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

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

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?