Conversion Graphs

Water freezes at $0$ °C and $32$ °F. It boils at $100$ °C and $212$ °F. Work out the ratio of the Fahrenheit value to the Celsius value at the boiling point. Now try the same at the freezing point. Say what goes wrong, and what that tells you about the type of relationship.

A conversion graph is a straight line drawn so that a value in one unit can be read off against the equivalent in another. Because most unit conversions are direct variation, the line passes through the origin and its gradient is the conversion factor.

A conversion graph puts one unit on each axis and draws the line relating them. Once drawn, converting a value takes no arithmetic: go up from one axis to the line, then across to the other.

Most unit conversions are direct variation, and it is worth seeing why. Converting kilometres to miles multiplies by a fixed factor, so doubling the distance doubles both readings, and zero kilometres is zero miles. Both conditions from Lesson 3 hold, so the graph is a straight line through the origin.

The same is true of kilograms to pounds, litres to gallons, dollars to another currency at a fixed rate, and centimetres to inches. In each case:

$$\text{second unit} = k \times \text{first unit}$$

and the conversion factor $k$ is the gradient of the line.

For kilometres to miles the factor is about $0.62$, so $50$ km reads as about $31$ miles, which is the reading shown in the diagram. Reading it the other way, $31$ miles reads back as about $50$ km, using exactly the same line.

A conversion graph needs one known equivalence and, for a proportional conversion, the origin.

1. Choose which unit goes on each axis, and label both with their units.
2. Mark the origin, which is on the line for any proportional conversion.
3. Plot the known equivalence, choosing a large value so the point sits well away from the origin.
4. Draw the straight line through both and extend it across the useful range.

To build a kilogram-to-pound graph from the equivalence $10$ kg $\approx 22$ lb: mark $(0,0)$, mark $(10, 22)$, and join. The gradient is $2.2$ pounds per kilogram, which is the conversion factor.

Two choices worth making deliberately. Pick scales that use most of the page, since a cramped graph cannot be read accurately. And choose the known point far from the origin, for the reason given in Lesson 3: a small plotting error near the origin swings the whole line, while the same error far out barely moves it.

Converting between Celsius and Fahrenheit is not direct variation, and it is the standard example of the distinction that has run through this whole focus area.

$$F = 1.8C + 32$$

The graph is a straight line, so a conversion graph still works perfectly well. But it crosses the vertical axis at $32$ rather than at the origin, since $0$ °C is $32$ °F rather than $0$ °F.

The consequences are exactly those from Lesson 1. The ratio $\tfrac{F}{C}$ is not constant: at $100$ °C it is $\tfrac{212}{100} = 2.12$, at $10$ °C it is $\tfrac{50}{10} = 5$, and at $0$ °C it is undefined. Doubling the Celsius value does not double the Fahrenheit value: $10$ °C is $50$ °F but $20$ °C is $68$ °F, not $100$ °F.

So a Celsius-to-Fahrenheit graph needs two known points to draw, because the origin is not one of them. Any two equivalences will do, and the freezing and boiling points of water are the usual choice.

Linear is not the same as proportional, and temperature is the clearest everyday case. Currency with a fixed fee, and taxi fares with a flag fall, are the same phenomenon.

A value read off a graph is an estimate, and saying so is part of a complete answer.

The precision depends on the scale. On a graph where one centimetre represents $10$ km, a reading is good to perhaps half a kilometre, and quoting an answer to three decimal places would be dishonest. Round to a precision the graph can actually support, and use words such as "about" or "approximately".

When exactness matters, use the equation. The graph gives $50$ km as about $31$ miles; the factor $0.6214$ gives $31.07$ miles. Both are right for their purpose, and the graph is faster for repeated rough conversions while the equation is better for a single precise one.

Interpolating, reading a value from inside the drawn range, is reliable. Extrapolating, extending the line beyond its range, is less so. For a pure unit conversion extrapolation is safe, because the relationship really does continue unchanged. For a conversion fitted to data, or one with a stepped structure such as the printing charges in the previous lesson, it may not, and the honest answer is to say over what range the graph is trusted.

Five items on conversion graphs. Work each one, then reveal the answer.

1. 1 $1$ kg is about $2.2$ lb. Convert $15$ kg to pounds.

   Show answer
   Multiply by the conversion factor.$33$ lb

2. 2 $1$ kg is about $2.2$ lb. Convert $11$ lb to kilograms.

   Show answer
   Divide by the factor, or read the graph the other way.$5$ kg

3. 3 Does a kilometres-to-miles conversion graph pass through the origin?

   Show answer
   Zero of one is zero of the other.Yes

4. 4 Does a Celsius-to-Fahrenheit graph pass through the origin?

   Show answer
   $0$ °C is $32$ °F.No

5. 5 How many known points are needed to draw a proportional conversion graph?

   Show answer
   The origin is free.One, plus the origin