Transcript Mathematical Tasks: The Study of Equivalence November 18

Supporting Rigorous Mathematics
Teaching and Learning
Making Sense of the Expressions and
Equations Standards via a Set of Tasks
Tennessee Department of Education
Middle School Mathematics
Grade 8
© 2013 University Of Pittsburgh
Rationale
Tasks form the basis for students’ opportunities to learn what
mathematics is and how one does it, yet not all tasks afford
the same levels and opportunities for student thinking.
[They] are central to students’ learning, shaping not only their
opportunity to learn but also their view of the subject matter.
Adding It Up, National Research Council, 2001, p. 335
By analyzing instructional and assessment tasks that are for
the same domain of mathematics, teachers will begin to
identify the characteristics of high-level tasks, differentiate
between those that require problem-solving, and those that
assess for specific mathematical reasoning.
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Session Goals
Participants will:
• make sense of the Expressions and Equations
Common Core State Standards (CCSS);
• determine the cognitive demand of tasks and make
connections to the Standards for Mathematical
Content and the Standards for Mathematical
Practice; and
• differentiate between assessment items and
instructional tasks.
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Overview of Activities
Participants will:
• analyze a set of tasks as a means of making sense of
the Expressions and Equations Common Core State
Standards (CCSS);
• determine the Standards for Mathematical Content and
the Standards for Mathematical Practice aligned with
the tasks;
• relate the characteristics of high-level tasks to the
CCSS for Mathematical Content and Practice; and
• discuss the difference between assessment items and
instructional tasks.
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The Data About Students’
Understanding of Expressions
and Equations
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Linking to Research
Students often misinterpret the equals sign as meaning
“the answer is.” In order to make sense of equations
and understand equality, students must recognize that
the equals sign indicates that two expressions have the
same value.
(Knuth, Stephens, McNeil, & Alibali, 2006)
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Linking to Research
Students with a weak conceptual understanding of
equality often incorrectly perform procedural operations.
For example, students who do not think of the equals
sign as an indicator of balance between the terms on
either side often delete or move the equals sign, or
perform operations to only one side of the equation.
(Booth & Koedinger, 2008)
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Linking to Research/Literature
Research has shown that children who have difficultly
translating a concept from one representation to another
are the same children who have difficulty solving
problems and understanding computations.
Strengthening the ability to move between and among
these representations improves the growth of children’s
concepts.
Lesh, Post, & Behr, 1987
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Analyzing Tasks as a Means of
Making Sense of the CCSS
Expressions and Equations
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Linking to Research/Literature:
The QUASAR Project
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, Smith, Henningsen, & Silver, 2000
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Linking to Research/Literature:
The QUASAR Project
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, Smith, Henningsen, & Silver, 2000
Setting Goals
Selecting Tasks
Anticipating Student Responses
Accountable Talk® is a registered trademark of the
University of Pittsburgh
Orchestrating Productive Discussion
• Monitoring students as they work
• Asking assessing and advancing questions
• Selecting solution paths
• Sequencing student responses
• Connecting student responses via Accountable
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Talk® discussions
Linking to Research/Literature:
The QUASAR Project
• Low-level tasks
• High-level tasks
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Linking to Research/Literature:
The QUASAR Project
• Low-level tasks
– Memorization
– Procedures without Connections
• High-level tasks
– Doing Mathematics
– Procedures with Connections
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The Cognitive Demand of Tasks
(Small Group Discussion)
Analyze each task. Determine if the task is a high-level
task. Identify the characteristics of the task that make it
a high-level task.
After you have identified the characteristics of the task,
then use the Mathematical Task Analysis Guide to
determine the type of high-level task.
Use the recording sheet in the participant handout to
keep track of your ideas.
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The Mathematical Task Analysis Guide
Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000) Implementing standards-based mathematics instruction:
A casebook for professional development, p. 16. New York: Teachers College Press
.
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The Cognitive Demand of Tasks
(Whole Group Discussion)
What did you notice about the cognitive demand of the
tasks?
According to the Mathematical Task Analysis Guide,
which tasks would be classified as:
• Doing Mathematics Tasks?
• Procedures with Connections?
• Procedures without Connections?
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Analyzing Tasks: Aligning with the CCSS
(Small Group Discussion)
Determine which Content Standards students would
have opportunities to make sense of when working on
the task.
Determine which Mathematical Practice Standards
students would need to make use of when solving the
task.
Use the recording sheet in the participant handout to
keep track of your ideas.
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Analyzing Tasks: Aligning with the CCSS
(Whole Group Discussion)
How do the tasks differ from each other with respect to
the content that students will have opportunities to
learn?
Do some tasks require that students use Standards for
Mathematical Practice that other tasks don’t require
students to use?
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The CCSS for Mathematical Content − Grade 8
Expressions and Equations
8.EE
Analyze and solve linear equations and pairs of simultaneous linear
equations.
8.EE.C.7
Solve linear equations in one variable.
8.EE.C.7a Give examples of linear equations in one variable with one
solution, infinitely many solutions, or no solutions. Show which of
these possibilities is the case by successively transforming the
given equation into simpler forms, until an equivalent equation of
the form x = a, a = a, or a = b results (where a and b are different
numbers).
8.EE.C.7b Solve linear equations with rational number coefficients, including
equations whose solutions require expanding expressions using
the distributive property and collecting like terms.
Common Core State Standards, 2010, p. 54, NGA Center/CCSSO
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The CCSS for Mathematical Content − Grade 8
Expressions and Equations
8.EE
Analyze and solve linear equations and pairs of simultaneous linear
equations.
8.EE.C.8
Analyze and solve pairs of simultaneous linear equations.
8.EE.C.8a Understand that solutions to a system of two linear equations in two
variables correspond to points of intersection of their graphs, because
points of intersection satisfy both equations simultaneously.
8.EE.C.8b Solve systems of two linear equations in two variables algebraically, and
estimate solutions by graphing the equations. Solve simple cases by
inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution
because 3x + 2y cannot simultaneously be 5 and 6.
8.EE.C.8c
Solve real-world and mathematical problems leading to two linear
equations in two variables. For example, given coordinates for two pairs
of points, determine whether the line through the first pair of points
intersects the line through the second pair.
Common Core State Standards, 2010, p. 55, NGA Center/CCSSO
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The CCSS for Mathematical Practice
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, 2010, p. 6-8, NGA Center/CCSSO
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A. Same or Different?
Explain in words and equations why each of the
following equations is or is not equivalent to
2x – 6 = 15
a. -4x + 12 = -30
a. x – 3 = 7.5
a. 10.5 = x
2
3
a. -2 = 5 – x
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B. A Friendly Walk
Lois, Emile, and Shantay are students at Smith Street
School. They all live on Smith Street. Lois is waiting in
front of her home. Emile and Shantay plan to meet her
there so the three of them can walk to school to watch
the school basketball team play.
• Lois lives x blocks east of the school;
• Emile lives 2 blocks east of Lois; while
• Shantay lives 3 times as many blocks east of the
school as does Emile.
If the total amount of blocks walked by all three students
is 23, how many blocks does each student walk?
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C. Saving Money
Sisters Aya and Jun keep track of the amount they have saved using
the tables below.
Jun’s Savings
Aya’s Savings
Month
Amount in
Bank in
Dollars
Month
Amount in
Bank in
Dollars
1
$10
2
$18
2
$12.50
34
$23
3
$15
6
$28
4
$17.50
8
$33
Write an equation to describe the amount of money in dollars that each
sister has in the bank after any number of months.
After how many months will the sisters have the same amount of money
in the bank? Explain in words how you know.
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D. To Meet or Not to Meet
Many algebra books contain the following message to students:
When 2 lines are graphed on the same set of axes, one of 3
situations will occur: the lines will meet in exactly one point, or
they will be parallel, or they will coincide.
a. Predict the conditions under which the 2 lines will meet in one
point, will be parallel, or will coincide. Use drawings, equations,
and words to write about the thinking behind your predictions.
Exploration
b. Using graph paper, graph 2 lines that meet in exactly one point.
Find the equation of each line. Share your graphs and
equations with a partner, then with another pair. What do you
notice? Using what you have discovered, revise or add on to
your prediction.
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D. To Meet or Not to Meet cont.
c. The graph below shows 2 parallel lines. Find the
equation of each line. Determine the equation of
another line parallel to the 2 below. Share your
equation with a partner, then with another pair. What
do you notice? Using what you have discovered,
revise or add on to your prediction.
-15
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-12
-9
-6
45
40
35
30
25
20
15
10
5
0
-3 -5 0
-10
-15
-20
3
6
9
12
15
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D. To Meet or Not to Meet cont.
d. Using graph paper, graph the following 2 lines:
1
2
2x + 4y = -10 and x =
−5
2
–y
Write about what you notice about the graphs of the 2
lines. Discuss your noticings with a partner, then with
another pair. Using what you have discovered, revise
or add on to your prediction.
e. If you are given 2 equations with the same
independent and dependent variable, how can you
decide (without graphing) whether the equations will
coincide, be parallel, or meet in exactly one point
when graphed on the same axes?
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E. Calling Plans 1
Long-distance Company A charges a base rate of $5
per month, plus 4 cents per minute that you are on the
phone. Long-distance Company B charges a base rate
of only $2 per month, but they charge 10 cents per
minute used. How much time per month would you have
to talk on the phone before subscribing to Company A
would save you money?
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F. Storage Tanks
Two large storage tanks, T and W, contain water. T is losing water. W is
gaining water. The graph below shows the amount of water in each tank
over a period of time; assume that the rates of water loss and water gain
continue as shown.
1000
900
Gallons of water
800
700
600
500
400
300
200
100
0
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15
Number of hours
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When will the two tanks
contain the same amount of
water?
Explain how you found your
solution and how your
solution relates to the
problem.
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G. To Meet Again!
Maya and her brother Devonte both belong to a local
fitness club. The club offers several membership plans.
• Under Maya’s plan, 10 visits cost $150,
while 20 visits cost $250.
• Under Devonte’s plan, 6 visits cost $100,
while 9 visits cost $145.
Explain to Maya and Devonte how they can decide,
without graphing, whether or not the graphs containing
the data above from the two plans will meet.
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Reflecting and Making Connections
• Are all of the CCSS for Mathematical Content in this
cluster addressed by one or more of these tasks?
• Are all of the CCSS for Mathematical Practice
addressed by one or more of these tasks?
• What is the connection between the cognitive
demand of the written task and the alignment of the
task to the Standards for Mathematical Content and
Practice?
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Differentiating Between Instructional
Tasks and Assessment Tasks
Are some tasks more likely to be assessment tasks
than instructional tasks? If so, which and why are you
calling them assessment tasks?
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Instructional Tasks Versus Assessment Tasks
Instructional Tasks
Assessment Tasks
Assist learners to learn the CCSS for
Mathematical Content and the CCSS for
Mathematical Practice.
Assesses fairly the CCSS for Mathematical
Content and the CCSS for Mathematical
Practice of the taught curriculum.
Assist learners to accomplish, often with
others, an activity, project, or to solve a
mathematics task.
Assess individually completed work on a
mathematics task.
Assist learners to “do” the subject matter
under study, usually with others, in ways
authentic to the discipline of mathematics.
Assess individual performance of content
within the scope of studied mathematics
content.
Include different levels of scaffolding
Include tasks that assess both developing
depending on learners’ needs. The
understanding and mastery of concepts and
scaffolding does NOT take away thinking from skills.
the students. The students are still required to
problem-solve and reason mathematically.
Include high-level mathematics prompts.
(The tasks have many of the characteristics
listed on the Mathematical Task Analysis
Guide.)
Include open-ended mathematics prompts as
well as prompts that connect to procedures
with meaning.
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Reflection
• So, what is the point?
• What have you learned about assessment tasks and
instructional tasks that you will use to select tasks to
use in your classroom next school year?
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