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?

Please Share!

Send an email to:

sgardiner@nasd.k12.pa.us

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