using transformations to make connections between …...14 . functions and sliders ... transform...

Using Transformations to Make Connections between Polygons and

Circles: Linking College- and Career-Readiness Standards to Effective Mathematical Teaching Practices

Benjamin J. Sinwell Anderson School District 4

Pendleton High School Pendleton, SC

[email protected]

What We Are Going To Do? • Build (Sketch) Isosceles Triangles and

Rhombuses in the Coordinate Plane • Build Equilateral Triangles in the

Coordinate Plane • Transform those Polygons and Analyze the

Equations of the Lines That Form Them • Make Connections to Algebra How Are We Going To Do It?

• Using a series of sequenced problems • Exploration • Focus on Effective Mathematics Teaching Practices

Principles to Action • Establish mathematics goals to focus learning • Implement tasks that promote reasoning and

problem solving • Use and connect mathematical representations • Facilitate meaningful mathematical discourse • Pose purposeful questions • Build procedural fluency from conceptual

understanding • Support productive struggle in learning

mathematics • Elicit and use evidence of student thinking

Geometry Standards

• Represent Transformations in the Plane • Describe Rotations and Reflections That

Carry a Polygon Onto Itself • Develop Definitions of Transformations • Use Transformations to Carry a Polygon

Onto Another Polygon • Define Congruence Using Transformations

Standards: G.CO.1 through 6.

Algebra Standards

• Graph Functions (show intercepts, domain and range of piecewise functions) (F.IF.7)

• Create and Graph Equations (A.CED.1,2) • Solve Systems of Linear Equations

(A.REI.6) • Relate the Domain of a Function to its

Graph (F.IF.5) • Interpret the Equation y=mx+b. (8.F.3)

Part A: Build An Isosceles Triangle That Has The X-axis As Its Base Its

Vertex On The Y-axis

• What is the equation of the line (blue)?

• Describe the mathematical relationship of the two lines.

• If the original line was y = 3x + 12, what would the other line be?

Geometry: Domain and Range? • Domain: -4 < x < 4 (x-intercepts) • Range: 0 < y < 8 (y-intercepts)

Part B: Make An Isosceles Triangle Congruent To The Last One That

Has The Segment Connecting (0,0) to (8,0) as its base.

• What are the equations of the lines?

• How do they compare to the lines in the previous problem?

• What is going on with the slopes and y-intercepts?

What changed? What stayed the same? What if I use y = 4x and y = -4x + 16?

Compare and Contrast

• Domain/Range? • Slope/y-intercept? • Perimeter?

• Area? • Vertex?

VIDEOS

As you watch the videos, think about the nature of the discussion that is going on in the class.

13

Water Tower

• Discussing the rate of change (slope) of the water flowing into a water tower using a graph, in particular, that the rate of change of the slope at 2.5.

WHAT ARE THE STUDENTS TALKING ABOUT?

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Functions and Sliders

• High School Algebra class changing parameters the slope of a linear equation.

• What are the students talking about?

15

Discourse

Talk At Your Table About What

Makes A Good Discussion

16

Partner 1: What equations will form a rhombus for Part A?

Partner 2: What equations will form a rhombus for Part B?

Rules: One Elbow Partner Does It For Part A and One Does It For Part B

• How are those equations related to the original ones?

• What transformations are happening? • Turn to your partner(s), compare and contrast your

results. • What would happen for a rhombus made from the

equations: y = (-1/2)x + 4 and y = (1/2)x + 4? (all can do together)

Geometry Software?

Given The Base AB Of An

Equilateral Triangle With A(-1,0) And B(0,1)

• What is the height of the triangle?

• What are the equations of the lines that make an equilateral triangle?

• What is the vertex? • What is going on with

the slopes and y-intercepts?

Practice Standards

• Look For Structure • Regularity in Repeated Reasoning • Tools • Precision • Modeling • Arguments/Reasoning • Abstract/Quantitative • Make Sense and Persevere

Content Standards • Graphing Functions • Domain/Range, Piecewise • Applying the Pythagorean Theorem • Systems of Equations • Transformation – Translations and Reflections • Area/Perimeter • Distance Formula • Special Right Triangles

Technology

Transformations WHAT CHANGES? WHAT STAYS THE

SAME? • Reflections – slopes, intercepts? • Translations – slopes, intercepts? How does centering using the x-axis, y-axis and origin help us? What about dilations?

An Equilateral Triangle

• What Equations Make An Equilateral Triangle With Base AB at A(0,0) and B(1,0)? Draw The Triangle.

Questions?

Exit Ticket

• What are you wondering about related to discourse?

• Related to any other MATHEMATICS TEACHING PRACTICES

• Other comments:

28

Disclaimer The National Council of Teachers of Mathematics is a public voice

of mathematics education, providing vision, leadership, and professional development to support teachers in ensuring equitable mathematics learning of the highest quality for all students. NCTM’s Institutes, an official professional development offering of the National Council of Teachers of Mathematics, supports the improvement of pre-K-14 mathematics education by serving as a resource for teachers so as to provide more and better mathematics for all students. It is a forum for the exchange of mathematics ideas, activities, and pedagogical strategies, and for sharing and interpreting research. The Institutes presented by the Council present a variety of viewpoints. The views expressed or implied in the Institutes, unless otherwise noted, should not be interpreted as official positions of the Council.

29

Using Transformations to Make Connections Between Polygons and

Circles: Linking the CCSSM Content and Practice Standards, NCTM Institute, 2015

Benjamin J. Sinwell

Anderson School District 4 Pendleton High School

Pendleton, SC [email protected]

What We Are Going To Do? • Build (Sketch) Squares and Equilateral Triangles in the Coordinate Plane • Transform those Polygons Using Dilations • Make Connections to The Unit Circle • Make Connections to Compass and Straightedge

Constructions

How Are We Going To Do It? • Using a series of sequenced problems

• Exploration • Focus on Mathematics Teaching Practices

Geometry Standards

• Represent Transformations in the Plane • Describe Rotations and Reflections That

Carry a Polygon Onto Itself • Develop Definitions of Transformations • Use Transformations to Carry a Polygon

Onto Another Polygon • Define Congruence Using Transformations

Standards: G.CO.1 through 6.

…Geometry Standards • Understand Similarity Using Transformations

(G.SRT.1-3) • Prove Geometric Theorems Algebraically

(G.GPE.4,5,7) • Construct Inscribed Triangles, Heaxagons, etc.

(G.C.3,12,13) • Derive The Equation of a Circle Using

Pythagorean Thm. (G.GPE.1) • Triangle Congruence (G.CO.8) • Experiment With Transformations in the Plane

Algebra Standards

• Graph Functions (show intercepts, domain and range of piecewise functions) (F.IF.7)

• Create and Graph Equations (A.CED.1,2) • Solve Systems of Linear Equations

(A.REI.6) • Relate the Domain of a Function to its

Graph (F.IF.5) • Interpret the Equation y=mx+b. (8.F.3)

…Algebra Standards

• Unit Circle and Trigonometry (F.TF.2) • Geometric Description of a Circle vs.

Equation of a Circle (G.GPE.1-3) • Use 2-D Shape to Generate 3-D Shapes

(G.GMD.4)

5 Practices Anticipating what students will do--what strategies they will use--in solving a problem Monitoring their work as they approach the problem in class Selecting students whose strategies are worth discussing in class Sequencing those students' presentations to maximize their potential to increase students' learning Connecting the strategies and ideas in a way that helps students understand the mathematics learned

Adding It Up

Questions for Both Partners

• What shape is it? How do you know? • What is its perimeter? area? • How do the two graphs compare? • What about y = 4, y = -4, x = 4 and x = -4

Partner A: Graph y = x – 5 y = -x – 5 y = x + 5 y = -x + 5

Partner B: Graph y = x – 10 y = -x – 10 y = x + 10 y = -x + 10

• Does this give any insight about 45-45-90 triangles? • How do we transform the smaller one so that a vertex is on the origin?

• Square with floating points • Square with axes fixed

Using Technology to Build A Square

Extend: What are the equations of those circles?

Extend: What are the equations of the circles below?

Another Way To Ask… • What Equations Make An Equilateral Triangle

With Base AB at A(0,0) and B(1,0)? Draw The Triangle.

• Reflect This Triangle Across Both The X And Y Axes.

• Reflect One of Those Triangles Into Quadrant 3.

• Do You See A Hexagon? • Which Equations Form The Regular Hexagon? • What Are Its Vertices?

Another Starting Place?

What questions could we ask?


• Area? • Vertex? • Transformations?

How Can We Use Technology?

Geometry Software?

Compass and Straightedge?

String?

Paper?

How may equilateral triangles can you find that are bigger than the shaded one using the points labeled on the left?

How may equilateral triangles can you find that are bigger than the shaded one using the points labeled on the left? Name them. How do you know they are equilateral?

How do we know we have a regular hexagon?

Hexagon: Compass and Straightedge Construct these circles: center (0,0) and radius 1 center (1,0) and radius 1 center (-1,0) and radius 1

Proof Using Radii

Sidewalk Chalk and String

Transformative Thinking

• Graph the equation x2+y2=1 • Translate the graph 1 unit to the right.

What is the equation of the circle? • Translate the original graph 1 unit to the

left? What is the equation of the circle?

• Graph the equation x2 + y2 = 1 • Translate the graph 1 unit to the right.

What is the equation of the circle? • Translate the original graph 1 unit to the

left? What is the equation of the circle? Answers: (x - 1)2 + y2 = 1, (x + 1)2 + y2 = 1

Flowery Fun: Math, Nature, Science and History

How Many Equilateral Triangles Are There With Sides Lengths Greater Than 1?

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=r-r-fC3PWCAleM&tbnid=sSYbBGuYC37LxM:&ved=0CAUQjRw&url=http://www.bibliotecapleyades.net/geometria_sagrada/esp_geometria_sagrada_6.htm&ei=SOnGUtSgKc_jsATC-IGgDQ&bvm=bv.58187178,d.eW0&psig=AFQjCNF6xSFEFswwqFJLly0LnJP0uzz2vA&ust=1388853952824625


http://www.google.com/url?sa=i&rct=j&q=saturn%20hexagon&source=images&cd=&cad=rja&docid=t9I0x5ghCT8TCM&tbnid=oYmC50wmdW4bJM:&ved=0CAUQjRw&url=http://newswatch.nationalgeographic.com/2013/12/04/cassini-photo-stunning-new-views-of-saturns-hexagon-storm/&ei=qyn1Uqf2FMTWkQfK54HoBQ&bvm=bv.60799247,d.eW0&psig=AFQjCNHj7fLd88aZGyztjqy5a0x6chSwjA&ust=1391885092031435

• Read Practices • Discuss where you saw them today • Discuss where you use them in your practice

41

What questions could we ask?


• Area? • Vertex? • Transformations?

How Can We Use Technology?

More Equilateral Triangles

Build an equilateral triangle with base CD with C(0,0) and D(2,0)? Build an equilateral triangle with base EF with E(0,0) and D(1,0)? Build an equilateral triangle with base GH with G(-4,0) and H(4,0)?

5 Practices Anticipating what students will do--what strategies they will use--in solving a problem Monitoring their work as they approach the problem in class Selecting students whose strategies are worth discussing in class Sequencing those students' presentations to maximize their potential to increase students' learning Connecting the strategies and ideas in a way that helps students understand the mathematics learned

Strength In Numbers: Selecting And Setting Up A Task

• Mathematical Goals • Prior Knowledge, Knowledge Needed, What

Questions • Ways To Solve (Student ways)? What Misconceptions? What Errors? • Expectations: Resources/Tools, Classroom

Structure, Recording/Reporting • Access to ALL, Ensuring Understanding

Strength In Numbers: Sharing and Discussing the Task

How will you orchestrate classroom discussion? • What solution paths will be shared? What order?

Why? • How will this help with the goals of the lesson? • What specific questions will you ask so that students

will: 1. make sense of the mathematical ideas? 2. expand on, debate and question the solutions being shared?

Strength in Numbers, Ilana Seidel Horn, 2012

More Sharing and Discussing… What specific questions will you ask so that students will: 3. make connections among the different strategies that are presented? 4. look for patterns? 5. begin to form generalizations? How will you ensure all students have the opportunity to share their reasoning and thinking?

More Sharing and Discussing… What will you see or hear that lets you know that all students in the class understand the mathematical ideas that you intended for them to learn? What will you do tomorrow that will build on this lesson?

Adding It Up: Intertwined Stands of Proficiency

Questions?






http://www.google.com/url?sa=i&rct=j&q=saturn%20hexagon&source=images&cd=&cad=rja&docid=t9I0x5ghCT8TCM&tbnid=oYmC50wmdW4bJM:&ved=0CAUQjRw&url=http://newswatch.nationalgeographic.com/2013/12/04/cassini-photo-stunning-new-views-of-saturns-hexagon-storm/&ei=qyn1Uqf2FMTWkQfK54HoBQ&bvm=bv.60799247,d.eW0&psig=AFQjCNHj7fLd88aZGyztjqy5a0x6chSwjA&ust=1391885092031435

• Tasks have to be justified in terms of the learning aims they serve and can work well only if opportunities for pupils to communicate their evolving understanding are built into the planning (Black & Wiliam, 1998)

51

Disclaimer The National Council of Teachers of Mathematics is a public voice

of mathematics education, providing vision, leadership, and professional development to support teachers in ensuring equitable mathematics learning of the highest quality for all students. NCTM’s Institutes, an official professional development offering of the National Council of Teachers of Mathematics, supports the improvement of pre-K-14 mathematics education by serving as a resource for teachers so as to provide more and better mathematics for all students. It is a forum for the exchange of mathematics ideas, activities, and pedagogical strategies, and for sharing and interpreting research. The Institutes presented by the Council present a variety of viewpoints. The views expressed or implied in the Institutes, unless otherwise noted, should not be interpreted as official positions of the Council.

52

Mathematical Discourse in the Classroom

Benjamin J. Sinwell Mathematics Teacher

Pendleton High School Anderson 4 Public Schools, SC [email protected]

Is it Worth Talking About?

2

Ask/Write each of the two questions below in a different way that might help students to “look for and make use of structure.” 1. Find the zeroes of f(x) = x2 + 6x – 16 ? 2. What is the vertex of y = x2 + 6x + 8

• A Rich Mathematical Task • Sequencing • Exploration • Multiple Representations • By Using The Effective Mathematical Teaching

Practices Standards

What are we going to do?

How are we going to do it?

Looking For Patterns

Ask a question or provide directions that might help students to “look for and make use of structure.”

• 5, 8, 11, 14, … • 4, 6, 8, 10, … • 10, 20, 30, 40 …

Patterns defined Recursively

To get the next term, double the previous term and subtract the term before that.

An= 2An-1 – An-2

• 5, 8, 11, 14, … An= An-1 + 3 • 4, 6, 8, 10, … An= An-1 + 2 • 10, 20, 30, 40, … An= An-1 + 10

More Patterns Defined Recursively

To get the next one, multiply the previous term by two

An= 2An-1 • 3, 6, 12, 24, … • 4, 8, 16, 32, … To get the next one, multiply the previous term by

three An= 3An-1 • 5, 15, 45, 135, …

Patterns from Trains

Car of Length 1

Car of Length 2

Car of Length 3

Car of Length 4

Car of Length 5

Two Different TRAINS of Length 5 built from a car of length 2 and a car of length 3

Step 1 Step 2 Step 3

What Is Changing As the steps increase the _______________ changes.

Step 1 Step 2 Step 3

What Is Changing With your partner, pick one of the attributes in our list and investigate how it changes. Make a conjecture and try to prove it. How would a graph, a table, and/or an equation support your conclusion?

With your partner, pick one of the attributes in our list and investigate how it changes. Make a conjecture and try to prove it. How would a graph, a table, and/or an equation support your conclusion? (If time, explore a 2nd or 3rd property)

11

Possible learning goals

• Distinguish between recursive and closed form rules for sequences

• Distinguish between linear and exponential relationships

• Describe a pattern with an expression or a rule • Recognize and be able to describe the

components of an geometric sequence • Explain what rate of change means in different

situations • Construct a viable argument

12

• In what order would you choose to discuss the solutions? Why?

• What connections would you want students to discuss? How would you help them see those connections?

13

Processing the activity • What answers do you expect to see?

(Anticipating) • What are students doing? (Monitoring) • What responses are worth discussing?

(Selecting) • How will you sequence the responses?

(Sequencing) • What is the mathematical punchline?

(Connecting)

14 Smith & Stein, 2011

Questions Engage Students in Mathematical Practices

How many trains are made from exactly 1 rod? 2 rods? 3 rods? 4 rods?


1

1 1

1 2 1

1 3 3 1


(H + T) (H + T)2 (H + T)3


1

1 1

1 2 1

1 3 3 1

Build all the Trains of Length 1,2,3,4, and 5 can you Make Using

Only Cars of Length 1 and 2 Train

Length (n)

Number of Ways

(An)

1 2 3 4 5

A train of Length 3

A train of Length 4

Trains of Length 1,2,3,4, and 5 Using Only Cars of Length 1 and 2?

An = An-1 + An-2

Train Length

(n)

Number of Ways

(An) 1 1

2 2

3 3

4 5

5 8

Organizing it…

An =15

1+ 52

n

−15

1− 52

n

An explicit formula?

An= 2n-1

came from An= 2An-1

11− 2n

B *2n

Explicit Formula From Earlier

came from An = An-1 + An-2

11− n − n2

1−1+ 5

2n

1−1− 5

2n

C * 1+ 52

D* 1− 52

An =15

1+ 52

n

−15

1− 52

n


How many trains are made from exactly 1 rod? 2 rods? 3 rods? 4 rods?


1

1 1

1 2 1

1 3 3 1


1

1 1

0 2 1

0 1 3 1

0 0 3 4 1


1 1 1

0 2 1 0 1 3 1

0 0 3 4 1


1 1 1

0 2 1 0 1 3 1

0 0 3 4 1

Why did I “circle” that piece?


1 1 1

0 2 1 0 1 3 1

0 0 3 4 1

Why did I move it?


1 1 1

0 2 1 0 1 3 1

0 0 3 4 1 0? 0? 1? 6? 5?1?


How Do We Write Explicit Formulas When A0 = x ?

• An= 2An-1 When A0 = 1 and A1 = x ?

• An= An-1 + An-2 • An= 2An-1 - An-2

Asking Questions

• Ask simple questions such as: – “What else do you notice?” – “What happens when…?” – “How do you know?”

Inside the Black Box, Assessment for Learning in the Classroom. Black and Wiliam, 2004 pg 12

35

Sentence Starters

• “Sentence starters can help students find the language for discussing their thinking together.”

• “The goal of sentence starters is to give students a way to have deeper mathematical conversations.”

Strength in Numbers, Ilana Seidel Horn, 2012, pg 52 36

Sentence Starters

Examples: • “How did you know how to ______________?” • “What does _______________ mean?” • “_____________ because ______________.” • “Why did you _____________?” • Why are our ____________ different?”

Strength in Numbers, Ilana Seidel Horn, 2012, pg 52 37

Participation quiz

• High school algebra class working on factoring

• You have expectations about the way discussions should happen in your classroom. Do your students know what they are?

• As you watch, what norms are being established to encourage discussion?

38

Why are norms important?

39

Norms for students

• Take turns • Listen to others ideas • Disagree with ideas not people • Be respectful • Helping is not the same as giving answers • Confusion is part of learning • Say your “becauses” • “I can’t do that yet?”

Horn, 2012

Norms have a purpose and need to be clear about this purpose. The enable students to achieve the math goals of learning content and how to think mathematically. Help students grow as listeners and as questioners Enable students to can take charge of their own learning and that of their peers.

41

Norms for teachers

• Listen for what can be learned about students' thinking rather than for correct answers

• Identify & check a “hinge point” in the lesson where student understanding is critical for moving on

• “no hands up, except to ask a question” Leahy et al, 2005

• Be relentless in asking what does it mean/why it works • Maintain neutral stance with respect to answers • Record responses so everyone can think about them • Wait time before responses/after response • Deflect questions to students • Offer examples/counterexamples to test understanding • Plan questions/discussion in advance

Things to get “math talk” going in your classroom

• Sentence starters • Gallery walk • Questioning framework • Favorite no • Dyads • Pair-share • ……

43

Mathematical Discourse

• Where did we see it today?

• Where did we see it yesterday?

• How does it look in your classroom?

• What can you do to get it there more often?

44

Exit ticket

• What was your biggest take away? • What ideas will you take from the institute to

do in your classroom?

45

Further Reading/Resources/Contact Info

Benjamin, A.T. and J.J. Quinn, Proofs That Really Count: The Art of Combinatorial Proof, Dolciani Mathematical Expositions, Volume 27, Mathematical Association of America, 2003.

Sinwell, B.J. “The Chebyshev Polynomials: Patterns and

Derivation.” Mathematics Teacher. August 2004, Volume 98, Issue 1, p.20.

Driscoll, Mark, “Fostering Algebraic Thinking: A Guide for

Teacher, Grades 6-10” http://illuminations.nctm.org/LessonDetail.aspx?ID=U184 Benjamin J. Sinwell, Pendleton High School, Anderson 4 Public Schools, South Carolina [email protected]

• Kader, G., & Memer, J. (2008). Contemporary curricular Issues: Statistics in the middle school: Understanding center and spread, pp. 38-43

47

Disclaimer

The National Council of Teachers of Mathematics is a public voice of mathematics education, providing vision, leadership, and professional development to support teachers in ensuring equitable mathematics learning of the highest quality for all students. NCTM’s Institutes, an official professional development offering of the National Council of Teachers of Mathematics, supports the improvement of pre-K-14 mathematics education by serving as a resource for teachers so as to provide more and better mathematics for all students. It is a forum for the exchange of mathematics ideas, activities, and pedagogical strategies, and for sharing and interpreting research. The Institutes presented by the Council present a variety of viewpoints. The views expressed or implied in the Institutes, unless otherwise noted, should not be interpreted as official positions of the Council.

48

using transformations to make connections between …...14 . functions and sliders ... transform...

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