common core state standards mathematical practice …
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COMMON CORE STATE STANDARDS
MATHEMATICAL PRACTICE #4
MODEL WITH MATHEMATICS
Mathematically proficient students can apply the mathematics they
know to solve problems arising in everyday life, society, and the
workplace. In early grades, this might be as simple as writing an
addition equation to describe a situation. In middle grades, a student
might apply proportional reasoning to plan a school event or analyze a
problem in the community. By high school, a student might use
geometry to solve a design problem or use a function to describe how
one quantity of interest depends on another.
Mathematically proficient students who can apply what they know are
comfortable making assumptions and approximations to simplify a
complicated situation, realizing that these may need revision later.
They are able to identify important quantities in a practical situation
and map their relationships using such tools as diagrams, two-way
tables, graphs, flowcharts and formulas.
Mathematically proficient students can analyze those relationships
mathematically to draw conclusions. They routinely interpret their
mathematical results in the context of the situation and reflect on
whether the results make sense, possibly improving the model if it has
not served its purpose.
KEY DATES FOR COMMON CORE TEST
IMPLEMENTATION
DATE ACTIVITY
SPRING
2014
PA STANDARDS
ALIGNED PSSA TESTS
GRADES 3 – 8
SPRING
2015
COMMON CORE
ALIGNED PSSA TESTS
GRADES 3 – 8
VOLUME 1 ISSUE 4
401 N. Whitehall Road
Norristown, PA 19403
610.630.5000 office
www.nasd.k12.pa.us
NORRISTOWN AREA SCHOOL DISTRICT CURRICULUM & INSTRUCTION
SEPTEMBER/OCTOBER 2013
8 M A T H E M A T I C A L
P R A C T I C E S
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
WHAT DOES THE TASK LOOK LIKE?
WHAT DOES THE TEACHER DO?
Task
Illustrates the relevance of the mathematics
involved.
Requires students to identify extraneous or missing
information.
Requires students to identify variables, compute
and interpret results, report findings, and justify
the reasonableness of their results within the
context of the task.
Teacher
Facilitates the discussion in evaluating the
appropriateness of the model.
Expects students to justify their choice of variables
and procedures.
STUDENTS ESTABLISH A BASE OF KNOWLEDGE ACROSS A WIDE RANGE OF SUBJECT MATTER BY
ENGAGING WITH WORKS OF QUALITY AND SUBSTANCE.
–COMMON CORE STATE STANDARDS
“Mathematics is not a careful march down a well-cleared highway, but
a journey into a strange wilderness, where the explorers often get lost.”
-W.S. Anglin
WHAT ARE STUDENTS DOING?
Apply the mathematics they know to
everyday life, society, and the workplace.
Write equations to describe situations.
Are comfortable in making assumptions
and approximations to simplify
complicated situations.
Analyze relationships to draw
conclusions.
Improve their model if it has not served
its purpose.
VOLUME 1 ISSUE 4
SEPTEMBER/OCTOBER 2013
Modified from: Institute for Advanced Study/Park City Mathematics Institute
-Hancock (2012)
MATHEMATICAL PRACTICE #4
- Jordan School District (2011)
WHAT ARE TEACHERS DOING?
Provide problem situations that apply to
everyday life.
Provide rich tasks that focus on
conceptual understanding, relationships,
etc.
WHAT DOES IT REALLY MEAN?
One intent of this standard is to ensure that children see, even at the
earliest ages, that mathematics is not just a collection of skills whose
only use is to demonstrate that one has these skills.
Another intent is to ensure that the mathematics students engage in
helps them see and interpret the world—the physical world, the
mathematical world, and the world of their imagination—through a
mathematical lens. One way, mentioned in the standard, is through
the use of simplifying assumptions and approximations. Children
typically find “estimation” pointless, and even confusing, when they
can get exact answers, but many mathematical situations do not
provide the information needed for an exact calculation.
What’s important here is not the context that’s used, but the kind of
thinking it requires. Using “approximations to simplify a
complicated situation” can be valuable even within mathematics and
even when exact answers are required.
WHAT ARE SOME EXAMPLES OF TASKS?
About how many children are
in our school? 50? 200?
1000? To that figure that out,
we could count, but that’s a
lot of work. Besides, we
don’t need to know exactly.
How can we come
reasonably close, just sitting
in our classroom?
How many blades of grass
are there on our soccer field?
How can we use estimation
to get reasonably close?
The Iditarod & Math
(Elapsed Time)
https://www.teachingchannel.org/videos/technology-and-math
“Mathematics is not a careful march down a well-cleared highway, but
a journey into a strange wilderness, where the explorers often get lost.”
-W.S. Anglin
QUESTIONS TO
ASK STUDENTS
Why is that a good
model for this
problem?
How can you use a
simpler problem to
help you find the
answer?
How would you
change your model
if…?
VIDEO EXAMPLE
VOLUME 1 ISSUE 4
SEPTEMBER/OCTOBER 2013
-GO Math! Houghton
Mifflin Harcourt (2012)
MATHEMATICAL PRACTICE #4
-www.curriculuminstitute.org (2012)
-Understanding the Mathematical Practices
(2012)
-Understanding the Mathematical Practices (2012)
STUDENTS ESTABLISH A BASE OF KNOWLEDGE ACROSS A WIDE RANGE OF SUBJECT MATTER BY
ENGAGING WITH WORKS OF QUALITY AND SUBSTANCE.
–COMMON CORE STATE STANDARDS
-Little (1999)
VOLUME 1 ISSUE 4
SEPTEMBER/OCTOBER 2013
“Mathematics is not a careful march down a well-cleared highway, but
a journey into a strange wilderness, where the explorers often get lost.”
-W.S. Anglin
Write captions for the selected photos.
WHAT ARE STUDENTS DOING?
WHAT IS THE TEACHER DOING?
Students
Apply mathematics to solve problems that arise in
everyday life.
Demonstrate understanding using a variety of appropriate
tools and strategies.
Are comfortable attempting challenging problems.
Reflect on their attempt to solve problems and make
revisions to improve their model as necessary.
Teachers
Select problems that are challenging and reflect everyday
situations.
Make connections between mathematics and everyday
life.
Focus students on the process rather than the solution.
-Tompkins Seneca Tioga BOCES (2012)
WHAT DO PROFICIENT
STUDENTS DO?
Model with Mathematics
Initial
Use models to
represent and solve a
problem, and
translate the solution
to mathematical
symbols.
Intermediate
Use models and
symbols to represent
and solve a problem,
and accurately
explain the solution
representation.
Advanced
Use a variety of
models, symbolic
representations, and
technology tools to
demonstrate a
solution to a problem.
-Hull, Balka, and Harbin Miles (2011)
mathleadership.com
MATHEMATICAL PRACTICE #4
-Lewis, Morgan, Wallen, and Younger (2012)
STUDENTS ESTABLISH A BASE OF KNOWLEDGE ACROSS A WIDE RANGE OF SUBJECT MATTER BY
ENGAGING WITH WORKS OF QUALITY AND SUBSTANCE.
–COMMON CORE STATE STANDARDS
VOLUME 1 ISSUE 4
SEPTEMBER/OCTOBER 2013
“Mathematics is not a careful march down a well-cleared highway, but
a journey into a strange wilderness, where the explorers often get lost.”
-W.S. Anglin
Write captions for the selected photos.
References
Curriculum Institute (2013). Standards for Mathematical Practice Posters. Available at
http://www.curriculuminstitute.org/indiana/materials/Standards%20of%20Mathematica
l%20Practice%20Student%20Posters.pdf
GO Math! Houghton Mifflin Harcourt (2012). Supporting Mathematical Practices
Through Questioning. Orlando, FL: Houghton Mifflin Harcourt.
Hancock, Melissa (2011). Practice Standards Walk-Through Document. Available at:
http://katm.org/wp/common-core/
Hausman, Todd (2013). Teaching Channel: The Iditarod & Math. Available at:
https://www.teachingchannel.org/videos/technology-and-math
Hull, Balka, and Harbin Miles (2011). Standards of Student Practice in Mathematics
Proficiency Matrix. Available at http://mathleadership.com/ccss.html
Institute for Advanced Study/Park City Mathematics Institute (2011). Rubric-
Implementing Standards for Mathematical Practice. Available at
http://ime.math.arizona.edu/2011-
12/FebProducts/Mathematical%20Practices%20Rubric.pdf
Jordan School District (2011). Mathematical Practices by Standard Posters. Available
at http://elemmath.jordandistrict.org/mathematical-practices-by-standard/
Lewis, S.; Morgan, T.; Wallen, K.; and Younger, J. (2012). Focusing on the
Mathematical Practices of the Common Core Grades K – 8. Available at
http://www.sevier.org/CommonCore/FocusingMathPracticices_CCSS.pdf
Little, Catherine (1999). Geometry Projects Linking Mathematics, Literacy, Art, and
Technology. Mathematics Teaching in the Middle School; v4 n5 p332-35 Feb.
Tompkins Seneca Tioga BOCES (2012). Mathematical Practices and Indicators.
Available at http://tst-math.wikispaces.com/Mathematical+Practices
Understanding the Mathematical Practices (2012). Practice Standard 4: Model with
Mathematics. Available at
http://www.cesu.k12.vt.us/modules/groups/homepagefiles/cms/1556877/File/PracticeSt
d4.pdf
MATHEMATICAL PRACTICE #4
Norristown Area
School District
401 N. Whitehall Road
Norristown PA 19403
Administration Office:
610.630.5000
www.nasd.k12.pa.us
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Practices in your
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STUDENTS ESTABLISH A BASE OF KNOWLEDGE ACROSS A WIDE RANGE OF SUBJECT MATTER BY
ENGAGING WITH WORKS OF QUALITY AND SUBSTANCE.
–COMMON CORE STATE STANDARDS