(page 30)

Eighth graders in Monroe City, MO

A second group in Monroe City, MO responds

Pamela (Brake Middle School) disagrees with the first response, and then corrects her own...

Work on the limit from some seventh graders in Indiana

Work on the limit from some seventh graders in Maine

Here is some brand new work on the limit from some students; I don't know where they're from!

Dear Tim,
Hi. We deleted this mesage about 5 times but here it is. We found a pattern. You have your basic numbers 1-12 using only the letters A & P. then, the numbers 13-24 you start using the letter E .At twenty-five you do E+1=P=EP=26 and 27=E+2=PP=EPP and so on until 63. At 63 You take 125 - 63 that equals 62 so you do Geeaapp. This is 125 - 63=62 the eeaapp is -62. After 63 for 64 you have 125-64=63 or Geeaap and so on until you reach 125 which is just G. Then for 126 you do G+1=P=GP=126. For 127 you do G+2=PP=GPP=127. You follow this pattern as we explained up to 250. Two fifty is GG and 251 is GG+1=P=GGP=251 . You follow this pattern and you find that you cannot do 313. From Jen, Tim, Leah, Andrea & Robby

Here's a solution from a group of eighth graders at Monroe City Middle School in Monroe City, MO:

Megan, Jennifer, Ashley, Joslynn, and Matthew of Monroe City, MO thought that this problem, GEAP!, was a little tougher than Znorlian. It took us about thirty minutes to complete this problem. Here is what we came up with:

					G= 125   g= -125
					E=  25   e=  -25
					A=   5   a=   -5
					P=   1   p=   -1

The first clue told us A= 5, AA= 10. The second clue told us PA= 6 and pA= 4. And since A was 5, P had to be 1 and p=-1 because clue third says lower case letters are negative and upper case letters are positive. The third clue also says GAA= 135, AA= 10, so G= 125. It also says no letter can be used more than twice in a number. Clue four says it doesn't matter what order the letters are in and that Ep= 24, since p= -1, E has to equal 25. The only way to make 13 is AApAp, but you can't do that because A is used more than twice. Here is a list of numbers up to 12:

1= P
2= PP
3= pAp
4= pA
5= A
6= AP
7= APP
8= pApA
9= AAp

teacher: Mrs. Barbara Carson, Monroe City Middle School [1997, I think]

eeps comments: impressive work. Do people out there agree with these kids in Missouri? Do the numbers -- 1, 2, 25, 125 -- ring any bells for you? Also: the second part of the problem is deciding what the smallest number is that you can't make in GEAP. A different group of students (from the same class) wrote about that... (on the left)

...and on the right, a group of students from Hermann Missouri find a typo.

The Eighth Grade Monroe City Middle School R.E.A.C.H. class including Kimberly, Micah, Erica, Allie, and Tina, found a solution for Geap. We decided that if you get a solution for the numbers 1-10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, and 300, then you can find a solution for the numbers between 1 and 300. Since GGEEAAPP is 312, then it is impossible to represent a number higher than 312, so the answer must be 313. Here is how we got that:

1-P 50-EE
2-PP 60-EEAA
3-App 70-AA
4-Ap 80-GeeA
5-A 90-Geaa
6-AP 100-Ge
7-APP 200-GGee
8-AApp 300-GGEE
10-AA G=125
20-Ea E=50
30-EA A=5
40-EEaa P=1

teacher: Mrs. Barbara Carson, Monroe City Middle School [1998]

Dear Tim,

We are a group of enriched 6th grade students at Hermann Middle School in MO.

We've discovered a small typographical error in the following question you've posted on your website: Do the numbers -- 1, 2, 25, 125 -- ring any bells for you? Where you listed "1, 2, 25, 125," the number 2 should actually be 5. With this change, the numbers are in a base 5 system. We figured this out by realizing that 25 times 5 = 125, but 1 times 5 is not 2. That's when we realized the mistake.

Emma, Jamie, and Andrew


eeps comments: Cool ! Interesting reasoning. Anybody out there want to refute it?

Meanwhile, Pamela, from Brake Middle School disagrees with the first answer about 13 being the smallest un-make-able counting number. I especially like that she showed how to make 13, which is a great proof! What do you think of HER reasoning?

The eighth grade students Monroe County have come up with the wrong solution. Here is my answer.

The question was, "What is the smallest counting number that you cannot represent on Geap?" They said that the answer is 13 and that the only way to reach 13 was with AApAp. But that is not true. My answer is 187. I say this because, you can also do Eaapp to reach 13. This is my solution: GGEEAAPP=312. So I figure if you add in the biggest - [that is, negative] number you will come up with the smallest counting number you cannot get. After adding -g my new solution is GGgEEAAPP. This = 187. The answer is 187.


I wrote back to Pamela, inquiring about her reasoning and wondering where (geographically) she was. She replied:

This is Pamela. I am an eighth grader at Brake Middle School in Taylor, Michigan. I would like to withdraw my previous submission and submitt a new one.

After writing you I thought long and hard about my response and it occured to me that I was wrong. But I do have the correct answer now. It is GGEEAAPPP. This =313. It is this because GGEEAAPP=312. So if you add one to 312 you get the smallest counting number that can not be represented on GEAP.

So I say I'm wrong with my first answer because if that were the way it is to be done that you could say 1 is the smallest counting number because pPP=1. So my final answer is 313.

Thank You ,


Comments on the limit from Amy Goff's 7th grade math applications class in Cloverdale, Indiana...

Rachel Jones: I believe that every number can be made until you run out of letters. You can use each letter twice, so you would put G,E,A,and P twice. Then 1 number above that is impossible to make correctly in the geap method. All the letters put twice is equal to 312. So 312 + 1 = 313--Impossible: if A=5, a= -5, P=1, p= -1, E=25, so e= -25. John Strader: We found that you can make up the biggest number possible GGEEAAPP = 312 because if there is a number that's a multiple of 5, there is no P needed. You can make every multiple of 5 up to 310 using geap. You can make a number ending in 3 or 4 by making number end in 5 and add a certain number of little p's. For a number ending in 1 or 2, make a number ending in 0 then add a certain number of big P's. The highest is 312 so the next one you can't get is 313. You can't go bigger than 312 because you are only allowed to use 2 of the same Geap letters. (example: Geapp is fine; however Geappp is not fine).

Here's more (November 2000) work on the limit, this time from Maine

Note from the teacher: My students noticed immediately that they were unable to make 313 in GEAP. The difficult part was when I asked how they knew this was the smallest number they couldn't make. They stuggled for several class periods with this. Here is their final explanation. -Karen Jagolinzer, Harrison Middle School, Yarmouth, ME.

G-E-A-P (From Mrs. Jagolinzer's 7th Grade Math Think Lab Class)

G (g) = 125 E (e) = 25 A (a) = 5 P (p) = 1

Two numbers next to each other is addition. Capitalized letters are positive numbers and lowercase letters are negative. Therefore, "AA" is adding together two positive numbers, "aa" is adding together two negative numbers, "Aa" is adding a negative number to a positive number, and "aA" is adding a positive number to a negative number. It is a given that "A" = 5. Since "PA" = 6, "P" has to be 1. And also, if pA = 4, then "p" has to be -1. If "Ep" = 24, knowing that "p" = -1, "E" has to be 25. Since "PEG" = 151, and "PE" = 26, "G" has to be 125.

We found that we could make the numbers 1-12 using just "P's" and "A's". Then, if you add an "E", you can get any number between, 25-12 and 25+12 using the "P's" and "A's" (positive and negative), because "E" is 25 and if you can get all of the numbers 1-12 using "A's" and "P's" than you can make 25 + (#'s 1-12) using just "A's", "P's", and "E". By adding another "E" you can get the numbers 50 - (#'s 1-12) and 50 +(#'s 1-12). From there, you can add a "G" to get "G"-( #'s 1-62) and "G"+(#'s 1-62). This gives you all of the numbers from 1 to 162. By adding another "G" you can get any number up to 312. If you put all of the numbers together (in positive form, "GGEEAAPP") the highest number you can is 312 and if you can get all of the numbers from 1-312, but nothing higher, the lowest number you can't make is 313.

Here is some work on the same issue by a different group of students in the same class:

To find the answer we first found that the highest number you can make on GEAP is GGEEAAPP (312), because you may only use two of each letter. Then we found how to make the numbers 1-12. P=1, PP=2, App=3, Ap=4, A=5, AP=6, APP=7, AApp=8, Aap=9, AA=10, AAP=11, AAPP=12, After trying several strategies we came up with a chart that by using the numbers 1-12 we could prove what the lowest number you couldn't make on GEAP is.

We had already made 12 so basically itās 1-12 is the highest you can go with just A and P so we added E (25). Then E(25)aapp(-12)=13 and if we can make 1-12 then we can make 13-25 and if we do uppercase instead of lowercase then that gives us the numbers through 37, then we added the other E so itās EEaapp to get us 38 then we do the same thing we did with the first E set bring us up to 62(EEAAPP). That is the highest number we are able to make with E, A, P so we then added the first G(125). We then repeated the process like what we did with Eās and 12 only we did it with 62 giving us the numbers 1-187. Finally we added the last G and we did 188(Ggeeaapp)-312(GGEEAAPP).

					312 GG+62
					250 GG+62188 GG-62
					187 G+62
					125 G
					 63 G-62
					 62 EE+12
					 38 EE-12
					 37 E+12
					 25 E
					 13 E-12

So we are saying the lowest, positive, number you can't make on GEAP is 313 because you can only make the numbers 1-312.

Annie, Jon, and Adam

eeps comments: very interesting. John Strader's work, especially, is the first one I've seen that begins to grapple successfully with WHY it's possible to make all of the numbers up to some limit. The Mainiacs from Yarmouth try to tackle it too. Which arguments do you find to be the clearest? What would you have to do to make them even more clear and convincing??

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