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 • Start with the side that tells how many lines to draw (the box does not tell how many lines to draw) • Draw lines on that side • Draw lines on the other side—for numeral and lines under the box to make the sides equal • The lines under the box tell you what numeral to write in the box

10. Addition the “Fast Way”Format 7.4 • How is this different from the “slow way”?

11. Addition the “Fast Way”Format 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-skill 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 numbers—Format 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 errors—what do you look for? • How do you remedy missing the same fact? • How do you remedy inconsistent fact errors?

27. Pattern of Errors—Component 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 Errors—Strategies • 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 Stage—conceptual and simple problems • Multi-digit stage—3 types of column subtraction • without “borrowing”, • simple borrowing problems, and • complex with multiple borrowing and/or zero

32. Introducing the Concept of Subtraction • Concept—semi 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 (Kinder’s) • Hypothesis of error pattern, confirm with through student interview • Identify “class” of error—fact, 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 8.2—concept 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 C—computation 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: A—structured board, B—structured worksheet, C—less-structured worksheet

49. Complex Renaming Problems • Format 8.5: C—less-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