Thought for the day: Sometimes, you can change the things you notice or wonder by removing parts of a prompt. (For example, if you remove the numbers at the top of this one, you will focus more on things you see in the picture of circles instead.
Concepts: properties of addition; collecting and organizing data; finding and extending patterns
I notice a triangle made of circles with the numbers 1 through nine written above.
I notice that there is a circle missing in the middle.
I notice three groups of three circles all arranged in equilateral triangles.
I notice a hexagon and three circles.
I notice that the number of circles is the same as the number of triangles.
I notice that the number of circles is the same as the number of numbers (9).
I wonder I can make a puzzle or problem out of putting the numbers into the circles.
I wonder if I can put a different number into each circle so that the sum of the numbers on each side of the triangle is the same.
I wonder what the sums are possible to make when I do this.
I notice that it helps to spread the large numbers out between different sides.
I notice that the corners are important because each one belongs to two sides.
I notice when the corners numbers are smaller, the sums tend to be smaller.
I wonder if it would help to make small adjustments to attempts that didn’t work (like trading numbers.)
I wonder if it would help to choose a sum and make lists of 4 numbers that have that sum.
I notice that when I do this, certain addends appear more often than others.
I wonder how many solutions there are (for some sum that I chose).
I wonder what counts as a new solution. For example, is it a new solution if I just switch two numbers in the middle of a side?
As they notice and wonder, students may create and explore strategies, solutions, patterns, and related questions. Many of their creations may flow out of things that they have wondered about. For example, they may
Create multiple strategies for making the sums of the sides the same.
Create multiple solutions (and search for patterns between them).
Create new questions to explore (by changing the numbers, the number of circles per side, the shape, etc.)
Reflecting and Extending
I notice there are two solutions for a sum of 17 (assuming small changes don’t count as new solutions).
I notice that the corners must be 1, 2, and 3 in order to get a sum of 17.
I wonder why the corners must be 1, 2, or 3 in this case.
I wonder what the smallest and largest possible sums are.
I wonder what would happen if there were more or fewer circles on each side of the triangle.
I wonder if there are patterns to the smallest or largest sums for different numbers of circles on each side.
I wonder what would happen if the circles were arranged into a square instead of a triangle.
I wonder how many solutions there would be if every small change counted as a new solution.
Most people do not consider solutions to be new if they (1) exchange numbers in the middle of a side, (2) reflect the numbers in the triangles between a pair of sides, or (3) rotate the numbers in the triangles. In this case, there are two solutions for a sum of 17.
The corner numbers must be 1, 2, and 3 when the sum is 17. To see why, think of these two sums:
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45 and 17 + 17 + 17 = 51
At first, you may think that the sums should be the same, because they involve adding the numbers around the triangle. However, 17 + 17 + 17 counts each corner twice. Therefore, the corners must account for the difference 51 – 45 = 6. The only way to make 6 in this case is 1 + 2 + 3!
For a more structured version of the problem, along with detailed solutions and suggestions for facilitating conversation see Triangle Sums, a free activity from my book Advanced Common Core Explorations: Numbers and Operations.