Lesson: Area and
perimeter
Guiding
Question:
How to
find area and perimeter of Northwest Queens?
Description:
This
lesson is the part of a bigger math project – The Roadmap to a Healthier
Community. However, it can be taught as a stand-alone activity. During this
lesson, students convert
inches into miles using a scale, and measure area and perimeter of the
community using string, 1-inch tracing paper, ruler, and markers. Students are
encouraged to be creative in finding their solutions. However, they can use
formulas for finding area and perimeter where applicable.
Student’s prior knowledge to complete this project should include:
·
Beginning knowledge about conversions,
area, and perimeter. To introduce the concepts of area and perimeter, I used “Tiles
Activity.”
·
Ability to work successfully in
groups
Learning
Objectives/Competencies:
· Make sense of problems and persevere in solving them
· Model with mathematics
· Attend to precision
· Use formulas for area of a rectangle and triangle to find square footage
· Use a scale on a map to convert inches to miles
Materials:
Newsprint/Chart paper
Tape
Markers
String
Rulers
One-inch tracing paper
Steps:
Area
and Perimeter
1. Divide
students into groups of three and distribute materials.
2. As groups
begin to work, circulate the classroom to answer questions and guide the
groups. Groups can work on the questions all together, but should record the
answers in their handouts.
3. Let the groups
work for 20 minutes. Then, stop their work and invite students to share out
their ideas about finding area and perimeter. Record a few ideas on the board
and let students continue their work in groups. During the discussion, the
teacher can ask the following questions:
·
What
does the scale on the map mean?
·
What do you need to find in this problem?
·
Please define area and perimeter concepts and
explain what they mean in the context of the map?
·
What do you already know about solving this problem?
·
What tools might you need to measure area and
perimeter? Please explain your choice.
4. Ask students
to make a poster describing how they found area and perimeter of the community.
Ask each group to nominate a presenter and present their ideas to the class.
Assessment: The teacher
can assess students’ understanding of area and perimeter concepts based on
their work in groups, discussion, and presentations.
LESSON PLANNING AND REFLECTION
THE PROBLEM
Area and Perimeter of Northwest Queens
1. Please take a look at the map of the community. If you measure one
inch on the map, what is the measurement in real life? Where can you find that
information (Tip: scale)?
2. Please record the scale of the map in a ratio form (inches to miles).
3. Measure the perimeter of your community and record the measurement in
inches and miles. If needed, you can use the ruler, tracing paper, string, and
markers.
4. What steps did you take to find the perimeter of the community?
5. Find the area of your community and record your measurements in
inches and square miles. If needed, you can use the ruler, tracing paper,
string, and markers.
6. What steps did you take to find the area of the community?
7. What is the difference between area and perimeter? Use specific
examples to explain your answer.
PLANNING
How
does this problem meet our criteria for cognitively challenging tasks?
·
The problem can be solved in different ways
·
Students are unable to proceed directly towards a solution
·
The solution requires the use of mathematical ideas of area, perimeter,
and conversion of measurements
Which math practices are you
hoping to develop by using this problem?
·
Make sense of problems and persevere in solving
them
·
Model with mathematics
·
Use appropriate tools strategically
·
Attend to precision
What do you want students to get
from working on this problem?
I would like students to develop conceptual understanding of the
mathematical concepts such as area and perimeter. Applying these concepts to
the map of Northwest Queens will help students acquire deeper understanding of
what area and perimeter are and how they can be calculated other than using
formulas. I would like students to develop flexibility and creativity in terms
of practical application of mathematical formulas. For example, students should
be able to find a perimeter of an odd shape (not simply a rectangle). In order
to do that, students need to understand that perimeter is essentially a length
of a border, no matter what shape that border takes on. Similarly, area is a
surface of an object that not always has perfect measurements. This
understanding can be developed by working with tracing paper or simply breaking
up the surface of the map into smaller shapes and then adding them up to
calculate the area of the community.
How did you solve the problem?
Perimeter: In
order to find the perimeter of the community I taped a string along the border
of the community and then measured it with a ruler. The length of the string
gave me the perimeter of Northwest Queens = 22 inches
Area: I
traced the map on 1-inch tracing paper and found how many inches squared fits
inside the borders of the community =28 inches squared
Miles: I used
a proportion to convert inches to miles:
a) 2 inches
= 22 inches x= 1x22/2=11 miles
1 mile x miles
b) 2 inches =1 mile 4 in sq = 28
in sq x= 28x1/4=7 ml sq
4 inches squared = 1 mile squared
1 ml sq x ml sq
What are other ways to solve the
problem?
·
To find perimeter, students can draw straight lines
around the community on the map and add up the length.
·
To find area students can break down the map into
rectangles and triangles and find the area of each using formulas. Then, the
area of different shapes can be added up to get the area of the community.
·
To find area, students can cut up 1-inch trace
paper into squares and fit them onto the map.
What are some questions you can
ask to help students unpack the problem?
·
What do you need to find in this problem?
·
Please define area and perimeter concepts and
explain what they mean in the context of the map?
·
What do you already know about solving this
problem?
·
What tools might you need to measure area and
perimeter? Please explain your choice?
·
Why does your calculations make sense?
·
How does one use the scale of the map?
·
How can you convert inches to miles? inches squared
to miles squared?
What are some challenges students
might have in solving this problem?
·
Students might be confused by the task of finding
area and perimeter in the real life context.
To help students with this
challenge, it might be helpful to ask the class to define area and perimeter
and explain what those words mean in the given context. For example, the
perimeter is the length of the border drawn around the community.
·
Students might not be sure what tools to use in
order to measure area and perimeter.
To help with this challenge, the
teacher can remind students the previous lesson when they measured the area of
the classroom and the molding around the floor. Also, the teacher can look at
the tools available for students (e.g., string, ruler, tracing paper, scissors)
and discuss with the class how they could use it. It is important to remind
students that there is no one way to approach this problem, and that trying out
different ideas people how might be a good way to go.
·
Students might have difficulties with converting
inches squared into miles squared.
1 mile 2 in =
1 ml 2 in x 2 in = 4 in sq, similarly 1 ml x1 ml =
1 ml sq. So 4 in sq = 1 ml sq
1 in
|
1 in
|
1 in
|
1 in
|
STUDENT WORK
This team successfully found perimeter by measuring the borders of the community with a ruler. They converted their findings of 24 inches into miles by using measurements data presented in the scale of the map: 24 in /2 = 12 miles
To find area, students traced the map of the community on tracing paper and counted inches squared. The showed whole squares with red and partial squares with green. Students added up all the squares and came up with 27 in sq.
This team made a mistake when converting 27 in sq into ml sq. They simply divided their answer by 2 as in the previous problem with the perimeter.
I liked that this team explained their thinking very precisely and was extremely clear with what steps they took and why they did it.
 |
To find perimeter, students drew straight lines around the rectangle and added them up. They converted the length of each line separately and organized the information into two columns. Then, they added up all measurements to get the perimeter – 22 in or 11 ml.
The group members had different ideas on how to measure the area. One student used the idea of the group working next to them and traced the map on tracing paper and counted squares. The other two members cut out 1-inch squares and taped them on the map and then counted them as well. They received the same answer – 27 in sq. The group made a mistake when converting in sq into ml sq.
The work of this group was interesting because the members worked independently at first and then shared their answers and came to consensus on solution.
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This team started their work by drawing a rectangle around the map of the community. First, they used it to find the perimeter, but realized that their answer is not very close to the actual perimeter as the sides of the rectangle are much longer than actual length of the border around the community. They revised their solution by measuring straight lines around the community and adding them up. They got 22 inches and converted them to 11 miles.
To find area, the team traced the community map on paper and divided it into 1-inch squares. Then, they multiplied the longest width by length and got 34. 875 in sq. To check their answer, students also counted the squares and got a bit smaller number – 28.5 in sq. They decided that the second answer is better as it is closer to the actual size. In the first method, by multiplying the longest sides the area did not account for the imperfect shape.
I liked how students tried different approaches and selected the ones that worked best.
|
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REFLECTION
How
did it feel doing this problem with your students?
I enjoyed working on this problem, as students were
really engaged and excited. They liked this activity because it was hands-on,
and they could work together in small groups to discuss their ideas and figure
out the ways to get answers. On the other hand, the activity was pretty
challenging, but students did not give up.
What
did you learn from using this problem with your students?
I learned that applying mathematical concepts to
solve real life problems requires deep understanding of what area and perimeter
are and how they are similar or different. I order to develop that
understanding, students need space and time for thinking, trying out different
ideas, and discussions. Therefore, I realized that it is essential that
students work in small groups and have about an hour to work on this activity.
I was trying to rush students a bit, but then realized that it was not
productive.
Also, I learned that students enjoy hands-on work
and deciding whose idea is the best. I found that it was important for me to
roam the room and guide students through the activity by asking a lot of
questions. Some groups needed more support than the others.
How
did your class benefit from their work on this problem?
·
Students modeled with mathematics by applying concepts of area and
perimeter to the map problem
·
They negotiated ideas and methods and developed language to explain
their thinking
·
Students persevered through the problem
·
Students received satisfaction from completing this activity, even if
they made mistakes
·
This problem allowed students with different skill levels master
understanding of math concepts
·
This activity fostered collaboration, creativity, and critical thinking
Did
all of your students get what you wanted from the problem? How do you know?
I think that most of the students developed a better
understanding of area and perimeter. A few students still struggled with
conversions, but this issue can be easily addressed in a different lesson. Each
group prepared a poster and described their solutions to the problem during the
whole-class discussion. The presentations helped me to assess where students
are with their understanding as well as I got a chance to ask clarifying
questions. On the other hand, other groups got a chance to learn from their
peers and ask questions as well.
Describe
one highlight from the class discussion of the problem and solution methods.
During the presentations, students compared the
answers between the team who converted their measurements before doing any
calculations and the other teams. The question of discussion was: Will the
answers still be the same and why? Why one team got 9 ml sq and the others
between calculated the area between 13 and 14 ml sq? Students said that by
using common sense the answers should be similar, so we started looking for
possible mistakes. Here the question of how to convert in sq into ml sq was
posed.
What
would you do differently if you used this problem again?
I would teach a lesson on conversions some time
before assigning this activity. I think it might help students to avoid
confusion and focus entirely on the ideas of area and perimeter.
What
advice do you have for a teacher who is considering using this problem with
their class?
·
This lesson requires about 1 h 30 min – 2 h of work, so it might be a
good idea to take a short break before presentations.
·
I found it useful to introduce the ideas of area and perimeter in a
previous lesson. Doing this lesson as an introduction to area and perimeter
might be too hard for both students and a teacher.
·
It is very important to work with different groups during the activity
and ask a lot of guiding questions. From time to time, it might be useful to
interrupt students’ work and clarify some questions as a class.
STUDENT REFLECTION
What
was easy and what was difficult for you?
I asked students the question above after their
presentations. Students mostly commented on the challenges they faced when
completing this activity:
·
It was difficult to connect formulas for finding area and perimeter of
rectangle to the map
·
It was hard to convert in sq into ml sq
·
“I was not sure what is area and what is perimeter”
·
It was hard to explain why
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