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Factor Arrays Make Abstract Ideas Concrete
Factor Arrays Make Abstract Ideas Concrete
Factor Arrays are models representing the number in a particular multiplication fact table visually.
Most arrays represent a whole multiplication problem (as pictured on the Multiplication Breakdown webpage, in the example of 8 x 7 as 8 rows of 7 dots in columns).
Factor Arrays are different, they only represent one of the numbers. They are a set of dots arranged in two columns. For odd numbers (see the Factor 7 image below) the last dot is below the others on the left.
Why left? Counting using Factor Arrays should go from left to right, not down the columns. By counting across the set, students are not as likely to rush and miscount. It's helpful to have them touch the dots as they count, which connects the visual with tactile-kinesthetic input. Counting aloud then adds in auditory input.
"It is essential to have good tools, but it is also essential that the tools should be used in the right way."
~ Wallace D. Wattles
Factor Array
of 6
Factor Array
of 7
One Factor Array is used to build out all the answers for that times table. Count the set once for, say, 6 x 1. Then count on from the top a second time for, say, 6 x 2. Continue counting on through 6 x 12, writing down each answer when that count of the set is completed.
It's helpful for students to be able to note those answers down while focusing only on the answers, so have the other factor written before starting writing each answer while counting.
Factor Arrays make repeated addition make sense.
They also enable students to work independently to learn a particular set of multiplication facts. Students absent during initial instruction can use them on their own to begin catching up.
Practice the table using multisensory methods.
Practice the table using multisensory methods.
Why? When information enters the brain through several sensory inputs, each sensory input strengthens the others – sort of like the cords of a braided rope.
Student directions:
Recite the table aloud three times. Read it as you recite so the practice is with right answers, not wrong ones. Listen to your voice saying the problem and the answers. Touch the problem, trace the answer on the table with your finger.
The student's brain is noting the pattern, the interval gap he or she is using even when not aware of it consciously. It is rhythmic.
If they have used the multisensory practice method above, they've used their eyes to read, ears to hear, touch and movement as they voice and mouth the answers with teeth, tongue, and lips, and movement as well as touch in tracing the answer on the table.
"Studies have shown that music and rhythmic sounds can enhance cognitive function, helping people learn and recall information more effectively."
~ The Science of Sound
Use both Factor Arrays and Multiplication Breakdown in practicing to get a particular answer not yet memorized. While multiplication breakdown is especially easy to use on the 7s and 12s, any multiplication problem can be solved using Factor Arrays. [add page links]
Next, students should practice the times table they just worked on with a deck of cards. Learn about that practice here. [add page link]
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