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