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Subject:   RE: data on which are the hardest multiplication facts
Author: ClassProf
Date: Sep 11 2010
Hi Daniel

Here is what I teach my students, who are preservice teachers (student

0x and 1x are not necessary as facts to be memorised, but are rather special
cases. These can be understood via discussion and modeling using physical

10x facts are also not really needed, since they are covered in place value
lessons ("3 tens is called 'thirty'", etc.).

1)  The easiest multiplication facts by far are the 2x facts. These are doubles,
which students will already have learned in addition lessons.

2)  Next, do 5x - since 5 is half of 10, this family of facts has the
well-known pattern of 5, 0, 5, 0 in the ones, and each fact can be derived
from working out half the number of tens, especially the even multiples.

3)  Now, 3x and 4x. Multiples of 3 and 4 can be related back to doubles. 3x is
the same as "double plus one more set", as in 3x6 = double 6 + 6, or 12 + 6 =
18. Four, of course, is the same as "double double": eg, 4x7 = double double 7 =
double 14 = 28.

4)  Now skip over to 9x. Although 9 is one of the largest multipliers, its
proximity to 10 makes multiplying it quite straight forward. Consider 10x the
multiplicand, then subtract one to find 9x. For example, 9x6 = 60 - 6 = 10x6 - 6
= 54. The number of tens is always one less than the multiplicand, and the sum
of the digits is always 9, up to 9x9. There is a "finger trick" you can use for
9x also: hold up all fingers and thumbs, with palms facing down. Bend the finger
corresponding to the second term away from you, so if the question is 9x5, bend
down the fifth finger (left thumb). The number of digits to the left of the bent
digit is the number of tens (4), digits to the right equal the number of ones
(5) - the answer is 45. Children love this one.

5)  I like to cover the square numbers at this point, though there isn't a
single strategy that is easy to apply to these facts. Nevertheless, these facts
are so useful it is worth committing them to memory as a special set of

6)  The remaining facts up to 9 are the hardest, in my opinion: 6x, 7x and 8x.
However, there is no need to learn all of them from 6x0 to 8x10, since most of
these facts have already been learned by this stage, when learning the previous
sets. It turns out that the only remaining facts to be learned are: 6x7, 6x8 and
7x8. Thus the hardest fact, in my opinion, is 7x8.

You will notice, of course, that I haven't mentioned 11x or 12x. On the one
hand, in countries such as the USA that still use "Imperial" or "British" units
of measurement (even though the Brits stopped using them a while ago!), students
will have to know facts up to 12x12 for working with feet and inches. On the
other hand, most nations now use Metric units, the conversions for which are all
based on powers of 10. Therefore many students around the world have no need to
memorize facts beyond 9x9; gifted students may be challenged with harder facts
up to 12x12 or even 20x20. The other reason for not stressing if students don't
know their 12x facts is that each one can be derived quickly from the basic
facts. For example, 12x8 (one of the hardest facts of all up to 12x12) can be
thought of as 10x8 + 2x8 = 80 + 16 = 96.

I realize that many teachers will be used to teaching multiplication facts up to
12x facts. In that case, I will include 11x early in the sequence, since they
are easy up to 11x10, due to 11 being equal to 10 + 1. I would then make 12x
facts the last ones to be learned, and talk to students about how to think of
them as being in two parts, as described above.

Daniel, I hope this rather long post helps. You may like to check out the free
eBook at, which contains a sequence of carefully
sequenced worksheets for teaching students multiplication facts based on the
same sequence I have described here.

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