Download Unit 4 - Long Beach Unified School District

Unit 4 Polygons
Geometry
Unit Goals – Stage 1
Number of Days: 14 days
1/30/17 – 2/17/17
Unit Description: Drawing on our knowledge of the definitions of special quadrilaterals and a few of their properties, students will now prove those
properties using a variety of proof formats: transformational, analytical, synthetic and paragraph. Students will begin this unit by deriving formulas for the
sum of the interior angles in a polygon and the sum of the exterior angles in a polygon. Drawing on our knowledge of triangle congruence from Unit 2,
students will now apply that knowledge to proofs involving quadrilaterals. Our family of special quadrilaterals includes parallelograms, rectangles, rhombi,
squares, kites, trapezoids and isosceles trapezoids.
Materials: Patty paper, rulers, compasses, protractors
Standards for Mathematical Practice
Transfer Goals
SMP 1 Make sense of problems and
Students will be able to independently use their learning to…
persevere in solving them.
• Make sense of never-before-seen problems and persevere in solving them.
SMP 2 Reason abstractly and
• Construct viable arguments and critique the reasoning of others.
quantitatively.
SMP 3 Construct viable arguments
Making Meaning
and critique the reasoning of
UNDERSTANDINGS
ESSENTIAL QUESTIONS
others.
Students will understand that…
Students will keep considering…
SMP 4 Model with mathematics.
• Triangle congruence criteria can be used to
• How can the properties of a triangle be used to prove
SMP 5 Use appropriate tools
prove properties of parallelograms.
the properties of a quadrilateral?
strategically.
• Simple theorems about coordinates, angles,
• What properties of a quadrilateral can you use to
SMP 6 Attend to precision.
segments and slope can be used to prove the
identify its shape precisely?
SMP 7 Look for and make use of
properties of the subsets of polygons.
• How are properties of geometric figures related to their
structure.
• Coordinates can be used to prove relationships
measurable attributes?
SMP 8 Look for and express
between the lines and angles of quadrilaterals
• How does graphing a quadrilateral on a coordinate
regularity in repeated
on a coordinate grid.
plane help to identify its shape precisely?
reasoning.
Standards for Mathematical Content
Clusters Addressed
[m] G-CO.C Prove geometric
theorems
[m] G-SRT.B Prove theorems
involving similarity.
[m] G-MG.A Apply geometric
concepts in modeling
situations.
KNOWLEDGE
Students will know…
• Properties of a parallelogram and
its subsets of quadrilaterals.
• Properties of trapezoids and kites.
• Properties of diagonals and
opposite angles of quadrilaterals.
LONG BEACH UNIFIED SCHOOL DISTRICT
2016-2017
1
Acquisition
SKILLS
Students will be skilled at and/or be able to…
• Find and use interior and exterior angle measures of polygons.
• Use properties of parallelograms and special parallelograms to find
side lengths and angles.
• Use the coordinate plane to identify quadrilaterals and solve for
their properties.
• Identify and verify that a quadrilateral is a parallelogram
• Identify and use properties of trapezoids and kites.
Posted 1/31/17
Unit 4 Polygons
Geometry
Assessed Grade Level Standards
Standards for Mathematical Practice
SMP 1 Make sense of problems and persevere in solving them.
SMP 2 Reason abstractly and quantitatively.
SMP 3 Construct viable arguments and critique the reasoning of others.
SMP 4 Model with mathematics.
SMP 5 Use appropriate tools strategically.
SMP 6 Attend to precision.
SMP 7 Look for and make use of structure.
SMP 8 Look for and express regularity in repeated reasoning.
Standards for Mathematical Content
[m] G-CO.C
G-CO.11
Prove geometric theorems. [Focus on validity of underlying reasoning while using variety of ways of writing proofs]
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.
[m] G-SRT.B
G-SRT.5
Prove theorems involving similarity.
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
[m] G-MG.A
G-MG.1
Apply geometric concepts in modeling situations.
Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a
cylinder).
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).
G-MG.3
Key: [m] = major clusters; [s] = supporting clusters; [a] = additional clusters

Indicates a modeling standard linking mathematics to everyday life, work, and decision-making
LONG BEACH UNIFIED SCHOOL DISTRICT
2016-2017
2
Posted 1/31/17
Unit 4 Polygons
Geometry
Evidence of Learning – Stage 2
Assessment Evidence
Unit Assessment
Claim 1: Students can explain and apply mathematical concepts and carry out mathematical procedures with precision and fluency.
Concepts and skills that may be assessed in Claim1:
[m] G-CO.C
•
Students will prove theorems about parallelograms, such as: opposite sides are congruent, opposite angles are congruent, the diagonals of a
parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
[m] G-SRT.B
•
Students will use congruence for triangles to solve problems.
•
Students will use congruence to prove relationships in polynomials.
[m] G-MG.A
•
Students will use geometric shapes, measures and properties to solve real-world problems.
•
Students will apply geometric methods to solve design problems.
Claim 2: Students can solve a range of wellposed problems in pure and applied
mathematics, making productive use of
knowledge and problem-solving strategies.
Standard clusters that may be assessed in
Claim 2:
•
None
Claim 3: The student can clearly and precisely
construct viable arguments to support their own
reasoning and critique the reasoning of others.
Standard clusters that may be assessed in
Claim 3:
•
G-CO.C
•
G-SRT.B
Claim 4: The student can analyze complex,
real-world scenarios and can construct and use
mathematical models to interpret and solve
problems.
Standard clusters that may be assessed in
Claim 4:
•
G-MG.A
Other Evidence
Formative Assessment Opportunities
•
Opening Tasks
•
Informal teacher observations
•
Checking for understanding using active participation strategies
•
Exit slips/summaries
•
Modeling Lessons (SMP 4)
•
•
•
•
•
Tasks
Formative Assessment Lessons (FAL)
Quizzes / Chapter Tests
Big Ideas Math Performance Tasks
SBAC Interim Assessment Blocks
Access Using Formative Assessment for Differentiation for suggestions. Located on the LBUSD website – “M” Mathematics – Curriculum Documents
LONG BEACH UNIFIED SCHOOL DISTRICT
2016-2017
3
Posted 1/31/17
Unit 4 Polygons
Geometry
Learning Plan – Stage 3
Days
1 day
3-4
days
Learning
Target
I will identify a
quadrilateral
based on the
given
diagonals.
I will investigate
the properties
and conditions
of polygons
by…
Suggested Sequence of Key Learning Events and Instruction
Big Ideas Math
Geometry
Expectations
(Activities and
Lessons)
OPENING TASK – Diagonals to Quadrilaterals
Curriculum Intranet
Conceptual Understanding:
• Diagonals to Quadrilaterals
This Opening Task provides the students with an opportunity to explore
the relationship between a pair of intersecting lines (diagonals) and the
resulting quadrilateral. After connecting the endpoints of intersecting
segments, the students will be asked to describe/identify the resulting
quadrilateral. Ask the students to make generalizations and conjectures.
While this activity is best when pairs of students have access to
computers/tablets, the teacher can demonstrate the applet for the class
or ask students to come up to the front and manipulate the figure.
The purpose of this task is to encourage student exploration and
hypothesizing. This task is an opportunity for the students to begin to
exploring and making connections with the properties of quadrilaterals.
• Finding and using interior angle measures to identify a polygon.
• Finding and using exterior angle measures to identify a polygon.
• Using properties to find side lengths and angles of parallelograms.
• Identifying and verifying parallelograms.
• Using a coordinate plane to show that a quadrilateral is a
parallelogram.
• Answering questions such as:
o Are the Polynomial Interior and Exterior Angles Theorems valid
for both convex and concave polynomials?
o Is it possible for a polygon to be equilateral and not equiangular?
o What is the measure of the central angle a regular n-gon?
o Is it possible for a pentagon to have four right angles?
o What is true about the consecutive angles, and sides, of a
parallelogram?
o How do you prove that a quadrilateral is a parallelogram?
o How can you differentiate between parallelograms and other
quadrilaterals (rectangles, rhombuses, and/or rectangles)?
o How can you construct a parallelogram?
o How can you construct a regular hexagon?
o How can you use coordinates to write a proof?
LONG BEACH UNIFIED SCHOOL DISTRICT
2016-2017
4
• Section 7.1
• Section 7.2
• Section 7.3
Conceptual Understanding:
• Perplexing Parallelograms
• Parallelograms Conjectures
and Proofs
• Guess My Parallelogram
• Is this a Parallelogram?
Procedural Skills and
Fluency:
• Square Graphs
• Cutting Corners
• Geogebra: Polygon Interior
Angles
• Geogebra: Car Drives
Through the Exterior Angles
of a Polygon
• Geogebra: Polygons –
Exterior Angles
• Is this a Rectangle?
• Mathematics VisionProject:
Under Construction
Posted 1/31/17
Unit 4 Polygons
Geometry
Learning Plan – Stage 3
Days
Learning
Target
I will investigate
the properties
of special
parallelograms,
trapezoids and
kites by…
•
•
•
•
•
•
•
•
3-4
days
2-3
days
I will check my
knowledge
about the
properties of
quadrilaterals
by participating
in the FAL.
Suggested Sequence of Key Learning Events and Instruction
Big Ideas Math
Geometry
Expectations
(Activities and
Lessons)
Using properties of special parallelograms to prove whether or not
• Section 7.4
the figure is a rhombus, rectangle or square.
• Section 7.5
Using coordinate geometry to identify special types of
• STEM Video:
parallelograms.
Diamonds
Using properties of trapezoids to find angle measures and side
lengths.
Using the Trapezoid Midsegment Theorem to find segment lengths.
Using the properties of kites to find angle measures, diagonals and
side lengths.
Proving properties of kites and trapezoids.
Using properties to classify a quadrilateral.
Answering questions such as:
o How can you differentiate between parallelograms and other
quadrilaterals (rectangles, rhombuses, rectangles, kites and/or
trapezoids)?
o Compare and contrast the diagonal properties of the
quadrilaterals.
o Always, sometimes or never true: A rectangle is a rhombus; a
square is a rectangle; a trapezoid is a parallelogram?
o Which quadrilateral could be called a regular quadrilateral?
o Is SSASS a valid congruence theorem for kites?
o How can you use coordinates to help write a proof?
FORMATIVE ASSESSMENT LESSON
LONG BEACH UNIFIED SCHOOL DISTRICT
2016-2017
Curriculum Intranet
Conceptual Understanding:
• Geogebra: Special
Quadrilaterals
Procedural Skills and
Fluency:
• Geogebra: Properties of
Trapezoids
• Geogebra: Properties of a
Kite
• MathOpenRef: Kite
Applications:
• Floor Pattern Task
• STEM Performance Task:
Diamonds
Procedural Skills and
Fluency:
• FAL: Guess My
Parallelogram
5
Posted 1/31/17
Unit 4 Polygons
Geometry
Learning Plan – Stage 3
Days
1-2
days
Learning
Target
I will prepare
for the unit
assessment on
polygons by...
Suggested Sequence of Key Learning Events and Instruction
Big Ideas Math
Geometry
Expectations
(Activities and
Lessons)
• Incorporating the Standards for Mathematical Practice (SMPs) along
with the content standards to review the unit.
1-2
days
LONG BEACH UNIFIED SCHOOL DISTRICT
2016-2017
Curriculum Intranet
Conceptual Understanding:
• Which One Doesn’t Belong:
Polygons
Procedural Skills and
Fluency:
• Sorting Quadrilaterals
Unit Assessment (LBUSD Math Intranet, Assessment)
6
Posted 1/31/17