The 6 million question
How long would it take to count to 6 million?
Big numbers present us with a real challenge! How do we understand their size and the significance of what they represent? Once numbers get past a certain size this gets harder and harder. One, two, three, .....big! There are numerous instances of billions and millions being incorrectly used in journalism and nobody even notices because they are just seen as big numbers. How far is a light year? How big is the national debt of the country you live in? How much money are 'apple' worth? In this activity, the aim is to give some meaning to a big number by working out the length of time it would take to count to it. The task involves calculating a considered estimate of how long it would take you to count up to six million. Take note, that it says calculate and considered. This is quite different from guessing. The task has real value when you consider what is really required to make such an estimate and leads us to a greater understanding of one large number in particular.
Important note of credit - I was first asked to do this during my teacher training by my inspirational tutor Tony Cotton who regularly provoked us to think deeply about mathematical ideas and how to share them with students - Thanks Tony
The task explained!
This task really doesn't need much in the way of resources. In its best form, students are simply given the question and challenged to make necessary considerations. The teacher can then prompt students with ideas and things to check as and when the think it is appropriate. You can also use the video, slideshow and worksheets below to help run the activity...
Here are the slides used to make the video above.
Here is the 6 million question activity sheet
This description of the activity mainly targets teachers. However, it is carefully worded so as not to give anything key away that may diminish the interest or excitement of the activity for the students.
- Students are presented with the question and the task
- Students are left to make considerations on their own
- Teachers can offer prompts as they see fit or hand out the activity sheet included above.
- Groups or individuals take tuns to justify their estimations and compare answers with each other.
- The class might try to make a compromise estimate.
- The teacher then shows the video example to the class as part of explaining why the number six million was used.
Teacher Notes - Why six million?
VERY IMPORTANT - It is important that this task is attempted by students before they read this section or watch the video example.
Things to think about for teachers....
This is not an intuitive task and it is really difficult for students to move from 'guessing' to 'considered, calculated estimate'. Of course this is a key point in understanding number and estimation and very relevant in a world that relies so heavily on the latter! This is the key point for student and teachers.
Help students understand the multiple stages of the exercise - estimating is not guessing and involves some measurement and calculation. The trick, though, is to do this without doing it for them and making it too formulaic. It is best if students do as much of this as they can for themselves.
- How long does it take you to say a number?
- Do some numbers take longer to say than others?
- How can you make a good estimate at the average length of time it takes you to say each of the numbers from 1 to 6 million?
- How long does it take you to say each of the numbers from 2,533,231 to 2, 533,240? How does this help answer the above
- How many seconds are there in a minute, hour, day?
- Have you converted your answer to days or weeks or months?
- Have you considered how many hours a day you should be able to count for?
- How do you take the above into account trying to come up with a total?
- How well laid out and justified is your estimate?
- Have you double checked your calculations?
Converting between units of time is notoriously difficult, probably because of the different base - just don't assume that your students will be fluent in doing this.
Ask students to produce a detailed breakdown of the estimate (I suggest in poster form) because without this they will skip steps. This also provides an excellent tool for comparison and reflection as students can simply go and read the estimates of other groups and see what things others might have considered that they didn't.
'Why 6 million?'
In the next section, there is a video of an example estimate that you might show students along with a gallery of the slides used to make it. This should be used at the end of the activity when students have produced their own. It is not intended in any way as a 'correct answer' but as an example answer that covers many of the details. If students disagree with some of the steps then great - that will make for some good discussion.
On a serious note, the choice of the number 6 million is not accidental. I was fist shown this activity during teacher training by one of my tutors, Tony cotton, and it has stayed with me ever since. We did this activity and came up with a variety of estimates and when Tony reminded us at the end that 6 million Jews were killed in the Holocaust there was a profound silence in the room as we understood the importance of putting numbers in to perspective. Obviously, teachers will manage this their own way. Perhaps you will just show the video that concludes with that statistic, or perhaps you will prefer to show the slide on the hidden image icon below....
An example estimate
Please add comments below if you have used or reflected on this activity....