LEARNING TO LOVE THE NUMBER LINE!

FRACTIONS AND DECIMALS

CMC - South ConferenceOrchestrating the Common Core ClassroomPalm Springs, CANovember 6, 2015Janeal Maxfield, NBCT andCristina Charney, NBCT

OUR PURPOSE TODAYOrigins of the number line in CCSS-MFoundational work in Grades K-2 with whole number operationsIntegral to the work in Grades 3-5 with decimal and common fraction operationsUniversal struggle

LEARNING TARGETSUnderstand the purpose of and difference between structured and open number linesIdentify big ideas and key strategies students use with number linesGain skills with using an open number line for operations (addition and subtraction)Consider classroom implications for your setting

COMMON COREUse place value understanding and properties of operations to add and subtract2.NBT.5 – fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction

Relate addition and subtraction to length2.MD.6 – …represent whole-number sums and differences within 100 on a number line diagram

COMMON CORE:NUMBER AND OPERATIONS - FRACTIONS3.NF.2 – Understand a fraction as a number on a number line; represent fractions on a number line

4.NF.3d and 5.NF.1– Solve word problems involving addition and subtraction of fractions…using visual fraction models

CCSS Glossary: visual fraction model – a tape diagram, number line diagram, or area model

WHY NUMBER LINES?A visual representation for recording and sharing students’ thinking strategies during mental computationClose alignment with children’s intuitive mental strategiesPotential to foster the development of more sophisticated strategiesWe can see the level of thinking and any errors that might occurEnhances communication in the math classroom (SMP 3)Supports development of special strategiesAn enduring model throughout students’ education

STRUCTURED NUMBER LINES:RELATIVE POSITION

Model for comparing and rounding numbers The distance between marks is important

(increments)

0 1,000

STRUCTURED NUMBER LINES:RELATIVE POSITION

Where is 325 on each number line?

0 1,000

RELATIVE POSITION

FRACTION MODELS

Length model

Number line model

WHOLE NUMBER OPERATIONS

ON A NUMBER LINEAdditionSubtraction

Complete number line

Partial number line

Open number line

BIG IDEAS & STRATEGIESCommutative Property – Think BIG, count small 13 + 48 might be easier as 48 + 13Special strategies – 57 + 34 might be easier as 60 + 31

compensation

64 – 47, add 3 to both to get 67 – 50constant difference

These apply to common and decimal fractions!

BIG IDEAS & STRATEGIES: WHOLE NUMBERSBridging across tens or hundredsNumbers can be composed and decomposed by tens and ones to make operations easier 47 is 4 tens and 7 onesNon-standard decomposing 7 can be 5 and 2, or 3 and 4, or even 5 and 1 and 1

BIG IDEAS & STRATEGIES:DECIMAL AND COMMON FRACTIONS

Bridging across whole numbers or place valueNumbers can be composed and decomposed by whole numbers and fractional parts 4.73 is 4 ones + 7 tenths + 3 hundredths

Non-standard decomposing – unit fractions 5/8 can be 3/8 and 2/8 or 4/8 and 1/8

BIG IDEAS & STRATEGIES: WHOLE NUMBERSCount On and Count BackSplitting or breaking apart by place valueCounting on and back in jumps of 10, both on and off the decade

BIG IDEAS & STRATEGIES:DECIMAL AND COMMON FRACTIONSCount On and Count BackSplitting or breaking apart by whole number and fractional partCounting on and back in jumps of whole numbers or fractions, from a whole number or fraction

USING THE NUMBER LINE

SHARE YOUR THINKING -Tell why…you made the size of your jumpsyou landed on certain numbersyou started with one number versus the otheryou counted on or counted backyou used any special strategies

SUBTRACTION 65 - 38 Where is the answer?

Take away - mark 65, jump back 38the answer is the number you land onDifference – mark 38 and 65, jump forward or backthe answer is the total of your jumpsThis is the same for common and decimal fractions!

OPERATIONS WITH FRACTIONS AND DECIMALS ON A

NUMBER LINE

AdditionSubtraction

BIG IDEAS & STRATEGIESCommutative Property – Think BIG, count small 13 + 48 might be easier as 48 + 13Special strategies – 57 + 34 might be easier as 60 + 31

compensation

64 – 47, add 3 to both to get 67 – 50constant difference

These apply to common and decimal fractions!

COMPENSATION & CONSTANT DIFFERENCE

BIG IDEAS & STRATEGIES:DECIMAL AND COMMON FRACTIONS

Bridging across whole numbers or place valueNumbers can be composed and decomposed by whole numbers and fractional parts 4.73 is 4 ones + 7 tenths + 3 hundredths

Non-standard decomposing – unit fractions 5/8 can be 3/8 and 2/8 or 4/8 and 1/8

NUMBER LINE STRATEGIES:DECIMAL AND COMMON FRACTIONSCount On and Count BackSplitting or breaking apart by whole number and fractional partCounting on and back in jumps of whole numbers or fractions, from a whole number or fraction

USING THE NUMBER LINE

SHARE YOUR THINKING -Tell why…you made the size of your jumpsyou landed on certain numbersyou started with one number versus the otheryou counted on or counted backyou used any special strategies

SUBTRACTION 65 - 38 Where is the answer?

Take away - mark 65, jump back 38the answer is the number you land onDifference – mark 38 and 65, jump forward or backthe answer is the total of your jumpsThis is the same for common and decimal fractions!

FAMILIAR WAYS TO SUBTRACT MIXED NUMBERSImproper Fractions Regroup one whole

2 + 1 + +

LEARNING TARGETSUnderstand the purpose of and difference between structured and open number linesIdentify big ideas and key strategies students use with number linesGain skills with using an open number line for operations (addition and subtraction)Consider classroom implications for your setting

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REFERENCES

Bobis, Janette, The Empty Number Line: A Useful Tool or Just Another Procedure?, Teaching Children Mathematics, April 2007.

Diezmann, Carmel, Tom Lowrie, and Lindy A. Sugars, Primary Students’ Success on the Structured Number Line, APMC (Australian Primary Mathematics Classroom), April 2010.

Klein, Anton S., Meindert Beishuizen and Adri Treffers, The Empty Number Line in Dutch Second Grade: Realistic Versus Gradual Program Design, Journal for Research in Mathematics Education, 1998, Volume 29 Number 4, pages 443-464.