Fractions

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
• OperationsAdding and Subtracting
• OperationsMultiplying
• OperationsDividing

4
Fractions Chapter

• Terminology (page 241)
• Numerator
• Denominator
• Proper fraction
• Mixed number
• GCF (Greatest Common Factor)
• LCD (Least Common Denominator)

5
Fractions Chapter

• Why are fractions so difficult for students?

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 ,
• 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 AnalysisPart-Whole

• Number line introductionFormat 12.1
• Diagram introductionFormat 12.2

12
Fractions AnalysisWriting 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

13
Fractions AnalysisDrawing Diagrams

• Drawing diagrams to represent fractions
• Teacher models how to divide circles into equal
  parts
• Worksheet, dividing circles and shading parts used

14
Fractions AnalysisDecoding Fractions

• Decoding fractions (traditional)
• Format 12. 4
• Teacher models reading fractions and tests
  students

15
Fractions AnalysisMore, Less, Equal to One

• Fractions that are more than one, less than one
  and equal to one
• Format 12. 5
• Part A Pictures
• Part B Rules
• Part C Structured Worksheet
• Model 12.5, B

16
Fractions AnalysisMixed Numbers

• Reading and writing mixed numbers
• Format 12. 6
• Part A Picture demonstration
• Part B Teacher models and tests reading mixed
  fractions
• Part C Writing numbers

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?

19
Rewriting FractionsEquivalent Fractions

• What is the basic strategy?
• What are the specific preskills?

3 4 12
3 3 9 4 3 12
x
20
Rewriting FractionsEquivalent 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 FractionsEquivalent Fractions

• Format 12.8 teaches
• Part A shows the concept of equivalent fractions
• Part B teaches the ruleWhen 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 FractionsEquivalent Fractions

• What are the example selection guidelines for
  12.8?

23
Rewriting FractionsReducing Fractions

• Two stages
• 1. Introducing greatest-common-factor (What does
  GCF mean?)
• Reducing fractions when GCF is difficult to
  determine

24
Rewriting FractionsReducing Fractions

• What are the preskills?

25
Rewriting FractionsReducing Fractions

• What are the factors of
• 15
• 12
• 9
• 36

26
Rewriting FractionsReducing Fractions

• Greatest-common-factor
• Format 12.10define GCF and lead students in
  finding

27
Rewriting FractionsReducing 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?

28
Rewriting FractionsReducing Fractions

• What strategy do we teach for reducing fractions
  with big numbers (those with difficult GCF)?

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

31
Rewriting Fractions Converting Mixed Numbers and
Improper Fractions

• Format 12.12
• What are the example selection guidelines?

32
Rewriting Fractions Converting Mixed Numbers and
Improper Fractions

• What is the procedure for converting mixed
  numbers to improper fractions?
• Format 12.13
• Part A is converting a whole number
• Part B is converting a mixed number

33
OperationsAdding 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

34
OperationsAdding and Subtracting

• Like denominators
• Format 12.14 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
  denominatorswhy?

35
OperationsAdding and Subtracting

• Fractions with unlike denominators
• What are the preskills?

36
OperationsAdding and Subtracting

• Fractions with unlike denominators
• Format 12.15 teaches students to find the least
  common multiple by skip counting for each
  denominator and selecting the smallest common
  number.
• 3 1
• 4 5

37
OperationsAdding 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.

38
OperationsAdding and Subtracting

• Adding and subtracting fractions with unlike
  denominators, Format 12.16
• What are the example selection guidelines?

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

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)

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 DecimalsReading 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 DecimalsReading 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 DecimalsWriting 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 DecimalsRepresenting
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
  remainderstudents 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, 13.9multiplying 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 cant 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