Magic Square Puzzles | Original Math Composition
Magic square puzzles are a great way to work on math facts and reasoning skills, but composing one’s own magic square offers even more benefit.
Today, I look at how to create magic square puzzles with the help of Cuisenaire rods (or any math rod really) and the habit of observation and manipulation.
Here are the rules to magic square puzzles:
- All the rows equal to the same sum.
- All the columns equal to the same sum
- Both diagonals equal to the same sum.
- All the sums are the same.
Notice and Solve Magic Square Puzzle
First, I like to set up a magic square grid, using orange rods. You could also use paper. Then start with a puzzle that has 4 clues: the magic number, the middle number of the puzzle, a number to the right or left of the middle number and a number adjacent to the middle number.
The puzzle pictured below has a magic number of 15 and the middle number has to be 5. The other two numbers are six and one.
The next step is review the rules and notice what clues are given. How are the clues helpful? How do they relate to the rules?
With four clues, the puzzle is relatively easy to solve, and most of the observation happens afterwards. However, encourage the student to begin by building as many 3 car trains that equals the magic number.
When students solve the puzzle, have them record the trains that they built that were used in the puzzle. It is unnecessary to record all the trains.
Notice the Solved Magic Square Puzzle
In a very similar fashion to the number puzzle studies before, it helps to notice more and explore the puzzle through manipulation of the puzzle. In doing so, the student discovers the workings of the magic square puzzles. But what should they notice?
First, let them notice what they want to notice. There are quite a few interesting patterns going on in the puzzle. Some resistant to writing may opt to use symbols and arrows. The more important pattern to emphasize is "how the middle number compares to each corner number?"
This pattern returns again and again as the student is required to manipulate the puzzle. Students familiar with Cuisenaire rods may identify this pattern as a plus one staircase and a plus three staircase. (4, 5, 6 and 2, 5, 8)
You may emphasize it by creating the staircase with rods.
Explore Magic Square Puzzles thru Manipulation
The next part is to change the puzzle. First, solve another variation of the puzzle with the same magic number. Observe again. Then create a similar puzzle with the same magic number, but with a new middle number.
In trial and error, they may find this is an impossible task to use any other middle number except 5. And so it is. But why? Here is a great time to explore student thinking. Here, they can practice justifying their answers and working towards a clear answer.
At first discussion, my kids didn't know why, but later in the discussion they built on what they did know. They did know that an odd plus an odd plus even equals even. Well, that would never work because the magic number is odd. They also notice that the outside numbers paired up to equal ten and paired in either all even or all odd (1 and 9, 2 and 8, 7 and 3, etc). In the end, they got very close to why it had to be five.
But do not linger forever. There is still more to explore. Add the same value to all the numbers in the squares to produce a new puzzle and by consequence, a new magic number is formed.
Students should again notice more. Don't be afraid to spread this out over a couple of weeks. It took us over 2 weeks (4 days a week) to work through this unit.
How did it change the puzzle? What remained the same? Did the relationship between the middle number and the corner numbers remain the same? That is, did the staircases remain the same?
The student uses the knowledge of the staircase pattern to solve the puzzle with just the middle number. But is one clue really enough? They will discover that it is not for chances are when the students compare puzzles each will have created a different puzzle.
Creating Their Own Magic Square Puzzles
I believe the work of creating is the strongest active learning state, but it must be built upon some sort of knowledge gained through previous experience. All the activities previous are designed to lay a strong experience, that is foundation, for creating their own magic square puzzle.
With that said, the work of creating will test that foundation. If students struggle, I suggest two things may be happening. One, the experience of playing with the puzzles and manipulating them may not be sufficient to ground them in the knowledge that is available in the activities.
This depends on the age, maturity, experience and the will of the child. My six year old managed only to replicate the previous exercises in his first attempt at creating his own magic square puzzle. He then promptly left for some legos. I expected this because of his age and maturity. I note that he will benefit from returning to this activity down the road.
He did gain a lot from the activities, but he needs more time playing and manipulating the puzzles. My motto quit before they want too. In such situations, he may put up with continued exploration but I risk losing his interest. It is better to come back to it another time.
The second thing that may be happening is the student is ignoring everything they have noticed. This happened to me as well. I had one kid creating intentionally based on what they had learned. I had another kid just plugging in numbers randomly. Guess who was struggling?
Yes, the kid just plugging randomly away. I let him struggle for a few minutes as a little failure is the best remedy for poor habits. Then I asked him to tell me some key things he remembered when we changed parts of the previous magic square puzzles.
I also suggested not to constrain himself with a magic number. I had him focus on picking any number for the middle square and then build staircases as he had seen in previous puzzles.
He soon saw success when he built the puzzle from this knowledge he gained through previous observations. His experience taught him that building on what he learned makes things easier. He also learned that the magic number is not what makes the puzzle work. It is the staircases.
Interestingly, the other child discovered that the staircases don't always have to be plus one and plus three. She tried a variety of staircases when she created her magic square puzzles. This discovery helped to determine which clues are necessary to figure out a magic square puzzle.
Children don't need to be Challenged
In the beginning, the children began with magic square puzzles that had 4 clues. In the end, the children decided that you only needed 3 clues. Well, you tell them if they are wrong. Here are their puzzles. Can you solve them?
Starting with an easy puzzle makes the process of exploring magic square puzzles more accessible to children. While a struggle is great, as educators we need to be careful in administering challenges.
Children prove to us that they are more than eager to challenge themselves. We are better suited to make things accessible to children rather than make things challenging.
When I mean accessible, I don't mean give the answers to the child. Rather, we should develop in them the good habits needed to solve challenges. We do this through planned exercise by which they gain skill to tackle their own challenges. We don't send a kid up a mountain on their first day of hiking. Instead, we provide exercises that build their skills so that the mountain becomes accessible to them.
In the exercises described above, the workings of the puzzle becomes accessible through the child's experience and the honing of the student's observation skills through good questions. In the end, the students are equipped by the knowledge gained in their experience to challenge themselves.
Time-Saving Value of Observation and Consideration
There is more to the study then I am able to convey, but I would like to delve deeper into one particular benefit we gained in our exploration, the time-saving benefit.
The children realized how observations and consideration of the knowledge gained saved them time. When using that knowledge, they found creating the puzzles easier.
Without such habits of pausing and remembering what we have learned and experienced, we fall victim to inefficiency. We just randomly throw things at the wall hoping something sticks.
This process of trial and error is a long process, and we risk losing the interest of a child by providing this as the only process by which to learn. The mind is such a heavy consumer of energy that it is always working towards the easiest path.
Time-saving is energy saving. If they never experience the time-saving path of observation and manipulation, there is the risk that they will abandon the pursuits of anything remotely challenging. They will prefer to be spoon-fed.
Children learn the time-saving value of pausing to observe and consider their experience by guided mentorship. It isn't done in a vacuum. Charlotte Mason was not wrong in her emphasis on building the habit of observation. It is only that some have limited it to nature study when it applies more than ever to math.
That is why this is called a number puzzle "STUDY" because the study is very similar to that of a nature study. Students observe in detail by the gentle questions of a careful mentor and by the mentor's provision of necessary vocabulary to clarify and enhance students' observations.
Like a scientist would a leaf, students manipulate the puzzle to discover the deeper mechanics. Then they try to emulate that which they observed and explored. This process is found throughout Gattegno's textbooks.
It is time-saving because the child isn't asked to build on the teacher's experience or the textbook's experience. Instead, it's the child's own observations that takes the stage.
As mentors, our job is just to ask questions that force the student to notice more and explore more. It is not for the student to memorize and apply our experience (knowledge) but to remember and apply their own.
I believe Gattegno's textbooks are brief because the experience of the child should do the most talking. Not the textbook and certainly not the teacher. This is the hard but necessary transformation that must take place in the teacher in order for the student to truly thrive.
It is one that has challenged me daily and it is an art. It takes practice to be silent, to take a backstage. To put the student's observations as center stage takes away a lot of our control. At first, it was scary for me, but now I find it liberating. I can do what I do best, facilitate play, ask questions and provide language when needed. They just have to tell me what they see and by seeing, the door of understanding is open to them.
More Information on Magic Square Puzzle Study
If you want to pick up this study, you can purchase it here at my teacherspayteachers story or check out the previous post for a free sample of the puzzle study
For a more detailed exploration of puzzles, I highly recommend getting on Sonya's email list (Arithmophobia No More) for her free online classes. There is nothing like experiencing Gattegno first hand. We did the number puzzles in her class recently and hopefully, she will have that class available for purchase at her next sale.
This activity complements chapter 4 of Gattegno’s textbook one (FREE HERE) and the magic square puzzles are a great way to extend Chapter 4 for older children who need a challenge.
In all, the child composes their own math which frees most children and adds the joy of mystery to the fun. I hope your kids have fun exploring and creating math puzzles!