## IM Commentary

The purpose of this task is to use scientific notation in the context of choosing units to report quantities. The teacher may wish to have students reflect on how the choice of units in each part either helps or hinders our ability to grasp the quantity under consideration. There is always a wide choice of units available and we should try to choose units which are most appropriate for communicating the size of an object without further calculations or conversions. Scientific notation is particularly useful when the quantities being described are very small or very large (though a good choice of units can often remove the need for these small or large quantities).

In part (a), students have an intuition, based on experience, for how much 1 gigabyte of storage is and so expressing the computer memory as 128 gigabytes is helpful. When we use scientific notation and write this in terms of bytes or kilobytes, the number is very large and does not carry much useful information except for an expert. For the situations presented in (b) and (c), neither seconds nor teaspoons is an appropriate unit of measure to give a good intuition for the quantities in question. There are many reasonable choices. For (b), hours is probably the best choice but even minutes or days would be better than seconds. For part (c), students will likely need a conversion factor. In metric units, there are about 4.9 ml in a teaspoon. In metric units, the best choice for giving an intuition for the size of the swimming pool is probably cubic meters. In standard units, gallons are too small: cubic feet or yards would be appropriate but the conversion will be difficult.

In extreme situations like the helium atom example in part (d), any choice of units will have both advantages and disadvantages. Picometers are a natural choice for measuring an atom in the sense that the number of picometers needed is a reasonable size number. On the other hand, we have little physical experience that allows us to grasp what a picometer is. If we choose the more comfortable unit of meters, then we can see that it takes many, many helium atom diameters to make a meter: unfortunately, it is difficult to imagine billions of atoms and so this also has a downside. When dealing with very large or very small quantities, this dilemma always exists and it is only by gaining familiarity with the given context that the units and quantities become more familiar and comfortable.