It was not my intention to teach logarithms to a seven year old, but somehow it is happening. I think it started when he had a math worksheet that involved charting the growth of a mythical creature that grows exponentially compared to one that grows two inches a day. I wanted to look up logs and found that the log2 height = day. Then we watched a little youtube video, that included mention of how the ph scale is logarithmic, and we got distracted by trying to understand the definition of acid.
(Side note: did it ever bother any of you in elementary school to be taught “acids and bases do this” but not the why? It used to bother me. Or magnetism. I hate it that so many children’s books about magnetism seem to think its amazing that two opposite poles attract without bothering to attempt an explanation of why. But that’s a rant for a different day.)
The next day we returned to the topic of logarithms. The seven year old made a chart showing what log10 10, log10 100, etc, etc is. We looked at how log10 230 is going to start with 2, just as log10 100 is going to, but it will have more numbers after the decimal point. The integer number is the characteristic, the fractional part is the mantissa. Then we could talk about how the characteristic of log10 relate to the number of digits to the left of the decimal.
I gave him a worksheet asking him graph the distances. I choose one relatively short distance, two medium ones and then one of of this planet distance. Of course if we labelled the vertical axis with hundreds or even thousands of kilometers the final distance would be off the chart. If we labelled the vertical axis with hundreds of thousands, then the first three would all be squished at the bottom. So instead we used log10. This worked.
We then turned to this comic by xkcd and a couple of other visuals for understanding log scales.
I know slide rules are based on logs, so I googled and found a video on making your own slide rule. The first part of the video I really enjoyed as it shows how the times table can be drawn so as to straighten out the repeating answers. I was disappointed though that the instructions for making a slide ruler were more for making a frame on which you paste his print-out of the ruler.
Now I could have looked closely at the slide rule my husband gave me for Christmas but I wanted to discover this on my own. So I drew out a times table with straightened lines. I did this by numbering a sheet of graph paper 1 to 100, with each square representing two. I placed the numbers 1 to 10 according to the first two digits of the log10. This means 2 was at 30, 3 was at 47, etc, etc. Once I had two axis drawn I could make myself a times table like the one in the video. But how do translate this into a slide rule?
I used my times-table edge as a ruler and copied it onto a long strip of paper. Then I copied it a second time. Using the times-table against my strip I could see multiples of the lower numbers, but what happens when I get past ten? I used the times-table edge ruler and started writing the numbers where they should be if it was to convey the times table. Presto! A slide-rule, and I can see visually that the same space that is equivalent to two units in one location is now representing 20 units.
But I didn’t feel like I was at a point I could explain the slide rule to the kids. So I kept exploring. Multiplication can be done by adding logs and it’s fairly easy to see with two normal rules how they can be used for adding. I made a little “addition slide rule” for the children so they could see how to add with it.
You could look up the logs of two numbers in a log table, then use two rulers to add those numbers together. Or to make it easier, you could write on the rulers what the logs of different numbers are so you don’t have to look in the chart, and that is of course what a slide rule involves.
To make a very basic slide-rule: Cut a piece of card paper six inches by 11. Fold it in half and draw two lines on one side so that you can see three strips each an inch long. Mark off a centimeter or so at the top and bottom of the middle strip and cut the middle of that you so you can see through it to where you’re going to put the second ruler. Tape the edge. Cut an almost 3 inch strip of cardpaper in a contrasting color and slide it in between the two other strips.
Now the optional step is to mark all the logs right now on one side of the ruler-case. Use a ruler and put little marks one centimeter apart. The bottom one should be 0.1, the next 0.2, and so on.
Take your calculator or a log chart and look up the log10 for each of the numbers. Multiply the log by 10 and that’s how many centimeters in to mark the number. The distances between each number gets smaller and 10 will be at 10 cm, and 100 at 20 cm. Copy the same marks on both the inside ruler and the outer ruler.
To multiply slide your ruler up so that the bottom edge of the inside ruler is lined up with the number you want to multiply by (x). Then your inner ruler should be lined up so your 2 is next to the outer-rulers 2x, the three is next to the outer-ruler’s 3x, and so on and so forth.