# Stick Times

## Aim / Overview

This activity introduces the idea that “multiplication is simply a warped version of addition” and helps grow numerical intuition through the building and use of a twentieth century calculating device – the slide rule.

## Materials

1. Two rulers
2. Paper
3. Stickytape
4. Pen
5. Calculator
6. Demonstration Slide Rule

## Ages

10 to 210

Lower limit depends highly on the child. Some as young as 7 can do this activity.

## The Experiment

Ask the children how one can use two rulers to do addition with. Guide them if need be to the idea that a ruler is like a number line, so that lining up two rulers with the same scale as shown below and measuring sums and differences of/between lengths is a physical embodiment of the operations addition and subtraction:

Figure 1: Summing with Two Ruled Sticks

Now: Can one do the same thing with rulers for multiplication and division?

To see that one can, cut two paper strips to fit the rulers, and make equal tick intervals on them as below. Instead of the regular number line, we make our ticks whole number powers of two, and we must make sure that the same tick spacing is used for both scales:

Figure 2: Multiplying Whole Number Powers of Two with Two Ruled Sticks

Your children likely won’t have met the idea of exponentiation before, so ask them whether they can describe the pattern of numbers on the equal ticks. Some prompts if they are having trouble:

1. Each number is double the number on its left;
2. To jump one place to the right of a number, multiply it by two, to jump two places, multiply it by $2\times 2$, for three places, multiply it by $2\times2\times2$ and so on.
3. The number of places you wish to jump to the right is the number of times we must multiply two by itself.

To explain how the division works, it might be enlightening to introduce the idea (if it isn’t already wonted to the children) that division can undo multiplication, subtraction can undo addition. So that, for example,  $\left(5\div 3\right)\times 3 = 5$ and $\left(5+3\right)- 3 = 5$.

Otherwise, do all this with products of ten, as in my drawing below. The idea that 1 with a number of noughts after it is $10\times10\times10\times\cdots$ with the number of noughts equal to the number of times ten multiplies itself by will be well wonted to the child, even if they don’t know it. So this may be a better beginning point for some children. If you go this way, I suggest you only do two or three tick marks.

Figure 3: Multiplying Whole Number Powers of Ten with Two Ruled Sticks

Maybe even do both exercises. The power of two one gives a reasonable number of powers to experiment with, the power of ten may be much easier for them to grasp, so the latter can strengthen understanding of the former.

Ask the children why they think this multiplication and division with sticks works. Hopefully they can see that things like what we have below happen:

$$\overbrace{2\times2\times2}^{\text{Thrice}}\,\times\,\underbrace{2\times2}_{\text{Twice}} = \overbrace{2\times2\times2\times2\times2}^{\text{Five Times}} = 32$$

So in adding number of jumps with our sticks, we’re adding the number of times we multiply two by itself.

Again, this may be a less thorny idea if done with powers of ten.

So now a question. If we put a tick mark exactly in the middle of 1 and 2 (or 1 and 10 in the tenfold example), what number should go here? Give this thought some time to be talked over.

Now most likely you will get the guess of 1.5. If you don’t get this answer, ask the child who doesn’t think it is 1.5 why they think it should be something else. 1.5 is not the right answer as we shall see, but it is a good answer for the children to give – it’s a good, intuitive guess, so let them go with it. Most importantly, we can work out how to test whether our answer is right.

So now ask: how can we test whether we are right?

Think back to what our sticks are meant to do, and then think of the meaning of the following drawing:

Figure 4: Halfway between Two Marks

If the answer were 1.5, then the above drawing would mean $1.5\times1.5=2$ , because our sticks are meant to be multiplying sticks. (This is where the calculator will come in handy – freeing the child’s mind from the details of the arithmetic so that they can think hard about meaning). Is this right? So the mystery number can’t be 1.5 if our sticks are meant to be multiplying sticks, because then they would give a wrong answer. So, try some other numbers for the middle tick:  $1.5\times1.5$ is too big, so let’s try 1.4. At this point you might like to ask the children to find the number for the middle tick that will keep our sticks good multiplying ones. Otherwise, if they are having difficulty, work through a procedure like this one:

$$\begin{array}{1c11} 1.4\times1.4 &=& 1.96 &\text{too small, although we’re mighty near}\\ 1.45\times1.45 &=& 2.1025 &\text{too big}\\ 1.41\times1.41 &=& 1.9881 &\text{too small – wow we’re near to the right answer now though!}\\ 1.42\times1.42 &=& 2.0164 &\text{too big – we’re still near to the right answer}\\ 1.415\times1.415 &=& 2.002225 &\text{too big – but we’re ever so near now!} \end{array}$$

And so forth. If they’re really keen they can find that:

$$1.4142135\times1.4142135=1.99999998$$

Depending on how your audience it taking all this, an interesting fact to mention is that the right answer has an unending number of decimal places. But watch that wonder is not slipping into overwhelmedness! This exercise may be better done in several sessions.

OK. Now: What number should go at the position of the tick halfway between two and three, shown by the red arrow in my drawing below?

Again, we need to keep in mind that we want our sticks to be good multipliers, so what is the meaning of the two rulers lined up as in my drawing below?

Figure 5: Halfway between Two and Three

It has to be two times the mystery number we’ve just found. So it must be 2.82842, or thereabouts. Again, this is a number with an unending number of decimal places. Reasoning likewise, we can now mark off all the ticks halfway in between our numbers:

Figure 6: All the Halfway Numbers

So now we can think of the same process again. What number will go on the tick halfway between 1 and our newly found 1.4142135. Hopefully you can see this from my drawing below. We ask what the meaning of the two sticks is, and recall that we want our sticks to be good multiplying sticks. So the number at that tick has to be a number that when multiplied by itself is 1.4142135. The same process as above should find that 1.18921 works rather well.

Figure 7: Slicing a Notch in Half Again

and that working as before we can now mark off all the ticks halfway between the ones we already have.

Figure 8: All the Quarter Notch Numbers

I’ve put all these workings in a spreadsheet which you can download from the link below. What I’ve done here in the second column is put the first mystery number 1.414214 below 1. Then I’ve generated all the numbers in that column by multiplying each one by this 1.414214 to get the number straight below it. Then  in the third column, I’ve written the number 1.189207 (the one that multiplies itself to get 1.414214) below the 1 and I’ve generated all the numbers in that column by multiplying each one by this 1.189207 to get the number straight below it. Likewise for the fourth column – here I replace 1.189207 by the number 1.090508 that multiplies itself to get 1.189207.

Figure 9: Spreadsheet layout for working out notch positions

Keeping on in this way with more and more columns, we can find numbers that belong to more and more finely spaced ticks: each column finds the ticks halfway in between the ticks of the foregoing column.

Now all this gives us awkward numbers to multiply with. We’d rather have ones like 1.2, 1.3, 1.4, 1.5 and so on so that we could simply find the numbers we want to multiply together easily. But we notice that after we do the column dividing trick in my spreadsheet enough times, we get numbers that are awfully near to “convenient” ones. So, for example, we can find a number very near to 2.6 in the fourth column.

So it should be fairly easy to see that, if we went on doing the column dividing trick enough, eventually we’d know exactly where to put all the “convenient” numbers on a scale. The result as found by a computer is below:

Figure 10: Computer Generated Slide Rule Scales

Before computers, people worked all these numbers out by hand! The results were published in big tables of numbers.

Print out the scales, cut along the middle line and then sticky tape them to your rulers. You should find that you can multiply with them.

Notice how the scales are not evenly spaced. They are stretched or “warped” scales. Multiplitcation truly is a “warped version” of addition and division a “warped version” of subtraction!

Hopefully a real demonstration slide rule will be at hand for the children to try out. This is how many people did calculations before about 1972, when calculators began to become widespread. Most manufacturers left off making slide rules about 1975.

Some real slide rules are shown in the photograph below and hopefully some demonstration slide rules can be arranged for children to experiment with.

Figure 11: Two Real Slide Rules