### Days and Dates

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

### 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?

### Always the Same

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?

# Number Pyramids

## Number Pyramids

Choose three single-digit numbers and write them in the bottom row of the pyramid.
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Try some different numbers.
Can you work out how the numbers in the upper layers are generated?

Here are some questions to consider:

• Given the numbers on the bottom layer in order, can you find a quick way to work out the number at the top?
• If you change the order of the numbers on the bottom layer, will the top number change?
• If you can rearrange the numbers on the bottom layer, can you find a quick way to work out the largest possible number that could go at the top?
• Given the number at the top, how can you come up with possible numbers to go at the bottom?
Test out your observations and insights. You could use big numbers, small numbers, negative numbers, decimals...

Can you explain what is happening?
Can you explain why it is happening?
Can you explain it algebraically

Can you adapt your insights so that they apply to pyramids with more than three layers?

Here is a number pyramid with four layers so that you can test some of your ideas:

Full Screen Version

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To experiment with 5 and 6 layer pyramids, you may find this spreadsheet useful.
You could adapt it to work on even larger pyramids!

For other problems that use this idea go to More Number Pyramids.

### Why do this problem?

This problem requires students to draw firm mathematical conclusions following a period of experimentation. It provides an interesting context for students to begin to appreciate the value of using algebraic notation.

The problem encourages students to ask more general questions beyond the initial context.

### Possible approach

Demonstrate how number pyramids work. This could be done with the teacher remaining silent throughout, and expecting the students to make sense of what is happening.

"What do you notice?"
"What questions do you think a mathematician might ask next?"
Give students time to discuss and suggest. If no suggestions are forthcoming, share the questions suggested in the problem:
• Given the numbers on the bottom layer in order, can you find a quick way to work out the number at the top?
• If you change the order of the numbers on the bottom layer, will the top number change?
• If you can rearrange the numbers on the bottom layer, can you find a quick way to work out the largest possible number that could go at the top?
• Given the number at the top, how can you come up with possible numbers to go at the bottom?
Give students time to work on the questions that have been raised. Encourage them to experiment before trying to draw more general conclusions. As the lesson draws on, make it clear that they are expected to be able to explain and justify any generalisations they make.

If students notice patterns but can't explain them, it may be helpful to introduce algebraic representation.

The class could then move on to More Number Pyramids

### Key questions

How are these numbers generated?
How does the position of a number on the bottom row affect the total at the top?

### Possible extension

With the starting numbers $10$, $1$, $6$, $4$, why is it impossible to make a top total which is a multiple of $3$?
If a 100-layer pyramid had $1$s in every cell on the bottom layer, how could you work out the number at the top?
Students could have a go at Top-heavy Pyramids

### Possible support

"Increase the first number on the bottom layer by $1$. What happens to the total at the top?
Now try with the other numbers. What do you notice?"