PA MDC GRADE 3 PERIMETER & AREA

Transcription

1 PA MDC GRADE 3 PERIMETER & AREA This lesson is intended to measure student understanding of perimeter and area by solving a real-world problem. Problem Solving Formative Assessment Lesson This lesson is modeled from unit taken from Everyday Mathematics and based on lesson revised by PA-MDC Writing Committee.

2 With Sincere thanks and appreciation for the effort and work of the Members of the PA- MDC Writing Committee and for the unwavering support from their home districts: Camp Hill School District, Cumberland Valley School District, Lower Dauphin School District and Shippensburg School District. Joan Gillis, State Lead MDC Dan Richards, Co- Lead MDC PA MDC Writing Committee Jason Baker Heather Borrell Susan Davis Carrie Tafoya Richard Biggs Carrie Budman Miranda Shipp Review Committee Katherine Remillard, Saint Francis University Carol Buckley, Messiah University Josh Hoyt, Berks IU Renee Yates, MDC Specialist, Kentucky Dr. Karla Carlucci, NEIU Kate Lange, CCIU If you have any questions please contact Joan Gillis at

3 UNDERSTANDING PERIMETER and AREA This Formative Assessment Lesson in problem solving is used following an instruction unit s completion. The task should be implemented a few weeks after the unit to assess if students apply the skills / standards that have been addressed. The teacher should seek out multiple representative ways to complete the problem and share the thinking of their students to broaden insight into approaching problem solving. Mathematical Goals: This lesson is intended to measure student understanding of perimeter and area by: Solving a real-world problem that will show understanding of area and perimeter Creating rectangles with a given perimeter Identifying the area of rectangles (by counting squares, using addition, or applying multiplication) Using mathematical reasoning to explain their understanding of area Common Core Standards: This lesson involves a range of mathematical content from the standards, with emphasis on: 3. MD.C.7 - Relate area to the operations of multiplication and addition. 3. MD.D.8 - Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters. This lesson also relates to all of the Standards for Mathematical Practice, with particular emphasis on: MP.1 Make sense of problems and persevere in solving them MP.4 Model with mathematics MP.6 Attend to precision PA Core Standards: PA CC A.5 Determine the area of a rectangle and apply the concept to multiplication and to addition. PA CC A.6 Solve problems involving perimeters of polygons and distinguish between linear and area measures. 1

4 Introduction: This lesson is structured in the following way: A few days before the Formative Assessment Lesson (FAL), students work individually on the pre-assessment task that is designed to reveal their current understandings and difficulties. The teacher then reviews their work and creates questions for students to answer in order to improve their understanding. At the beginning of the FAL, whole-class introduction reviewing perimeter/area concepts will include defining perimeter/area, finding perimeter and area of rectangles, and defining a rectangle (all squares are rectangles, but not all rectangles are squares). After the whole class introduction, students will revisit their own pre-assessment task. Then they work collaboratively in pairs to produce a better solution. In this lesson, it is best to work in pairs and then have teams of 4 combine to discuss solutions and again choose the best and final solution to the problem. Throughout their work, students justify and explain their decisions to their peers. Groups present their solutions, have a whole-class discussion about the work, and evaluate sample solutions. Finally, students will self-reflect on improvements they have made during the FAL by completing a Ticket Out the Door summarizer. Time Needed Estimated minutes Timings are approximate: exact timings will depend on the needs of your students Pre-assessment: no more than 15 minutes Whole Class Introduction: minutes Collaborative Activity: minutes Whole Class Discussion: minutes Collaborative Error Analysis: minutes Reflection: 5 minutes Materials Required: Each student will need a pen, pencil, eraser, dry erase board, dry erase marker, copy of the pre-assessment sheet (Building a Rabbit Pen), copy of Ticket Out the Door, copy of Dog Pens, Perimeter sheet (optional), and Quadrilateral sheet Each small group of students will need a large sheet of paper for making a poster and Student Response Sheets 1-4 Have grid paper, base-ten block cubes, and rulers accessible 2

5 Before the Lesson Pre-Assessment Task: Building a Rabbit Pen (no more than 15 minutes) Your students should complete the pre-assessment task using a pen. This will allow you to see the changes they make during the FAL lesson without the students erasing their previous response. Have students work on the pre-assessment task a few days before the formative assessment lessons. This will give you an opportunity to assess the work, find out about students different problem-solving approaches, and notice any difficulties students experience with the task. You should then be able to focus your instruction more effectively in the lessons that follow. Framing the Task: Give each student a copy of the pre-assessment task Building a Rabbit Pen. Introduce the task and help students to understand what you are asking them to do. Today you are going to help James create a rabbit pen for his rabbit. Read the directions on the Building a Rabbit Pen pre-assessment task. There are different ways to design a rectangular pen using the fence that is available. You are allowed to use any manipulatives available to help you solve the problem. Read through the task carefully and answer the questions. Make sure you record all your work and use mathematical reasoning to explain your answer. It is important that students answer these questions without assistance. Some students may find it difficult to get started; be aware that if you offer help too quickly, students will not show their understanding of the task. If students are still struggling after several minutes, reread the directions and prompt the students with the questions on the Common Issues chart. Assessing Students Responses Collect students responses to the task. Make some notes on what their work reveals about their current levels of understanding and their different problem solving approaches. We suggest that you do not score students work. Research shows that this will be counterproductive, as it will encourage students to compare their scores and distract their attention from what they can do to improve their mathematics. Instead, help students to make further progress by summarizing their 3

6 difficulties as a series of questions. Some suggestions for these are given in the Common Issues table below. These have been drawn from common difficulties observed in trails of this unit. We suggest you make a list of your own questions, based on your students work. It is recommended that you either: Write one or two questions on each student s work Give each student a printed version of your list of questions and highlight the questions for each individual student Add additional questions based on your student s misconceptions to the questions posted Identify most common misconceptions by highlighting the question on the list of questions posted If you do not have time to do this, you could select a few questions that will be of help to the majority of students and write these on the board when you return the work to the students at the start of the lesson. Common Issues Suggested Questions and Prompts Student has difficulty getting started. The student is drawing the pen with an area of 24 feet instead of perimeter of 24 feet. What are you being asked to do? Explain the task in your own words. What do you know? What do you need to find out? Refer to posted questions. Was the correct amount of fence used for each rectangular pen? Can you define perimeter and/or area? What is the difference between perimeter and area? Student is miscounting the area instead of the pen. What key word is used when identifying area (square)? When we find area, what do we count? Student did not provide area measures for each pen. What is the area of each pen? Student drew two pens with the same dimensions but different orientations. Do the rectangular pens have different lengths and widths? Do the rectangles model two different rabbit pens? 4

7 Student drew a polygon with a perimeter of 24, but did not draw a rectangle. What shape of pen did Michael want to build for his rabbit? Student is not tracing the lines provided on the grid. Are your drawings precise? Student is struggling with identifying the pen (A, B, etc.) on the mathematical reasoning question. Student did not label pens. Student is struggling with answering the mathematical reasoning question. How do I know which pen you are referring to? Did you include area measures with square units for each of your rectangular pens? Which type of mathematical reasoning will you use: words, pictures, numbers, or a combination? Student response to mathematical reasoning question is not welldeveloped. Did you use mathematical language to solve the open response question? Please use the blanks to add your class misconceptions and directed questions. 5

8 Extension Activity Find all the possible rectangular pens that Michael could build using 36 feet of fence. Find the area of each rabbit pen. This can also be used when a team completes the assignment earlier than their peers do, or if a student has completed the pre-assessment with ease. Suggested Lesson Outline Whole Class Introduction (15 20 minutes) Students will identify the perimeter and of their desk using their hands to show understanding of perimeter and area. Teacher will give examples of real life situations of using perimeter and area. Students will identify whether to use perimeter/area. Examples: Would you need to know the perimeter or area of a room if you were purchasing carpet? Explain. Would you need to know the perimeter or area of a window if you were purchasing wood for the frame? Explain. 1. Review vocabulary: perimeter, area Perimeter the distance around the boundary of a 2-dimensional shape Area the amount of surface inside the boundary of a 2-dimensional shape 2. Hand out the Rectangle sheet or display the following rectangles on a projector. Students will find the perimeter and area of each rectangle using the lengths provided. You can choose a unit (ft, yd, cm, in.) or have students use unit. (See Page 23) Teacher Key 6

9 3. Hand out the Quadrilaterals sheet or display the following rectangles on a projector. Identify the attributes of the quadrilaterals (number of sides, all sides equal in length/not equal in length, number of angles, number of vertices, sets of parallel lines). Teacher Key 4. Define rectangles All quadrilaterals have 4 sides, 4 angles, and 4 vertices. Rectangle a four-sided shape that is made up of two pairs of parallel lines and has four right angles. 5. Identify which of the quadrilaterals shown are rectangles. Note: All squares are rectangles, but not all rectangles are squares. Review Individual Solutions to the Task Teacher: Recall the Building a Rabbit Pen problem we were working on previously. Today you are going to work together to try to expand on your initial attempts at this task. First, I have looked at your work and have some questions for you to think about. On your own, carefully read through the questions I have written. I would like you to use the questions to help you think about ways of improving your own work. Return your students work on Building a Rabbit Pen. Before we break into collaborative pairs, I am providing you time to make any changes to your paper you would like to make. Post the following questions (Display on board or projected) while students are revising their work: Was the correct amount of fence used for each rectangular pen? Do the rectangular pens have different lengths and widths? 7

10 Do the rectangles model two different rabbit pens? Are your drawings precise? How do I know which pen you are referring to? Did you include area measures for each of your rectangular pens? Did you use mathematical language to solve the open response question? Collaborative Activity: Producing a Joint Solution (25 35 Minutes) Organize students into groups of two and/or a single group of three (using the pre-assessment task to sort students into groups). Grouping students who have taken different rectangles and mathematical reasoning may lead to more profitable discussions. Allow minutes for pair discussion, then pair two groups together making a group of 4. Allow 20 minutes for groups of 4 to share ideas and write their response on chart paper to create posters. The teacher will be looking for the following characteristics: Two labeled rectangular rabbit pens Areas of each pen Which pen would be best for Michael s rabbit Explanation using mathematical reasoning why the pen would be best for Michael s rabbit Planning a Joint Solution Create a slide, laminate handout for tables or keep posted where students can follow the steps of planning a joint solution. A full-page poster of this is found later in the workbook. Planning a Joint Solution 1. Take turns to explain your original method. How do you think your work could be improved having considered my feedback? 2. Listen carefully to each other and ask questions if you don t understand. 3. Once you understand each other s work agree together in your group on the best approach for completing the problem. 4. Make sure that everyone in the group can explain the reason for your chosen method. 5. Outline on your large sheet of paper the approach you are going to use. 8

11 Support Student Problem Solving If students are struggling to produce a joint solution to the task, try not to make suggestions that move them towards a particular approach. Instead, ask questions that help them to clarify their thinking, focusing on the strategies rather than the solution. Encourage students to justify decisions they make. What have you found out so far? What decision do you need to make? What is your strategy? What do you need to do next? What can you do to better organize your thinking? You may want to use the questions in the Common Issues table to support your questioning. If the whole class is struggling on the same issue, you could highlight/identify one or two relevant posted questions or hold a brief whole-class discussion. Whole Class Discussion (10-15 minutes) When students have had sufficient time to work on their posters, give them the opportunity to compare their work, by one person from each group visiting the poster of another group. The visiting student should try to make sense of what is on the poster and the assumptions the group has made. The group being visited should explain their work to the visitor. Next, choose 2 or 3 groups to present their work. Include the challenge question for all students to see and hear. One person per group will share the poster with the class. Students will find similarities and differences of responses on posters. Which groups had similar responses? Did everyone solve the problem the same way? Which rectangular pen would provide the most room for the rabbit? Encourage students to focus on evaluating the strategies and math in the student work, not on superficial features, such as whether or not the student has neat handwriting. Hold a whole-class discussion to consider the different approaches used in the sample work/partner work/group work. Focus the discussion on parts of the task students found difficult. Ask the students to compare the different solution methods. How did others draw a rectangle using the perimeter of 24? How did others find the area of their pen? Try to focus the discussion on any common misconceptions you noticed in the collaborative work. You may want to draw on the questions in the Common Issues table to support your own questioning. Try to resist simply explaining the mistakes students have made. 9

12 Collaborative Error Analysis (10-15 minutes) Each group of 4 will receive one copy of Student Sample Responses 1-4. Each group will identify the errors on each student response. Once ample time has been provided, follow up with a brief whole class share of the findings. Reflection (5 minutes) Ticket Out the Door What did you do to improve your drawing? What did you do to improve your explanation? 10

13 Follow Up - Take Home Task Students can complete Dog Pens as a follow up assignment or take home task. 11

14 Building a Rabbit Pen Answer Key Michael wants to build a rectangular pen for his rabbit. He has 24 feet of fence that he can use to make the pen. He plans to use all 24 feet of fence to make the best pen he can for his rabbit. Step 1: Use the grid to draw at least 2 pens using different lengths and widths that Michael could build. Step 2: Label the pens. Step 3: Find the area of each pen and record it inside the pen. Pen A: Area = 11 sq ft Pen B: Area = 20 sq ft Pen E: Area = 35 sq ft Pen C: Area = 27 sq ft Pen D: Area = 32 sq ft Pen F: Area = 36 sq ft 12

15 Mathematical Reasoning Which pen do you think would be the best for Michael s rabbit? Pen F Use mathematical reasoning (words, pictures, and numbers) to explain the reason for your choice. Pen F would be the best for Michael s rabbit. The area of Pen F can be found by multiplying 6 x 6 = 36 and I know that a square is a rectangle. After finding the area of the other pens, 36 square feet is the largest area. Therefore, Pen F would give Michael s rabbit the most room. NOTE: The task does not require students to find the largest possible area. The purpose of this FAL is to show understanding of drawing a rectangle with a given perimeter, connecting the relationship of perimeter and area, and identifying which area drawn covers the most surface area. Students are being asked to identify the largest area of the two rectangular pens that they drew. 13

16 Grade 3 Math: BUILDING A RABBIT PEN ANNOTATED STUDENT WORK This section contains annotated student work. The student work shows examples of student understandings and misunderstandings of the task. Your students will do an error analysis and make suggestions for misconceptions or errors they find. Some problems in this section were taken from: Unit 4 Measurement and Geometry. (2015). In Everyday Mathematics (Vol. 1, pp ). Columbus, OH: McGraw Hill Education. 14

17 STUDENT RESPONSE 1 15

18 STUDENT RESPONSE 2 16

19 STUDENT RESPONSE 3 17

20 STUDENT RESPONSE 4 18

21 STUDENT RESPONSE 1: Teacher Key 19

22 STUDENT RESPONSE 2 Teacher Key 20

23 STUDENT RESPONSE 3 Teacher Key 21

24 STUDENT RESPONSE 4 Teacher Key 22

25 Grade 3 Math: BUILDING A RABBIT PEN LESSON MATERIALS This section contains materials that can be printed, copied, and dispersed as instructional tools for the FAL lesson. 23

26 Some problems in this section were taken from: Unit 4 Measurement and Geometry. (2015). In Everyday Mathematics (Vol. 1, pp ). Columbus, OH: McGraw Hill Education. Perimeter Quadrilateral 24

27 Planning a Joint Solution 1. Take turns to explain your original method and how you think your work could be improved having considered my feedback. 2. Listen carefully to each other and ask questions if you don t understand. 3. Once you understand each other s work agree together in your group on the best approach for completing the problem. 4. Make sure that everyone in the group can explain the reason for your chosen method. 5. Outline on your large sheet of paper the approach you are going to use. 25

28 Name: Building a Rabbit Pen Michael wants to build a rectangular pen for his rabbit. He has 24 feet of fence that he can use to make the pen. He plans to use all 24 feet of fence to make the best pen he can for his rabbit. Step 1: Use the grid to draw at least 2 pens using different lengths and widths that Michael could build. Step 2: Label the pens. Step 3: Find the area of each pen and record it inside the pen. (front) 26

29 (back) Mathematical Reasoning Which pen do you think would be the best for Michael s rabbit? Use mathematical reasoning (words, pictures, and numbers) to explain the reason for your choice. 27

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31 Dog Pens Nicole made drawings of her 2 dog pens. She measured the total length of each pen s fence in feet. In her drawings, each square represents 1 square foot. Pen A Pen B 1 ft 1 sq ft 1) Calculate the perimeter and area for each pen. Record the measures using appropriate units. Pen A Perimeter = (Unit) Area = (Unit) Pen B Perimeter = (Unit) Area = (Unit) 2) Are both pens rectangles? Use mathematical reasoning (words, pictures, and numbers) to explain your answer. 29

32 Reflection: Building a Rabbit Pen How Did You Do? Name Group My answer was because: I remembered to state my answer clearly I remembered to provide an equation I remembered to explain my answer Using a Scale of 1 4 with 1 being not so good and 4 being, we were awesome! How well did your group do the following? Allow for conversation Respect each other Listen to each other Work as a team As a Group: Did your answer improve? Did it change? Explain Reviewing the Sample Student Work we realized our answer was most like: Student 1 Student 2 Student 3 Student 4 The best response was Student because 30

33 Tear-Off Sheet with Suggestions for Helping Students Access Information Barrier to Learning Suggested Strategy Student lacks understanding of math language Review domain specific vocabulary; create a picture dictionary with student. Student lacks basic perimeter/area knowledge Review prior lessons in perimeter and area; use manipulatives to explore perimeter/area concepts. Use key words and "picture stories" to help students identify the appropriate operation. Students have problems in understanding math word problems (reading comprehension) Build vocabulary through repeated classroom use and picture dictionary. Work on reading and understanding problems through modeling in small groups and peer-to- peer situations. Student struggles with multi-step problems Student struggles with writing explanations and math reasoning Student struggles with creating a rectangle with a given perimeter. Break the problem into smaller tasks, an understandable sequence. Continued use of Math Journaling and Share Time in which classmates critique each other can help strengthen this. Explain to the student that written explanation takes the place of verbal communication and the reader needs to understand how you solve the problem. Identify the perimeter of objects by measuring with a ruler. Develop skill through use of geoboard. Have student count each line drawn creating a rectangle. Student struggles with identifying the area of a rectangle. Provide multiple methods of find area: counting squares, repeated addition, composite units, multiplying the length and width. 31

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