ADDITION
Terminology
Be sure to know the following: Addend Missing Addend Commutative Property of Addition Associative Property of Addition Identity Element for Addition Equality Rule
What are the Preskills?
Beginning Stage? Multi-digit Addition?
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
Beginning Addition
What are these? Addition the “Slow Way” Missing Addend Addition Addition the “Fast Way”
Addition the “Slow Way”
What are the preskills?
Addition the “Slow Way”
How? Format 7.21. Students read the equation2. Students state the equality rule3. Students draw lines for first and second
addend4. Students count all the lines on “this” side5. Students use the equality rule and draw
the same number of lines on the “other” side
6. Students write the numeral for the lines on the “other side”
Addition the “Slow Way”
What examples should one include?
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 side3. Draw lines on the other side—for
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
Addition the “Fast Way”Format 7.4 How is this different from the “slow
way”?
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?
Sequencing
When can you begin subtraction (concept)?
When can you start addition facts instruction?
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)
Diagnosis and Remediation
What is a component skill pattern of errors?
How, in general, do you remedy a component error?
Diagnosis and Remediation
What is a pattern of strategy errors? What is the remedy for a pattern of
strategy errors?
Multi-digit Addition
Multi-digit addition without renaming Multi-digit addition with renaming More that 2 multi-digit addends with
renaming
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
Multi-digit Addition with Renaming What are the preskills?
Multi-digit Addition with Renaming Adding three single-digit numbers—
Format 7.5 What are the example selection
guidelines for these problems?
Multi-digit Addition with RenamingFormat 7.61. Students read the problem2. Identify where to start adding (ones)3. Add the ones and determine if they must
rename 4. Use expanded notation to determine the
number for the tens and ones column5. Write the renamed number and ones
number6. Add the first two numbers in then tens,
then add the next number to the sum7. Write the tens number
Multi-digit Addition with RenamingFormat 7.6What is the common error?What should the teacher do?
Multi-digit Addition with RenamingFormat 7.6Example selection for Structured Board
and Structured Worksheet?Example selection for Less Structured
Worksheet?
3 or More Multi-digit Addends with Renaming Why are these particularly difficult?
3 or More Multi-digit Addends with Renaming How are the complex addition facts
sequenced?
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)
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?
Pattern of Errors—Component SkillExample: “Carrying” the wrong numberRemediation (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
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
Diagnosis and Remediation
Figure 7.2 What are the 4 steps?
SUBTRACTION
Subtraction
First Stage—conceptual and simple problems
Multi-digit stage—3 types of column subtraction1. without “borrowing”, 2. simple borrowing problems, and 3. complex with multiple borrowing and/or
zero
Introducing the Concept of Subtraction Concept—semi concrete Strategy— “subtracting” lines
Introducing the Concept of Subtraction How do students use the “crossing-out”
strategy?6 – 4 = 1)2)3)
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?
Introducing the Concept of Subtraction Missing Subtrahend Problems
What are they? When do you teach them? How do you teach them?
Introducing the Concept of SubtractionMissing Subtrahend Problems
9 – = 41) Read the problem2) Draw lines under minuend (first number)3) Students figure out what number them must end
up with4) Students circle the number of lines that they
must end up with5) Students cross out (minus) the lines that are not
circled6) Students count the number of crossed outlines
and put that number in the box
Diagnosis and Remediation
What are the three “classes” of error diagnoses?
Diagnosis and Remediation
What are 4 steps in diagnosis and remediation (Kinder’s)
1. Hypothesis of error pattern, confirm with through student interview
2. Identify “class” of error—fact, component, strategy
3. Identify how you would reteach 4. Describe the examples that you would use
when reteaching and after (return to original worksheet problem types)
Diagnosis and Remediation
What is a common component error on worksheets?
How do you remediate this?
Multi-digit Subtraction Stage Column subtraction without renaming Subtraction with renaming Complex renaming problems
Subtraction with Renaming
Preskills?
Subtraction with Renaming
Format 8.2—concept of regrouping (semi concrete)
Subtraction with Renaming
Format 8.3 Part A:
What is the rule? Example selection?
Part B: Borrowing component skill
Subtraction with Renaming
Format 8.3 C—computation summary1. Read the problem2. Determine if we must rename3. Borrow the ten and put it with the ones4. Subtract the ones column5. Subtract the tens column
Subtraction with Renaming
Format 8.3 What types of problems should one
include on less structured?
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
Complex Renaming Problems Problems requiring renaming more than
once (without zeros) Possible errors?
Complex Renaming Problems Problems with zeros: Strategy? Preskill? Format 8.5: A—structured board, B—
structured worksheet, C—less-structured worksheet
Complex Renaming Problems Format 8.5: C—less-structured
worksheet What are the example selection guidelines?
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
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
MULTIPLICATION
Review
What is the difference between a correction and a diagnosis and remediation?
What are the 3 “types or classes” of diagnoses?
Describe each.
Two Stages of Multiplication
What are they? What are the preskills for introducing
multiplication? What are the preskills for the second
stage?
Multiplication Introducing the Concept Single-digit Multiplication Missing-Factor Multiplication Diagnosis and Remediation
Multiplication Introducing the Concept Preskills? Format 9.1
Multiplication Introducing the Concept
Steps in Format 9.11. Picture demonstration2. Reading problems (as count bys)3. Structured board solving problem—
counting by a number x times—and structured worksheet
4. Less structured worksheet (What type of problems are included?)
Format 9.1
What predictable problems will students have with saying the numbers as they touch their extended fingers?
What do you do?
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 extended—put
that in the box
Format 9.2 Missing-factor Multiplicaton
Structured Board and Structure Worksheet—What types of problems?
Independent Worksheet—What types of problems?
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)
Multiplication Introducing the Concept
Remediation for component errors?1. Skip counting incorrectly2. Consistently off by one count-by
number
Multiplication Introducing the Concept
Remediation for strategy errors?1. Confuse addition and multiplication2. Confuse regular multiplication and
missing factor multiplication
Multi-digit Multiplication
Algorithms based on distributive property of multiplication.
5 x 67 =
What are the long and short algorithms?
Multi-digit Multiplication
What are the preskills?How is each preskill taught?
Multi-digit Multiplication
Sequence1. Single digit x multiple digit without
renaming, 24
x 22. Single digit x multiple digit with renaming,
24 x 3
Format 9.3
Multi-digit Multiplication
Sequence cont.3. Two-digit x two-digit4. Two-digit x three-digit
Multi-digit Multiplication
Format 9.4 StepsPart A—Order of multiplicationPart B—Structure board—modeling the
algorithm (What is critical in this model?)
Part C—Structured worksheetPart D—Less structured worksheet (What
problem types?
Multi-digit Multiplication
Diagnosis and Remediation Can we have fact errors? Why? When do you remediate fact errors?
How? What are common component errors?
DIVISION
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?
Division
What are the two stages of instruction? What are the preskills for introducing
division?
Division Stage One
Problems without remainders Format 10.1
A: Translation of problem (How do you translate problems?)
B: Structured board—working the problem by dividing lines and writing the answer in the correct place
C & D: Worksheets with lines drawn
Division Stage One
Problems with remainders Why are these important? Format 10.2
A: Demonstrate with lines when another group cannot be formed—other lines are the remainder
B & C: Worksheets with line showing students where to write “stuff” (that is what they call it in higher mathematics!)
Division Stage One
Remainder Facts—mentally 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
Division Stage One
Remainder Facts—mentally computing facts including remainders
Format 10.3 C: Worksheet—students 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 remainders—WHY?—and some with quotients of zero.
Division Stage OneDiagnosis and Remediation Fact errors Component errors
Quotient that is too small or too large Subtraction error Placing remainder and quotient wrong
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
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
Division Stage OneDiagnosis and Remediation Subtraction error
Reteach subtraction (with regrouping)—provide division problems partially completed—have 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
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?
Multi-digit Division ProblemsTwo-digit Quotients
What are the steps in the short form algorithm?
Multi-digit Division ProblemsTwo-digit QuotientsWhat are the steps in the short form algorithm?1. S read the problem2. S underline the part they work first3. S determine and write answer to first part4. S multiply, subtract and bring down5. S read “new” problem and determine
answer6. S write answer over digit just brought down7. S multiply and subtract to determine
remainder8. S say the problem and answer
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?
Multi-digit Division ProblemsTwo-digit Divisors Lengthy and complex algorithm! What are the steps in the algorithm
suggested by our text?
Multi-digit Division ProblemsTwo-digit Divisors1. S read the problem2. S underlines the part to work first3. S writes the “rounded-off” problem4. S computes the division problem using the
answer from the rounded-off problem5. S multiplies and subtracts (if possible)6. S adjusts the quotient if needed (if you
can’t 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
Multi-digit Division ProblemsTwo-digit Divisors What additional preskills (in addition to
single-digit divisor problems) do students need?
10.8—A: Multiplying horizontally Model 10.8—B (rounded-off problems) &
C (entire strategy)
Multi-digit Division ProblemsTwo-digit Divisors What do you do when the estimated
quotient does not yield the correct answer?
Multi-digit Division ProblemsTwo-digit Divisors Format 10.9Rule: If you can’t subtract, make the
answer smaller; if the remainder is too big, make the answer bigger
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