Suggested Instructional Strategies:
Resources:
· Challenge students to find real-world examples of
· Textbook Correlation:
similar triangles on their own. Encourage them to prove
o 7.3 Proving Triangles are Similar
how they were able to conclude that the triangles were
o 7.4 Similarity in Right Triangles
similar.
· Videos:
o “Section A: Proving the Similarity of Triangles”
located at www.discoveryeducation.com
http://player.discoveryeducation.com/index.cfm?guidAssetId=
F6D5649D-C548-40B4-9546DB85CA892F02&blnFromSearch=1&productcode=US
Sample Assessment Tasks
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Math iii
Skill-based task
Unit # 2
Unit Title: Geometry
Problem Task
1. At 4:00 pm, Karl stands next to his house and measures
his shadow and the house’s shadow. Karl’s shadow is 8
ft. long and the house’s shadow is 48 ft. long. If Karl is 6
ft tall, how tall is his house?
2. From www.pearsonsuccessnet.com, 7.4 Enrichment
assignment (specifically #6-7)
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Math iii
Conic Sections: (part 3)
Unit # 2
Unit Title: Geometry
Enduring understanding (Big Idea): Students will understand that concepts related to circles and conic sections are
applicable in real world scenarios as they explore properties of tangent lines, represent circles as equations based on the
center and radius, calculate the measures of central angles, inscribed angles, and their intercepted arc measures and
lengths.
Essential Questions:
1. Discuss what it means to “go off on a tangent”?
2. How do you find the equation of a circle in a coordinate plane?
3. When lines intersect a circle or intersect within a circle, how do you find the measure of resulting angles, arcs, and
segments?
4. How can you prove relationships between angles and arcs in a circle?
5. What is the intersection of a cone and a plane parallel to a line along side of the cone?
6. How can you derive the equation for a parabola, given a focus and directrix?
BY THE END OF THIS UNIT:
Students will know…
· Properties of tangent lines as it relates to a circle
· Concepts of chords, arcs, and angle measures as it
relates to a circle
· Arc Length and Segment Lengths as it relates to
circles
· Equations of a Circle and Parabola
· Conic Sections
Vocabulary: arc measure, arc length, inscribed angle,
intercepted arc, chord, point of tangency, tangent line
(tangent to a circle), secant, standard form of the equation
of a circle, conic sections, directrix, focus, parabola,
ellipse, hyperbola
Unit Resources
Learning Task:click on Circle Formulas – download file, print, and
copy
http://www.mathworksheetsgo.com/sheets/geometry/circles/circl
e-formula-graphic-organizer.php
Performance Task:Have students view the power point
presentation. In writing, allow students to describe how well the
presentation reflects what was learned in class. Be sure to include
what concepts were discussed and which were left
out.www.btinternet.com/~mathsanswers/CircleTheorems.ppt
Unit Review Game:Jeopardy Review
Gamehttp://www.superteachertools.com/jeopardyx/jeopardy-reviewgameconvert.php?gamefile=../jeopardy/usergames/May201221/jeopardy133
7974033.txt
Students will be able to…
· Identify a tangent and use properties of tangent as it
relates to a circle
· Compute chord, arc, and angle measures
· Find arc length given the arc’s central angle and the
circle’s diameter or radius
· Find lengths of segments related to circles and its
intersecting lines
· Write the equation of a circle given its center and
radius
· Identify conic sections
· Write the equation of a parabola given its directrix
and focus.
Mathematical Practices in Focus:
1. Make sense of problems and persevere in solving
them.
2. Reason abstractly and quantitatively.
4. Model with mathematics.
6. Attend to precision.
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
NOTE: For Unit Resources, the Performance
Task Activity can
also be a Project.
Course Name: Math iii
Unit # 2
Unit Title: Geometry
Cluster Title: Find arc lengths and areas of sectors of circles
Standard: G.C.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the
radius, and define the radian measure of the angle as the constant of proportionality
Concepts and Skills to Master
•
•
•
Identify major and minor arcs and semicircles
Find the measure of a central angle and its intercepted arc
Compute the circumference of a circle and arc length (i.e. distances along circular paths)
SUPPORTS FOR TEACHERS
Critical Background Knowledge
• Circumference of a Circle
• Exact Circumference (leave your answer in terms of pi)
• Congruent circles have congruent radii
Academic Vocabulary
circle, center, diameter, radius, congruent circles, central angle, semicircle, minor arc, major arc, adjacent arcs,
intercepted arc, circumference, pi, concentric circles, arc length, congruent arcs, exact circumference
Suggested Instructional Strategies
Resources
· Be sure to highlight for students that an arc is
· Textbook Correlation: 10-6 Circles and Arcs
measured by the central angle that defines it. The
central angle captures within its rays the
· Online Teacher Resource Center:
intercepted arcs.
www.pearsonsuccessnet.com
· Error Prevention: Students may benefit from
Activities, Games, and Puzzles (10-6 Circles and Arcs
tracing the cited arc(s) of the figure(s) with colored
Crossword)
pencils
· Explain to students that as it relates to standard
· Commonly Confused: Arc Measure & Arc Length
G.C.5, the length of an arc can be found by
Bright storm Video – use the link below
multiplying the ratio of the arc’s measure to 360
degrees by the circle’s circumference.
www.brightstorm.com/math/geometry/.../arc-length/
· Students often confuse arc measure with arc
length. Be sure to note that one is measured in
degrees and the other is measured in units.
Sample Formative Assessment Tasks
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Math iii
Skill-based task
Find the arc measure and arc length of each
darkened arc.
Leave your answer in terms of π.
Unit # 2
Unit Title: Geometry
Problem Task
Task: It is 5:00. What is the measure of
theminor arc formed by the hands of
an analog clock hanging on a
classroom wall? What is the arc
length if the radius of the clock is 6
inches?
Sketch a wall clock to support your answer.
Cluster Title: Understand and apply theorems about circles
Standard: G.C.2Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship
between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a
circle is perpendicular to the tangent where the radius intersects the circle.
Concepts and Skills to Master
·
·
·
·
Tangent Lines
Chord and Arc Measures
Central and Inscribed Angles
Angle Measures and Segment Lengths
SUPPORTS FOR TEACHERS
Critical Background Knowledge
· Students will use understanding of congruent triangles and right triangles to prove statements about tangent lines.
· Prior knowledge of a circle and its common features are needed: center, radius, diameter, chord, arc.
· Triangle Angle Sum Theorem
· Pythagorean Theorem
· Perimeter of Polygons
· Congruence
Academic Vocabulary
tangent to a circle, point of tangency, inscribed circles, chord, arc, semicircle, inscribed angles, circumscribed polygons,
secant
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Math iii
Suggested Instructional Strategies
• Students sometimes get confused identifying
segments of a circle. Have students create a
vocabulary sheet that includes definitions and
diagrams of each type of segment.
• Students sometimes get confused identifying
central and inscribed angles and, therefore, use
the wrong formula to compute angle measures.
Perhaps making a connection that a central
angle has its vertex in the center of the circle
will help students distinguish between the two.
• Paper folding activities offer students a good
way to develop key concepts related to central
angles, chords, and arcs.
• Have students to organize all the theorems
taught in sections 12.1 to 12.4 in an effort to
increase learning.
Unit # 2
Unit Title: Geometry
Resources
•
Cluster Review
http://library.thinkquest.org/20991/geo/circles.html
• Circle Concept Interactive Math Site
http://www.mathopenref.com/chordsintersecting.html
(Explore circle concept by scrolling down and clicking from the
selection on the bottom left of the screen)
•
Concept Byte Exploration Activity: p.770 - Paper Folding
With Circles
Critical Background Knowledge
· Domain and Range
· Graphing on a Coordinate Plane
· Lines of Symmetry
Academic Vocabulary
conic sections (parabolas, circles, ellipses, and hyperbolas), lines of symmetry, focus, directrix, focal length
Suggested Instructional Strategies
Resources
· At this point, do not make graphing the conic
· Textbook Correlation: Algebra II Textbook
sections a more difficult task by having students
10-1 Exploring Conic Sections
solve for x and or y. Instead, simply have
(www.pearsonsuccessnet.com)
student graph conic sections using a table of
10-2 Parabolas (www.pearsonsuccessnet.com)
values that range from -5 to +5; substituting for
· Conic Sections Explained
whichever variable is easier. [Note: If you have
http://math2.org/math/algebra/conics.htm
a classroom set of graphing calculators, you
· Parabolas and Their Equations Powerpoint
https://docs.google.com/viewer?a=v&q=cache:epOo8GE
may want students to practice solving for y in
order to use the equation editor and table of
PeOIJ:princemath.wikispaces.com/file/view/parabolas.pp
values.] (Also, note that more emphasis will
t+parabola+and+its+equations+powerpoint&hl=en&gl=us
be placed on conic sections in further math
&pid=bl&srcid=ADGEESh5fKhyjqpZxcMuqaQOU5kouLH
LYDR4TuYHy5eWBU8yqGviMzQqb_iESTO7MRFVXhc3
courses.)
mKlAOn· Be sure that students understand that a conic
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section is simply the intersection of a plane and
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4HTR3RP-aRLQ
a cone. (Use resource: Conic Sections
Explained as a teaching aid if needed.)
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Math iii
Unit # 2
Unit Title: Geometry
Cluster Title: Understand and apply theorems about circles (i.e. Circle Similarity)
Standard: G.C.1 Prove that all circles are similar.
Concepts and Skills to Master
•
Prove Similarity in Circles
SUPPORTS FOR TEACHERS – NOTE:This concept is not in the textbook and limited information appropriate for HS
students is available online.
Critical Background Knowledge
• Definition of Similarity
• Applications with Circle Formulas and Right Triangles
Academic Vocabulary
similarity of circles
Suggested Instructional Strategies
Resources
· Recall: being similar means having corresponding
· Textbook Correlation: none
congruent angles but proportional corresponding
· Online Resource A
sides. See Online Resource A.
Core Challenge – Standard G.C.1 – Prove all circles
· In general, two figures are similar if there is a set of
are similar. Click on ‘download file’.
transformations that will move one figure exactly
http://app.corechallenge.org/learningobjects/7878
covering the other. To view proof, see Online
Resource B.
· Online Resource B – All Circles are Similar
· To prove any two circles are similar, only a
Examples.pdf
translation (slide) and dilation (enlargement or
www.cpm.org/pdfs/state_supplements/Similar_Circle
reduction) are necessary. Using the differences in
s.pdf
the center coordinates to determine the translation
and determining the quotient of the radii for the
· Online Resource C – YouTube Video –
dilation can always do this. For further
All circles are similar demonstration
explanation, see Online Resource C.
http://www.youtube.com/watch?v=jTvlvLFZQPY
· Problem Task:Take students to the lab if possible
to view the you-tube video that teaches the lesson
on circle similarity. If students do not have access
to the site, save the link elsewhere so that students
can view it – or make it a homework assignment.
(Honor and IB Classes only) If the video is used for
Standard classes, teacher explanation and
modeling is necessary.
Sample Formative Assessment Tasks
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Math iii
Skill-based task
Unit # 2
Unit Title: Geometry
Problem Task(see suggested instructional strategies – item
4)
Use the link below to view the 32min 20 sec you-tube video
that discusses circle similarity. Take notes during the video.
After viewing the video, complete a written assignment,
documenting what you have learned.
Link: http://www.youtube.com/watch?v=2QOj02EKDTE
CORE CONTENT
Cluster Title: Apply geometric concepts in modeling situations
Standard: G.MG.3- Apply geometric methods to solve design problems (e.g. designing an object or structure to satisfy
physical constraints or minimize cost; working with typographic grid systems based on ratios.
Concepts and Skills to Master
Students will be able to solve design problems by designing an object or structure that satisfies certain constraints.
SUPPORTS FOR TEACHERS
Critical Background Knowledge
Parts of a right triangle and congruent corresponding parts
Academic Vocabulary
congruent triangles, hypotenuse, legs of a right triangle
Suggested Instructional Strategies:
Resources:
· Textbook Correlation: Pearson Chapter 11.2, 3, 4,
5, 6
· Perfume Packaging –Dana Center chapter 5 pg 1
Sample Assessment Tasks
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Math iii
Skill-based task
Unit # 2
Problem Task
Pearson Additional Problems
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Unit Title: Geometry