Beginning

I notice three rows and three columns.
I notice that all of the shapes are circles except for one square.
I notice that the "+" and "–" symbols go back and forth.
I notice that the numbers get smaller (if you read them like a book).
I wonder if there is reason that the numbers get smaller.
I wonder what happens if I do the operations shown by the arrows.
I wonder if I can replace the question marks by operations.
I wonder if the operations should still keep going back and forth when I do this.
I wonder if there is something special about the number that goes in the square.

Exploring

I notice that the number in the square (the magic number) is both a sum (vertically) and a difference (horizontally).
I wonder if this will still happen if I start with other numbers.
I wonder why the pattern always works.
I notice that the magic number for this puzzle is 12.
I wonder how many other ways I can fill in the grey circles to make the magic number equal 12.
I wonder if I can make the magic number equal 0.
I wonder what happens when all four shaded numbers are the same.
I wonder what happens if I swap numbers in the grey circles.
I wonder what happens to the magic number if I increase the top-left circle by 1.
I wonder what happens to the magic number when I make other small changes to the shaded numbers.
I notice that when all of the shaded numbers are even, the magic number is also even.
I wonder why this happens.
I wonder what happens when all four shaded numbers are odd.
I wonder if there is a faster way to calculate the magic number without doing the whole puzzle.
I wonder if I could write a formula for the magic number.

Creating

As they notice and wonder, students may create and explore their own questions and puzzles 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 patterns in the four numbers they choose to start with.
Create patterns in even and odd numbers.
Create examples that cause numbers at the right or bottom to be negative.
Create new puzzles by changing the order of the + and – operations (and, of course, watching for new patterns).
Choose a magic number, and create puzzles that have that magic number.
Try to create puzzles with more rows and columns.
Create rules for predicting the magic numbers.

Reflecting and Extending

I notice that it helps to keep track of the things I have already tried, especially when I work on long problems.
I notice that when I change numbers in an addition or subtraction problem, I can predict how the answer will change.
I notice that I can predict the magic number without completing the whole puzzle.
I wonder if I can make a puzzle like this using multiplying and dividing.
I wonder if I can make a puzzle like this with more rows and columns. (How would the subtraction work?)
I wonder if the patterns I discovered will still work when some of the numbers are negative.

Notes

The possibilities for exploration with this prompt are endless. Students should discover that when they add the numbers in the right column and subtract the numbers in the bottom row, the answer is always the same. In the case of this particular prompt, the magic number is 12.

It can be fun to fiddle with the given numbers to find new sets of four numbers that have the same magic number (or to increase or decrease the magic number by 1 or 2, etc.).

For a more structured activity based on the ideas in this prompt, see the Magic Numbers activity on the Deep Math Projects page. The activity also gives lots of detail about specific things you can explore.

For those who are curious about algebra connections, the prompt illustrates a pair of equivalent algebraic expressions. If you represent the numbers in the four grey circles as

a b

c d

then the expressions look like

(a – b) + (c + d) = (a + c) – (b – d).

The following expressions are also equivalent, and they represent possible “shortcuts” for calculating the mystery number.

a – b + c + d
a + c + d – b
a + c – b + d

or even
-b + a + c + d
etc.

All of these will give the same result as each other no matter which four numbers you choose! Students may experiment (search for patterns!) to find other expressions that are equivalent to these. As long as they subtract the b and either start with or add the other three variables, they will get an equivalent expression.

Students do not need formal algebra training in order to explore and try to understand these expressions! In fact, they may make some of their own algebraic discoveries just by playing around and observing closely!

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