Cannon Balls
Identify a pattern and describe this using their own words
Use equipment appropriately when exploring mathematical ideas; devise and use problem solving strategies to explore situations mathematically (guess and check, make an list).
In this problem students can start to develop a feel for 2- and 3-dimensional objects (or can reinforce what they already know). The basic shape of each of the layers of the pyramid of cannon balls is a triangle (actually an equilateral triangle as all sides are equal). So the problem will give the students the opportunity to discuss triangles.
And that’s the cue for triangular numbers. The numbers of cannon balls in each layer are triangular numbers. These become of more interest further up the school, where students my look at square numbers, pentagonal numbers and so on. The pictures below show why these numbers are named after geometric objects.
The first three triangular numbers The first three square numbers
The triangular numbers take on even more importance in the secondary school, where they take on new roles as part of the family of Binomial Coefficients. These numbers have a major part to play in counting, and so are vital to probability and statistics generally.
In order for the cannon balls to sit on top of each other, the students will have to realise that one ball will comfortably fit on top of three others. They may need to be shown this using tennis balls or oranges.
This problem will fit nicely into the Pirate unit of the Ministry of Education Publications: Teaching Problem Solving in Mathematics: Years 1 - 8, reference to this publication is available from Problem Solving References page
Problem
There is a pyramid of cannon balls on a pirate ship. The first layer looks like this:
How many cannon balls are there in the first layer?
How many cannon balls will there be in the second layer?
How many cannon balls will there be in the third layer?
How many cannon balls in the top layer?
How many cannon balls do you need to complete the pyramid?
Teaching Sequence
- Introduce the problem using a number of cannon balls (tennis balls or similar). Ask the students to think of ways that they couild stack the balls.
- Read problem. Check that the students understand the meaning of the word 'layers' and also know how the pirates piled up their cannon balls.
- Ask the students to guess how many cannon balls they will need. Record the estimates to check against later.
- Brainstorm for ways to solve the problem. (Link these to problems that they have solved before.)
What strategies could you use?
What equipment will you need?
How will you record your information?
What do you have to find out? - As the students work ask questions that focus on the patterns they are using to solve the problem.
What can you tell me about the cannon balls?
How are you keeping track of the number of cannon balls? - Share solutions
Other contexts
Sometimes fruit is piled this way at supermarkets.
Extension to the problem
If the pirates wanted to put another layer of cannon balls on their pile they would need to lift it up and put another triangle on the bottom. How many cannon balls would there be in such a layer?
Solution
In the first layer there are 1 (across the top) + 2 and 3 (across the middle two rows) + 4 (four at the bottom) = 10 cannon balls.
In the second layer there are 1 + 2 + 3 = 6 cannon balls.
In the third layer there are 1 + 2 = 3 cannon balls.
There is only one cannon ball in the top layer.
All together there are 10 + 6 + 3 + 1 = 20 cannon balls.
Solution to the extension:
If a layer of cannon balls is put underneath the present bottom one it would need 1 + 2 + 3 + 4 + 5 = 15 cannon balls.
| Attachment | Size |
|---|---|
| Cannon.pdf | 70.15 KB |
| CannonMaori.pdf | 79.84 KB |
Similar Resources
Can Stack
Find the rule for summing consecutive numbers
Identify the pattern of triangular numbers.
Devise and use problem solving strategies to explore situations mathematically (be systematic, use algebra).
Counting Pills
Express rules in words
See more than one rule for a given pattern
Devise and use problem solving strategies to explore situations mathematically
Toothpick squares
State the general rule for a practical situation.
Devise and use problem solving strategies to explore situations mathematically (guess and check, be systematic, make a drawing, use equipment, make a table).
Shaking Hands
Describe in words the rules for the pattern
Identify the pattern of triangular numbers
Devise and use problem solving strategies to explore situations mathematically (systematic list, draw a picture, use equipment).
Triangular Number Links
Use algebra to simplify expressions
Use geometry to assist their algebra.
