La-STEM Math Academies for ENFA and LA Educators “Transforming Numbers & Operations and Algebra Instruction in Grades 3-5” DAY 2 of 8 “My heart is singing for joy this morning. A miracle has happened! The light of understanding has shone upon my little pupil’s mind, and behold, all things are changed.” Anne Sullivan (A teacher of the blind and deaf)

GOALS: Understand fractions and decimals represented as parts of a unit or area, parts of a collection of objects, and locations on a number line Use concrete materials to represent fractions and decimals as parts of a unit or area, parts of a collection of objects, and locations on a number line Name fractional and decimal parts of a unit or area, of a collection of objects, and of a number line

Fair Shares Children must be aware of two components of fractions: 1.The number of parts 2.The equality of the parts (equal size, but not necessarily shape)

Rational Numbers A rational number is a number that can be written in the form a/b where a and b are integers and b≠0. In the set of rational numbers, a is called the numerator (Latin word meaning number) and b is called the denominator (Latin word meaning namer), of the fraction. QUESTION: Is every fraction a rational number and is every decimal a rational number?

5 Main Interpretations of Fractions 1.Fractions as parts of wholes or parts of sets ▫A unit is partitioned equally into equivalent parts. 2.Fractions as the result of dividing two numbers ▫The quotient- meaning results when a number of objects are shared by a set number. 3.Fractions as the ratio of two quantities ▫Compares a part to a whole. 4.Fractions as operators ▫A fraction acts on another number by stretching or shrinking it. 5.Fractions as measures ▫The length marked on a number line or subunits. NOTE: equal parts and equal size pieces (but not necessarily identical shapes) are ESSENTIAL when dealing with fractions.

Different ways to express: “I have $3 I’d like to share with 4 of you.” …..with 4 of you. Or

Research indicates: Using models is critical in understanding fractions. Younger grades are better at this than later grades. Models help clarify what is being written symbolically. Sometimes it helps to do the same activity with different models.

1. Area or Region Model This is the place we usually begin for MOST students but students have to understand what we mean by AREA. Area models involve sharing something that can be cut into smaller parts. Circular models are good about emphasizing the amount that remains but not very good when the fractions move beyond ½, 1/3, ¼, 1/5 or when we have to operate with fractions.

Area Model

An AREA model using pattern blocks

Use your Pattern Blocks: Activity 1

2. Measurement or Length Model Length is the critical factor in this model-- instead of the area of the unit The number line is significantly more sophisticated that most other models Each number represents a distance to the labeled point from zero

Length Model Navigating Through Number and Operations

Cuisenaire Rods-- a LINEAR model

More practice with a linear model…

Number Line-- Activity 2 1.Find the Cuisenaire Rod that allows you to partition the second number line into halves. Mark ½ on the number line. What is the rod color? 2.Find the Cuisenaire Rod that allows you to partition the third number line into thirds. Mark 1/3 and 2/3 on the number line. The rod color is? 3.Continue in this manner until you have partitioned and labeled the final three number lines into fourths, sixths, and twelfths.

Number Line Fraction Division

Helping Students Reason About Fractions as Numbers 1.What number is halfway between zero and one? 2.What number is halfway between zero and one-half? 3.What other numbers are the same as one- half? 4.What number is ¼ more than ½? 5.What number is 1/6 more than ½? 6.What number is 1/6 less than one? cont.

More questions for understanding… 7.What number is 1/3 more than one? 8.What number is halfway between 1/12 and 3/12? 9.What number is closest to zero? 10.What number is closest to one? 11.What would you call a number halfway between zero and 1/12?

3. Set or Discrete Model The whole is understood to be a set of objects and subsets make the fractional parts Good at establishing real world uses of fractions and ratio concepts Most common manipulative is two-color counters We also use set models in showing data. [1/4 of boys are blond]

Set Model Navigating Through Number and Operations

Draw a small square on your paper. If the square = 2/5, draw 1 whole Lynne Tullos, LDOE 2010

Look at your Pattern Pieces If this piece = 3/5 unit How much of a unit is this piece =? Draw the unit piece. Lynne Tullos, LDOE 2010

Using Fraction Circles, Strips, or Fraction Bars

Understanding Size with Fraction Bars ▫What do you notice about fraction bars? ▫Compare fraction bars to circle fractions. Lynne Tullos, LDOE 2010

Understanding Size with Fraction Bars ▫What do you notice about these bars? ▫Get out the 3 rd and 4 th bars: without finding a common denominator, which is larger, 1/3 or 1/4? ▫Which is larger, 1/5 or 1/8? How do you know? Lynne Tullos, LDOE 2010

Understanding Size with Fraction Bars ▫Which is larger 7/8 or 8/9? ▫Which is larger 2/3 or 11/12? ▫Which is larger 5/8 or 5/7? ▫NOTE: if the numerators are the same, you must look at the size pieces! Lynne Tullos, LDOE 2010

REMEMBER To Ask Children How can you tell which fraction is larger? What must you consider? What strategies can you use? Lynne Tullos, LDOE 2010

Modeling Decimals Notations for money are the first thing to come to mind. $ means 127 dollars and some part of another dollar. The decimal point separates the dollars from the parts of a dollar.

Modeling $ Because 95 ¢ is 95/100 of a dollar, we have $ = /100 dollars Because 95 ¢ is 9 dimes and 5 cents; one dime =1/10 of a dollar and 1 cent = 1/100 of a dollar, 95 ¢ is 9 x 1/ x 1/100 Consequently: $ = 1 x x x x 1/ x 1/10 2

GOALS: Compare fractions and decimals by using concrete models Compare fractions and decimals by using benchmarks Order fractions and decimals by using concrete models, benchmarks, and parallel number lines

Which Comes First? Lynne Tullos, LDOE 2010

Sort these 12 fractions into the categories below and write what you notice: 2/47 4/5 1/14 7/8 4/9 10/13 6/14 1/35 5/9 3/100 6/7 5/12 Close to 0: Close to ½: Close to 1: Lynne Tullos, LDOE 2010

Sort these 12 fractions into the categories below and write what you notice: 2/47, 4/5, 1/14, 7/8, 4/9, 10/13, 6/14, 1/35, 5/9, 3/100, 6/7, 5/12 ▫Close to 0: 2/47, 1/14, 1/35, 3/100  (numerator is small compared to denominator) ▫Close to ½: 4/9, 6/14, 5/9, 5/12  (numerator is close to ½ of the denominator) ▫Close to 1: 4/5, 7/8, 6/7, 10/13  (numerator and denominator are about the same size) Lynne Tullos, LDOE 2010

Looking for Fractions: Find a fraction between 3/7 and 4/7; Lynne Tullos, LDOE 2010

Looking for Fractions: Find a fraction between 3/7 and 4/7; Find another fraction between 3/7 and 4/7 that isn’t ½. Lynne Tullos, LDOE 2010

Denseness Property of Rational Numbers: Between every 2 rational numbers, there is another rational number!! Lynne Tullos, LDOE 2010

More/Less Fraction Tree Choose 1 fraction Choose a fraction smaller (left) and larger (right) than the first fraction Continue taking turns until someone misses their turn with an incorrect fraction. Lynne Tullos, LDOE 2010 LESS MORE L M L M

GOALS: Represent fractions and terminating decimals by using visual models Determine fraction and decimal equivalents by using tenths and hundredths grids, parallel number lines, clocks, money, and calculators Understand relationships among fractions and decimals

Taking an Hour for Clock Fractions Engage students by asking questions such as: ▫How many minutes in an hour? ▫How many minutes after the hour is it when the minute hand is pointing to the 6?

Clock Fractions ▫What are some ways you can use to find this?  This leads to discussing 30 minutes out of 60 is ½ of an hour ▫ How many minutes after the hour is it when the minute hand is pointing to the 3?  Since we were ½ way around the clock showed 30 min., we must be ½ of 30, or 15 minutes.  The clock hand divides the clock into 4 parts so 15 minutes must be ¼ of the clock.

Decimals Make Cents!

Ten Tenths Make A Whole The definition of equivalent is an equal amount..1 = 10 cents.2 = 20 cents.3 = 30 cents.4 = 40 cents.5 = 50 cents.6 = 60 cents.7 = 70 cents.8 = 80 cents.9 = 90 cents Of course, we usually see a zero (0) in the hundredths place to show we are dealing with money..10 = 10 cents.20 = 20 cents.30= 30 cents.40 = 40 cents.50 = 50 cents.60 = 60 cents.70 = 70 cents.80 = 80 cents.90 = 90 cents

The Hundredths Place Hundredths place is the last digit to the right and represents coins smaller than a dime..01 = 1 cent.02 = 2 cents.03 = 3 cents.04 = 4 cents.05 = 5 cents.06 = 6 cents.07 = 7 cents.08 = 8 cents.09 = 9 cents Look at the following numbers. Can you identify the number in the hundredths place? What coins can be used? $2.12 $1.47 $0.94 $5.55 $0.83 $20.06

Fraction and Decimal Equivalents Simplified Number of Shaded Numerator Written Fraction Written Fraction Squares (Numerator) W/Denominator of as a Decimal Number 100 ½ 50 50/ (or 0.5)

Contact Information STEM: Science, Technology, Engineering, and Mathematics Goal Office Lynne Tullos Jenny Foot Dr. Guillermo Ferreyra