How to Teach Multiplication Table Factors through Nine

One of the most irritating phenomena that I confront in teaching math to elementary, middle school, and high school students enrolled in the New Salem Educational Initiative is the failure of teachers in the Minneapolis Public Schools to have imparted knowledge of basic multiplication table factors, those including the numbers 0 through 9. In my observation, this unfortunate circumstance results from two causes: 1) education professors in general, and math education professors in particular, who see learning basic multiplication factors as rote memorization of the kind that they abhor; and 2) classroom teachers who typically suffer from having matriculated at university departments of education and are all too ready to retreat into lazy pedagogy that exalts the calculator.

All of this is a great shame, because full memorization of multiplication factors through 9 is necessary efficiently to find common denominators, reduce fractions, calculate decimals and percentages, solve proportions, find ratios and probability, solve algebraic equations, and perform a variety of mathematical operations in geometry, trigonometry, and calculus. And the shame is further intensified when one discovers that young children love to learn multiplication table factors when properly taught.

So consider this approach:

I always start with "0" and "1" as multiplied by the factors through 9, even when students have long since learned that "0" times any number is "0," and "1" times any number is that latter number. I rarely go through the whole group of factors through 9, but I do give students a chance to do one or two of each of these, upon which I say, "Wow, look at that--- you already know 20 of these cards. We just have to look at the ones involving 2 through 9 now."

So then I move on to the "2's," which tend to go very fast. I don't have a problem with students using their fingers to count by "2's," although after all factors "0" through "9" have been memorized or gained quick responses, I do begin to insist that students move so fast as to make counting on the fingers impossible. By this time, the "2's" have in any case typically been memorized.

I then skip to the "5's," which involve the counting exercise that comes easily to most students, so that I can then say, "See there, we don't have that far to go now. You already have mastered the table with factors using '0,' '1,' '2,' and '5.'" The student's confidence builds and what might have once seemed a daunting task now seems very manageable.

Then I back up to the "3's" and "4's." These are not quite so easily countable as the "2's" and "5's," but most students do not have much trouble first counting by
"3's" and then performing the corresponding multiplication tasks. And "4's" are still low enough that this holds true for this set, as well.

I have by this time also introduced my students to the pneumonic potential of rhyme by pointing out that "5 times 5 is 25" (vocal stress on the 5) and "6 times 4 is 24" (vocal stress on the 4). Particularly in this age of hip-hop, students love this sort of word play.

After the "3's" and "4's," I skip ahead to the "9's," at which point I teach them my version of the "9 trick." To my astonishment, few students are taught this easy pneumonic device in today's classrooms. I get "0" and "1" as multiplied by "9" out of the way and then focus on the trick, which is simply that the first digit in the product is the number sequentially just before the number by which "9" is multiplied, then the second number is that first digit subtracted from "9." So in
6 times 9, the first digit is "5" (just before "6"), and the second digit is "4"
("5"subtracted from "9") to yield "54." Similarly, the trick reveals the first digit in the product of 8 and 9 to be the number right before "8" and the second digit to result from subtracting that "7" from "9" to get "2" and thus yields a product of "72." The trick works throughout the numbers multiplied by "9" from "0" through "9."

So all of this usually goes quite quickly, allowing us to focus on the three numbers in the "0" through "9" sequence that tend to come most slowly for students: "6,"
"7," and "8." But by this time, there really are just a few individual sets of factors that the student has not seen in earlier sets of factors, because she or he has already seen these numbers multiplied by "0," "1," "2," "3," "4," "5," and "9." And the "6's" come rather quickly after I point out that in addition to the rhyming value of "6" times "4" is "24," that same sort of rhyming value exists in "6" times "6" is "36."

Students can then settle in to master the remaining "7's" and "8's," focusing especially on the most difficult five pairings for most students: "6" times
"7"; "7" times "7"; "7" times "8"; "6" times "8"; and "8" times "8." But "6 times 8 is 48" features another rhyme, and many students remember that "7" times
"8" is "56" after I tell them that this is the hardest one in the deck; interestingly, this observation (based on my experience in teaching multiplication over the years) helps students remember the correct product when they are presented with "7" times
"8."

I have found that this method incrementally builds student mastery and confidence, enabling young people to master the multiplication table with factors "0" through
"9" quickly. All of my Grade 2 students master this fundamental skill and knowledge set, and I have a few Grade 1 students who have already mastered the multiplcation table through "9" by the end of that academic year.

My students to a person love to learn multiplication in the way described above, and in the knowledge of this kind of response deserve to have this skill and knowledge set imparted to them as early in their K-12 years as possible.