welcome tq summer 2011 workshop: proportionality june 6-17, 2011
TRANSCRIPT
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WELCOME
TQ SUMMER 2011 WORKSHOP:
PROPORTIONALITY
JUNE 6-17, 2011
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GROUP NORMS
Be an active learner
Be an attentive listener
Be a reflective participant
Be conscious of your needs
and needs of others
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Success in Algebra
What mathematics do students
need to know to be successful in
Algebra I?
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Pre Test
Work on test items individually.Write your participant number on the test.
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Our Focal Point for Proportional Reasoning
Students will be able to identify, analyze, and represent proportional
and non-proportional linear relationships, to transition
from proportional models a/b = c/d
to
direct variation models y = kx.
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Understanding Proportionality
What do students need to know and be able to do to understand proportionality?
Chart ideas and be prepared to share your ideas with the whole group
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Texas Response to Curriculum Focal Points for
Grade 8 MathematicsRepresenting, applying, and analyzing
proportionality
Students extend their understanding of proportionality to include representations on a coordinate plane and applications, including proportional changes.
TEA 2009
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Fish is Fish
Let’s read a story by Leo Lionni.
Read the story individually.
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Fish is Fish
What 2 or 3 messages does this story convey to you?
Discuss these at your table. Select someone who will share at least
one message from your table with the whole group.
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PRINCIPLES OF LEARNING
Principle One:
Building on Prior Knowledge
Principle Two:
Understanding requires factual knowledge and conceptual framework
Principle Three: Metacognition
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Principle One
Students come to the classroom with preconceptions about how the world works. If their initial understanding is not engaged, they may fail to grasp the new concepts and information that are taught or they may learn them for purposes of a test but revert to their preconceptions outside the classroom.
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Principle Two
Achieving goals in mathematical understanding requires teachers to develop students’ knowledge networks, address students’ learning paths, and use multiple methods.
This suggests the importance of conceptual understanding, procedural fluency, and an effective organization of knowledge.
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Principle Three
Metacognition, getting students to think about their thinking, can help students learn to take control of their own learning by defining learning goals and monitoring their progress in achieving them.
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Atlantis 2008 Unit on Proportional
Reasoning
What is a unit?
Structure of the unit
Planning and Development of the Unit
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Unit Framework
Pre-Assessmen
t
Post-Assessmen
t
Unit Map/
Learning Pathway
Unit Project/ Introductory
Activity
Unit Objectives
Language and Communication
Prior Knowledge/
Pre-Lesson
Generalization
Debugging
Application
Extension
Connection
Unit Core
Introductory Elements
Applied Elements
Core Elements
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Outcomes
Identify and incorporate the three Principles of Learning
Understand proportional reasoning and connect to linear relationships
Transition from a single lesson lens to a unit learning lens
Experience the unit through teacher and student lenses
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Unit Pre-Lesson
Student Hat: Complete the activity individually
Write your answers on the extra copy
Consider the following as you work:
What content is new and what is familiar to you as a student?
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Unit Pre-Lesson
Teacher Hat: Table group discussion
What preconceptions might your students have? (misconceptions, good knowledge, and missing knowledge)
How does the pre-lesson connect with the three principles?
How will the pre-lesson inform classroom instruction?
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Pre-Lesson: Teacher Notes
Review the teacher notes.
Highlight areas you want to remember.
What else would you add to the teacher notes?
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Looking at Student Work
Analyze the student work on the
Pre-LessonWhat misconceptions did you find?How would you address these
misconceptions?
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Another ProblemI went to the store and bought the same number of books as records. Books cost two dollars each and
records cost six dollars each. I spent $40 altogether.
Assuming that the equation 2B+6R=40 is correct, what is wrong if anything, with the following student
reasoning?
“2B+6R=40, since B=R, I can write 2B+6B=40, then 8B=40. This last equation says 8 books is equal to
$40. So, one book costs $5.”
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REFLECTIONS
In your memo book, list two or three advantages to having a pre-lesson.
How does this connect to principle one?
(refer to summary pages in first tab of notebook)
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Time for Reading
“From Words to Algebra: Mending Misconceptions ”
by Jack Lochhead and Jose P. Mestre
(Readings are in your binder under the Resources tab.)
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Mark your Reading
! Insight
? Questions
√ Affirmation
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From Words to Algebra: Mending Misconceptions
Table Group Discussion:
How does reading the article assist you in addressing misconceptions you identified in the student work?
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From Words to Algebra: Mending Misconceptions
Table Group Discussion:
How will this impact your teaching?
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BIG MATHEMATICAL IDEA
Goal is to transition students from proportional models to direct variation models.
rational numbers proportional relationships direct variation models linear functions
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BIG IDEAS - NCSM A ratio is a multiplicative comparison of
quantities
Ratios give relative sizes of quantities being compared
Ratios can be expressed as units by finding an equivalent ratio where the second term is one.
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BIG IDEAS – NCSM, continued
A proportion is a relationship between relationships.
If two quantities vary proportionally, the ratio of corresponding terms is constant.
If two quantities vary proportionally, the constant ratio can be expressed in lowest terms ( a composite unit) or as a unit amount; the constant ratio is the slope of the related linear function.
Ratio is a reflectively abstracted constant ratio.
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LEARNING PATHWAY
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REFLECTIONS
In your memo book, describe one thing you will do differently to identify and address students’ misconceptions.
Share an insight that you have identified in the past two days, that will help you clarify misconceptions.
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Atlantis Mission One
Student Hat: Complete the activity
and applications in groups of two
Write your answers on the extra
copy.
Use your resources as needed.
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Atlantis Mission One
Teacher Hat
What concepts and skills are used in this mission?
What misconceptions might your students have?
How did the applications support student learning?
How does the pre-lesson connect with mission one and the three principles?
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Atlantis Mission Two
Teacher Hat
What concepts and skills are used in this mission?
What misconceptions might your students have?
How does the Mission 1 connect with Mission 2 and the three principles?
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Mission One & Two Teacher Notes
Take five minutes to review the teacher notes.
Highlight areas you want to remember.
What else would you add to the teacher notes?
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Reconnecting with Missions 1 & 2
As a teacher, what connections do you make in regard to mathematical content in the Mission 1&2 activities?
How will these impact your teaching?
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Time for Reading
Proportional Reasoning: Student Misconceptions and
Strategies for Teaching
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Discussion Questions
1. Why do you think 90% of adults do not reason proportionally, according to Lamon 2007?
2. How are multiplicative situations different from additive situations?
3. What are the key steps to proportional reasoning?
4. What is the difference between ratio & rate?
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Implications for Teaching
In what ways will the PCK tool on proportional reasoning impact
your teaching?
As a table group, discuss, chart, and post your ideas on the wall
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Atlantis Mission Three
Student Hat: Complete the activity in groups of two
Think about different ways
students might solve the problems.
Write your answers on the extra copy.
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Atlantis Mission Three
Teacher HatWhat concepts and skills are used in
this mission?What misconceptions might your
students have?How does Mission Three connect with
the three principles?
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Mission Three Teacher Notes
Take five minutes to review the teacher notes.
Highlight areas you want to remember.
What else would you add to the teacher notes?
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Reflections
What are the conceptual ideas in
Mission Three?
In your memo book, answer:
How might transitioning from a single lesson lens to a unit (sequence of lessons) learning lens impact teaching and learning in your classroom?
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Time for Reading
Proportional Reasoning: Student Misconceptions and
Strategies for TeachingPp. 7- 13
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Discussion QuestionsWhat are the common student error
strategies in solving proportional reasoning tasks?
Does the way a problem is presented affect how student interact with the problem? Explain how.
Could additive methods be used in proportional situation?
What are the most common correct strategies in solving proportional tasks?
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Mission Four
Student Hat:
Complete the activity in groups of two at your table
Write your answers on the extra copy
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Atlantis Mission Four Teacher Hat
How is Mission Four different than the other missions?
What different levels of thinking, cognitive demands, are required to complete Mission Four?
What are the common characteristics of proportional relationships? How do they appear in each representation?
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Teacher Notes
Take five minutes to review the teacher notes.
Highlight areas you want to remember.
What else would you add to the teacher notes?
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Mission Five
Student Hat:
Complete the activity in groups of two at your table
Write your answers on the extra copy
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Atlantis Mission Five
Teacher HatWhat concepts and skills are used in
this mission?What misconceptions might your
students have?How does Mission Five connect with
the three principles?
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Mission Six
Student Hat:
Complete the activity in groups of two at your table
Write your answers on the extra copy
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Atlantis Mission Six
Teacher HatWhat concepts and skills are used in
this mission?What misconceptions might your
students have?How does Mission Six connect with
the three principles?
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What are Key Characteristics
of a Proportional Thinker?
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Key Characteristics of a Proportional Thinker
Knowing and understanding the idea of relationship
Ability to recognize multiplicative situations and distinguish it from additive situations
Ability to recognize and explain the difference
between proportional (y=kx) and non-proportional (y=kx+b) situations
Ability to recognize and distinguish different types
of proportionality: direct and inverse
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Ability to use proportionality as a mathematical model in real-world contexts
Knowing and use of the language of proportionality Understanding the concept of function to express the co-
variation Knowing that the graph of a direct proportional situation is a
straight line that passes through the origin Knowing that the graph of non-proportional situation is a
straight line intersecting the y axis b units from the origin Knowing that the graph of an inversely proportional situation is
a hyperbola Understanding that k is the constant ratio in a direct
proportional situations Understanding that k is the constant product in an inversely
proportional situations
Key Characteristics of a Proportional Thinker (cont.)
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Metacognition and Self-Monitoring
Restate Principle Three in your own words. What does it mean to you?
How does the sequence of Mission activities address Principle Three? List at least two examples.
In using metacognition, who is doing the work? What is the teacher doing? What is the student doing?
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Post Test
Work on test items individually.Write your participant number on the test.
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ReflectionDescribe one significant event, theme or idea from your summer professional development. How is the idea you selected connected to your teaching practice? How are you planning to implement in your classroom these ideas from your professional development experience for your own benefit and the benefit of your students?
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Reflection Questions
Where does this unit fit in my curriculum?
What support will I need to implement this unit?
What will I do differently as a result of my experience in the institute?