A Story of Functions

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NYS COMMON CORE MATHE MATICS CURRI CULUM A Story of Functions

A Story of FunctionsA Close Look at Grade 10 Module 2


N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M A Story of Functions

Participant Poll• Classroom teacher• Math trainer• Principal or school leader• District representative / leader• Other



Session Objectives

• Examination of:

• The development of mathematical understanding across the module using a focus on concept development within the lessons.

• Examples that highlight themes and changes according to the Common Core State Standards.



Agenda• Orientation to Materials (if needed)

• Review of Module 1: Congruence, Proof, and Constructions- Themes and Examples

• Module 2: Similarity, Proof, and Trigonometry- Themes and Examples



What’s In a Module?Teacher Materials• Module Overview• Topic Overviews• Daily Lessons• AssessmentsStudent Materials• Daily Lessons with Problem SetsCopy Ready Materials• Exit Tickets • Assessments



Types of Lessons1. Problem Set

Students and teachers work through examples and complete exercises to develop or reinforce a concept.

2. SocraticTeacher leads students in a conversation to develop a specific concept or proof.

3. ExplorationIndependent or small group work on a challenging problem followed by debrief to clarify, expand or develop math knowledge.

4. ModelingStudents practice all or part of the modeling cycle with real-world or mathematical problems that are ill-defined.



What’s In a Lesson?Teacher Materials Lessons• Student Outcomes and Lesson Notes (in select lessons)• Classwork

• General directions and guidance, including timing guidance• Bulleted discussion points with expected student responses• Student classwork with solutions (boxed)

• Exit Ticket with Solutions• Problem Set with SolutionsStudent Materials• Classwork• Problem Set



Key Changes in CCSS that Impact M1 & M2

Common Core State Standards for Mathematics, p74

“The concepts of congruence, similarity, and symmetry can be understood from the perspective of geometric transformation.

Fundamental are the rigid motions: translations, rotations, reflections, and combinations of these, all of which are here assumed to

preserve distance and angles (and therefore shapes generally)…

In the approach taken here, two geometric figures are defined to be congruent if there is a sequence of rigid motions that

carries onto the other.”



Transformations

• How they are first introduced• The manner in which they are described and studied• Their use in the definition of congruence and similarity• Other uses to which transformations are put, e.g.,

reasoning and steps in proofs

Key Changes in CCSS that Impact M1 & M2



January 2014 Geometry Regents



January 2014 Geometry Regents



A Checklist of “rules”:



Module 1: Congruence, Proof, and ConstructionsTopic A: Basic ConstructionsTopic B: Unknown AnglesTopic C: Transformations/Rigid MotionsTopic D: CongruenceTopic E: Proving Properties of Geometric FiguresTopic F: Advance ConstructionsTopic G: Axiomatic Systems



Topic A: Basic Constructions

• Standards • G-CO.1, G-CO.12, G-CO.13

• Constructions• How to:

• construct an equilateral triangle• copy an angle• bisect an angle• construct a perpendicular bisector• determine the incenter and circumcenter of a triangle



Lesson 1: Construct an Equilateral Triangle

Flow of the lesson:• Opening Exercise: (1) A question that jogs students ideas on distance (2)

key vocabulary review • Example 1: Guide students to discovering how to construct an equilateral

triangle (with the use of a compass and straightedge).• Example 2: Read an excerpt from Elements that describes Euclid’s approach

to constructing an equilateral triangle; have students develop their own set of steps on how to construct an equilateral.

• Geometry Assumptions*: Introduce the notion that geometry is an axiomatic system- everything in the subject can be traced back to a short list of basic assumptions we take for granted.



Lesson 1: Construct an Equilateral Triangle

segment

radius

circle



Lesson 1, Example 1



Lesson 1, Example 1



Lesson 1, Example 2



Lesson 1, Example 2



Topic B: Unknown Angles

• Standards • G-CO.9

• Breakdown of Lessons 6-11:• Lessons 6-8: Unknown angle problems involving angles and lines at

a point, transversals, and angles in a triangle. Students use simple justification in non-algebraic steps.

• Lessons 9-11: Unknown angle proofs, non-numeric problems emphasizing justification of a relationship, sometimes a known existing relationship.




Lesson 8, Exercise 2:

Lesson 9, Discussion: What can be established about x, y, and z?




What can be established about x, y, and z?

w



Lesson 10, Unknown Angle Proofs- Proofs with ConstructionsProblem Set, 1

AB DE and . Prove that

m ABC = m DEF.

BC EF



Topic C: Transformations/Rigid Motions

• Standards • G-CO.2, G-CO.3, G-CO.4, G-CO.5, G-CO.6, G-CO.7 G-CO.12

• Key concepts in Lessons 12-21:• Formal definition of each rigid motion (rotation, reflection,

translation) and how each rigid motion is related to constructions• Compositions of rigid motions• Applications of rigid motions: discussion regarding

correspondence, how congruence is defined



Rigid Motions in Grade 8

• What knowledge do students arrive with?• Rigid motions first introduced in Grade 8, Module 2• G8 focus:

• Understanding of what a transformation is• Intuitive understanding and experimental verification of the

properties of the rigid motions (translation, reflection, and rotation).

• Descriptive definitions• Hands-on exploration with the use of transparencies



Grade 8, Module 2, Lesson 4

• Grade 8 introduction to reflections:



(Grade 10) Lesson 14: Reflections

• Exploratory Challenge



Reflections and the Perpendicular Bisector Construction



Lesson 14: Reflections

• Example 1




• Example 1




Definition of Reflection:




Definition of Reflection:




Application of Reflection:



Lesson 19: Construct and Apply a Sequence of Rigid Motions• We define two figures in the plane as congruent if there exists

a finite composition of basic rigid motions that maps one figure onto the other.

• The idea of “Same size, same shape” only paints a mental picture; not specific enough.

• It is also not enough to say that two figures are alike in all respects except position in the plane.

• A congruence gives rise to a correspondence.



Topic D: Congruence• Standards • G-CO. 7, G-CO.8

• Key theme in Topic D:



Lesson 22: Congruence Criteria for Triangles- SAS












Lesson 26: Triangle Congruency Proofs – Part I

Exercise 5



Topic G: Axiomatic Systems

• Standards • All

• A mathematician’s story:Many years ago I taught calculus for business majors. I started my class with an offer: "We have to cover several chapters from the textbook and there are approximately forty formulas. I may offer you a deal: you will learn just four formulas and I will teach you how to get the rest out of these formulas." The students gladly agreed.



Example from Mid-Module Assessment



Example from Mid-Module Assessment



Key Themes of Module 1: Congruence, Proof, and Constructions

• Module is anchored by the definition of congruence• Emphasis is placed on extending the meaning and use of

vocabulary in constructions • There is an explicit recall and application of facts learned over

the last few years in unknown angle problems and proofs• Triangle congruence criteria are indicators that a rigid motion

exists that maps one triangle to another; each criterion can be proven to be true with the use of rigid motions.



Module 2: Similarity, Proof, and Trigonometry

Topic A: Scale DrawingsTopic B: DilationsTopic C: Similarity and DilationsTopic D: Applying Similarity to Right TrianglesTopic E: Trigonometry



Topic A: Scale Drawings

• Standards • G-SRT.1, G-SRT.4

• Key Concepts• Two ways two construct scale drawings: Ratio Method and Parallel

Method• Triangle Side Splitter Theorem• Dilation Theorem



Lesson 2: Ratio Method



Lesson 2: Ratio Method, Example 1



Lesson 2, Example 2



Lesson 3: Parallel Method, Example 2



Lesson 3, Example 2



Consequences of the Ratio and Parallel Methods • Students prove that the Ratio and Parallel Methods produce the same scale

drawing for a give figure• Resulting theorems used frequently in Topic B:

• Triangle Side Splitter Theorem. A line segment splits two sides of a triangle proportionally if and only if it is parallel to the third side.

• Dilation Theorem. If a dilation with center O and scale factor r sends point P to P’ and Q to Q’, then P’Q’ = r PQ . Furthermore, if r≠1, and O, P, and Q are the vertices of a triangle then PQ P’Q’.



Topic B: Dilations

• Standards • G-SRT.1, G-SRT.4

• Key Concepts• Proving properties of dilations, including how it is that

dilations map segments to segments, lines to lines, rays to rays, and angles to angles.



Lesson 7: How Do Dilations Map Segments?

Example 3• To show that dilations map segments

to segments, call on:• The definition of a dilation• The Triangle Side Splitter Theorem• Construction of a ray

For r < 1



Topic C: Similarity and Dilations

• Standards • G-SRT.2, G-SRT.3, G-SRT.4, G-SRT.5

• Key Concepts• Similarity Transformation. The composition of a finite number of

dilations and/or rigid motions.• Two figures are said to be similar if a similarity transformation exists

maps one figure onto another. A congruence is a similarity with scale factor 1.



Lesson 12: What are Similarity Transformations and Why Do We Need Them?Example 1• Describe a similarity transformation that maps figure Z onto

figure Z’.



Lesson 15: Angle-Angle Criterion for Two Triangles to be Similar

Theorem. Two triangles with two pairs of equal corresponding angles are similar.

Discussion




Theorem. Two triangles with two pairs of equal corresponding angles are similar.



Lesson 15: Angle-Angle Criterion for Two Triangles to be SimilarExercise 9



Lesson 15: Angle-Angle Criterion for Two Triangles to be SimilarProblem Set 4






Lesson 17: SAS and SSS Criteria for Two Triangles to be Similar



Lesson 17: SAS and SSS Criteria for Two Triangles to be Similar



Lessons 19 and 20: Modeling Lessons using Similarity

Eratosthenes’ calculation of the Earth’s

circumference

Aristarchus’ calculation of the distance

from the Earth to the Moon



Topic D: Applying Similarity to Right Triangles

• Standards • G-SRT.6, G-SRT.7, G-SRT.8

• Key Concepts• The special relationships that result when an altitude is drawn in a

right triangle from the vertex of the right angle to the hypotenuse.• Proving the Pythagorean Theorem using similarity• Simplifying radical expressions



Lesson 21: Special Relationships with Right Triangles- Dividing into Two Similar Sub-Triangles

Discussion

• We can use the AA criterion to showthat each of the smaller two right triangles is similar to the large right triangle;similarity is transitive, so all three trianglesare similar.












Lesson 22: Proving the Pythagorean Theorem by SimilarityDiscussion• How can the ratios in right

triangles (e.g. shorter leg:hypotenuse) be used to prove the Pythagorean Theorem?



Lesson 22: Proving the Pythagorean Theorem by Similarity• Which right triangle ratios can be used to express a in

terms of other variables?



Lesson 22: Proving the Pythagorean Theorem by Similarity

longer leg: hypotenuse shorter leg: hypotenuse



Lessons 23 and 24:

Multiplying and Dividing Expressions with Radicals Adding and Subtracting Expressions with Radicals



Topic E: Trigonometry

• Standards • G-SRT.6, G-SRT.7, G-SRT.8

• Key Concepts• Definition of sine, cosine, tangent • Relationship between sine and cosine• Application of the trig ratios to solve missing value, real world, and area

problems.



Lesson 25: Incredibly Useful Ratios

Quote from Lesson 25:

“In fact if, since we know the values of these ratios are constant for a given angle measure, we could compile a table of every acute angle between 0˚ and 90˚ and list the values of the adj/hyp and opp/hyp ratios with respect to each acute angle measure.”



Lesson 27: Sine and Cosine of Complementary Angles and Special Angles

• What conclusions can be drawn based on each of the ratios?









Lessons 28-29: Solving missing value problems

• Selected examples to try:• Lesson 28: Problems using Sine and Cosine, Exercise 6 • Lesson 29: Applying Tangents, Example 2



L28, Exercise 6



L29, Example 2



Key Themes of Module 2: Similarity, Proof, and Trigonometry

• Just as rigid motions are used to define congruence, so dilations are added to define similarity.

• A close look at the way dilations are constructed yields information about how dilations behave.

• Right triangle similarity is built on the concepts of within-triangle ratios and between-triangle ratios leading

to an understanding of trigonmetric ratios.



Biggest Takeaway• What is your biggest takeaway with respect to Modules 1

and 2?• How can you support successful implementation at your

school/s given your role?

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A Story of Functions