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MAKING RATIONAL NUMBERS MORE MEANINGFUL ATMIM WINTER CONFERENCE BOSTON COLLEGE JANUARY 9 2014 Margaret J. Kenney (firstname.lastname@example.org) 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
1 3 6 2 3 9
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
Fractions. ADDING FRACTIONS Build each fraction so that the denominators are the same ADD the numerators Place the sum of the two numerators on.
Adding a Sequence of numbers (Pairing Method ) 1. Find the sum of …… Problem Solving Strategy :- Step 1 – Divide the last number by.
Fractions Review. Fractions A number in the form Numerator Denominator Or N D.
Operations with Fractions REVIEW CONCEPTS. Fractions A number in the form Numerator Denominator Or N D.
6.3 Least Common Denominators. Objective 1 Find the least common denominator for a group of fractions. Slide
Fractions Continued. Fractions A number in the form Numerator Denominator Or N D.
Warm Up Simplify:. Adding and Subtracting with Unlike Denominators.
+ Fractions. + Part of a whole + + Numerator How many pieces The number on the top of a fraction.
Chapter 7 Section 3 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.
Fractions Addition, Subtraction, Multiplication and Division June 25, 2016June 25, 2016June 25, 2016.
Adding, Subtracting, Multiplying, & Dividing Rational Expressions Objective: Add, subtract, multiply, divide, and simplify rational expressions.
+ Adding and Subtracting. + How do you add or subtract fractions?
Sums and Differences of Rational Expressions Objective: to add and subtract rational expressions.
Chapter 7 Section 3. Objectives 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Least Common Denominators Find the least common denominator for.
Simplifying Radical Expressions Objective: Add, subtract, multiply, divide, and simplify radical expressions.
Fraction Operations Using Pattern Blocks. Fractions CCSS 4.NF.3c Add and subtract mixed numbers with like denominators. 4.NF.4 Apply and extend previous.
Solving Equations Inverse operations. INVERSE = Opposite If I am solving an equation using inverses operations, I am solving it using opposite signs.
Objective Standard 15.0 I can use the rules of exponents and factorization to simplify the multiplication and division of rational expressions.
Fractions Adding & Subtracting Whole Number and Fractions.
Rational Expressions To add or subtract rational expressions, find the least common denominator, rewrite all terms with the LCD as the new denominator,
How to Simplify Rational Expressions How to Simplify Complex Fractions.
Working With Common Denominators By: Larry Wilson.
12-6 Rational Expressions with Like Denominators Objective: Students will be able to add and subtract rational expressions with like denominators.
Solving Inequalities. ● Solving inequalities follows the same procedures as solving equations. ● There are a few special things to consider with inequalities:
Adding and Subtracting Fractions Review. When adding fractions you MUST have a common denominator! _ + x 5 = = x Find the smallest.
Adding and Subtracting Rational Expressions with Different Denominators Find the LCD of two or more rational expressions. 2.Given two rational expressions,
Measurement Adding and Subtracting Fractions with Different Denominators.
Section 8.8. In this lesson you will learn to add, subtract, multiply, and divide rational expressions. In the previous lesson you combined a rational.
Aug. 12, 2013 Sixth Grade Math. The sum of two addends multiplied by a number equals the sum of the product of each addend and that number 4(3 + 7) =
Dear Power point User, This power point will be best viewed as a slideshow. At the top of the page click on slideshow, then click from the beginning.
Adding and Subtracting Rational Numbers. The term, Rational Numbers, refers to any number that can be written as a fraction. This includes fractions that.
RATIONAL EXPRESSIONS. EVALUATING RATIONAL EXPRESSIONS Evaluate the rational expression (if possible) for the given values of x: X = 0 X = 1 X = -3 X =
Adding and Subtracting Rational Expressions. WARM UP 1) Suppose y varies inversely with x, Write an equation that includes the point (4,2) 2) The length.
Operations with Rational Numbers Any number that can be written in the form, where m and n are integers and n 0, is called a rational number In other.
ADDING AND SUBTRACTING RATIONAL EXPRESSIONS: TO ADD OR SUBTRACT RATIONAL EXPRESSIONS USE THE ADDITION PROPERTY:
Complex fractions. Objective Simplify complex fractions Lets Review fraction rules first…………..
MATH 2A CHAPTER EIGHT POWERPOINT PRESENTATION FRACTIONS AND ALGEBRA.
Solving 2 step equations. Two step equations have addition or subtraction and multiply or divide 3x + 1 = 10 3x + 1 = 10 4y + 2 = 10 4y + 2 = 10 2b +
Adding and Subtracting Fractions with Like Denominators.
Adding, Subtracting, Multiplying, and Dividing Rational Numbers.
Unit 4 Day 4. Parts of a Fraction Multiplying Fractions Steps: 1: Simplify first (if possible) 2: Then multiply numerators, and multiply denominators.
One step equations Add Subtract Multiply Divide When we solve an equation, our goal is to isolate our variable by using our inverse operations. What.
Objective SWBAT simplify rational expressions, add, subtract, multiply, and divide rational expressions and solve rational equations SWBAT simplify rational.
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 1 1 Chapter 7 Rational Expressions and Equations.
OPERATIONS USING FRACTIONS. 1. Add, subtract, multiply and divide fractions with and without a calculator. 2. Convert between equivalent forms of fractions.
5.1 Fractions, Mixed Numbers, and Rational Expressions.
Chapter 8 – Rational Expressions and Equations 8.2 – Adding and Subtracting Rational Expressions.
Chapter 12 Final Exam Review. Section 12.4 “Simplify Rational Expressions” A RATIONAL EXPRESSION is an expression that can be written as a ratio (fraction)
Chapter Complex Numbers What you should learn 1.Use the imaginary unit i to write complex numbers 2.Add, subtract, and multiply complex numbers 3.
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