Magic Square Strategies

Transcription

1 Magic Square Strategies Teacher Knowledge The Magic Square is a classic problem in mathematics: Place the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 in a box of 3 rows and 3 columns so that the sum of the numbers in each row, column, and diagonal is the same. Mathematical Knowledge We, as teachers, need to know three key pieces of information: 1. The sum of each row, column, and diagonal MUST equal The center number MUST be The even numbers MUST go in the corners. Instructional Knowledge Although we, as teachers, know those three key pieces of information, we do not tell students. Instead, we help students make the discoveries. We start students with a simpler problem: Place the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 in a box of 3 rows and 3 columns so that the sum of the numbers in each row is the same. This is a much easier problem to solve. After students solve this simpler problem (of making the sums of the three rows equal): We help them clarify the magic number. Once students have discovered that the magic number must be 15, we then help them discover the center number. We continue giving hints until students have solved the problem. 1

2 Hints Helping Students Discover the Magic Number Do NOT tell students that the magic number is 15. Instead, ask them to make the three rows equal. This is something that they can do (There are many possible arrangements that gives the rows all equal to the same sum). Then ask, could that sum be 20? Ask, what would the total be if each of the rows sums were 20? [Since there are 3 rows: = 3 x 20 = 60] Ask, could the grand total be 60? What is the sum of the numbers 1 through 9? In this way, students can see that the sum of each row must be 15 because the grand total is 45. Note the important distinction: The students discover that the magic number is 15. The teacher never mentions this fact. The teacher does not say 15 until a student says 15. Help students make this discovery using as few hints as possible. Once the students realize that M = 15, Ask, Why do you think the magic number is 15? Listen to their responses. However, also be ready to help clarify their answers: The Total sum of all 9 numbers is 45. There are 3 rows so: = 45 3 x 15 = = 15 2

3 Helping Students Discover the Center Number Make certain that students are very comfortable with the discovery that M = 15: Ask: Could we put 9 in the center and still solve the problem? If they need another hint, ask: where would we put the 8? What is 9 + 8? What is the number of any row column or diagonal? Use these hints (more if needed) to help them see that: 9 cannot go in the center any number in the center must make a sum of 15 (using two other numbers) = 17, which is already too big. After students have discovered that 9 cannot go in the center: Ask: Could we put 1 in the center and still solve the problem? [See if students can reply without hints.] If hints are needed, ask: Ask: Where would we put the 2? [See if students realize that = 3; that we would need 12 to make a sum of 15. And that the largest number is 9.] Ask, What number do you think we should put in the center? [Do NOT say 5 ] If students do not realize that 5 must go in the center: Ask: Could we put 8 in the center? Could we put 7 in the center? Could we put 6 in the center? If needed, ask any number EXCEPT 5. When a student suggests 5: Say: Go ahead and try that in the center. Ask: What is the magic number? [Make certain students know that the magic number is 15.] Allow students time to solve the problem. 3

4 Showing how to solve the problem Say: Let s solve the problem together. Ask: What is the magic number? [The sum of each row, column and diagonal] Make certain that students can say the magic number is 15 Because the total of all 9 numbers is 45 There are three rows 45 3 = 15 Once students have explained WHY the magic number is 15: Ask: What number do we put in the center? Make certain that students can say the center number is 5 Because it can t be 6, 7, 8 or 9 (they are too big) it can t be 1, 2, 3 or 4 (they are too small) 5 is the only number that works Also, 5 is the middle number, so it makes sense to try that in the center. On the overhead, place 5 in the center. Place 2 in the upper right corner. Ask: Do we know where to place any number? [Students may say that 8 goes in the lower left corner] If students do not know the next step: Point to the lower left corner; Ask: What number do you think would go here? At this point, clarify the reasoning: = 7 We need 8 to make a sum of 15: = 15 4

5 At this point, the square shows: Ask: Where can we place the 9? Can we place it here: Wait for student response. The sum of the last column is already = 17. Ask: Where can we place the 9? Can we place it here: Ask: What must happen? Wait for student response = 14 we need 1 to make 15: Continue step by step until the solution is reached. 5

6 When the solution is reached: Ask: What patterns do you see? Wait for student responses: Even numbers in corners: The large numbers (7, 8, and 9) are in different rows and columns: Likewise, the small numbers (1, 2, and 3) are on different rows and columns: 6

7 Show again: Ask: Do you think the even numbers must go in the corners? [The answer is Yes. Hints: Is the Magic Number an odd or even number? Is even + odd + even an odd or even number?] Show again: Ask: Do you think the large numbers must go in different rows and columns? [Yes. Add any two and the sum is already too large.] Likewise, the small numbers (1, 2, and 3) are on different rows and columns: Ask: Do you think the small numbers must go in different rows and columns? [Yes. Add any two and the sum is too small.] 7

8 References Schoenfeld, Alan H. (ed). (1994). Mathematical Thinking and Problem Solving. Lawrence Erlbaum Associates. Hillsdale, New Jersey. Magic Square Websites [This site presents interesting 4 x 4 examples; does not show how to solve.] 8