Analog clock (preferably with a second hand) Post-Assessments Paper Pencil Accommodations for Learners with Special Needs (ELL, Special Ed, 504, GT, learning styles, etc.): None provided below, though this could be added. 5Es ENGAGEMENT 1 Time: 5 Minutes What the Teacher Will Do Probing/Eliciting Questions Student Responses and Misconceptions Use a pencil to draw a square on a piece of paper. Then divide that square into 9 smaller squares. [Draws on board.] Now use the numbers 1 through 9 to fill in the square. Here are the rules: You can only use each number once. And you have to place the numbers so that, when you add the numbers on each row, you get the same answer. For example, look at this square. Don’t write this down on your paper. [Writes on board] Did I use each number once? Do the rows have the same sum? Students draw blank magic squares on paper. Yes! No! 1+2+3 = 6, but 4+5+6=15 and 7+8+9=24. Evaluation/Decision Point Assessment Student Outcomes Once students understand the row rule for magic squares, we can continue. Students understand the rules.
Good. So I want you to start with the square that we’ve made and try swapping rows and swapping columns. See if you can make a square so that the diagonals add up to 15. Please write it on the board. This square is an example of a magic square. [Writes on board.] All the rows, all the columns, and all the diagonals have the same sum. For this square, we’re going to call 15 the magic sum. [Writes on board.] What do the rows sum to? And the columns? And the diagonals? [Students experiment. The answer will be that the rows and/or columns have to be swapped so that the 5 ends up in the middle square… give students hints along these lines if necessary.] I got it! Student writes it on the board. [A sample is below; there are only eight squares that work]: 15! 15! 15! Wow! [Students write magic square and magic sum on their vocabulary sheets.] Evaluation/Decision Point Assessment Student Outcomes When students are comfortable with the definition of a magic square, we’ll continue. Students are able to correctly identify magic squares.
EXPLANATION 1 Time: 10 Minutes What the Teacher Will Do Probing/Eliciting Questions Student Responses and Misconceptions Let’s see what modifications we can make to a magic square. First, let’s turn the square 90 degrees clockwise. [In the following, we illustrate with the sample square given on the previous page. Naturally, the teacher should use the square that was constructed by the class, so that the squares written by the class could be different than what’s presented here.] [A clock with a second hand should be prominently placed in the class.] Good. Turning the square like this is called a rotation. [Writes on board.] Please write these terms down. What happens to the numbers on the square? What does 90 degrees mean? [Writes on board.] What does clockwise mean? [Writes on board.] Now the big question. Is this new square a magic square? Let’s try it again. What happens If we rotate the square clockwise by another 90 degrees? How many degrees have we turned it? And is this new square a magic square? [Some smart aleck will probably say that the digits 1-9 should also be sideways if the square is turned 90 degrees.] A quarter-turn. In the same direction that the hands of a clock turn. [Students write 90 degrees, clockwise, and rotation on vocabulary sheet.] Yes! 180 degrees. Yes!
Have we seen this square before? How did we get this square before? So that means that rotating by 180 degrees produced the same squares as…. Let’s review. How many different magic squares have we found so far? No... wait, yes! Rotating by 180 degrees. ...as making two reflections. 6. Evaluation/Decision Point Assessment Student Outcomes Students are comfortable with rotations and reflections. Students are able to correctly produce new magic squares by rotations and reflections.
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.]
Doing mathematics with these letters, called variables, is called algebra. It’s a short-cut way of writing magic squares. It turns out that every 3x3 magic square can be written like this, though you may need to apply a rotation and/or a reflection. In other words, using variables allows us to quickly write down a pattern. [Students write algebra on their vocabulary sheets.] Evaluation/Decision Point Assessment Student Outcomes Students have been introduced to the idea of using variables to efficiently represent a pattern, even if it requires it a little imagination [or abstract thinking]. Students understand how letters (variables) can be used to represent a pattern. EVALUATION 2 Time: 10 minutes What the Teacher Will Do Probing/Eliciting Questions Student Responses and Misconceptions Post-Assessment 2.