#### A Math Forum Project

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

Student Version

### Llama II - posted December 8, 1999

(Try Llama I before you begin this problem.)

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? Experiment with many different widths to find the greatest area, and then summarize what you discover:

1. What does the graph say about the area of the rectangle?

2. How does your graph help you answer the following question: Is there just one solution for the largest area? (From question 3 in Llama, Part I.)

3. What are the tradeoffs for using the area calculator (from Llama, Part I) and the graphing tool (from this problem)?

Bonus: What is the shape of the graph?

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In part II of the Llama question, we wanted students to see the idea of a maximum graphically. Three groups of students submitted correct answers that described how the shape of the graph is related to the area of the rectangle.

We should have been more specific in posing the question, and in answering math questions such as these students need to make an effort not to be so brief! When students encounter open-ended questions such as, "What does the graph say about the area of the rectangle?" they should make an effort to think the question through and should try to give a detailed answer. Many of the incorrect submissions were so brief that they missed the heart of the question. Students should try to imagine that they are writing to someone with no knowledge of the problem, and should supply the level of detail necessary for their audience to understand their answers.

Congratulations to all of you who knew that the shape of the graph was a parabola!

### Highlighted solutions:

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

```Page Workspace
The graph says that the area of the square is the sum of the height and width
of the pen. The graph helps me because you could keep moving the line to get
the sample and it makes some kind of a shape.

The curve I see is a parabola. I found it because I knew it started with a P
and ended with an A. So I went to the back of my pre-algebra book and looked
at all the words that started with a P and ended with an A and then I went to
that page and told my teacher I had found it.
------------------------------------
```

 From: Katie C., age 13Amber B., age 13 School: Issaquah Middle School, Issaquah, WA

```Page Workspace
1.  What does the graph say about the area of the rectangle? The graph tells
you what the area of the rectangle is going to be.
one solution for the largest area? (from question #3 in Llama, part 1)  the
graph shows you whether or not there is more than one solution. no there is
not more than one solution.
3.  What are the tradeoffs for using the area calculator (from Llama, part 1)
and the graphing tool (from this problem)? you can use your calculater to plug
in the numbers on your graph to see what area is going to be.
BONUS
4.  What type of geometric curve do you see in the graph? a parabola
------------------------------------
```

### 4 students received credit this week.

Amber B., age 13 - Issaquah Middle School, Issaquah, WA
Miguel B., age 13 - Frisbie Middle School, Rialto, CA
Katie C., age 13 - Issaquah Middle School, Issaquah, WA
Jessica D., age 13 - Frisbie Middle School, Rialto, CA