Vista Middle Vista Middle School’s GardenSchool’s GardenLas Cruces, New Las Cruces, New

MexicoMexico

A Lesson Study On A Lesson Study On Perimeter and Area in the Perimeter and Area in the

77thth Grade Grade

IntroductionIntroduction

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

The ProcessThe Process

Deciding on a TopicDeciding on a TopicOur 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

The melding of ideasThe 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

The Math ProblemThe Math ProblemOriginally 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 PlanInitial 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

11stst teach (Pat – 7 teach (Pat – 7thth grade) 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 planRevised planFocused 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

22ndnd teach (Lisa – 8 teach (Lisa – 8thth grade) 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.

Final Revisions For Public Final Revisions For Public LessonLesson

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 Revisions For Public LessonLesson

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 LessonThe Lesson

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

Setting the StageSetting 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.

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

Presenting AnswersPresenting 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 LessonAdding to the Lesson• Now see what other shapes your

group can make.• What other areas can the garden

have?

Wrapping it all up!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 WorkPoster of the Work• Taping the shapes down to show

what the students learned.• This goes into a follow-up lesson.

The Changes The Changes and Extensionsand Extensions

ChangesChanges• The use of string

– Hindered most students– Not a thinking tool

• Presentation of Solutions– Students apprehensive about solutions– Inhibited multiple solutions

ChangesChanges• 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

ExtensionsExtensions• 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 DebriefingThe Debriefing

Debriefing Process…Debriefing Process…• The questions we ask

– The answers we obtained

• The new questions we received• The possible solutions we got

Questions we askedQuestions 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…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!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/stringUse 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.

The math used by the The math used by the studentsstudents

• 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…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 New questions from the debriefing sessiondebriefing 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…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!Thanks for all!