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3x3 Magic Squares and Similarity TransformationsAuthor(s): John QuintanillaDate/Time Lesson to be Taught: August 6, 2012Technology Lesson: YesNoCourse Description: Name: Summer Mathematical Enrichment Class for Girls 2012Grade Level: Mostly 3rd gradersHonors or Regular: HonorsLesson Source: Inspired by Exploration 1.5 in Mathematics for Elementary School Teachers Explorations (3rd edition), by Tom Bassarear (Houghton Mifflin, New York, 2005).Objectives:SWBAT identify magic squares. SWBAT correctly rotate and reflect magic squares. SWBAT identify patterns in magic squares.SWBAT create their own magic squares.SWB introduced to the ideas of using variables and thinking algebraically. Texas Essential Knowledge and Skills:§111.16. Mathematics, Grade 4. (b)??Knowledge and skills(9) Geometry and spatial reasoning. The student connects transformations to congruence and symmetry. The student is expected to (A) demonstrate translations, reflections, and rotations using concrete models(15) Underlying processes and mathematical tools. The student communicates about Grade 4 mathematics using informal language. The student is expected to:(A) explain and record observations using objects, words, pictures, numbers, and technology; and(B) relate informal language to mathematical language and symbols.(16) Underlying processes and mathematical tools. The student uses logical reasoning. The student is expected to:(A) make generalizations from patterns or sets of examples and nonexamples; and(B) justify why an answer is reasonable and explain the solution process.Materials List and Advanced Preparations:Handout of 9 magic squaresPaper for writing vocabulary wordsAnalog clock (preferably with a second hand) Post-AssessmentsPaperPencilAccommodations for Learners with Special Needs (ELL, Special Ed, 504, GT, learning styles, etc.): None provided below, though this could be added.5EsENGAGEMENT 1Time: 5 MinutesWhat the Teacher Will DoProbing/Eliciting QuestionsStudent Responses and MisconceptionsUse 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]123456789Did 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 AssessmentStudent OutcomesOnce students understand the row rule for magic squares, we can continue. Students understand the rules.EXPLORATION 1Time: 20 MinutesWhat the Teacher Will DoProbing/Eliciting QuestionsStudent Responses and MisconceptionsOK, that didn’t work. So I want you to try it out. This is like a puzzle. See if you can find a way to fill in the square so that each number is used only once, and each row has the same sum.Good observation. It’d be helpful if we knew what the sum of each row was. OK, very good. Now try to fill in the numbers so that each row adds up to 15.OK, very good. Now I’m going to give you a trickier puzzle. See if you can make a square so that each row has the same sum and each column has the same sum.C’mon, it won’t be that bad. Try it out.OK, let me give you a hint. Let me put the 9 in the upper-left corner.So the top row has to be either 1 and 5 or 2 and 4, and the left column has to be other pair of numbers. Let’s fill those numbers in.Well, let’s fill in something.OK, let’s figure out the rest. Let’s think about where the 7 has to go. OK, so wherever the 7 goes, there has to be a 2 and 6 or 3 and 5 in the same row and column. But wait --- we’ve already put down the 2 and the 5.Good. Let me write that down. [The square should now look something like this:915274][If by chance the square produced is actually a magic square, then skip to the end. But that probably won’t happen.]OK. So now let me give you a really tricky puzzle. Let’s make a square so that the rows all add to 15, the columns all add to 15, and both diagonals add to 15.Let me give you some good news: we won’t have to start from the beginning. We’ll use the square that we’ve already made as our starting point.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.]OK, let’s think about it. If I add the numbers 1 through 9, what do I get?And how many rows are there?So what does the sum of each row have to be?If you think you’ve got it, write your answer on the board.Do all of these work?[Only do the part in italics if they’re stuck.]Having trouble?What do the rest of the top row and left column have to sum to?And how can we get a sum of 6? You choose: where would you like me to put the numbers?If I put 7 someplace, what do the other two numbers in its row have to sum to?And how can we get a sum of 8?So where should I put the 7?Now how should we get the rest of the squares?Good. So figure out the remaining three squares.Let’s check. Do all of the rows and columns of this square add to 15?Now this is tricky. If I add the diagonals, what two sums do we get?Suppose we switch the top two rows. Will all of the sums still be 15?And if we switch two of the columns?What do the rows sum to?And the columns?And the diagonals?[Students experiment for 2-3 minutes.][Frustrated.] This would be a whole lot easier if we know what the numbers were supposed to add up to.45!3!Oooh, I get it. 45 ÷ 3 = 15.[Students experiment for a few minutes. If someone gets it early, ask them to quiet move on to the next puzzle. If everyone is absolutely stuck, place 9 in the northwest square and ask them to figure out what the rest of top row and left column have to be.][Students write answers on board.]Yes! [hopefully]Oh, man.[Students experiment for about 5 minutes. If someone gets it, wonderful... have him/her share with the class, then skip the italics and move on to the next puzzle. However, if everyone gets stuck, see italics.]YES!15 – 9, so 6.1 and 5.2 and 4.3 and 3… oops, that won’t work since we can only use 3 once.Which numbers go where?[Give directions.]15 – 7, so 8.1 and 7… oops, that won’t work since we can only use 7 once.2 and 6.3 and 5.4 and 4… oops, that won’t work since we can only use 4 once.Oooh… in the same row/column as 2 and 5!All the rows and columns add up to 15![Students figure out the remaining squares.]Yes![Gives answers.]Oh, man.Yes!Yes![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]:81635749215!15!15! Wow![Students write magic square and magic sum on their vocabulary sheets.]Evaluation/Decision Point AssessmentStudent OutcomesWhen students are comfortable with the definition of a magic square, we’ll continue.Students are able to correctly identify magic squares. EXPLANATION 1Time: 10 MinutesWhat the Teacher Will DoProbing/Eliciting QuestionsStudent Responses and MisconceptionsLet’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.Good. Turning the square the other direction is called rotating counterclockwise. [Writes on board.] That means that the rotation is in the opposite direction of the way the hands of a clock turn. Please write this on your vocabulary sheet.In England, they don’t say counterclockwise. They say anticlockwise.Good. This new square was made by a reflection. [Writes on board.] The middle row didn’t move, but the other rows were reflected through the middle row.Excellent. Rotations and reflections are two examples of transformations, or ways that a figure can be changed to get a new figure. [Writes on board.] Please write those words 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?What happens If we rotate the square clockwise by another 90 degrees? How many degrees have we turned it?Is this new square a magic square?What happens If we rotate the square clockwise by another 90 degrees? How many degrees have we turned it?Have you ever heard of the terms “180 degrees” or “360 degrees” before?Can anyone suggest a way of getting the 270-degree square directly from the original square?So we’ve seen that we can rotate a magic square to get a new magic square. Can anyone think of anything else we can do to get a new magic square?Let’s try something different. What happens if we take the original square and flip the top and bottom rows?Did we see this square before?Is this new square a magic square?Can anyone think of a real-life example of a reflection… perhaps in your bathroom?Can anyone think of another way to make a new magic square?Good. Start with the original square. What do you get if we reflect through the middle column?Have we seen this square before?Is it also a magic square?Now let’s take this square and reflect through the middle row. What do we get?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?438951276[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.294753618Yes!270 degrees.672159834Yes!360 degrees.Hey, we get the original square!Basketball (slam dunks).Gymnastics.Snowboarding.Sure, just turn the original square in the other direction by 90 degrees.[Students write counterclockwise on vocabulary sheet.][Stunned silence.]492357816No.Yes!A mirror!The top of a swimming pool![Students write reflection and transformation on vocabulary sheet.]Reflect the first and third columns!618753294No.Yes!294753618No... wait, yes!Rotating by 180 degrees....as making two reflections.6.Evaluation/Decision Point AssessmentStudent OutcomesStudents are comfortable with rotations and reflections.Students are able to correctly produce new magic squares by rotations and reflections. ELABORATION 1Time: 10 MinutesWhat the Teacher Will DoProbing/Eliciting QuestionsStudent Responses and MisconceptionsOK, I want you to try to make a new magic square. Start with one of the squares we’ve already found, and then either perform a rotation or a reflection. Here are the rules:Make a new square that we haven’t seen before. Make sure that the new square is a magic square.Write down how you got your new square… which square you started with, and which transformation you used.It turns out that there are no other magic squares using 1-9 besides the ones we just made. There are only eight such magic squares.It turns out that these rotations and reflections make up something called a dihedral group. Maybe later this summer we’ll study this a little further.OK, how did you get your squares?Wow, we had some different answers for that one. Can anyone explain why that happened?[Students start experimenting.]834159672276951438[Answers should vary.]Oooh. A rotation and a reflection can end up the same as a different reflection and a different rotation.EVALUATION 1Time: 10 MinutesWhat the Teacher Will DoProbing/Eliciting QuestionsStudent Responses and MisconceptionsStudents complete Post-Assessment 1.ENGAGEMENT 2Time: 15 minutesWhat the Teacher Will DoProbing/Eliciting QuestionsStudent Responses and Misconceptions[Passes out sheet with several different magic squares.]Last time, we made magic squares using only the numbers 1 through 9. Now, let’s take a look at some magic squares that use other numbers.For each magic square, let’s call the sum of any row, column or diagonal the magic sum. And it’s OK if the magic sum isn’t 15.You’ll see that three of the magic squares on the sheet have blank spaces. Figure out the magic sum, and then figure out the blanks.Please pick one of the new magic squares.Is that also a magic square?[Students pick one.][Students add to confirm that it’s a magic square.][Students figure out the missing squares.]Evaluation/Decision Point AssessmentStudent OutcomesReinforce definitions of magic square.Students can correctly identify a magic square. EXPLORATION 2Time: 15 minutesWhat the Teacher Will DoProbing/Eliciting QuestionsStudent Responses and MisconceptionsNow 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 elseOK, 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 write3, 5, 71, 5, 91, 18, 3525, 18, 114, 10, 1618, 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.A group of numbers like this is called an arithmetic sequence.[Writes 10, 13, ____ on board.][Writes ____, 11, 18 on board.][Writes 4, ____, 10 on board.]Good. The middle number is the average of the two outside numbers.So we’ve seen that the middle row of a magic square is an arithmetic sequence, and the middle column is also an arithmetic sequence.Let’s now find another pattern. Take a look at the edge squares, which are the four outside squares that aren’t corner squares.All of these groups of three numbers have something in common. Can you figure out what it is?If these numbers make an arithmetic sequence, what’s the next number?How about this one?How about this one?How did you get 7?What do you notice? What’s true about all four of those numbers?[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.][Students write arithmetic sequence on their vocabulary sheets.]16!4!7!It’s the average of 4 and 10.[Students write average on their vocabulary sheets.]Hey! They’re all even or they’re all odd!For the teacher’s reference, here are some more possible student observations:1. Each diagonal is an arithmetic sequence.2. If I multiply the middle number by three, I get the magic sum.3. Opposite corners are either both even or both odd.4. It’s possible for a precocious student to come up with the pattern for how the corner squares can be obtained from the other five squares, but that’s doubtful.Evaluation/Decision Point AssessmentStudent OutcomesStudents find the three important properties of 3x3 magic squares --- and perhaps some others that turn out to be consequences of these three properties, though they probably won’t see the connection[s].Students can articulate mathematically (not necessarily algebraically) the patterns that they find.EXPLANATION 2Time: 25 minutesWhat the Teacher Will DoProbing/Eliciting QuestionsStudent Responses and MisconceptionsAs you may have noticed, the middle number of the magic square is really important. [The board should end up with eight magic squares, preferably in two rows and four columns. So be sure to leave space appropriately when writing.]Let’s see if we can figure out a way of writing down any 3x3 magic square. Let’s begin by looking at this magic square:492357816Good observations. So I could write those entries like this:5+45-255+25-4Now let’s look a different magic square.92347121720115Now let’s try to build our own magic square that different from all of these others. [Draws blank magic square on board.]OK, I’ll put that number in the middle.OK, I’ll put that number to the right of the middle one. [These two squares are written on the board, one above the other.]#1 here#2 here#1#1 +/-diff1#3 hereCom-puted#1 here#2 hereCom-puted#1 +/- diff2#1 -/+ diff1#1#1 +/-diff1#1 -/+ diff2OK, let’s now turn to the corners (pardon the pun), starting with the first magic square on the board.4923578165+45-35-255+25+35-4OK, very good. Let’s now go to the magic square that we made. [Point to the square between #1 + diff1 and #1 – diff1; this could be in any of the four corners, depending on the numbers that the class chose.]Why is the middle number important?Let’s take a look at the middle row. What do we have to do to start with the middle number and end up with 7?And what do we have to do to start with the middle number and end up with 3?Now let’s look at the middle column. What do we have to do to start with the middle number and end up with 9?And what do we have to do to start with the middle number and end up with 1?We’ll worry about the corners later. But so far, does everyone agree with this?Let’s again take a look at the middle row. What do we have to do to start with the middle number and end up with 16?And what do we have to do to start with the middle number and end up with 3?Now let’s look at the middle column. What do we have to do to start with the middle number and end up with 18?And what do we have to do to start with the middle number and end up with 9?So how could I write the middle row and column using addition and subtraction?Does anyone see a pattern?What will you like the middle number to be?What would like the number to the right of the middle to be?So far, so good. Now, how will we figure out the number to the left of the middle?So what is it?And what is the magic sum?And how did you get that?Now, what do you want the top number to be? I make one request: choose an even [or odd] number because the left and right numbers are both even [or odd].Now, how will we figure out the bottom number?So what is it?And does the magic sum still work? How do we go from 5 to 2?How do we go from 5 to 8?How about the second magic square? Can we do the same thing with the top right and bottom left corners?92347121720115Does anyone see a pattern with the numbers we’re adding and subtracting?What about whether to add or subtract?Does the same pattern hold for this magic square?OK, we have two corners left. How do we figure out what goes in those corners?OK, so let’s figure out what goes in the corners of our magic square.Finally, let’s double-check. Is this square really a magic square?[Teacher then asks the class to make a fourth magic square, repeating the same steps that were used to make the third, to ensure that everyone gets the procedure.]Possible answers:It’s the average of the numbers to the left and right.It’s the average of the numbers above and below.It’s the average of the opposite corners.It’s one-third of the magic sum.Add 2!Subtract 2!Add 4!Subtract 4!Yes!Add 6!Subtract 6!Add 8!Subtract 8!12+1112-51212+512-11Sure. To get the middle row, we add and subtract the same number from the middle number. Same for the middle column, but with a different number.The middle row and columns are arithmetic sequences.[Student volunteers a number. Make sure it’s a positive integer between 5 and 100, for simplicity.][Student volunteers a different number. Make sure that it’s positive and less than twice the middle number so that the left-of-middle number is also positive.]Subtract the two numbers to figure out the difference. Then use that difference to figure out the left number.[Students give the answer.][Students give the answer.]Adding the three numbers together.Multiplying the middle number by three.[Student volunteers a different number. Make sure that it’s positive and less than twice the middle number so that bottom number is also positive. Also, the number must have the same parity as the left and right numbers.]Subtract the top and middle numbers to figure out the new difference. Then use that difference to figure out the bottom number.[Students give the answer.]Yes!Possible student question: but why don’t the top and right squares have +’s like the first two? [Answer: rotations and reflections.][Groans.]Subtract 3!Add 3!12+1112-812-51212+512+812-11[Silence.]Yes! The numbers 2,3,4 are in a sequence [adding 1], and the numbers 5,8,11 are in a sequence [adding 3].Between 12+5 and 12+11, we enter 12-8.Between 12-5 and 12-11, we enter 12+8.Yes!Possible student question: But why aren’t we looking at the top right corner [a 75% chance of happening].Answer: We’re looking at the square between the two +’s.Possible answers:1. Use the magic sum, which we already know from the middle row, the middle column, and the two diagonals.2. A precocious student may see that you need to subtract the row difference and column difference and then divide by 2.[Students figure out the two remaining corners.]Yes!Evaluation/Decision Point AssessmentStudent OutcomesWhen students are comfortable producing their own magic squares, we can proceed to developing algebraic thinking.Students can recognize arithmetic sequences of three integers and can produce either the next term, the previous term, or a middle term.ELABORATION 2Time: 15 minutesWhat the Teacher Will DoProbing/Eliciting QuestionsStudent Responses and MisconceptionsOK, let’s make another magic square.OK, I’m going to write all of your ideas down at once. Let’s call m the middle number. mThe letter m is called a variable. That’s because it can change (or vary) with different magic squares.Let me enter those on the middle row:m – hmm + hSo here’s our magic square so far:m + vm – hmm + hm - vGood. The average of h and v is written as h+v2. So we can now fill in the top right and bottom left corners:m + v m-h+v2m – hmm + h m+h+v2m - vIt turns out that the other two corners also have a pattern: m+h-v2m + v m-h+v2m – hmm + h m+h+v2m - v m-h-v2Doing 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. [Polls class.]What would you like the middle number to be?How about you?How about you?I need a new variable for the differences on the middle row. Any ideas?In the first magic square, what were the values of m and h?How about the second magic square?Let’s go back to the magic square we were making. So what should the other entries in the middle row be?Does the m + h absolutely have to go to the right?I need a new variable for the differences on the middle column. Any ideas?So what should the other entries in the middle column be?So what do I have to do get the top-right corner?Good. It’s good to be m minus something. How do I get the something?Did this pattern work for the squares that we had before?Here’s another pattern. What do we get if we add the middle row. What is m plus m + h plus m – h?How about the middle column? What is m plus m + v plus m – v?So the magic sum is always equal to….So how does the magic sum compare to the middle number? [Students give suggested middle numbers.]Huh?[Students write variable on their vocabulary sheets.][Student volunteers a letter --- perhaps h for horizontal or a for across.]5 and 2.12 and 5.m + h and m – h.No, it’d be OK to put it on the left side.[Student volunteers a letter --- perhaps v for vertical.]m + v and m – v. [Use the letter given by the class.]Subtract something from m.It’s going to be the average of h and v.[Students confirm that for the first square, m = 5, h = 2, v = 4.][Students confirm that for the second square, m = 12, h = 5, v = 11.]3 times m.3 times m.3 times m!Oooh. The magic sum is always equal to 3 times the middle number.[Students write algebra on their vocabulary sheets.]Evaluation/Decision Point AssessmentStudent OutcomesStudents 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 2Time: 10 minutesWhat the Teacher Will DoProbing/Eliciting QuestionsStudent Responses and MisconceptionsPost-Assessment 2. ................
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