# Strategies for x8 multiplication

I am creating an app for multiplication learning/practice. During practice students get hints if they do not answer the question within some time or if they provide a wrong answer. These hints are tied into learning strategies presented separately via a number of learning activities; the goal is to remind the student of a strategy they learned about before.

I was wondering if anybody has recommendations for x8 multiplication strategies (I am looking for things anchored in research, not reciting/memorization tricks). Any pointers to papers, resources, etc. are highly appreciated.

As an example, a strategy for multiplying by x9 could be to get to the closest known answer, in this case by multiplying by 10 and then subtracting the other factor (e.g. 6*9 = 6 * (10 - 1) = 6*10 - 6 = 60 - 6 = 54.

EDIT 1: Focus is on single-digit multiplication. As for the comment about the research, perhaps let me restate that I am NOT interested in memorization tricks or anything that's not based in math (e.g. reciting, etc. I've seen a number of things that may make student memorize the fact, but they may not have a clue why that's the result). (Given the audience on stackexchange maybe that comment was not needed)

• Double it, double it again, double it again. – mweiss Aug 10 '15 at 18:17
• You could modify your x9 strategy. $8x = 10x - 2x = 5x+3x$. You could also double three times. $7x8 = 7x2x2x2$. So you would think $14$, $28$, $56$. I have no research to support this. – Steven Gubkin Aug 10 '15 at 18:17
• @pjs36 not sure. I know there is quite a bit of research about number fluency, and the strategy of coming up with an answer based on the closest known information/fact is mentioned for sure. I clarified in the original question though that I am looking for something that's based on math (number fluency), and not some quick tricks on how to memorize without really understanding what's going on – Ognjen Todic Aug 11 '15 at 4:25
• I have seen that many kids love doubling. So double, double, double is enjoyable and sensible. – Sue VanHattum Aug 11 '15 at 15:15
• The double double double strategy of decomposing $8$ as $2^3$ is rooted in research: Check for papers on "doubles facts" (related to early additive learning) and consider the foundational importance of viewing numbers multiplicatively (see, for example, research of Zazkis and Campbell on number theory from the mid-1990s). – Benjamin Dickman Aug 19 '15 at 19:51

At the risk of being cheeky, my own strategy for multiplying things by 8 is that I've internalized (memorized) the answers. $4 \times 8$ reads simultaneously to me as both the multiplication statement and as the value $32$, and I expect that's the case for most of us.

I think that a successful training regimen for the multiplication tables will aim to have its users ultimately converge on the same strategy.

That said, employing intermediate strategies such as the repeated doubling can:

• Serve as a bridge techniques that both informs and motivates the eventual internalization (ie, after some number of times working out a particular product, students are likely to simply recall the answer, and students are likely to be pleased with themselves when the answer does start 'popping' into their heads, since they've saved themselves some time and demonstrated some hard-earned knowledge)

• Provide high grade intellectual fodder for students to chew on: Why is it that doubling a number three times is the same as multiplying it by 8? (An especially valuable line of questioning if the students also use repeated doubling for multiplication by 4). Similar questions arise from any such technique, and I think that they're all potentially valuable in their own ways.

Similar to the double it, double it again, double it again, but a little more advanced. If students know their 4x tables, they can use those as well. Example: I don't remember 8×7, but I know 4x7=28. So, 8×7=2(4×7)=2×28=40+16=56.

Also, like the 9 strategy, students can use their 10's. Example: I don't remember 8x7, but I know 10x7=70, and I know 2x7=14. So, 8×7=(10×7)-(2×7)=70-14=60-4=56.

• Does this answer have any support from research? Are these methods known to work for students? The OP asked for answers telling what research has to say about the issue, not what tricks and methods people like. – Joonas Ilmavirta Sep 25 '15 at 18:56

For the $9\times$ table, there is the well known technique of using one's hands. Drop the $k^{th}$ digit (finger/thumb), and count fingers/thumbs to the left as tens, and to the right as units. In this example, we have lowered the $7^{th}$ finger, in order to calculate $9\times 7$. We count $6$ fingers/thumbs to the left and $3$ to the right, giving $6\times10+3=63$.

This is extensible beyond $10$ if we use slightly different methods.

We can also emulate different bases, and hence multiplication by $b-1$, by pretending we have only $b$ fingers.

In this example, we pretend the right thumb is not there, to work in base $9$ for the $8\times$ table. Now, fingers to the left count as nines, and fingers to the right remain as units. We lower the $4^{th}$ finger in order to calculate $8\times 4$. We have $3$ fingers to the left, and $5$ to the right, and so $8\times 4=3\times9+5=32$.

Again, this technique can be extended for larger multiplications. And working in base $2$ is a feat in itself!

• I think you are missing the point: " I am NOT interested in memorization tricks", wrote the OP. – Benoît Kloeckner Sep 15 '15 at 7:06