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**Algorithms for Multiplication and Division**

Section 3.4 Algorithms for Multiplication and Division Mathematics for Elementary School Teachers - 4th Edition O’DAFFER, CHARLES, COONEY, DOSSEY, SCHIELACK Linda Roper

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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?

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**Developing Algorithms for Multiplication: Using Paper-and-Pencil**

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**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

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13 x 24

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x 24 13

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x 24 13

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x 24 13

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x 24 13

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**Add the partial products**

10 x 20 10 x 4 3 x 20 3 x 4 24 x 13 12 3 x 4 60 3 x 20 40 10 x 4 200 10 x 20 312 Add the partial products

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**Developing Algorithms for Multiplication: Using Paper-and-Pencil**

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**Use the area model to solve the multiplication problem.**

15 x 21 Use the area model to solve the multiplication problem.

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Example: 6 × 345 Expanded algorithm: Standard algorithm:

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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.

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**Example Use lattice multiplication to find 247 × 681.**

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

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**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.

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**Developing Algorithms for Division: Using Paper-and-Pencil**

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**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.

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**Standard Algorithm for Division Step 1: Set up the problem**

Model Algorithm

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**Standard Algorithm for Division Step 2: Decide where to start**

Model Algorithm

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**Standard Algorithm for Division Step 3: Divide the hundreds**

Model Algorithm

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**Standard Algorithm for Division Step 4: Divide the tens**

Model Algorithm

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**Standard Algorithm for Division Step 5: Divide the ones**

Model Algorithm

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312 ÷ 2

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312 ÷ 2

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312 ÷ 2

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312 ÷ 2

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312 ÷ 2

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312 ÷ 2

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312 ÷ 2

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312 ÷ 2 312 ÷ 2 = 156

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The End Section 3.4 Linda Roper

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