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Counting on Letters

The letters of the word ABACUS have been arranged in the shape of a triangle. How many different ways can you find to read the word ABACUS from this triangular pattern?

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Pair Sums

Five numbers added together in pairs produce: 0, 2, 4, 4, 6, 8, 9, 11, 13, 15 What are the five numbers?

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Summing Consecutive Numbers

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

Cinema Problem

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Why do this problem?

This problem involves a significant 'final challenge' which can either be tackled on its own or after working on a set of related 'building blocks' designed to lead students to helpful insights. It requires a lot of mental calculations involving money, and could provide good practice of these skills while students also work on problem solving strategies.

Initially working on the building blocks then gives students the opportunity to work on harder mathematical challenges than they might otherwise attempt.

The problem is structured in a way that makes it ideal for students to work on in small groups.

Possible approach

Hand out a set of building block cards (Word, PDF) to each group of three or four students. (The final challenge will need to be removed to be handed out later.) Within groups, there are several ways of structuring the task, depending on how experienced the students are at working together.

Each student, or pair of students, could be given their own building block to work on. After they have had an opportunity to make progress on their question, encourage them to share their findings with each other and work together on each other's tasks.

Alternatively, the whole group could work together on all the building blocks, ensuring that the group doesn't move on until everyone understands.

When everyone in the group is satisfied that they have explored in detail the challenges in the building blocks, hand out the final challenge.

The teacher's role is to challenge groups to explain and justify their mathematical thinking, so that all members of the group are in a position to contribute to the solution of the challenge.

It is important to set aside some time at the end for students to share and compare their findings and explanations, whether through discussion or by providing a written record of what they did.
 
A teacher comments:
Working on this problem reminded students that maths is sometimes best done by guessing and trying to improve on that guess. Could be used for sequences as well as Trial and Improvement.

Key questions

What important mathematical insights does my building block give me?
How can these insights help the group tackle the final challenge?
How many £10 shall I include?
If I swap a 50p for a 10p, what does that do to the total amount of money?

Possible extension

Of course, students could be offered the Final Challenge without seeing any of the building blocks.

What happens when the prices change to:
£10 for adults
£1 for pensioners
50p for children
How many solutions are there this time?

Possible support

Encourage groups not to move on until everyone in the group understands. The building blocks could be distributed within groups in a way that plays to the strengths of particular students.