Vista Middle Schools Garden Las Cruces, New Mexico

1
Vista Middle Schools GardenLas Cruces, New
Mexico

• A Lesson Study On Perimeter and Area in the 7th
  Grade

2
Introduction

• Team Members
• Claudia Matus
• Lisa Hufstedler
• Patricia Carden-Harty
• Michelle Sterling-Rodriguez

3
The Process
4
Deciding on a Topic

• Our Group Focus
• Making connections between area and perimeter
  maximums and minimums
• Making connections between formulas and physical
  representations

• Extended Lesson Study Community Focus
• Students will actively construct, utilize and
  communicate mathematical concepts.
• Algebra

5
The melding of ideas

• Connecting WHAT students are learning with
  HOW they learn it.
• How did we connect these areas in our planning of
  the lesson?

Geometry
Algebra
Process
6
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

7
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

8
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

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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

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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 discourse and reasoning.

11
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.

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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

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The Lesson
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Materials

• String
• Ruler
• Tape
• Paper
• Pencil
• Butcher Paper
• Markers

15
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.

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Vocabulary

• Perimeter
• Area
• Right angles (T-post)

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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.

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Adding to the Lesson

• Now see what other shapes your group can make.
• What other areas can the garden have?

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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?

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Poster of the Work

• Taping the shapes down to show what the students
  learned.
• This goes into a follow-up lesson.

21
The Changes and Extensions
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Changes

• The use of string
• Hindered most students
• Not a thinking tool
• Presentation of Solutions
• Students apprehensive about solutions
• Inhibited multiple solutions

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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

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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

25
The Debriefing
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Debriefing Process

• The questions we ask
• The answers we obtained
• The new questions we received
• The possible solutions we got

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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?

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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

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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

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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 perimeter.

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The math used by the students

• Guess and check (subtract one side and adapt to
  get 100cm)
• Divide by 4 (100cm/425cm) square!
• No one used 2L2W100cm
• Formula area of square (As2)
• Formula area rectangle (ALxW) with some
  miscalculations

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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

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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?

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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 students 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

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Thanks for all!