The Lesson by jennyyingdi


									  Vista Middle
School’s Garden
 Las Cruces, New
   A Lesson Study On
Perimeter and Area in the
       7th Grade
       – 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
• Making connections
  between formulas and   • Algebra
 The melding of ideas
                    • Connecting
                      “WHAT” students
                      are learning with
                      “HOW” they learn
Algebra   Process
                    • How did we
                      connect these
                      areas in our
                      planning of the
      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
• 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
• 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
         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
    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
• 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.
• Perimeter
• Area
• Right angles (T-post)
      Presenting Answers
• The students presented on their
  – 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
       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
      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
• The use of string
  – Hindered most students
  – Not a thinking tool

• Presentation of Solutions
  – Students apprehensive about solutions
  – Inhibited multiple solutions
• Perimeter of Garden
  – Encourage students to use decimals and
  – Change the outcomes of solutions
    • Not all will choose 25cm by 25 cm
    • Less obvious
• 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
• Why did students find the same solution?
• Was the “manipulative” helpful? the
• 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
• 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
  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
• 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
• 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
• Guess and check (subtract one side
  and adapt to get 100cm)
• Divide by 4 (100cm/4=25cm)
• 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
• 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?
– 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
– 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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