Teacher Notes for Forty Squares puzzle
If the students haven't had many opportunities for problem solving and reasoning the square puzzle is a good puzzle to start with for students in or above fourth grade.
While younger students will attempt to solve the puzzle with guess and check non systematic strategies they may discover (or the teacher may facilitate the idea) to systematically count the squares according to size and to construct an illustrations to show where each is. Yet, if someone hasn't solved similar problems and operationalized this kind of procedure, it is challenging.
Decide if students will work alone, in pairs, triads, or groups of four. I wouldn't recommend group sizes larger than four. Distribute to each group a work sheet and review the directions.
The time limit is not for the purpose of saying that everyone should find all the squares within the time, but is just to provide a guide for when to begin to share. Be willing to be flexible as you see fit.
When students ask if they have found all the squares, You may reply "You (plural meaning the group) are within criteria." Criteria might be within two or 38-42 of the number of squares in the puzzle. Or, "You (the group) are not within criteria." I really try to down play the "right answer" and just say that in 15 minutes we will start to share and look at the different patterns, squares... that we have collectively found.
I try never to say how many squares there are. When students ask, I turn the question back to them. How many do you think there are? If they say they aren't sure, then I ask them if they have found any pattern and if so how that might help them to have confidence in the number of solutions they have found.
For example: How many 1x1 squares?
Do you think there are more? and repeat that line of questioning and reasoning for 2x2, 3x3, 4x4, 1/4x1/4.
Then ask if there are 5x5? Why not? and If there are any other sizes? Why? Why not?
Finally, try to summarize with, then I don't have to tell you if you are "right" you seem pretty confident that you are.
The only other hurdle is for students to discover that some squares overlap other squares. Usually this is discovered and passed around the class fairly quickly. However, with younger students it might be a hurdle that students might need help to cross.
Representation - Each square can be outlined to represent each square, each square can be counted to find the total number, could color code different sizes, could order squares from small to large to account for each one, could draw each square on a separate sheet of paper and number each sheet for each square.
Reasoning and proof - I can find all examples of each sized square, count the total number for each shape, add the different sizes and find the total. The proof is based on the fact that only a finite number of squares exists in the diagram and they can be counted. This conclusion is based on the definition of a square, the limited number of lines, and how and where they intersect in the diagram.
Problem solving - strategy guess and check, systematically organize, break down into small problems, solve each and put together.
Communication - I can communicate by logically and systematically arranging the possible sizes of squares and illustrating where each is to communicate and convince others that our totals match.
Connections - All five process are used in soving this problem as well as geomerty, spacial reasoning, number value, and algebra patterns
Dr. Robert Sweetland's Notes ©