Summer Mathematical Enrichment Class for Girls 2012
3x3 Magic Squares and Similarity Transformations
Author(s): John Quintanilla
Date/Time Lesson to be Taught: August6, 2012
Technology Lesson: YesNo
Course Description:
Name: Summer Mathematical Enrichment Class for Girls 2012
Grade Level: Mostly 3rd graders
Honors 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
- 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 1Time: 5MinutesWhat 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 Outcomes
Once students understand the row rule for magic squares, we can continue. / Students understand the rules.
EXPLORATION 1Time: 20Minutes
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 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:
9 / 1 / 5
2 / 7
4
]
[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, thenskip 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]:
8 / 1 / 6
3 / 5 / 7
4 / 9 / 2
15!
15!
15! Wow!
[Students write magic squareand 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 1Time: 10Minutes
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.
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? / 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!
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.]
4 / 9 / 2
3 / 5 / 7
8 / 1 / 6
No.
Yes!
A mirror!
The top of a swimming pool!
[Students write reflection and transformation on vocabulary sheet.]
Reflect the first and third columns!
6 / 1 / 8
7 / 5 / 3
2 / 9 / 4
No.
Yes!
2 / 9 / 4
7 / 5 / 3
6 / 1 / 8
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.
ELABORATION 1Time: 10Minutes
What the Teacher Will Do / Probing/Eliciting Questions / Student Responses and Misconceptions
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:
- 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 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.
EVALUATION 1Time: 10Minutes
What the Teacher Will Do / Probing/Eliciting Questions / Student Responses and Misconceptions
Students complete Post-Assessment 1.
ENGAGEMENT 2Time:15 minutes
What the Teacher Will Do / Probing/Eliciting Questions / Student 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 Assessment / Student Outcomes
Reinforce definitions of magic square. / Students can correctly identify a magic square.
EXPLORATION 2Time: 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
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.
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.