Pyramids

Further practice

In pairs, use these templates to create questions for each other.  Start easy then make the questions more difficult.  Try to include the following:

• Negative numbers

• Decimals

• Fractions (including improper fractions and mixed numbers)

• A mixture of the above

Make sure you can do the questions yourselves, if you are unsure about any of the topics, click on the hyperlinks above for a video of how to do them.

It is important that you show steps!  If you need further help checking that your steps are correct, use this.

When your partner has completed the questions you made for them, you need to check their working and make sure they are correct.  If they are not correct, you need to teach them.

Now try the questions below:

2. Now take some blank pyramids (you may want to start with a size 3 and work up to a size 5) and answer the following questions:

• If all the numbers on the bottom row of a pyramid go up in a regular amount (arithmetic sequence), what is the relationship with the top number?

• If all the numbers on the bottom row of a pyramid are multiplied by a common multiplier (geometric sequence), what is the relationship with the left-hand diagonal and the top number?

• If all the numbers on the bottom row of a pyramid are consecutive triangular numbers, establish that the top number will always be the sum of two squares.

3. Take a look at the many pyramids below that involve manipulating algebraic expressions:

Further Questions and Challenges

Part 1

• Let's now move on to a Size 5 Pyramid.  Can you work out what the top number is in terms of the variables on the bottom row (WITHOUT working out the numbers on the middle row)? What is your rule?

• Look at your Sizes 3, 4 and 5 pyramids, what pattern do you see?  How might this pattern continue?  

1. • Test your conjecture with size 6, size 7 pyramid.

   • What is the rule for a size n pyramid?

• How many numbers do you need to be given to solve a pyramid of a given height i.e. a Size, 3, 4, 5...? Does it matter where the numbers are?

• Back to the questions in the introduction section.  For a Size 3 Pyramid, if we have 3 different numbers and place them randomly on the bottom row, what is the probability that we will get the highest possible total? 

• Repeat for a Size 4, 5, 6, ... n Pyramid.  What rules can you spot?

• Explain in terms of the variables on the bottom row how you would make the top expression as large or as small as possible. 

Part 2

Now take some blank pyramids (you may want to start with a size 3 and work up to a size 5) and answer the following questions:

• If all the numbers on the bottom row of a pyramid go up in a regular amount (arithmetic sequence), what is the relationship with the top number?

• If all the numbers on the bottom row of a pyramid are multiplied by a common multiplier (geometric sequence), what is the relationship with the left-hand diagonal and the top number?

• If all the numbers on the bottom row of a pyramid are consecutive triangular numbers, establish that the top number will always be the sum of two squares.

Part 3 - Game

Have a go at this Two-player game Pyramid Game on a Size 3 Pyramid with your partner:

1. Each player starts with 500 points.

2. Player A completes a 3-storey pyramid without showing Player B 

3. Player B then “buys” information from Player A to try to complete the pyramid – the less Player B spends, the better.

• The top number costs 5 points, 

• A number of the middle row costs 10 points, 

• And a number on the bottom row costs 20 points. 

• Think carefully: Which numbers should you buy? Why is that a good strategy?

4. When Player B has enough information, complete the pyramid and show it to Player A. 

• If it’s completely correct, Player A pays Player B 50 points; 

• If there are any mistakes, Player B pays Player A 50 points.

5. Now swap over.  The winner is the one with the most points after the round.

Round 2:

If you have time, repeat the game on a Size 4 Pyramid

• • The top number costs 5 points, 

  • A number of the second row costs 10 points, 

  • A number of the third row costs 20 points, 

  • And a number on the bottom row costs 40 points. 

  • If it’s completely correct, Player A pays Player B 100 points; 

  • If there are any mistakes, Player B pays Player A 100 points.

Part 4

Take a look at the many pyramids that involve manipulating algebraic expressions below:

Extension

The rules in Part 3 above are linked to something called Yang Hui’s Triangle (楊輝三角). Pascal Triangle after the 13th Century Chinese scholar of that name.

• Mathematicians in Persia (Iran) and South Asia were using the triangle as early as the 9th and 10th centuries. In Iran, it is known as the Khayyam triangle, after Omar Khayyam.

• In the Western world it is known as Pascal’s Triangle after 17th Century French mathematician and philosopher, Blaise Pascal.

• To find out more about this very interesting triangle, take a look at "Alice in Fractaland" here on mathigon.