1. LEARNING TO LOVE THE NUMBER LINE 53 rd NW Math Conference Portland, Oregon October 11, 2014 Janeal Maxfield, NBCT and Cristina Charney, NBCT

3. OUR PURPOSE TODAY  Origins of the number line in CCSS-M  Reflects Grade 2 standards  Foundations begin in K and Grade 1  Integral to the work in Grades 3-5 and beyond  Whole numbers but fractions and decimals, as well  Universal struggle

4. LEARNING TARGETS  Identify prerequisite skills for using number lines  Understand the differences between structured and open number lines  Gain skills with using an open number line for operations (addition and subtraction)  Identify key strategies to look for when working with students  Consider classroom implications for your setting

5. COMMON CORE Use place value understanding and properties of operations to add and subtract  2.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 length  2.MD.6 – …represent whole-number sums and differences within 100 on a number line diagram

6. 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

7. WHY NUMBER LINES?  A visual representation for recording and sharing students’ thinking strategies during mental computation  Close alignment with children’s intuitive mental strategies  Potential to foster the development of more sophisticated strategies  We can see the level of thinking and any errors that might occur  Enhances communication in the math classroom (SMP 3)  Supports development of special strategies (general strategies)

8. STRUCTURED NUMBER LINES  A measurement model  The distance between marks is important (increments)

9. STRUCTURED NUMBER LINES

10. Complete number line Partial number line Open number line

11. PREREQUISITE UNDERSTANDINGS  Number Tracks  Relative position  Properties of operations: commutative and associative  Place value  Composing and decomposing

12. 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.

19. RELATIVE POSITION

20. 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

21. PLACE VALUE UNDERSTANDING  Numbers 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

22. NUMBER LINE STRATEGIES  Count on  Count back  Splitting or breaking apart by place value  Counting on and back in jumps of 10, both on and off the decade  Bridging across tens

23. USING THE NUMBER LINE without bridgingwith bridging 42 + 2765 + 29 36 + 5213 + 48 87 - 2346 - 28 79 - 6162 - 17

24. SHARE YOUR THINKING…EXPLAIN WHY  you made the size of your jumps  you landed on certain numbers  you started with one addend versus the other  you added on one addend versus the other  you counted on or counted back  you used any special strategies

25. SUBTRACTION 65 - 38  Where is the answer?  Take away - mark 65, jump back 38  the answer is the number you land on  Difference – mark 38 and 65, jump forward or back  the answer is the total of your jumps

26. STUDENT WORK

28. 24 + 35

30. 87 - 24

32. 52 - 17

34. 52 – 39?

35. 52 - 39

36. OPERATIONS WITH FRACTIONS

37. LEARNING TARGETS  Identify prerequisite skills for using number lines  Understand the differences between structured and open number lines  Gain skills with using an open number line for operations (addition and subtraction)  Identify key strategies to look for when working with students  Consider classroom implications for your setting

38. 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.