Strategies for Accessing Algebraic Concepts (K-8)
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Strategies for Accessing Algebraic Concepts (K-8) Access Center September 20, 2006 Agenda Introductions and Overview Objectives Background Information Challenges for Students with Disabilities Instructional and Learning Strategies Application of Strategies Objectives:

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Strategies for Accessing Algebraic Concepts (K-8)

Access Center

September 20, 2006

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  • Introductions and Overview

  • Objectives

  • Background Information

  • Challenges for Students with Disabilities

  • Instructional and Learning Strategies

  • Application of Strategies

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  • To identify the National Council of Teachers of Mathematics (NCTM) content and process standards

  • To identify difficulties students with disabilities have with learning algebraic concepts

  • To identify and apply research-based instructional and learning strategies for accessing algebraic concepts (grades K-8)

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How Many Triangles?

Pair off with another person, count the number of triangles, explain the process, and record the number.

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Why Is Algebra Important?

  • Language through which most of mathematics is communicated (NCTM, 1989)

  • Required course for high school graduation

  • Gateway course for higher math and science courses

  • Path to careers – math skills are critical in many professions (“Mathematics Equals Equality,” White Paper prepared for US Secretary of Education, 10.20.1997)

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NCTM Goals for All Students

  • Learn to value mathematics

  • Become confident in their ability to do mathematics

  • Become mathematical problem solvers

  • Learn to communicate mathematically

  • Learn to reason mathematically

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Numbers and Operations



Data Analysis and Probability



Problem Solving

Reasoning and Proof




NCTM Standards:

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“Teachers must be given the training and resources to provide the best mathematics for every child.”


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Challenges Students Experience with Algebra provide the best mathematics for every child.”

  • Translate word problems into mathematical symbols (processing)

  • Distinguish between patterns or detailed information (visual)

  • Describe or paraphrase an explanation (auditory)

  • Link the concrete to a representation to the abstract (visual)

  • Remember vocabulary and processes (memory)

  • Show fluency with basic number operations (memory)

  • Maintain focus for a period of time (attention deficit)

  • Show written work (reversal of numbers and letters)

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At the Elementary Level, Students with Disabilities Have Difficulty with:

  • Solving problems (Montague, 1997; Xin Yan & Jitendra, 1999)

  • Visually representing problems (Montague, 2005)

  • Processing problem information (Montague, 2005)

  • Memory (Kroesbergen & Van Luit, 2003)

  • Self-monitoring (Montague, 2005)

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At the Middle School Level, Students with Disabilities Have Difficulty:

  • Meeting content standards and passing state assessments(Thurlow, Albus, Spicuzza, & Thompson, 1998; Thurlow, Moen, & Wiley, 2005)

  • Mastering basic skills(Algozzine, O’Shea, Crews, & Stoddard, 1987; Cawley, Baker-Kroczynski, & Urban, 1992)

  • Reasoning algebraically(Maccini, McNaughton, & Ruhl, 1999)

  • Solving problems(Hutchinson, 1993; Montague, Bos, & Doucette, 1991)

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Therefore, instructional and learning strategies should address:

  • Memory

  • Language and communication

  • Processing

  • Self-esteem

  • Attention

  • Organizational skills

  • Math anxiety

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Instructional Strategy address:

  • Instructional Strategies are methods that can be used to deliver a variety of content objectives.

  • Examples: Concrete-Representational-Abstract (CRA) Instruction, Direct Instruction, Differentiated Instruction, Computer Assisted Instruction

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Learning Strategy address:

  • Learning Strategies are techniques, principles, or rules that facilitate the acquisition, manipulation, integration, storage, and retrieval of information across situations and settings (Deshler, Ellis & Lenz, 1996)

  • Examples: Mnemonics, Graphic Organizers, Study Skills

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Best Practice in Teaching Strategies address:

1. Pretest

2. Describe

3. Model

4. Practice

5. Provide Feedback

6. Promote Generalization

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Effective Strategies for Students with Disabilities address:

Instructional Strategy: Concrete-Representational- Abstract (CRA) Instruction

Learning Strategies: Mnemonics

Graphic Organizers

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Concrete-Representational-Abstract Instructional Approach ( address:C-R-A)

  • CONCRETE: Uses hands-on physical (concrete) models or manipulatives to represent numbers and unknowns.

  • REPRESENTATIONAL or semi-concrete: Draws or uses pictorial representations of the models.

  • ABSTRACT: Involves numbers as abstract symbols of pictorial displays.

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Example for K-2 address:Add the robots!

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Example for K-2 address:Add the robots!








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Example for 3-5 address:

Tilt or Balance the Equation!

  • 3 *4 =2* 6

    • ?

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Example 3-5 address:Represent the equation!

3 * 4 = 2 * 6


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Example for 6-8 address:

Balance the Equation!

3 * +=2 * -4

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Example for 6-8 address:

Represent the Equation

3 * + = 2 * - 4

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Example for 6-8 address:


3 * + =2 * - 4

3 *1+7 =2 * 7-4

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Case Study address:

Questions to Discuss:

  • How would you move these students along the instructional sequence?

  • How does CRA provide access to the curriculum for all of these students?

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Mnemonics address:

  • A set of strategies designed to help students improve their memory of new information.

  • Link new information to prior knowledge through the use of visual and/or acoustic cues.

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3 Types of Mnemonics address:

  • Keyword Strategy

  • Pegword Strategy

  • Letter Strategy

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Why Are Mnemonics Important? address:

  • Mnemonics assist students with acquiring information in the least amount of time (Lenz, Ellis & Scanlon, 1996).

  • Mnemonics enhance student retention and learning through the systematic use of effective teaching variables.

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D address:iscover the sign

Read the problem

Answer or draw a representation of the problem using lines, tallies, or checks

Write the answer and check

DRAW: Letter Strategy

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DRAW address:

  • D iscover the variable

  • R ead the equation, identify operations, and think about the process to solve the equation.

  • A nswer the equation.

  • W rite the answer and check the equation.

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DRAW address:

4x + 2x = 12

Represent the variable "x“ with circles.


By combining like terms, there are six "x’s." 4x + 2x = 6x

6x = 12

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DRAW address:

Divide the total (12) equally among the circles.

6x = 12

The solution is the number of tallies represented in one circle – the variable ‘x." x = 2

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STAR: Letter Strategy address:

The steps include:

  • Search the word problem;

  • Translate the words into an equation in picture form;

  • Answer the problem; and

  • Review the solution.

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STAR address:

The temperature changed by an average of -3° F per hour. The total temperature change was 15° F. How many hours did it take for the temperature to change?

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STAR: address:

  • Search: read the problem carefully, ask questions, and write down facts.

  • Translate: use manipulatives to express the temperature.

  • Answer the problem by using manipulatives.

  • Review the solution: reread and check for reasonableness.

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Activity: address:

  • Divide into groups

  • Read Preparing Students with Disabilities for Algebra (pg. 10-12; review examples pg.13-14)

  • Discuss examples from article of the integration of Mnemonics and CRA

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Example K-2 address:Keyword Strategy

More than & less than (duck’s mouth open means more):


5 > 2

(Bernard, 1990)

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Example Grade 3-5 address:Letter Strategy

  • O bserve the problem

  • Read the signs.

  • D ecide which operation to do first.

  • Execute the rule of order (Many Dogs Are Smelly!)

  • R elax, you're done!

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ORDER address:

Solve the problem

(4 + 6) – 2 x 3 = ?

(10) – 2 x 3 = ?

(10) – 6 = 4

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Parentheses, brackets, and braces;

Exponents next;

Multiplication and Division, in order from left to right;

Addition and Subtraction, in order from left to right.

Example 6-8 Letter Strategy

Please Excuse My Dear Aunt Sally

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P address:lease Excuse My Dear Aunt Sally

(6 + 7) + 52 – 4 x 3 = ?

13 + 52 – 4 x 3 = ?

13 + 25 - 4 x 3 = ?

13 + 25 - 12 = ?

38 - 12 = ?

= 26

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Graphic Organizers (GOs) address:

A graphic organizer is a tool or process to build word knowledge by relating similarities of meaning to the definition of a word. This can relate to any subject—math, history, literature, etc.

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GO Activity: Roles address:

  • #1 works with the figures (1-16)

  • #2 asks questions

  • #3 records

  • #4 reports out

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GO Activity: Directions address:

  • Differentiate the figures that have like and unlike characteristics

  • Create a definition for each set of figures.

  • Report your results.

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GO Activity: Discussion address:

  • Use chart paper to show visual grouping

  • How many groups of figures?

  • What are the similarities and differences that defined each group?

  • How did you define each group?

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Why are Graphic Organizers Important? address:

  • GOs connect content in a meaningful way to help students gain a clearer understanding of the material (Fountas & Pinnell, 2001, as cited in Baxendrall, 2003).

  • GOs help students maintain the information over time (Fountas & Pinnell, 2001, as cited in Baxendrall, 2003).

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Graphic Organizers: address:

  • Assist students in organizing and retaining information when used consistently.

  • Assist teachers by integrating into instruction through creative


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Graphic Organizers: address:

  • Heighten student interest

  • Should be coherent and consistently used

  • Can be used with teacher- and student- directed approaches

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Coherent Graphic Organizers address:

  • Provide clearly labeled branch and sub branches.

  • Have numbers, arrows, or lines to show the connections or sequence of events.

  • Relate similarities.

  • Define accurately.

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How to Use Graphic Organizers in the Classroom address:

  • Teacher-Directed Approach

  • Student-Directed Approach

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Teacher-Directed Approach address:

  • Provide a partially incomplete GO for students

  • Have students read instructions or information

  • Fill out the GO with students

  • Review the completed GO

  • Assess students using an incomplete copy of the GO

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Student-Directed Approach address:

  • Teacher uses a GO cover sheet with prompts

    • Example: Teacher provides a cover sheet that includes page numbers and paragraph numbers to locate information needed to fill out GO

  • Teacher acts as a facilitator

  • Students check their answers with a teacher copy supplied on the overhead

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Strategies to Teach Graphic Organizers address:

  • Framing the lesson

  • Previewing

  • Modeling with a think aloud

  • Guided practice

  • Independent practice

  • Check for understanding

  • Peer mediated instruction

  • Simplifying the content or structure of the GO

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Types of Graphic Organizers address:

  • Hierarchical diagramming

  • Sequence charts

  • Compare and contrast charts

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A Simple Hierarchical Graphic Organizer - example address:






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Another Hierarchical Graphic Organizer address:





List examples of each type

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Hierarchical Graphic Organizer – example address:




6y ≠15

14 < 3x + 7

2x > y

10y = 100

2x + 3 = 15

4x = 10x - 6

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Compare and Contrast address:


What is it?




Irregular set

What are some examples?

What is it like?

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Compare and Contrast - example address:


What is it?



6, 17, 25, 100

Positive Integers

Whole Numbers

-3, -8, -4000

Negative Integers




What are some examples?

What is it like?

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Venn Diagram address:

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Prime Numbers address:

5 7

11 13



Even Numbers

4 6

8 10

Multiples of 3

9 15 21


Venn Diagram - example

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3 sides address:

3 sides

3 angles

3 angles

3 angles = 60°

1 angle = 90°

3 sides

3 angles

3 angles < 90°

Multiple Meanings – example







3 sides

3 angles

1 angle > 90°

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Series of Definitions address:

Word = Category + Attribute

= +

Definitions: ______________________



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Series of Definitions – example address:

Word = Category + Attribute

= +

Definition: A four-sided figure with four equal sides and four right angles.

4 equal sides &

4 equal angles (90°)



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Four-Square Graphic Organizer address:

1. Word:

2. Example:

4. Definition

3. Non-example:

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Four-Square Graphic Organizer – example address:

1. Word: semicircle

2. Example:

4. Definition

3. Non-example:

A semicircle is half of a circle.

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Matching Activity address:

  • Divide into groups

  • Match the problem sets with the appropriate graphic organizer

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Matching Activity address:

  • Which graphic organizer would be most suitable for showing these relationships?

  • Why did you choose this type?

  • Are there alternative choices?

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Problem Set 1 address:

Parallelogram Rhombus

Square Quadrilateral

Polygon Kite

Irregular polygon Trapezoid

Isosceles Trapezoid Rectangle

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Problem Set 2 address:

Counting Numbers: 1, 2, 3, 4, 5, 6, . . .

Whole Numbers: 0, 1, 2, 3, 4, . . .

Integers: . . . -3, -2, -1, 0, 1, 2, 3, 4. . .

Rationals: 0, …1/10, …1/5, …1/4, ... 33, …1/2, …1

Reals: all numbers

Irrationals: π, non-repeating decimal

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Problem Set 3 address:

Addition Multiplication

a + b a times b

a plus b a x b

sum of a and b a(b)


Subtraction Division

a – b a/b

a minus b a divided by b

a less b b) a

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Problem Set 4 address:

Use the following words to organize into categories and subcategories of


NUMBERS, OPERATIONS, Postulates, RULE, Triangles, GEOMETRIC FIGURES, SYMBOLS, corollaries, squares, rational, prime, Integers, addition, hexagon, irrational, {1, 2, 3…}, multiplication, composite, m || n, whole, quadrilateral, subtraction, division.

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Graphic Organizer Summary address:

  • GOs are a valuable tool for assisting students with LD in basic mathematical procedures and problem solving.

  • Teachers should:

    • Consistently, coherently, and creatively use GOs.

    • Employ teacher-directed and student-directed approaches.

    • Address individual needs via curricular adaptations.

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Resources address:

  • Maccini, P., & Gagnon, J. C. (2005). Math graphic organizers for students with disabilities. Washington, DC: The Access Center: Improving Outcomes for all Students K-8. Available at

  • Visual mapping software: Inspiration and Kidspiration (for lower grades) at http:/

  • Math Matrix from the Center for Implementing Technology in Education. Available at

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Resources address:

  • Hall, T., & Strangman, N. (2002).Graphic organizers. Wakefield, MA: National Center on Accessing the General Curriculum. Available at

  • Strangman, N., Hall, T., Meyer, A. (2003) Graphic Organizers and Implications for Universal Design for Learning: Curriculum Enhancement Report. Wakefield, MA: National Center on Accessing the General Curriculum. Available at

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How These Strategies Help Students Access Algebra address:

  • Problem Representation

  • Problem Solving (Reason)

  • Self Monitoring

  • Self Confidence

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Recommendations: address:

  • Provide a physical and pictorial model, such as diagrams or hands-on materials, to aid the process for solving equations/problems.

  • Use think-aloud techniques when modeling steps to solve equations/problems. Demonstrate the steps to the strategy while verbalizing the related thinking.

  • Provide guided practice before independent practice so that students can first understand what to do for each step and then understand why.

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Additional Recommendations: address:

  • Continue to instruct secondary math students with mild disabilities in basic arithmetic. Poor arithmetic background will make some algebraic questions cumbersome and difficult.

  • Allot time to teach specific strategies. Students will need time to learn and practice the strategy on a regular basis.

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Wrap-Up address:

  • Questions

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Closing Activity address:

Principles of an effective lesson:

Before the Lesson:

  • Review

  • Explain objectives, purpose, rationale for learning the strategy, and implementation of strategy

    During the Lesson:

  • Model the task

  • Prompt students in dialogue to promote the development of problem-solving strategies and reflective thinking

  • Provide guided and independent practice

  • Use corrective and positive feedback

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Concepts for Developing a Lesson address:

Grades K-2

  • Use concrete materials to build an understanding of equality (same as) and inequality (more than and less than)

  • Skip counting

    Grades 3- 5

  • Explore properties of equality in number sentences (e.g., when equals are added to equals the sums are equal)

  • Use physical models to investigate and describe how a change in one variable affects a second variable

    Grades 6-8

  • Positive and negative numbers (e.g., general concept, addition, subtraction, multiplication, division)

  • Investigate the use of systems of equations, tables, and graphs to represent mathematical relationships