EDCP 342: Reflection – How many squares in an 8 x 8 chessboard?

 A chessboard itself is aesthetically appealing due to its repetition of a simple colour and square combination. From a student perspective, my immediate reaction is to then see how I can use this pattern to assist me in counting the number of squares. I tried to first use the obvious, and to me simplest approach, before trying to explore more in depth and complex solutions. Breaking down the pattern into one simple sequence of shaded and unshaded squares then allows me as a student to build another pattern after first completely understanding the base. Start by counting all squares of size 1 x 1 that line the width and length of the board, follow by counting all squares of size 2 x 2 that line the length and width again (this gives 7 squares by 7 squares), and so on for the larger squares until we arrive at a single 8 x 8 square. (A pattern emerges that the chessboard is composed of the sum of the squares of square sizes inclusive of 1 and 7.)

However, that being said, it is important to acknowledge from a teacher’s perspective that the problem itself may be overwhelming. Students may become stressed and not even feel as if they are able to begin the problem since the idea of having to imagine and count various square sizes appears to be such a large task. For these students, it is important to then simplify the problem and make it more relatable. Breaking down the beginning of the solution into simple steps and presenting it in a few different ways may help to get those troubled students to become comfortable with exploring further on their own. A teacher could begin by drawing the chessboard and highlighting how different square sizes can appear along the board. Another approach could be to remind students about previously discussed topics such as perfect squares and areas and see if they can make connections between the concepts. Thirdly, a teacher could suggest writing tables that record the size of the square found in the chessboard and how many squares are found (starting with the two largest being 8 x 8 and 7 x 7). Each of these approaches appeals to slightly different learning styles and can then help students to be able to more confidently find a solution.

While it is important to prepare for the possibility of students experiencing challenges with the activities, it is equally important to account for those students who are not challenged enough. To increase the level of reasoning and promote further investigation into the topics introduced through the activity, a teacher could add an increased challenge of counting rectangles with specific width to length ratio. Having students count the number of cubes found in a large cube, a 3-D version of a chessboard, such as a Rubik’s cube, could also push students to now think through volumes. To bring the activity to an even more challenging level, students could attempt to count the number of squares found in a cube – requiring an in depth understanding of surface areas of 3-D shapes.

When providing activities for students, teachers must be able to prepare for the possible troubles their diverse students may have and the ways to assist them through these problems. They must also prepare for the lack of challenge that some students may experience and again have additional suggestions to ensure continuous learning. Overall, this activity reinforces the idea that although we may be working on developing skill sets to become effective and meaningful teachers, we must never lose the part of us that remains a student.