Teacher Notes for Forty Squares puzzle
The forty square puzzle is an excellent opportunity to focus learners on team building, social skills, for problem solving and reasoning . I have use this activity numerous times with groups from fourth grade to adults.
If I were using it with learners who may not have had many experiences with systematic organizing or visualizing over lapping squares, then I would consider using a less challenging puzzle like the triangles puzzle. It can be used for the same purposes of team building, group work, social skills, and introduction to problem solving hueristic and strategies.
Decide if students will work alone, in pairs, triads, or groups of four. I would recommend groups of four if one of the goals is to explore team building or social skills. 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, 1/4 x 1/4.
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 or a reason for starting with thetriangles puzzle.
Connections to the five process categories of the NCTM
(National Council of Mathematics)
- 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