Study Group 3 - High School
Math (Algebra 1 & 2, Geometry)
Welcome Back!
Let’s spend some quality time discussing what we learned
from our Bridge to Practice exercises.
© 2013 UNIVERSITY OF PITTSBURGH
Let’s Go Over Bridge to Practice #2:
Time to Reflect on Our Learning
Part 1:
For Algebra 1, Using the Bike and Truck Task:
For Algebra 2, Using the Missing Function Task:
For Geometry, Using the Building a New Playground Task:
a. Choose the Content Standards from the relevant pages in your
module 2 handout (or view the standards on the following slides for
each subject area Alg 1: 6-9, Alg 2: 16-19, Geometry: 25-28)
b. Choose the Practice Standards students will have the opportunity
to use while solving this task and find evidence to support them.
© 2013 UNIVERSITY OF PITTSBURGH
For Algebra 1: 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
For Algebra 1 - 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
Algebra 1 - Reflecting on Our Learning
• 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
Algebra 1 Task
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
a star,OFeach
standard in that domain is a modeling standard.
© 2013with
UNIVERSITY
PITTSBURGH
Common Core State Standards, 2010, p. 65, NGA Center/CCSSO
The CCSS for Mathematical Content
Algebra 1 Task
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),
© 2013 UNIVERSITY OF PITTSBURGH
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
Algebra 1 Task
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.
© 2013 UNIVERSITY OF PITTSBURGH
Common Core State Standards, 2010, p. 65, NGA Center/CCSSO
The CCSS for Mathematical Content
Algebra 1 Task
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
© 2013
UNIVERSITY
OF PITTSBURGH
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 Common
a modeling
standard.
Core
State Standards, 2010, p. 69, NGA Center/CCSSO
For Algebra 1 Task:
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
Common Core State Standards for Mathematics, 2010
reasoning.
© 2013 UNIVERSITY OF PITTSBURGH
Part 3 - Underlying Mathematical Ideas
Related to the Lesson – For Algebra 1
(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 – Algebra 1
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.
© 2013 UNIVERSITY OF PITTSBURGH
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 – Algebra 1
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.
© 2013 UNIVERSITY OF PITTSBURGH
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.
For Algebra 2: 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
Algebra 2 - Reflecting on Our Learning
• 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 – Alg 2 Task
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 - Alg 2 Task
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 – Alg 2 Task
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 – Alg 2 Task
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
For Algebra 2 Task:
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
Part 3 - Underlying Mathematical Ideas
Related to the Lesson – For Algebra 2
(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
For Geometry:
Building a New Playground Task
The City Planning Commission is considering building a
new playground. They would like the playground to be
equidistant from the two elementary schools,
represented by points A and B in the coordinate grid
that is shown.
© 2013 UNIVERSITY OF PITTSBURGH
For Geometry:
Building a New Playground
PART A
1. Determine at least three possible locations for the park
that are equidistant from points A and B. Explain how you
know that all three possible locations are equidistant from
the elementary schools.
2. Make a conjecture about the location of all points that are
equidistant from A and B. Prove this conjecture.
PART B
3. The City Planning Commission is planning to build a third
elementary school located at (8, -6) on the coordinate
grid. Determine a location for the park that is equidistant
from all three schools. Explain how you know that all
three schools are equidistant from the park.
4. Describe a strategy for determining a point equidistant
from any three points.
© 2013 UNIVERSITY OF PITTSBURGH
Geometry - Reflecting on Our Learning
• 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 – Geometry
Congruence
(G-CO)
Understand congruence in terms of rigid motions.
G-CO.B.6 Use geometric descriptions of rigid motions to transform
figures and to predict the effect of a given rigid motion
on a given figure; given two figures, use the definition of
congruence in terms of rigid motions to decide if they
are congruent.
G-CO.B.7 Use the definition of congruence in terms of rigid
motions to show that two triangles are congruent if and
only if corresponding pairs of sides and corresponding
pairs of angles are congruent.
G-CO.B.8 Explain how the criteria for triangle congruence (ASA,
SAS, and SSS) follow from the definition of congruence
in terms of rigid motions.
© 2013 UNIVERSITY OF PITTSBURGH
Common Core State Standards, 2010, p. 76, NGA Center/CCSSO
The CCSS for Mathematical Content
CCSS Conceptual Category – Geometry
Congruence
(G-CO)
Prove geometric theorems.
G-CO.C.9
Prove theorems about lines and angles. Theorems include:
vertical angles are congruent; when a transversal crosses
parallel lines, alternate interior angles are congruent and
corresponding angles are congruent; points on a
perpendicular bisector of a line segment are exactly those
equidistant from the segment’s endpoints.
G-CO.C.10 Prove theorems about triangles. Theorems include: measures
of interior angles of a triangle sum to 180°; base angles of
isosceles triangles are congruent; the segment joining
midpoints of two sides of a triangle is parallel to the third side
and half the length; the medians of a triangle meet at a point.
Prove theorems about parallelograms. Theorems include:
opposite sides are congruent, opposite angles are congruent,
the diagonals of a parallelogram bisect each other, and
conversely, rectangles are parallelograms with congruent
diagonals.
© 2013 UNIVERSITY OF PITTSBURGH
G-CO.C.11
Common Core State Standards, 2010, p. 76, NGA Center/CCSSO
The CCSS for Mathematical Content
CCSS Conceptual Category – Geometry
Similarity, Right Triangles, and Trigonometry
(G-SRT)
Define trigonometric ratios and solve problems involving right
triangles.
G-SRT.C.6
Understand that by similarity, side ratios in right triangles are
properties of the angles in the triangle, leading to definitions
of trigonometric ratios for acute angles.
G-SRT.C.7
Explain and use the relationship between the sine and
cosine of complementary angles.
G-SRT.C.8
Use trigonometric ratios and the Pythagorean Theorem to
solve right triangles in applied problems.★
★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.
© 2013 UNIVERSITY OF PITTSBURGH
Common Core State Standards, 2010, p. 77, NGA Center/CCSSO
The CCSS for Mathematical Content
CCSS Conceptual Category – Geometry
Expressing Geometric Properties with Equations
(G-GPE)
Use coordinates to prove simple geometric theorems algebraically.
G-GPE.B.4
G-GPE.B.5
G-GPE.B.6
G-GPE.B.7
Use coordinates to prove simple geometric theorems algebraically.
For example, prove or disprove that a figure defined by four given
points in the coordinate plane is a rectangle; prove or disprove that
the point (1, √3) lies on the circle centered at the origin and containing
the point (0, 2).
Prove the slope criteria for parallel and perpendicular lines and use
them to solve geometric problems (e.g., find the equation of a line
parallel or perpendicular to a given line that passes through a given
point).
Find the point on a directed line segment between two given points
that partitions the segment in a given ratio.
Use coordinates to compute perimeters of polygons and areas of
triangles and rectangles, e.g., using the distance formula.★
★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,UNIVERSITY
each standard
in that
© 2013
OF PITTSBURGH
domain is a modeling standard.
Common Core State Standards, 2010, p. 78, NGA Center/CCSSO
For Geometry Task:
Which 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
© 2013 UNIVERSITY OF PITTSBURGH
Part 3 - Underlying Mathematical Ideas
Related to the Lesson - For Geometry
(Essential Understandings)
• Coordinate Geometry can be used to form and test conjectures
about geometric properties of lines, angles and assorted
polygons.
• Coordinate Geometry can be used to prove geometric theorems
by replacing specific coordinates with variables, thereby showing
that a relationship remains true regardless of the coordinates.
• The set of points that are equidistant from two points A and B lie
on the perpendicular bisector of line segment AB, because every
point on the perpendicular bisector can be used to construct two
triangles that are congruent by reflection and/or Side-Angle-Side;
corresponding parts of congruent triangles are congruent.
• It is sometimes necessary to prove both 'If A, then B' and 'If
B, then A' in order to fully prove a theorem; this situation is
referred to as an "if and only if" situation; notations for such
situations include <=> and iff.
© 2013 UNIVERSITY OF PITTSBURGH
Part 2 - Research Connection:
Findings by Tharp and Gallimore
This slide pertains to Alg 1, Alg 2, & Geometry
• 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
Tharp & Gallimore, 1991
happen in conversation.
End of review of Bridge to Practice #2
Now we will move into our new Study Group
Module 3 which is divided into two parts:
1. The impact of teacher implementation of a high level
task on student learning
2. Using assessing and advancing questions to support
student learning
© 2013 UNIVERSITY OF PITTSBURGH
Supporting Rigorous Mathematics
Teaching and Learning
Part 1
Enacting Instructional Tasks:
Maintaining the Demands of the Tasks
Tennessee Department of Education
High School Mathematics
© 2013 UNIVERSITY OF PITTSBURGH
Using the Assessment to Think About
Instruction
In order for students to perform well on the Constructed
Response Assessments (CRAs), what are the
implications for instruction?
• What kinds of instructional tasks will need to be
used in the classroom?
• What will teaching and learning look like and sound
like in the classroom?
© 2013 UNIVERSITY OF PITTSBURGH
Rationale
Effective teaching requires being able to support
students as they work on challenging tasks without
taking over the process of thinking for them
(NCTM, 2000). By analyzing the classroom actions
and interactions of six teachers enacting the same
high-level task, teachers will begin to identify
classroom-based factors that are associated with
supporting or inhibiting students’ high-level
engagement during instruction.
© 2013 UNIVERSITY OF PITTSBURGH
Session Goals
Participants will:
• learn about characteristics of the written task that impact
students’ opportunities to think and reason about
mathematics
• learn about the factors of implementation that contribute
to the maintenance and decline of thinking and reasoning
• analyze student work to determine what students know
and can do
• develop assessing and advancing questions based on
student work (this will be part of the Bridge to Practice #3)
© 2013 UNIVERSITY OF PITTSBURGH
The Mathematical Tasks Framework
TASKS
TASKS
TASKS
as they
appear in
curricular/
instructional
materials
as set up by
the teachers
as
implemented
by students
Student
Learning
Stein M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards-based mathematics
instruction: A casebook for professional development, p. 4. New York: Teachers College Press.
© 2013 UNIVERSITY OF PITTSBURGH
The Enactment of the Task
(Private Think Time)
• Read the vignettes.
• Consider the following question:
What are students learning in each classroom?
Scenario A – Mrs. Fox
Scenario B – Mr. Chambers
Scenario C – Ms. Fagan
Scenario D – Ms. Jackson
Scenario E – Mr. Cooper
Scenario F – Ms. Gorman
© 2013 UNIVERSITY OF PITTSBURGH
The Enactment of the Task
(Small Group Discussion)
Discuss the following questions and cite evidence from
the cases:
What are students learning in each classroom?
What made it possible for them to learn?
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The Enactment of the Task
(Whole Group Discussion)
What opportunities did students have to think and
reason in each of the classes?
© 2013 UNIVERSITY OF PITTSBURGH
Research Findings:
The Fate of Tasks
TASKS
TASKS
TASKS
as they
appear in
curricular/
instructional
materials
as set up by
the teachers
as
implemented
by students
© 2013 UNIVERSITY OF PITTSBURGH
Student
Learning
Linking to Research/Literature:
The QUASAR Project
How High-Level Tasks Can Evolve During a Lesson:
• Maintenance of high-level demands.
• Decline into procedures without connection to
meaning.
• Decline into unsystematic and nonproductive
exploration.
• Decline into no mathematical activity.
© 2013 UNIVERSITY OF PITTSBURGH
Factors Associated with the Maintenance
and Decline of High-Level Cognitive
Demands
Decline
• Problematic aspects of the
task become routinized.
• Understanding shifts to
correctness, completeness.
• Insufficient time to wrestle
with the demanding aspects
of the task.
• Classroom management
problems.
• Inappropriate task for a given
group of students.
• Accountability for high-level
products or processes not
expected.
© 2013 UNIVERSITY OF PITTSBURGH
Factors Associated with the Maintenance
and Decline of High-Level Cognitive
Demands
Maintenance
• Scaffolds of student thinking
and reasoning provided.
• A means by which students
can monitor their own
progress is provided.
• High-level performance is
modeled.
• A press for justifications,
explanations through
questioning and feedback.
• Tasks build on students’ prior
knowledge.
• Frequent conceptual
connections are made.
• Sufficient time to explore.
© 2013 UNIVERSITY OF PITTSBURGH
Linking to Research/Literature:
The QUASAR Project
Task Set-Up
Task Implementation Student Learning
A.
High
High
High
B.
Low
Low
Low
C.
High
Low
Moderate
Stein & Lane, 1996
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Mathematical Tasks:
A Critical Starting Point for Instruction
There is no decision that teachers make that has a
greater impact on students’ opportunities to learn and on
their perceptions about what mathematics is than the
selection or creation of the tasks with which the
teacher engages students in studying mathematics.
Lappan & Briars, 1995
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Supporting Rigorous Mathematics
Teaching and Learning
Part 2
Illuminating Student Thinking: Assessing and
Advancing Questions
Tennessee Department of Education
High School Mathematics
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Rationale
Effective teaching requires being able to support
students as they work on challenging tasks without
taking over the process of thinking for them
(NCTM, 2000).
• Asking questions that assess student
understanding of mathematical ideas, strategies or
representations provides teachers with insights
into what students know and can do.
• The insights gained from these questions prepare
teachers to then ask questions that advance
student understanding of mathematical ideas,
strategies or connections to representations.
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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
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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.
Small Groups based on subject area
(Algebra 1, Algebra 2, or Geometry)
Participants will:
• analyze given student work for their subject area to
determine what the students know and what
they can do based only on the evidence from
student work
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Bike and Truck Task – Algebra 1
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)
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Bike and Truck Task - Algebra 1
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.
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What Does Each Student Know?
Algebra 1
Now we will focus on three pieces of student work.
Individually examine the three pieces of student work
A, B, and C for the Bike and Truck Task in your
participant handout.
What does each student know?
Be prepared to share and justify your conclusions.
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Response A - Algebra 1
54
Response B - Algebra 1
55
Response C - Algebra 1
56
Missing Function Task – Algebra 2
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
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What Does Each Student Know?
Algebra 2
Now we will focus on three pieces of student work.
Individually examine the three pieces of student work
A, B, and C for the Missing Function Task in your
Participant Handout.
What does each student know?
Be prepared to share and justify your conclusions.
© 2013 UNIVERSITY OF PITTSBURGH
Response A – Algebra 2
59
Response B – Algebra 2
60
Response C – Algebra 2
61
Building a New Playground Task Geometry
The City Planning Commission is considering building a
new playground. They would like the playground to be
equidistant from the two elementary schools,
represented by points A and B in the coordinate grid
that is shown.
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Building a New Playground Geometry
PART A
1. Determine at least three possible locations for the park
that are equidistant from points A and B. Explain how you
know that all three possible locations are equidistant from
the elementary schools.
2. Make a conjecture about the location of all points that are
equidistant from A and B. Prove this conjecture.
PART B
3. The City Planning Commission is planning to build a third
elementary school located at (8, -6) on the coordinate
grid. Determine a location for the park that is equidistant
from all three schools. Explain how you know that all
three schools are equidistant from the park.
4. Describe a strategy for determining a point equidistant
from any three points.
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What Does Each Student Know?
Geometry
Now we will focus on three pieces of student work.
Individually examine the three pieces of student work
A, B, and C for the Building a New Playground Task
in your Participant Handout.
What does each student know?
Be prepared to share and justify your conclusions.
© 2013 UNIVERSITY OF PITTSBURGH
Response A - Geometry
65
Response B - Geometry
66
Response C - Geometry
67
Group D - Cannot Get Started
Imagine that you are walking around the room, observing
your students as they work on the task for either Algebra 1
or 2, or Geometry.
Group D has little or nothing on their papers.
Consider an assessing question and an advancing question
for Group D. Be prepared to share and justify your
conclusions.
Reminder: You cannot TELL Group D how to start.
What questions can you ask them?
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Before Beginning Bridge to Practice #3:
As you complete your next Bridge to Practice, reflect on
the Content Standards and Essential Understandings as
needed to help focus our discussion
• For Algebra 1, Using the Bike and Truck Task:
F-IF.B.4; F-IF.B.5; F-IF.B.6
• For Algebra 2, Using the Missing Function Task:
A-APR.A.1; A-APR (cluster); F-BF.A.1b
• For Geometry, Using the Building a New Playground Task:
G-GPE.B.4; G-GPE.B.5; G-GPE.B.6
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Bridge to Practice #3
Part A:
Use the list developed of what the students know and
what they can do from the Student Work A-D to
develop questions to be asked during the Explore
Phase of the lesson
– Develop at least one assessing question for
Students A-D for your subject area
– Develop at least one advancing question for
Students A-D for your subject area
© 2013 UNIVERSITY OF PITTSBURGH
Bridge to Practice #3
Part B:
Now that you have solved the task, examined some student
work, and developed your assessing and advancing questions,
facilitate this task with your students and record your
assessing and advancing questions during the small group
explore phase of the lesson.
Note: Record could be audio or video using a device such as
your phone, or have a colleague script your questions for you.
Come prepared to share the questioning from your lesson.
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