How Many
Blades
of Grass
Are on a
Football Field?
282
Teaching Children Mathematics / February 2006
Copyright © 2006 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.
This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
H
ave you ever wondered how many blades of
grass are on a football field? The students in
my fifth-grade class did and in the process
learned a lot about estimation strategies, multiplication, measurement, problem solving, and communication.
Photograph by Gary Olsen; all rights reserved
Getting Started
I posed the question “How many blades of grass do
you think are on a football field?” The students
made all kinds of guesses: a million, a billion, even
a googol! We talked about the reasonableness of
the guesses. One student brought up the point that
an estimation of a googol pieces of grass would not
make sense when not even a googol grains of sand
are found on all the beaches in the whole world, a
fact that we had discussed at the beginning of the
year.
The next step was trying to determine how
many pieces of grass were on the school football
field. Would counting them all be possible? The
students decided that it would not. Counting would
create all kinds of issues. First, it would take far too
long. Second, we doubted that any football team
would let us crawl around on their field counting
pieces of grass. Because of that, we decided that
we would not be able to come up with an exact
answer but would try for a good estimate.
Next we discussed strategies for making a good
estimate. One strategy the students came up with
was to first determine how much one piece of grass
weighs. Then they thought that whoever cuts the
grass at the football field could collect it in bags.
We could then weigh the bags, subtract the weight
of the bags, and divide by the weight of a piece of
grass to determine an approximate number. An
interesting strategy, I thought. However, students
were concerned about the accuracy of weighing a
blade of grass. Could they weigh just one blade, or
would that mass be too small to measure using the
tools at our disposal? Would they have to weigh
more than one blade? Also, when we contacted the
local high school and asked if it was possible to
collect all the blades of grass in bags when they cut
the football field, the administration was not overly
receptive to the idea. Therefore, we needed to find
another strategy.
Another idea the students had was to count all
the blades of grass in a yard and multiply by 100
yards. A good strategy, but the students were missing the concept of area. A football field does not
Teaching Children Mathematics / February 2006
have just 100 yards, but 100 yards by 50 yards,
making 5000 square yards. For me to just tell them
that information would not be as meaningful as if
they discovered it for themselves, so off we went
with a couple of yardsticks to the school playground to count pieces of grass. When we got to
the playground, we put the yardsticks down and
started counting. When the students saw what we
were doing, one student decided that counting the
pieces of grass in a yard was not going to give us
the number of pieces in the whole football field.
As a teacher I could see the understanding
reflected in the students’ eyes. For some, it was a
definite light bulb going on, an “aha!” moment. For
others, I could see a glimmer of light, but full
understanding had not yet dawned. My hope was
that the light would get
brighter as we went
through the other activities associated with
answering the question.
Another student who
made the connection
came up with the idea of
area and using square
yards. So we decided to
count the pieces of grass
in a square yard and
multiply by the number
of square yards in a football field. This decision led
to another problem. A square yard seemed to contain lots of pieces of grass. Too many to count!
Someone said maybe we should try square feet.
Out came the rulers. However, the number of
blades of grass in a square foot seemed to be too
many pieces to count, as well. Finally, someone
came up with using a square inch.
To count the pieces of grass in a square inch, we
needed to know the exact size of a square inch. The
students formed pairs to measure a square inch on
a piece of paper and cut it out. That way, when they
went outside they had a square-inch frame that they
could put over the grass and count the blades
inside. Off we went outside. However, we still had
difficulties. Should we count the weeds or just the
I could see the
understanding in the
students’ eyes.…
It was an “aha!”
moment
By Christina M. Nugent
Christina Nugent, [email protected], is the mathematics instructional strategist for the Dubuque Community School District in Dubuque, Iowa.
She is interested in problem-based instructional tasks, the integration of mathematics and language arts, and success for all students. She enjoys helping
teachers implement standards-based mathematics.
283
Photographs by Gary Olsen; all rights reserved
grass? We decided that a football field would not
have weeds, so we should try to pick a spot on the
playground that had grass and not weeds. Another
question was “Can we pull the grass out of the
ground and count the blades?” We decided that this
was not a very good idea, because if everyone
pulled out a square inch of grass, the playground
would have a lot of bare spots! The students tried
their best to accurately count the blades of grass.
With that task finished, back into the classroom we
went.
When we got inside, the students were surprised
to discover that they had gotten many different
answers for the number of blades of grass in a
square inch. We talked about the possible reasons
for the differences. Since students are not perfect,
some had not followed the directions accurately
284
and had chosen areas with weeds; others had
selected areas with thick pieces of grass, leading to
a smaller total; and still others had picked areas
with lots of thin pieces of grass, giving a larger
total.
Our next dilemma was how to know which
number to use. The students decided that finding
the average would give the best number to use to
estimate the number of blades of grass in a football
field. Since the students had studied the different
measures of central tendency, we discussed
whether we should use the mean, median, or mode.
Since we had more than one mode, that measure
would not give us a good representation. The mean
and median were close together, so some students
decided to use the mean and some, the median.
Letting the students decide which measure to use
allowed them some control over the outcome.
Next we needed to decide how many square
inches are in a football field. This calculation was
easier for some students than for others. Some students could see right away that a square foot has
144 square inches, a square yard has 1296 square
inches, and a football field has 1296 × 5000 square
inches. However, some students still had a difficult
time with the concept of area. To help those students, we constructed a model out of graph paper.
Each small graph-paper grid represented a square
inch. First we constructed a square foot, then a
square yard. It would have been nice to construct
the whole football field, but when we tried, it
turned out much larger than we thought. However,
constructing just a square foot and a square yard
helped students understand the concept of area and
the multiplication involved in computing the area.
More light bulbs went on.
Teaching Children Mathematics / February 2006
Writing to Explain
Mathematical Thinking
The activity continued with the question “Who
cares how many blades of grass are on a football
field?” We talked about the fact that groundskeepers for professional football teams might care. A
small number of blades might indicate that they
were not doing their job well. Also, football players
might wear different shoes depending on the number of blades of grass. We therefore decided to write
to a professional football team, tell them what we
did, and see what they had to say. This excellent
writing project gave students a purpose for writing;
it also motivated them by giving them a reason to
ask their favorite players for an autograph!
Common complaints that I hear from colleagues
are “I can’t get my students to write about their
thinking!” or “They never write enough!” Students
do not automatically know how to write about their
thinking any more than they know, without instruction, how to write a descriptive paragraph or a
compare-contrast paragraph. As teachers, we have
to use the writing process to teach students to write
mathematically just as we do to teach students to
write descriptively or technically. When students
are taught using the writing process, few of them
produce incomplete writing or fail to include their
thinking.
Our first step in writing an explanation was to
brainstorm the process we had gone through to
solve the problem. Students accomplished this step
first as individuals and then as a large group. Next
we matched the steps of the process with the reasons we did those steps. Getting students to explain
the reasoning behind their choices is the most difficult part of the process. Brainstorming as a class
helps some students with this phase. Then the students wrote their rough drafts.
To revise their rough drafts, we used the Telling
Rubric from our mathematics textbook series,
Math Trailblazers (University of Illinois 2003). At
this point in the year, the students had used the
Telling Rubric as the basis for their writing on
many previous occasions. They were familiar with
the criteria: include all the steps, state the reasons
for choosing the steps, include pictures or diagrams
that explain their thinking, and incorporate number
sentences in their explanations. First the students
Teaching Children Mathematics / February 2006
revised their own writings individually as I guided
them in analyzing one criterion at a time using different colors. Students used one color to number all
the steps they had listed in their writing; a second
color, to underline the whys; a third color, to circle
the number sentences that were included; and a
fourth color, to circle the picture or diagram.
Because we did one criterion at a time, students
were able to go back through their writing and add
things that they had left out. Doing this phase in a
large group helps students focus on one component
at a time.
The next part of the writing process involves
peer revising. To accomplish this phase, the students were assigned to groups of four. Each member of a group was in charge of checking group
members’ writing for one of the four criteria. Once
students had completed this part of the process,
they met with me for a writer’s conference. I made
suggestions based on the rubric. Finally, students
Photograph by Gary Olsen; all rights reserved
Once we figured the area, we could compute the
estimated number of blades of grass in the football
field. The students were very surprised at the number!
285
recopied their letters, and they were ready to be
sent.
Assessment
Many teachers struggle with how to assess openended problems such as this. An investigative
activity such as the blades-of-grass problem could
take a week or more to complete. How do you
assess students on their learning during this time?
First, teachers constantly assess student learning
and make decisions throughout their teaching. All
through the blades-of-grass activity, I was assessing students’ learning and making adjustments to
my teaching. For example, because I observed that
making the writing easier for them and the assessment easier for me.
Another advantage was that the textbook also
presented an open-ended individual assessment to
be used after the group activity. The assessment
gave students various options, including that of
estimating either the number of holes in a screen or
the number of loops in a towel. Students could also
pose their own similar problems as part of this
assessment. Through this individual assessment, I
could evaluate a student’s progress in several areas,
including her or his ability to solve an open-ended
problem and to use data and multiplication to solve
problems.
In addition to their attainment of the mathematics concepts involved, students can be evaluated on
their participation in the group activity itself. Using
a teamwork rubric, a teacher can grade students on
how well they worked together and participated in
groups during this activity.
Photograph by Gary Olsen; all rights reserved
Differentiating for a Variety
of Learning Needs
some students were still struggling with the concept of area, I had them make examples of a square
foot and a square yard. Since I knew that the students were struggling with writing their letters, we
revised them individually within the large group.
However, the difficulty comes when students
need to be evaluated and given a grade for a report
card. To determine this grade, I could evaluate the
students’ letters on the basis of the Telling Rubric.
Although I have done this activity for several years,
this year a similar activity also appeared in our
Math Trailblazers textbook, giving us the advantage of using the Telling Rubric earlier in the
school year. By the time we did this investigation,
the students had become familiar with the rubric,
286
This activity can be taught successfully to a class
of diverse learners. I teach in an at-risk school with
a high percentage of students with special needs.
My classroom also contains a couple of students
classified as talented and gifted. I use a variety of
strategies to meet the needs of all students.
The first strategy I use is small groups. Sometimes the groups are divided by ability, with the
higher achieving students working together and the
lower achieving students working together. However, for this activity I used heterogeneous groupings. Having students work in small groups helps
them come up with the answers themselves instead
of waiting for the teacher to supply them. What one
student in a group thinks of, another might not if
working independently. Also, students can piggyback on the ideas of others in small groups, enabling
them to solve problems and come up with solution
strategies that they would not be able to devise on
their own. For example, when students were deciding to count the blades of grass in a one-yard length,
one student did not think that approach would work;
however, another student came up with using a
square yard. From there the students went to the idea
of a square foot and then to a square inch. These
ideas came from students working together and
building on the thinking of others. Some students
who are intimidated in large-group settings are less
reticent about participating and asking questions in a
Teaching Children Mathematics / February 2006
small-group setting. Small-group work also holds
students accountable. Shirking responsibilities is
much more difficult for students in a small group
than in a large one. Students do not want to let their
peers down, and peers are quick to call a group
member on lack of participation.
I also did different activities with different
groups. Some groups needed my assistance to
complete the model of the square foot and the
square yard. Other groups were able to rely on the
knowledge of the members of their group to do the
explaining.
The jobs I gave students for writing their letters
were also based on ability. Students who were better able to include reasons were given the job of
checking group members’ papers for that aspect.
Other students were assigned the easier jobs of
checking for number sentences or for inclusion of
a picture.
Posing a high-quality question is absolutely crucial
to problem solving in the classroom. The bladesof-grass problem evidences high quality on many
levels. First, it is meaningful to students. Many students like football, and even those who do not may
know someone who plays the game or will at least
have seen a game on television. Second, it is connected with real life, not conjured up only for a
textbook. It piques students’ curiosity. Because no
set solution path exists, students can go about solving the problem in many ways.
Many concepts can be taught and practiced
through this problem. First, students learn about
the vastness of numbers. I think that the students
described here were very surprised that they had
estimated so high. Large numbers like a million, a
billion, and a googol have little meaning for students at this age. The fact that a football field has
only about 194 million blades of grass is surprising
to them because prior to doing the activity, they do
not realize how large a million actually is. Second,
students learn about measurement and develop a
deep understanding of the concept of area. When
calculating the number of square feet in a square
yard, some of my students were surprised that the
total is 9. At first they thought that the total is only
6, length plus width. They had forgotten what was
in the middle, thus missing the whole point of area.
This activity also helps students practice estimation strategies as they determine the number of
blades of grass in a football field. They can apply
Teaching Children Mathematics / February 2006
Photograph by Gary Olsen; all rights reserved
Reflections
the same strategy to other instances in which estimation is necessary.
Another component of the successful use of
open-ended problem solving, such as the grass
problem, is not telling students how to do the problem, but guiding them in finding the answers themselves. The teacher’s job becomes one of asking
clarifying questions such as the following to get
students to think about how to solve the problem:
• What do you know?
• How did you solve similar problems?
• Would it help to draw a picture, make a diagram, or use other tools?
Questions such as these can help students think for
themselves rather than always rely on the teacher
to provide the answers.
Many problem-solving opportunities arose
287
within the central question of how many blades of
grass are on a football field. Students had to
decide how they were going to answer the question. They also had to explain why they got different totals for the number of blades of grass in a
square inch. They needed to choose whether to
use the mean, median, or mode. Pursuing all these
questions reinforces the idea that mathematics is
connected. Rarely in real life do we do a single
type of mathematics.
Writing to explain thinking can be a very difficult activity for students. Having a real audience
for their writing, as in this activity, boosts the quality of students’ writing. I told my students that I
would not mail their letters unless they were of
high quality and matched the rubric. The rubric
gave the students the standards by which to measure the quality of their letters. They knew that
each letter must contain a picture or diagram, number sentences, all the steps in the solution process,
and the reasons for the steps. Students typically do
not have difficulty drawing a decorative picture to
illustrate their letters, but they do have trouble
drawing a picture that helps explain their thinking.
288
Students wanted to draw a picture of themselves
counting the blades of grass instead of a picture
that helped explain the mathematics they did to
compute the number of blades of grass in a football
field. Students also struggle to include number sentences as a meaningful part of their explanation,
wanting to tack them on the end instead. Giving
reasons for their thinking is also difficult for students. This activity helped students with this aspect
of their writing, and having opportunities to solve
more problems and write other explanations will
further increase their ability to effectively explain
their thinking.
You no longer need to wonder about how many
blades of grass are on a football field. My students have determined that the number is about
194 million!
Bibliography
Keystone Area Education Association. Developing
Mathematical Thinking with Effective Questions.
Unpublished.
University of Illinois at Chicago. Math Trailblazers.
Dubuque, IA: Kendall Hunt Publishing Co., 2003. ▲
Teaching Children Mathematics / February 2006