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 Stage Introducing 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 Students read the equation 2. Students state the equality rule 3. Students draw lines for first and second addend 4. Students count all the lines on “this” side 5. 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 Addition Format 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 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 Remediation 4 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 Renaming Format Students read the problem 2. 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 column 5. Write the renamed number and ones number 6. Add the first two numbers in then tens, then add the next number to the sum 7. Write the tens number

Multi-digit Addition with Renaming Format 7.6 What is the common error? What should the teacher do?

Multi-digit Addition with Renaming Format 7.6 Example 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 Remediation 4 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 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

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 subtraction 1. 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 Subtraction Missing Subtrahend Problems 9 –  = 4 1) Read the problem 2) Draw lines under minuend (first number) 3) Students figure out what number them must end up with 4) Students circle the number of lines that they must end up with 5) Students cross out (minus) the lines that are not circled 6) 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 summary 1. Read the problem 2. Determine if we must rename 3. Borrow the ten and put it with the ones 4. Subtract the ones column 5. 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

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 Picture demonstration 2. 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 incorrectly 2. Consistently off by one count-by number

Multiplication Introducing the Concept Remediation for strategy errors? 1. Confuse addition and multiplication 2. 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 Sequence 1. Single digit x multiple digit without renaming, 24 x 2 2. Single digit x multiple digit with renaming, 24 x 3 Format 9.3

Multi-digit Multiplication Sequence cont. 3. Two-digit x two-digit 4. Two-digit x three-digit

Multi-digit Multiplication Format 9.4 Steps Part A—Order of multiplication Part B—Structure board—modeling the algorithm (What is critical in this model?) Part C—Structured worksheet Part 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 One Diagnosis and Remediation  Fact errors  Component errors  Quotient that is too small or too large  Subtraction error  Placing remainder and quotient wrong

Division Stage One Diagnosis 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 One Diagnosis 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 One Diagnosis 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 Problems Two-digit Quotients What are the steps in the short form algorithm?

Multi-digit Division Problems Two-digit Quotients What are the steps in the short form algorithm? 1. S read the problem 2. S underline the part they work first 3. S determine and write answer to first part 4. S multiply, subtract and bring down 5. S read “new” problem and determine answer 6. S write answer over digit just brought down 7. S multiply and subtract to determine remainder 8. 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 Problems Two-digit Divisors  Lengthy and complex algorithm!  What are the steps in the algorithm suggested by our text?

Multi-digit Division Problems Two-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 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 Problems Two-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 Problems Two-digit Divisors  What do you do when the estimated quotient does not yield the correct answer?

Multi-digit Division Problems Two-digit Divisors  Format 10.9 Rule: If you can’t subtract, make the answer smaller; if the remainder is too big, make the answer bigger