Addition

1
Addition
2
Terminology

• Be sure to know the following
• Addend
• Missing Addend
• Commutative Property of Addition
• Associative Property of Addition
• Identity Element for Addition
• Equality Rule

3
What are the Preskills?

• Beginning Stage?
• Multi-digit Addition?

4
Beginning StageIntroducing the Concept

• Why use semi-concrete objects (lines)?
• Why teach the equality rule?
• How do you teach the equality rule?
• Format 7.1

5
Beginning Addition

• What are these?
• Addition the Slow Way
• Missing Addend Addition
• Addition the Fast Way

6
Addition the Slow Way

• What are the preskills?

7
Addition the Slow Way

• How? Format 7.2
• Students read the equation
• Students state the equality rule
• Students draw lines for first and second addend
• Students count all the lines on this side
• Students use the equality rule and draw the same
  number of lines on the other side
• Students write the numeral for the lines on the
  other side

8
Addition the Slow Way

• What examples should one include?

9
Missing Addend AdditionFormat 7.3

1. Start with the side that tells how many lines to
   draw (the box does not tell how many lines to
   draw)
2. Draw lines on that side
3. Draw lines on the other sidefor numeral and
   lines under the box to make the sides equal
4. The lines under the box tell you what numeral to
   write in the box

10
Addition the Fast WayFormat 7.4

• How is this different from the slow way?

11
Addition the Fast WayFormat 7.4

• When are the students ready for addition the fast
  way?
• What potential pattern of errors might the
  students make?
• How do you remedy this error?

12
Sequencing

• When can you begin subtraction (concept)?
• When can you start addition facts instruction?

13
Diagnosis and Remediation4 Steps

• Diagnosis Analyze pattern of errors if
  necessary ask student to solve a problem
  thinking aloud
• Determine type of pattern of errors(component-skil
  l or strategy) (Later fact errors)
• Determine how to re-teach/remedy
• Determine examples (problems)

14
Diagnosis and Remediation

• What is a component skill pattern of errors?
• How, in general, do you remedy a component error?

15
Diagnosis and Remediation

• What is a pattern of strategy errors?
• What is the remedy for a pattern of strategy
  errors?

16
Multi-digit Addition

• Multi-digit addition without renaming
• Multi-digit addition with renaming
• More that 2 multi-digit addends with renaming

17
Multi-digit Addition without Renaming

• When can these problems be introduced?
• How?
• Students read the problem
• Teacher tells students that we start adding in
  the ones column and then the tens (Why?)
• Students write the answer in each column

18
Multi-digit Addition with Renaming

• What are the preskills?

19
Multi-digit Addition with Renaming

• Adding three single-digit numbersFormat 7.5
• What are the example selection guidelines for
  these problems?

20
Multi-digit Addition with Renaming

• Format 7.6
• Students read the problem
• Identify where to start adding (ones)
• Add the ones and determine if they must rename
• Use expanded notation to determine the number for
  the tens and ones column
• Write the renamed number and ones number
• Add the first two numbers in then tens, then add
  the next number to the sum
• Write the tens number

21
Multi-digit Addition with Renaming

• Format 7.6
• What is the common error?
• What should the teacher do?

22
Multi-digit Addition with Renaming

• Format 7.6
• Example selection for Structured Board and
  Structured Worksheet?
• Example selection for Less Structured Worksheet?

23
3 or More Multi-digit Addends with Renaming

• Why are these particularly difficult?

24
3 or More Multi-digit Addends with Renaming

• How are the complex addition facts sequenced?

25
Diagnosis and Remediation4 Steps

• Diagnosis Analyze pattern of errors if
  necessary ask student to solve a problem
  thinking aloud
• Determine type of pattern of errors (fact,
  component, or strategy)
• Determine how to re-teach/remedy
• Determine examples (problems)

26
Pattern of Errors--Facts

• Most common
• Pattern of errorswhat do you look for?
• How do you remedy missing the same fact?
• How do you remedy inconsistent fact errors?

27
Pattern of ErrorsComponent Skill

• Example Carrying the wrong number
• Remediation (Go back to teaching the component
  skill)
• Reteach expanding notation for the total in the
  ones column
• Practice examples can have a box for the carried
  number and ones number
• Practice examples should include problems with
  and without renaming

28
Pattern of ErrorsStrategies

• Example Not regrouping
• Reteach For all strategy errors reteach the
  format for that particular strategy
• Examples Structured board, structured
  worksheet, then less structured.
• Then a worksheet similar to original

29
Diagnosis and Remediation

• Figure 7.2
• What are the 4 steps?

30
Subtraction
31
Subtraction

• First Stageconceptual and simple problems
• Multi-digit stage3 types of column subtraction
• without borrowing,
• simple borrowing problems, and
• complex with multiple borrowing and/or zero

32
Introducing the Concept of Subtraction

• Conceptsemi concrete
• Strategy subtracting lines

33
Introducing the Concept of Subtraction

• How do students use the crossing-out strategy?
• 6 4 ?
• 1)
• 2)
• 3)

34
Introducing the Concept of Subtraction

• Example selection
• Format 8.1 What is the difference between the
  examples in the structured worksheet and the less
  structured worksheet? Why?

35
Introducing the Concept of Subtraction

• Missing Subtrahend Problems
• What are they?
• When do you teach them?
• How do you teach them?

36
Introducing the Concept of Subtraction

• Missing Subtrahend Problems
• 9 ? 4
• Read the problem
• Draw lines under minuend (first number)
• Students figure out what number them must end up
  with
• Students circle the number of lines that they
  must end up with
• Students cross out (minus) the lines that are not
  circled
• Students count the number of crossed outlines and
  put that number in the box

37
Diagnosis and Remediation

• What are the three classes of error diagnoses?

38
Diagnosis and Remediation

• What are 4 steps in diagnosis and remediation
  (Kinders)
• Hypothesis of error pattern, confirm with through
  student interview
• Identify class of errorfact, component,
  strategy
• Identify how you would reteach
• Describe the examples that you would use when
  reteaching and after (return to original
  worksheet problem types)

39
Diagnosis and Remediation

• What is a common component error on worksheets?
• How do you remediate this?

40
Multi-digit Subtraction Stage

• Column subtraction without renaming
• Subtraction with renaming
• Complex renaming problems

41
Subtraction with Renaming

• Preskills?

42
Subtraction with Renaming

• Format 8.2concept of regrouping (semi concrete)

43
Subtraction with Renaming

• Format 8.3
• Part A
• What is the rule?
• Example selection?
• Part B
• Borrowing component skill

44
Subtraction with Renaming

• Format 8.3 Ccomputation summary
• Read the problem
• Determine if we must rename
• Borrow the ten and put it with the ones
• Subtract the ones column
• Subtract the tens column

45
Subtraction with Renaming

• Format 8.3
• What types of problems should one include on less
  structured?

46
Subtraction with renaming

• Renaming from tens
• ¾ subtraction ½ require renaming
• ¼ addition
• Renaming from 100s
• Mostly subtraction
• ½ rename from 100s
• ¼ rename from 10s
• ¼ no renaming

47
Complex Renaming Problems

• Problems requiring renaming more than once
  (without zeros)
• Possible errors?

48
Complex Renaming Problems

• Problems with zeros
• Strategy?
• Preskill?
• Format 8.5 Astructured board, Bstructured
  worksheet, Cless-structured worksheet

49
Complex Renaming Problems

• Format 8.5 Cless-structured worksheet
• What are the example selection guidelines?

50
Diagnosis of Errors

• First, specify the error pattern
• Next, identify if this is a fact, component, or
  strategy error
• See examples on page 129-131

51
Remediation of Errors

• Specify specifically what the teacher would
  do/say in reteaching (remediation)
• Determine examples that would be used in
  reteaching (remediation)
• See page 131

52
Multiplication
53
Review

• What is the difference between a correction and a
  diagnosis and remediation?
• What are the 3 types or classes of diagnoses?
• Describe each.

54
Two Stages of Multiplication

• What are they?
• What are the preskills for introducing
  multiplication?
• What are the preskills for the second stage?

55
Multiplication Introducing the Concept

• Single-digit Multiplication
• Missing-Factor Multiplication
• Diagnosis and Remediation

56
Multiplication Introducing the Concept

• Preskills?
• Format 9.1

57
Multiplication Introducing the Concept

• Steps in Format 9.1
• Picture demonstration
• Reading problems (as count bys)
• Structured board solving problemcounting by a
  number x timesand structured worksheet
• Less structured worksheet (What type of problems
  are included?)

58
Format 9.1

• What predictable problems will students have with
  saying the numbers as they touch their extended
  fingers?
• What do you do?

59
Missing-factor Multiplication

• What is this a preskill for?
• Steps 5 x ? 30
• Count by 5
• Hold up a finger as you count until you get to 30
• Count the number of fingers extendedput that in
  the box

60
Format 9.2 Missing-factor Multiplicaton

• Structured Board and Structure WorksheetWhat
  types of problems?
• Independent WorksheetWhat types of problems?

61
Multiplication Introducing the Concept

• Diagnosis and Remediation
• Will there be fact errors? Why?
• What types of component errors might we expect?
  (Figure 9.3, page 148)

62
Multiplication Introducing the Concept

• Remediation for component errors?
• Skip counting incorrectly
• Consistently off by one count-by number

63
Multiplication Introducing the Concept

• Remediation for strategy errors?
• Confuse addition and multiplication
• Confuse regular multiplication and missing factor
  multiplication

64
Multi-digit Multiplication

• Algorithms based on distributive property of
  multiplication.
• 5 x 67
• What are the long and short algorithms?

65
Multi-digit Multiplication

• What are the preskills?
• How is each preskill taught?

66
Multi-digit Multiplication

• Sequence
• Single digit x multiple digit without renaming,
• 24
• x 2
• Single digit x multiple digit with renaming,
• 24
• x 3
• Format 9.3

67
Multi-digit Multiplication

• Sequence cont.
• Two-digit x two-digit
• Two-digit x three-digit

68
Multi-digit Multiplication

• Format 9.4 Steps
• Part AOrder of multiplication
• Part BStructure boardmodeling the algorithm
  (What is critical in this model?)
• Part CStructured worksheet
• Part DLess structured worksheet (What problem
  types?

69
Multi-digit Multiplication

• Diagnosis and Remediation
• Can we have fact errors? Why?
• When do you remediate fact errors? How?
• What are common component errors?

70
Division
71
Review

• What are common instructional features across the
  operations (addition, subtraction,
  multiplication, and division)?
• What is the identity property?
• What is the commutative property?
• What is the associative property?
• What is the distributive property?

72
Division

• What are the two stages of instruction?
• What are the preskills for introducing division?

73
Division Stage One

• Problems without remainders
• Format 10.1
• A Translation of problem (How do you translate
  problems?)
• B Structured boardworking the problem by
  dividing lines and writing the answer in the
  correct place
• C D Worksheets with lines drawn

74
Division Stage One

• Problems with remainders
• Why are these important?
• Format 10.2
• A Demonstrate with lines when another group
  cannot be formedother lines are the remainder
• B C Worksheets with line showing students
  where to write stuff (that is what they call it
  in higher mathematics!)

75
Division Stage One

• Remainder Factsmentally computing facts
  including remainders
• Format 10.3
• A Teacher presents a diagram circling multiples
  and models how many times the multiple goes into
  various numbers with a remainder
• B Teacher tests students using the diagram

76
Division Stage One

• Remainder Factsmentally computing facts
  including remainders
• Format 10.3
• C Worksheetstudents determine the quotient,
  multiply and subtract to determine the remainder
• Worksheet follows the sequence of fact
  introduction, includes earlier sets and some
  problems that do not have remaindersWHY?and
  some with quotients of zero.

77
Division Stage OneDiagnosis and Remediation

• Fact errors
• Component errors
• Quotient that is too small or too large
• Subtraction error
• Placing remainder and quotient wrong

78
Division Stage OneDiagnosis and Remediation

• How do you remediate these component errors?
• Quotient that is too small or too large
• Subtraction error
• Placing remainder and quotient wrong

79
Division Stage OneDiagnosis and Remediation

• Quotient that is too small or too large
• Format 10.4, compare remainder to the divisor or
  Format 10.5, showing that if they cannot subtract
  the answer is too big, then return to original
  worksheet

80
Division Stage OneDiagnosis and Remediation

• Subtraction error
• Reteach subtraction (with regrouping)provide
  division problems partially completedhave
  subtract. Then return to the original worksheet
  and complete full problems
• Placing remainder and quotient wrong
• Reteach where to put answer and remainder,
  structured worksheet focusing on placement of
  quotient and remainder, then return to original
  worksheet

81
Multi-digit Division Problems

• What are the long and short forms?
• Which is used most commonly?
• What are the preskills?
• What determines the difficulty of these problems?

82
Multi-digit Division ProblemsTwo-digit Quotients

• What are the steps in the short form algorithm?

83
Multi-digit Division ProblemsTwo-digit Quotients

• What are the steps in the short form algorithm?
• S read the problem
• S underline the part they work first
• S determine and write answer to first part
• S multiply, subtract and bring down
• S read new problem and determine answer
• S write answer over digit just brought down
• S multiply and subtract to determine remainder
• S say the problem and answer

84
Multi-digit Division Problems

• Demonstrate Format 10.6
• What is the critical part when there is a zero in
  the quotient?
• How can students self-check their division?

85
Multi-digit Division ProblemsTwo-digit Divisors

• Lengthy and complex algorithm!
• What are the steps in the algorithm suggested by
  our text?

86
Multi-digit Division ProblemsTwo-digit Divisors

1. S read the problem
2. S underlines the part to work first
3. S writes the rounded-off problem
4. S computes the division problem using the answer
   from the rounded-off problem
5. S multiplies and subtracts (if possible)
6. S adjusts the quotient if needed (if you cant
   subtract make the answer smaller, if the
   remainder is too big, make the answer bigger
7. S completes the problem and reads the problem and
   answer

87
Multi-digit Division ProblemsTwo-digit Divisors

• What additional preskills (in addition to
  single-digit divisor problems) do students need?
• 10.8A Multiplying horizontally
• Model 10.8B (rounded-off problems) C (entire
  strategy)

88
Multi-digit Division ProblemsTwo-digit Divisors

• What do you do when the estimated quotient does
  not yield the correct answer?

89
Multi-digit Division ProblemsTwo-digit Divisors

• Format 10.9
• Rule If you cant subtract, make the answer
  smaller if the remainder is too big, make the
  answer bigger