Transcript Document
Division of Fractions: Help All Students Achieve Proficiency Steve Klass and Nadine Bezuk California Mathematics Council – South, 49th Annual Fall Conference, Nov. 2008 Today’s Session Welcome and introductions What students should know before operating with fractions Meanings for division Models for division of fractions Contexts for division of fractions Discussion © 2008 Professional Development Collaborative 2 What Students Need to Know Well Before Operating With Fractions Meaning of the denominator (number of equalsized pieces into which the whole has been cut); Meaning of the numerator (how many pieces are being considered); The more pieces a whole is divided into, the smaller the size of the pieces; Fractions aren’t just between zero and one, they live between all the numbers on the number line; A fraction can have many different names; Understand the meanings for whole number operations 3 © 2008 Professional Development Collaborative Solving a Division Problem With Fractions 1 1 ? 3 How would you solve How would you solve 11 1 ? 2 3 How might a fifth or sixth grader solve these problems and what answers might you expect? How can pictures or models be used to solve these problems? © 2008 Professional Development Collaborative 4 What Does Elliot Know? What does Elliot understand? What concepts is he struggling with? How could we help him understand how to model and reason about the problem? © 2008 Professional Development Collaborative 5 What Do Children Need to Know in Order to Understand Division With Fractions? 6 © 2008 Professional Development Collaborative What Does Elliot Know? What does Elliot understand? What concepts is he struggling with? How could we help him understand how to model and reason about the problem? © 2008 Professional Development Collaborative 7 Reasoning About Division Whole number meanings for division 6÷2=3 • Sharing / partitive • What does the 2 mean? What does the 3 mean? • Measurement / repeated subtraction • Now what does the 2 mean and what does the 3 mean? 8 © 2008 Professional Development Collaborative Now Consider 6 1 ÷2 What does this mean? How can it be modeled? What contexts make sense for – Sharing interpretation – Measurement interpretation © 2008 Professional Development Collaborative 9 Reasoning About Division With Fractions Sharing meaning for division: 1 1 3 • One shared by one-third of a group? • How many in the whole group? • How does this work? © 2008 Professional Development Collaborative 10 Reasoning About Division With Fractions Measurement / repeated subtraction meaning: 1 31 • How many times can one-third be subtracted from one? • How many one-thirds are contained in one? • How does this work? • How might you deal with anything that’s left? © 2008 Professional Development Collaborative 11 Materials for Modeling Division of Fractions How would you use these materials to model 1 1 2 1 3 • Paper strips • Fraction circles You could also use: • Pattern blocks • Fraction bars / fraction strips/ paper tape 12 © 2008 Professional Development Collaborative Using a Linear Model With a Measurement Interpretation 1 12 1 3 How many one-thirds are in one and one-half? 1 0 1 1 3 1 3 1 3 1 3 1 2 ? © 2008 Professional Development Collaborative 13 Using an Area Model With a Measurement Interpretation 1 1 Representation of 1 with fraction 2 3 circles. © 2008 Professional Development Collaborative 14 How Many Thirds? ? 1 2 0 1 3 1 3 1 1 1 3 © 2008 Professional Development Collaborative 12 1 3 ? 1 3 15 Helping All Students Achieve Proficiency A Context For Division of Fractions You 1 have 1 2 1 takes 3 cups of sugar. It cup to make 1 batch of cookies. How many batches of cookies can you make? • How many cups of sugar are left? • How many batches of cookies could be made with the sugar that’s left? © 2008 Professional Development Collaborative 17 Another Context For Division of Fractions 1 1 You have yards of licorice rope. It 2 takes 2 yard to make one package of 3 licorice. How many packages can be made? • How much of a yard of licorice is left? • How much of the of the original amount of licorice is left? © 2008 Professional Development Collaborative 18 Model Using Your Materials Use your materials to model 1 1 2 2 3 © 2008 Professional Development Collaborative 19 Thinking More Deeply About Contexts for Division of Fractions What contexts work best for division of fractions? What aspects of these contexts allow them to work better than others? Develop your own new context for the problem we just modeled. © 2008 Professional Development Collaborative 20 Thinking More Deeply About Division of Fractions Estimating and judging a reasonable answer. Recognizing situations involving division of fractions. Considering and creating other contexts where the division of fractions occurs. Making thoughtful number choices when considering examples. © 2008 Professional Development Collaborative 21 Questions/Discussion 22 Contact Us [email protected] [email protected] www.sdsu-pdc.org/ © 2008 Professional Development Collaborative 23 24