Thought for the day: Consider leaving a prompt in front of the classroom for a few days before discussing it—especially for complex prompts. This gives students more time to think. It also generates more ideas and may help quieter or less confident kids get involved in the discussion.
Concepts: multiplication facts; patterns in multiplication and addition; multiples (possibly using algebra to describe and understand patterns)
I notice numbers in a multiplication table surrounded by rings of color.
I wonder if the different colors mean something.
I notice that squares of the same color are always on opposite sides of the ring.
I notice that the sum of the blue (or purple) numbers is always double the middle number.
I notice that the sum of the green numbers is always two less than double the middle number.
I notice that the sum of the brown numbers is always two greater than double the middle number.
I wonder if these patterns will be true no matter where I put the ring of color in the table.
I wonder what causes the patterns.
I wonder if the pattern will be true if the ring goes off the table (into "0" columns/rows or into larger numbers).
I wonder what will happen if I add all eight numbers in a ring. Will there be another pattern?
I wonder if there are patterns for larger rings? Other shapes?
As they notice and wonder, students may create and explore their own patterns in the multiplication table. Many of their ideas will flow from things that they have wondered about. For example, they may
Create new rings on the table. See if the patterns they discovered continue to hold.
Create new versions of the table. For example, extend it or include only even numbers, etc. Draw the rings on these tables.
Create new shapes on the table.
For each new drawing they create, students should find patterns, test them, extend them, and try to explain what causes them.
Reflecting and Extending
I notice that the sum of the blue and purple squares is always 4 times the middle square.
I notice that the sum of the green and brown squares is also 4 times the middle square.
I notice that the sum of all eight numbers in ring is 8 times the middle square.
I notice that I can shift amounts between the eight numbers so that they are all the same.
I wonder what would happen if I made the ring larger so that every number in it was two squares away from the middle.
I wonder what other kinds of shapes I could draw that would have patterns.
I wonder if I can write algebraic expressions for the patterns I find.
I wonder if I can use algebraic expressions to understand what causes the patterns.
The multiplication table is a treasure trove of patterns for all ages and abilities. I created this particular image to suggest the idea of adding all eight numbers surrounding a single number in the table. It turns out that the the sum of the eight numbers is always equal to 8 times the middle number! One way for students to see this is to shift amounts between the numbers.
For example, if you shift a certain amount from the larger blue number into the smaller blue number, you can make both numbers equal the middle number! In the upper-left ring;
Shift 3 from the 12 to the 6.
12 – 3 = 9 and 6 + 3 = 9.
9 is the middle number.
You can do the same with the purple numbers. The corners are trickier, but with some creativity, you can make them work, too. By the time you are finished, all 8 numbers will be the same as the middle number—and shifting amounts between them will not have changed the sum. Lots to think about for both children and adults!
Once they get going, your students are almost certain to discover new patterns of their own. Definitely follow up on these—and never worry if you don't know why—or even if—they always work. It is a great thing for students to see that teachers don't know all of the answers. Become their partner in exploration. As well-known math educator, James Tanton, says “Be your honest true self in front of students. Wonder about math; be honest if you don’t know the answer and then help students try to find it. Model what it means to do and learn mathematics.”