Exemplars
Billy’s Goat
Billy’s family goat liked to jump their fence and eat
grass in their neighbor’s yard. Billy’s dad told the kids
that he would have a contest to see which child could
solve this problem most efficiently. He asked his
children to find the best possible place to tie the goat so
that it would eat the most grass in their square yard. He
reminded the kids that the total length of their fence
was 600 feet. He gave the children some rope and a
stake to which to tie the goat. Please help Billy win the
contest by answering and explaining the following
questions:
Where should the stake be placed so that the goat will
eat the most grass?
How long should the rope be, and how much space will
the goat have to eat grass?
How much grass will have to be mowed?
Bonus: If the lawn mower cuts 15,000 square inches in
one and a half minutes, about how many hours are
needed to cut the remainder of the grass?
Exemplars
TM
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Billy’s Goat
- Page 1-
Exemplars
Grade Level 6–8
Billy’s Goat
Billy’s family goat liked to jump their fence and eat grass in their neighbor’s yard. Billy’s dad
told the kids that he would have a contest to see which child could solve this problem most
efficiently. He asked his children to find the best possible place to tie the goat so that it would
eat the most grass in their square yard. He reminded the kids that the total length of their
fence was 600 feet. He gave the children some rope and a stake to which to tie the goat.
Please help Billy win the contest by answering and explaining the following questions:
Where should the stake be placed so that the goat will eat the most grass?
How long should the rope be, and how much space will the goat have to eat grass?
How much grass will have to be mowed?
Bonus: If the lawn mower cuts 15,000 square inches in one and a half minutes, about how
many hours are needed to cut the remainder of the grass?
Context
This task was written to culminate a geometry and measurement unit. The students knew
how to find the area of rectangles and circles. They were challenged to convert the square feet
to square inches.
What This Task Accomplishes
This task measures the students’ ability to solve a multi–step problem involving several
geometry and measurement concepts as well as critical thinking.
Time Required for Task
One to two 40 minute periods.
Interdisciplinary Links
This task can be linked to other geometry and measurement tasks, especially the task
entitled An Architect Needed on the Best of Math Exemplars II. .
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Billy’s Goat (cont.)
- Page 2-
Exemplars
Teaching Tips
Have students practice or discuss the idea of maximizing space. They need to understand
that by using a stake and a rope, the shape made is a circle. Therefore, knowing the formula
for the area of a circle is vital to solving the task. Students should be familiar with finding
areas of both rectangles and circles. Some of my students got to the point of knowing the area
in which the goat ate grass, but had trouble determining what was left. The greatest challenge came when solving the bonus. Some students who found the area in square feet tried
unsuccessfully to convert to square inches. See the teaching tips in the task An Architect
Needed for ideas on how to teach students this concept. Finally, the last question in the task
asks for an estimate. Many students wanted to give me an exact number. It is a good idea to
discuss what “about how many” means.
Suggested Materials
I suggest pencil and graph paper so students can draw a diagram. For calculations, I let them
use calculators.
Possible Solutions
The stake should be placed in the middle of the square yard.
The rope should be 75 feet long, and the goat will be able to eat grass in an area of 17,662.5
square feet.
4,837.5 square feet will still have to be mowed.
It will take about an hour.
FYI:
Area of yard: 22,500 square feet;
Area of circle: 17,662.5 square feet;
Area of leftover: 4,837.5 square feet;
3,240,000 square inches
2, 543,400 square inches
696,600 square inches
The exact time needed to cut the grass is 1.16 hours.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Billy’s Goat (cont.)
- Page 3-
Exemplars
Benchmark Descriptors
Novice
The novice will draw a diagram but will make errors in labeling it, showing little
understanding of the task. The length of the rope will not reach the fence, or the space will
not be circular. There will be so many errors in calculations that the problem cannot be solved
correctly.
Apprentice
The apprentice will show some understanding of the task. The diagram will be accurate, but
the strategy will lack direction toward solving the problem. There will be errors in
calculations, and little math language will be used.
Practitioner
The practitioner will show a thorough understanding of the task. The diagram will be
accurately labeled. The approach will be appropriate for solving all parts of the problem. The
explanation will be clear, and mathematical language will be used to communicate correct
answers.
Expert
The expert will use a logical, clear approach to solve the problem. The response will be
sophisticated in using mathematical language to explain the steps necessary to solve the
problem. Communication of ideas will be clear and effective. The expert will make
mathematically relevant comments or observations about the solution.
Author
Kimberly Butler teaches gifted 5th graders reading and math at a Department of Defense
school on Fort Bragg, North Carolina. This will be her seventh year in education. She has
come to enjoy math much more by teaching it, even though she was never an avid
mathematician in her early years. She earned her bachelor’s degree from the University of
North Carolina at Chapel Hill and her master’s degree from Fayetteville State University.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Billy’s Goat (cont.)
- Page 4-
Exemplars
Novice
Diagram lacks labels
and accuracy.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Few or no parts
are correct.
Billy’s Goat (cont.)
Math language used
lacks accuracy.
- Page 5-
Exemplars
Apprentice
The diagram lacks some
accuracy and labels.
Parts of the solution
are correct.
The student shows
his/her work.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Some parts lack
mathematical reasoning.
Billy’s Goat (cont.)
- Page 6-
Exemplars
Practitioner
An accurate and appropriate
representation is drawn.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Billy’s Goat (cont.)
- Page 7-
Exemplars
Practitioner (cont.)
The student explains
his/her approach.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
The student uses
math language.
Billy’s Goat (cont.)
The student achieves
correct answers.
- Page 8-
Exemplars
Expert
All parts of the
problem are correct.
The student’s
approach and
reasoning are
explained.
Precise math language
is used throughout.
The answer to the
bonus question is
accurately
determined.
The diagram is accurate
and appropriate.
Exemplars
271 Poker Hill Rd., Underhill, VT 05489
Phone 800-450-4050
Billy’s Goat (cont.)
- Page 9-