## IM Commentary

The goal of this task is to study conversion between some volume and weight units. There are many important aspects to these conversions, including:

- Gaining an intuition for the meaning of different units of measure. How big, in every day terms, is a cubic centimeter or cubic millimeter? This intuition can then be used to
- Assess the reasonableness of an answer: is it reasonable that there are about 30 centimeters in a foot?
- Understanding the relationship between multiplication, linear measurements, area, and volume.

The focus of this task is on the last of these but the teacher may wish to take time to let the students experiment with an online conversion program if computer resources are available. For part (a) of this task http://www.asknumbers.com/VolumeConversion.aspx is one online resource while http://www.asknumbers.com/WeightConversion.aspx was used for part (b). Students will perhaps enjoy learning about some of the different measures which they have probably not heard of before, such as 'hogshead' and 'jigger' for volume or 'parsec' and 'dyne' for mass. The picture for part (a) of the solution is taken from a document created by the Dana Center: http://www.utdanacenter.org/k12mathbenchmarks/elementary/measurement.php. The actual conversion from pounds to kilograms is indeed rather subtle, as one of the units takes into account the strength of gravity, and so varies based on your location on earth. Nonetheless, the most standardly quoted conversion factor is that 1 kilogram is approximately 2.20462 pounds.

The first part of the question builds on student knowledge of 6.G.2, relating volume to multiplication of length, width, and height. After finding out how many cubic inches are in a cubic foot, students then apply knowledge of ratios to make additional conversions from cubic inches to cubic feet. The second question gives students an opportunity to think about precision in calculation. They may also wonder what exactly a centimeter or an inch is and might be surprised to find out how technical the definitions of these lengths are.