1. Learning to love the number line! Fractions and decimals
Introduce ourselves
Explain our role as instructional coaches and our interest in promoting the use of the number line
Norms: working presentation, please participate. Leave if this is not a fit for you today. Stay engaged!
Learning to love the number line! Fractions and decimals
CMC - South Conference
Orchestrating the Common Core Classroom
Palm Springs, CA
November 6, 2015
Janeal Maxfield, NBCT and
Cristina Charney, NBCT

2. This is what is being distributed on social media.

3. Our purpose today Origins of the number line in CCSS-M
Foundational work in Grades K-2 with whole number operations
Integral to the work in Grades 3-5 with decimal and common fraction operations
Universal struggle
In the upper grades, when working with decimal fractions and common fractions, especially when adding and subtracting, we want to connect what students already know from working with whole numbers.
We want students to experience and trust that what they know applies just as well when operating with decimal and common fractions.
Strategies like count on and count back, bridging to friendly numbers, decomposing by place value or other ways, special strategies like constant difference and compensation ALL apply to decimal and common fraction addition and subtraction.
Just like with whole numbers, we want students to pause and consider the numbers in the problem. Which strategy would be the best fit given the numbers and operations involved?
For some problems, using a number line as a model would be an efficient strategy.
Consider: adding and subtracting with decimal fractions that do not have the same place value positions (one number with tenths and another with tenths and hundredths), OR subtracting mixed numbers when regrouping is necessary (convert both mixed numbers to improper fractions OR regroup one whole into a fraction – borrowing – in order to subtract.
Goal: identify and encourage a wide range of strategies so that students have many tools to use when solving problems. Efficiency lies in the mind of the learner. Promote number sense and mental math.

4. Learning targets
Understand the purpose of and difference between structured and open number lines
Identify big ideas and key strategies students use with number lines
Gain skills with using an open number line for operations (addition and subtraction)
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
This is where is really hit us! Second grade, using the number line to add and subtract within 100

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
We knew it was worth pursuing as the number line is a model that endures into the upper grades and beyond. We had to figure this out.
What are visual fraction models?
Click to make the glossary definition appear…

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
An enduring model throughout students’ education
This is what we eventually realized and have since tried to communicate to teachers.
Much of these points are made in the article, The Empty Number Line…. (show article)

8. Structured number lines: relative position
Model for comparing and rounding numbers
The distance between marks is important (increments)
CCSS
3.NBT.A.1 – round numbers to nearest 10 or 100
3.NF.A.2 – understand a fraction on a number line; represent fractions on a number line diagram
4.NBT.A.1-3 – digits are ten times more as we move place value; compare and round multi-digit numbers
4.NF.C.6 – use decimal notation for fractions with denominators of 10 or 100….locate 0.62 on a number line diagram
5.NBT.A.1 – digits are ten times more or ten times less as we move left or right with place value; round decimals to any place
What is the interval in each of these number lines?
Where is 325 on each number line?
1,000

9. Structured number lines: relative position
Where is 325 on each number line?
What is the interval in each of these number lines?
Where is 325 on each number line?
1,000

10. Relative position Grade 2 Module 1 Lesson 5
Have participants do the Relative Position task at their tables. Have a short debrief discussing how they determined where to place numbers or how they decided the value of the mark.

11. Fraction models Length model Number line model
Debrief if needed from Relative Position task of locating ½
Length model: ribbon, licorice, string, lumber, pepperoni stick
One dimension – students impose/visualize cuts or partitions in one way (top to bottom) compared to area models with multi-dimensional cuts (a cake can be cut top to bottom and left to right)
Number line: convention that we look at distance from zero to one whole. We then partition how we want (halves, thirds, fourths, etc.) and each increment is equal to that distance from zero. We can make repeated hops of ½ along a number line but there is only one unique location for the fraction ½. Same as with whole numbers – 1 and 10 have one unique location but we make jumps of 1 and 10 from any point.

12. Whole number operations on a number line
Addition
Subtraction

13. Complete number line Partial number line Open number line
Notice the 3 versions of the number line, turn to talk with your partner, what do you notice is the difference? What kind of understanding will be necessary for students to be able to use each one? How do they support each other as they evolve?
It is important to move students past the complete number line…otherwise some depend on count- all.
We would argue that the open number line is the best model for operations - more flexible (fine motor) and promotes more strategies (beyond count by one). However, efficiency is in the mind of the learner – listen carefully to your students’ descriptions of their mental strategies. Which type of number line will best support them?
Open number line

14. Big ideas & strategies Commutative Property – Think BIG, count small
might be easier as
Special strategies –
might be easier as
compensation
64 – 47, add 3 to both to get 67 – 50
constant difference
These apply to common and decimal fractions!
Special strategies:
compensation (in addition) , take 3 from 34 and add to 57 to make
Constant difference (add three to each number) 64 – 47 is the same as 67 – 50

15. Big ideas & strategies: whole numbers
Bridging across tens or hundreds
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

16. Big ideas & strategies: decimal and common fractions
Bridging across whole numbers or place value
Numbers 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

17. Big ideas & strategies: whole numbers
Count On and Count Back
Splitting or breaking apart by place value
Counting on and back in jumps of 10, both on and off the decade
Explain bridging – composing a new place value (i.e., tens or hundreds). For example, in 47 add 5, a new ten is composed when three ones are added to the seven ones (we’ve bridged into a new decade)
No bridging – 13 add 5, or 52 add 37 (think add 2 + 7, , 89)

18. Big ideas & strategies: decimal and common fractions
Count On and Count Back
Splitting or breaking apart by whole number and fractional part
Counting on and back in jumps of whole numbers or fractions, from a whole number or fraction
Explain bridging – composing a new place value (i.e., tens or hundreds). For example, in 47 add 5, a new ten is composed when three ones are added to the seven ones (we’ve bridged into a new decade)

19. Using the number line
Just try these three problems. We’ll move to the others in just a little while.

20. Share your thinking - Tell why… you made the size of your jumps
you landed on certain numbers
you started with one number versus the other
you counted on or counted back
you used any special strategies

21. 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
This is the same for common and decimal fractions!
Solve 65 – 38 both ways on number line in page protector

22. Operations with fractions and decimals on a number line
Addition
Subtraction

23. Big ideas & strategies Commutative Property – Think BIG, count small
might be easier as
Special strategies –
might be easier as
compensation
64 – 47, add 3 to both to get 67 – 50
constant difference
These apply to common and decimal fractions!
Special strategies:
compensation (in addition) , take 3 from 34 and add to 57 to make
Constant difference (add three to each number) 64 – 47 is the same as 67 – 50

24. Compensation & constant difference
Try these out on the back of your packet
3 5/ /8, build up to 4 by adding 3/8 from the other addend, /8 = 6 4/8, or take 1/8 to build up to 3, 3 4/8 + 3 = 6 4/8
6 1/3 – 1 2/3, add 1/3 to each, 6 2/3 – 2 = 4 2/3

25. Big ideas & strategies: decimal and common fractions
Bridging across whole numbers or place value
Numbers 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

26. Number line strategies: decimal and common fractions
Count On and Count Back
Splitting or breaking apart by whole number and fractional part
Counting on and back in jumps of whole numbers or fractions, from a whole number or fraction
Explain bridging – composing a new place value (i.e., tens or hundreds). For example, in 47 add 5, a new ten is composed when three ones are added to the seven ones (we’ve bridged into a new decade)

27. Using the number line

28. Share your thinking - Tell why… you made the size of your jumps
you landed on certain numbers
you started with one number versus the other
you counted on or counted back
you used any special strategies

29. 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
This is the same for common and decimal fractions!
Solve 65 – 38 both ways on number line in page protector

30. An example of where the answer is – where you land
Started at 3.9, jumped back the 1.2 to land on the answer of 2.7

31. An example of where the answer is – the total of your jumps
Grade 4 Lesson 7.10
Finding the difference in weight. The answer is in the total of the jumps:
2/ ¾ = 1 5/4 = 2 ¼

32. familiar ways to subtract mixed numbers
Improper Fractions
Regroup one whole
Let’s look at one of the problems you worked on….
How did we learn this?
Knowing your fractions as numbers (in regrouping) Builds off whole number thinking
Improper fraction is following a procedure

33. Grade 4 Module 7 Lesson 11
Difference between two mixed numbers
Consider the way we typically teach this
3 4/8 – 1 7/8
28/8 – 15/8 = 13/8 = 1 5/8
Where can kids go wrong?
Or, decompose one whole
2 12/8 – 1 7/8 = 1 5/8

34. More visuals – in your packet
Grade 5 Module 5 Lesson 5

35. More visu
Grade 5 Module 7 Lesson 5

36. 2 + 1 +
More visuals in your packet

37. Learning targets
Understand the purpose of and difference between structured and open number lines
Identify big ideas and key strategies students use with number lines
Gain skills with using an open number line for operations (addition and subtraction)
Consider classroom implications for your setting

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40. 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
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