every student every step of the way - colorado department of

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Specific Learning Disability Math Webinar SeriesWebinar #3: Let’s Be Rational- Learning Integers, Fractions & Decimals

Presented by Dr. Brad Witzel, Ph.D.April 3, 2017

4:00-5:00 pmSponsored by The Exceptional Student Services Unit

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VisionAll students in Colorado will become educated and productive citizens capable of succeeding in society, the workforce, and life.

MissionThe mission of the CDE is to ensure that all students are prepared for success in society, work, and life by providing excellent leadership, service, and support to schools, districts, and communities across the state.

Every studentevery step of the way

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2017 SLD Math Webinar Series

CDE‐SLD website: http://www.cde.state.co.us/cdesped/SD‐SLD

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• This is a recorded webinar and your microphones will not be activated during the webinar.

• Please sign into the Chat Box with your name, role, administrative unit/agency, and the grade level(s) you serve.

• Please type any questions into the Question and Answer window or Chat Box and we will address them as we are able during the presentation or at the end of the session.

• If you have difficulty accessing the webinar, or have technical issues, please call Amanda Timmerman at 303.866.6969 or email her at [email protected], or contact Jill Marshall at [email protected]

• Please also feel free to email us after the webinar with any additional questions.

Zoom Webinar Info

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• There will be brief event evaluation survey at the end of this webinar. The link will be provided on the last slide of this presentation.

• After you complete the evaluation survey your certificate of attendance for one CDE training hour will open automatically. Please print your certificate for your records.

• The recording of this webinar will be made available to participants in the future. Please contact Jill Marshall or Amanda Timmerman for viewing requests.

Webinar Evaluation

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• Brad Witzel, Ph.D.Office: 803-323-2453Fax: 803-323-2585Email: [email protected]

Our Presenter

http://coe.winthrop.edu/witzelb/

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Let’s Be Rational: Learning Integers, Fractions, and

DecimalsFor Colorado Educators

Dr. Brad Witzel, Ph.D.

[email protected]

[email protected]

© Witzel, 2017 7

Procedures and Concepts are Equally Important!

© Witzel, 2017 8

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Our Path

© Witzel, 2017

MayMathese:

The Language of Mathematics

AprilLet’s Be Rational: Learning Integers, Fractions, DecimalsMarch

Learning Whole Number Operations

June:Bridging the Arithmetic

to Algebra Gap

FebruaryFocusing on the

Nonstrategic Learner

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Understanding and Solving Rational Numbers

© Witzel, 2017 10

Witzel & Little (2016)

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Agenda

• Frequent Errors

• Modeling decimals (Core instruction)

• Fractions and decimals on a number line (Core instruction)

• Integer computation (Core and intervention)

• Fractions using Multisensory Math (Intervention)

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Which is most important when it comes to learning fractions: Concepts or procedures?

© Witzel, 2017 12

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Fractions concepts v procedures (Bailey et al., 2015)Investigated China vs US 6th and 8th grade level students’ fractions procedural and conceptual understanding

Results:

• Procedural knowledge is connected to conceptual understanding

• For low achievers – better understanding of fractions magnitude helped build procedural knowledge. Procedural knowledge facilitated a better conceptual understanding.

Conclusion:

• It is important not to separate procedures and concepts when it comes to fractions computation.

• This finding replicated a similar finding concerning computation (Siegler & Mu, 2008) and decimals (Rittle‐Johnson, et al., 2001)

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Avoid Tricks: Converting Fractions

Convert this mixed fraction into a fraction with a larger numerator.

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How did you know how to do it?

Did you…

a. 4x5

b. + 2 = 22

c. The 5 slides over

d. 22/5

Why?

Say, “Four and two – fifths”

= =

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Avoid Tricks:Division of Fractions

• Why is it that when you divide fractions, the answer

might be larger? Moreover, why do you invert and

multiply?

2/3 ÷1/4 =

2/3 (4/1) =

8/3

2/3 (4/1)

8/38/3

8/3

1/4 (4/1)

4/41/1

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“Just flip it!”

Why Understand Fractions?

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Why Teach the Basics Correctly

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Adding with unlike denominators

Division of fractions

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Fractions as a Predictor

Carnegie Mellon (http://www.cmu.edu/news/stories/archives/2012/june/june15_mathsuccess.html)

• Siegler et al (2013) found that 5th graders' facility with fractions and division predicted high school students' knowledge of algebra and overall math achievement

• The prediction was even after statistically accounting for parents' education and income and for the children's own age, gender, I.Q., reading comprehension, working memory, and knowledge of whole number addition, subtraction and multiplication.

© Witzel, 2017 18

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Fractions Research

National Math Panel (2008)

• "Difficulty with fractions (including decimals and percents) is pervasive and is a major obstacle to further progress in mathematics, including algebra“

• Tom Loveless stated, "Students don't know how to translate fractions into decimals or into percentages and they can't locate fractions on a number line“

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Fractions are a Common Difficulty (Sanders, Riccomini, & Witzel, 2005)

Code Category Entering Math Tech 1

Entering Algebra 1

FRAC Fractions and their Applications 3(3.6%)

43(44.8%)

DECM Decimals, their Operations and Applications: Percent

11(13.1%)

64 (66.7%)

EXPS Exponents and Square Roots; Scientific Notation

27(21.1%)

62(64.6%)

GRPH Graphical Representation 13(15.5%)

59(61.5%)

INTG Integers, their Operations & Applications 27(32.1%)

83(86.5%)

Total Number of Students per course 84 96

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Common Difficulties with Fractions(Riccomini, Hughes, Morano, Hwang, & Witzel, 2015)

Fraction Item Error Analysis of Low, Middle, and High Achieving Groups

Achievement Groups

Low Middle High

Item Categories M SD Rank M SD Rank M SD Rank

Division .03 .15 1 .16 .31 1 .64 .44 2

Ordering .04 .23 2 .25 .44 2 .55 .50 1

Multiplication .51 .39 8 .42 .40 3 .88 .23 5

Word Problems .09 .15 5 .5 .33 4 .87 .20 4

Addition D(1) .04 .17 3 .51 .42 5 .82 .30 3

Subtraction D(1) .08 .26 4 .62 .43 6 .91 .24 6

Transform L(2) .30 .33 6 .8 .29 7 .94 .15 7

Transform E(3) .43 .47 7 .91 .25 8 .99 .10 9

Subtraction S(4) .67 .47 9 .95 .22 9 .98 .13 8

Addition S(4) .73 .37 10 .97 .15 10 .99 .08 10

Note. (1) Different denominator, (2) Least form, (3) Equivalent form, (4) Same denominator. Rank numbers from 1‐10 signify greatest frequency of errors ‘1’ to smallest frequency of errors ‘10’.

© Witzel, 2017

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Research in the Area of Fractions and Its Application to Classroom Practices

Errors in Interpreting Fractions(Beckman, 2012)

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What is Mathematically Challenging About Decimals?

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Adapted from Boerst & Shaughnessy, 2015; Irwin, 2001

Example of error

• .34 > .5

• .75 > .7500002

• .20 is ten times greater than .2

• Hundreds, tens, ones, oneths, tenths, hundredths...

Underlying misconception

• Longer decimals are greater (overgeneralizing from whole numbers)

• Longer decimals are lesser (overgeneralizing new insights into decimals)

• Adding a zero to the right makes a number ten times larger(overgeneralizing from whole numbers)

• Lack of understanding of the “specialness” of one in the place value system

© Witzel, 2017

Common Misunderstandings About Connecting Fractions and Decimals

Why might a student believe this is the answer?

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Common Misunderstandings About Connecting Fractions and Decimals

How might a students produce these answers?

Dividing the denominator by the numerator

(and not knowing how to express the

remainder)

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Agenda

• Frequent Errors





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When you really know place value…

2.43How many ones are in this

number?

How many tenths are in this

number?

How many hundredths are in

this number?

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How do teachers represent decimal to students?

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(Glasgow et. al, 2000)

© Witzel, 2017

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Commonplace/Typical Decimal Representations Used in TeachingUse several representations to show the following numbers:

1 .6 .007 .60 2.8

What are the affordances and limitations of the representations?

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Race to One

•Partner work ‐ Repeat each new number as a total amount and place value increments

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Ones Tenths Hundredths

© Witzel, 2017

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CRA as Effective Instruction(Gersten et al, 2009; NMP, 2008; Riccomini & Witzel, 2010; Witzel, 2005)

• Concrete to Representational to Abstract Sequence of Instruction (CRA)

Concrete (expeditious use of manipulatives)

Representations (pictorial)

Abstract procedures

2. Excellent for teaching accuracy and understanding

Example:http://engage.ucf.edu/v/p/2wKBsbBfcit.usf.edu/mathvids/strategies/cra.html

© Witzel, 2017 32

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Use Place Value to Add Within 100

33Adapted from (Witzel, et al, 2013)

0.26 + 0.18

(4.3)(2.4) Using CRA

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Ones times ones are ones. There are 8 ones.

Ones times tenths are tenths. There are 16 tenths.

Tenths times ones are tenths. There are 6 tenths.

Tenths times tenths are hundredths. There are 12 hundredths.

Total = 8 ones; 22 tenths; 12 hundredths

8.0_2.2_0.1210.32_

© Witzel, 2017

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(4.3)(2.4) using CRA

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X X X X X X X

X X x x x x x X

X X X X x X x X

X X X X X X X X

X X X X X X X X

X X X X X X X X

X x x x X x X x

Ones times ones are ones. There are 8 ones.

Ones times tenths are tenths. There are 16 tenths.

Tenths times ones tenths are tenths. There are 6 tenths.

Tenths times tenths are hundredths. There are 12 hundredths.


8.0_2.2_0.1210.32_

© Witzel, 2017

(4.3)(2.4) using CRA

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multi 4 .3

2 8 0.6

.4 1.6 0.12


8.0_2.2_0.1210.32_

© Witzel, 2017

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.7 x .1 using repeated addition (REL‐SE, 2015)

.7 x .1how many size of

groups one group

© Witzel, 2017 37

.7 x .5 using repeated addition (REL‐SE, 2015)

.7 x .5how many size of

groups one group

© Witzel, 2017 38

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© Witzel, 2017 39

Expand to Fractions Arrays (2/5) (2/4)

Agenda

• Frequent Errors





© Witzel, 2017 40

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IES Practice Guide on Fractions (Siegler et al., 2010)

http://ies.ed.gov/ncee/wwc/pdf/practiceguides/fractions_pg_093010.pdf

1. Build on students’ informal understanding of sharing and proportionality to develop initial fraction concepts

2. Help students recognize that fractions are numbers and that they expand the number system beyond whole numbers.

3. Help students understand why procedures for computations with fractions make sense

4. Develop students’ conceptual understanding of strategies for solving ratio, rate, and proportion problems before exposing them to cross-multiplication as a procedure to use to solve such problems

5. Professional development programs should place a high priority on improving teachers’ understanding of fractions and of how to teach them

© Witzel, 2017 41

Fraction Strips as an Introduction to Number Lines with Fractions

• Use the term “whole” rather than “one” so that students understand the proportionality of fractions per a whole.

• In pairs, have students communicate relationships.

• List all relationships on chart paper and have students confirm or deny these relationships.

http://www.cpalms.org/Public/PreviewResource/Preview/30161

42© Witzel, 2017

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Whole to Part Thinking: Fraction Strips

• Students cut, fold, and color strips of paper to create length‐based models of fraction lines.

• Stack lines in order to make comparisons

• Ask questions such as, “Which strip is one‐fourth of the whole?” and “Which strip is one‐half of one‐fourth?”

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1/

1/ /

1/

1/ 1/1/ 1/

1/ 1/ 1/

1/ 1/ 1/ 1/ 1/ 1/ 1/1/ 1/ 1/ 1/ 1/

1/

1/ /

1/

1/ 1/1/ 1/

1/ 1/ 1/

1/ 1/ 1/ 1/ 1/ 1/ 1/1/ 1/ 1/ 1/ 1/

© Witzel, 2017

Whole to Part Thinking: Making and Investigating Fraction Decimal Strips

• Students cut, fold, and color strips of paper to create length‐based models of decimals.

• Strips are stacked in order to make comparisons

• Ask questions such as, “Which strips show six‐tenths of the whole?” and “Which strip is two‐tenths?”

• “How many two‐tenths does it take to make six‐tenths?”

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0.2

0.5

0.2

0.1 0.1 0.1

0.2 0.2 0.2

0.1 0.1 0.1 0.1 0.1 0.1 0.1

0.2

0.5

0.2

0.1 0.1 0.1

0.2 0.2 0.2

0.1 0.1 0.1 0.1 0.1 0.1 0.1

0.5

© Witzel, 2017

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Number Line Connections

• Relevance ‐ Connect use of fractions to other areas

2. Use formative assessment approaches to identify errors within rational number calculations

3. Implement practice techniques, such as Incremental Rehearsal

© Witzel, 2017

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Fractional Clothesline: Pinning Cards on the Line

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0 112

13

25

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Variations include:• Clothesline versus tape and sticky‐notes• Using key fraction benchmarks to assist students• Graduating from whole class to small group or

individual

© Witzel, 2017

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Fractional Clothesline

• Stretch a clothesline across the room.

• Pin cards to indicate location on a number line

• Vary cards showing fractions, decimals, percents, and combinations

• Vary the objective from ordering to comparing

• Ask students to explain their reasoning

http://www.cpalms.org/Public/PreviewResourceUrl/Preview/5109

47© Witzel, 2017

Decimal Clothesline: Extension

© Witzel, 2017

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0 1

Next Steps:

a) fractions to decimals to percent

b) combinations of cards

c) change the representation of a whole

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Next Steps:

a) fractions to decimals to percent

b) combinations of cards

c) change the representation of a whole

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0 112

© Witzel, 2017

Connect Decimals and Fractions Using the Number Line

Write as a decimal.

50

610

.

© Witzel, 2017

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Progressions:

(2/3)(1/2)Verbally say, “2/3rds of 1/2”

(2/3) (1/2) = 2/6

How could this be translated to 1/3?

1/2

© Witzel, 2017 51

(Witzel & Little)Elementary Math

Progressions:

¾ ÷ ½Verbally say, “How many one‐halves go into three‐fourths?” What is the logical answer?

Answer: 1+ 1/2 = 3/2

Check: 1(1/2) + 1/2(

1/2) = 3/4

© Witzel, 2017 52

¾ X

(Witzel & Little)Elementary Math

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Intervention Aspects of Number Lines

• Visual for conceptual and procedural memory

• Intuitive approximations connect to number sense

• Progresses from early to more difficult concepts

• Physical manipulation possibilities

• Increments to Open Number Lines

© Witzel, 2017 53

Agenda

• Frequent Errors





© Witzel, 2017 54

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Integer computation

• Start with knowledge of negatives on a number linea) small number subtraction problems

b) Larger numbers

13 ‐ 34

‐20‐1 = ‐21

© Witzel, 2017 55

100‐10‐20

21‐10 3

Field model – connected to number lines (Witzel & Little, 2016)

1) Provide a choice of computation options by creating a field

2) Focus on the computation signs, ex. +‐

3) Choose based on the rule:a) ++ I am addingb) +‐ I am subtractingc) ‐+ I am not addingd) ‐‐ I am not subtracting

© Witzel, 2017 56

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Subtract within 100 using models and strategies based on place value33 ‐ 18

+ 30 + 3

‐ 10 ‐ 8

20 ‐5

Reorganize (regroup/borrow)

+ 20 + 13

‐ 10 ‐ 8

+ 10 + 5

10+5 = 15

© Witzel, 2017 57

Agenda

• Frequent Errors





© Witzel, 2017 58

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Fractions using Multisensory Math

• Equivalent fractions and simplification of fractions

• Addition or subtraction

• Multiplication and division

• For more help applying the Multisensory Math approach, don’t hesitate to contact Dr. Witzel ([email protected])

© Witzel, 2017 59

Complex fractions start early

Consider the algebra expression

It shouldbeviewedas

Why do students erroneously renameitas ?

This instruction starts early = 3

© Witzel, 2017 60

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Mixed to improper and back: concrete

• 8 ÷ 2

• 8 ÷ 3

• 13 ÷ 4

© Witzel, 2017 61

Equivalent fractions

• Find an equivalent fraction using repeated addition of numerator and denominator

• Set‐up the proportional look of a fraction: numerator over denominator

=

• Warning! Do Not say, “add another two‐thirds”

• Say, “Another numerator, another denominator

© Witzel, 2017 62

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Aim interventions at procedural processes (Witzel & Riccomini, 2009)

1/3 –2/3

‐+ =‐

© Witzel, 2017 63

Intervention with Fractions procedures (Witzel & Riccomini, 2009)

2/3 + 1/2

++ =

(2 + 2) (1 + 1 + 1) + 7+ =

(3 + 3) (2 + 2 + 2) 6

+

© Witzel, 2017 64

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Connection to Decimals (Witzel & Little, 2016)

• Fractions show the denominator while for decimals it is verbally interpreted.

• 0.23+0.4

65

+ +

© Witzel, 2017

Connection to Decimals• Fractions show the denominator while for decimals

it is verbally interpreted.

• 0.23+0.4

66

+ +

© Witzel, 2017

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Planful Prep

• CRA must be planned

• Small #s first

• Focus on the process and steps

• Share reasoning with the process

© Witzel, 2017 67

Non‐example

CRA delivery (From Witzel, Riccomini, & Schneider, 2008)

• Choose the math topic to be taught;

• Review procedures to solve the problem;

• Adjust the steps to eliminate notation or calculation tricks;

• Match the abstract steps with an appropriate concrete manipulative;

• Arrange concrete and representational lessons;• Teach each concrete, representational, and abstract lesson to student mastery; and

• Help students generalize what they learn through word problems.

© Witzel, 2017 68

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Agenda

• Frequent Errors





© Witzel, 2017 69

3‐2‐1 Take Home

3 things you learned

2 things you can implement

with ease

1 question you still have

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Thank you!

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2017 SLD Math Webinar Series – Upcoming Dates and Topics

CDE‐SLD website: http://www.cde.state.co.us/cdesped/SD‐SLD

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• National Center on Intensive Intervention (NCII) has a number of math instructional videos available: http://www.intensiveintervention.org/resource/place-value-computation-instructional-videos

• Progress Monitoring Tools Chart to help select PM tools: http://www.intensiveintervention.org/chart/progress-monitoring

• Academic Intervention: http://www.intensiveintervention.org/chart/instructional-intervention-tools

• National Council of Teachers of Mathematics: http://www.nctm.org/• CDE Mathematics website: https://www.cde.state.co.us/comath• LD online: http://www.ldonline.org/indepth/math• CLD Strategies and Interventions to Support Students with Mathematics

Disabilities: http://www.council-for-learning-disabilities.org/wp-content/uploads/2014/12/Math_Disabilities_Support.pdf

Additional Resources

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“This material was developed under a grant from the Colorado Department of Education. The content does not necessarily represent the policy of the U.S. Department of Education, and you should not assume endorsement by the Federal Government.”

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Jill MarshallSLD Specialist

Colorado Department of Education

[email protected]

http://www.cde.state.co.us/cdesped/SD‐SLD

Contact Information

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Webinar evaluation:

https://www.surveymonkey.com/r/SLD_Math_Web3

Event Evaluation Link

every student every step of the way - colorado department of

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