10/19/2011 Division of Fractions: A Learning Trajectory Designed to Promote Conceptual Understanding Amanda Geist Centennial Middle School amanda.geist@bvsd.org Background • At the time I implemented this project I taught honors 7 th grade math students • In general, many of them caught onto math “rules” quickly, but sometimes lacked the Do students really understand what this means? understanding behind them • These students were exposed to division of fractions the previous year in 6 th grade • Can they draw a picture to represent this problem? • I began asking students questions like draw a picture or give a real life situation in class and • Can they give a real life situation that could be realized they couldn’t do either task represented by this problem? Project Exploration Project Implementation • Can students gain conceptual understanding • Pre-Assessment of dividing fractions, even if they’ve already • Daily number talks (around 10 minutes at the been exposed to the tip of the iceberg (formal start of class) for a period of 6 weeks algorithms)? – Pose division situations in context, students share – Is it possible for students to truly remember the strategies and ideas, teacher introduces other formal algorithms (if they don’t have floating strategies capacity for the iceberg)? • Post-Assessment • Can the iceberg work backwards? 1
10/19/2011 Pre-Assessment Conclusions Division Pre-Assessment � Students who solved problems correctly used a wide Problem % of Success range of strategies ranging from informal to formal � Many students were unsure of what to do with the Fraction division (out of context) 20% remainder in fraction division situations Cookie problem (fraction division in context) 50% � Many students couldn’t remember exactly how to implement the formal division of fractions algorithm Paint problem (fraction division in context) 70% � Students did better solving fraction division Draw a picture of 1% problems in context rather than out of context. Many of them resorted to informal methods due to Give a real life situation to represent 15% the context. 20 ÷ 5 Common Denominator Multiply by the Algorithm Reciprocal “If you equally “How many divide 20 into 5 groups of 5 fit Give a Real Life Situation that could groups, how inside of 20?” be represented by the problem many will be in one group?” • Quotative Number Line Ratio Table • Grouping • Partitive • Measurement Drawing Pictures Unifix Cubes • Fair-Share Exploration of Division Problems in Context “If you equally divide 20 into 5 “How many groups, how groups of 5 fit “If you equally “How many many will be in inside of 20?” divide 4 among ¾ groups of ¾ fit one group?” of a group, how inside of 4?” “How many much will be in one “If you equally groups of ¾ fit group?” • Quotative divide 4 among ¾ inside of 4?” • Grouping of a group, how • Partitive • Measurement much will be in • Fair-Share one group?” 2
10/19/2011 You have three bars of cheese. If it takes 2/3 Quotative Pathway bar of cheese to make one pizza, how many (Also known as grouping or measurement) pizzas can you make? Eventually leads to the common denominator algorithm You have three bars of cheese. If it takes 4/3 bar of cheese to make one pizza, how many 20 ÷ 5 = “How many groups of 5 fit inside of 20?” pizzas can you make? = “How many groups of ¾ fit inside of 6?” Drawing a picture. Drawing a picture. I have 3 bars of cheese. It takes 2/3 bar to make I have 3 bars of cheese. It takes 4/3 bar to make one pizza. How many pizzas can I make? one pizza. How many pizzas can I make? Underlying Question Number Line “How many groups of ____ bars of cheese can How many times does 2/3 fit inside of 3? be made from ____ bars of cheese?” “How many times does ____ fit inside of ____?” 3
10/19/2011 Number Line Ratio Table How many times does 4/3 fit inside of 3? How many times does 2/3 fit inside of 3? Groups of 2/3 Total Ratio Table These are Examples of Division How many times does 4/3 fit inside of 3? You have 5 bars of cheese. It takes 2/3 bar of cheese to make a pizza. How many pizzas can you make? Groups of 4/3 • How many groups of 2/3 fit inside of 5? Total • Bows require ¾ yard of ribbon. How many • A serving size is 2/3 cookies. How many bows can you make from 2 yards of ribbon? servings can be made from 4 cookies? • How many groups of 3/4 fit inside of 2? • How many groups of 2/3 fit inside of 4? • Division number sentence_______________ • Division number sentence_______________ (Use unifix cubes) (Draw a Picture) 4
10/19/2011 How many groups of ____ fit How many groups of ____ fit inside of ____? (Number Line) inside of ____? (Ratio Table) Groups of 3/4 Total Ask students to give a real life context Ask students to give a real life context A serving size of rice is 1/6 cup. How many A serving size of rice is 1/2 cup. How many servings can be made from 1/2 of a cup? servings can be made from 1/6 of a cup? • In other words, how many times does 1/6 fit • In other words, how many times does 1/2 fit inside of 1/2? inside of 1/6? A serving size of cereal is 3/8 cup. How many A serving size of cereal is 1/2 cup. How many servings can be made from 1/2 of a cup? servings can be made from 3/8 of a cup? • In other words, how many times does 3/8 fit • In other words, how many times does 1/2 fit inside of 1/2? inside of 3/8? 5
10/19/2011 A serving size of cereal is 1/3 cup. How many • If we split the bars into the same size pieces, servings can be made from 3/4 of a cup? we’re asking how many times does 4/12 fit • In other words, how many times does 1/3 fit inside of 9/12? inside of 3/4? Tip of the Iceberg: Common • If we split the bars into the same size pieces, Denominator Algorithm we’re asking how many times does 4/12 fit inside of 9/12? • A serving size of rice is 2/3 cup. How many servings can you make from 5/6 cup? • How many times does 2/3 fit inside of 5/6? Tip of the Iceberg: Common Partitive Model Pathway Denominator Algorithm (Also known as fair sharing) • A serving size of rice is 2/7 cup. How many servings can you make from 4/5 cup? Eventually leads to multiplying by the reciprocal • How many times does 2/7 fit inside of 4/5? 20 ÷ 5 = “If you equally divide 20 into 5 groups, how many will be in one group?” = “If you equally divide 6 among ¾ of a group, how much will be in one group?” 6
10/19/2011 • 6 strawberries (equally sized) fill ¾ of a cup. • 4 cookies fill 2/3 of a cookie jar (a very small How many strawberries will fill one cup? cookie jar!) How many cookies will fill one jar? • If you equally share 6 among ¾ of a group, • If you equally share 4 among 2/3 of a group, how much will be in one group? how much will be in one group? • 5 kiwis weigh 4/5 of a pound. How many kiwis • If you equally share 1/3 cake among 3 people, will weigh one pound? how much cake does each person get? • If you equally share 5 among 4/5 of a group, • If you equally share 1/3 among 3 groups, how how much will be in one group? much is in one group? • 1/2 of a cake fills 2/3 of a container. How • If you equally share 2/3 cake among 5 people, much cake will fill one container? how much cake does each person get? • If you equally share 1/2 among 2/3 of a group, • If you equally share 2/3 among 5 groups, how how much is in one group? much is in one group? 7
10/19/2011 • 1/2 of a cake fills 2/3 of a container. How • 1/2 of a cake and fills 2/3 of a container. How much cake will fill one container? much cake will fill one container? Cake 1/2 1/4 3/4 Container 2/3 1/3 1 Cake Container Tip of the Iceberg: • If it takes 5/8 hours to pain 2/3 of a room, Multiply by the Reciprocal how many hours will it take to paint 1 room? • 3/4 of a pizza fills 2/3 of a container. How Hours much pizzas will fill one container? Pizza Painted Room Container • If you equally share 4 among 2/3 of a group, how much will be in one group? “If you equally divide 6 among ¾ of a group, how much will be in one group?” 8
10/19/2011 Summative Exit Ticket (Post Test) • If you equally share 5 among 4/5 of a group, how much will be in one group? 1) 2) Draw a picture to represent this problem 3) Give a real life situation that could be represented by this problem Pre-Test Post-Test Common Denominator Multiply by the 20% 84% Algorithm Reciprocal Draw a Give a Real Life Situation that could 1% 81% be represented by the problem Picture to Represent Number Line Ratio Table Give a Real 15% 70% Life Situation Drawing Pictures Unifix Cubes Exploration of Division Problems in Context 9
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