In the pursuit of patterns, don’t be afraid to give kids a calculator and jump into some complex stuff! Having a calculator allows kids to explore mathematical ideas and look for patterns in situations that they might normally be barred from without that handy tool.
Take powers (aka exponents) for example. Have your students or kids make a list of the powers of 2 like this:
2×2* = 4
2x2x2 = 8
2x2x2x2 = 16
2x2x2x2x2 = 32
2x2x2x2x2x2 = 64
2x2x2x2x2x2x2 = 128
2x2x2x2x2x2x2x2 = 256
Depending upon how old the kids are, how much time you have and how much you want to go into it, you can show them exponential notation, i.e. 26, but listing out the 2x2x2 helps all kids to understand what we are doing mathematically, which is that we are multiplying a string of the same numbers.
Now, take a look at the ones column in the powers of 2. Do you notice anything? More importantly, do your kids notice anything? One of the many cool things about powers is that the ones column will have a repeating pattern. In this case it goes: 4-8-6-2-4-8-6-2 and so on. You can actually predict the ones digit of the power product based on this pattern. So cool!
Why does this happen? Well looking at some other numbers (particularly powers of 5 or powers of 6) may help you to see why:
5×5 = 25
5x5x5 = 125
5x5x5x5 = 625
5x5x5x5x5 = 3125
So, look at what’s happening with the 5’s: they all end in 5!! Why, well, what do you get when you multiply 5 by 5? You get 25. So, then when you multiply 25 by 5 again, the ones digits will still be 5 times 5 which is 25.
Now try 6:
6×6 = 36
6x6x6 = 216
6x6x6x6 = 1296
6x6x6x6x6 = 7776
Same thing with the 6’s: they all end in 6!!! When you multiply 6 by 6 what do you get? 36! Again, so then when you multiply 36 by 6 the ones digits will still be 6 times 6 which is 36. The other numbers are not quite so clear cut as 5 and 6 but the same principal still applies. Basically I think of it as every number has a set “pool” of multiples to work with. Because you are multiplying the same number by itself over and over again, that “pool” of multiples will get repeated.
Try more powers and see what you can find! There are fascinating patterns in the ones digits on all the powers and really interesting connections between some of the powers, like the 2’s and the 8’s.
Challenge yourself and your kids to see just how many patterns you can find!
Have fun with math every day!