# Bigger or smaller?

When you change the units, do the numbers get bigger or smaller?

When measuring some property of an object the number we measure depends on the units chosen. For example, 1cm = 0.01m, so converting from cm to m makes the number get smaller; we need more small units to make up the number of big units. In each case below, does the number get bigger or smaller following a change in units? Can you estimate without a calculator an approximate factor by which the
numbers would change in each case?

- 1 cm $^2\rightarrow ??$ m $^2$
- 1 foot $\rightarrow ??$ inches
- 1 mile $\rightarrow ??$ kilometers
- 1 litre $\rightarrow ??$ cm $^3$
- 1 foot $^3\rightarrow ??$ inches $^3$
- 1 m s $^{-1}\rightarrow ??$ miles / hour
- 1 mm $^3\rightarrow ??$ m $^3$
- 1 degrees C $\rightarrow ??$ degrees K
- 85 degrees $\rightarrow ??$ radians
- 1 Pa $\rightarrow ??$ cm$^{-1}$ g s$^{-2}$
- 1 W $\rightarrow ??$ cm$^2$ g s$^{-3}$
- 1 Hz $\rightarrow ??$ per minute
- 1 Mol $\rightarrow ??$ trillion
- Molarity of 1$\rightarrow ??$ per cm$^3$
- 1 katal $\rightarrow ??$ million million per picosecond

Can you make up some of your own similar problems?

**Other problems**

Try the fun Zin Obelisk task from the main NRICH site

- 1 cm $^2\rightarrow $ 10$^{-4} m ^2$
- 1 foot $\rightarrow $ 12 inches
- 1 mile $\rightarrow $ 1.6 kilometres
- 1 litre $\rightarrow $ 1000 cm $^3$
- 1 foot $^3\rightarrow $ 1728 inches $^3$
- 1 m s $^{-1}\rightarrow$ 2.25 miles / hour
- 1 mm $^3\rightarrow$ 10$^{-9} m ^3$
- 1 degrees C $\rightarrow $ 1 degrees K
- 85 degrees $\rightarrow $ 1.48 radians
- 1 Pa $\rightarrow $ 10 cm$^{-1}$ g s$^{-2}$
- 1 W $\rightarrow $ 10$^{7} $cm$^2$ g s$^{-3}$
- 1 Hz $\rightarrow $ 60 per minute
- 1 Mol $\rightarrow$ 6$\times$10$^{11}$ trillion
- Molarity of 1$\rightarrow$ 6$\times$10$^{17}$ per cm$^3$
- 1 katal $\rightarrow$ 6$\times$10$^{-1}$ million million per picosecond

### Why do this problem ?

Conversion between, and use of, units is a critical skill in the sciences, and one which often leads to confusion. This problem will encourage students to understand the relationships between various types of units, as well as possibly introducing them to new important scientific units.It will also help to embed the important skill of checking numerical answers to see if they make sense in terms of orders of magnitude.### Possible approach

This question works well through discussion in pairs. Remind
the students that common sense works well when dealing with
units.

### Key questions

In each case which is the small unit and which is the large
unit?