Thousands and Millions
Problem
Thousands and Millions printable worksheet
Do human beings live for as long as a million hours?
If you have been alive for a million seconds, how many birthdays have you had?
What year was it one billion minutes ago?
How long would it take to count to a million?
Suppose you were worth your weight in £1 coins. How much would you be worth?
Could you fit the population of London into one hundred thousand double-decker buses?
Could you run one thousand metres in one minute?
Could you eat exactly one tonne of food in a year without altering your weight dramatically?
Could you walk as much as one hundred thousand miles during your lifetime?
Could one thousand drink cans fit into one cubic metre?
The questions are taken from Tony Gardiner's Maths Challenge, Book 1, Oxford University Press 2000.
Getting Started
You could start by selecting some of the questions you consider to be easier.
You could try changing the 1000000 to 100 to make the problem simpler while you establish a correct method.
You might want to do a little research or experimentation to get some numbers to work with.
Student Solutions
Here are a few of the solutions we received:
Do human beings live for as long as a million hours?Amy from Frances Bardsley School for Girls concluded that most of us don't:
24 hrs = 1 day365 days = 1 year
24 x 365 = 8760 hrs in a year
8760 x 79 (that's the rough age humans live for) = 692,040 hours, so we do not live for 1million hours.
And students from Jumeirah Primary School came to the same conclusion:
No, because the average human lives for 80 years. This equals 700800 hours, which is less that 1000000.
Saad from JPS in Dubai worked out that just a few of us do:
24 hours = 1day.24 hours x 365 = 8760 hours per year.
10 years = 87600 hours
50 years = 43800 hours
100 years = 876000 hours
105 years = 919800 hours
110 years = 963600 hours
112 years = 981120 hours
114 years = 998640 hours
114.10 years = 999516 hours
114.13years = 999788.8 hours
114.1551 years = 999998.676 hours
114.1552 years = 999999.52 hours
114.1553 years = 1000000.428 hours
If you have been alive for a million seconds, how many birthdays have you had?
Linden from HighdownSchool worked out that there would be no birthdays:
1000000 seconds / 60 =16667 mintues16667 mintues / 60 = 278 hours
278 hours / 24 = 12 days
Birthdays = 0
Alex from Welsey Collegecame to the same conclusion:
How long would it take to count to a million?
Alex from Toot Hill's Junior Maths Academy (Mathletes) had a go at this problem:
Thank you all for your clear explanations.
Teachers' Resources
Why do this problem?
To give students an opportunity to work with large numbers.
To give students an opportunity to work on problems that require more than one simple step.
Possible approach
This printable worksheet may be useful: Thousands and Millions
- Discuss the kinds of thinking required for these types of questions.
"If I were to ask you to work out answers to these questions, what information would you need to know?"
"Some of these things you can work out from things you already know, some of them need an estimate."
- Let pairs/groups of students select the questions they want to do, and then work together, on paper.
"You're going to be comparing answers with other groups, so make sure that you have written your final thinking carefully, to make it easier to spot differences."
"Each question should be done on a separate sheet of paper."
- Students stick finished questions on the board, next to other solutions to the same problem, they can check whether theirs is the same/different to what other groups have done.
"If two groups work on the same question and get different answers, it may be because their estimates are different, or it may be because one method is wrong."
- Teacher to monitor board and organise "case conferences" where answers or methods are different - i.e. representatives from the different groups come together and troubleshoot each others' working, then feed back to their own group.
Key questions
How can I ensure that I make reasonable estimates?
Possible support
The process of adapting difficult problems, "making sense" of them in simpler cases, is a powerful technique for dealing with hard questions. This may be an appropriate moment at which to model this:
Pick one question to work on as a class task.
Ask the students to pose related, smaller questions that they can answer, and build up a body of work (display work?) around the theme, for example:
Then encourage students to work towards the original question.
Possible extension
Which of these questions is the hardest, and why? Have a go at answering it.
Suggest 10 more questions using large numbers, decide whether they are one star, two star or three star in difficulty. What do you think comprises a three star answer? Solve a couple, or swap with another fast student and solve some of theirs.