Thought for the day: The multiplication table contains many amazing patterns that can lead to surprisingly deep discoveries.

Concepts: adding, subtracting, multiplying, and factoring polynomial expressions, sums of cubes (possibly triangular numbers)

Examples of noticing and wondering

I notice two pictures that show identical portions of the upper-left corner of a multiplication table.
I notice that each picture highlights a different pattern within the tables.
I notice that the corners of the "backward Ls" in the left picture are always square numbers.
I notice that the sum of the numbers in each "L" is always a perfect cube.
I notice that the sum of each row in the right picture is a multiple of the sum of the first row.

I wonder why the sum of each number in an "L" pattern is always a cube number.
I wonder what the sum of all 25 numbers in each table is.
I wonder if there is a fast way to calculate this sum.
I wonder if the patterns I am seeing will hold up when I extend the multiplication table.
I wonder what would happen if I modified the table to include fractions and mixed numbers.
I wonder if I can find other interesting patterns by looking at other rectangular regions of the multiplication table.

Creating something new

Students may use some of the things that they wondered as inspirations to create new pictures and ideas. For example, they may:

Extend the tables and test that their patterns continue to hold.
Insert "halves" fractions and numbers in the table and see which patterns continue to hold.
Create different patterns in the tables and use them to discover, develop, and test new algebraic relationships.

Notes

For a more complete exploration of these images and other patterns in the multiplication table, see the activity Multiplication Table Algebra on the Deep Algebra Projects page of this site.

In the left picture, the sum of the numbers within each "backwards L" is a cube number. The sum of all twenty-five numbers is

You may check that this pattern continues by understanding what causes it. By reordering and regrouping the numbers in each "L," you may rewrite the sum as n groups of n-squared, which is equal to n-cubed.

In the right picture each row is a multiple of the sum 1 + 2 + 3 + 4 + 5. The sum of all twenty-five numbers is

1(1 + 2 + 3 + 4 + 5 ) + 2(1 + 2 + 3 + 4 + 5 ) + 3(1 + 2 + 3 + 4 + 5 ) + 4(1 + 2 + 3 + 4 + 5 ) + 5(1 + 2 + 3 + 4 + 5 ) =
(1 + 2 + 3 + 4 + 5)(1 + 2 + 3 + 4 + 5) =
(1 + 2 + 3 + 4 + 5) ^ 2

This pattern generalizes to larger square regions of the multiplication table, leading to the conclusion:

Students who are familiar with triangular numbers may rewrite this as