Contexts for Learning

Muffles Truffles: Multiplication and Division with the Array

The focus of this math unit is the development of the open array as a model for multiplication and division. This unit uses a series of investigations based on the context of Muffles’ Truffles shop. The questions posed in the first investigation gives students an opportunity to explore place value, the multiplicative structure of our base-ten system and quotative division. In the second and third investigations, students build two-dimensional blueprints and use these arrays to explore some big ideas in multiplication, including the distributive, associative and commutative properties. In the fourth and final investigation, students apply their knowledge with open arrays in the context of labeling and pricing wrapped boxes of truffles. To figure out the dimensions of the wrapped boxes (open arrays) and the cost, students need to apply a number of big ideas previously developed in this unit.

Partial Products Use a fact you know to make another. For example, 5 packages of paper towels (4 rolls in each package) contains 20 rolls, and 3 packages contain 12, then 8 packages contain 32.

Five-times and Ten-times Once students begin to make use of partial products, this strategy becomes helpful. Knowing the ten times can be helpful for nine times (just subtract one group from the ten-times). 6 ten-times is 60, take away one group of 6 is 54 ( 6 nine-times). The same goes for five-times to help with four-times and six-times.

Doubling and halving As student’s multiplicative reasoning becomes stronger, the develop the ability to group more efficiently. They begin to realize that if they double the number of groups and want the total to be the same, they need to have halve the amount in each group.

4 x 6 = 8 x 3

Open Array Model

Students will manipulate open arrays in order to group efficiently. Students will use open array models to further develop their understanding of the various strategies as well the properties of multiplication, such as, the associative, commutative, and distributive properties. i.e. Students may apply the distributive property to help them figure out the dimensions of a larger open array.