1. Fractions

2. Proportional Reasoning
Number Sense
Are 1.7 and 1/7 the same or are they different?
Are 0.5 and 6/12 the same or different?
Order the following numbers from largest to smallest: 0.48, 5/8, 14/13, 0.99.
What is 5 + ½ +0.5 = ?
Are there any fractions between 2/5 and 3/5?
Are there any decimals between 2/5 and 3/5?
Are there any decimals between 0.46 and 0.47?
Are there any fractions between 0.46 and 0.47?

3. Fractions Chapter Fractions Analysis Rewriting Fractions
Operations—Adding and Subtracting
Operations—Multiplying
Operations—Dividing

4. Fractions Chapter
Terminology (page 241) Numerator Denominator Proper fraction Mixed number GCF (Greatest Common Factor) LCD (Least Common Denominator)
Fractions - Definitions
Fractions - division of units into equal-sized segments
numerator - top number in a fraction
denominator - bottom number in a fraction
proper fraction - numerator is less than denominator
improper fraction - numerator is equal or greater than denominator
mixed number - improper fraction expressed as a whole number and a fraction
greatest common factor - largest factor of both numerator and denominator
lowest common denominator - lowest common number a set of denominators can be multiplied to reach (e.g., least common multiple)

5. Fractions Chapter Why are fractions so difficult for students?
unfamiliar
applies to a range of operations
incompatibility of units
requires new strategies

6. Today Jigsaw activity 4 groups Fraction analysis Rewriting fractions
Adding/fubtracting fractions
Multiplying/dividing fractions

7. Today Jigsaw cont. 30 minutes to identify Preskills
General teaching procedures
Diagnosis and remediation
Corrections
Example selection
Develop a 5 min. activity for the group

8. Today
Present to group
10 minutes to present
5 minutes for activity

9. Fractions Analysis
Instruction begins around second grade and includes:
Part/whole
Writing a faction for a diagram
Reading fractions
Determining if fraction is >, <, = to one whole
Reading mixed fractions

10. Fractions Analysis Important features:
Proper and improper introduced concurrently
Initial instruction is to interpret a fraction
Initially fraction instruction is figures divided into parts

11. Fractions Analysis Part-Whole
Number line introduction—Format 12.1
Diagram introduction—Format 12.2
Distinguish between number of whole units and number of parts in each unit
Example selection
Include different number of circles (wholes)
Vary the number of parts in each whole
Practice several days to mastery before moving on

12. Fractions Analysis Writing Fractions
Format 12.3
Part A: Students learn parts of fraction
B & C: Daily practice for several weeks
Model Format 12.3
Example selection guidelines
Vary the number of parts in each whole, number of wholes, and number of parts shaded
Include proper and improper fractions
Translating diagram into numerical fractions
Figure 13.4
In Part A, students learn that top number tells how many parts are used, bottom number tells how many parts in the whole (total)
Guidelines
Vary the number of parts and number of wholes across examples
less than one unit
more than one unit
equal to one unit
units that are not divided (teach & introduce a week later)
Preskill for converting whole numbers to fractions
Include improper fractions
Daily practice for several weeks followed by intermittent review

13. Fractions Analysis Drawing Diagrams
Drawing diagrams to represent fractions
Teacher models how to divide circles into equal parts
Worksheet, dividing circles and shading parts used
Translate numerical fractions to diagrams to reinforce conceptual understanding
Introduce a week or two after fraction analysis
Teacher models and guides for several days then introduces worksheets (fig. 13.5)
Guidelines
Limit fractions to numbers 2, 3, or 4
Keep whole numbers constant to avoid misrule that number of wholes has anything to do with the fraction

14. Fractions Analysis Decoding Fractions
Decoding fractions (traditional) Format Teacher models reading fractions and tests students
Read fractions in traditional way
Figure 13.6
Teacher models and then tests
Errors are corrected with model, test, alternating test
Examples
Introduce irregular fractions (2, 3, 5) later and separately
Include 1/4 examples where 1 is the numerator to show plurality
(e.g., one-fifth versus two-fifths)

15. Fractions Analysis More, Less, Equal to One
Fractions that are more than one, less than one and equal to one Format Part A: Pictures Part B: Rules Part C: Structured Worksheet Model 12.5, B
Preskill to rewriting improper fractions
Introduce after students can decode fractions traditionally
Figure 13.7
Part A: Pictorial demonstrations
Part B: Series of rules for determining if fraction is more, less, or equal to a whole.
Example Selection
1/3 numerator and denominator the same
1/3 numerator greater than denominator
1/3 numerator less than denominator

16. Fractions Analysis Mixed Numbers
Reading and writing mixed numbers Format Part A: Picture demonstration Part B: Teacher models and tests reading mixed fractions Part C: Writing numbers
Introduce after students can identify whether fraction is more, less, or equal to whole
Figure 13.8
Part A: Pictorial demonstration - express diagram of improper fraction as a mixed number
Part B: Read mixed fractions
Part C: Write mixed fractions

17. Rewriting Fractions
Identify missing numerator in an equivalent fraction
What are/is
Equivalent fractions
Reducing fractions
Converting mixed fractions to improper and vice versa

18. Rewriting Fractions What are the general preskills?
Students must learn
Computational strategies needed to rewrite fractions
Conceptual underpinning of “equivalency” governing the strategies

19. Rewriting Fractions Equivalent Fractions
What is the basic strategy? What are the specific preskills?
Basic strategy is to multiply number by a fraction that is equal to one and will result in a denominator of the size given.
Preskills
Terms numerator and denominator
Multiplication of two fractions
Introduced in third grade
Constructing fractions equal to one whole
Introduce two weeks before equivalent fraction problems
See Figure 13.9
3 =
= 9 x

20. Rewriting Fractions Equivalent Fractions
Format 12.7 teaches the rule for factions equal to 1: When the top number is the same as the bottom number, the fraction equals 1.

21. Rewriting Fractions Equivalent Fractions
Format 12.8 teaches:
Part A shows the concept of equivalent fractions
Part B teaches the rule—When you multiply by a fraction that equals 1, the answer equals the number that you start with.
Part C is the structured presentation of the strategy

22. Rewriting Fractions Equivalent Fractions
What are the example selection guidelines for 12.8?
Example Selection
denominator of first number must be a number that can be multiplied by a whole number to end with the second denominator
numbers appearing in parenthesis (to equal 1) should vary across examples
all numbers should require multiplication (2nd denominator should be bigger)

23. Rewriting Fractions Reducing Fractions
Two stages:
1. Introducing greatest-common-factor (What does GCF mean?)
Reducing fractions when GCF is difficult to determine
Two stages
Greatest Common Factor (GCF)
Reduce fraction pulling out GCF
Only works well when its easy to determine the GCF
Reducing when GCF is hard to determine

24. Rewriting Fractions Reducing Fractions
What are the preskills?
Preskills
Determine all possible factors of a number
Determine GCF

25. Rewriting Fractions Reducing Fractions
What are the factors of:

26. Rewriting Fractions Reducing Fractions
Greatest-common-factor Format 12.10—define GCF and lead students in finding
Part A: Defines term factor
Part B: Presents strategy for figuring out all possible factors
Part C: Worksheet
Practice each target number daily for several weeks
“What is the greatest common factor of x and y?”
Present format for a week, then move to worksheets
Common error -- identifying factors that are not the GCF
Example Selection
In 1/2 problems GCF should be the smaller of the two target numbers (e.g., 6, 18; 4, 8) to prevent misrule that smaller number is never the GCF
Limit examples to factors that students know
Initially use factors under 20

27. Rewriting Fractions Reducing Fractions
Format 12.11
Part A, teacher presents the strategy (model)
Part B, the structured worksheet
What are the example selection guidelines for Format 12.11?
Introduced when students can determine GCF for numbers under 20
Figure 13.13
Steps:
Pull out GCF
Turn GCF into fraction = 1
Use missing factor multiplication
Example Selection
Begin with numbers students can factor
1/3 numerators = GCF
1/3 fractions already in simplest terms
After several weeks introduce larger numbers by repeated factoring

28. Rewriting Fractions Reducing Fractions
What strategy do we teach for reducing fractions with big numbers (those with difficult GCF)?
Teach students to reduce fractions by repeatedly pulling out common factors
Provide students with clues
If both numbers are even, reduce
If numbers end in 5 or zero, reduce
45 = (5) 9 = (3) 3 = 3
75 (5) 15 (3) 5 5

29. Rewriting Fractions: Converting Mixed Numbers and Improper Fractions
What is a mixed number? What is an improper fraction?

30. Rewriting Fractions: Converting Mixed Numbers and Improper Fractions
What is the procedure for converting an improper fraction to a mixed number? Format 12.12: Part A shows the concept in pictures Part B teaches the strategy Part C is a worksheet
Convert improper fraction to mixed number by dividing
13/4 = 3 1/4
Introduce improper to mixed conversion early 4th grade
Figure 13.14
Part A: Pictorial demonstration showing how to construct a diagram to figure out how many whole units an improper fraction equals
Part B: Structured board presenting strategy
Part C: Structured worksheet with division prompt
Part D: Less structured

31. Rewriting Fractions: Converting Mixed Numbers and Improper Fractions
Format 12.12: What are the example selection guidelines?
Example Selection
1/2 fractions requiring translation to a mixed number
1/4 fractions requiring translation to a whole number
1/4 proper fractions
After several weeks introduce problems in which they must first reduce and then translate to a mixed number

32. Rewriting Fractions: Converting Mixed Numbers and Improper Fractions
What is the procedure for converting mixed numbers to improper fractions? Format Part A is converting a whole number Part B is converting a mixed number
Convert mixed number to improper fraction by multiplying denominator by whole number and adding the numerator
3 x 4 = = 13/4
Introduce mixed to improper several months after improper to mixed to avoid confusion
Figure 13.15
Part A: Component skill of translating any whole number into an improper fraction
Part B: Structured board for converting mixed number to improper fraction
Part C: Structured worksheet
Example selection similar to previous conversion (improper to mixed)

33. Operations—Adding and Subtracting
Three basic problem types of addition/subtraction problems
With like denominators
With unlike denominators with easy lowest- common denominators
With unlike denominators and difficult lowest- common denominators
Three types
With like denominators
Introduce in primary grades
Students taught to work problem only across numerators
With unlike denominators - lowest common denominator easy to determine
Introduce in 4th grade
Three part strategy
Figure out lowest common denominator
Rewrite each fraction as an equivalent fraction
Work problem
With unlike denominators - lowest common denominator not easy to determine
Usual involve large numbers
Introduced in junior high
Requires factoring skills
Not covered in DI Math

34. Operations—Adding and Subtracting
Like denominators: Format teaches students (at this point) only to add or subtract fractions in which the whole has the same number of parts. Worksheets should include problems with unlike denominators—why?
Introduce after fraction analysis skills have been taught (2nd, 3rd grade)
Figure 13.16
Part A: Pictorial demonstration demonstrating adding fractions
Part B: Structured board presenting rule that students can only add and subtract fractions where the whole has the same number of parts
Parts C & D: Structured and Less Structured Worksheets
Example Selection
1/2 like denominators
1/2 unlike denominators
Initially written horizontally
In two weeks include problems written vertically

35. Operations—Adding and Subtracting
Fractions with unlike denominators: What are the preskills?
Preskills
finding the least common multiple of two numbers (Figure 13.18)
Example Selection
In 1/2 numbers, larger number should be a multiple of the smaller number
In 1/2 numbers, larger numbers should not be multiples of the smaller number, but both target numbers are below 10
rewriting a fraction as an equivalent fraction with a given denominator

36. Operations—Adding and Subtracting
Fractions with unlike denominators: Format teaches students to find the least common multiple by skip counting for each denominator and selecting the smallest common number
Figure Basic Strategy
Decide whether can add or not (are denominators the same?)
Determine lowest common multiple
Determine fraction to multiply by to equal least common multiple
Rewrite as equivalent fractions and solve

37. Operations—Adding and Subtracting
Adding and subtracting fractions with unlike denominators, Format 12.16: Part A, teacher demonstrates finding the LCM, multiplying both fractions by a fraction of 1, and then adding. Part B and C are worksheets.
Figure 13.19
Part A: Structured board to teach strategy
Parts B & C: Structured and less structured worksheets

38. Operations—Adding and Subtracting
Adding and subtracting fractions with unlike denominators, Format 12.16: What are the example selection guidelines?
Example selection
1/2 problems in which only one number needs to be rewritten (i.e., smaller number is a multiple of the larger number)
1/2 problems in which both numbers need to be rewritten (i.e., smaller number is not a multiple of larger number)
Include problems with like denominators
Initially present problems horizontally, later add vertically aligned problems

39. Operations--Multiplication
Three problem types: 1. Multiplying proper fractions 2. Multiplying a fraction and a whole number 3. Multiplying one or more mixed numbers
Three Types
Multiply two proper fractions
Figure 13.22
Multiply top numbers, then bottom numbers
Multiply a proper fraction by a whole number (Figure 13.23)
Convert whole number to fraction and solve
Multiply one or more mixed problems (Figure 13.24)
Early strategy - convert to improper fraction and solve
Later strategy - use distributive property (multiply whole numbers, then multiply fractions)

40. Operations--Multiplication
Multiplying proper fractions: Students are taught the simple rule: Work the top times the top and the bottom times the bottom. Include multiplying proper fractions with addition and subtraction of fractions on worksheets.

41. Operations--Multiplication
Multiplying fractions and whole numbers: Format 12.18: Part A teaches changing a whole number to a fraction: 5 = 5/1 Part B and C worksheets, students change whole number to a fraction, multiply, and covert the answer to a mixed number.

42. Operations--Dividing
Model of the strategy that illustrates the rationale as well as the procedures (not included in the book)
Introduce in 5th grade
Three types
Proper fraction divided by proper fraction
Fraction divided by a whole number
Problems involving mixed numbers
In early grades, teach a rote rule
To divide we invert the second fraction and change the sign to multiplication

43. Operations--Dividing
Preskills: Identity element: 1 x a = a Fraction of one has same number top and bottom: a/a Any number divided by 1 equals that number Multiplying fractions Reciprocals: a x b = ab = 1 b a ab

44. Diagnosis and Remediation
What are the following patterns of errors? What are examples? (Page 268)
Computational
Component-skill
Strategy

45. Diagnosis and Remediation
Expand the remediation for Summary Box 12.4
What skills would you teach (often an isolated skill)
What types of problems would you include in your remediation practice problems?
What types of problems would you include in the final problem set?

46. dECIMALS

47. Decimals 7 areas: Reading and writing decimals and mixed decimals
Converting decimals to equivalent decimals
Adding and subtracting decimals
Multiplying decimals
Rounding off decimals
Dividing decimals
Converting between decimals and fractions

48. Decimals What are the preskills for all decimal areas?
What decimals skills are preskills for all other decimal areas?

49. Reading and Writing Decimals: Reading Tenths and Hundredths
Format 13.1:
Structured board: Teaches the rule that one digit after the decimal tells about tenths and two digits after the decimal tell about hundredths
Structured work sheet: Given a decimal students identify the fraction and visa versa

50. Reading and Writing Decimals: Reading Tenths and Hundredths
Format 13.1:
What example sets should be used for this format?
What critical behavior must the teacher emphasize?

51. Reading and Writing Decimals: Writing Tenths and Hundredths
Format 13.2
Part A: Teacher demonstrates how to write a fraction as a decimal
Part B: Less structured worksheet, given a fraction, writing a decimal
What examples should we include?

52. Reading and Writing Mixed Decimals: Tenths and Hundredths
What must the students be able to do (preskills)?
Format 13.3:
Part A: Teacher demonstrates reading mixed numbers emphasizing and between whole number and decimal
Part B: Given a mixed fraction student writes decimal and/or given words students write the mixed decimal

53. Reading and Writing Decimals: Representing Thousandths
Same procedures as Formats 13.1 and 13.2:
What should the initial examples be like?
After several lessons what examples should be added?

54. Equivalent Decimals What are equivalent decimals?
What are equivalent decimal skills needed for?
Format 13.4:
Part A: Rationale for adding zeros (.3 = .30)
Part B: Structured board for rewriting decimals
Part C: Worksheet with column chart

55. Adding and Subtracting Decimals and Mixed Decimals
Types of problems:
Both numbers have the same number of decimal places
Numbers have different numbers of digits after the decimal point

56. Adding and Subtracting Decimals and Mixed Decimals
Both numbers have the same number of decimal places:
Teach students to align problems vertically
Students write the decimal place directly below other decimal points
Problems oriented horizontally are rewritten vertically and worked as above

57. Adding and Subtracting Decimals and Mixed Decimals
Numbers have different numbers of digits after the decimal point:
What does this require?

58. Adding and Subtracting Decimals and Mixed Decimals
Numbers have different numbers of digits after the decimal point:
Format 13.5
What examples should we include?

59. Rounding Off Decimals What is rounding off a preskill for? Format 13.6
1. Determine how many digits will be after the decimal place
Count the number of digits and draw a line
If the number after the line is 5 or more, “round up”

60. Rounding Off Decimals What are the 3 example selection guidelines?
What type of numbers are particularly hard to round off?

61. Multiplying Decimals
What is the “tricky” part of multiplying decimals?
Format 13.7:
Part A: Teaching students the rule for determining where the decimal point goes in the answer
Part B: Structured worksheet, students determine where to put the decimal point in completed multiplication problems
What examples could be confusing?

62. Dividing Decimals Complex skill
What are the preskills? What must be emphasized in the preskills?

63. Dividing Decimals 4 categories of problems, what are examples of:
No remainder & no conversion
Conversion required to solve without remainder
A remainder that requires rounding
Divisor is a decimal

64. Dividing Decimals No remainder & no conversion:
Rule: Decimal in answer goes directly above the decimal in the number being divided
Teacher leads students
What are the example selection guidelines?

65. Dividing Decimals
Conversion required to solve without remainder— students must rewrite the dividend as an equivalent decimal to avoid a remainder.
Tell students will work until no remainder
Model
What are the example selection guidelines?

66. Dividing Decimals A remainder that requires rounding Format 13.8
Part A: Students read the directions, determine how many digits in the answer, work the problem until they have that many digits, draw a line, work one more digit so that they can round off
Special consideration is given to what type of problems?

67. Dividing Decimals Divisor is a decimal
What must be done to solve these problems?
What is the preskill?

68. Dividing Decimals Divisor is a decimal
Preskill, 13.9—multiplying decimals by multiples of 10
Part A: Rule about moving the decimal
Part B: Structured worksheet
What types of examples should be included?

69. Dividing Decimals Divisor is a decimal Format 13.10
Part A: Teaches that you can’t divide by a decimal, then shows how to divide
Part B: Structured worksheet
What are the two example selection guidelines?

70. Converting Fractions and Decimals
Why teach this?
What are the two preskills?
What is the rule for converting fractions into decimals?
What examples should be included?

71. Diagnosis and Remediation
Review Summary Box 13.1