# The Lesson by jennyyingdi

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```									  Vista Middle
School’s Garden
Las Cruces, New
Mexico
A Lesson Study On
Perimeter and Area in the
7th Grade
Introduction
– Team Members
•   Claudia Matus
•   Lisa Hufstedler
•   Patricia Carden-Harty
•   Michelle Sterling-Rodriguez
The Process
Deciding on a Topic
Our Group Focus           Extended Lesson
Study Community
• Making connections               Focus
between area and       • Students will actively
perimeter –              construct, utilize and
maximums and             communicate
minimums                 mathematical
concepts.
• Making connections
between formulas and   • Algebra
physical
representations
The melding of ideas
• Connecting
“WHAT” students
Geometry
are learning with
“HOW” they learn
it.
Algebra   Process
• How did we
connect these
areas in our
planning of the
lesson?
The Math Problem
Originally we wanted students to:

Compare different lengths of string to
make a final decision on where to cut
a wire into two pieces to form a
circle and a square with maximum
and minimum combined area
Initial Plan
• Teach it to students the way “we”
experienced it as adults – how this
affected our plan

• Guidance from a knowledgeable
other (Dr. Takahashi) – how this affected
our plan

• Our understanding of the complexity
behind this mathematical relationship
– how this affected our plan
1st teach (Pat – 7th grade)
What did we learn?
• Wire was a problem (accuracy)
• Kids decided on length of wire
(revealed student thinking)
• How to organize data so it is useful
for students/Time to analyze the
data
• Tools students used
Revised plan
Focused our Goal
• utilize prior knowledge to actively
construct a conceptual understanding
of the relationship between area and
perimeter – and to understand and
communicate how one is used to
compute the other.
• The task is to find the largest
possible perimeter with 100 meters
of fence
2nd teach (Lisa – 8th grade)
What did we learn?
• How we ask/word the question is
essential!!!
___________________________________
_____
• Right angles
• Students attention to details of real
context situation
• How to create a “need” in the students to
prove their dimensions are truly the
largest areas – promote mathematical
Final Revisions For Public
Lesson
Crafting The Question

“You have been hired by Farmer John to
build a fence around his future garden.
He has 100 meters of fencing already and
four t-posts for the corners. He wants
only four right angles in his garden but
does not care about the length of the
sides. You are to make a model with
100cm string to represent his garden.
You must find the area.”
Final Revisions For Public
Lesson
Final Instructional Decisions
• String vs. Wire

• Make measuring tools available

• Context of companies competing for
employment based on “The Largest
Area” possible for the garden
The Lesson
Materials
• String
• Ruler
• Tape
• Paper
• Pencil
• Butcher Paper
• Markers
Setting the Stage
• Farmer John is going to hire your
group to design his garden. He has
100 feet of fencing to use. He wants
your group to design the largest
garden possible. The requirements
are to make a four sided garden
using four T-post and all the fencing.
Vocabulary
• Perimeter
• Area
• Right angles (T-post)
Presenting Answers
• The students presented on their
findings.
– The largest area was a square that
measures 25ft by 25ft. The total area
was 625 sq.ft.
– Students found this first.
Adding to the Lesson
• Now see what other shapes your
group can make.
• What other areas can the garden
have?
Wrapping it all up!
• Discussing all the different shapes
that meet the requirements and
different areas
• How did they find area for their
garden when they were given
perimeter?
Poster of the Work
• Taping the shapes down to show
what the students learned.
• This goes into a follow-up lesson.
The Changes
and Extensions
Changes
• The use of string
– Hindered most students
– Not a thinking tool

• Presentation of Solutions
– Students apprehensive about solutions
– Inhibited multiple solutions
Changes
• Perimeter of Garden
– Encourage students to use decimals and
fractions
– Change the outcomes of solutions
• Not all will choose 25cm by 25 cm
• Less obvious
Extensions
• Make a table
– Identify patterns and relationships between
side lengths, area and perimeter
– Provide a proof for the largest area
• Proof of Largest area
– Algebraically
– Using table
• Allow students to choose their perimeter
The Debriefing
Debriefing Process…
• The questions we ask
– The answers we obtained
• The new questions we received
• The possible solutions we got
Questions we asked
• What the students learn out of this
lesson?
• Why did students find the same solution?
• Was the “manipulative” helpful? the
Board?
• Did students using “algebra/math” for
solving the problem?
• Did students “prove” their answer?
What students learned…
• Make a rectangle/square given the
perimeter (drawing, using the wire)
• Calculate the side length given the
perimeter
• Apply the formula of area of
rectangle/square given the perimeter
• Know the special case when a rectangle
and a square have same perimeter but
different area
• Discuss that the square hold the biggest
area
The same solution… plop!
• Students did not think in rectangles.
• Students found, in fact, the same solution,
“the square of 25cm” at the beginning
• We wanted them to find rectangles first
• Students change answers for squares?
• Students were pushed to think in squares
by the context (100cm)
• The board influenced each other answers
Use of the wire/string
• Polemic utensil. The assembly did not
agree!
• Some students used it to figure out the
answer, while others did not.
• Some experience difficulties trying to
use it.
• A group used for trying different
solutions.
• At the end, made sense for some
students to stretch it to see which
rectangle holds smaller area but same
The math used by the
students
• Guess and check (subtract one side
and adapt to get 100cm)
• Divide by 4 (100cm/4=25cm)
square!
• No one used 2L+2W=100cm
• Formula area of square (A=s2)
• Formula area rectangle (A=LxW)
with some miscalculations
The proof…
• No one presented a proof for the square
being the biggest area
• Students used their intuition/perception to
figure out that the square holds the
largest area.
• Make rectangles was a difficult task for
them
• Students limited to do measurements and
compute the areas of rectangles.
• They are not used to prove
New questions from the
debriefing session
•   Is   100cm such simple for 7th grade?
•   Is   the string a tool of thinking?
•   Is   the board used right?
•   Is   expected to have kids making proofs
in   7th grade?
Suggestions…
– Change the dimension of the perimeter
to facilitate to make rectangles (96)
– Clear instructions for the use of the
string/ use rulers/ or do not use string
– Post some of the student’s answers/
have prepared some other solutions to
discuss
– Use tables to find a pattern that relates
perimeter and area of rectangles,
instead asking for a proof
– Change that question from the lesson
Thanks for all!

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