This is a pattern even school kids could discover (when gently pointed to). I never did conciously, and cannot remember to have been pointed to explicitly, neither at school nor later:

$$\color{red}{\mathbf{2}}\cdot 9 = 1\color{red}{\mathbf{8}}$$ $$\color{red}{\mathbf{8}}\cdot 9 = 7\color{red}{\mathbf{2}}$$

$$\color{blue}{\mathbf{3}}\cdot 9 = 2\color{blue}{\mathbf{7}}$$ $$\color{blue}{\mathbf{7}}\cdot 9 = 6\color{blue}{\mathbf{3}}$$

$$\color{green}{\mathbf{4}}\cdot 9 = 3\color{green}{\mathbf{6}}$$ $$\color{green}{\mathbf{6}}\cdot 9 = 5\color{green}{\mathbf{4}}$$

which may come as kind of a miracle when first discovering it.

In mathematical terms

$$\boxed{a\cdot (10-1) \equiv b \mod 10\ \ \ \ \Leftrightarrow\ \ \ \ \ b\cdot (10-1) \equiv a \mod 10 \\ a\cdot (10-1) \equiv b \mod 10\ \ \ \ \Leftrightarrow\ \ \ \ \ a + b = 10 \equiv 0 \mod 10}$$

This holds not only for $10$ but for every $p \in \mathbb{N}$, i.e. in every "number system":

$$\boxed{a\cdot (p-1) \equiv b \mod p\ \ \ \ \Leftrightarrow\ \ \ \ \ b\cdot (p-1) \equiv a \mod p \\ a\cdot (p-1) \equiv b \mod p\ \ \ \ \Leftrightarrow\ \ \ \ \ a + b = p \equiv 0 \mod p}$$

and is responsible for the fact that the graphical multiplication tables of $\mathbb{Z}/p\mathbb{Z}$ always looks the same for $p-1$:

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I wonder if there are attempts (in educational research and literature) to make use of the simple observability of the pattern above to explain to (clever) school kids that the observed regularity is not by pure coincidence, why it is so, and what it does "mean".

  • 1
    $\begingroup$ There's plenty of advanced math clubs for middle to high school students all over the world. This is sure to be a popular topic there. Not that I'm familiar with the area $\endgroup$ – Yuriy S Nov 12 '18 at 10:11
  • $\begingroup$ What do you mean by "math clubs"? $\endgroup$ – Hans-Peter Stricker Nov 12 '18 at 10:13
  • $\begingroup$ Organized meetings of students in their free times with a tutor or tutors. I'm not sure how is it called in English $\endgroup$ – Yuriy S Nov 12 '18 at 10:21
  • $\begingroup$ I'm not sure if there are plenty such clubs e.g. in Germany. Where do you live, possibly Russia? What about other countries? $\endgroup$ – Hans-Peter Stricker Nov 12 '18 at 10:34
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    $\begingroup$ Maybe this question is better suited for the Mathematics Educators SE site? $\endgroup$ – YiFan Nov 12 '18 at 10:38

After having the class talk about the quote

And yet patterns exist, and we slowly discover them. Seasons, migrations, moons: the template is there. Consciously or unconsciously, most people accept certain components of cycle theory. We seek and see patterns in things. It is the way our minds work, presumably for the purpose of survival.

--- Nick Paumgarten

the teacher should then gently explain to their students that to survive in his class, they MUST memorize the 1-digit multiplication tables; with $\bar n = \{0, 1, 2, 3,4,5,6,7,8\}$,

$\tag 1 (m,n) \mapsto m \times n \quad \text{ for } m,n \in \bar n$

She can then discuss an interesting pattern:

If one of the digits in $(m,n)$, say $n$, is a nine, and $m$ is non-zero then

$\tag 2 m \times n = string(m - 1) \text{ || } string(k), \text{ where } k \text{ is chosen so that } m - 1 + k = 9$

Notice how the concept of string concatenation from computer languages is being introduced!

Example 1: $7 \times 9$: Since $6 + 3 = 9$, ANSWER: $63$.

Example 2: $9 \times 9$: Since $8 + 1 = 9$, ANSWER: $81$.

Example 3: $1 \times 9$: Since $0 + 9 = 9$, ANSWER: $09$ (but tell them to drop the leading $0$).

The teacher can also explain why (2) works. Whey you multiply a digit $n$ by $9$ it has to be under that nice round number $n \times 10$, and the pattern describes by how much.

Of course the students who don't care about understanding patterns can keep working the flash cards.

Interestingly, this cycle repeats. Students who can grasp patterns really don't have to memorize a bunch of stuff to do mathematics. The majority are always trying memorize all the formulas, when, if in fact, they UNDERSTAND just a couple of things, they are much better off.

  • $\begingroup$ (2) is quite hard stuff - to read, to understand, and to apply. Why not just memorizing the 1-digit multiplication table for $n=9$, too? $\endgroup$ – Hans-Peter Stricker Nov 12 '18 at 12:54
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    $\begingroup$ Actually first the the students memorize the table, but then they can learn to understand the pattern that makes it easier to recall. $\endgroup$ – CopyPasteIt Nov 12 '18 at 12:58

Not sure about this pattern. But the computation (addition and multiplication) using the clock modulo 12 or modulo 24 is something that kids are capable of to understand. I'd start with say a clock modulo 4 to explain addition (say $2+3=1$) and multiplication (say $2\cdot 3 = 2$).

  • $\begingroup$ Right, that's the standard way to do it - and it surely is a good one. This is how you would introduce modular arithmetic from scratch. $\endgroup$ – Hans-Peter Stricker Nov 12 '18 at 11:18
  • $\begingroup$ Alternatively one could ask: Where do you find modularity in the decimal system? (Of course when you add two decimal numbers digit-wise. But you would not mention "modularity" as a teacher - just teach the technique and use (mental) addition tables with entries like $7+8 = 15$ to be used like "write $5$, carry $1$".) $\endgroup$ – Hans-Peter Stricker Nov 12 '18 at 11:34
  • $\begingroup$ Conceptually it's two tables: $7+8\operatorname{mod}10 = 5$ and $\lfloor(7+8)/10\rfloor=1$ the students do learn in one. $\endgroup$ – Hans-Peter Stricker Nov 12 '18 at 11:42

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