Vista Middle School’s Garden Las Cruces, New Mexico

1. Vista Middle School’s 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. 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 Deciding on a Topic

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

9. Revised plan Focused our Goal • utilizeprior knowledge to actively constructa conceptual understanding of the relationship between area and perimeter – and to understand and communicatehow one is used to compute the other. • The task is to find the largest possible perimeter with 100 meters of fence

10. 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.”

12. 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

13. The Lesson

14. 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.

16. Vocabulary • Perimeter • Area • Right angles (T-post)

17. 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.

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

19. 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?

20. 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

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

23. 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

24. 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

26. Debriefing Process… • The questions we ask • The answers we obtained • The new questions we received • The possible solutions we got

27. 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?

28. 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

29. 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

30. 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.

31. 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

32. 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

33. 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?

34. 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

35. Thanks for all!