Section 3.4

Algorithms for Multiplication and Division

Mathematics for Elementary School Teachers - 4th EditionO’DAFFER, CHARLES, COONEY, DOSSEY, SCHIELACK

Linda Roper

9 x 12 = ?

How does a child who does not know the multiplication fact 9 x 12, but knows some other facts, figure out the answer?

Developing Algorithms for Multiplication:

Using Paper-and-Pencil

Developing Algorithms for Multiplication:

Using the Area Model Factors are the length and width of the

rectangle. The product is the area of the rectangle,

possibly found using partial products. Example: 13 × 24 = 312

13x 24

13

x 24

x 24

13

x 24

13

x 24

13

24x 13

126040

200312

3 x 43 x 20

10 x 410 x 20

Add the partial products

3 x 43 x 20

10 x 4

10 x 20

Developing Algorithms for Multiplication:

Using Paper-and-Pencil

2. 15 x 21

Use the area model to solve the multiplication problem.

Example: 6 × 345

Expanded algorithm: Standard algorithm:

Other Ways to Multiply

A spreadsheet is a powerful way to find the product of a large set of numbers and a single factor.

Lattice multiplication is an algorithm that reduces multidigit calculations to a series of basic multiplication facts followed by a series of simple sums. The diagonals correspond to place values. Partial products are found using the

distributive property.

Example

Read the final product from the top down and to the right: 168,207.

Use lattice multiplication to find 247 × 681.

Developing Algorithms for Division: Using Paper-and-Pencil

The expanded algorithm for division features repeated subtraction to find the quotient, which is simple to use but can be quite inefficient.

The standard algorithm for division has several steps and is based on the sharing interpretation of division.

Developing Algorithms for Division: Using Paper-and-Pencil

Developing Algorithms for Division: Using Models as a Foundation Use base-ten blocks to model the sharing interpretation for division:

105 ÷ 15

Trade 1 hundred for 10 tens, then trade 10 tens for 100 ones.

There are 105 ones, which we can divide into 15 equal groups. Seven ones can go into each of the groups, so 105 ÷ 15 = 7.

Standard Algorithm for DivisionStep 1: Set up the problem

Model Algorithm

Standard Algorithm for DivisionStep 2: Decide where to start

Model Algorithm

Standard Algorithm for DivisionStep 3: Divide the hundreds

Model Algorithm

Standard Algorithm for DivisionStep 4: Divide the tens

Model Algorithm

Standard Algorithm for DivisionStep 5: Divide the ones

Model Algorithm

312÷2

2÷312

2÷312

2÷312

2÷312

2÷312

2÷312

2÷312 =156312÷2

The EndSection 3.4

Linda Roper