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Thousands and Millions

Age 11 to 14
Challenge Level

 

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?"

Make a list on the board.

"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?

How can I set out my working most clearly?

 

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:

Your age if you are 100 seconds old, 1000 seconds old, 10000 seconds old, etc.
The age in months/days/hours/minutes or seconds of people in the class, their siblings, etc.
The time since key moments in history (or sport, etc) measured in different units.

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.