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Summer Mathematical Enrichment Class for Girls 2012

3x3 Magic Squares and Similarity TransformationsAuthor(s): John QuintanillaDate/Time Lesson to be Taught: August 6, 2012Technology Lesson: Yes No

Course Description: Name: Summer Mathematical Enrichment Class for Girls 2012Grade Level: Mostly 3rd gradersHonors or Regular: Honors

Lesson 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 squares Paper for writing vocabulary words

Page 1 of 13


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.

5EsENGAGEMENT 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]

1 2 3

4 5 6

7 8 9

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 OutcomesOnce students understand the row rule for magic squares, we can continue.

Students understand the rules.

Page 2 of 13


Page 3 of 13

EXPLORATION 1 Time: 20 Minutes

What the Teacher Will Do Probing/Eliciting Questions Student Responses and Misconceptions

OK, 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

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?

[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.


Page 4 of 13


Page 5 of 13

EXPLANATION 1 Time: 10 Minutes



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?

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

4 3 8

9 5 1

2 7 6

[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.

2 9 4

7 5 3

6 1 8

Yes!

270 degrees.

6 7 2

1 5 9

8 3 4

Yes!

360 degrees.

Hey, we get the original square!


Page 6 of 13

ELABORATION 1 Time: 10 Minutes



OK, 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:

1. Make a new square that we haven’t seen before.

2. Make sure that the new square is a magic square.

3. 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.]

8 3 4

1 5 9

6 7 2

2 7 6

9 5 1

4 3 8

[Answers should vary.]

Oooh. A rotation and a reflection can end up the same as a different reflection and a different rotation.


ENGAGEMENT 2 Time: 15 minutes


[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 Assessment Student OutcomesReinforce definitions of magic square. Students can correctly

identify a magic square.

Page 7 of 13

EVALUATION 1 Time: 10 Minutes



Students complete Post-Assessment 1.


Page 8 of 13


Page 9 of 13

EXPLORATION 2 Time: 15 minutes



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, 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.]

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?

[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.


Page 10 of 13


Page 11 of 13

EXPLANATION 2 Time: 25 minutes


As 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:

4 9 2

3 5 7

8 1 6

Good observations. So I could write those entries like this:

5+4

5-2 5 5+2

5-4

Now let’s look a different magic square.

9 23 4

7 12 17

20 1 15

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?


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?


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

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!


Page 12 of 13

ELABORATION 2 Time: 15 minutes



OK, 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.

m

The 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 – h m m + h

[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?

[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.]


Page 13 of 13

EVALUATION 2 Time: 10 minutes



Post-Assessment 2.

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Description - Web viewIf someone gets it early, ask them to quiet move on to the next puzzle. If everyone is absolutely stuck, ... Title: Description Last modified by: John Company: