# 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 w*hat 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.