Fractions Addressing a Stumbling Block for

Developmental Students

Wade Ellis, Jr.

West Valley College (retired)

A Problem

There are 135 students in a class. There are 25% more boys than girls. How many boys and how many girls are in the class?

Possible Solutions

135

0.25

1.25

1.25 135

2.25 135

135

2.2560

135 60 75

x y

x x y

x y

x x

x

x

x

y

135

0.25

1.25

5

45 4

4 4 4 135

4 5 4 135

9 4 135

4 5135 60 135 75

9 9

x y

x x y

x y

x y

x y

x y

x x

x

x x y

A PROBLEM

There are 135 students in a class. There are 25% more boys than girls. How many boys and how many girls are in the class?

100% 135

1% 1.35

5:4 5/9 4/91/9 55.5

%44.4%

OUTLINE

Setting the stage AMATYC Crossroad and Beyond Crossroads MathAMATYC Educator

What is a Fraction? and Equivalent Fractions Using Technology ― Action/Consequence

Principle Questions that Advance Student Learning A Progression for Learning Fractions Ratios and Proportions & Percents Comments and Suggestions

1988 NCTM Yearbook on Algebra Common Mistakes in Algebra (Marquis, 1988)

10 of 22 were related to fractions

72 7a b ab

x y y

x z z

2 24 16x x

x r x r

y s y s

LEARNING FRACTIONS

If you are training someone to be a retail clerk, and you believe that that person will never need to know much more math than a retail clerk knows, then you can teach fractions using standard algorithms for doing common fraction problems.  But, if you think that the person you are teaching might need to know more advanced mathematics later, then you should teach fractions in a different way. 

 Jim Pellegrino Distinguished Professor of

Cognitive Psychology at the University of Illinois at Chicago

LEARNING FRACTIONS (CONT’D)

In math, you can teach arithmetic by simply teaching the most efficient arithmetical algorithms or you can teach it in a way that greatly facilitates the learning of algebra – so you understand the idea of equivalence . . . , not just what you need to do to execute procedures.  . . . Research shows what kids understand and what they don’t understand depends very much on how we teach the material.

Jim Pellegrino

CROSSROADS IN MATHEMATICS

First, technology can be used to aid in the understanding of mathematical principles.

Second, students will use technology naturally and routinely as a tool to aid in the solution of realistic problems.

BEYOND CROSSROADS

Inquiry. Effective mathematics instruction should require students to be active participants. Students learn through investigation. Advances in neuroscience confirm that students’ active involvement in learning mathematics is important in the process of building understanding and modifying the structure of the mind.

JAMES STIGLER IN THE MATHAMATYC EDUCATOR

Students who have failed . . .[might succeed] if we can first convince them that mathematics makes sense . . .

. . . key concepts in the mathematics curriculum . . . included comparisons of fractions, placement of fractions on the number line, operations with fractions/decimals/percents, ratio, . . .

. . . the ability to correctly remember and execute procedures . . . is a kind of knowledge that is fragile without deeper conceptual understanding of fundamental mathematical ideas.

Finally, when students are able to provide conceptual understanding, they also produce correct answers.

TECHNOLOGY: WHAT’S A FRACTION?

TECHNOLOGY: EQUIVALENT FRACTIONS

FRACTIONS IN THE COMMON CORE Grade 3

Develop understanding of fractions as numbers. Grade 4

Extend understanding of fraction equivalence and ordering.

Build fractions from unit fractions. Understand decimal notation for fractions, and compare

decimal fractions. Grade 5

Use equivalent fractions as a strategy to add and subtract fractions.

Apply and extend previous understandings of multiplication and division.

Grade 6 Apply and extend previous understandings of

multiplication and division to divide fractions by fractions.

CCSS MATHEMATICAL PRACTICES

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.

A PROGRESSION FOR LEARNING FRACTIONS (PROF. WU)

1. What is a Fraction?*2. Equivalent Fractions*3. Fractions and Unit Squares4. Creating Equivalent Fractions5. Adding & Subtracting Fractions with Common Denominators*6. Adding Fractions with Unlike Denominators7. Fractions as Division8. Mixed Numbers*9. Multiplying Whole Numbers and Fractions10. Fraction Multiplication*11. Dividing a Fraction by a Whole Number12. Division of Whole Numbers by a Fraction13. Dividing a Fraction by a Fraction14. Units Other Than Unit Squares15. Comparing Units

WHAT IS A FRACTION?

Teacher Guidance Document I. The Mathematical Focus

References Common Core Standards Covered

II. About the File III. Possible Objectives IV. Sample Questions

Engaging in a concrete experience

Observing reflectivelyDeveloping an abstract conceptualization based upon the reflection

Actively experimenting/testing based upon the abstraction

People learn by

Zull, 2002

Conceptual Knowledge: Makes connections visible, enables reasoning about the mathematics, less susceptible to common errors, less prone to forgetting.

Procedural Knowledge: strengthens and develops understanding allows students to concentrate on relationships

rather than just on working out results

NRC, 1999; 2001

Take an action on a mathematical object

Observe the mathematical consequences and

Reflect on the mathematical implication of those consequences

Conceptual Understanding

Action Consequence Principle

Interactive Dynamic Technology

DYNAMIC INTERACTIVE TECHNOLOGY: ACTION CONSEQUENCE PRINCIPLE

Students take an action on a mathematical object, observe the consequences of that action,

and reflect on the mathematical implications of those

consequences

Burrill & Dick, 2008

A/C DOCUMENTS & LEARNING

Take an action on a mathematical object

Observe the consequences

Reflect on the mathematical implications

Engage in concrete experience

Observe reflectively

Develop abstract conceptualization

Experiment and test concepts

TECHNOLOGY AS ATOOL FOR DEVELOPING UNDERSTANDING

Key is asking good questions Predict consequence in advance of action (what would happen if…?) Consider action that would produce a given

consequence (what would make … happen?) Conjecturing/Testing/Generalization

(When…?) Justification (Why…?)

The only reasons to ask questions is to: (Black et al., 2004)

Probe to uncover students’ thinking discover misconceptions that

exist

Push to advance students’ thinking make connections justify or prove their thinking

WHAT TEACHERS DO:

POSSIBLE QUESTIONS

Handout

WHAT IS A RATIO?

RATIOS AND FRACTIONS

RATIO TABLE

RATIO AND SLOPE

PERCENT

QUESTIONS FOR WHAT IS A FRACTION?

Describe where three fifths will be. How will three fifths differ from seven fifths? Explain your thinking, then check your answer using the tns file.

Where will 4/8 be? b) 0/8? c) Is eleven eighths closer to one or to two? How do you know?

 If the number of 1/5’s is larger than the 5, what can you say about the size of the fraction? Explain.

Suppose the unit fraction was 1/5 and the numerator was between 11 and 14. Where is the fraction?

 If the unit fraction were 1/6, where would fractions with a numerator between 25 and 29 be?

QUESTIONS FOR WHAT IS A FRACTION? (CONT’D)

How many copies of ½ are in 2? Use the file to make a conjecture about

whether the following sentences are correct.

a) 0 is a fraction. b) A whole number cannot be a fraction. c) A fraction can have many names.

PROBLEM

At a dance, 2/3 of the girls dance with 3/5 of the boys. What proportion of the students are dancing?

• A constant way to think: k/p is k copies of 1/p - the length of the concatenation of k segments each of which has length 1/p .

• Behavior similar to whole numbers: k/3 is a multiple of 1/3Larger fraction is to the right on the number line

• Connection of whole number to fractions.

• One number has many names and none more important than another.

• No difference between proper and improper fractions

What does fraction as a point on a number line buy us?

CLOSING DISCUSSION

Questions Comments

REFERENCES Burrill, G. & Dick, T. (2008). What state assessments

tell us about student achievement in algebra. Paper presented at NCTM 2008 Research Presession

Dick, T. & Burrill, G. (2009). Technology and teaching and learning mathematics at the secondary level: Implications for teacher preparation and development. Presentation at the Association of Mathematics Teacher Educators, Orlando FL.

National Research Council. (1999). How People Learn: Brain, mind, experience, and school. Bransford, J. D., Brown, A. L., & Cocking, R. R. (Eds.). Washington, DC: National Academy Press

Zull, J. ( 2002). The Art of Changing the Brain: Enriching the Practice of Teaching by Exploring the Biology of Learning. Association for Supervision and Curriculum Development, Alexandria, Virginia.

REFERENCES

What Does it Really Mean to be College and Work Ready?: The Mathematics Required of First Year Community College Students, National Center on Education and the Economy, 2013.

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wellis@ti.com