Predict Tens Digit in Multiplication Table. 7*8 = predict 5 for tens. Rule for finding the pattern + Easy-Guess Trick for kindergarten kids.

Solution 1:

As I remarked in your prior question, we can intuitively represent such periodicity via star polygons (represented in toys like Spirograph). Since you are working $\!\bmod 10\,$ we use a "$10$ hour clock" of $10$ points placed equidistant along a circle. To get all multiples of $n$ modulo $10$ we start at $\,0\,$ then repeatedly add $\,n,\,$ by taking "big" steps of length $n$ along the circle. The path of this walk inscribes the $\{10/n\}$ star-polygon in the circle. Conveniently there is a nice YouTube video Star Polygons, by Linda Roper animating this case $\{10/n\}$ so please see there for further details on the basics (the image below is from there).

Let's construct the star polygon $\{10/6\}$ above. We obtain all multiples of $6$ by starting with $0$ then successively adding $6\pmod{\!10},\,$ yielding $\,0,\,6,\,12\!\equiv\! 2,\, 8,\, 14\!\equiv\! 4,\, 0.\,$ The $5$-point aqua star within $\{10/6\}$ is a graph of this process: starting at topmost point $(=0)$ draw an aqua line to the point $6,\,$ then draw a line from $6$ to $12\equiv 2,\,$ etc. The resulting star polygon is the path traced by taking a walk on this $10$-point circle by taking steps of size $6$. The journey visits the following points $\bmod 10\!:\ 6\Bbb Z = 6\Bbb Z + 10\Bbb Z = \gcd(6,10)\Bbb Z = 2\Bbb Z\,$ i.e. all multiples of $2$, i.e. all evens. The pink star is the coset $\,1+6\Bbb Z = 1+2\Bbb Z = $ all odds, obtained by rotating the aqua star by one.

The Spirograph toy works the same way except it uses curves (vs. straight lines) to connect the successive points in the star polygon.

This (and related methods) provide great (visual) motivation for many results about cyclic groups - with the benefit they they can be understood long before one learns group theory. I've had success explaining such ideas to bright grade school students. It may prove helpful in your endeavor.

You can find an introduction to star polygons (and polytopes) in Coxeter's classic book Regular Polytopes. Below is an excerpt.