common core state standards mathematical practice #5 · common core state standards mathematical...
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COMMON CORE STATE STANDARDS
MATHEMATICAL PRACTICE #5
USE APPROPRIATE TOOLS
STRATEGICALLY
Mathematically proficient students consider the available tools when
solving a mathematical problem. These tools might include pencil and
paper, concrete models, a ruler, a protractor, a calculator, a
spreadsheet, a computer algebra system, a statistical package, or
dynamic geometry software.
Proficient students are sufficiently familiar with tools appropriate for
their grade or course to make sound decisions about when each of
these tools might be helpful, recognizing both the insight to be gained
and their limitations. For example, mathematically proficient high
school students analyze graphs of functions and solutions generated
using a graphing calculator. They detect possible errors by
strategically using estimation and other mathematical knowledge.
When making mathematical models, they know that technology can
enable them to visualize the results of varying assumptions, explore
consequences, and compare predictions with data.
Mathematically proficient students at various grade levels are able to
identify relevant external mathematical resources, such as digital
content located on a website, and use them to pose or solve problems.
They are able to use technological tools to explore and deepen their
understanding of concepts.
KEY DATES FOR COMMON CORE TEST
IMPLEMENTATION
DATE ACTIVITY
SPRING
2014
PA STANDARDS AND
PA CORE ALIGNED
PSSA TESTS
GRADES 3 – 8
SPRING
2015
PA CORE ALIGNED
PSSA TESTS
GRADES 3 – 8
VOLUME 1 ISSUE 5
401 N. Whitehall Road
Norristown, PA 19403
610.630.5000 office
www.nasd.k12.pa.us
NORRISTOWN AREA SCHOOL DISTRICT CURRICULUM & INSTRUCTION
NOVEMBER/DECEMBER 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
Requires multiple learning tools (i.e.,
manipulatives, calculator, graph paper)
Requires students to demonstrate fluency in mental
computations.
Teacher
Allows students to choose appropriate learning
tools.
Creatively finds appropriate alternatives where
tools are not available.
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
“The important thing is to not stop questioning, curiosity has its
own reason for existing.”
-Albert Einstein
WHAT ARE STUDENTS DOING?
Consider available tools when solving a
mathematical problem.
Are familiar with a variety of
mathematics tools and use them when
appropriate to explore and deepen their
understanding of concepts.
VOLUME 1 ISSUE 5
NOVEMBER/DECEMBER 2013
Modified from: Institute for Advanced Study/Park City Mathematics Institute
-Hancock (2012)
MATHEMATICAL PRACTICE #5
- Jordan School District (2011)
WHAT ARE TEACHERS DOING?
Provides a variety of tools and
technology for students to explore to
deepen their understanding of math
concepts.
Provides problem solving tasks that
require students to consider a variety of
tools for solving. (Tools might include
pencil/paper, concrete models, ruler,
protractor, calculator, spreadsheet,
computer algebra system, statistical
package, or dynamic geometry software)
WHAT DOES IT REALLY MEAN?
Essential, and easily overlooked, is the call for students to develop the
ability “to make sound decisions about when each of these tools might be
helpful, recognizing both the insight to be gained and their limitations.”
This certainly requires that students gain sufficient competence with the
tools to recognize the differential power they offer; it also requires that
their learning include opportunities to decide for themselves which tool
serves them best.
It also requires curricula and teaching to include the kinds of problems that
genuinely favor different tools. It may also require that, from time to time,
a particular tool is prescribed—or proscribed—until students develop a
competency that would allow them to make “sound decisions” about which
tool to use.
Teachers who are developing students' capacity to "use appropriate
tools strategically" make clear to students why the use of manipulatives,
rulers, compasses, protractors, and other tools will aid their problem
solving processes. A middle childhood teacher might have his students
select different color tiles to show repetition in a patterning task. A
teacher of adolescents and young adults might have established norms for
accessing tools during the students' group "tinkering processes," allowing
students to use paper strips, brass fasteners, and protractors to create and
test quadrilateral "kite" models.
ONLINE RESOURCES
NCTM Illuminations: This website
has many resources and lesson
ideas that allow for the integration
of various teaching tools into
functions lessons.
http://illuminations.nctm.org/
National Library of Virtual
Manipulatives: Features online
manipulatives that can be used as
learning tools.
http://nlvm.usu.edu/en/nav/vlibrary
.html
Smart Skies: This game was
developed by NASA to help
students with their understanding
of linear functions.
http://www.smartskies.nasa.gov/
Fluently Add and Subtract
within 1,000
https://www.teachingchannel.org/videos/counting-collections-lesson?fd=1
“The important thing is to not stop questioning, curiosity has
its own reason for existing.”
-Albert Einstein
QUESTIONS TO
ASK STUDENTS
What strategy could
you use to make that
calculation easier?
How would
estimation help you
solve that problem?
Why did you decide
to use…?
VIDEO EXAMPLE
VOLUME 1 ISSUE 5
NOVEMBER/DECEMBER 2013
-GO Math! Houghton
Mifflin Harcourt (2012)
MATHEMATICAL PRACTICE #5
-www.curriculuminstitute.org (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
-Resources to Supplement Rubric (2012)
VOLUME 1 ISSUE 5
NOVEMBER/DECEMBER 2013
“The important thing is to not stop questioning, curiosity has its
own reason for existing.”
-Albert Einstein
Write captions for the selected photos.
WHAT ARE STUDENTS DOING?
WHAT IS THE TEACHER DOING?
Students
Detect possible errors when using tools by strategically
using estimation and other mathematical knowledge.
Make sound decisions about tool selection.
Consider the available tools when solving a problem (i.e.
manipulatives, ruler, protractor, calculator)
Are able to use technological tools.
Teachers
Provide a variety of tools daily during mathematics
instruction.
Teaching and modeling appropriate use of tools.
Facilitating discussion regarding tool selection.
Modeling the use of technological tools to explore and
deepen student understanding.
-Tompkins Seneca Tioga BOCES (2012)
WHAT DO PROFICIENT
STUDENTS DO?
Model with Mathematics
Initial
Use the appropriate
tool to find the
solution.
Intermediate
Select from a variety
of tools the ones that
can be used to solve a
problem, and explain
their reasoning for
the selection.
Advanced
Combine various
tools, including
technology, explore
and solve a problem
as well as justify their
tool selection and
problem solution.
-Hull, Balka, and Harbin Miles (2011)
mathleadership.com
MATHEMATICAL PRACTICE #5
-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 5
NOVEMBER/DECEMBER 2013
“The important thing is to not stop questioning, curiosity has
its own reason for existing.”
-Albert Einstein
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/
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
Todd, L. (2013). Sharing Strategies for Counting Collections. Available at
https://www.teachingchannel.org/videos/counting-collections-lesson?fd=1
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 #5
Norristown Area
School District
401 N. Whitehall Road
Norristown PA 19403
Administration Office:
610.630.5000
www.nasd.k12.pa.us
Are you integrating
the Mathematical
Practices in your
lessons?
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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