by Dave Youngs
Seeing Squares is designed to exercise your students' spatial visualization and organizational skills. The latter skill is one that is crucial in many problem-solving situations, including this one. While the puzzle is not very difficult --it simply asks how many squares there are in the figure -- it is quite easy to miss some squares if the counting is not done in a systematic manner.
After students think they have found all the squares in the figure, they can share their answers. This will inevitably lead to an interesting discussion with several different answers being given. I find that students usually miss some of the squares in this figure, so many of the answers are too low. On occasion, students are careless and count several squares more than once, giving them a total that is too high. As different answers are shared, even students who have come up with the correct total tend to lose confidence in their answers.
The value of this activity comes when students are challenged to organize their counting procedures so that they are sure their count is correct. This takes the activity out of the puzzle realm and places it directly into the realm of mathematical problem solving.
Watching students as they try to organize their counting processes in a logical way should provide some valuable insight into their mathematical thinking and processing skills. Resist the temptation to solve the problem for them. If students haven't any idea of how to approach the problem, a few carefully chosen questions can get them started. This technique of using questioning to get students pointed in the right direction without telling exactly what to do is well worth learning. Remember students can become problem-solvers if they are given many opportunities to practice problem solving. This puzzle provides just such an opportunity.
We hope that you enjoy this puzzle/problem solving activity!