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 1
© 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
Bike and Truck Task
Distance from start of road (in feet)
A bicycle traveling at a steady rate and a truck are
moving along a road in the same direction. The graph
below shows their positions as a function of time. Let
B(t) represent the bicycle’s distance and K(t) represent
the truck’s distance.
Time (in seconds)
© 2013 UNIVERSITY OF PITTSBURGH
Bike and Truck Task
1. Label the graphs appropriately with B(t) and K(t).
Explain how you made your decision.
2. Describe the movement of the truck. Explain how
you used the values of B(t) and K(t) to make
decisions about your description.
3. Which vehicle was first to reach 300 feet from the
start of the road? How can you use the domain
and/or range to determine which vehicle was the first
to reach 300 feet? Explain your reasoning in words.
4. Jack claims that the average rate of change for both
the bicycle and the truck was the same in the first 17
seconds of travel. Explain why you agree or disagree
with Jack.
© 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 Bike and Truck Task.
• Work with a partner and then others at your table.
• Consider the information that can be determined
about the two vehicles.
© 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
Bike and Truck Task
Distance from start of road (in feet)
A bicycle traveling at a steady rate and a truck are
moving along a road in the same direction. The graph
below shows their positions as a function of time. Let
B(t) represent the bicycle’s distance and K(t) represent
the truck’s distance.
Time (in seconds)
© 2013 UNIVERSITY OF PITTSBURGH
Bike and Truck Task
1. Label the graphs appropriately with B(t) and K(t).
Explain how you made your decision.
2. Describe the movement of the truck. Explain how
you used the values of B(t) and K(t) to make
decisions about your description.
3. Which vehicle was first to reach 300 feet from the
start of the road? How can you use the domain
and/or range to determine which vehicle was the first
to reach 300 feet? Explain your reasoning in words.
4. Jack claims that the average rate of change for both
the bicycle and the truck was the same in the first 17
seconds of travel. Explain why you agree or disagree
with Jack and why.
© 2013 UNIVERSITY OF PITTSBURGH
Discuss the Task
(Whole Group Discussion)
• How did you describe the movement of the truck,
as opposed to that of the bike? What information
from the graph did you use to make those
decisions?
• In what ways did you use the information you
determined about the two vehicles to determine
which vehicle was first to reach 300 feet from the
start of the road?
• When, if ever, is the average rate of change the
same for the two vehicles?
© 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 – Algebra
Creating Equations*
(A–CED)
Create equations that describe numbers or relationships.
A-CED.A.1 Create equations and inequalities in one variable and use them to
solve problems. Include equations arising from linear and quadratic
functions, and simple rational and exponential functions.
A-CED.A.2 Create equations in two or more variables to represent
relationships between quantities; graph equations on coordinate
axes with labels and scales.
A-CED.A.3 Represent constraints by equations or inequalities, and by systems
of equations and/or inequalities, and interpret solutions as viable or
nonviable options in a modeling context. For example, represent
inequalities describing nutritional and cost constraints on
combinations of different foods.
A-CED.A.4 Rearrange formulas to highlight a quantity of interest, using the
same reasoning as in solving equations. For example, rearrange
Ohm’s law V = IR to highlight resistance R.
*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. 65, NGA Center/CCSSO
The CCSS for Mathematical Content
CCSS Conceptual Category – Algebra
Reasoning with Equations and Inequalities
(A–REI)
Solve equations and inequalities in one variable.
A-REI.B.3
Solve linear equations and inequalities in one variable,
including equations with coefficients represented by letters.
A-REI.B.4
Solve quadratic equations in one variable.
A-REI.B.4a Use the method of completing the square to transform any
quadratic equation in x into an equation of the form (x – p)2 = q
that has the same solutions. Derive the quadratic formula from
this form.
A-REI.B.4b Solve quadratic equations by inspection (e.g., for x2 = 49),
taking square roots, completing the square, the quadratic
formula and factoring, as appropriate to the initial form of the
equation. Recognize when the quadratic formula gives
complex solutions and write them as a ± bi for real numbers a
and b.
Common Core State Standards, 2010, p. 65, NGA Center/CCSSO
The CCSS for Mathematical Content
CCSS Conceptual Category – Algebra
Reasoning with Equations and Inequalities
(A–REI)
Represent and solve equations and inequalities graphically.
A-REI.D.10 Understand that the graph of an equation in two variables is the set of
all its solutions plotted in the coordinate plane, often forming a curve
(which could be a line).
A-REI.D.11
Explain why the x-coordinates of the points where the graphs of the
equations y = f(x) and y = g(x) intersect are the solutions of the
equation f(x) = g(x); find the solutions approximately, e.g., using
technology to graph the functions, make tables of values, or find
successive approximations. Include cases where f(x) and/or g(x) are
linear, polynomial, rational, absolute value, exponential, and
logarithmic functions.★
A-REI.D.12
Graph the solutions to a linear inequality in two variables as a halfplane (excluding the boundary in the case of a strict inequality), and
graph the solution set to a system of linear inequalities in two
variables as the intersection of the corresponding half-planes.
★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. 65, NGA Center/CCSSO
The CCSS for Mathematical Content
CCSS Conceptual Category – Functions
Interpreting Functions
(F–IF)
Interpret functions that arise in applications in terms of the context.
F-IF.B.4
For a function that models a relationship between two quantities, interpret key
features of graphs and tables in terms of the quantities, and sketch graphs
showing key features given a verbal description of the relationship. Key
features include: intercepts; intervals where the function is increasing,
decreasing, positive, or negative; relative maximums and minimums;
symmetries; end behavior; and periodicity.★
F-IF.B.5
Relate the domain of a function to its graph and, where applicable, to the
quantitative relationship it describes. For example, if the function h(n) gives the
number of person-hours it takes to assemble n engines in a factory, then the
positive integers would be an appropriate domain for the function.★
F-IF.B.6
Calculate and interpret the average rate of change of a function (presented
symbolically or as a table) over a specified interval. Estimate the rate of
change from a graph.★
★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. 69, NGA Center/CCSSO
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
What standards for mathematical
practice 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 language of change and rate of change (increasing,
decreasing, constant, relative maximum or minimum) can
be used to describe how two quantities vary together over
a range of possible values.
• 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 two variables
(NCTM, EU 2b).
• The average rate of change is the change in the
dependent variable over a specified interval in the
domain. Linear functions are the only family of functions
for which the average rate of change is the same on every
interval in the domain.
© 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.
Cooney, T.J., Beckmann, S., & Lloyd, G.M., & Wilson, P.S. (2010). Developing essential understanding of functions for teaching mathematics in grades 9-12 (p. 8-10).
Reston, VA: National Council of Teachers of Mathematics.
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.
Cooney, T.J., Beckmann, S., & Lloyd, G.M., & Wilson, P.S. (2010). Developing essential understanding of functions for teaching mathematics in grades 9-12 (p. 8-10).
Reston, VA: National Council of Teachers of Mathematics.