#### A Math Forum Project

 ESCOT Problem of the Week: Archive of Problems, Submissions, & Commentary

Student Version

### Llama I - posted December 1, 1999

Lester has a big red barn. He just got a llama named Latisha, but Lester doesn't think the neighbors would appreciate the little presents she might leave on their lawns, so Latisha needs a pen so she won't wander the neighborhood.

Lester buys 36 meters of fencing from Orchard Depot. Because of your reknowned mathematical skills, he hiresyou to design the perfect pen with this fencing.

Lester wants you to use the fencing for three sides of the pen, and to use the long side of his barn as the fourth side. He wants the pen in the shape of a rectangle, but he needs you to figure out just how wide or narrow it should be.

Latisha the llama isn't picky, but she is ravenous! She needs the largest possible area in which to graze. Use the activity on the next page to help Lester design the pen with the greatest possible area.

What combination of height and width will give Latisha the greatest area to graze?

1. What did you expect to happen?

2. What was the height and width that gave the largest area? Explain how you figured it out.

If you are not using the Escot Runner, you must use it to work on this problem.
Other browsers will not display Escot activities

If you are using the Escot Runner, you may use the link below to download and try this problem.
Want to try it?

For Llama I, we were expecting answers for the first question to reflect students' expectations about what shape of pen would have the greatest area. Because of the way we asked the question, we also received predictions about how the simulation would work, and about students being surprised when they "squashed" Latisha. Next time we'll try to ask the question more clearly. Congratulations to those of you who made a prediction about the area, and condolences for those of you whose llama didn't survive. :)

Many of you expected the greatest area to occur when the pen was a square, and were surprised that it was a long rectangle.

In part two, we received many answers in which students had the correct maximum, but didn't answer the question that was asked. These students wrote that the answer was 162 square meters, which was the maximum area, but did not give the dimensions. We also received many responses in which students had the correct dimensions for the pen, but didn't explain how they reached the answer. I hope that through the problems of the week, we can encourage students to answer the question asked, and answer all parts of the question.

In part three, we wanted some explanation of why there was only one greatest area. Students were able to explain this in a number of ways, the most common being that the area decreased when they moved the dot to either side of the maximum.

### Highlighted solutions:

 From: Landon M., age 14 School: Issaquah Middle School, Issaquah, WA

```Page Workspace
1. What did you expect to happen?I expected the largest area to be a rectangle
with the long end to the barn because that the barn is free fence space!
2. What was the width and height that gave the largest area?  Explain how you
figured it out.9 height, and 18 width. I fiqured it out by making estamates
with large spaces in between. When the area started to get smaller I went
knoch by knoch till I got the highest area.
3. Is there just one answer?  Explain your thinking. No, because you can
always get more specific with decimals so the area could always get slightly
bigger but with a fence you probably want to stick with .5's.

------------------------------------
```

 From: Scott S., age 14 School: Issaquah Middle School, Issaquah, WA

```Page Workspace
1. What did you expect to happen?   I didn't expect that it would be a perfect
square, but not a long, skinny rectangle.
2. What was the width and height that gave the largest area?  Explain how you
figured it out.  The height was 9 meters, the width was 18 meters, and the
area from this was 162 meters squared.  I used the tool that changed the size
of the pen to make a pen sized near to what I thought the pen would look like,
then checked values near my first answer.
3. Is there just one answer?  Explain your thinking.   Yes, there is one
answer because if the width was 9 meters and the length was 18, there would
not be enough fence. The total length  would be 45 meters.  All other answers
would be less than 162 meters squared.

------------------------------------
```

 From: Keturah B., age 13 School: Frisbie Middle School, Rialto, CA

```Page Workspace
1. What did you expect to happen?

I thought about LatishaŐs owner having 36 meters of fencing.  Since the
barn was going to be one of the sides for the pen, I tried to figured out what
times what equaled 36.  Right a way I knew it was 12, I predicted that each
side of the pen would be 12 meters long.

2. What was the width and height that gave the largest area?  Explain how you
figured it out.

I got my answer by trying different measurements.  The first thing I tried
is 36, because it was my prediction.  36 was impossible because either the
width or height would have to be zero.  I finally realized that the father I
went to the right the smaller the area got.  By inching the mouse I figured
out that 9 was the widht and 18 was the height.  The area was 162.

I think so because while I was inching the mouse I only found 162 once

------------------------------------
```

 From: Ciara D., age 13 School: Frisbie Middle School, Rialto, CA

```Page Workspace
1. What did you expect to happen?

I think the area will come out to be a square.

2. What was the width and height that gave the largest area?  Explain how you
figured it out.

The width is 18 meters.The height is 9 meters.The square meter is 162
metters.I figered it out by moving the red dot in and out and 162 was the
highest square meters there was for her pen.

Yes,there is just one answer because if you go any higher than 9 meters as the
hight, then there will be less square meters for her pen.So you canŐt go any
higher than 9

------------------------------------
```

 From: Jeni B., age 12 School: Issaquah Middle School, Issaquah, WA

```Page Workspace
1. What did you expect to happen? I thought that the farther out you go with
the ball, the skinnier the fence would get. The more in you got, the taller it
would be.
2. What was the width and height that gave the largest area?  Explain how you
figured it out. Height: 9 meters Width: 18 meters I got this by carefully
moving the dot until the space was getting smaller and smaller and harder to
move the ball. I also did guess and check. I would guess "coordinates" check
them, and if i could go bigger, i would guess again.
3. Is there just one answer?  Explain your thinking. I think that there s only
one answer because if you need to find the BIGGEST possible area, then there
should be only one. The key is in "biggest" meaning the ONLY space I want that
can fit my llama.
Also, let's say you make it 18 height, and 9 width. it squishes the llama. I
tried all of the other possible ways and it only goes to 162.(9 by 18)
------------------------------------
```

 From: Norie B., age 13 School: Issaquah Middle School, Issaquah, WA

```Page Workspace
1. What did you expect to happen? I expected that the pen would look sort of
like a square or around the same length for each side.
2. What was the width and height that gave the largest area? The width is 18m
and the height is 9m. Explain how you figured it out. I tried to move the
lines so it sort of looked like a square. When I found the area that I thought
would be the biggest, then I checked the areas around it by moving it again,
but the area kept getting smaller. So I took the largest area.
3. Is there just one answer? Yes, there is only one answer. Explain your
thinking. Yes, because each time I moved the red dot and checked the
calculation of the area, and it got smaller.
------------------------------------
```

### 58 students received credit this week.

Cara A., age 12 - Issaquah Middle School, Issaquah, WA
Chris A., age 12 - Issaquah Middle School, Issaquah, WA
Erica A., age 13 - Frisbie Middle School, Rialto, CA
Mark A., age 13 - Frisbie Middle School, Rialto, CA
Amy B., age 13 - Issaquah Middle School, Issaquah, WA
Andrea B., age - Issaquah Middle School, Issaquah, WA
Jeni B., age 12 - Issaquah Middle School, Issaquah, WA
Jessie B., age 13 - Frisbie Middle School, Rialto, CA
Keturah B., age 13 - Frisbie Middle School, Rialto, CA
Madison B., age 12 - Issaquah Middle School, Issaquah, WA
Miguel B., age 13 - Frisbie Middle School, Rialto, CA
Norie B., age 13 - Issaquah Middle School, Issaquah, WA
Bryan C., age 12 - Issaquah Middle School, Issaquah, WA
Douglas C., age 12 - Issaquah Middle School, Issaquah, WA
Jonathan (Jon) C., age 13 - Issaquah Middle School, Issaquah, WA
Chanelle D., age 13 - Frisbie Middle School, Rialto, CA
Ciara D., age 13 - Frisbie Middle School, Rialto, CA
Jessica D., age 13 - Frisbie Middle School, Rialto, CA
Michael D., age 12 - Issaquah Middle School, Issaquah, WA
Nick D., age 12 - Issaquah Middle School, Issaquah, WA
Griffonzo de Day, age 4 - Durban Girls College, Durban, South Africa
Enrique de Farmer, age 4 - Durban Girls College, Durban, South Africa
Kelly E., age 13 - Issaquah Middle School, Issaquah, WA
Christian G., age 13 - Issaquah Middle School, Issaquah, WA
Consuelo G., age 13 - Frisbie Middle School, Rialto, CA
Kellyn G., age 12 - Issaquah Middle School, Issaquah, WA
Bryant H., age 13 - Frisbie Middle School, Rialto, CA
Jennifer H., age 12 - Issaquah Middle School, Issaquah, WA
Karly H., age 14 - Issaquah Middle School, Issaquah, WA
Marvin H., age 13 - Frisbie Middle School, Rialto, CA
Lindsay J., age - Issaquah Middle School, Issaquah, WA
Mark J., age 12 - Issaquah Middle School, Issaquah, WA
Octavius J., age 13 - Frisbie Middle School, Rialto, CA
Ian K., age 12 - Issaquah Middle School, Issaquah, WA
Kristi K., age 14 - Issaquah Middle School, Issaquah, WA
La Shanette K., age 13 - Frisbie Middle School, Rialto, CA
Weston K., age 12 - Issaquah Middle School, Issaquah, WA
Alison L., age - Issaquah Middle School, Issaquah, WA
Brian L., age 13 - Issaquah Middle School, Issaquah, WA
Akira M., age 13 - Frisbie Middle School, Rialto, CA
Eric M., age 12 - Issaquah Middle School, Issaquah, WA
Landon M., age 14 - Issaquah Middle School, Issaquah, WA
Nick N., age 13 - Issaquah Middle School, Issaquah, WA
Alexandra P., age 13 - Issaquah Middle School, Issaquah, WA
Holly R., age 12 - Issaquah Middle School, Issaquah, WA
Kathy R., age 13 - Frisbie Middle School, Rialto, CA
Leah R., age 13 - Frisbie Middle School, Rialto, CA
Darbi S., age 13 - Issaquah Middle School, Issaquah, WA
Jasmine S., age 13 - Frisbie Middle School, Rialto, CA
Joshua S., age 12 - Issaquah Middle School, Issaquah, WA
Kris S., age 12 - Issaquah Middle School, Issaquah, WA
Sarah S., age 12 - Issaquah Middle School, Issaquah, WA
Scott S., age 14 - Issaquah Middle School, Issaquah, WA
Shambria S., age 13 - Frisbie Middle School, Rialto, CA
Eric T., age 13 - Issaquah Middle School, Issaquah, WA
Louis V., age 13 - Frisbie Middle School, Rialto, CA
Robert V., age 13 - Frisbie Middle School, Rialto, CA
Eric W., age 13 - Issaquah Middle School, Issaquah, WA