Arrays are arrangements of numbers in rows. They are created with objects or by rectangles and squares on graph paper. Arrays provide graphic representations of basic facts and help students who are visual learners. Students can construct rectangles consisting of six rows of seven squares to model the multiplication fact 6 x 7. By turning the array, students can reinforce their understanding of the commutative property and the concept that order does not change the outcome of multiplication. Instruction should focus on extending the number of rows to model multiplication as repeated addition.
A hundred chart displays the numbers between one and 100 in rows of 10. Students can circle the multiples of given numbers using color coding. Students will see that many of the numbers will have more than one color circle. Patterns will begin to emerge. For example, the multiples of three alternate between even and odd numbers. Every other multiple of two is also a multiple of four. Every other multiple of three is also a multiple of six. The digits in the multiples of nine add up to nine. Using graphic representations will help reinforce fact mastery and understanding of common multiples.
A fact family is a group of numbers that are related through multiplication and division. The numbers 4, 9 and 36 can be used to create two multiplication and two division facts (4 x 9 = 36, 9 x 4 = 36; and 36 divided by 9 = 4, 36 divided by 4 = 9). Using fact families will help students associate numbers in groups to enhance fact mastery. Students learn that division is the inverse or undoing of multiplication. Square numbers will stand out as a fact family that creates only two facts instead of four.
Without number sense, fact mastery is simply memorization of facts without the understanding of how to apply the skill of multiplication. Students should be able to construct a problem-solving story that uses multiplication. The use of the word "each" is essential in grasping the concept of multiplication. A typical story states that there are six vases. Each vase has eight flowers. How many flowers are there altogether? A story without the use of the word "each" shows that the concept has not yet been understood and that further instruction is required.