by Michelle Pauls
In Select-a-Square, students are presented with square grids of differing sizes - 2 x 2 and 3 x 3. On the first student sheet, they are challenged to select two squares from the 2 x 2 grid and three squares from the 3 x 3 grid so that none of the squares are in the same row or column. In order to assist students in discovering solutions, they should be given small marking chips such as Area Tiles or Math Chips that they can manipulate in the large grids. Once they find a solution using the marking chips, the solution can be recorded in one of the smaller grids by coloring in the appropriate squares.
A second student sheet follows with questions that challenge students to identify the number of solutions for each level and justify why they believe they have found them all. Students are also challenged to look for patterns, to organize their solutions based on these patterns, and to try to predict the number of solutions for 4 x 4 grid. (As an extension, advanced students can be challenged to discover and record the more than 20 solutions for the 4 x 4 grid.)
As the teacher, it is important for you to have a good understanding of the patterns, even if your students can only grasp them at the most basic level. For this reason, we will break our tradition of not giving the solutions to the Puzzle Corner activity until the following month, and discuss them here.
As you can see, the number of solutions increases very quickly from one level to the next. While there are only two possible solutions for a 2 x 2 grid, that number triples to six possible solutions for a 3 x 3 grid, and-as previously mentioned-there are more than 20 solutions for a 4 x 4 grid. (The pattern that governs the number of solutions for a given level is fairly advanced, and should not be addressed unless students are very proficient at pattern discovery. The number of solutions for an n x n grid is n! [read "n factorial"]. n! is the product of all of the positive whole numbers between 1 and n.)
The fourth question on the second student sheet asks students to describe three different ways in which they could organize their solutions based on the visual patterns that they discover. These patterns are very useful in helping students determine if they have found all of the possible solutions and should not be overlooked. Encourage students to share their methods with the class and to explain why they chose to organize their solutions the ways they did. Several different methods of organization that your students might use are given below.
Each of these methods provides a check to help students determine if they have discovered all of the solutions. As you can see in the first set of solutions, each grid has a mirror image. Once this is discovered, students can quickly generalize that the total number of solutions must be even, and that each solution should have a pair. This allows missing solutions to be easily identified and recorded.
Another pattern that can be observed involves the number of times a given square is selected. For example, in the 3 x 3 grids, each of the squares is selected in a total of two solutions. The last three sets of solutions on the previous page are organized based on this pattern. Again, when this pattern is observed, it provides a simple way for students to fill in any missing solutions.
Student Activity Page 1 | Student Activity Page 2 | Puzzle Corner