MULTIPLICATION & DIVISION

 Disjoint subsets:  Multiplication: Making 3 party cups, 5 favors in each cup – how many favors would you need?  Division: Given 15 favors, 3 cups, how many favors in each cup? Multiplication/Division (CRA) Concrete  Representational  Abstract

 Equal group multiplication models w/ repeated addition  5 x 3 =  Repeated subtraction 15 ÷ 3 = 15 – 3 – 3 – 3 – 3 – 3 Multiplication/Division (CRA) Concrete  Representational  Abstract

Introducing Division (DI Formats 10.1 & 10.2) Division (CRA) Concrete  Representational  Abstract l l l l l l l

 Recall of math facts  Multi-digit operations Operations (CRA) Concrete  Representational  Abstract 8 x x 6 2

MULTIPLICATION

Preskills  What are the preskills for introducing multiplication?  What are the preskills for the multidigit problems?

Multiplication Introducing the Concept  Single-digit Multiplication  Missing-Factor Multiplication

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?)

Predictable Problems  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) How will you remediate?  What types of strategy errors might we expect? How will you remediate?

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—Structured 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?

Other Algorithms for Multiplication  Partial Products 6 7 x x x x x Uses distributive property + understanding of place value (expanded notation)

Other Algorithms for Multiplication  Lattice Multiplication 53 x

Other Algorithms for Multiplication  Lattice Multiplication 53 x

Other Algorithms for Multiplication  Lattice Multiplication 53 x

Other Algorithms for Multiplication  Lattice Multiplication 53 x

Other Algorithms for Multiplication  Lattice Multiplication 53 x

Other Algorithms for Multiplication  Lattice Multiplication 53 x

Other Algorithms for Multiplication  Lattice Multiplication 53 x

Other Algorithms for Multiplication  Lattice Multiplication 53 x

Other Algorithms for Multiplication  Lattice Multiplication 53 x

Other Algorithms for Multiplication  Lattice Multiplication 53 x =3551

Alternative Strategies: Multiplication Table Calculators

Factors to consider for alternative strategies…  Goals  Conceptual development v. procedural fluency and accuracy?  Age of the student  Instructional priorities  Other strategies tried  Ease of use  Availability

DIVISION

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

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 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?

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