MATE 6059 Rational Numbers and Learning Trajectories:

Module 1Rational and Irrational Numbers

August 27, 2011

Warm-Up TaskLife is full of numerical relationships. Think

about your own experiences, professional and personal. Find two things that you can say about yourself that are based on numerical relationships. Consider multiple ways to express that relationship.

You will be asked to introduce yourself and your two ‘number’ descriptions.

Course OverviewSyllabusUse of BlackboardUse of CentraHow does this course fit within the EMAoL

Program of Study?What types of assessment will be included by

DPI at the end of the series of courses?

Professional PracticeExamining practice: Wondering versus

certaintyJustification: Provide evidence and reasoning

for statementsCollegiality: Listen, be open-minded, share

ideasKeeping records

This means… Be conscious of how much you speak (not

too much, not too little)Actively listen to your colleagues and be

willing to share ideas.Behave professionally (be on time, be

respectful, etc.)Be positive. Specifically, do not talk bad

about teachers or students - it’s not what teaching is about.

Some goals…The mathematics that a teacher needs

to know is different than the mathematics that a student needs to know

Teaching mathematics involves getting others to learn and do mathematics.

What are rational numbers?

Let’s start by talking about Counting or natural numbersWhole numbersIntegersRational numbersIrrational numbersReal numbers

Find three numbers that belong in each region of the Venn Diagram.

Exploring Irrational NumbersIf irrational numbers represent decimal

expansions that do not terminate and do not repeat, can they be used to indicate measurements?Use geoboards or dot paper to construct and draw

fourteen line segments of different lengths with dots as endpoints.

Determine the lengths of the line segments in units without using a measuring tool.

Which of the lengths represent rational numbers? Which represent irrational numbers? Explain how you know.

Without using the square root key on your calculator, use your calculator to determine the approximate length of one of the irrational number line segments to three decimal places.

Line Up CardsYou have been given a card with a number

written on it.Study your number carefully.Compare your number with your peers.

Order yourselves around the room by numerical order (small to large).

Be ready to describe your number and tell how it fits within the hierarchy.

Representing Concepts

Representing Concepts

Representing Concepts

Representing Concepts

Word

Definition (in own words)

Non-Example

s

Visual Representation

Facts & Characteristic

s

Personal Associatio

n

Examples

Rotational

Symmetry

A shape matches itself at

least one time before

rotating 360°

All shapes that have 2

or more lines of symmetry

have rotational symmetry

Graphic Art

Nature (flowers)

Word

Definition (in own words)

Non-Example

s

Visual Representatio

n

Facts & Characterist

ics

Personal Associatio

n

Examples

EQUIPARTITIONING/SPLITTING AS A FOUNDATION OFRATIONAL NUMBER REASONING USING LEARNING TRAJECTORIES

Read the article and then in small groups discuss the following:

1. What are the meanings of a/b as a: ratio, fraction-as-number, and operator? Give examples.

2. Distinguish between ideas of many-to-one, many-as-one, and times-as-many from fair sharing and equipartitioning/splitting. Illustrate with examples.

3. Consider the Learning Trajectories Map for Rational Number. Be ready to discuss how the map explicates key concepts such as equipartioning or fraction as number.

Rational Number as RatioRatio is a relation that conveys the notion of

relative magnitude; therefore, it is more correctly considered as a comparative index rather than as a number.

When two ratios are equal they are said to be in proportion to one another. A proportion is simply a statement equating two ratios. The use of proportions is a very powerful problem-solving tool in a variety of physical situations and problem settings that require comparisons of magnitudes.Behr, M., Lesh, R., Post, T., & Silver E. (1983). Rational Number Concepts. In R. Lesh & M. Landau (Eds.),

Acquisition of Mathematics Concepts and Processes, (pp. 91-125). New York: Academic Press.

Fraction-as-numberTo develop understanding of the concept of fraction,

it is important for teachers to attend to the following two aspects.

1. Fractions are useful to express an amount smaller than a standard unit of measure.

2. Like integers and decimal numbers, fractions have the characteristics of numbers.… If, as a result of third grade instruction, students

develop a rigid way of thinking about fractions as so many parts of a partitioned whole, it can be difficult to then nurture an understanding of fraction as number (Tokyo Shoseki, 2000, p.85)Learning Fractions in a Linear Measurement Context: Development and Fieldtests of A Lesson Study Intervention

Catherine C. Lewis, Rebecca R. Perry, Shelley Friedkin and Elizabeth K. Baker, Mills College

Rational Number as OperatorThe subconstruct of rational number as operator imposes on

a rationalnumber p/q an algebraic interpretation; p/q is thought of as a function that transforms geometric figures to similar geometric figures p/q times as big, or as a function that transforms a set into another set with p/q times as many elements.

When operating on continuous object (length), we think of p/q as a stretcher-shrinker combination. Any line segment of length L operated on by p/q is stretched to p times its length and then shrunk by a factor of q.

A multiplier-divider interpretation is given to p/q when it operates on a discrete set. The rational number p/q transforms a set with n elements to a set with np elements and then this number is reduced to np/q.

Behr, M., Lesh, R., Post, T., & Silver E. (1983). Rational Number Concepts. In R. Lesh & M. Landau (Eds.), Acquisition of Mathematics Concepts and Processes, (pp. 91-125). New York: Academic Press.

One-as-many and Many-as-one• The one-as-many and many-as-one schemes are complementary to

each other.

• A many-to-one correspondence exists when a fixed number of target objects (greater than 1) is associated with each of a set of referents, as in putting 3 flowers in each of several vases (Sophian & Madrid, 2003).

• The one-as-many scheme allows a child to coordinate a unit with a multiplicity of another unit. For example, when a child counts one dark green rod as two light green rods, he/ she is said to have employed the one-as-many scheme. On the other hand, when a child counts three white rods as one light green rod, it is an example of the many-as-one coordination (Watanabe, 1995).

• The recursive character of the operation can be described by replacing a single fair share or part or size m, by n times as many of that part, to produce the original collection or whole (later, the product) mn (Confrey, Maloney, Nguyen, Mojica, and Myers).

Equipartitioning/splittingFair SharingEquipartitioning is defined as the cognitive

behaviors that lead to the creation of equal sized groups from a collection, or equal sized pieces from a continuous whole, and which result in fair shares (Confrey, 2008).

Fair Sharing is based in early experiences of young children.

Mathematical Tasks

Solve and then consider:What could the following tasks tell us about students’

understanding of rational number?

MURDOCK-STEWART, 2005

Jose ate ½ of a pizza. Ella ate ½ of another pizza. Jose said that he ate more pizza than Ella, but Ella said they both ate the same amount. Use words and pictures to show that Jose could be right.

What do the following responses tell us about the children’s understanding of rational numbers?

1992 ResultsReason (Meaning of Fraction)

Minimal Partial Satisfactory

Extended Omitted

National 18 2 8 15 57

North Carolina 17 3 8 12 60

Module 1 HomeworkSee homework in Module 1 folder in

Blackboard