Title of the Lesson Finding a prism that has the same measure 的教訓(xùn)尋找一個棱鏡具有相同的測量的標(biāo)題

1、.Lesson Plan for GeometryDate: February 1, 2008Place: Las Cruces High School LibraryInstructor: Martha Campos or Connie JaramilloLesson plan developed by: Mary Andrews Martha Campos Connie Jaramillo Sandy PetersonTitle of the Lesson: Comparing the Volume between rectangular prisms.Goal of the Lesson

2、: To increase students proficiency in solving open-ended questions, and to explain the process of how to solve the problem through writing. Prior Knowledge: Students have constructed nets and know how to find the surface area and volume of prisms, and cylinders. Class: Geometry 9th and 10th Grade. (

3、Inclusion Class)Relationship of the Lesson to the Standards: I9-12 Benchmark: Represent and analyze mathematical situations and structures using algebraic symbols.Represent and analyze relationships using written and verbal expressions, tables, equations and graphs, and describe the connections amon

4、g those representations.II9-12 Benchmark: Analyze characteristics and properties of two-and three-dimensional geometric shapes and develop mathematical arguments about geometric relations.Interpret and draw three-dimensional objects an find the surface area and volume of basic figures (e.g. spheres,

5、 rectangular solids, prisms, polygonal cones) and calculate the surface area and volumes of these figures as well as figures constructed from unions of rectangular solids and prisms with faces in common, given the formulas for these figures.Rationale for this lesson: After reviewing the results for

6、the NMSBA (New Mexico Standards Based Assessment), we noticed that students performed poorly in the following two areas:1) Standard: Algebra 1C - Modeling relationships.2) Standard: Geometry 2A- Analyzing shapes.Instruction of the Lesson: New Mexico high school geometry students are expected to use

7、area formulas of basic figures such as rectangles, triangles, and circles to develop strategies to find surface area and volume of three dimensional figures such as pyramids, cones, prisms, and cylinders. Results from the NMSBA (New Mexico Standards Based Assessment) show that our students struggle

8、with modeling mathematical relationships and analyzing shapes. Students that do develop the necessary strategies and make the connections to find surface area and volume of three dimensional objects often struggle with the written process of communicating their strategies and connections.Prior to th

9、e research lesson, students have been introduced to isometric drawings (three dimensional corner views of various cubic shaped figures). Students have also practiced identifying solids from nets and sketching nets for three-dimensional figures. They have investigated surface area and volume of pyram

10、ids, cones, prisms, and cylinders.If students understand the difference between surface area and volume, they can develop strategies and make connections that will help facilitate their ability to find the surface area and volume of any of the above mentioned three-dimensional figures. A clear under

11、standing will also enable them to model and communicate their strategies and connections through pictures, diagrams and most importantly for this research lessonthrough writing. Rectangular prisms are the three-dimensional figures chosen to meet the goal of the lesson.Materials: Isometric dot paper,

12、 ruler, pencil, color pencils, calculator, 1-cubic inch blocks and 1 cubic cm blocks, optical illusion prints, rectangular prisms (one per group of two)Plan of the Lesson:Steps, Learning ActivitiesTeachers questions and Expected Students Reactions.(This column shows the major events and flow of the

13、lesson.)Teachers Support(This column shows additional moves, questions, or statements that the teacher may need to make to help students.)Points of Evaluation(This column identifies what the teacher should look for to determine whether to proceed, and what observers should look for to determine the

14、effectiveness of the lesson.)1. HookTeacher will demonstrate different pictures that portray optical illusions.Continue to ask students to comment on what they see.Teacher will demonstrate three versions of the box and block illusion.Teacher will discuss the relationship between optical illusions an

15、d geometric art.Present the M.C. Escher Poster.Individual pictures of the M.C. Escher picture will be given to all of the studentsAllow students to look at the poster and make comments.Students will then be asked to write about the following:First writing assignment:Select something in this picture

16、that is structurally impossible to construct or physically impossible to occur? Students will be given a worksheet and will be expected to complete all writing assignments on the worksheet. Students will be given time to write.Teacher will talk about businesses and how they design their products to

17、lure the people into purchasing their products.Students will be given 8 one inch cubes. Two students per group. Students will be asked to build as many different rectangular prisms possible.Teacher will walk around and look at the different prisms.Students will be asked to display their rectangular

18、prisms.All of the different prisms are displayed while the teacher asks the students to think about the different prisms and their volume.Give students time to write their conjecture.Students will be given 24 one inch cubes and be asked to build three different rectangular prisms. The students will

19、be asked to use their three prisms to prove or disprove their conjecture.Second writing assignment:Was your conjecture correct or incorrect, explain.Teacher will ask students to write their conjecture on their worksheet. Students will then be asked to prove or disprove their conjecture by building t

20、hree different prisms and use their prisms to prove or disprove their conjecture.Teacher will ask specific students to share their conjecture and reasoning.Teacher will give each student a specific rectangular prism and ask the students to find the volume of the prism.Teacher will walk around classr

21、oom and observe students. If there is a student who has an incorrect process for finding the volume, the teacher will ask the student or students to explain how they found their solution.Once the correct volume is established, teacher will ask the students to find as many different prisms possible t

22、hat have a volume of 240 cubic centimeters.Third writing assignment:Explain in writing the process you went through to find another prism or prisms with the same volume.Emphasize what changed from the original.Students will be asked to read what they wrote. Teacher will select specific groups to pre

23、sent.After student sketches are posted on the white board, the teacher will ask students to select one based on the following:Fourth Writing Assignment:In your own words, how would you explain volume to a younger brother or sister so that he/she could understand what volume means.If Time:Teacher wil

24、l say: If you were a company, which of these prisms would you choose to package your merchandize and why?.Sometimes things are not what they appear to be at first glance.Teacher begins to show optical illusions.What do you see?Does somebody see something different? How about now?#1 rabbit/duck#15 va

25、se/face#20 cylinders # 9-BlocksWhich one is the top block?Are you sure? How about now.What do you see? Where is the block? #6 box & block Why are we exploring Optical Illusions? Mathematically they contain geometric shapes such as angles, segments, lines or recurring patterns. You might also not

26、ice symmetry within the pictureIt may even seem like our brain is playing tricks on us. Objects sometimes appear to be bigger, smaller, longerDo you recognize this poster? This poster has been in my room taped to the back of the podium. Have you had the chance to look closely at this poster? This is

27、 a drawing created by MC Escher. Escher was an artist that used geometry in creative ways to make some interesting works of art. Take a moment to look at the picture.What do you see? Do you think this is structurally possible? Could this really be constructed? I would like you to select something in

28、 this picture that is structurally impossible to construct or physically impossible to occur and then write about what you see. Write on your worksheet where it says first writing assignment. *Intro/transitionBusinesses design their products purposefully to entice their customers into purchasing the

29、ir products. Think about various drink containers that you have seen or maybe even purchased. Why did you choose a particular drink? Businesses package their products to appear bigger, wider, longerand we buy it! We automatically think we are getting more or a better bargain. Compare the bottles nex

30、t time you go to a grocery store to buy bottled water or a power drink. Today we are going to explore volume by building rectangular prisms.Before we begin, I would like us to review the properties of a rectangular prism. What are the properties of a rectangular prism?In your groups, using the 8 cub

31、es for each prism, I would like you to build as many different prisms as you think are possible.(Remember the prism must meet the conditions we just talked about.)Would someone like to show us their prism?Did anyone build a different prism?How many different prisms are there?What is a conjecture?I w

32、ould like you to take a moment to think about this question and write a conjecture. This is the question: What can you tell me about these prisms and their volume. Take your 24 cubes and build three different rectangular prisms using 8 cubes for each prism. Make a sketch of each prism on the dot pap

33、er and then use these drawings to show why your conjecture is correct or incorrect. Please explain in detail.(Was your conjecture correct or incorrect, explain.)What was your conjecture?Was your conjecture correct? Please explain.Find the volume of this prism?Tell me as much as you can about a recta

34、ngular prism(s) with volume 240 cubic centimeters.How many different prisms can you find that have the same volume as this rectangular prism of 240 cubic centimeters? Find as many different prisms as you can.Write down your strategy for finding the new prism and write down what changed from the orig

35、inal to the new one.In your own words, write on about the following: How would you explain volume to a younger brother or sister so that he/she could understand what volume means.If you were a company, which of these prisms would you choose to package your merchandize and why? .Students might commen

36、t: Duck, RabbitStudents perspective will shift.Students make comments about the work of art, such as; crazy, neat, weird, etc.Students might make comments such as;Water is going up.Stairs go in both directions.This is making me dizzy.Its confusing.This is our first writing assignment. We would like

37、students to begin to feel comfortable with writing.The container was attractive.Possible responses.Two bases, 6 faces, 8 vertices, 12 edges, 4 lateral faces, lateral faces are rectangles, closed solid.Six prisms, eight prisms?Comments students might make.Statement. Theorem. Guess. Hypothesis.Conject

38、ure: Given 8 one inch cubes and building a rectangular prism the volume of the three prisms remains the same.Students might calculate volume of the three prisms using the formula for volume. Students may conclude that it doesnt matter how you change the cubes, the volume remains the same.Students wi

39、ll probably measure the length of the sides and use the formula for volume as a way to calculate the volume.Volume is 240 cubic centimeters. Students might work with the formula for volume.Other students might take half of the prism and double the height and reduce the length.This is our way to asse

40、ss if the student understands the meaning of volume.This question has multiple solutions depending on your product.Group Reflections: Public Lesson January 31, 2008The first part of the report will include the questions that were raised during the post discussion by the observers of the lesson. The

41、question will be stated and then a short response will be included.Question #1: Why centimeters and inches?Context of question: Students were given one inch cubes to work with as one activity. For another activity, students were asked to find another rectangular prism that had a volume of 220 cubic

42、centimeters. Response: The New Mexico standards based assessment requires students to use proper units in their answers. However, we find that students do not do this consistently. By the time students get to high school, they should have experience working with various units. It is important for st

43、udents to continue to work with different units, be flexible, and be able to transfer their knowledge when presented with new concepts.Question #2: Why optical illusion and art as hook? What did it have to do with lesson?Context of question: Students were shown different optical illusions and were a

44、sked to discuss what they saw in each picture. Response: This was considered our hook. By capturing their attention with the optical illusions, we wanted students to recognize that things arent always the way they appear. We talked about companies that package their products in certain containers to

45、 attract customers. For instance, there may be two different containers that contain the same volume, yet one bottle may appear to have a greater volume than another container. However, the container with the lesser volume may be the popular brand due to its eye-catching packaging. With this example

46、 fresh in their mind, our expectations were to have them transfer that information when presented with the problem of volume. Question/Comment #3: The lesson seemed rushed, students were just getting into it.Response: As a group, we agree with this statement. We feel that this lesson can be extended

47、 to two days. The first time the lesson was taught, the lesson ended at least 10 minutes before the end of class. Therefore, we did not anticipate that time would be an issue in our public lesson. As a result of Akis suggestion, we presented the lesson a third time to another class after the public

48、lesson, and implemented Aki suggestion to eliminate/shorten the introduction and use of blocks and allow students more time to explore volume get to the “meat of the lesson”. Although the time fit the lesson better, we found that the results were the same. In fact, we felt the students may have gras

49、ped the concept more easily had they been presented with the original lesson. We know our students and understand their needs; in these classes, we have students with various learning styles, and a good majority of our students benefit from manipulating, visualizing and drawing new concepts. Questio

50、n #4: Volume? What was their background knowledge?Response: Students had calculated the volume for prisms and cylinders using the volume formula prior to this class.Question #5: Prism- Why different? Just because you rotate the prism, it isnt a different prism?Context of question: Students were give

51、n 8 one inch cubes and asked to build as many prisms possible. Response: When the students in the previous lesson were presented with the task of building different prisms, the students considered rotated prisms to be different prisms. This made for great discussion amongst our lesson study members.

52、 After much discussion, we decided that a rotated prism is a different prism than the original. However, when we did the public lesson, the students did not consider a rotated prism as a different prism. Thus, when the students presented all of their different prisms, it was mentioned that they had

53、missed the prisms that were rotated. This may have actually confused the students later in the lesson, because when they were presented with a different task (students were given a prism with a volume of 220 cubic centimeters and asked to find another prism with the same volume) some of the students

54、 simply rotated the prism. As a team, we have continued to discuss this issue. We realize that both results/responses may be correct, the answer will depend on the problem and how it is stated. This is something that we learned; as educators we need to specify our expectations by providing clear ins

55、tructions and expected outcomes. Furthermore, if we offer students the opportunity to explore and present their findings when given open-ended questions, we need to be open-minded and flexible with student responses, asking students to support their findings. These types of lessons are more time consuming for teachers but better for the students. As instructors, we need to take time to review