Using Transformations to Make
Connections between Polygons and
Circles: Linking College- and CareerReadiness 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 yintercepts?
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?
14
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 yintercepts?
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
30
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
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.
…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
Partner A: Graph Partner B: Graph
y = x – 10
y=x–5
y = -x – 10
y = -x – 5
y = x + 10
y=x+5
y = -x + 10
y = -x + 5
•
•
•
•
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
• 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?
Using Technology to Build A Square
• Square with floating points
• Square with axes fixed
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?
• Domain/Range?
• Slope/y-intercept?
• Perimeter?
• 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?
• Read Practices
• Discuss where you saw them today
• Discuss where you use them in your practice
41
What questions could we ask?
• Domain/Range?
• Slope/y-intercept?
• Perimeter?
• 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?
• 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
53
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?
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
2
What are we going to do?
• A Rich Mathematical Task
• Sequencing
How are we going to do it?
• Exploration
• Multiple Representations
• By Using The Effective Mathematical Teaching
Practices Standards
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
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
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?
Step 1
Step 2
Step 3
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?
Questions  Engage Students in
Mathematical Practices
1
1 1
1 2 1
1 3 3 1
Questions  Engage Students in
Mathematical Practices
(H + T)
(H + T)2
(H + T)3
Questions  Engage Students in
Mathematical Practices
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
A train of Length 3
A train of Length 4
Train
Length
(n)
1
2
3
4
5
Number
of Ways
(An)
Trains of Length 1,2,3,4, and 5 Using
Only Cars of Length 1 and 2?
Trains of Length 1,2,3,4, and 5 Using
Only Cars of Length 1 and 2?
Organizing it…
An = An-1 + An-2
Train Number
Length of Ways
(n)
(An)
1
1
2
2
3
3
4
5
5
8
An explicit formula?
1 1− 5 
1 1+ 5 
An =



 −
5 2 
5 2 
n
n
Explicit Formula From Earlier
An= 2n-1
came from
An= 2An-1
1
1− 2n
B*2
n
1 1+ 5 
1 1− 5 
An =

 −


5 2 
5 2 
n
came from
An = An-1 + An-2
1
2
1− n − n
 1+ 5 
n
1−
2


 1− 5 
n
1−
2


n
1+ 5 
C *

 2 
1− 5 
D *

 2 
Questions  Engage Students in
Mathematical Practices
How many trains
are made from
exactly 1 rod? 2
rods? 3 rods? 4
rods?
Questions  Engage Students in
Mathematical Practices
1
1 1
1 2 1
1 3 3 1
Questions  Engage Students in
Mathematical Practices
Questions  Engage Students in
Mathematical Practices
1
1 1
0 2 1
0 1 3 1
0 0 3 4 1
Questions  Engage Students in
Mathematical Practices
1
1 1
0 2 1
0 1 3 1
0 0 3 4 1
Questions  Engage Students in
Mathematical Practices
1
1 1
0 2 1
0 1 3 1
0 0 3 4 1
Why did I “circle” that piece?
Questions  Engage Students in
Mathematical Practices
1
1 1
0 2 1
0 1 3 1
0 0 3 4 1
Why did I move it?
Questions  Engage Students in
Mathematical Practices
1
1 1
0 2 1
0 1 3 1
0 0 3 4 1
0? 0? 1? 6? 5?1?
Questions  Engage Students in
Mathematical Practices
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”
Horn, 2012
“I can’t do that yet?”
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
•
•
•
•
•
•
•
43
Sentence starters
Gallery walk
Questioning framework
Favorite no
Dyads
Pair-share
……
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
49