Chapter 10 IDEA ShareDeveloping Fraction Concepts

Jana KienzleEDU 307 Math Methods

3rd Grade StandardsCluster: Develop understanding of fractions as numbers. Code Standards Annotation 3.NF.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is

partitioned into b equal parts; understand a fraction a/b as the quantity formed by a

parts of size 1/b.

Example: ¼ is the quantity formed by 1 part when a whole is partitioned into 4 equal parts. A fraction ¾ is the quantity formed by 3 parts of size ¼. (ND)

3.NF.2 Understand a fraction as a number on the number line; represent fractions on a

number line diagram.

a. Represent a fraction 1/b on a number line diagram by defining the interval

from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates

the number 1/b on the number line.

Example: A whole is partitioned into 4 equal parts. Recognize that each part is

equal to ¼. (ND)

1 4 1 4 1 4 1 4

0 1 4 2 4 3 4 1

b. Represent a fraction a/b on a number line diagram by marking off a lengths

1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.

Example: Students will be able to draw a number line from 0 to 1 using intervals

representing the denominators 2, 3, 4, 6, 8. Students will be able to label the number line with coordinating fractions (see number line above). (ND)

3.NF.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.

a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.

Example (ND): Are 24 and 1 2 equivalent fractions?

0 1/4 2/4 3/4 1

0 1/2 1

b. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3. Explain why the fractions are equivalent, e.g., by using a visual fraction model.

c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form

3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.

d. Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when

the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

Example (ND): When numerators are the same, the fraction with the larger

denominator is smaller

4th Grade StandardsCluster: Extend understanding of fraction equivalence and ordering. Code Standards Annotation 4.NF.1 Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual

fraction models, with attention to how the number and size of the parts differ even

though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

Example (ND):

1 4 = 3 12 because 1 × 3 = 3

4 × 3 = 12

4.NF.2 Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark

fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

Example: Compare 6/14 to 8/12 using <,>,= , and justify your conclusion. (ND)

Solution: 6/14 < 8/12 because the numerator of the first fraction is less than ½ of the denominator thus the fraction is less than ½; in the second fraction the numerator is greater than ½ of the denominator thus the fraction is greater than ½.

Cluster: Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. Code Standards Annotation 4.NF.3 Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

a. Understand addition and subtraction of fractions as joining and separating

parts referring to the same whole.

b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 +

1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.

c. Add and subtract mixed numbers with like denominators, e.g., by replacing

each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.

Example (ND):

11

4+ 2

1

4= 3

2

4

5

4+

9

4=

14

4

d. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.

4.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.

a. Understand a fraction a/b as a multiple of 1/b. For example use a visual

fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).

b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding

to multiply a fraction by a whole number. For example, use a visual fraction

5th Grade Math StandardsDomain: Number and Operations - Fractions 5.NF

Cluster: Use equivalent fractions as a strategy to add and subtract fractions. Code Standards Annotation 5.NF.1

Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3

+ 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)

5.NF.2 Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and

number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7 by observing that 3/7 < 1/2.

What are fractions?• The fractions studied in elementary school are

rational numbers that can be written as a/b where aand b are integers with b not equal to zero.

• Fractions are numbers representing objects that havebeen “broken” into parts.• “parts of a whole”

4 Principles to Help Children Understand Fractions

• Children learn best through active involvement with avariety of concrete models.

• Most children need extended experiences withmanipulative materials in order to develop mental imagesof fractions in order to reason and think conceptuallyabout fractions.

• Children benefit from opportunities to talk about theirfraction understandings with each other and with theirteacher.

• Learning experiences should begin with helping childrendevelop conceptual knowledge of fractions before movingto more formal work with symbols and computation.

Understanding Sharing Situations• Understanding fraction concepts builds on

familiarity with situations involving sharing.Children use what they already understand tobuild their understandings of new concepts.

4 Children want to share 3 candy bars equally. How much can each child have?

Problem:

(How might a child solve this problem?)

• One way is to start by cutting the first 2 candybars in half, which produces 1 piece (1/2 of acandy bar) for each child. Then the remainingcandy bar is cut into 4 equal parts, creating 1more piece (1/4 of a candy bar) for each child.So each child gets two pieces: 1 bigger piece(1/2) and 1 smaller piece (1/4).

• Another way to solve the problem is to start bycutting each candy bar into 4 equal pieces.Each child would get 1 piece from each candybar, which is ¾ of a candy bar although.

Solutions:

Number Sense with Fractions• Assessing Fraction Number Sense: teachers

ask children to model fractions concretely,pictorially, and symbolically.

• Developing the Meaning of “Half”: half isone of two equal parts.• Activities:1. Sharing for two2. Cutting in half3. Partitioning a square in half

Different Interpretation of Fractions• Part-Whole Interpretations: a region (an

object to be shared or an area to be divided), a set of objects, or a unit of linear measure.• Region Model• Equality of parts• Part-of-a-Set Model• Measurement Model• Area Model

• Other Interpretations of Fractions: ration, quotient, and multiplicative operator• Ratio Interpretation of Fractions• Quotient Interpretation of Fractions• Operator Interpretation of Fractions

Fraction Names• Fraction Symbolism: should be introduced only when

children understand the meaning of the terms one-half, one-third, one-fourth, and so on, and when children can use fractions in problem situations involving regions and parts of a set and in measurement.

• Different Units: generally are represented by continuous quantities, such as regions, and discrete quantities, such as a set of distinct objects.• Continuous but divisible (ex. a cake cut into squares to be shard among 3 siblings).• A discrete set with divisible elements (ex. six cookies to be shared among four children).• A discrete set with separate subsets (ex. 5 boxes of candy, 12 candies per box, to be shared among

4 people).

Developing Comparison and Ordering Fractions

• Comparing and Ordering Fractions• Using a calculator to compare fractions

• Relative Size of Fractions• Improper Fractions and Mixed Numbers

Understanding Equivalent Fractions

• Dealing with Equivalent Fractions• Renaming and Simplifying Fractions

Literature and Internet Resources

http://www.kidsnumbers.com/http://www.kidsmathgamesonline.com/http://www.fuelthebrain.com/search/?search=fractionshttp://www.mathsisfun.com/fractions-menu.html

Activity from Textbook (Page 213)• Activity 10-1 Determining Whether Parts Are the Same Size• Materials: Pairs of partitioned figures as shown• Procedure:1. A child is shown the partitioned figures in pairs as in the diagram.2. Look at these two figures. Are parts (a) and (b) the same size? [or, Do parts (a) and (b)

show the same amount?] Explain how you know.

Hershey’s Fraction Book Activity• Split children up into groups and have each

group come up with some fractions using their chocolate bar.

• Have the groups of students draw their fractions with a brown crayon or marker on a sheet of paper.

Additional Activities• Lego Fractions• Colored Marshmallow Fractions• Dominos Games• Card Games

• Fraction War• Compare Fractions

• Dice Games