Transcript PowerPoint Presentation - Supporting Student Learning of

Supporting Rigorous Mathematics
Teaching and Learning
Engaging In and Analyzing Teaching and
Learning
Tennessee Department of Education
High School Mathematics
Algebra 2
© 2013 UNIVERSITY OF PITTSBURGH
Rationale
Asking a student to understand something means asking a teacher
to assess whether the student has understood it. But what does
mathematical understanding look like? One hallmark of
mathematical understanding is the ability to justify, in a way
appropriate to the student’s mathematical maturity, why a particular
mathematical statement is true….…Mathematical understanding
and procedural skill are equally important, and both are assessable
using mathematical tasks of sufficient richness.
Common Core State Standards for Mathematics, 2010
By engaging in a task, teachers will have the opportunity to
consider the potential of the task and engagement in the task for
helping learners develop the facility for expressing a relationship
between quantities in different representational forms, and for
making connections between those forms.
Session Goals
Participants will:
• develop a shared understanding of teaching and
learning; and
• deepen content and pedagogical knowledge of
mathematics as it relates to the Common Core State
Standards (CCSS) for Mathematics.
© 2013 UNIVERSITY OF PITTSBURGH
Overview of Activities
Participants will:
• engage in a lesson; and
• reflect on learning in relationship to the CCSS.
© 2013 UNIVERSITY OF PITTSBURGH
Looking Over the Standards
• Look over the focus cluster standards.
• Briefly Turn and Talk with a partner about the
meaning of the standards.
• We will return to the standards at the end of the
lesson and consider:
 What focus cluster standards were addressed in
the lesson?
 What gets “counted” as learning?
© 2013 UNIVERSITY OF PITTSBURGH
Missing Function Task
If h(x) = f(x) · g(x), what can you determine about g(x)
from the given table and graph? Explain your reasoning.
x
-2
-1
0
1
2
f(x)
0
1
2
3
4
© 2013 UNIVERSITY OF PITTSBURGH
The Structures and Routines of a Lesson
Set Up of the Task
The Explore Phase/Private Work Time
Generate Solutions
The Explore Phase/Small Group Problem
Solving
1. Generate and Compare Solutions
2. Assess and Advance Student Learning
Share, Discuss, and Analyze Phase of the Lesson
1. Share and Model
2. Compare Solutions
3. Focus the Discussion on
Key Mathematical Ideas
4. Engage in a Quick Write
MONITOR: Teacher selects
examples for the Share,
Discuss, and Analyze Phase
based on:
• Different solution paths to the
same task
• Different representations
• Errors
• Misconceptions
SHARE: Students explain their
methods, repeat others’ ideas,
put ideas into their own words,
add on to ideas and ask for
clarification.
REPEAT THE CYCLE FOR EACH
SOLUTION PATH
COMPARE: Students discuss
similarities and difference
between solution paths.
FOCUS: Discuss the meaning
of mathematical ideas in each
representation
REFLECT: By engaging
students in a quick write or a
discussion of the process.
Solve the Task
(Private Think Time and Small Group Time)
• Work privately on the Missing Function Task.
• Work with others at your table. Compare your
solution paths. If everyone used the same method to
solve the task, see if you can come up with a
different way.
• Consider what each person determined about g(x).
© 2013 UNIVERSITY OF PITTSBURGH
Expectations for Group Discussion
• Solution paths will be shared.
• Listen with the goals of:
– putting the ideas into your own words;
– adding on to the ideas of others;
– making connections between solution paths;
and
– asking questions about the ideas shared.
• The goal is to understand the mathematics and to
make connections among the various solution paths.
© 2013 UNIVERSITY OF PITTSBURGH
Missing Function Task
If h(x) = f(x) · g(x), what can you determine about g(x)
from the given table and graph? Explain your reasoning.
x
-2
-1
0
1
2
f(x)
0
1
2
3
4
© 2013 UNIVERSITY OF PITTSBURGH
Discuss the Task
(Whole Group Discussion)
• What do we know about g(x)? How did you use the
information in the table and graph and the
knowledge that h(x) = f(x) · g(x) to determine the
equation of g(x)?
• How can you use what you know about the graphs
of f(x) and g(x) to help you think about the graph of
h(x)?
• Predict the shape of the graph of a function that is
the product of two linear functions. Explain from the
graphs of the two functions why you have made
your prediction.
© 2013 UNIVERSITY OF PITTSBURGH
Reflecting on Our Learning
• What supported your learning?
• Which of the supports listed will EL students benefit
from during instruction?
• Which CCSS for Mathematical Content did we
discuss?
• Which CCSS for Mathematical Practice did you use
when solving the task?
© 2013 UNIVERSITY OF PITTSBURGH
Linking to Research/Literature
Connections between Representations
Pictures
Manipulative
Models
Written
Symbols
Real-world
Situations
Oral
Language
Adapted from Lesh, Post, & Behr, 1987
Five Different Representations of a Function
Language
Context
Table
Graph
Van De Walle, 2004, p. 440
Equation
Reflecting on Our Learning
• What supported your learning?
• Which of the supports listed will EL students benefit
from during instruction?
• Which CCSS for Mathematical Content did we
discuss?
• Which CCSS for Mathematical Practice did you use
when solving the task?
© 2013 UNIVERSITY OF PITTSBURGH
The CCSS for Mathematical Content
CCSS Conceptual Category – Number and Quantity
The Real Number System
(N-RN)
Extend the properties of exponents to rational exponents.
N-RN.A.1 Explain how the definition of the meaning of rational
exponents follows from extending the properties of
integer exponents to those values, allowing for a
notation for radicals in terms of rational exponents. For
example, we define 51/3 to be the cube root of 5
because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must
equal 5.
N-RN.A.2 Rewrite expressions involving radicals and rational
exponents using the properties of exponents.
Common Core State Standards, 2010, p. 60, NGA Center/CCSSO
The CCSS for Mathematical Content
CCSS Conceptual Category – Algebra
Seeing Structure in Expressions
(A–SSE)
Write expressions in equivalent forms to solve problems.
A-SSE.B.3
Choose and produce an equivalent form of an expression to
reveal and explain properties of the quantity represented by
the expression.★
A-SSE.B.3c Use the properties of exponents to transform expressions for
exponential functions. For example the expression 1.15t can
be rewritten as (1.151/12)12t ͌ 1.01212t to reveal the approximate
equivalent monthly interest rate if the annual rate is 15%.
A-SSE.B.4
★
Derive the formula for the sum of a finite geometric series
(when the common ratio is not 1), and use the formula to solve
problems. For example, calculate mortgage payments.★
Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and
specific modeling standards appear throughout the high school standards indicated with a star (★). Where
an entire domain is marked with a star, each standard in that domain is a modeling standard.
Common Core State Standards, 2010, p. 64, NGA Center/CCSSO
The CCSS for Mathematical Content
CCSS Conceptual Category – Algebra
Arithmetic with Polynomials and Rational Expressions (A–APR)
Understand the relationship between zeros and factors of
polynomials.
A-APR.B.2
Know and apply the Remainder Theorem: For a polynomial
p(x) and a number a, the remainder on division by x – a is
p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).
A-APR.B.3
Identify zeros of polynomials when suitable factorizations are
available, and use the zeros to construct a rough graph of the
function defined by the polynomial.
Common Core State Standards, 2010, p. 64, NGA Center/CCSSO
The CCSS for Mathematical Content
CCSS Conceptual Category – Functions
Building Functions
(F–BF)
Build a function that models a relationship between two quantities.
F-BF.A.1
Write a function that describes a relationship between two
quantities.★
F-BF.A.1a
Determine an explicit expression, a recursive process, or
steps for calculation from a context.
F-BF.A.1b
Combine standard function types using arithmetic operations.
For example, build a function that models the temperature of
a cooling body by adding a constant function to a decaying
exponential, and relate these functions to the model.
F-BF.A.2
Write arithmetic and geometric sequences both recursively
and with an explicit formula, use them to model situations,
and translate between the two forms.★
★Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific
modeling standards appear throughout the high school standards indicated with a star (★). Where an entire domain is
marked with a star, each standard in that domain is a modeling standard.
Common Core State Standards, 2010, p. 70, NGA Center/CCSSO
Reflecting on Our Learning
• What supported your learning?
• Which of the supports listed would EL students
benefit from during instruction?
• Which CCSS for Mathematical Content did we
discuss?
• Which CCSS for Mathematical Practice did you use
when solving the task?
© 2013 UNIVERSITY OF PITTSBURGH
What math practices made it possible for
us to learn?
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning
of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Common Core State Standards for Mathematics, 2010
Research Connection: Findings by
Tharp and Gallimore
• For teaching to have occurred - Teachers must “be aware of
the students’ ever-changing relationships to the subject
matter.”
• They [teachers] can assist because, while the learning process
is alive and unfolding, they see and feel the student's
progression through the zone, as well as the stumbles and
errors that call for support.
• For the development of thinking skills—the [students’] ability to
form, express, and exchange ideas in speech and writing—the
critical form of assisting learners is dialogue -- the questioning
and sharing of ideas and knowledge that happen in
conversation.
Tharp & Gallimore, 1991
Underlying Mathematical Ideas
Related to the Lesson
(Essential Understandings)
• The product of two or more linear functions is a polynomial
function. The resulting function will have the same xintercepts as the original functions because the original
functions are factors of the polynomial.
• Two or more polynomial functions can be multiplied using
the algebraic representations by applying the distributive
property and combining like terms.
• Two or more polynomial functions can be added using their
graphs or tables of values because given two functions f(x)
and g(x) and a specific x-value, x1, the point (x1,
f(x1)+g(x1)) will be on the graph of the sum f(x)+g(x). (This
is true for subtraction and multiplication as well.)
© 2013 UNIVERSITY OF PITTSBURGH
Essential Understandings
EU #1a:
Functions are single-valued mappings from one set—the domain of
the function—to another—its range.
EU #1b:
Functions apply to a wide range of situations. They do not have to
be described by any specific expressions or follow a regular pattern.
They apply to cases other than those of “continuous variation.” For
example, sequences are functions.
EU #1c:
The domain and range of functions do not have to be numbers. For
example, 2-by-2 matrices can be viewed as representing functions
whose domain and range are a two-dimensional vector space.
EU #2a:
For functions that map real numbers to real numbers, certain
patterns of covariation, or patterns in how two variables change
together, indicate membership in a particular family of functions and
determine the type of formula that the function has.
EU #2b:
A rate of change describes how one variable quantity changes with
respect to another—in other words, a rate of change describes the
covariation between variables.
EU #2c:
A function’s rate of change is one of the main characteristics that
determine what kinds of real-world phenomena the function can
model.
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Essential Understandings
EU #3a: Members of a family of functions share the same type of rate of
change. This characteristic rate of change determines the kinds of
real-world phenomena that the function can model.
EU #3c:
Quadratic functions are characterized by a linear rate of change, so
the rate of change of the rate of change (the second derivative) of a
quadratic function is constant. Reasoning about the vertex form of a
quadratic allows deducing that the quadratic has a maximum or
minimum value and that if the zeroes of the quadratic are real, they
are symmetric about the x-coordinate of the maximum or minimum
point.
EU #5a: Functions can be represented in various ways, including through
algebraic means (e.g., equations), graphs, word descriptions, and
tables.
EU #5b: Changing the way that a function is represented (e.g., algebraically,
with a graph, in words or with a table) does not change the function,
although different representations highlight different characteristics,
and some may only show part of the function.
EU #5c:
Some representations of a function may be more useful than others,
depending on the context.
EU #5d: Links between algebraic and graphical representations of functions
are especially important in studying relationships and change.
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