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MAKING RATIONAL NUMBERS MORE MEANINGFUL ATMIM WINTER CONFERENCE BOSTON COLLEGE JANUARY 9 2014 Margaret J. Kenney ([email protected]) Boston College, Chestnut Hill MA 02467

MAKING RATIONAL NUMBERS MORE MEANINGFUL

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MAKING RATIONAL NUMBERS MORE MEANINGFUL. ATMIM WINTER CONFERENCE BOSTON COLLEGE JANUARY 9 2014. Margaret J. Kenney ([email protected]) Boston College, Chestnut Hill MA 02467. Factors Common Core Treatment Of the Rationals Pre-service teachers Bias about Number L ine Use. - PowerPoint PPT Presentation

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Page 1: MAKING  RATIONAL NUMBERS MORE MEANINGFUL

MAKING RATIONAL NUMBERS

MORE MEANINGFUL

ATMIM WINTER CONFERENCEBOSTON COLLEGE

JANUARY 9 2014

Margaret J. Kenney ([email protected])Boston College, Chestnut Hill MA 02467

Page 2: MAKING  RATIONAL NUMBERS MORE MEANINGFUL

Factors Common Core Treatment

Of the RationalsPre-service teachers

Bias about Number Line Use

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Unit Fractions are key to the development of

rational numbers in CCSSM

What of unit fractions in math history?

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Solve: Divide L loaves Among P persons

Equally using ‘largest unit fractions’ .

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OUR SOLUTION ?

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OUR SOLUTION ?4 ÷ 7

4 / 7

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EGYPTIAN SOLUTION

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EGYPTIAN SOLUTION

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TRYDivide 3 loaves Among 7 persons

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EGYPTIAN SOLUTION

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EGYPTIAN SOLUTION

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EGYPTIAN SOLUTION

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EGYPTIAN SOLUTION

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Equality

Given: a /b = c /d

if and only if

ad = bc

Key relation forRational numbers

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WHY ??

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a /b = c /d

if and only if

ad/bd = bc/bd

implies ad = bc

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a /b = c /d

if and only if

ac/bc = ac/ad

implies ad = bc

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WHY ??SAME

DENOMINATORSOR

NUMERATORS

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A Ready-to-Use Fraction Tool

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TryNumber tiles

Useful inProblem Solving

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1

3 6

2 3

9

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How Many like

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How many like

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Page 29: MAKING  RATIONAL NUMBERS MORE MEANINGFUL

Points to Ponder:

What is largest/smallest sum?

Closest means?

How many pairs ? Sums?

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a/b c/d

The Power of the Number Line

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Find a Rational Number between two Rational

Numbers

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Find a Rational Number between two Rational

NumbersNAME THREE WAYS TO

ACCOMPLISH THIS

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AveragingHalve the distance

Add /Subtract

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Given: 1 /4 and 2 /3

Between: (1+ 2) / (4 +3) ? ?

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Given: 1 /4 and 2 /3

Is (1+ 2) / (4 +3) between them ? ?

Try other examples

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1 /5 and 3 /4

Between: (1+ 3) / (5 +4) ? ?

What about

3 /8 and 5 /4

Between: (3+ 5) / (8 + 4) ? ?

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CONJECTURESWith

Rational Numbers

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Given: a /b and c /d

then (a+ c) / (b + d) is between them

CONJECTURE -

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Less than Key relation to apply

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Less than

Given: a /b < c /d

if and only if

ad < bc

Key relation to apply

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Given: a /b < c /d

Show a /b < (a+ c) / (b + d)

and (a+ c) / (b + d) < c /d

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Algorithms /OperationsWith

Rational Numbers

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a cad bc

b dbd bd

ad + bc

bd

=

=+ +

Addition

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a ca x c

b db x d

=

Multiplication

x

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a ca d

b db c

=

DivisionInvert & Multiply

÷

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a ca ÷ c

b db ÷ d

=

DivisionIs this valid? ÷

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a ÷ c a/c b ÷ d b/d

(a/c )(d/b)

(b/d)(d/b)

ad

bc

Is this valid?

=

=

=

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Page 52: MAKING  RATIONAL NUMBERS MORE MEANINGFUL

How would you find all rational numbers with a specified denominator be- tween 2 given rationals?

Describe procedure

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Suppose the two rational numbers are 1/4 and 5/6 and the denominator is 15.

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Relation < is key

1/4 < 4/15 as 15 <16

12/15 < 5/6 as 72 < 75

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Relation < is key

1/4 < 4/15 as 15 <16

12/15 < 5/6 as 72 < 75

4/15, 5/15, …. 11/15, 12/15

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A Mathematical SystemOrganizer

ElementsDefinitions & RelationsOperationsUnProved Statements/

AssumptionsProved Statements/ Theorems

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A Mathematical SystemRational Numbers

Elements : Numbers of form a/b where a,b are integers and b≠0.

Definitions & Relations:Include =, <, >, …Operations: +, −, x, ÷

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A Mathematical SystemRational Numbers

UnProved Statements/Assumptions for +,

xClosure, Commutative, Associative, Identity,Inverses, DistributiveProved Statements/ Theorems

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