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### Number and algebra

### Geometry and measure

### Probability and statistics

### Working mathematically

### Advanced mathematics

### For younger learners

# Top-heavy Pyramids

### Why do this problem?

This problem is simple to explain yet involves quite a complicated
solution process. This problem will hone skills of addition of two
digit numbers whilst challenging the organised mathematical
thinking of students. The problem may be done by trial and error or
with some appeal to algebra.

### Possible approach

### Key questions

### Possible extension

### Possible support

Students who struggle with the level of addition might be provided with a simple spreadsheet to do the calculations. They could also be asked simply to work out $5$ pyramids with different numbers to see who can get the largest number or the answer closest to $200$.

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Age 11 to 14

Challenge Level

- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

It is possible for this problem to be done entirely
individually but a group discussion may lead to more insights about
the strucure of the number pyramid. There are a great number of
possible combinations of base numbers; ideally students should be
encouraged to understand some of the structure of the pyramid in
order to reduce the number of possibilities that they have to try
out.

You might initially discuss the problem as a group. Can anyone
see any structure or offer a solution strategy? Students could then
experiment individually with various combinations of numbers.
Encourage students to devise a clear recording system. Encourage
them to decide sensibly on the next combination of numbers to try
rather than randomly. For example, if a top number is too small
then how can the numbers at the bottom be altered to increase
this?

You could use a simple
spreadsheet to model the pyramid. Could students construct one
of these themselves? This is an interesting structural challenge
which allows the creation and investigation of larger
pyramids.

Key questions should lead to understanding the strucutre of
the pyramids

- What is the total for the order $1, 2, 4, 8, 9, 12$? Would we get the same total with a different order? Why?
- What is the largest possible top number for the pyramid?
- What is the smallest possible top number total for the pyramid?
- Which pairs of numbers can be switched without changing the value at the top of the pyramid?

Once an student has found a solution they could be asked these
questions:

- Are there any other possible combinations of $1, 3, 4, 8, 9, 12$ which lead to the answer?
- What other top numbers are possible? Can you find top numbers which are not possible?
- Are there any other combinations of 6 base numbers which lead to the top number being $200$?

Students who struggle with the level of addition might be provided with a simple spreadsheet to do the calculations. They could also be asked simply to work out $5$ pyramids with different numbers to see who can get the largest number or the answer closest to $200$.

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

Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34?

The well known Fibonacci sequence is 1 ,1, 2, 3, 5, 8, 13, 21.... How many Fibonacci sequences can you find containing the number 196 as one of the terms?