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Calculation 2000

Weekly Problem 49 - 2013
What is the value of 2000 + 1999 × 2000?

Going Round in Circles

Stage: 3 Challenge Level: Challenge Level:1

Why do this problem?

This problem encourages students to consider a range of alternative methods of calculation. There is an opportunity for students to appreciate the importance of the quotient and the remainder when using division to solve a problem. When using calculators, students can explore the relationship between the remainder and the decimal part of the answer.

Possible approach

This printable worksheet may be useful: Going Round in Circles.

Work on this problem complements the ideas introduced in Shifting Times Tables.

"If today is Monday, it will be Monday again in 7 days, 770 days, 140 days."
Pause, and ask students to suggest other numbers of days. Keep a record on the board as they are suggested, without any comment. Encourage them to suggest some large numbers. If none are forthcoming, perhaps ask "Does 35035 belong here?" and other similar questions.

"It will be Wednesday in 2 days, 72 days, 702 days, 772 days."
Pause, and ask students to suggest other numbers of days. Keep a record on the board as they are suggested, without any comment.

Once a large collection of correct examples have been recorded, point to some of them and ask the class to explain how they know they are correct.

Ask the class to devise a way of working out what day it will be in 1000 days' time, initially without a calculator and then with a calculator. Share strategies, and if Ann's or Luke's methods (below) have not been suggested, show them the half-completed methods and ask them to finish them off.

Ann's Method:

"It will be Monday in $700$ days, $770$ days, $840$ days... "
Can you suggest how Ann might continue?

Luke's Method:

"On my calculator, I can work out that $1000 \div 7 = 142.8571429$.
Then I can work out $142 \times 7$..."
Can you suggest how Luke might continue?

Hand out this worksheet, which has space for students to keep a record of their methods. Perhaps organise students to work in pairs, and challenge them to find the most efficient mental, written, and calculator methods for each question.

Once the seven set questions have been answered, ask students to generate similar questions (perhaps using random numbers between 100 and 10000), and to suggest efficient methods for answering their questions.

Round and Round and Round encourages similar ways of working with division and remainders on a calculator, in the context of positions around a circle.
Students could use the insights from this problem to analyse the game GOT IT.

Key questions

How can I use what I know (small multiples of 7, 12, 360...) to generate number facts that are not at my fingertips (large multiples of 7, 12, 360...)?

When I divide using a calculator, how does the answer on the screen help me to work out the quotient and remainder?


Possible extension

Days and Dates begins in the same way as this problem, but then encourages exploration of the algebra behind modular arithmetic.

Modular Arithmetic is an article introducing the notation and uses of modular arithmetic.

Possible support

Colour Wheels provides a simple context for exploring repeating patterns.