Developing Operational Sense: Multiplication & Division

Guides to Effective Instruction: Volume 3: Multiplication and Volume 4: Division

NOTE: Factor x factor = product

Dividend ¸ divisor = quotient

Multiplication can be thought of as:

1. Repeated Addition (introduced in Grade 2)

2. Equal Groupings

3. Array model: Arrays are rectangular arrangements of countable objects in equal columns and rows (similar to repeated addition model).

4. Area model: The area model represents multiplication as covering a rectangular area (length x width) using Base Ten Blocks.

5. Cross Product/Cartesian Product: involves multiplication through matching each member of one set with each member of a second set (e.g.) “How many different outfits can you make with 2 shirts and 3 pairs of pants (2x3=6)?

Try a Problem

Imagine you are a Grade 5 student who has not learned the two-digit by two-digit algorithm for multiplication.

You are asked to solve the following problem:

Jeff works for a window replacement company. His company has been asked to replace the windows in a number of different buildings.

Jeff has replaced the windows on three sides of a building. He needs to order windows for the last side of the building. Jeff looks at the windows on the last side of the building. and discovers that there are 15 floors, and on each floor, there are 12 windows. How many windows will Jeff need to order?

Division Models

Equal sharing model (Partitive division): The number of groups is known and the task is to determine how many are in each group. (e.g.,) “There are 3 children (number of groups) and 12 balloons (total), how many balloons (number in each group) does each child get?”

Measurement model (Quotitive division): The number in each group is known, and the task is to find how many groups there are. (e.g.) “12 balloons (total) were tied together in bunches of 4 (number in each group). How many children (number of groups) could get a bunch of balloons?

What About Remainders?

Explore problems with remainders at the same time as problems without, so that students do not develop the misconception that division always leads to a “nice answer”… which, in real life applications is less likely to happen.

If presented in a problem solving context remainders can have multiple meanings:

• As a “Left Over”

• As a “Fraction” (partition/shared among the groups)

• Discard (e.g., extra wood or rope)

• Forces the answer UP to the next WHOLE number (e.g., bus problem)

Connecting a Concrete Representation to the “Traditional Algorithm”

Steps for solving 435 ÷ 3 using Base 10 Blocks:

• How many are in each group? (record answer above the division sign)

• How many, in total, did you put in each group? (the quantity subtracted)

• How much is left to still share/divide? (the remainder after you subtract).

Developing Operational Sense: Multiplication & Division