Thousands and Millions

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.