Math Detours · Grade 6 · Around the corner · 35

Add with a ruler

Before calculators, engineers got answers by laying a ruler across printed scales. You can build a chart that adds any two numbers on it, with no arithmetic at all.

Name ____________________Date ____________

Tonight's idea

Some sums can be done by drawing. A chart you read with a ruler is called a nomogram. Tonight you build the simplest one. It adds.

Draw three upright lines, equally spaced. Number the left and right lines 0 to 10, one number per square, with both zeros level. Lay a ruler from \(a\) on the left to \(b\) on the right. The middle line is halfway across, so when the ruler reaches it, it has done exactly half of its climb (or drop). Its height there is halfway between \(a\) and \(b\), which is \(\tfrac{a+b}{2}\) squares.

Half the sum is not the sum. So number the middle line twice as fast: two numbers per square, 0 to 20, starting level with the other zeros. Now the point \(\tfrac{a+b}{2}\) squares up carries the label \(a + b\).

Three things must be right: the middle line exactly halfway, equal steps on the two outside lines, and all three zeros level. Then a ruler from \(a\) to \(b\) crosses the middle line at \(a + b\).

Watch one: 3 + 5

a a + b b 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 0 2 4 6 10 12 14 16 18 20 8 zeros level
The ruler from 3 to 5 crosses the middle line 4 squares up, at 8. Short ticks on the middle line are the odd numbers.

Put one end of the ruler on 3 on the left line and the other on 5 on the right line. Going across, the ruler climbs \(5 - 3 = 2\) squares. Halfway across it has climbed 1 square, so it crosses the middle line \(3 + 1 = 4\) squares up. That matches \(\tfrac{3+5}{2} = 4\).

The middle line has two numbers per square, so 4 squares up the label is 8. The ruler has worked out \(3 + 5 = 8\). Swap the ends and the ruler drops from 5 to 3, but it still crosses at 8.

Your turn

  1. Build it. On squared paper draw three upright lines, each 10 squares tall, with 4 squares between neighbours. Number the left and right lines 0 to 10 from the bottom, one number per square. Number the middle line 0 to 20, two numbers per square. Check the three zeros are level. Test it on \(3 + 5\), \(10 + 10\) and \(0 + 0\). Then use it for \(2 + 7\) and \(4.5 + 1.5\). For those two, say how many squares up the ruler crosses the middle line, and what label is there.
  2. Run it backwards to subtract. For \(13 - 6\), put the ruler on 13 in the middle and 6 on the left, and read the right line. Then try \(4 - 7\). What must you do to the right line to read the answer?
  3. Sam's nomogram has one fault. The middle line is 2 squares from the left line and 6 squares from the right line. Everything else is correct. What does Sam's ruler read for \(3 + 5\)? For \(4 + 4\)? What is the lesson about testing?
  4. Drawing can solve equations too. To solve \(x^2 = x + 1\) (here \(x^2\) means \(x \times x\)), draw axes using 4 squares for each unit, with \(x\) from \(-1\) to 2 and \(y\) from 0 to 4. Plot \(y = x^2\) at \(x = -1, -0.5, 0, 0.5, 1, 1.5, 2\) and join the points with a smooth curve. Rule the straight line \(y = x + 1\) through \((-1, 0)\) and \((2, 3)\). Read off \(x\) at each crossing, to one decimal place.

Stretch: make it multiply

Use the bottom 5 squares and change the numbers. On the outside lines write 1 at the bottom and double with every square: 1, 2, 4, 8, 16, 32. On the middle line write 1 at the bottom and double with every half square, up to 1024. Lay the ruler from 4 to 8. What does it read? Try 2 and 16. What is the machine doing now, and why? Roughly where on an outside line should 3 go?

Answers

  1. \(3 + 5\): the ruler crosses 4 squares up, at 8. \(10 + 10\): the ruler lies level across the tops and reads 20. \(0 + 0\): it lies along the bottom and reads 0. If your readings are a little out, check that the middle line is exactly 4 squares from each outside line and that the zeros are level. \(2 + 7\): the ruler climbs 5 squares, so halfway across it is \(2 + 2.5 = 4.5\) squares up, where the label is 9. \(4.5 + 1.5\): the ruler drops 3 squares, so halfway across it is \(4.5 - 1.5 = 3\) squares up, where the label is 6. Numbers between the marks work as well as whole numbers.
  2. \(13 - 6 = 7\). The 13 is 6.5 squares up, half a square above the 6 on the left, so the ruler climbs another half square and hits the right line at 7. This works because \(6 + 7 = 13\): a subtraction is an addition with a different number missing. For \(4 - 7\) the ruler starts at 7 and has dropped 5 squares by the middle (the 4 is 2 squares up), so it drops 5 more and ends 3 squares below the zero. Extend the right line downwards and number it \(-1, -2, -3\). The answer is \(-3\).
  3. Sam's middle line is a quarter of the way across, so the ruler has done only a quarter of its climb. For \(3 + 5\) that is a quarter of 2, which is 0.5, so it crosses 3.5 squares up and reads 7. Wrong. For \(4 + 4\) the ruler is level, crosses 4 squares up and reads 8. Right. Sam's nomogram is right only when the two numbers are equal. The lesson: one test that passes does not prove a tool works. Test with different kinds of numbers.
  4. The curve goes through heights 1, 0.25, 0, 0.25, 1, 2.25 and 4. The line crosses it twice, at about \(x = 1.6\) and \(x = -0.6\). Check: \(1.6 \times 1.6 = 2.56\) and \(1.6 + 1 = 2.6\). Also \((-0.6) \times (-0.6) = 0.36\) and \(-0.6 + 1 = 0.4\). Close, not equal. A drawing gives approximate answers. The true values are \(1.618\ldots\) and \(-0.618\ldots\), with decimals that never end.

Stretch. On the outside lines, 4 is 2 squares up and 8 is 3 squares up, so the ruler crosses the middle line 2.5 squares up. That is 5 half squares, so 5 doublings from 1, which is 32. And \(4 \times 8 = 32\). For 2 and 16 the heights are 1 and 4, the ruler crosses 2.5 squares up again, and \(2 \times 16 = 32\). The machine multiplies. Heights now count doublings, the nomogram still adds heights, and multiplying two numbers adds their doublings: 2 doublings and then 3 more make 5. The 3 goes between 2 and 4, a little above halfway, about 1.58 squares up. Scales like these are called logarithmic. A slide rule multiplies in the same way, by adding lengths on logarithmic scales.

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