This printable worksheet may be useful: Half a Minute.
Half a Minute
Why do this problem?
This problem offers students the opportunity to apply their understanding of mean, mode, median and range. You could use it to introduce some or all of the mean, mode, median and range, or to build understanding of the differences between them in context.
The problem also challenges students to work systematically whilst promoting reasoning and discussion between the group.
This problem can also provide an opportunity for a class to carry out their own estimations and analyse their own data.
You could begin by having the students estimate a length of time. You could time how long it takes them to settle down as they arrive in the classroom and then ask them for their estimates of the amount of time it took. Alternatively, you could set a timer once you have their full attention and ask them to raise their hands when they think 30 seconds have passed. Praise the most accurate estimations.
Now, project the ‘Half a Minute’ times onto the board and get students to share their initial thoughts on who they think was the best at estimating and why. Once you have a variety of ideas, put students into small groups and challenge them to agree on who was the best at estimating. What maths can they use to back up their answers?
If the students have seen the mean, median, mode and range before, then you could circulate, gently pushing them towards using these statistics and thinking about which are most useful. If they haven’t, then you could circulate and encourage students to come up with their own measures of central tendency and spread. Some good ideas are likely to emerge organically.
Bring the students together for a group discussion. As well as sharing and taking notes on the board about the maths they used, be very clear about what they are measuring with each calculation and how that relates to being ‘better’ at estimating. Note that measures of central tendency (averages) give an idea of how much each character generally over- or under-estimates the time, and measures of spread (e.g. range) give an idea of how consistent they are. Which is more important here? If the students haven’t seen all of the mean, median, mode and range before, then make sure you introduce them now and get students to work in their groups again, incorporating the new measures into their work.
Finally, get students to try it out themselves. Depending on the availability of stop watches, get students to work in pairs and time each other for 30 seconds. You could use the timer here. They could agree on their criteria before starting (e.g. “Whoever has the smallest range is the winner”) and then refine their criteria as they generate more data.
Can they decide who is the best at estimating in the class? Can they agree on it?
How many trials are necessary to see who the best estimator really is?
Does getting exactly 30 seconds once mean that you are the best at estimating?
Which of the summary statistics did you use, and why?
Which is more important, the range or the average?
You could invent some statements for students to agree or disagree with when they initially see the problem, such as “Charlie is the best because he guessed 30 seconds exactly”, “Ben is the worst because he made the worst guess of 19 seconds”, “Anna is the best because her answers are all close to 30 seconds”, etc.
You could revise or introduce the different types of averages and how to find each one (verbally or written) before you begin.
‘Darren’ was even better at estimating 30 seconds. Can students produce a set of times for Darren to show this? How many different sets of times can they produce for Darren?
Students could try the investigations suggested in Estimating Time.