Transcript for the Math Jam "MOEMS Teachers Math Jam" on Jan 3.

Math Jam hosted by sumtchr (Mary Altieri ).

rrusczyk (19:27:16)
Greetings and welcome to another MOEMS Math Jam.

rrusczyk (19:27:22)
MOEMS is an outstanding way for students in grades 4-8 to get started with problem solving mathematics. Therefore, we have invited the people from MOEMS to host Math Jams to discuss how to use MOEMS to inspire students to tackle challenging problems.

rrusczyk (19:27:28)
For those of you who have showed up to see what AoPS classes are like, please view transcripts of other Math Jams from the past:

rrusczyk (19:27:33)
http://www.artofproblemsolving.com/Community/AoPS_Y_MJ_Transcripts.php

rrusczyk (19:27:37)
Today's Math Jam will be considerably different than AoPS classes, as it is both for a program that focuses on an earlier age than most of our classes, and because it is aimed at teachers involved with MOEMS, instead of being aimed more at students.

rrusczyk (19:27:49)
Now, on with the show!

rrusczyk (19:27:54)
I'll now turn things over to your instructor for today.

sumtchr (19:28:05)
Nice to be here. I hope nobody catches my cold!

sumtchr (19:28:29)
My name is Mary Altieri and I am lucky to be here!

sumtchr (19:29:04)
Tonight we are going to do a series of problems that are somewhat related

sumtchr (19:29:19)
The relationship may not be obvious at first.

sumtchr (19:29:42)
Here is the first one:

sumtchr (19:29:48)
Consider the set of triangles shown. They form a pattern from left to right.

sumtchr (19:30:26)


sumtchr (19:30:34)
Can everyone see it?

sumtchr (19:31:16)
Consider the set of triangles shown. They form a pattern from left to right.

sumtchr (19:31:22)
Can you describe the next triangle in the pattern?

sbanerjee (19:31:32)
it will have 16 triangles

sumtchr (19:32:13)
I asked you to describe it

bkhatchell (19:31:59)
The base of the triangle has 4 triangles

sumtchr (19:32:52)
Some may see it as more than four

tsalerno (19:32:45)
It has a row of seven triangles on the bottom.

sumtchr (19:33:22)
Why are we getting 4 and 7?

lscott (19:33:18)
I would say each "row" adds two triangles to the # of the row above it.

sumtchr (19:33:48)
nice way to describe it

sumtchr (19:33:57)
Does anyone see it a different way?

sbanerjee (19:33:34)
because some triangles are upside down

lscott (19:34:44)
4 vs 7 has to do with the way the triangle "sits"

british8985 (19:34:50)
# of bottom triangles of each triangle increases by2

sumtchr (19:35:22)
nice!

sumtchr (19:35:39)
So how many of the smallest triangles will be in the fifth one?

tsalerno (19:35:49)
23

sbanerjee (19:35:52)
25

bkhatchell (19:36:28)
25

sumtchr (19:36:56)
How did you get your numbers? First 23

british8985 (19:36:57)
bottom 9=4+5 then 4+3 then 3+2 then 2+1 then 1

sumtchr (19:37:55)
That's 25

british8985 (19:37:17)
so it's 25

sumtchr (19:38:21)
Anybody still think it is 23?

tsalerno (19:38:24)
My posts are taking a long time to show up. 25 is right. My initial answer was wrong.

sumtchr (19:39:30)
I am sorry to be so slow

tsalerno (19:37:29)
25 is right...add 9

Amigo (19:38:49)
# of triangles in bottom row squared

sumtchr (19:40:08)
Does anyone else see that

bkhatchell (19:36:51)
9+7+5+3+1=25

rrusczyk (19:40:33)
I will also note that the classroom is moderated - therefore your comments will only appear when the intructor passes them into the room. (This is why it may seem like a long time between your post and it appearing in the room, or why some of your comments won't appear at all.)

sbanerjee (19:40:21)
that's how i got the answer also

sumtchr (19:41:02)
How about the 10th one?

conartist (19:41:22)
100

sbanerjee (19:41:26)
100

bkhatchell (19:41:30)
Yes - so the sixth one would have 36 and so on, and the 10th would have 100 triangles

sumtchr (19:42:02)
Great

sumtchr (19:42:14)
So how about the nth one?

conartist (19:42:20)
n^2

lscott (19:42:26)
n squared

sbanerjee (19:42:36)
or N squared

tsalerno (19:42:37)
n squared

PenguinIntegral (19:42:49)
n^2

sumtchr (19:43:30)
What is the difference between trying to figure out the 4th or 5th one and the nth one?

Amigo (19:42:39)
n squared

PenguinIntegral (19:43:33)
Note the difference of consecutive squares is an odd number

sumtchr (19:44:02)
Remember that for later?

sumtchr (19:44:49)
What will our students say when we ask for the nth of anything

lscott (19:44:09)
4th and 5th are small enough to "add", with N you need to think of it abstractly

tsalerno (19:44:17)
You can draw the 4th and 5th, or add odd numbers to the last one, but you can't do that for the nth.

sumtchr (19:45:20)
Good!

sumtchr (19:45:41)
So what kind of a relationship do you have to draw?

PenguinIntegral (19:45:15)
With known nubers, you reduce it to a computation exercise. With N, you need to think in terms on a general solution.

lscott (19:45:21)
kids will say "what does that mean?"

sumtchr (19:46:07)
Right!

sumtchr (19:46:33)
So we have to think about the term and the answer rather than recursively--what happened before

sumtchr (19:47:54)
Does anyone find it odd that a series of triangles gives a series of square numbers?

sumtchr (19:48:27)
You might want to try (on your own) using pattern blocks to build the next larger similar figure for all of the pattern block shapes--which ones work, which ones don't and what patterns become evident. It might be a surpising discovery.

Amigo (19:48:32)
2 triangles in a square

sbanerjee (19:48:33)
i dont think this is the reason but-half a square is a triangle

sumtchr (19:49:22)
And?

sbanerjee (19:46:31)
what do you mean by kids?

british8985 (19:49:16)
how can we prove it's n^2

PenguinIntegral (19:49:40)
Note they are equal triangles. I could just draw some lines and say "Three triangles".

RichKal-MOEMS (19:51:08)
How can prove WHAT is n^2?

RichKal-MOEMS (19:51:26)
What does N represent?

sumtchr (19:51:51)
What do you think might happen with a trapezoid?

sumtchr (19:52:33)
The next larger similar trapezoid to the red pattern block

sumtchr (19:52:59)
You might want to try that with the blocks.

sbanerjee (19:52:07)
i think it would be N^3

british8985 (19:53:57)
I saw that on text book ...it's like one block then add three blocks to form 2^2 ...add 5 blocks to form 3^2..etc

sumtchr (19:54:34)
So the series is once again of square numbers?

sumtchr (19:55:07)
I've given you plenty of food for thought. Have your students play with it!

sumtchr (19:55:44)
Now to the next problem:

sumtchr (19:55:49)
Some of you are familiar with the handshake problem.

sumtchr (19:56:10)
For those who aren't, the problem goes like this:

sumtchr (19:56:18)
There are 11 of people in a room. Everyone greets each other person with a handshake. After everyone has shaken hands, how many shakes have occurred?

Does everyone understand?

sbanerjee (19:56:04)
ah yes...the handshake problem

sbanerjee (19:56:13)
i've seen it before

sumtchr (19:56:54)
What is yoour method of solution

sbanerjee (19:45:35)
i didnt think about that

bkhatchell (19:56:51)
Yes, a basic combinations problem

PenguinIntegral (19:57:24)
It is helpful to start out with small cases to get a feel for the problem. Making 11 dots and connecting them to represent handshakes is also a idea.

sumtchr (19:57:49)
Nice!

sumtchr (19:58:23)
And the solution begins, how?

bkhatchell (19:58:00)
Most kids will just looks a how many times one person shakes hands, multiply by the number of people, and then divide by two.

sumtchr (19:59:00)
Is that the right answer?

sbanerjee (19:58:04)
I saw in a chapter in a text book. The answer is 11! (factorial)

sbanerjee (19:58:06)
i think

sumtchr (19:59:36)
You have to think your own way through it.

sumtchr (19:59:50)
Remembering what is in a book might not be the best.

tsalerno (19:59:10)
Person 1 shakes hands with 10 people. Person 2 has already shaken hands with person 1, so he shakes hands with 9 people, etc.

british8985 (19:59:43)
11*10/2*1

british8985 (19:59:13)
we say A shakes with B..but B also shake with A..so divide by two

british8985 (19:56:38)
55

sumtchr (20:01:21)
Good job!

preya (20:00:07)
thats not the right answer

sumtchr (20:01:44)
What do you think is the right answer?

sbanerjee (20:01:09)
[img id=em-2]

sumtchr (20:02:08)
Don't be so sad

sumtchr (20:02:21)
Has anyone approached this problem "geometrically"?

preya (20:01:55)
never mind

british8985 (20:00:29)
10+9+8+7+6+5+4+3+2+1

Amigo (20:02:37)
what about diagonals with the 11 dots?

sumtchr (20:03:39)
Yes, but I was thinking of triangles.

bkhatchell (20:04:02)
11C2 = 11!/(2!*(11-2)!)

sumtchr (20:05:04)
Right, but let's remember who our audience is. Students in grades 4 - 8.

sumtchr (20:05:24)
Consider the triangular numbers as a right triangle. I have used a grid because it is easier to display it this way.


sumtchr (20:06:50)



sumtchr (20:07:12)
How does this relate to handshakes?

PenguinIntegral (20:01:03)
general formula: n(n-1)/2

PenguinIntegral (20:07:49)
Notice you can use this triangle to explain the general formual n(n-1)/2

sbanerjee (20:07:52)
the first row is 1 the 2nd row has 2 blocks etc.

british8985 (20:07:59)
eleven people stand on row and columns

british8985 (20:08:07)
no, 10 ppl

sumtchr (20:08:55)
What if we cut the triangle halfway down horizontally?

sumtchr (20:09:11)
And rotated it to fit in with the other half?

sumtchr (20:09:32)
It becomes a rectangle

sumtchr (20:09:44)
With what as its dimensions?

smueller (20:08:44)
There is one less square on each stairstep in the triangle. There is one less person with whome to shake hands?

sumtchr (20:10:26)
yes

sbanerjee (20:09:51)
it would be a square I think

sumtchr (20:10:35)
Why a square?

bkhatchell (20:10:02)
11*5=55

sumtchr (20:11:06)
Those are the dimensions of the rectangle!

sumtchr (20:11:30)
So that the general formal can be gotten from the picture!

british8985 (20:10:59)
rextangle, with sides 10*11

sumtchr (20:11:48)
Not exactly.

sumtchr (20:12:10)
Once we cut in halfway down, the height is only 5.

sbanerjee (20:11:31)
it is a rectangle

british8985 (20:11:54)
must be a rectangle coz u don't shake with urself

sumtchr (20:13:05)
Why does that make it a rectangle?

british8985 (20:12:35)
yep.....sorry....half of 10*11

sumtchr (20:13:33)
good

sumtchr (20:13:55)
Does anyone know about "triangular numbers"

sbanerjee (20:14:00)
yes i do!

tsalerno (20:14:02)
I do.

sumtchr (20:14:17)
Isn't that what we have been doing?

sumtchr (20:15:11)
The number of handshakes was 1, then 3, then 6, then 10, then 15, 21, then 28, etc.

british8985 (20:15:09)
what triangle numbers??? 1, 2 , 3, 4, ??

sumtchr (20:15:41)
I have a picture to show you.

sumtchr (20:16:14)


tsalerno (20:12:18)
5 * 11

british8985 (20:15:52)
oh I understand

sumtchr (20:17:02)
How is this for an extension?
For those of you who know the other problem, this one may still present a challenge.

british8985 (20:16:02)
1+2+3+4+5..

sbanerjee (20:17:45)
then 0+1=1,1+2+3,3+3=6,6+4,et.

sumtchr (20:18:17)
Suppose in a room there are n people. When they have each finished shaking everyone's hand, 91 handshakes have occurred. What number is represented by n?

sumtchr (20:18:36)
This is one you can play with on your own!

sumtchr (20:18:56)
Here is another related problem.

sumtchr (20:19:49)


sumtchr (20:20:10)
Consider this triangle of balls.

sumtchr (20:20:21)
What if another layer were added.

sumtchr (20:20:32)
The layer would sit in the spaces

sumtchr (20:20:53)
And layer upon layer would become a pyramid

sumtchr (20:21:05)
Do you see?

sumtchr (20:21:35)
So how many layers for the pyramid on this base?

sbanerjee (20:20:52)
making it part of a tetrahedron sort of thing?

sumtchr (20:22:12)
It is a triangular pyramid, with the first layer of balls.

sumtchr (20:22:21)
as the base.

tsalerno (20:22:05)
I'm confused.

sumtchr (20:22:38)
Still?

tsalerno (20:22:36)
Would there be 10 balls in the second layer?

sumtchr (20:23:03)
How did you get that?

bkhatchell (20:21:37)
15+10+6+3+1

british8985 (20:21:47)
five

sbanerjee (20:22:52)
i get it but it's hard to visualize in your mind

sumtchr (20:23:43)
I know it is hard to visualize, but pay attention to the numbers.

sumtchr (20:23:49)
Are they familiar?

sumtchr (20:24:10)
Think about it from bkhatchell's response

british8985 (20:23:24)
triangle number

sbanerjee (20:24:16)
15 in the base

sbanerjee (20:24:24)
10 in the next layer

sbanerjee (20:24:33)
6 in the next layer

sumtchr (20:24:50)
Great!

sumtchr (20:25:02)
So now, what if I ask a new question

sumtchr (20:25:28)
There are 100 in the bottom two layers. How many layers?

sbanerjee (20:25:54)
that's an interesting twist

sumtchr (20:26:28)
One way to think about it is to think about consecutive triangular numbers>

sumtchr (20:26:53)
What is the sum of every consecutive pair?

british8985 (20:26:43)
1 3 6 10 15 21 28 36 45 55

RichKal-MOEMS (20:27:06)
Find 2 consecutive triangular nos. whose sum is 100?

british8985 (20:26:47)
45+55=100

sumtchr (20:27:23)
How many layers?

tsalerno (20:27:30)
10

british8985 (20:27:34)
10

bkhatchell (20:27:46)
55+45=100, and there are 10 layers, 100= 10^2

sumtchr (20:28:32)
What are you answering?

british8985 (20:27:12)
n^2

sbanerjee (20:28:59)
55 is the 10th triangleular number so 10 layers

tsalerno (20:29:00)
The sum of every consecutive pairs is a square number.

sumtchr (20:29:43)
Good thinking!

RichKal-MOEMS (20:29:45)
bk, are you saying that the sum of two triangular nos is the square of the larger layer?

british8985 (20:30:01)
that's right

sbanerjee (20:30:08)
if thats true it woudl help...

Amigo (20:30:09)
I agree

sumtchr (20:30:25)
Great.

RichKal-MOEMS (20:30:31)
Great insight!

sumtchr (20:30:44)
Now, what is it that was the theme of tonight's problems?

sbanerjee (20:30:54)
patterns

sumtchr (20:31:16)
True, any special kinds?

bkhatchell (20:31:19)
Triangular numbers, but is there a closed formular for them?

sumtchr (20:31:57)
Yes there is, and it is the one mentioned several times.

british8985 (20:18:53)
(1+n)n/2=91

tsalerno (20:31:21)
triangular and square numbers and their relationship

sumtchr (20:32:18)
Thanks,

sumtchr (20:32:33)
Good insights everyone.

sumtchr (20:32:47)
Remember that patterns don't always have to come from formulas

sumtchr (20:32:59)
They can come from great pictures!

tsalerno (20:31:47)
Thanks, all. Great class!

sumtchr (20:33:26)
And I thank you all.

sumtchr (20:33:39)
The next session is the 31st of January!

sumtchr (20:33:50)
Marshalyn Baker will be the host

sumtchr (20:34:01)
Good night all!

RichKal-MOEMS (20:34:08)
Great work, Mary!

bkhatchell (20:33:32)
Thanks alot - it was fun