Chapter 10 IDEA Share Developing Fraction Concepts Jana Kienzle EDU 307 Math Methods
3 rd Grade Standards Cluster: 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 3.NF.3 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. 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: 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) Example (ND): Are 2 4 and 1 2 equivalent fractions? 0 1/4 2/4 3/4 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. 0 1/2 1 Example (ND): When numerators are the same, the fraction with the larger denominator is smaller
4 th Grade Standards Cluster: 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): 1 1 4 + 2 1 4 = 3 2 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. 5 4 + 9 4 = 14 4 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
5 th Grade Math Standards Domain: Number and Operations - Fractions 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 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 a and b are integers with b not equal to zero. Fractions are numbers representing objects that have been broken into parts. parts of a whole
Learning experiences should begin with helping children develop conceptual knowledge of fractions before moving to more formal work with symbols and computation. 4 Principles to Help Children Understand Fractions Children learn best through active involvement with a variety of concrete models. Most children need extended experiences with manipulative materials in order to develop mental images of fractions in order to reason and think conceptually about fractions. Children benefit from opportunities to talk about their fraction understandings with each other and with their teacher.
Understanding Sharing Situations Understanding fraction concepts builds on familiarity with situations involving sharing. Children use what they already understand to build their understandings of new concepts. Problem: 4 Children want to share 3 candy bars equally. How much can each child have? (How might a child solve this problem?)
Solutions: One way is to start by cutting the first 2 candy bars in half, which produces 1 piece (1/2 of a candy bar) for each child. Then the remaining candy bar is cut into 4 equal parts, creating 1 more 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 by cutting each candy bar into 4 equal pieces. Each child would get 1 piece from each candy bar, which is ¾ of a candy bar although.
Number Sense with Fractions Assessing Fraction Number Sense: teachers ask children to model fractions concretely, pictorially, and symbolically. Developing the Meaning of Half : half is one of two equal parts. Activities: 1. Sharing for two 2. Cutting in half 3. 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, onethird, 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=fractions http://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