Multiplication can be thought of in terms of putting equal groups together (e.g., we can interpret the expression 3 x 7 as 3 groups of 7. When we interpret a multiplication in terms of equal groups, the first factor tells us the number of groups and the second factor tells us the size of each group.
We can use an arrangement of equal rows of dots (i.e., an array) to model a multiplication expression. The first factor tells us the number of rows in the array and the second tells us the number of dots in each row.
We can find the total number of dots in an array by multiplying the two factors in the related expression.We can use fluency with doubles to multiply 3 by another factor (3 x n). We can interpret the expression as 3 groups of a number and draw an array of 3 rows with n dots in each row. We can see the related array of 2 rows of n inside this array. We can find the total number of dots in the larger array by adding the product of 2 x n to the number of dots in the additional row.
Knowing about the commutative property and putting that property to work can cut students' work in half!
Students can develop fluency by working on subsets of facts at a time. They should progress through the facts by using known facts to derive unknown products.