Approximately certain

Estimate these curious quantities sufficiently accurately that you can rank them in order of size

Age

14 to 18

Challenge level

Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving

Being curious Being resourceful Being resilient Being collaborative

Problem

In this problem we look at several sets of physical quantities. Some of the quantities are very precisely stated and will be possible to work out exactly. Others might be clearly stated, yet will defy an exact calculation. Others might not be clearly stated: you will need to state some more assumptions or do some research before a meaningful approximation might be made (be scientific about this process).Whilst it might not be possible to calculate exactly each value, each does have a value: you are required to arrange each of these in order of magnitude . Be sure to justify your ordering with scientific and mathematical rigour.

The energy:

Used to walk up the steps of the Burk Dubai skyscraper ($818$m)
Contained in a full-sugar can of coke
Contained in a single atom of lead (according to Einstein's equation $E=mc^2$)
Needed to boil a kettle of tap water

The time taken:

For a radio wave to travel halfway around the world
For a top sprinter to run $1$mm at top speed
For the end of the second hand on a watch to move a distance of $1$ micron
For a test tube of hydrogen gas to fully combust when exposed to a flame

The distance:

You could jump vertically up on the surface of the moon
You could throw a tennis ball-sized lump of lead
Between peaks of two sound waves caused by two successive hand claps in your fastest possble round of applause
You can run in $1$ second

The mass:

Of the atmosphere
Of all of the people in the world
Of the north polar ice cap
Of all of the living bacteria presently living on Earth

Can you make up other similar quantities? Be inventive!

Teachers' Resources

Why do this problem?

This problem gives an excellent workout in estimation and calculation using a wide range of physical equations and situations. It is rather open, and will particularly benefit those students used to following recipes in their work. It also highlights the fact that in science it is rather hard to calculate anything without making some sort of assumptions. Good science will clearly state and be aware of these assumptions; bad science will ignore them.

Possible approach

This problem could be approached in two stages. First of all the problem could be discussed as a group without any calculations being made (except in students' heads or on the back of an envelope). Once the issues are uncovered, students might wish to begin calculation or might need to turn to the internet or other resources for more information.

It is likely that students' answers and approximations will vary. Once the task is finished, groups could feed back to the rest of the class. Is their reasoning and explanation clear? They will need to convince the rest of the class that their ordering is correct. This could be done with each group attempting each of the four different sets of data. Alternatively, different groups could attempt different parts of the task, in which case the explanations might need to take on a more detailed focus.

There are two different levels at which the problem might be approached

Basic: Produce a means of calculating each part with various estimated values of the data. Order the answers.

Advanced: Produce definite upper and lower bounds for the quantities, using upper and lower bounds for the input data. The ordering is only guaranteed when these intervals do not overlap.

Don't lose sight of the fact that the ordering is important. If a very crude approimation shows that one of the quantities is clearly largest or smallest, then that is sufficient. Of course, students might be interested in computing a more accurate answer out of general interest.

This worksheet has all the quantities.

Key questions

What is precisely stated and what is not precisely stated?
What factors would complicate the most accurate calculation? How can we deal with these? Which factors can we neglect and which are important?
Can you give quick, sensible lower and upper bounds on the quantities before attempting a computation?
Is a detailed computation necessary for all of the parts of the problem?

Possible extension

The most able students should be required to approach the task with the most rigour. They might also consider the best way to represent their results. What accuracies are most relevant? Is a linear measurement scale suitable?

Possible support

Focus on the basic method of approaching the task, as mentioned in the possible approach.

Or search by topic

Number and algebra

Geometry and measure

Probability and statistics

Working mathematically

Advanced mathematics

For younger learners