LEARNING TO LOVETHE NUMBER LINE

53rd NW Math ConferencePortland, OregonOctober 11, 2014Janeal Maxfield, NBCT andCristina Charney, NBCT

OUR PURPOSE TODAY

Origins of the number line in CCSS-MReflects Grade 2 standardsFoundations begin in K and Grade 1Integral to the work in Grades 3-5 and beyondWhole numbers but fractions and decimals, as wellUniversal struggle

LEARNING TARGETS

Identify prerequisite skills for using number lines Understand the differences between structured and open number linesGain skills with using an open number line for operations (addition and subtraction)Identify key strategies to look for when working with studentsConsider classroom implications for your setting

COMMON CORE

Use 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 - Fractions

3.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 strategies (general strategies)

STRUCTURED NUMBER LINES A measurement model

The distance between marks is important (increments)

STRUCTURED NUMBER LINES

Complete number line

Partial number line

Open number line

PREREQUISITE UNDERSTANDINGSNumber TracksRelative positionProperties of operations: commutative and associativePlace valueComposing and decomposing

Students need experiences with number tracks to fully understand the abstract idea of a number line.

Number Tracks

Number tracks serve to bridge discrete set models and the continuous number line model.

RELATIVE POSITION

PROPERTIES OF OPERATIONS

Commutative Property – Think BIG, count small

13 + 48 might be easier as 48 + 13

Special strategies – 64 - 47 might be easier as 64 - 50, then add 3

PLACE VALUE UNDERSTANDINGNumbers can be composed and decomposed by tens and ones to make operations easier 47 is 4 tens and 7 ones

Non-standard decomposing 7 can be 5 and 2, or 3 and 4, or even 5 and 1 and 1

NUMBER LINE STRATEGIES

Count onCount backSplitting or breaking apart by place valueCounting on and back in jumps of 10, both on and off the decadeBridging across tens

USING THE NUMBER LINE

without bridging with bridging

42 + 27 65 + 2936 + 52 13 + 4887 - 23 46 - 2879 - 61 62 - 17

SHARE YOUR THINKING…EXPLAIN WHYyou made the size of your jumpsyou landed on certain numbersyou started with one addend versus the otheryou added on one addend 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 on

Difference – mark 38 and 65, jump forward or backthe answer is the total of your jumps

STUDENT WORK

24 + 35

24 + 35

87 - 24

87 - 24

52 - 17

52 - 17

52 – 39?

52 - 39

OPERATIONS WITH FRACTIONS

LEARNING TARGETS

Identify prerequisite skills for using number lines Understand the differences between structured and open number linesGain skills with using an open number line for operations (addition and subtraction)Identify key strategies to look for when working with studentsConsider classroom implications for your setting

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.