Download 2015-2016 Geometry 3rd Quarter Mathematics Scope and Sequence

2015-2016 Geometry 3rd Quarter Mathematics Scope and Sequence
Unifying Concept: Similarity
Mathematical Practice Focus
Mathematically Proficient Students…
1. Make sense of problems and persevere in solving
Students apply these ideas to triangles in order to
them.
develop trigonometric ratios. They will use
2. Reason abstractly and quantitatively.
trigonometric ratios to solve right triangles.
3. Construct viable arguments and critique the reasoning
of others.
Students will also explore properties and conditions of 4. Model with mathematics.
polygons and parallelograms. They will use the
5. Use appropriate tools strategically.
Pythagorean Theorem to solve missing side lengths
6. Attend to precision.
of similar triangles
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Students explore the idea of similar figures and the
ratios they contain. They use these ideas to
understand the relationships between similar
triangles.
Target Standards
G.CO.C.10
Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum
to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two
sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at
a point.
G-CO.C.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.
G-SRT.A.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.
G-SRT.A.3
Use the properties of similarity transformations to establish the AA criterion for two triangles to be
similar.
Mathematics Content Focus:
(Chapters 5, 8,6,7)
G-SRT.B.4
G-SRT.B.5
G-SRT.C.6
G-SRT.C.7
G-SRT.C.8
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.
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in
geometric figures.
Understand that by similarity, side ratios in right triangles are properties of the angles in the
triangle, leading to definitions of trigonometric ratios for acute angles.
Explain and use the relationship between the sine and cosine of complementary angles.
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied
problems. ★
Find the point on a directed line segment between two given points that partitions the segment in a
given ratio.
Quarter Major Clusters
Arizona considers Major Clusters as groups of related standards that require greater emphasis than some of the
others due to the depth of the ideas and the time it takes to master these groups of related standards.
G-CO.C Prove geometric theorems
Essential Concepts:
Essential Questions:
 A theorem is a statement that can be proven from
 How is a theorem different from an axiom?
previously known facts, including postulates, axioms
 What is the difference between an axiom and a
and definitions.
postulate?
 A proof is a logical argument that shows that a
 How do you know when a proof is complete and
8/28/2015 9:58 AM
Curriculum Instruction and Professional Development Math Department
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G-GPE.B.6
2015-2016 Geometry 3rd Quarter Mathematics Scope and Sequence
theorem is true based on given information.
 Properties of the sides and angles of geometric figures
can be stated as theorems and proven. Several
examples are listed in the standards below.
valid?
Examples of essential questions for particular
theorems:
 How do you prove that the interior angles of a
triangle add up to 180°?
 What method would you use to prove theorems
about parallelograms, and why?
G-SRT.A Understand similarity in terms of similarity transformations
Essential Concepts
Essential Questions
 A similarity transformation is a combination of rigid
 What is the difference between similar and
motion and dilation.
congruent figures?
 Triangles are similar if all corresponding angles are
 How can you describe the relationship between
congruent and all corresponding sides are
sides of similar figures?
proportional.
 How can you show that two triangles are similar?
 It can be shown using properties of similarity
transformations that triangles are similar if two pairs of
corresponding angles are congruent.
G-SRT.B Prove theorems involving similarity
Essential Concepts:
Essential Questions:
 A theorem is a statement that can be proven from
 Why does a line cutting a triangle parallel to one
previously known facts, including postulates, axioms
side of the triangle divide the other two
and definitions.
proportionally?
 A proof is a logical argument that shows that a
 How many pieces of information do you need in
theorem is true based on given information.
order to solve for the missing measures of a
triangle?
 The Pythagorean Theorem can be proven using
 How can you use congruence or similarity of
triangle similarity.
pairs of figures to determine missing information
 Congruence and similarity can be used to determine
about the figures?
missing angle measures and side lengths in geometric
figures.
 Why does SSA not work to prove triangle
congruence?
G-SRT.C Define trigonometric ratios and solve problems involving right triangles
Essential Concepts:
Essential Questions:
 The ratios of the sides in any right triangle are
 What role does similarity play in defining
properties of the angles in the triangle; these are
trigonometric ratios?
called trigonometric ratios.
 What is the relationship between the sine and
 The sine and cosine of complementary angles are
cosine of complementary angles?
related.
 How can right triangles be used to solve real Trigonometric ratios and the Pythagorean Theorem
world problems?
can be used to solve right triangles in applied
 Why do we need trigonometric ratios to solve
problems.
some real-world problems involving right
triangles?
G-GPE.B Use coordinates to prove simple geometric theorems algebraically
Essential Concepts:
Essential Questions:
 Coordinate geometry can be used to solve simple
 How do you find the point on a directed line
geometric theorems algebraically.
segment between two given points that partitions
the segment in a given ratio?
8/28/2015 9:58 AM
Curriculum Instruction and Professional Development Math Department
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