CARLISLE AREA SCHOOL DISTRICT

Carlisle, PA 17013

Math

Grade 3

Date of Board Approval: August 16, 2007

1

CARLISLE AREA SCHOOL DISTRICT PLANNED INSTRUCTION COVER PAGE

Title of Course: Math______ Subject Area:_Math________ Grade Level: 3________________

Course Length: (Semester/Year): Year Duration: _20-25 minutes Frequency: 5 times per cycle____

Prerequisites: __Not Applicable Credit: Not Applicable Level: Not Applicable

Course Description/Objectives: The district shall provide for attainment of the academic standards per Chapter 4,

Section 4.12. Each student shall demonstrate proficiency in the following area: numbers, number systems and number

relationships; computation and estimation; measurement and estimation; mathematical reasoning and connections,

mathematical problem solving and communication; statistics and data analysis; probability and predictions; algebra and

functions; geometry; trigonometry; and concepts of calculus.

Major Text(s)/Resources:

Harcourt Math 2002

Name of Writing Committee:

Mary Ann Brenneman Mary Kay Durham Denise Eschenmann

Cindy Birdwell Deb Trozzo

2

Strand 2.1 Numbers, Number Systems and Number Relationships Subject Area: Math Grade: 3

PA Academic Standards Performance Indicators Assessment

A. Count using whole numbers (to 10,000) and by 2’s, 3’s, 5’s, 10’s, 25’s and

100’s.

 Count using whole numbers (to 10,000) by: 1’s (e.g., 8987,----,----,8990), 2’s,

3’s, 5’s, 10’s, 25’s, 100’s (all M)

(PSSA).

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Assessment (PSSA) – Grade 3

B. Use whole numbers and fractions to

represent quantities

 Use whole numbers to represent quantities. (R)

 Use fractions (1/2 (R), 1/3, 1/4, 1/5 (M)) to represent quantities.

 Use mixed numbers to represent quantities. (I)

C. Represent equivalent forms of the same

number through the use of concrete

objects, drawings, word names and

symbols.

 Represent equivalent forms of the same number (1-10,000) through the use of:

concrete objects (R), drawings (R),

word names (M), symbols. (M)

(PSSA).

C. Represent equivalent forms of the same

number through the use of concrete

objects, drawings, word names and

symbols.

 Write several number sentences that represent a given number (1-

10,000).(M)

D. Use drawings, diagrams or models to

show the concept of fraction as part of a

whole.

 Use models (geometric shapes) to show fractions as part of a whole: (M)

Identify a fraction of a geometric shape,

Identify a fraction of a set (PSSA).

3

Strand 2.1 Numbers, Number Systems and Number Relationships Subject Area: Math Grade: 3

PA Academic Standards Performance Indicators Assessment

D. Use drawings, diagrams or models to

show the concept of fraction as part of a

whole.

 Use drawings or diagrams to show fractions as part of a whole. (M)

(PSSA)

 Identify the numerator and denominator of a fraction. (M)

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Assessment (PSSA) – Grade 3

E. Count, compare and make change using

a collection of coins and one-dollar

bills.

 Match the name and value of a penny, nickel, dime, and quarter. (R)

 Match the name and value of a half- dollar, one-dollar, and five-dollar bill.

(M)

E. Count, compare and make change using

a collection of coins and one-dollar

bills.

 Count a collection of coins and one- dollar bills. (M) (PSSA)

 Compare collections of coins and one- dollar bills. (M) (PSSA)

 Make change up to $5.00. (M) (PSSA)

F. Apply number patterns (even and odd)

and compare values of numbers on the

hundred boards.

 Apply number patterns (even and odd). (M) (PSSA)

 Compare values of numbers on a hundred chart. (M)

G. Use concrete objects to count, order and

group.

 Use concrete objects to count. (R)

 Classify groups by the number of items in each group. (R)

4

Strand 2.1 Numbers, Number Systems and Number Relationships Subject Area: Math Grade: 3

PA Academic Standards Performance Indicators Assessment

H. Demonstrate an understanding of one-

to-one correspondence.

 Demonstrate one-to-one correspondence. (R)

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Assessment (PSSA) – Grade 3

I. Apply place –value concepts and

numeration to counting, ordering and

grouping.

 Apply place-value concepts and numeration to counting, ordering and

grouping to 10,000. (M) (PSSA)

 Write numbers in expanded notation to 5 places. (M)

I. Apply place –value concepts and

numeration to counting, ordering and

grouping.

 Read and use whole numbers through 10,000. (M) (PSSA)

 Apply knowledge of place value to read and write numbers in expanded form

(1000+100+10+1=1111). (M)

J. Estimate, approximate, round or use

exact numbers as appropriate.

 Round numbers to the nearest 10 or 100 or 1000 (up to 9000). (PSSA)

K. Describe the inverse relationship between addition and subtraction.

 Describe and use the commutative property of addition. (M)

 Describe and use the commutative property of multiplication. (M)

5

Strand 2.1 Numbers, Number Systems and Number Relationships Subject Area: Math Grade: 3

PA Academic Standards Performance Indicators Assessment

K. Describe the inverse relationship between addition and subtraction.

 Write fact families. (R)

 Demonstrate the inverse relationship between addition and subtraction using

fact families.

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Assessment (PSSA) – Grade 3

L. Demonstrate knowledge of basic facts

in four basic operations.

 Demonstrate knowledge of basic facts in addition. (R)

 Demonstrate knowledge of basic facts in subtraction, multiplication (0-10) and

division (in a reasonable amount of

time). (M) (PSSA)

L. Demonstrate knowledge of basic facts

in four basic operations.

 Apply addition strategies of counting on, making ten and adding doubles.

(M)

 Apply subtraction strategies of counting up. (M)

L. Demonstrate knowledge of basic facts

in four basic operations.

 Add fractions having like denominators with sums less than one. (R)

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Strand 2.2 Computation and Estimation Subject Area: Math Grade: 3

PA Academic Standards Performance Indicators Assessment

A. Apply addition and subtraction in

situations involving concrete objects.

 Apply addition situations using concrete objects. (R)

 Apply subtraction situations using concrete objects. (R)

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Assessment (PSSA) – Grade 3

B. Solve single and double-digit addition

and subtraction problems with

regrouping in vertical form.

 Add 2-digit numbers with regrouping in vertical form and horizontal form. (R)

(PSSA)

 Add 3-digit numbers with regrouping in vertical form and horizontal form. (M)

(PSSA)

B. Solve single and double-digit addition

and subtraction problems with

regrouping in vertical form.

 Subtract 2- and 3-digit numbers with regrouping in vertical form and

horizontal form. (M) (PSSA)

 Solve addition and subtraction problems that involve zero. (M)

(PSSA)

B. Solve single and double-digit addition

and subtraction problems with

regrouping in vertical form.

 Check subtraction with addition. (M)

 Explain and use the associative property of addition (e.g., column

addition). (I)

C. Demonstrate the concept of multiplication as repeated addition.

 Demonstrate the concept of multiplication using the following

strategies: repeated addition, doubling,

and skip counting. (M) (PSSA)

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Strand 2.2 Computation and Estimation Subject Area: Math Grade: 3

PA Academic Standards Performance Indicators Assessment

C. Demonstrate the concept of multiplication as repeated addition.

 Demonstrate concept of arrays: Identify, make and describe arrays in

terms of factors and products. (M)

(PSSA), use arrays to write

multiplication sentences. (M),

(continued)

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Assessment (PSSA) – Grade 3

C. Demonstrate the concept of multiplication as repeated addition.

 Recognize that the product of 1 and any number is that number. (M), explain the

zero property. (M)

C. Demonstrate the concept of multiplication as repeated addition.

 Multiply using 1-digit multipliers. (M)

 Multiply mentally with multiples of 10. (M)

C. Demonstrate the concept of multiplication as repeated addition.

 Explain and use the associative property of multiplication. (M)

 Solve problems involving multiplication through the 9’s tables

through 9 x 5.

D. Demonstrate the concept of division as

repeated subtraction and as sharing.

 Demonstrate the concept of division as repeated subtraction and as sharing.

(M)

 Recognize and describe the relationship between multiplication and division

using fact families. (M)

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Strand 2.2 Computation and Estimation Subject Area: Math Grade: 3

PA Academic Standards Performance Indicators Assessment

D. Demonstrate the concept of division as

repeated subtraction and as sharing.

 Solve division problems with remainders. (M)

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Assessment (PSSA) – Grade 3

E. Use estimation skills to arrive at

conclusions.

 Use rounding skills to arrive at conclusions. (M) (PSSA)

F. Determine the reasonableness of

calculated answers.

 Determine the reasonableness of calculated answers in numbers up to

hundreds. (M)

G. Explain addition and subtraction

algorithms with regrouping.

 The student will explain addition and subtraction algorithms with reqrouping,

both orally or in written form up to

hundreds. (M)

9

Strand 2.3 Measurement and Estimation Subject Area: Math Grade: 3

PA Academic Standards Performance Indicators Assessment

A. Compare measurable characteristics of

different objects on the same

dimensions.

 Compare measurable characteristics of different objects on the same

dimensions (e.g., time, temperature,

area, length, weight, capacity,

perimeter, and volume). (M) (PSSA)

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Pennsylvania System of School

Assessment (PSSA) – Grade 3

A. Compare measurable characteristics of

different objects on the same

dimensions.

 Order objects on basis of weight, capacity, and length. (M) (PSSA)

B. Determine the measurement of objects

with non-standard and standard units

 Measure to the nearest ½ inch or centimeter. (M) (PSSA)

 Read and write temperature to nearest degree (Fahrenheit and Celsius). (M)

C. Determine and compare elapsed times.

 Find elapsed time to increments of 5 minutes (limited to 2 adjacent hours)

(PSSA)

 Identify times of the day and night as A.M. and P.M. (PSSA)

C. Determine and compare elapsed times.

 Determine elapsed time between two dates on a calendar. (M)

10

Strand 2.3 Measurement and Estimation Subject Area: Math Grade: 3

PA Academic Standards Performance Indicators Assessment

D. Tell time (analog and digital) to the

minute.

 Read and write time to the minute on digital and analog clocks. (M) (PSSA)

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Pennsylvania System of School

Assessment (PSSA) – Grade 3

E. Determine the appropriate unit of measure.

 Determine the appropriate unit of measure, using US customary or metric

units (meters, yards, kilometers, miles,

ounces, pounds, tons, grams,

kilograms). (M) (PSSA)

F. Use concrete objects to determine area

and perimeter.

 Measure area by counting or multiplying squares. (M) (PSSA)

 Measure perimeter and area of rectangles and write dimensions using

correct labels. (M) (PSSA)

G. Estimate and verify measurements.

 Estimate and verify measurements (lengths). (R) (PSSA)

 Estimate and verify measurements (weights and capacity). (I) (PSSA)

G. Estimate and verify measurements.

 Estimate line segments to the nearest ½ inch; to the nearest centimeter and

verify by measuring the line. (M)

(PSSA)

11

Strand 2.3 Measurement and Estimation Subject Area: Math Grade: 3

PA Academic Standards Performance Indicators Assessment

H. Demonstrate that a single object has

different attributes that can be

measured in different ways (e.g.,

length, mass, weight, time, area,

temperature, capacity and perimeter.)

 Select appropriate units of measure for length, weight and capacity. (M)

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Pennsylvania System of School

Assessment (PSSA) – Grade 3

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Strand 2.4 Mathematical Reasoning and Connections Subject Area: Math Grade: 3

PA Academic Standards Performance Indicators Assessment

A. Make, check and verify predictions

about the quantity, size and shape of

objects and groups of objects.

 Verbalize estimation strategies. (I)

 Compare estimations and results. (I)

 Revise estimates. (I)

 Use a calculator to check estimation. (M)

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Pennsylvania System of School

Assessment (PSSA) – Grade 3

B. Use measurements to determine the

geography of the school building.

 Use measurements in everyday situations (e.g., determine the size of

the gym, the length, width, height of

hallways or classroom, size of

doorways). (M)

13

Strand 2.5 Mathematical Problem Solving and Communication Subject Area: Math Grade: 3

PA Academic Standards Performance Indicators Assessment

A. Use appropriate problem-solving

strategies such as guess and check and

working backwards.

 Use appropriate problem-solving strategies: Guess and check (M), Work

backwards (M), Draw pictures or

diagrams (M), Make lists, charts, or

tables (M), Use patterns. (R)

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Pennsylvania System of School

Assessment (PSSA) – Grade 3

B. Determine when sufficient information is present to solve a problem and

explain how to solve a problem.

 Determine when sufficient information is present to solve a problem. (M)

 Explain how to solve a problem and show the process. (M)

C. Select and use an appropriate method,

materials and strategy to solve

problems, including mental

mathematics, paper and pencil, and

concrete objects.

 Select and use appropriate method to solve problems: Mental math (M),

Paper and pencil (M), Concrete objects

(M), Choose the correct operation(s) to

solve word problem (no more than 2

operations using +, - and/or x).

14

Strand 2.6 Statistics and Data Analysis Subject Area: Math Grade: 3

PA Academic Standards Performance Indicators Assessment

A. Gather, organize and display data using

pictures, tallies, charts, bar graphs and

pictographs.

 Make & interpret a chart. (M) (PSSA)

 Make a pictograph. (M) (PSSA)

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Pennsylvania System of School

Assessment (PSSA) – Grade 3

B. Formulate and answer questions based

on data shown on graphs.

 Formulate questions and draw conclusions from charts, pictographs,

and bar graphs and explain them orally

and in writing. (M) (PSSA)

C. Predict the likely number of times a

condition will occur based on analyzed

data.

 The student will make predictions about the probability of events

occurring based on analyzed data. (M)

(PSSA)

D. Form and justify an opinion on whether

a given statement is reasonable based

on a comparison to data.

 Form and justify an opinion on whether a given statement is reasonable based

on a comparison of data. (M)

15

Strand 2.7 Probability and Predictions Subject Area: Math Grade: 3

PA Academic Standards Performance Indicators Assessment

A. Predict and measure the likelihood of

events and recognize that the results of

an experiment may not match predicted

outcomes.

 Determine if an event is equally likely or not equally likely to occur. (M)

(PSSA)

 Recognize that the results of an experiment may not match predicted

outcomes. (M)

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Pennsylvania System of School

Assessment (PSSA) – Grade 3

B. Design a fair and an unfair spinner.

 Design a fair and unfair spinner. (M)

C. List or graph the possible results of an

experiment.

 The student will list or graph the possible results of an experiment and

explain the data. (M)

D. Analyze data using the concepts of

largest, smallest, most often, least often

and middle.

 The student will use the concept of middle to analyze data. (M)

16

Strand 2.8 Algebra and Functions Subject Area: Math Grade: 3

PA Academic Standards Performance Indicators Assessment

A. Recognize, describe, extend, create and

replicate a variety of patterns including

attribute, activity, number and

geometric patterns.

 Introduce function tables in addition and multiplication tables grade

appropriate. (I)

 Investigate and create arrays. (M)

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Pennsylvania System of School

Assessment (PSSA) – Grade 3

A. Recognize, describe, extend, create and

replicate a variety of patterns including

attribute, activity, number and

geometric patterns.

 Describe attributes of attribute blocks, geometric figures, etc.

 Investigate, create, and extend patterns. (M)

A. Recognize, describe, extend, create and

replicate a variety of patterns including

attribute, activity, number and

geometric patterns.

 Extend or find a missing element in a pattern of numbers or shapes (pattern

must show 3 repetitions-if multiples are

used, limit to 2, 3, or 5.) (PSSA)

A. Recognize, describe, extend, create and

replicate a variety of patterns including

attribute, activity, number and

geometric patterns.

 Identify/describe the rule for a pattern shown (pattern must show 3 repetitions

– if multiples are used, limit to 1, 3, or

5.) (PSSA)

B. Use concrete objects and trial and error

to solve number sentences and check

whether solutions are sensible and

accurate.

 Use concrete objects and trial and error to solve number sentences involving

addition, subtraction, multiplication,

and division. (M)

 Check to see if solutions are sensible and accurate. (M)

17

Strand 2.8 Algebra and Functions Subject Area: Math Grade: 3

PA Academic Standards Performance Indicators Assessment

C. Substitute a missing addend in a

number sentence.

 Substitute a missing addend or in a number sentence beyond 20. (M).

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Pennsylvania System of School

Assessment (PSSA) – Grade 3

C. Substitute a missing addend in a

number sentence.

 Find a missing number that makes a number sentence true (1 digit or 2 digit

numbers up to 18 using +, - or x

through 9 x 5.) (PSSA)

C. Substitute a missing addend in a

number sentence.

 Identify the missing symbol (+, -, =, <, >) that makes a number sentence true.

(PSSA)

D. Create a story to match a given

combination of symbols and numbers.

 The student will create and solve a story to match a given combination of

symbols and numbers. (M) (+, -, x, <,

>, =) (PSSA)

E. Use concrete objects and symbols to

model the concepts of variables,

expressions, equations and inequalities.

 The student will use concrete objects and symbols to model the concepts of

variables, expressions, equations and

inequalities. (M)

18

Strand 2.8 Algebra and Functions Subject Area: Math Grade: 3

PA Academic Standards Performance Indicators Assessment

F. Explain the meaning of solutions and

symbols.

 Explain the meaning of solutions and symbols. (M)

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Pennsylvania System of School

Assessment (PSSA) – Grade 3

F. Explain the meaning of solutions and

symbols.

 Compare numbers reading the symbols

>, <, =,  (e.g., Use the words greater

than, less than, equal to and not equal

to (to thousands). (M) (PSSA)

F. Explain the meaning of solutions and

symbols.

 Identify the ordinal position of an object (e.g., 1

st , 2

nd , and 3

rd ). (M)

 Read a mathematical number sentence created with <, > and =. (R)

F. Explain the meaning of solutions and

symbols.

 Write a mathematical number sentence created with <, > and = for 2-digit

numbers. (R)

G. Gather information and display it in the

form of a table or a chart.

 Gather information and display it in the form of a table or a chart.(M)

19

Strand 2.8 Algebra and Functions Subject Area: Math Grade: 3

PA Academic Standards Performance Indicators Assessment

H. Describe and interpret the data shown

in tables and charts.

 Describe and interpret the data shown in tables and charts. (M) (PSSA)

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Pennsylvania System of School

Assessment (PSSA) – Grade 3

I Translate information from one type of

display to another.

 Translate information from one type of display to another (e.g. convert tally

chart to bar graph). Limit to tally

charts, bar graphs, tables and

pictographs. (PSSA)

J. Demonstrate simple function rules.

 Demonstrate simple function rules (one step functions +, -, x). (M) (PSSA)

J. Demonstrate simple function rules.

 Determine the missing element in a function table (functions may use +, -,

or x; allowable multiples are 2, 3, or 5.)

Tables must have 3 Ins and 3OUTs

listed. (PSSA)

K. Analyze simple functions and

relationships and locate points on a

simple grid.

 Analyze simple functions and relationships. (e.g., What do you do to

3 to get 6?) (M) (PSSA)

 Locate points on a simple grid. (M) (PSSA)

20

Strand 2.9 Geometry Subject Area: Math Grade: 3

PA Academic Standards Performance Indicators Assessment

A. Name and label geometric shapes in

two and three dimensions

 Name and label geometric shapes in two and three-dimensions. (R)

Circle/sphere, Square/cube,

Triangle/pyramid, Rectangle/prism.

(PSSA)

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Pennsylvania System of School

Assessment (PSSA) – Grade 3

A. Name and label geometric shapes in

two and three dimensions

 Describe the difference between an open figure and a closed figure. (R)

 Explain the number of sides and corners of shapes. (R)

B. Build geometric shapes using concrete

objects

 Build geometric shapes using concrete objects. (R)

C. Draw two and three-dimensional

geometric shapes and construct

rectangles, squares and triangles on the

geoboard and on graph paper satisfying

specific criteria.

 Draw two (R) and three-dimensional geometric shapes. (M)

 Construct rectangles, squares, and triangles and on graph paper satisfying

specific criteria. (M)

C. Draw two and three-dimensional

geometric shapes and construct

rectangles, squares and triangles on the

geoboard and on graph paper satisfying

specific criteria.

 Identify parallel, perpendicular, line segments, and congruent lines. (M)

21

Strand 2.9 Geometry Subject Area: Math Grade: 3

PA Academic Standards Performance Indicators Assessment

D. Find and describe geometric figures in

real life.

 Find and describe geometric figures in real life. (R)

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Pennsylvania System of School

Assessment (PSSA) – Grade 3

E. Identify and draw lines of symmetry in

geometric figures.

 Identify and draw all lines of symmetry in two-dimensional geometric figures.

(M)

F. Identify symmetry in nature.

 Identify symmetry in Nature and in two-dimensional figures

G. Fold paper to demonstrate the

reflections about a line.

 Fold paper to demonstrate the reflections about a line. (M)

H. Show relationships between and among

figures using reflections.

 Show relationships between and among figures using reflections (e.g.,

tesselations). (M)

22

Strand 2.9 Geometry Subject Area: Math Grade: 3

PA Academic Standards Performance Indicators Assessment

I. Predict how shapes can be changed by

combining or dividing them.

 Predict how shapes can be changed by combining or dividing them. (M)

 Recognize slides, flips and turns of a simple two-dimensional figure. (M)

(PSSA)

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Pennsylvania System of School

Assessment (PSSA) – Grade 3

23

Strand 2.10 Trigonometry Subject Area: Math Grade: 3

PA Academic Standards Performance Indicators Assessment

A. Identify right angles in the

environment.

 Identify right angles in the environment. (R) (PSSA)

 Determine whether an angle is <, >, or = to a right angle. (M)

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Performance assessments

Portfolios

Research papers

Essays

Oral presentations

Multi-medial presentations

Experiments

Pennsylvania System of School

Assessment (PSSA) – Grade 3

B. Model right angles and right triangles

using concrete objects.

 The student will model right angles and right triangles using concrete objects.

(PSSA)

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Strand 2.11 Concepts of Calculus Subject Area: Math Grade: 3

PA Academic Standards Performance Indicators Assessment

A. Identify whole number quantities and

measurements from least to most and

greatest value.

 Identify whole number quantities and measurements from least to most and

greatest value from 1 to 10,000. (M)

(PSSA)

Teacher made tests and quizzes

Chapter tests and quizzes

Curriculum-based assessments

Standardized tests

Demonstrations

Performance assessments

Portfolios

Research papers

Essays

Oral presentations

Multi-medial presentations

Experiments

Pennsylvania System of School

Assessment (PSSA) – Grade 3

B. Identify least and greatest values

represented in bar graphs and

pictographs.

 Identify least and greatest values in pictographs with variables. (M)

C. Categorize rates of change as faster and

slower.

 Categorize rates of change as faster and slower. (M)

D. Continue a pattern of numbers or

objects that could be extended

infinitely.

 Continue a pattern of numbers or objects that could be extended

infinitely. (R)

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Adaptations/Modifications for Students with I.E.P.s

Adaptations or modifications to this planned course will allow exceptional students to earn credits toward graduation or develop skills necessary

to make a transition from the school environment to community life and employment. The I.E.P. team has determined that modifications to this

planned course will meet the student’s I.E.P. needs.

Adaptations/Modifications may include but are not limited to:

INSTRUCTION CONTENT - Modification of instructional content and/or instructional approaches - Modification or deletion of some of the essential elements

SETTING - Preferential seating

METHODS

- Additional clarification of content - Occasional need for one to one instruction - Minor adjustments or pacing according to the student’s rate of mastery - Written work is difficult, use verbal/oral approaches - Modifications of assignments/testing - Reasonable extensions of time for task/project completion - Assignment sheet/notebook - Modified/adjusted mastery rates - Modified/adjusted grading criteria - Retesting opportunities

MATERIALS - Supplemental texts and materials - Large print materials for visually impaired students - Outlines and/or study sheets - Carbonless notebook paper - Manipulative learning materials - Alternatives to writing (tape recorder/calculator)

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Strategies  and  Interventions  to  Support   Students  with  Mathematics  Disabilities  

  Brittany  L.  Hott,  PhD   Laura  Isbell,  PhD   Texas  A&M  University-­‐  Commerce     Teresa  Oettinger  Montani,  EdD   Fairleigh  Dickinson  University   (December  2014) In  the  absence  of  intensive  instruction  and   intervention,  students  with  mathematics   difficulties  and  disabilities  lag  significantly   behind  their  peers  (Jitendra  et  al.,  2013;   Sayeski  &  Paulsen,  2010).  Conservative   estimates  indicate  that  25%  to  35%  of   students  struggle  with  mathematics   knowledge  and  application  skills  in  general   education  classrooms,  indicating  the   presence  of  mathematics  difficulty   (Mazzocco,  2007).  Additionally,  5%  to  8%  of   all  school-­‐age  students  have  such  significant   deficits  that  impact  their  ability  to  solve   computation  and/or  application  problems   that  they  require  special  education  services   (Geary,  2004).  This  InfoSheet  provides  an   overview  of  strategies  and  resources  to   support  students  with,  or  at-­‐risk  for,   mathematics  learning  disabilities.  

Common  Core  Mathematics  Standards  

With  the  current  emphasis  on  the  Common   Core  State  Standards  (CCSS;  National   Governors  Association  Center  for  Best   Practices  [NGA  Center],  2010,  2014),  it  is  

essential  that  students  with  math  difficulties   and  disabilities  be  prepared  to  meet  with   success  on  these  newly  articulated  grade   level  expectations  in  mathematics.  Special   education  teachers  and  general  education   teachers  need  to  have  strategies  to  help   students  who  struggle  with  mathematics  to   gain  access  to  the  general  education   curriculum  and  to  meet  with  success  in  all   areas  of  math  including  math  literacy  and   conceptual  knowledge  (Gargiulo  &  Metcalf,   2013;  Powell,  Fuchs,  &  Fuchs,  2013).  

Although  the  CCSS  do  not  provide  a   curriculum,  they  do  specify  the  topics  within   standards  that  should  be  addressed  by  grade   level.  CCSS  included  two  major  components:   Standards  for  Mathematics  Practice  and   Standards  for  Mathematics  Content.  These   standards  indicate  that  students  should  be   able  to  (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  

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structure,  and  (8)  look  for  and  express   regularity  in  repeated  practices.  During  the   elementary  years,  focus  is  placed  on   mathematics  fundamentals  with  the  goal  of   moving  from  counting  skills  to  multiplying   and  dividing  fractions.  By  middle  school,   students  are  expected  to  understand   geometry,  ratios  and  proportions,  and  pre-­‐ algebra  skills.  During  high  school,  the  focus  is   on  more  advanced  algebra,  functions,   modeling,  advanced  geometry,  statistics,  and   probability  content.  For  a  complete  listing  of   grade  level  standards  download  the  complete   set  of  grade  specific  standards   (www.corestandards.org/the-­‐ standards/mathematics).  

The  Early  Learning  in  Mathematics   program  (Davis  &  Jungjohann,  2009)  is  an   example  of  a  core  mathematics  program  that   embodies  the  current  thinking  on  effective   instruction  in  math  (Doabler  et  al.,  2012).   Both  systematic  and  explicit  instruction  and   detailed  coverage  of  significant  areas  of   content  in  mathematics  are  addressed  in  this   program.  The  successful  elements  of  explicit   and  systematic  instruction  incorporated  in   this  program  that  can  also  be  utilized  in  other   core  mathematics  instruction  include  the   following:  

1. Specific  and  clear  teacher  models   2. Examples  that  are  sequenced  in  level  

of  difficulty   3. Scaffolding   4. Consistent  feedback   5. Frequent  opportunity  for  cumulative  

review  (NCEERA,  2009)   Fuchs  and  Fuchs  (2008)  identified  seven  

principals  of  effective  practice  for  primary  

students  with  math  disabilities.  In  their   article,  the  authors  stated  that  third  grade  is  a   time  when  mathematical  disabilities  tend  to   be  identified,  and  used  the  seven   interventions  to  illustrate  the  principles.  The   seven  principles  include  (1)  instructional   explicitness,  (2)  instructional  design  to   minimize  the  learning  challenge,  (3)  provide   strong  conceptual  knowledge  for  procedures   taught,  (4)  drill  and  practice,  (5)  cumulative   review,  (6)  motivation  to  help  students   regulate  their  attention  and  behavior  and  to   work  hard,  and  (7)  on-­‐going  progress   monitoring.   Strategies  for  Teaching  Problem  Solving  Skills     Strategy  training  has  been  helpful  to  students   with  LD  when  learning  mathematical   concepts  and  procedures.  The  following  are  a   few  examples  of  strategies  that  are  useful  to   teachers  when  instructing  students  with  LD   in  problem  solving.   RIDE  (Mercer,  Mercer,  &  Pullen,  2011)   RIDE  is  a  strategy  used  to  assist  students   with  solving  word  problems.  Students  who   experience  difficulty  with  abstract  reasoning,   attention,  memory,  and/or  visual  spatial   skills  may  benefit  from  the  strategy.  Ensure   that  steps  are  taught  through  demonstration   and  plenty  of  opportunities  for  practice  are   provided  before  asking  students  to   independently  use  the  strategy.  Visually   display  the  strategy  on  a  chart  or  class   website  as  a  reminder.  

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R-- Remember the problem correctly

I-- Identify the relevant information

D-- Determine the operations and unit for expressing the answer E-- Enter the correct numbers, calculate and check the answer

FAST  DRAW  (Mercer  &  Miller,  1992)   Like  RIDE,  FAST  DRAW  is  another  strategy   used  to  solve  word  problems.  Teach  each   step  in  the  sequence  allowing  sufficient  time   for  guided  practice  prior  to  asking  students  

to  independently  implement  the  strategy.   Create  a  visual  display  and  post  in  the   classroom  or  student  notebooks  to  assist   students.  

F— Find what you’re solving for. A— Ask yourself, “What are the parts of the problem?” S— Set up the numbers. T— Tie down the sign.

D — Discover the sign. R — Read the problem. A — Answer, or draw and check. W— Write the answer.

TINS  Strategy  (Owen,  2003)  The  TINS   strategy  allows  students  to  use  different   steps  to  analyze  and  solve  word  problems.  

T—Thought Think about what you need to do to solve this problem and circle the key words.

I— Information Circle and write the information needed to solve this problem; draw a picture; cross out unneeded information.

N— Number Sentence Write a number sentence to represent the problem. S-- Solution Sentence Write a solution sentence that explains your answer.

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Strategies  to  Support  Vocabulary   Development  

Strategies  that  can  help  students  improve   their  mathematic  vocabulary  include  (a)  pre-­‐ teach  vocabulary,  (b)  mnemonic  techniques,   and  (c)  key  word  approaches.  These   strategies  are  only  a  few  strategies  available   to  help  enhance  students’  mathematics   vocabulary  comprehension.   Pre-­‐teach  Vocabulary  

• Use  representations,  both  pictorial   and  concrete,  to  emphasize  the   meaning  of  math  vocabulary  (Sliva,   2004).  

• Pretest  students’  knowledge  of   glossary  terms  in  their  math  textbook   and  teach  vocabulary  that  is  unknown   or  incorrect.  

Mnemonic  Techniques   • Teach  mnemonic  techniques  to  help  

remember  word  meanings.   • Use  mnemonic  instruction  to  help  

students  improve  their  memory  of   new  information  (The  Access  Center,   2006).  

Key  Word  Approach   • Use  the  keyword  approach  (e.g.,  

visualize  a  visor  as  the  keyword  for    

• divisor;  visualize  quotation  marks  as   the  keyword  for  quotient  (Mastropieri   &  Scruggs,  2002).  

Strategies  to  Assist  with  Teaching    

Algebraic  Concepts    

Algebra  is  introduced  in  elementary  school  as   students  learn  algebraic  reasoning  involving   patterns,  symbolism,  and  representations.   Students  experience  difficulty  with  algebra   for  various  reasons  including  difficulty   understanding  the  vocabulary  required  for   algebraic  reasoning,  difficulties  with  problem   solving,  and  difficulties  understanding   patterns  and  functions  necessary  for   algebraic  reasoning.  Possible  strategies  to   assist  with  teaching  algebraic  concepts   include,  but  are  not  limited  to,  (a)  teaching   key  vocabulary  needed  for  algebra,  (b)   providing  models  for  identifying  and   extending  patterns,  (c)  modeling  “think   aloud”  procedures  for  students  to  serve  as   examples  for  solving  equations  and  word   problems,  (d)  incorporating  technology   usage  (e.g.,  graphing  calculators)  (Bryant,   2008),  and  (e)  implementing  Star  Strategy   described  below  (Gagnon  &  Maccini,  2001).  

S— Search the word problem.

T— Translate the words into an equation in picture form A— Answer the problem R— Review the problem.

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CRA  and  CSA  Instructional  Methods   Maccini  and  Gagnon  (2005)  stated  that  the   STAR  strategy  incorporates  the  concrete-­‐ Semiconcrete-­‐Abstract  (CSA)  instructional   sequence,  which  gradually  advances  to   abstract  ideas  using  the  following   progression:  (a)  concrete  stage,  (b)   semiconcrete  stage,  and  (c)  abstract  stage.  By   using  the  CSA  framework  teachers  can   incorporate  effective  teaching  components  to   teach  students  effectively  and  efficiently.   Students  progressively  move  through  each   stage  to  achieve  mastery  in  a  mathematic   concept.  

Using  multiple  representations,  beginning   with  the  concrete  level  and  moving  to  the   abstract  level,  is  an  effective  technique  in   helping  struggling  learners  solve  calculation   problems.  The  Concrete-­‐Representational-­‐ Abstract  (CRA)  teaching  sequence  has  been   found  to  help  students  with  LD  learn   procedures  and  concepts  (Flores,  Hinton,  &   Strozier,  2014).  During  the  concrete  stage   students  are  in  the  “doing”  stage,  during  the   representational  stage  students  are  in  the   “seeing”  stage,  and  during  the  abstract  phase   students  are  in  the  “applying”  stage.  Students   move  through  the  phases  fluidly.

C— Concrete: students use three-dimensional objects to represent math problems R— Representational: students use pictures to represent math problems A— Abstract: students represent the problem using numerical symbols

C— Concrete: students use three-dimensional objects to represent math problems

S— Semiconcrete: students use two-

dimensional representation to draw pictures of the math problem

A— Abstract: students represent the problem using numerical symbols

Strategies  to  Assist  with  the  Use  of  

Metacognitive  Skills   Metacognition  refers  to  individuals’   awareness  of  how  they  think  and  plan   activities.  Metacognition  also  involves   strategizing,  monitoring  success  and  effort,   and  knowing  when  to  change  directions  or  to   try  a  different  approach  to  problem  solving.   Many  students  with  learning  difficulties   benefit  from  the  use  of  metacognitive  skills  to   help  them  focus  on  what  they  are  doing  and   to  plan  for  how  to  employ  strategies  as   needed  and  change  directions  when   appropriate  (Mevarech  &  Amrany,  2008).    A   few  examples  of  how  to  incorporate   metacognitive  strategies  include:  

• Demonstrating  “think-­‐alouds”  so   students  become  aware  of  how  one   talks  oneself  through  a  learning  task.  

• Demonstrating  the  use  of  graphic   organizers,  schematics,  and  visual   imagery.  

• Explicit,  direct  instruction   accompanied  by  modeling  of  self-­‐ monitoring,  self-­‐talk,  and  self-­‐checks.  

Mathematics  Advisory  Panels  and  Their  

Reports   Developing  foundational  mathematics  skills   at  the  elementary  level  is  essential.   Maintaining  basic  skills  acquired  during  the   elementary  years  is  essential  as  students   move  toward  more  advanced  computational,   place  value,  and  fractional  concepts.  As  

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students  move  from  elementary  to  secondary   mathematics,  it  is  important  that  students   maintain  skills  mastered  and  that  teachers   continue  to  scaffold  instruction  and  provide   supports  to  ensure  that  foundational  skills  

are  addressed  while  affording  access  to  more   advanced  mathematics  concepts.  Below  are   links  to  various  advisory  panel   recommendations  for  effectively  teaching   mathematics.  

Panel Link National Commission on Mathematics and Science Teaching for the 21st Century— Before It’s Too Late

http://www.ptec.org/items/detail.cfm?ID=4059

National Research Council—Adding It Up: Helping Children Learn Mathematics

http://www.nap.edu/catalog.php?record_id=9822

RAND Mathematics Study Panel— Mathematical Proficiency for All Students

http://www.rand.org/pubs/monograph_reports/MR1643/index.ht ml

Foundations for Success: The Final Report of the National Mathematics Advisory Panel

http://www2.ed.gov/about/bdscomm/list/mathpanel/report/final- report.pdf

The Access Center’s Math Problem Solving for Primary Elementary Students with Disabilities

http://www.k8accesscenter.org/training_resources/mathprimaryp roblemsolving.asp

The Access Center’s Math Problem Solving for Upper Elementary Students with Disabilities

http://www.k8accesscenter.org/training_resources/MathPrblSlvi ng_upperelem.asp

Resources  

There  are  numerous  website  and  resources   available  to  assist  with  mathematics  

instruction.    Following  are  suggested   websites  with  a  summary  of  resources  for   teaching  a  variety  of  mathematics  concepts   across  levels.  

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Website Resource www.cast.org http://www.factmonster.com/math/flashcards.html www.aplusmath.com/flashcards www.flash-cardmachine.com www.academicskillbuilders.com www.sunburst.com www.intellitools.com www.tomsnyder.com www.illuminations.nctm.org www.internet4classrooms.com http://ncisla.wceruw.org/teachers/index.html

• expand learning opportunities through universal design

• increase math fluency with web- based flashcards

• enhance fluency through the use

of web-based flashcards

• enhance fluency with the use of web-based flashcards

• support math and vocabulary

fluency (Brownell, Smith, Crockett, Griffin, 2012)

• research based achievement

solutions; standards based products prescriptive web based instruction K-5, digital classroom

• technology for preK – 8

classrooms, free downloads available.

• software for fluency, word

problems, graphing, etc.

• activities, lessons, standards web links for math education; preK-12.

• Common Core State Standards

internet sites for teaching mathematics in culturally responsive ways

• resources related to supporting

understanding of science and math

References   The  Access  Center.  (2006).  Using  mnemonic  

instruction  to  teach  math.  Retrieved  from   http://www.k8accesscenter.org  

Brownell,  M.  T.,  Smith,  S.  J.,  Crockett,  J.  B.,   &  Griffin,  C.  C.  (2012).  Inclusive  instruction:   Evidence  based  practices  for  teaching  students   with  disabilities.  New  York,  NY:  The  Guilford   Press.  

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Bryant,  D.  P.  (2008).  Teaching  mathematics.  In  D.   P.  Bryant,  D.  D.  Smith,  &  B.  R.  Bryant  (Eds.),   Teaching  students  with  special  needs  in   inclusive  classrooms.  Boston,  MA:  Allyn  &   Bacon.  

Davis,  K.,  &  Jungjohann,  K.  (2009).  Early  learning   in  mathematics:  Level  K.    Unpublished   curriculum,  Center  on  Teaching  and  Learning,   University  of  Oregon,  Eugene,  OR.  

Doabler,  C.  T.,  Cary,  M.  S.,  Jungjohann,  K.,  Clarke,   B.,  Fien,  H.,  Baker,  C.,  Smolkowski,  K.,  &  Chard,   D.  (2012).  Enhancing  core  mathematics   instruction  for  students  at  risk  for   mathematics  disabilities.  Teaching  Exceptional   Children,  44(4),  48-­‐57.  

Flores,  M.  M.,  Hinton,  V.,  &  Strozier,  S.  (2014).   Teaching  subtraction  and  multiplication  with   regrouping  using  Concrete-­‐Representational-­‐ Abstract  sequence  and  the  Strategic   Instruction  Model.  Learning  Disabilities   Research  and  Practice,  29(2),  75-­‐88.  

Fuchs,  L.  S.,  &  Fuchs,  D.  (2008),  Mathematics   disabilities  in  the  primary  grades:  Seven   principles  of  effective  practice.  Retrieved  from   www.TeachingLD.org  

Gagnon,  J.,  &  Maccini,  P.  (2001).  Preparing   students  with  disabilities  for  algebra.  Teaching   Exceptional  Children,  34(1),  10-­‐17.  

Gargiulo,  R.  M.,  &  Metcalf,  D.  (2013).  Teaching  in   today’s  inclusive  classrooms:  A  universal  design   for  learning  approach  (2nd  ed.)  Independence,   KY:  Cengage  Learning.  

Geary,  D.  C.  (2004).  Mathematics  and  learning   disabilities.  Journal  of  Learning  Disabilities,  37,   4-­‐15.  

Jitendra,  A.  K.,  Rodriguez,  M.,  Kanive,  R.,  Huang,  J.,   Church,  C.,  Conrroy,  K.  A.,  &  Zaslofsky,  A.   (2013).  Impact  of  small-­‐group  tutoring   interventions  on  the  mathematical  problem   solving  and  achievement  of  third-­‐grade   students  with  mathematics  difficulties.   Learning  Disability  Quarterly,  36,  21-­‐35.  

Maccini,  P.,  &  Gagnon,  J.  (2005).  Mathematics   strategy  instruction  (SI)  for  middle  school   students  with  learning  disabilities.    The  Access   Center.  Retrieved  from   http://digilib.gmu.edu/jspui/bitstream/1920 /284/1/MathSIforMiddleSchoolStudentswith LD.2.pdf  

Mastropieri,  M.  A.,  &  Scruggs,  T.  E.  (2002).   Effective  instruction  for  special  education  (3rd   ed.).  Austin,  TX:  PRO-­‐ED.  

Mazzocco,  M.  (2007).  Defining  and  differentiating   mathematical  learning  disabilities  and   difficulties.  In  D.  Berch  &  M.  Mazzocco  (Eds.),   Why  is  math  so  hard  for  some  children?  The   nature  and  origins  of  mathematics  learning   difficulties  and  disabilities  (pp.  29-­‐47).   Baltimore,  MD:  Paul  H.  Brooks.  

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