Transcript for the Math Jam "MOEMS Teachers Math Jam" on Sep 20.

Math Jam hosted by RichKal-MOEMS (Richard Kalman ).

MOEMS MATH JAM

RichKal-MOEMS (19:39:14)
How many different numbers between 10 and 200 have their digit-sum =6?

oc918 (19:38:26)
While we wait, may I pose a question? How often do you meet with your students for team practice?

RichKal-MOEMS (19:40:41)
We recommend once a week

brendajay (19:40:26)
we meet once a week

winluo2 (19:39:59)
Isn't it any number that is 6 mod 9?

RichKal-MOEMS (19:42:14)
Would Pre=algebra kids know mods?

oc918 (19:41:49)
12 numbers

RichKal-MOEMS (19:44:37)
Explain, please in ways young children would follow.

Stefanie (19:43:26)
so the answer is : 12

sandra atmar (19:43:59)
16 numbers?

Hitchcock (19:44:45)
12

JSlupek (19:44:06)
Is the answer 12?

Hitchcock (19:45:08)
I used a list

RichKal-MOEMS (19:46:45)
Ereic tells me 12.

RichKal-MOEMS (19:48:34)
Eric says : 15,24,33,42,51,60,105,114,123,132,141,150

oc918 (19:45:47)
I began writing the numbers: 15, 24, 33, soon one notices that 60 will be the last with two digits. Then the first with 3 digits is 105, so 105, 114, 123, 132, 141, 150

brendajay (19:46:10)
the only digits you can use is 1,5,2,4,3

RichKal-MOEMS (19:49:44)
and 0

C-squared (19:47:22)
We looked at each group of tens (10-20 20-30,30-40 etc to find what would fit the criteria

Mary Ella Verdes (19:48:08)
if you look at the list a pattern is formed. 15 or 1+5 =6, then 24, 33,42,51,60 the witht he 3 digit numbers 104,114,123,132,141,150. The pattern is that the value of the ones place in one less than the one before. Each add up to 6 giving you the answer of 12 numbers

JSlupek (19:49:01)
I took the digits and started adding them. Between 10 and 60 there are are 6 numbers. You can't have sums of 6 for the 70's, 80's, and 90. Then you have the same for the sums of the digits from 100 through 160. (6)

Mary Ella Verdes (19:50:20)
if you look this is a pattern. The ones digit is always on less than the one before

RichKal-MOEMS (19:51:33)
Followups are a staple of our solution sheets on contests. Let's try one.

RichKal-MOEMS (19:52:30)
Extend the limits to 1 to 1000. How does affect the answer?

RichKal-MOEMS (19:53:43)
What does this change?

RichKal-MOEMS (19:54:26)
(It's more important to watch the nature of your patterns than to just get an answer.)

Stefanie (19:54:37)
26

RichKal-MOEMS (19:54:57)
Can you explain?

brendajay (19:54:52)
700 to 1000 there won't be any solutions

RichKal-MOEMS (19:55:45)
True

RichKal-MOEMS (19:55:51)
Good.

oc918 (19:55:15)
6 less than 100, 6 between 100 and 200, 5 between 200 and 300, 4 between 300 & 400, 3 between 400 & 500, 2 between 500 & 600, and 600 is the last. So, 27 in all.

sandra atmar (19:56:05)
I got 27

JSlupek (19:55:27)
There are 6 numbers in each set of 100 but you can only use this until you reach the 700. Then your digits will be too high. Is the answer 36?

RichKal-MOEMS (19:57:09)
Recheck your centuries

Jocelyn Chow (19:56:36)
We got 28

oc918 (19:56:43)
As the hundreds digits grows by one, that leaves us with one less each interval of 100.

Jocelyn Chow (19:57:03)
Wouldn't there be 7 number between 1-100, if you include the number 6?

brendajay (19:57:04)
there's only 7 in less than 100

DKING (19:57:17)
Since we went down to 1, do we include 6 in this? If so, I got 28.

C-squared (19:57:54)
there are 28 possibilities, Between 1-100 there are 7, 100-200 there are 6, 200-300 there are 5 numbers. There is a progression 7,6,5,4,3,2,1 which equals 28

brendajay (19:58:12)
my son says the answer is 28

RichKal-MOEMS (19:58:52)
Your son is super!

RichKal-MOEMS (19:59:55)
C-squared's progression of 7,6,5,4,3,2,1 leads you to triangular numbers, if you want.

Jocelyn Chow (19:59:07)
I agree your son is super! Fantastic work!

C-squared (20:00:02)
Jocelyn and I think we are going to use this first problem in our newsletter and then extend it at our first meeting

RichKal-MOEMS (20:00:45)
Great! That's how to stretch students.

RichKal-MOEMS (20:01:39)
How does changing 6 to 7, then 8, then 9, etc. change the outcomes?

RichKal-MOEMS (20:03:07)
Stefanie is referring to the sum of 6

Stefanie (20:01:23)
each time for the three digit numbers when the hundreds digit decreases, the # of numbers whoses digit's sum = 6 decreases by one Ex. 105,114,123,132,141,150, (6 of these) 204,213,222,231,240 (5 of these)

RichKal-MOEMS (20:04:39)
Do you agree with her? (That question's a fine way to develop critical thinking in kids)

RichKal-MOEMS (20:06:23)
I think she meant increasing the first time. Let's move back to the 7,8,9... question.

JSlupek (20:02:55)
It increases the outcomes because the possibilties are increased.

oc918 (20:02:57)
Patterns will be similar. Making organized lists should work for all students. Now, the answers are...?

Stefanie (20:05:35)
I agree with me

oc918 (20:05:37)
Yes, I agree. Something analogous will happen with other sums (7, 8, etc.).

C-squared (20:05:58)
There is an increase. The triangular progression becomes 8+7+6+5+4+3+2+1 which is equal to 36 possibilities.

RichKal-MOEMS (20:08:15)
This gives you an opening to help students build a formula.

Stefanie (20:06:46)
I meant to say increases *when the hundreds digit increases

brendajay (20:08:39)
For 7's the pattern might be 8 in under 100, then 8, 7, 6, 5, 4,3,2, 1 for a total of 36

Stefanie (20:08:31)
What is the 6,7,8,9,ect. question?

RichKal-MOEMS (20:09:42)
That was the flast followup: digit sum = 6, then 7, then 8, then ...

oc918 (20:08:47)
There will be one more per century as we increase the sum. For example, there are 9 numbers under 100 whose digits sum 8, 10 whose sum is 9, and something equivalent will occur with the next centuries.

RichKal-MOEMS (20:10:35)
Good conjecture. With your students focus on why it happens.

C-squared (20:09:52)
Is building formulas appropriate for 4th and 5th graders? I think I am more interested in seeing that they recognize a pattern

RichKal-MOEMS (20:11:59)
Formulas don't have to be formal and algebraic. Fourth graders can still generalize, just in words. Or Pictures.

C-squared (20:06:43)
Please explain triangular numbers. I can't explain it to Jocelyn

RichKal-MOEMS (20:14:11)
The triangular numbers (See our book Creative Problem Solving or several others) are the sum of the counting numbers 1 +2+3+4+5+6+7+8+... as high as you wish. They can be formed by dots in a triangular array.

RichKal-MOEMS (20:14:37)
For instance, Bowling Pins!

oc918 (20:12:41)
Depending on the students, in most cases it is appropriate. Recognizing patterns can lead to useful generalizations that can be expressed as formulas, although some scaffolding might be needed. Some people have used Excel with 5th graders to help them test their formulas.

oc918 (20:13:56)
Triangular numbers are those that have a triangular shape when expressed as collections of dots: 1, 3, 6, 10,...

Stefanie (20:13:37)
Is this a class for teachers or parents? I'm in 8th Grade

RichKal-MOEMS (20:15:50)
Actually, we intend it for our PICOs (coaches).

C-squared (20:10:30)
new problem please

RichKal-MOEMS (20:17:18)
Area of a rectangle is 36 sq cm. Length of its sides are whole numbers. What is the largest perimeter?

RichKal-MOEMS (20:18:33)
Again a reminder. Eric is locked out of his computer by a nasty firewall and I'm doing this as he dictates to me over the phone. Hold on, battery.

Stefanie (20:16:48)
coaches? What should I do?

RichKal-MOEMS (20:19:20)
Stay with us. You're very good.

Eric Ricketts (20:18:15)
40

RichKal-MOEMS (20:19:49)
How?

oc918 (20:19:12)
74, when sides are 1 and 36.

brendajay (20:19:33)
I would say the largest perimeter is 26

Stefanie (20:20:07)
74 36*1

DKING (20:20:12)
The largest perimeter will belong to the longest, skinniest rectangle. I got 74 sq. cm.

delbar (20:20:51)
1 X 36 rectangle gives you a perimeter of 74

DKING (20:21:01)
Oops! I meant 74 cm.

Eric Ricketts (20:20:03)
how did I get my answer?

andypoulsen (20:21:01)
well, one way to approach it is to find the two extremes -- 6x6 (square) and 36x1 (LONG rectangle) for the first, P= 4 sides times 6 = 24. for the second, P=2 sides times 36 plus 2 sides times 1 = 72 + 2 = 74. so the max is 74

C-squared (20:21:27)
The lengths of the sides are 36,1, 36, and 1. The perimeter is then 74

Sheila Nugent (20:21:53)
74

Eric Ricketts (20:22:08)
ok, 74 is correct, I was wrong, I did not think of 36 and 1

Sheila Nugent (20:22:00)
74

brendajay (20:21:20)
hurray for 74

Eric Ricketts (20:19:47)
I should see my questions appear on the screen?

RichKal-MOEMS (20:23:48)
This is moderated, so I get to pick and choose from what you submit. It's good to be the king!

andypoulsen (20:23:53)
though remember, the important thing is to make sure we can help the kids understand the *process* to get there! (though it's a fun little competition here, marred only by the time lag!)

RichKal-MOEMS (20:25:19)
A good followup is to ask for the least perimeter.

andypoulsen (20:24:27)
Hail to the king! :)

DKING (20:25:36)
With this problem, we can go back to lists again. Have the kids make a list of all the possible rectangles and then compute the perimeters.

oc918 (20:25:40)
Ahh! A classic!

Sheila Nugent (20:26:09)
well, my process was the same as another member's - it would be one extreme or the other, perfect square or very long

RichKal-MOEMS (20:26:40)
Why are those the extremes?

Stefanie (20:26:14)
36, a square has the least

andypoulsen (20:26:18)
so in this case, they'd want to find all the factors of 36 (i.e. which lengths are even possible), and then go from there... (i.e. the other extreme where the factors are closest together, in this case 6x6, so P(min) = 24.

Sheila Nugent (20:27:07)
because once you pass perfect square, you start heading back to long and lean

RichKal-MOEMS (20:28:12)
Tables make this clear.

RichKal-MOEMS (20:28:21)
So does graphing.

judi (20:27:39)
the answer is 24. all sides are 6. It is a square.

RichKal-MOEMS (20:29:09)
To the smallest perimeter.

RichKal-MOEMS (20:29:27)
Is a square a rectangle?

RichKal-MOEMS (20:29:37)
Explain.

Sheila Nugent (20:28:45)
the graph would be a circle, right?

RichKal-MOEMS (20:30:34)
Explain, please.

sandra atmar (20:30:07)
yes but a rectangle is not a square

RichKal-MOEMS (20:31:26)
Good! Good! Worth stressing with students.

judi (20:30:11)
yes, it is a special rectangle with congruent sides

andypoulsen (20:30:12)
Yup, a specialized case where both sets of sides just happen to be equal in length (with four 90 degree corners).

andypoulsen (20:29:27)
These are great -- I think the kids will have some good fun with them!

oc918 (20:29:38)
Students at this level would more easily use tables, or lists, but this problem is a nice one for graphs.

DKING (20:30:10)
Yes. The definition of a rectangle is opposite sides parallel and 4 right angles.

Stefanie (20:31:32)
a square is a rectangle

oc918 (20:30:21)
[Don't post this] We should wait for Stefanie's response...

Sheila Nugent (20:34:12)
If you graph the coordinates of the two side lengths - it would be (1,36) (2, 18) (3, 12) (4,9) (6,6) (9,4) (12,3)(2,18) (1,36) okay, not a circle. So it would be.... ::too add for this:::

RichKal-MOEMS (20:35:33)
It's a parabola, which they learn in high school. But they might appreciate the shape. (graphing)

sandra atmar (20:32:31)
a square has all 4 sides that are equal and 4 right angles a rectangle has 2 sets of parallel sides but they do not have to be equal lengths

RichKal-MOEMS (20:36:13)
But they might.

oc918 (20:33:19)
What about opposite sides parallel and *one* right angle?

RichKal-MOEMS (20:37:18)
Both pair of sides. What do you get?

RichKal-MOEMS (20:37:26)
AND WHY?

Sheila Nugent (20:36:45)
well, no it's not, it's an inverse curve, it doesn't go back up again. And it's not even a curve, really, it's a discrete set of points.

RichKal-MOEMS (20:39:45)
This is beyond our level. But I was wrong before. xy = 36 graphs as a hyperbola. Since these are whole numbers we have just separate points along one branch of the hyperbola.

Sheila Nugent (20:36:54)
Parabola goes back up (or down) like a cup, right?

RichKal-MOEMS (20:41:01)
It's either shaped like a bowl (think cord held loosely at both ends) or ""upside down.: cap or cup,

C-squared (20:40:42)
back to oc918: you would also need to clarify the length of the parallel sides. Without that condition you might have a trapezoid. If the parallel sides are equal, then the angles will all be right angles through a very long geometric proof.

Sheila Nugent (20:40:26)
well, wait, you can't have opposite sides parallel and *one* right angle. Nonsense.

RichKal-MOEMS (20:42:26)
Does one right angle force any others to be right angles?

sandra atmar (20:37:35)
if opposite sides are parallel you can't have one right angle

RichKal-MOEMS (20:43:07)
We didn't say ONLY one rt. angle.

C-squared (20:37:01)
Do any coaches have the students keep a journal or folder of the work done in ""club"" ? Does anyone give ""homework""?

RichKal-MOEMS (20:43:48)
Good question. Responses?

RichKal-MOEMS (20:44:48)
I always did when I coached.

RichKal-MOEMS (20:45:22)
Give homework, I mean. I also gave them the solutions and asked them to grade themselves.

oc918 (20:44:56)
I would like them to keep a notebook, but I don't require it. I ask them to solve at home the problems we didn't finish in ""club."" They present their solutions the following meeting.

Stefanie (20:45:19)
dont make them keep a folder they wont like keeping track of it

Stefanie (20:46:30)
give homework most people wont learn when they dont have to

C-squared (20:45:52)
did you allow them to take the work home each week? or is it best to keep the work at school?

RichKal-MOEMS (20:47:04)
Eric says his students feel the opposite way. They ask for folders and like being able to refer back to past problems.

RichKal-MOEMS (20:48:32)
My students took stuff home and had to show me how they did. They graded themselves and I recorded it for them, but never counted that.

Jocelyn Chow (20:48:38)
I like that idea. C-squared and I were thinking of having the students write down strategies that they learn in ""club"" in their notebook and be able to use them when doing the homework

sandra atmar (20:43:59)
when is the first competition

RichKal-MOEMS (20:49:14)
Nov 15-16.

judi (20:49:08)
I tried the suggestion in the Newsletter to let teams solve past tests by cutting up one test and putting it in envelopes for each team. As one question is answered correctly, they get another question to solve. First team to complete all five questions wins. Debrief each answer by calling on each team to explain one question .

Stefanie (20:49:44)
how do I compete in the competition

RichKal-MOEMS (20:50:32)
Only schools can join. See www.moems.org for more info.

C-squared (20:50:35)
Will the contest be posted early enough to do the contest on nov.10. We have early release days that week for parent conferences

RichKal-MOEMS (20:51:31)
Yes. At worst, email the office to have the contest sent as an attachment.

judi (20:51:02)
The March test is during our FCAT week. Can I give the test March 1?

RichKal-MOEMS (20:51:58)
Yes, but see the last response.

Jocelyn Chow (20:52:09)
Can we give our test a week later in March?

RichKal-MOEMS (20:53:21)
Yes, but enter the scores quickly. It's the end of the season. Also, I hope you're entering them online, because the mails slow things down.

sandra atmar (20:44:34)
how many students do you try to have in a club

Jocelyn Chow (20:52:36)
on March 16?

RichKal-MOEMS (20:54:35)
Sandra: The more the merrier. Everyone benefits from learning to think mathematically.

Stefanie (20:54:01)
what kind of competition r u talking about?

RichKal-MOEMS (20:55:07)
Math Olympiads: www.moems.org

RichKal-MOEMS (20:56:40)
Thank you all for your attendance and your input. Again, I apologize for the technical difficulty. We'll avoid that next week.

RichKal-MOEMS (20:57:06)
You really are a live, thoughtful group. Good night.

C-squared (20:57:36)
This was fun. Do we meet again next week

RichKal-MOEMS (20:58:37)
Yes See the schedule.