Teaching Fractions with Technology Wipe Away the Tears Arjan Khalsa Conceptual Calculations...

Teaching Fractions with

Technology

Wipe Away the Tears

Arjan KhalsaConceptual Calculations

akhalsa@comcast.net

Themes for this talk1. Fractions, rational numbers, are important

and hard for many people to understand.2. These must be taught through concept

development and systematic hierarchy.3. Software can play a critical role in making

this a success.

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.

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”?

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

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

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?

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

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

We MUST think in proportions to succeed!

This is basic literacy.

And the standards emphasize this greatly.

NCTM Standards

NCTM Focal PointsSignificant emphasis in grades 3-6Fractions, decimals, and ratios all to be understood by grade 6

(NCTM 2006)

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.

National Mathematics Advisory Panel (2008)

A major goal for K–8 mathematics education should be proficiency with fractions (including decimals, percents, and negative fractions).

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.

Leading Research Susan J. Lamon Karen Fuson Deborah Ball Rational Numbers Project Nadine Bezuk

Thought Leaders Marilyn Burns John Mighton, JUMP

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

Big Topics with Fractions Understanding fractions Comparing fractions Adding and subtracting fractions with common

denominators, and then with uncommon denominators

Multiplying fractions Dividing fractions

Understanding fractions

Parts of whole Parts of set Segment of distance Ratio Expression of division Portion of time

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

Fractions Models in Classroom Suite 4

Learning Fractions Models 1

Learning Fractions Models 2

Parts of a wholeParts of a set

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

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Connecting models to numbers Connecting concepts to numbers

Neufeld MathSoftware

The National Library of Virtual Manipulatives

Software

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

The pizza slice conundrum

1/8 + 1/8 = ?

Some people see this as 1/16 + 1/16. It’s all a matter of unitizing.

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.

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

Conceptual Calculations Software 5 Days mature! Using the RNP manipulative set Designed intentionally to address

many topics in fractions Excellent for unitizing and more

Software

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Comparing fractions Benchmark fractions Terminal fractions Equivalent fractions Simplest terms Common denominators Common numerators Classroom

Suite Tools

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

Classroom Suite and Fractions Sense Research

Learning Fractions Models 1

Learning Fractions Models 2

Equivalent Fractions Assess

Equivalent Fractions

Num

Improper Fractions

FS – Circle Model

FS - Stories

Equivalent Fractions

Den

Classroom Suite

Overview

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