Tree farm

Here is a view of a tree farm from above. How many ways can you determine the number of trees?

Approaches

counting one at a time

Can you find an easier way?

grouping in some way

for example, 6 x 4 + 1

adding rows

1 + 3 + 5 + 7 + 5 + 3 + 1 1 + 1 + 3 + 3 + 5 + 5 + 7

combining rows or columns

(1 + 7) + (3 + 5) + (5 + 3) + 1 = 3 x 8 + 1 2 x (1 + 3 + 5) + 7 = 2 x 9 + 7

adding diagonals

4 + 3 + 4 + 3 + 4 + 3 + 4

using squares

The 4 diagonals of 4 dots form one square and the 3 diagonals of 3 dots form another, to get 4 x 4 + 3 x 3.

Respond: Is this the same as adding diagonals? Elicit ideas of multiplication as repeated addition, commutativity

breaking into pieces

middle column (7) + middle row (6 more) + 4 x 3 in each corner)

using symmetry

by horizontal or vertical reflection symmetry, 2 x 9 + 7

by diagonal reflection symmetry, 2 x 11 + 3

introduce idea of reflective, or mirror, symmetry

Rewrite all approaches as number sentences.

Compare number sentences. What properties of arithmetic say that these sentences are equivalent?

Your experience

Have you used this problem with a class and seen approaches other than(or more specific than) those mentioned above? Or do you have other comments or criticisms or stories? If so, please tell us!

Last updated 30 November, 2004