FRACTIONS, DECIMALS, PERCENTS

Rational Numbers Integers Real Numbers Whole Numbers Natural Numbers Irrational Numbers π.333 1/

Activity - Rational Numbers  What’s the definition of a fraction?  What is a decimal (how is it related to fractions)?  What is a percent (how is it related to fractions and decimals)?

Teaching Rational Numbers Fractions  Decimals  Percent  Fraction instruction begins at grade 1 with conceptual understanding proceeds to fraction computation and equivalency in grades 3-6.  Decimal concepts are introduced in grade 4 and continues with computation through 6.  Percent concepts are introduced in grade 5 and continues with computation and problem solving through grade 6.

Fractions  Write a fraction in the box:  Write 5 examples to teach students to read fraction numbers:  How do you reduce ?  How do you divide these fractions?

Fractions are conceptually complex  Unfamiliar units: Which is the largest?  Equivalence

New vocabulary  Terms  Numerator  Denominator  Proper  Improper  Mixed number

Equal partitioning  Children’s background knowledge and experience of half is dividing into 2 parts  Fractional units require that the parts are equal in size  Fractional units require determining what the whole refers to  1 cookie or package of 12 cookies

Conceptual understanding - fair sharing - solve the following….  Share 5 sandwiches equally among 3 children. How much can each child have?  Share 4 pizzas with 6 children. How much can each child have?

Developing conceptual understanding  Fractions = relationship  Fractions ≠ specific amount  Defining the whole - use a variety of examples  Continuous quantities - single unit is divided into parts  Discrete quantities, collections - sets divided into parts  Vary examples and provide explicit instruction

Developing conceptual understanding  One whole; one unit  Continuous quantities  Discrete quantities - collections - sets

Importance of the unit

Activity

Comparing Magnitudes  Comparing whole numbers 2 and 5  Comparing fractions  Only compare same whole unit  1/5 of a cake is larger than 1/2 of a cupcake.

Conceptual Understanding of Fractions  Determine the unit - oral practice  1/2 of the students in class v. 1/2 of a pizza v. 1/2 glass of milk  Manipulatives  Paper foldingPaper strips Fraction tiles  Colored circlesRulers

Equivalent Fractions

Operations are conceptually complex  Range of operations are procedurally different than whole numbers

Adding and Subtracting Fractions with Like Denominators  Show conceptually (with pictures) why you can’t add different size units  Present procedural strategy for adding and subtracting fractions with like denominators  Provide examples with like and unlike denominators  Students work problems with like denominators  Students cross out problems with unlike denominators See DI Format 12.14

Activity

New Vocabulary and Strategies  Adding and Subtracting Fractions with Unlike Denominators  New Preskill: Lowest (least) common denominator or lowest common multiple  Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40  Multiples of 3:

New Vocabulary and Strategies  Adding and Subtracting Fractions with Unlike Denominators  New Preskill: Lowest (least) common denominator or lowest common multiple  Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40  Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30

New Vocabulary and Strategies  Adding and Subtracting Fractions with Unlike Denominators  New Preskill: Lowest (least) common denominator or lowest common multiple  Rewrite the fraction with like denominators (12)  Relies on knowing the identity element of multiplication is 1 - any number times 1 equals that number and fractions equivalent to 1.

New Vocabulary and Strategies  Adding and Subtracting Fractions with Unlike Denominators  New Preskill: Lowest (least) common denominator or lowest common multiple  Rewrite the fraction with like denominators (12)  Add  Change improper fraction to a mixed number

New Vocabulary and Strategies  Reducing Fractions  New Preskill: Greatest Common Factor  Factors of 24: 1x24, 2x12, 3x8, 4x6 = 1, 2, 3, 4, 6, 8, 12, 24  Factors of 36:

New Vocabulary and Strategies  Reducing Fractions  New Preskill: Greatest Common Factor  Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24  Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

New Vocabulary and Strategies  Reducing Fractions  New Preskill: Greatest Common Factor  Relies on knowing the identity element of multiplication is 1 - any number times 1 equals that number and fractions equivalent to 1.

Other Reducing Strategies  What would you do with: Repeated reducing with common factors

Conceptually complex  When you multiply fractions, is the answer bigger or smaller?  When you divide fractions, is the answer bigger or smaller?

Conceptual understanding of dividing fractions  Devin has 3 1/2 cookies. She will give 1/4 of a cookie to each student as they enter class. How many people will get a cookie?  Will the answer get bigger or smaller?

Conceptual understanding of dividing fractions  Devin has 3 1/2 cookies. She will give 1/4 of a cookie to each student as they enter class. How many students will get cookies?  Will the answer get bigger or smaller? = 14

Multiplying Fractions  Procedural Strategy is fairly easy  Critical features:  Differentiating strategy from addition and subtracting fractions  X sign means “of”: “What is one half of two thirds?”

New Vocabulary and Strategies  Dividing Fractions  Conceptual Understanding  Equivalent fractions  Reciprocals (1/2 x 2/1 = 1)  Identity property of division ( any number divided by 1 equals that number)

Decimals  Understanding of fractions is critical!  Another way of expressing fraction in base ten number system  Start with what’s familiar - money  Decimal concepts  Understanding decimal values and magnitudes  Decimal and fraction equivalency  Reading and writing decimal numbers

Conceptual Understanding of Decimals (whole units v. parts) = 1.4 = 2.5 = 1.25

Conceptual Understanding of Decimals (whole units v. parts) = 1.4 = 2.5 = 1.25

Conceptual Understanding of Decimals Equivalent Units.2 =.20.5 =.50

Decimals on a number line

Reading & Writing Decimal Numbers 3/10 3/100. = decimal point One digit after the decimal point tells about tenths Two digits after the decimal point tells about hundredths

Using place value prompt HundredsTensOnes. TenthsHundredths

Reading and Writing Decimals Critical Features”  Use minimally different examples /100 = = 8/10 8/100 1/8

Operations with Decimals  Conceptual Understanding  Using objects / pictures  Procedural Strategies  Given problems set up  Setting up own problems Involves place value concepts

Percent  Understanding of fractions and decimals is critical!  Start with what’s familiar  Percent used for grades  Percent for shopping discounts  Sales tax or tipping

Conceptual Understanding Percent on the Number Line

Conceptual Understanding of Percent  Circle Graphs 50% 20% 15%

Procedural Strategies for Percent  Changing percents to fractions and decimals  Changing fractions and decimals to percent

Problem Solving with Percent  Calculate % of whole or decimal value  What is 10% of 40?  What is 1% of 200?  Calculating discounts with %  Find total cost of a $85.00 coat that is on sale for 40% off.

Rational Numbers  Beware of misrules  Use concrete and pictorial representations to develop a solid conceptual understanding of rational numbers.  Make connections between fractions, decimals, and percent explicit!