Patterns in The Multiplication Table

The multiplication table is a treasure trove of mathematical patterns! From the time that students first begin learning about multiplication through high school and beyond, students can gain new insights by exploring these patterns.

I wonder what would happen if there were a row and a column for 0.
I notice that including 0 in the table would fit with patterns that I already see.
I wonder if I could extend the table by predicting what the numbers would be without multiplying.
I notice that I could check my answers by multiplying or using other strategies.

I notice that there is a pattern to how the numbers on the grey diagonal change.
I notice the grey diagonal contains a special kind of number.
I wonder if the numbers on the yellow diagonal change in a similar way.
I notice that the yellow numbers are always one less than the neighboring grey ones!
I wonder what causes this pattern.
I wonder if this pattern will continue forever.
I wonder if I can use pictures or algebra to prove my observation.
I notice a diagonal below the grey one that has the same numbers as the yellow diagonal.
I wonder why these numbers are the same.
I wonder if there are similar patterns for other diagonals.

What patterns do you see? What new questions do you have?

I notice that the shaded rows remind me of equivalent fractions!
I notice that I can find the simplest form in the left column.
I wonder if other pairs of rows would create equivalent fractions.
I wonder if I could see equivalent fractions using columns.
I wonder if I could find equivalent fractions or other fraction patterns by looking along diagonals.

I notice that this picture is an extension of the original picture.
I wonder how the orange numbers compare to the grey ones.
I wonder how about the other colors compare to the grey ones.
I wonder how the other colors compare to each other.

I wonder what causes these patterns.
I wonder if these patterns continue forever.
I wonder if I can use arrays to understand the patterns.
I wonder if I can use algebra to understand the patterns.
I wonder if I can use the patterns to invent mental math strategies.

I notice that the white numbers are all even.
I notice many other interesting patterns in the white numbers.

I wonder what happens when I add the two purple squares in one of the boxes.
I wonder what happens if I do this for the other colors.
I notice that the sum of two purple numbers equals the sum of the two brown numbers.
I notice that both of these sums are double the middle number.
I wonder if these patterns always happen.
I wonder what causes these patterns.

I notice that the brown sum is always four greater than the green sum.
I wonder what happens if I add all 8 squares around the middle number.
I wonder if this pattern continues when I start with other numbers in the center.
I wonder what causes these patterns.

I notice that the left side of each triangle is always three boxes tall.
I notice that the triangles on the right are longer.
I wonder if I can predict the exact size of a triangle by looking at the column that its left side is in.
I notice that the sum of the two numbers on the left vertices equals the number on the right vertex.
I wonder if this is always true.
I wonder what causes this pattern.

I wonder if I can draw arrays to show what causes this pattern.
I wonder if I can write an algebraic equation to describe this pattern.
I wonder if I can prove that this equation is always true.

Here are a couple more to think about!

I wonder how many squares of each color there are.
I notice a pattern when I count the squares of each color.
I wonder if this pattern will continue forever.
I wonder what causes this pattern.

I wonder what happens if I add the numbers in the squares for each color.
I notice that the sums are the cube numbers.
I wonder if this pattern continues forever.
I wonder what causes this pattern.
I wonder if I can explain the pattern in the sums by looking at patterns in the numbers I am adding.
I wonder if I can prove it algebraically.

I wonder if I can find my own ways to color in the multiplication table to find new patterns and make new discoveries!

I notice that this table includes 0.
I notice that the entries in the table match the ones in a regular table.
I notice rows and columns inserted between each row and column in the regular table.
I wonder if the new rows and columns will contain fractions or decimals.

I wonder how I can figure out the numbers that belong in the table.
I wonder if numbers in each row and column will increase at a steady rate.
I notice that some answers in the table are less than both of the factors.
I wonder if the patterns that I see elsewhere on this page will continue to work for this table.
I wonder if I could create a table with two or more rows and columns between each of the whole numbers.
I wonder if the new tables will show patterns that I did not see before.

See Multiplication Table Patterns under 5280 Math Resources  K–8 Activities to explore some of these ideas with students.