Teaching Fractions with Technology

1. Teaching Fractions with Technology Arjan Khalsa Conceptual Calculations akhalsa@comcast.net Wipe Away the Tears

2. Themes for this talk • Fractions, rational numbers, are important and hard for many people to understand. • These must be taught through concept development and systematic hierarchy. • Software can play a critical role in making this a success.

3. You face far more rational number problems than whole number problems in your life. Every day, you contemplate rational number situations. This is especially true on BIG days.

4. Big Decisions Require “Rational” Thinking • Like Thanksgiving dinner! • How many yams should you prepare? • Your cookbook says that each medium yam feeds one person? • Are the yams you find at the market “medium”? • Do you really think people will eat a whole yam with all of the other food available? • Your sister is bringing regular baked potatoes because her kids “don’t like yams.” So, which type will people choose? • How many people are coming? • Do kids count as “people”?

5. The classic turkey dilemma Your Thanksgiving turkey weighs 16 pounds, 10 ounces. Some cookbooks say is takes 20 minutes a pound to cook a stuffed turkey; others say it takes only 15 minutes a pound if the turkey is over 12 pounds. What time should you start cooking the turkey if you plan to serve the meal at 2:00 pm? Burns (paraphrased), 1998

6. A typical engineering problem Two gears, A and B, are arranged so that the teeth of one gear mesh with the teeth if another. Gear A turns clockwise and has 54 teeth. Gear B turns counterclockwise and has 36 teeth. If gear A makes 5.5 rotations, how many turns will gear B make? Lamon, 2005

7. Fractions and Rational Numbers in Real Life • Carbs per serving • Cooking: quantities of ingredients and cooking times for different amounts of people • Time required to: • Drive somewhere ( We’ll be there in 5 minutes!) • Finish watching the hockey game (There are just 5 minutes left!) • To finish homework (I have 21 problems to go!) • Mortgage payments • What is the interest? • How much principle are you paying each month? • Is it an adjustable rate? If yes, tied to what?

8. Compare gasoline costs – USA basis • Luxury car • Uses super at $2.70/gallon • Gets18 miles/gallon • That equals 15 cents/mile • $2,250 to drive 15,000/ year • $11,250 in five years • My Camry hybrid • Uses regular at $2.40/gallon, • Gets 38 miles/gallon • That equals 6 cents/mile • $900 to drive 15,000/year • $4,500 in five years • Convert this to liters, kilometers, and Canadian money, and this involves more rational thinking

9. Global considerations – USA basis • There are about 250,000,000 cars in the USA • If only 100,000,000 are being driven on a regular basis. . . • And if the hybrid saves $1,350/year • Then the total USA savings of hybrids over cars that use Super gas and get 18 miles per gallon is…. • $135,000,000,000 • The US federal education budget is about $62,000,000,000

10. We MUST think in proportions to succeed!This is basic literacy.And the standards emphasize this greatly.

11. NCTM Standards • NCTM Focal Points • Significant emphasis in grades 3-6 • Fractions, decimals, and ratios all to be understood by grade 6 • (NCTM 2006)

12. NCTM Focal Points Grade 5 • Number and Operations: Developing an understanding of and fluency with division of whole numbers. • Number and Operations: Developing an understanding of and fluency with addition and subtraction of fractions and decimals. • Algebra: Discover concepts related to the addition and subtraction of fractions as they use factors and multiples, including applications of common factors and common multiples.

13. National Mathematics Advisory Panel (2008) A major goal for K–8 mathematics education should be proficiency with fractions (including decimals, percents, and negative fractions).

14. NMAP Final Report • “Difficulty with fractions (including decimals and percents) is pervasive and is a major obstacle to further progress in mathematics, including algebra.” • Fluency with Fractions • By the end of Grade 4, ...identify and represent fractions and decimals, and compare them on a number line or with other common representations. • By the end of Grade 5, ...with comparing fractions and decimals and common percents, and with the addition and subtraction of fractions and decimals.

15. Leading Research • Susan J. Lamon • Karen Fuson • Deborah Ball • Rational Numbers Project • Nadine Bezuk • Thought Leaders • Marilyn Burns • John Mighton, JUMP

16. Fractions – Hard Stuff! • No longer a one-to-one correspondence between a numeral and a value • New academic terms: numerator and denominator, proper, improper • Ordinance is dependent upon the relationship between two numbers • ELL concerns are manye.g. “common denominator” ELL Alert

17. Big Topics with Fractions • Understanding fractions • Comparing fractions • Adding and subtracting fractions with common denominators, and then with uncommon denominators • Multiplying fractions • Dividing fractions

18. Understanding fractions • Parts of whole • Parts of set • Segment of distance • Ratio • Expression of division • Portion of time

19. How Software Helps • Conceptual Models • Bridging Concepts to Procedures • Rich Language Support • UDL • Strategic Feedback • Distributed Practice • Assessment and Assignment

20. Fractions Models in Classroom Suite 4 Parts of a whole Parts of a set Learning Fractions Models 1 Learning Fractions Models 2

21. Mixed Modeling In Visualize Math • Visualize Math, ST Math • The MIND Research Institute • Language independent! • Spatial, temporal math • Parts of a whole + a segment of distance or length Software 70

22. Connecting models to numbers • Connecting concepts to numbers • Neufeld Math Software • The National Library of Virtual Manipulatives Software

23. Unitizing – A big issue! • Marilyn Burns, 2004 • Karen Fuson, 2009 • Susan J. Lamon, 20004 • The Rational Numbers Project

24. The pizza slice conundrum 1/8 + 1/8 = ? Some people see this as 1/16 + 1/16. It’s all a matter of unitizing.

25. Unitizing • Fourth quarter of the game – 12 min? 15 min? • It’s half-price, today only! • We’re half way there. • I understand about a tenth of what you said. • Ever buy too many pizzas or sodas? Was there a problem with the math? • Unitizing: • Different sizes of pizzas and sodas • Cases, slices, six-packs, etc.

26. Unitizing – A big issue! • Marilyn Burns, 2004 • Karen Fuson, 2009 • Susan J. Lamon, 2004 • The Rational Numbers Project

27. Conceptual Calculations Software • 5 Days mature! • Using the RNP manipulative set • Designed intentionally to addressmany topics in fractions • Excellent for unitizing and more Software 90

28. Comparing fractions • Benchmark fractions • Terminal fractions • Equivalent fractions • Simplest terms • Common denominators • Common numerators Classroom Suite Tools

29. The National Library of Virtual Manipulatives • Free! • Many models • Some help in unitizing • Separate, small applets • Look at equivalent fractions,one aspect of comparing fractions NLVM

30. Classroom Suite and Fractions Sense Research Learning Fractions Models 1 Learning Fractions Models 2 Classroom Suite Overview Equivalent Fractions Num Improper Fractions Equivalent Fractions Assess FS – Circle Model FS - Stories Equivalent Fractions Den

31. What you can do • Understand how important fractions and rational thinking are to every learner • Teach fractions carefully and conceptually • Use Classroom Suite 4 and other software effectively • Work with leaders in the mathematics and technology arena • Conceptual Calculations, LLC • Bridges as a conduit for our work akhalsa@comcast.net