Thought for the day: Most prompts apply across a range of ages and grade levels. This prompt will be meaningful for the entire K – 12 span!
Concepts: connections between addition and subtraction, properties of addition and subtraction, visual patterns, counting patterns, equality and equivalent names for numbers, rules
I notice a square with four colored squares in inside it.
I notice that the colors are the primary colors and green.
I wonder if there is a reason to have colors in the picture.
I notice circles with numbers between neighboring colors.
I notice that one of the circles is missing a number.
I notice that all of the numbers are odd.
I wonder what number the empty circle should have in it. Will it be 11 (for the pattern 7, 9, 11, 13)?
I wonder if the colored regions could also have numbers in them.
I notice that two of the numbers are the sums of the number of letters in the colors that they touch. (This does not work for 13.)
I wonder if I could change the colors so that it works for all of them.
I wonder if I can put a number in each colored region so that the number in each circle is the sum of the numbers in the colors it touches.
I notice that there is more than one way to do this.
I notice that it helps to start by just picking a number for one of the colors.
I notice that certain numbers that I pick don’t work. (Or they would cause other numbers to be negative.)
I wonder how many ways there are to choose a number for each color.
I notice that the empty circle has 15 in it, no matter what numbers I choose.
I wonder what happens if I change the numbers in the circles.
I notice that I can use letters (variables) to make equations for the numbers in each square.
As they notice and wonder, students may create and explore their own pictures like the one in the prompt. Many of their creations may flow out of things that they have wondered about. For example, they may
Create new puzzles by changing the numbers in the circles.
Create new puzzles by removing or adding circles. (For example, put one in the middle—touching all four colors.)
Create new puzzles by beginning with other shapes (triangles, pentagons, random-looking shapes etc.).
Create new puzzles by expanding the existing prompt to make it include more parts.
Create puzzles that have only one solution (or any particular number of solutions that you choose).
Create new puzzles by using operations other than addition.
Create new methods for solving your puzzles.
Create methods to predict the number of solutions that a puzzle has.
Create stories to go with your pictures.
Try to create and discover new patterns in your puzzles.
Reflecting and Extending
I notice that the prompt (and new pictures that I create) are a way to visualize more than one equation at a time.
I notice that my solutions always have interesting patterns.
I notice that I can sometimes solve complicated equations by choosing numbers or making guesses.
I notice that it helps to keep track of the things I have tried.
I wonder if the number in the empty circle would always be the same if I started with a triangle (or pentagon, etc.).
I wonder if there is a formula to predict what the fourth number should be.
I wonder what happens if I create pictures where some of the circles touch more than two colors.
I wonder what happens if I add the numbers in the squares.
You may find ideas and puzzles similar to the one suggested by this prompt at The Greatest K to 12 Math Problem Ever by Sunil Singh and in Linda Jensen Sheffield’s excellent book Extending the Challenge in Mathematics.
Following up on the idea of putting a number in each smaller square so that the sum of the numbers in neighboring squares equals the number in the circle between them, the picture below shows one possible solution.
If you allow only whole number values, there are 8 solutions all together:
Y R B G
6 7 0 9
7 6 1 8
8 5 2 7
9 4 3 6
10 3 4 5
11 2 5 4
12 1 6 3
13 0 7 2
Notice the many beautiful patterns that appear when you organize the solutions! In fact, organizing solutions as you work will help you to know when you have them all. How many solutions would there be if you allowed negative numbers? Fractions or decimals? Can you find some of these solutions?
The table may also help you to understand why the number in the circle on the left is always 15. (Pay close attention to how each value increases or decreases by 1 as you look down the columns.) It is interesting to notice that the sum of numbers in the circles (44) is twice the sum of the numbers in the squares (22). Will this always happen? How do you know?
Those who are comfortable with variables may recognize that this prompt represents a system of equations.
Y + R = 13 R + B = 7 B + G = 9 G + Y = 15
Young students who play with this image by testing different combinations of numbers or changing the structure of the picture are actually exploring patterns that they will encounter in later algebra courses! Algebra students may be able to explain why this system is dependent: any one of the equations may be derived from the other three. In other words, once you solve any three of the equations, the fourth one will automatically be true! The activity, Go With the Flow, in my collection of Deep Algebra Projects provides older students with a challenging real-world application of these ideas and patterns.
The following images may give you a few more ideas about some of the countless ways you could play around with the original prompt.