3.4 Lecture Notes

Multiplication can be modeled in different ways and have different cognitive learning levels:
Level 1 - Repeated Addition: If I eat 2 cookies each day for three days, how many cookies have I eaten? 3x2 would be modeled as 2+2+2
Level 2 - Array(a discrete model): The room has three rows and each row has 4 chairs, how many chairs are in the room? This can be modeled with dots representing the chairs showing 4 dots in three rows. In this model we start with chairs in each row and find total chairs in the room.
Level 3 - Fundamental Counting Principle: You have two shirts and three pairs of pants, how many different outfits do you have? we use a tree diagram to solve this type of problem.
Level 4 - Area(a continuous model): If a room is 10 feet by 8 feet, what is its area? In this model the units change, we start with feet for width and length but end with square feet (difficult concept for students).
Level 5 - Fractional Part: I bought 6 pizzas for the party and ordered half of them as pepperoni, a third as plain, and the rest veggie. How many are pepperoni? This is a hard concept for students because it is solved by (1/2) x 6. Students have difficulty with this concept because they think about multiplication as repeated addition, which means the answer should be larger than what you start with if you use multiplication to solve.