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MAKING RATIONAL NUMBERS MORE MEANINGFUL ATMIM WINTER CONFERENCE BOSTON COLLEGE JANUARY Margaret J. Kenney Boston College, Chestnut Hill MA 02467
Factors Common Core Treatment Of the Rationals Pre-service teachers Bias about Number Line Use
Unit Fractions are key to the development of rational numbers in CCSSM What of unit fractions in math history?
Solve: Divide L loaves Among P persons Equally using ‘largest unit fractions’.
OUR SOLUTION ?
4 ÷ 7 4 / 7
TRY Divide 3 loaves Among 7 persons
Equality Given: a / b = c / d if and only if ad = bc Key relation for Rational numbers
a / b = c / d if and only if ad / bd = bc / bd implies ad = bc
a / b = c / d if and only if ac / bc = ac / ad implies ad = bc
WHY ?? SAME DENOMINATORS OR NUMERATO RS
A Ready-to-Use Fraction Tool
Try Number tiles Useful in Problem Solving
How Many like
How many like
Points to Ponder: What is largest/smallest sum? Closest means? How many pairs ? Sums?
a/bc/d The Power of the Number Line
Find a Rational Number between two Rational Numbers
NAME THREE WAYS TO ACCOMPLISH THIS
Averaging Halve the distance Add /Subtract
Given: 1 / 4 and 2 / 3 Between: (1+ 2) / (4 +3) ? ?
Given: 1 / 4 and 2 / 3 Is (1+ 2) / (4 +3) between them ? ? Try other examples
1 / 5 and 3 / 4 Between: (1+ 3) / (5 +4) ? ? What about 3 / 8 and 5 / 4 Between: (3+ 5) / (8 + 4) ? ?
CONJECTURES With Rational Numbers
Given: a / b and c / d then (a+ c) / (b + d) is between them CONJECTURE -
Less than Key relation to apply
Less than Given: a / b < c / d if and only if ad < bc Key relation to apply
Given: a / b < c / d Show a / b < (a+ c) / (b + d) and (a+ c) / (b + d) < c / d
Algorithms /Operations With Rational Numbers
acadbc bdbdbd ad + bc bd = = + + Addition
aca x c bdb x d = Multiplication x
aca d bdb c = Division Invert & Multiply ÷
aca ÷ c bdb ÷ d = Division Is this valid? ÷
a ÷ c a/c b ÷ d b/d (a/c )(d/b) (b/d)(d/b) ad bc Is this valid? = = =
How would you find all rational numbers with a specified denominator be- tween 2 given rationals ? Describe procedure
Suppose the two rational numbers are 1/4 and 5/6 and the denominator is 15.
Relation < is key 1/4 < 4/15 as 15 <16 12/15 < 5/6 as 72 < 75
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
A Mathematical System Organizer Elements Definitions & Relations Operations UnProved Statements/ Assumptions Proved Statements/ Theorems
A Mathematical System Rational Numbers Elements : Numbers of form a/b where a,b are integers and b≠0. Definitions & Relations: Include =,, … Operations: +, −, x, ÷
A Mathematical System Rational Numbers UnProved Statements/ Assumptions for +, x Closure, Commutative, Associative, Identity, Inverses, Distributive Proved Statements/ Theorems
Operations on Rational Expressions Review Test Tomorrow!!!
Variables and Expressions. Vocabulary Variable – A symbol, usually a letter of the alphabet, such as the letter n, that is used to represent a number.
Working With Common Denominators By: Larry Wilson.
HOW TO MULTIPLY FRACTIONS Introducing: factor product reciprocal multiplicative inverse identity invert.
Sums and Differences of Rational Expressions Objective: to add and subtract rational expressions.
Objective SWBAT simplify rational expressions, add, subtract, multiply, and divide rational expressions and solve rational equations SWBAT simplify rational.
How To Multiply Fractions Introducing: factor product reciprocal inverse identity inverse.
Revision Linear Inequations Algebraic and Graphical Solutions. By I Porter Natural Numbers N Integers J Rational Numbers Q Irrational Numbers I Real Numbers.
Exponent Rules. Parts When a number, variable, or expression is raised to a power, the number, variable, or expression is called the base and the power.
LESSON 1.1 INTEGERS MFM1P. Homework Check McGraw-Hill [Ch. 5.1]: pages Q# 5a, 6, 8, 10, 11, 12, 13, 14, 16.
3.1 Solving Linear Equations Part I A linear equation in one variable can be written in the form: Ax + B = 0 Linear equations are solved by getting “x”
Decimal Disconnects And, what to do about them… *which includes decimals, and percent…
Properties of Algebra By: Grayson W.. Commutative Property When you add or multiply two numbers, it does not matter what order they are in. The sum and.
Introducing: common denominator least common denominator like fractions unlike fractions. HOW TO COMPARE FRACTIONS.
Chapter 6 Review A ratio compares 2 quantities. There are three ways to write a ratio: For example: The ratio of boys to girls in the class is: 11 to 9.
1. How would you explain meaningfully to 5 th graders so that they can understand why the answer for 1 ÷ 2/5 is 2½. 1 2/5 1 ÷ 2/5 “1 ÷ 2/5 = ?” means“how.
A. CONGRUENT SEGMENTS b. MIDPOINT OF A SEGMENT CONGRUENT SEGMENTS & MIDPOINT OF A SEGMENT.
Introducing: first addend second addend sum HOW TO ADD FRACTIONS.
1 MA 1128: Lecture 15 – 6/17/13 Adding and Subtracting Rational Expressions (cont.)
Math Vocabulary. The answer to an addition (+) problem.
Linear Equation in One Variable. A linear equation in one variable is an equation that can be written in the form ax + b = 0 Where a 0 For example: 5x.
Simplify Radical Expressions What steps would you take in simplifying the following: (3+9 2 )
Empowering Learners through the Standards for Mathematical Practice of the Common Core Juli K. Dixon, Ph.D. University of Central Florida
Products and Factors of Polynomials Objective: To multiply polynomials; to divide polynomials by long division and synthetic division.
Fixing Common 27 Myth-Takes & Myth- Conceptions in 43 Myth-Tical Minutes: From Elementary through High School to College Mathematics Alan Zollman Northern.
Rational Numbers A PowerPoint for Ms Purcell’s 6 th grade. MCC6.NS.6.c: Find and position integers and other rational numbers on a horizontal or vertical.
ADDING FRACTIONS You will need some paper! Fractions The top number is the numerator The bottom number is the denominator Example:2 numerator 5 denominator.
Unit 3 Day 3 - Rational Exponents and Radicals. Quiz! Have desks cleared! When you finish your quiz, have out HW!
West Virginias Next Generation Mathematics Content Standards and Objectives Spring Conference for Federal Program Directors and Chief Instructional Leaders.
INEQUALITIES Brought To You By- Tutorial Services-The Math Center.
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