Daily Dollar: Cannon balls.

Cannons were used in battles from the 1300’s until 1890’s. The cannon balls which they fired were round and heavy! They were often stacked ready to be loaded into the gun in pyramids with square bases.

The 20 Dollar Question. In the picture the cannon balls are arranged as a pyramid with a square base. How many cannon balls are in the pile?

The 40 Dollar Question.

How many cannon balls are stacked here? The 60 Dollar Question.

In this photograph from the American Civil War, the cannon balls have been stacked as a pyramid. How many cannon balls are there?

The 80 Dollar Question.

This pyramid stack has a base which has 7 x 7 cannon balls. How many cannon balls are in the stack? The 100 Dollar Question.

Here is a pyramid stack of small cannon balls. There seems to be lots here! How many cannon balls are there in the stack?

Answers Daily Dollar Questions. Cannon balls. The 20 Dollar Question. Cannon balls in the pile are 42 + 32 + 22 + 12 = 30 The 40 Dollar Question. Cannon balls in the stack are 32 + 22 + 12 = 14 The 60 Dollar Question. Cannon balls in the stack are 52 + 42 + 32 + 22 + 12 = 55 The 80 Dollar Question. Cannon balls in the stack are 72 + 62 + 52 + 42 + 32 + 22 + 12 = 140 The 100 Dollar Question. Cannon balls in the stack are 152 + 142 + 132 + 122 + 112 + 102 + 92 + 82 + 72 + 62 + 52 + 42 + 32 + 22 + 12 = 1240 Teacher Notes. • The purpose of these questions is to get students to look for

patterns. • The first two questions are easily done by counting. • From these students will notice a pattern of square numbers. • Once they have confirmed a pattern with the third question they

will be able to answer the others. • The 100 dollar question is quite tricky. • There is a formula although it is a bit complicated for Level 3

students. • The formula is n = 1/6 k (k + 1) (2k + 1)

k is the number of cannon balls across the bottom layer. • Cannon ball questions have been around for centuries! • The numbers 1, 5, 14, 30, 55, 91, 140, …

are often called square pyramid numbers. • Triangle pyramid numbers have a triangle base. They are often

called tetrahedron numbers. 1, 4, 10, 20, 35, … They can be found in Pascal’s triangle.