EXPLORATION 2 Time: 15 minutes Probing/Eliciting Student Responses and What the Teacher Will Do Questions Misconceptions Now take a look at the different [Students work in groups for 3-4 magic squares. Please carefully minutes to look for patterns.] right down any patterns that you find in all of the magic squares. You might want to look for: Relationships between numbers in rows or columns or diagonals, Patterns in how the numbers are arranged, Which numbers are even and odd, Or something else OK, going to ask each group I’m about the patterns that you found. [Students start sharing patterns. Make sure you explain each There could be a lot of correct pattern that you’ve found. patterns. But most of the that are found probably “patterns” Let me give you a hint about a make much sense.] don’t really important pattern. take Let’s a good look at the middle rows and middle columns of each magic square. write down those Let’s numbers. [Teacher writes down a few middle All of these groups of [Students stare at the numbers to row and a few middle columns. three numbers have find a pattern.] For example, from the worksheet, something in the teacher could write common. Can you Ooh. You add or subtract figure out what it is? something to go from the first 3, 5, 7 number to the second, and do it 1, 5, 9 again to get to the third number. 1, 18, 35 25, 18, 11 [or: The middle number is the 4, 10, 16 average of the other two 18, 10, 2, etc.] numbers.] Excellent. For the first group, 3 + 2 = 5, and then 5 + 2 = 7. For the second group, 1 + 4 = 5, and then 5 + 4 = 9. Also, 18 8 = 10, and – then 10 8 = 2. –
EXPLORATION 2 Time: 15 minutes What the Teacher Will Do Probing/Eliciting Questions Student Responses and Misconceptions Now take a look at the different magic squares. Please carefully right down any patterns that you find in all of the magic squares. You might want to look for: Relationships between numbers in rows or columns or diagonals, Patterns in how the numbers are arranged, Which numbers are even and odd, Or something else OK, I’m going to ask each group about the patterns that you found. Make sure you explain each pattern that you’ve found. Let me give you a hint about a really important pattern. Let’s take a good look at the middle rows and middle columns of each magic square. Let’s write down those numbers. [Teacher writes down a few middle row and a few middle columns. For example, from the worksheet, the teacher could write 3, 5, 7 1, 5, 9 1, 18, 35 25, 18, 11 4, 10, 16 18, 10, 2, etc.] Excellent. For the first group, 3 + 2 = 5, and then 5 + 2 = 7. For the second group, 1 + 4 = 5, and then 5 + 4 = 9. Also, 18 – 8 = 10, and then 10 – 8 = 2. All of these groups of three numbers have something in common. Can you figure out what it is? [Students work in groups for 3-4 minutes to look for patterns.] [Students start sharing patterns. There could be a lot of correct patterns. But most of the “patterns” that are found probably don’t make much sense.] [Students stare at the numbers to find a pattern.] Ooh. You add or subtract something to go from the first number to the second, and do it again to get to the third number. [or: The middle number is the average of the other two numbers.]