Chapter 15 Developing Fraction Concepts Copyright 2015 Pearson

Chapter 15 Developing Fraction Concepts Copyright © 2015 Pearson Canada Inc.

Big Ideas 1. For students to really understand fractions, they must experience fractions across many constructs, including part of a whole, ratio, and division. 2. Three categories of models exist for working with fractions—area, length, and set or quantity. 3. Partitioning and iterating are ways for students to understand the meaning of fractions, especially numerator and denominator. 4. Students need many experiences estimating with fractions. 5. Understanding equivalent fractions is critical. Copyright © 2015 Pearson Canada Inc. 2

Content Connections n n Algebraic Thinking (Chapter 14) Fraction Computation (Chapter 16) Decimals and Percents (Chapter 17) Ratio and Proportion (Chapter 18) Copyright © 2015 Pearson Canada Inc. 3

Meanings of Fractions are a critical foundation for students as they are: n used in measurement across various professions n essential to the study of algebra and more advanced mathematics Copyright © 2015 Pearson Canada Inc. 4

Developmental Fraction Constructs n Part-whole. Two-thirds n of the class went on a field trip. $10. 00 with 4 people is 10/4 or 10 ÷ 4. n n n Measure. For 5/8 use five 1/8 unit fractions to count or measure. Division. Sharing Operator. 4/5 of 20 metres Ratio. The fraction 1/4 can mean the probability of an event is one in four. Copyright © 2015 Pearson Canada Inc. 5

Addressing Fraction and Fraction Part Misconceptions n n Thinking of numerator and denominator as separate and not as a single value Not recognizing equal parts; thinking 3/4 green instead of 1/2 green Thinking that fraction 1/5 is smaller than 1/10 because it has a smaller denominator Using the rules from whole numbers to compute Copyright © 2015 Pearson Canada Inc. 6

Models for Fractions Area models Copyright © 2015 Pearson Canada Inc. 7

Models for Fractions Length models Copyright © 2015 Pearson Canada Inc. 8

Models for Fraction Set models Copyright © 2015 Pearson Canada Inc. 9

Concept of Fractional Parts n n n Fraction size is relative Fraction language n The number of parts that make up the whole determines the name of the fractional parts or shares; halves, thirds, n fourths, fifths Equivalent size fractional pieces Copyright © 2015 Pearson Canada Inc. 10

Partitioning with Linear Models n n Partitioning is a strategy used in Singapore for problem solving Try This One A nurse has 54 bandages: 2/9 are white and the rest are brown. How many are brown? Partitioning a strip into nine parts can be used to figure out the problem. Copyright © 2015 Pearson Canada Inc. 11

Class Fractions: Try This One n n n Ask six students to stand up in front of the group. Ask students to identify the fractional parts of the group that have tennis shoes, glasses, earrings, brown eyes, or brown hair. Discuss how this supports conceptual understanding of fraction of a set. (Partitioning with set models) Copyright © 2015 Pearson Canada Inc. 12

Using Fraction Language n n n Counting fractional parts: Iteration Fraction notation is a convention and giving explicit attention to the meaning of numerator and denominator comes from iteration Fraction language n The top number counts (numerator) n The bottom number tells what is being counted (denominator) 3/4 is a count of three parts called fourths Try This One Count 4/5. What is being counted? How many of them do you have? Copyright © 2015 Pearson Canada Inc. 13

Fractions Greater Than One n Avoid using improper fraction n 12/5 is a fraction greater than 1 Terminology needs to be clear for students to recognize that improper does not indicate that it is wrong Models will support this understanding Copyright © 2015 Pearson Canada Inc. 14

Estimating with Fractions n n Benchmarks of zero, one-half, and one Using number sense to compare n n n n About how much Name a fraction for each drawing and explain why you chose that fraction. Focus on the infinite number of fractions that can be use to explain between 0, 1, and 1/2 Copyright © 2015 Pearson Canada Inc. 15

Equivalent-Fraction Concepts n Conceptual focus on equivalence Concept: Two fractions are equivalent if they are representations for the same amount or quantity—if they are the same number. Procedure: To get an equivalent fraction, multiply (or divide) the top and bottom numbers by the same nonzero number. n Equivalent-fraction length models Copyright © 2015 Pearson Canada Inc. 16

Equivalent-Fraction Area Models Copyright © 2015 Pearson Canada Inc. 17

Apples and Bananas: Try This One Provide 24 two-colour counters Task is to group the counters into different fractional parts of the whole and use the parts to create fraction names for the apples and bananas Example: Share your fractional part models. Copyright © 2015 Pearson Canada Inc. 18

Developing Equivalent-Fraction Algorithm Important the students understand that equivalent fractions can have different names Examine models of the equivalent fractions they made to see if the rule of multiplying (dividing) top and bottom numbers by the same number holds true. Copyright © 2015 Pearson Canada Inc. 19

Comparing Fractions Copyright © 2015 Pearson Canada Inc. 20

Comparing Equivalent Fractions Key Concept Connecting visuals with procedure and not rushing the algorithm provides a more sound understanding and skill. Discuss the understanding this student is demonstrating Copyright © 2015 Pearson Canada Inc. 21

Teaching and Learning Fractions 1. Emphasize number sense and the meaning of fractions, rather than on the rote procedures for manipulating them 2. Provide a varied models and contexts 3. Emphasize that fractions are numbers, making extensive use of number lines 4. Dedicate time for understanding of equivalence (concretely, symbolically) 5. Link to benchmarks and encourage estimation Copyright © 2015 Pearson Canada Inc. 22

Literature Connections Context takes students away from rules and encourages them to explore ideas in a more open and meaningful manner. The way that student approach fraction concepts in these contexts may surprise you: How Many Snails? A Counting Book (Giganti, 1998) The Doorbell Rang (Hutchins, 1986) The Man Who Counted: A Collection of Mathematical Adventures (Tohan, 1993) Can you think of others? Copyright © 2015 Pearson Canada Inc. 23