Download Regional Integrated Geometry Curriculum

Regional Integrated Geometry Curriculum
UNIT: Informal & Formal Proofs
TOPIC: Logic
Time frame for unit:
Time frame for topic: 10 days
Prior Knowledge
A.RP.2
Use mathematical strategies to reach a conclusion and provide
supportive arguments for a conjecture
A.RP.11 Use a Venn diagram to support a logical argument
Content Strand
G.G.24 Determine the negation of a statement and establish its
truth value
G.G.25 Know and apply the conditions under which a compound statement
(conjunction, disjunction, conditional, biconditional) is true
G.G.26 Identify and write the inverse, converse, and contrapositive of a given
conditional statement and note the logical equivalence
G.G.27 Write a proof (ex. indirect proof) arguing from a given hypothesis to a
given conclusion
Concepts
Logic
a. translate written expressions to symbolic representation (p, q, V, →, etc)
b. negation
c. conjunction
d. disjunction
e. conditional, hypothesis, conclusion, hidden conditional
f. biconditional
g. truth values
h. converse (change order)
i. inverse (insert negation)
j. contrapositive (change order and negate)
k. logically equivalent
Logic Proofs
a.
b.
c.
d.
e.
contrapositive
detachment
syllogism/chain rule (Modus Tollens)
deMorgan’s
disjunctive inference
Essential Questions
How will I use logical reasoning to help me make decisions in my life?
How could logic be used to determine if I’ve been tricked by someone? Or by an
advertisement? Or a commercial?
Process/skills
G.PS.4 Construct various types of reasoning, arguments, justifications and
methods of proof for problems
G.RP.3 Investigate and evaluate conjectures in mathematical terms, using
mathematical strategies to reach a conclusion
G.RP.7 Construct a proof using a variety of methods (e.g., deductive,
analytic, transformational)
G.RP.8 Devise ways to verify results or use counterexamples to refute incorrect
statements
G.CM.5 Communicate logical arguments clearly, showing why a result makes
sense and why the reasoning is valid
G.CM.12 Draw conclusions about mathematical ideas through decoding,
comprehension, and interpretation of mathematical visuals, symbols,
and technical writing
G.CN.6
Recognize and apply mathematics to situations in the outside world
G.R.7
Use mathematics to show and understand social phenomena
(e.g., determine if conclusions from another person’s argument have a
logical foundation)
Vocabulary
antecedent
argument
biconditional
chain rule (syllogism)
compound statement
conclusion
conditional statement
conjecture
conjunction
consequence
contradiction
contrapositive
converse
deductive reasoning
deMorgan’s
detachment
disjunctive inference
hypothesis
inverse
justify
logic
logical argument
logical equivalence
Modus Tollens
negation
reason
statement
tautology
truth value
valid/invalid argument
Suggested assessments







Pre-assessment
Regents/State
exams
Projects
Participation
Q-A Responses
Oral responses
Conversations
Resources




On-spot checks of
class work
Homework
Observations of
students’
work/behaviors
Students’
explanations


Draw a picture
Tests and
Quizzes
 T/F
 Multiple Choice
 Written Response
 Ticket out the
door
See Blackboard for
specific Assessments*
Regional Integrated Geometry Curriculum
UNIT: Informal/ Formal Proofs
TOPIC: Triangles - congruent
Time frame for unit:
Time frame for topic: 20 days
Prior Knowledge
5.G.6 Classify triangles by properties of their angles and sides
5.G.9 Identify pairs of congruent triangles
5.G.10 Identify corresponding parts of congruent triangles
7.G.5 Identify the right angle, hypotenuse, and legs of a right triangle
8.G.1 Identify pairs of vertical angles as congruent
8.G.2 Identify pairs of supplementary and complementary angles
8.G.4 Determine angle pair relationship with two parallel lines cut by a transversal
Content Strands
G.G.27 Write a proof arguing from a given hypothesis to a given conclusion
G.G.28 Determine the congruence of two triangles by using one of the five
congruence techniques (SSS, SAS, ASA, AAS, HL), given
sufficient information about the sides and/or angles of two
congruent triangles
G.G.29 Identify corresponding parts of congruent triangles
Concepts
Identify corresponding parts (sides, angles, appropriate naming)
Identify included side, included angle
Congruent () – define & recognize
Establish the format of a two-column proof
SSS
SAS
ASA
AAS
HL – right triangle
CPCTC—Corresponding Parts of Congruent Triangles are Congruent
Essential Questions
How do you analyze the given information to determine the strategy(s)
needed to complete a proof?
How much information do you need to know to determine whether two triangles
are congruent?
How can you show the logical progression of a proof?
Why would it be important to know that two triangles (polygons) are congruent?
Process/skills
G.PS.4
Construct various types of reasoning, arguments, justifications and
methods of proof for problems
G.PS.8
Determine information required to solve a problem, choose
methods for obtaining the information, and define parameters for
acceptable solutions
G.RP.3
Investigate and evaluate conjectures in mathematical terms,
using mathematical strategies to reach a conclusion
G.RP.7
Construct a proof using a variety of methods (e.g., deductive, analytic,
transformational)
G.CM.1
Communicate verbally and in writing a correct, complete, coherent,
and clear design (outline) and explanation for the steps used in solving
a problem
G.CM.5
Communicate logical arguments clearly, showing why a result makes
sense and why the reasoning is valid
G.CN.2
Understand the corresponding procedures for similar problems or
mathematical concepts
G.R.8
Use mathematics to show and understand mathematical phenomena
(e.g., use investigation, discovery, conjecture, reasoning, arguments,
justification and proofs to validate that the two base angles of an
isosceles triangle are congruent)
Vocabulary
Alternate interior angles
Altitude
Analytical proof
Angle
Angle addition &
subtraction
Angle bisector
Assumption statement
Bisector
Complementary angles
Corresponding parts
Deductive reasoning
Equidistant
Given
Hypotenuse
Included side & angle
Indirect proof
Inductive reasoning
Interior angles
Leg
Midpoint
Paragraph Proof
Parallel Lines
Perpendicular
Perpendicular bisector
Proof
Proof by Contradiction
Reflexive
Statement/Reason Two column proofs
Substitution
Supplementary angles
Transversal
Suggested assessments







Pre-assessment
Regents/State
exams
Projects
Participation
Q-A Responses
Oral responses
Conversations




On-spot checks of
class work
Homework
Observations of
students’
work/behaviors
Students’
explanations
Resources
Review & extend from the previous units
addition & subtraction properties (angles and sides)
altitude (GG21)
base angles of an isosceles triangle (GG31)
bisector & angle bisector (GG17)
median (GG21)
midpoint (GG18)
// lines, transversals, and angles (GG35)
perpendicular lines & right angles (GG18)
reflexive, symmetric, and transitive properties
substitution/transitive property
two angles forming a linear pair
vertical angles


Draw a picture
Tests and
Quizzes
 T/F
 Multiple Choice
 Written Response
 Ticket out the
door
See Blackboard for
specific Assessments*
Regional Integrated Geometry Curriculum
UNIT: Informal/ Formal Proofs
TOPIC: Triangles - similar
Time frame for unit:
Time frame for topic: 15 days
Prior Knowledge
5.G.2
Identify pairs of similar triangles
5.G.3
Identify the ratio of corresponding sides of similar triangles
6.G.1
Calculate the length of corresponding sides of similar triangles,
using proportional reasoning
7.G.5
Identify the right angle, hypotenuse, and legs of a right triangle
7.G.6
Explore the relationship between the lengths of the three sides of a
right triangle to develop the Pythagorean Theorem
7.G.8
Use the Pythagorean Theorem to determine the unknown length of
a side of a right triangle
7.G.9
Determine whether a given triangle is a right triangle by applying
the Pythagorean Theorem and using a calculator
A.G.1
Find the area and/or perimeter of figures composed of polygons and
circles or sectors of a circle Note: Figures may include triangles,
rectangles, squares, parallelograms, rhombuses, trapezoids, circles,
semi-circles, quarter-circles, and regular polygons (perimeter only).
A.A.25 Solve equations involving fractional expressions
Note: Expressions which result in linear equations in one variable.
A.A.26
Solve algebraic proportions in one variable which result in linear
or quadratic equations
Content Strands
G.G.42
Investigate, justify, and apply theorems about geometric relationships,
based on the properties of the line segment joining the midpoints of
two sides of the triangle.
G.G.44
Establish similarity of triangles, using the following theorems:
AA, SAS, and SSS
G.G.45
Investigate, justify, and apply theorems about similar triangles
G.G.46
Investigate, justify, and apply theorems about proportional
relationships among the segments of the sides of the triangle,
given one or more lines parallel to one side of a triangle and
intersecting the other two sides of the triangle
G.G.47
Investigate, justify, and apply theorems about mean proportionality:
--the altitude to the hypotenuse of a right triangle is the
mean proportional between the two segments along
the hypotenuse
--the altitude to the hypotenuse of a right triangle divides the
hypotenuse so that either leg of the right triangle is the
mean proportional between the hypotenuse and segment
of the hypotenuse adjacent to that leg
G.G.48
Investigate, justify, & apply the Pythagorean theorem &
its converse
Review & extend from the previous units
Various theorems on working with angles and segments
Concepts
Similar triangles - dilation
AA (~) Proofs
SSS (~) Proofs
SAS (~) Proofs
Pythagorean theorem & its converse
Mean proportional in right triangles (geometric mean)
CPSTP—Corresponding Parts of Similar Triangles are in Proportion
Essential Questions
How are similar triangles different from congruent triangles?
How are similar triangles the same as congruent triangles?
How can similar triangles be used in a career/real-life?
Why is the Pythagorean theorem useful in building a house?
Process/skills
G.PS.4
Construct various types of reasoning, arguments, justifications
and methods of proof for problems
G.PS.8
Determine information required to solve a problem, choose
methods for obtaining the information, and define
parameters for acceptable solutions
G.RP.2
Recognize and verify, where appropriate, geometric relationships
of perpendicularity, parallelism, congruence, and similarity,
using algebraic strategies
G.RP.5
Present correct mathematical arguments in a variety of forms
G.CM.12
Draw conclusions about mathematical ideas through decoding,
comprehension, and interpretation of mathematical visuals,
symbols, and technical writing
G.CN.2
Understand the corresponding procedures for similar problems
or mathematical concepts
G.R.4
Select appropriate representations to solve problem situations
G.R.8
Use mathematics to show and understand mathematical
phenomena (e.g., use investigation, discovery, conjecture,
reasoning, arguments, justification and proofs to validate that
the two base angles of an isosceles triangle are congruent)
Vocabulary
Alternate interior angles
Altitude
Analytical proof
Angle
Angle addition &
subtraction
Angle bisector
Area
Bisector
Complementary angles
Corresponding parts
Cross-products
Deductive reasoning
Equidistant
External angle
Hypotenuse
Included side & angle
Indirect measurement
Inductive reasoning
Interior angles
Leg
Mean proportional
Means
Midpoint
Mid-segment of a
triangle
Paragraph Proof
Parallel Lines
Perimeter
Perpendicular
Perpendicular bisector
Proof
Proof by Contradiction
Proportion
Proportional segments
Pythagorean theorem
Ratio
Reflexive
Remote angle
Scale factor
Similar
Statement/Reason –
2column proofs
Substitution
Extremes
Geometric mean
Given
Suggested assessments
 Pre-assessment
 Regents/State
exams
 Projects
 Participation
 Q-A Responses
 Oral responses
 Conversations
Resources
Fractals




On-spot checks of
class work
Homework
Observations of
students’
work/behaviors
Students’
explanations


Draw a picture
Tests and
Quizzes
 T/F
 Multiple Choice
 Written Response
 Ticket out the
door
See Blackboard for
specific Assessments*
Regional Integrated Geometry Curriculum
UNIT: Informal/ Formal Proofs
TOPIC: Triangles – properties
Time frame for unit:
Time frame for topic: 7 days
Prior Knowledge
5.G.6
Classify triangles by properties of their angles and sides
5.G.7
Know that the sum of the interior angles of a triangle is 180 degrees
5.G.8
Find a missing angle when given two angles of a triangle
Content Strands
G.G.30
Investigate, justify, and apply theorems about the sum of the
measures of the angles of a triangle
G.G.31
Investigate, justify, and apply the isosceles triangle theorem
and its converse
G.G.32
Investigate, justify, and apply theorems about geometric inequalities,
using the exterior angle theorem
G.G.33
Investigate, justify, and apply the triangle inequality theorem
G.G.34
Determine either the longest side of a triangle given the three angle
measures or the largest angle given the lengths of three sides
of a triangle
Concepts
Angle sum in a triangle (measuring angles: protractor, G.S. software?)
*review classification using acute, obtuse, right,
scalene, isosceles, equilateral
Isosceles triangles & properties
*investigate, apply,
*proof, indirect proof (assumption statement, proof by contradiction)
Exterior angle theorem
*investigate, apply, proof, indirect proof
Triangle inequality theorem
*investigate, apply, proof, indirect proof
Essential Questions
How are triangles used in the construction industry?
Why do all interior angles of a triangle add up to 180 degrees?
Why is the longest side of a triangle opposite the largest angle?
Why does the sum of any two sides of a triangle have to be greater
than the measure of the third side?
Process/skills
G.PS.5
Choose an effective approach to solve a problem from a variety
of strategies (numeric, graphic, algebraic)
G.RP.8
Devise ways to verify results or use counterexamples to refute
incorrect statements
G.RP.9
Apply inductive reasoning in making and supporting
mathematical conjectures
G.CM.3
Present organized mathematical ideas with the use of
appropriate standard notations, including the use of symbols
and other representations when sharing an idea in verbal
and written form
G.CM.12
Draw conclusions about mathematical ideas through decoding,
comprehension, and interpretation of mathematical visuals,
symbols, and technical writing
G.CN.3
Model situations mathematically, using representations to draw
conclusions and formulate new situations
G.R.1
Use physical objects, diagrams, charts, tables, graphs, symbols,
equations, or objects created using technology as
representations of mathematical concepts
G.R.8
Use mathematics to show and understand mathematical phenomena
(e.g., use investigation, discovery, conjecture, reasoning, arguments,
justification and proofs to validate that the two base angles of an
isosceles triangle are congruent)
Vocabulary
Acute
Angle
Angle bisector
Alternate interior angles
Altitude
Base angles
Bisector
Complementary angles
Exterior angle
Suggested assessments
 Pre-assessment
 Regents/State
exams
 Projects
 Participation
 Q-A Responses
 Oral responses
 Conversations
Resources
Hypotenuse
Indirect proof
Inductive reasoning
Interior angles
Isosceles
Leg
Median
Midpoint
Obtuse




On-spot checks of
class work
Homework
Observations of
students’
work/behaviors
Students’
explanations
Parallel Lines
Paragraph Proof
Perpendicular
Perpendicular bisector
Proof
Proof by Contradiction
Scalene
Statement/Reason 2column proofs
Substitution


Draw a picture
Tests and
Quizzes
 T/F
 Multiple Choice
 Written Response
 Ticket out the
door
See Blackboard for
specific Assessments*
Regional Integrated Geometry Curriculum
UNIT: Formal & Informal Proofs
Time frame for unit:
TOPIC: Centroid: Concurrent Lines of Triangles Time frame for topic: 1 Day
Prior Knowledge
8.G.0 Construct the following, using a straight edge and compass:
Segment congruent to a segment
Angle congruent to an angle
Perpendicular bisector
Angle bisector
G.G.21 Investigate and apply the concurrence of medians, altitudes, angle
bisectors, and perpendicular bisectors of triangles
Content Strand
G.G.43 Investigate, justify, and apply theorems about the centroid of a triangle,
dividing each median into segments whose lengths are in the ratio 2:1
Concepts
Proportions
Parallel lines
Median
Centroid
Essential Questions
Why is the centroid dividing the median into a 2:1 proportion?
Process/skills
G.PS.2 Observe and explain patterns to formulate generalizations and
conjectures
G.RP.2 Recognize and verify, where appropriate, geometric relationships of
perpendicularity, parallelism, congruence, and similarity, using algebraic
strategies
G.CM.5 Communicate logical arguments clearly, showing why a result makes
sense and why the reasoning is valid
G.CN.6 Recognize and apply mathematics to situations in the outside world
G.R.4 Select appropriate representations to solve problem situations
Vocabulary
Altitude
Centroid
Construct
Median
Perpendicular bisector
Point of Concurrency
Suggested assessments
 Pre-assessment
 Regents/State
exams
 Projects
 Participation
 Q-A Responses
 Oral responses
 Conversations
Resources




On-spot checks of
class work
Homework
Observations of
students’
work/behaviors
Students’
explanations


Draw a picture
Tests and
Quizzes
 T/F
 Multiple Choice
 Written Response
 Ticket out the
door
See Blackboard for
specific Assessments*
Regional Integrated Geometry Curriculum
UNIT: Formal & Informal Proofs
TOPIC: Proportional Parts of Triangles
Midsegments
Time frame for unit:
Time frame for topic: 1 day
Prior Knowledge
6.G.1 Calculate the length of corresponding sides of similar triangles,
using proportional reasoning
8.G.0 Construct the following, using a straight edge and compass:
a. Segment congruent to a segment
b. Angle congruent to an angle
c. Perpendicular bisector
d. Angle bisector
Content Strand
G.G.42 Investigate, justify, and apply theorems about geometric relationships,
based on the properties of the line segment joining the midpoints of two
sides of the triangle
Concepts
Proportions
Properties of the line segment joining the midpoints of 2 sides of the triangle
Similar Triangles
Essential Questions
How are parallel lines and triangles used in real life?
Why are the midsegments proportional to the parallel side?
Process/skills
G.PS.2 Observe and explain patterns to formulate generalizations and
conjectures
G.RP.2 Recognize and verify, where appropriate, geometric relationships of
perpendicularity, parallelism, congruence, and similarity, using algebraic
strategies
G.CM.5 Communicate logical arguments clearly, showing why a result makes
sense and why the reasoning is valid
G.CN.6 Recognize and apply mathematics to situations in the outside world
G.R.4 Select appropriate representations to solve problem situations
Vocabulary
Geometric relationship
Median
Midpoint
Mid-Segment
Parallel
Proportions
Similar triangles
Suggested assessments
 Pre-assessment
 Regents/State
exams
 Projects
 Participation
 Q-A Responses
 Oral responses
 Conversations
Resources




On-spot checks of
class work
Homework
Observations of
students’
work/behaviors
Students’
explanations


Draw a picture
Tests and
Quizzes
 T/F
 Multiple Choice
 Written Response
 Ticket out the
door
See Blackboard for
specific Assessments*
Regional Integrated Geometry Curriculum
UNIT: Formal and Informal Proofs
TOPIC: Polygons
Time frame for unit:
Time frame for topic: 2 Days
Prior Knowledge
7.G.7 Find a missing angle when given angles of a quadrilateral
8.G.2 Identify pairs of supplementary and complementary angles
8.G.3 Calculate the missing angle in a supplementary or complementary pair
8.G.4 Determine angle pair relationships when given two parallel lines cut by a
transversal
8.G.5 Calculate the missing angle measurements when given two parallel lines
cut by a transversal
8.G.6 Calculate the missing angle measurements when given two intersecting
lines and an angle
Content Strands
G.G.36 Investigate, justify, and apply theorems about the sum of the measures of
the interior and exterior angles of polygons
G.G.37 Investigate, justify, and apply theorems about each interior and exterior
angle measure of regular polygons
Concepts
Interior/Exterior Angles of Polygons
a. Exterior angle of a triangle = sum of remote interior angles
b. Sum of the measures of the interior angles
c. Sum of the measures of the exterior angles
d. Measure of an interior angle of a regular polygon
e. Measure of an exterior angle of a regular polygon
Polygons
a. Classify polygons by sides
b. Classify polygons by properties
Essential Questions
Where does the formula for the sum of the interior angles come from?
How does the sum of the interior/exterior angles relate to the number of sides?
How can we determine the measure of an interior angle of a regular polygon given
the number of sides?
What is the correlation between the number of sides to the sum of the interior
angle measures?
How many diagonals can be drawn in a polygon?
Process/skills
G.PS.2 Observe and explain patterns to formulate generalizations and
conjectures
G.RP.3 Investigate and evaluate conjectures in mathematical terms, using
mathematical strategies to reach a conclusion
G.CM.4 Explain relationships among different representations of a problem
G.CN.1 Understand and make connections among multiple representations of the
same mathematical idea
G.R.1
Use physical objects, diagrams, charts, tables, graphs, symbols,
equations, or objects created using technology as representations of
mathematical concepts
Vocabulary
Adjacent angles
Decagon
Diagonal
Dodecagon
Exterior angles
Heptagon
Suggested assessments
 Pre-assessment
 Regents/State
exams
 Projects
 Participation
 Q-A Responses
 Oral responses
 Conversations
Hexagon
Interior angles
Linear pair
Nonagon
Octagon
Pentagon




On-spot checks of
class work
Homework
Observations of
students’
work/behaviors
Students’
explanations
Polygon
Quadrilateral
Regular polygon
Supplementary angle
Triangle
Undecagon
Vertices


Draw a picture
Tests and
Quizzes
 T/F
 Multiple Choice
 Written Response
 Ticket out the
door
See Blackboard for
specific Assessments*
Resources
Review and Extend
Solve geometry problems algebraically
Regional Integrated Geometry Curriculum
UNIT: Formal & Informal Proofs
TOPIC: Quadrilaterals
Time frame for unit:
Time frame for topic: 15 days
Prior Knowledge
5.G.4 Classify quadrilaterals by properties of their angles and sides
8.G.2 Identify pairs of supplementary and complementary angles
8.G.3 Calculate the missing angle in a supplementary or complementary pair
8.G.4 Determine angle pair relationships when given two parallel lines cut by a
transversal
8.G.5 Calculate the missing angle measurements when given two parallel lines
cut by a transversal
8.G.6 Calculate the missing angle measurements when given two intersecting
lines and an angle
Content Strands
G.G.38 Investigate, justify, and apply theorems about parallelograms involving
their angles, sides, and diagonals
G.G.39 Investigate, justify, and apply theorems about special parallelograms
(rectangles, rhombuses, squares) involving their angles, sides, and
diagonals
G.G.40 Investigate, justify, and apply theorems about trapezoids (including
isosceles trapezoids) involving their angles, sides, medians, and
diagonals
G.G.41 Justify that some quadrilaterals are parallelograms, rhombuses,
rectangles, squares, or trapezoids
Concepts
Properties:
a. Parallelogram
b. Rectangle
c. Rhombus
d. Square
e. Trapezoid
f. Isosceles Trapezoid
Given a specific quadrilateral (parallelogram, rhombus, square, rectangle,
trapezoid), prove a specified property.
Given specific properties, prove a specific quadrilateral (parallelogram, rhombus,
square, rectangle, trapezoid).
Applications of properties of quadrilaterals
a. solving linear equations
b. solving quadratic equations
Essential Questions
Why is it important to compare and contrast different types of quadrilaterals?
Why is a square a rectangle but a rectangle not a square?
What are some essential steps/information that are needed to start a proof of a
specific quadrilateral?
Process/skills
G.PS.3 Use multiple representations to represent and explain problem situations
(e.g., spatial, geometric, verbal, numeric, algebraic, and graphical
representations)
G.PS.7 Work in collaboration with others to propose, critique, evaluate, and value
alternative approaches to problem solving
G.RP.4 Provide correct mathematical arguments in response to other students’
conjectures, reasoning, and arguments
G.CM.5 Communicate logical arguments clearly, showing why a result makes
sense and why the reasoning is valid
G.CN.2 Understand the corresponding procedures for similar problems or
mathematical concepts
G.R.7
Use mathematics to show and understand social phenomena (e.g.,
determine if conclusions from another person’s argument have a logical
foundation)
Vocabulary
Adjacent angles
Direct proof
Parallelogram
Adjacent sides
Equal
Perpendicular
Alternate interior angles
Formal Proof
Perpendicular bisector
Altitude
Given
Quadrilateral
Angle bisector
Height
Reason
Base angles of a
Indirect proof
Rectangle
trapezoid
Isosceles trapezoid
Rhombus
Complementary angles
Kite?
Segment bisector
Congruent
Legs of a trapezoid
Square
Congruent triangles
Consecutive angles
Corresponding angles
CPCTC
Diagonal
Suggested assessments
 Pre-assessment
 Regents/State
exams
 Projects
 Participation
 Q-A Responses
 Oral responses
 Conversations
Resources
Linear pair
Median
Midpoint
Opposite angles
Parallel




On-spot checks of
class work
Homework
Observations of
students’
work/behaviors
Students’
explanations
Statement
Supplementary angles
Transversal
Trapezoid
Vertical angles


Draw a picture
Tests and
Quizzes
 T/F
 Multiple Choice
 Written Response
 Ticket out the
door
See Blackboard for
specific Assessments*