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					Lesson Title: Spatial Reasoning – Three Dimensional Space Teacher: Steven Bernstein Subject: Geometry Grade Level: 10th Time Required: Five Periods (200 minutes) Topic: This unit is concentrated on teaching students how to apply different area and volume formulas to different three dimensional objects. Essential Question: What is the difference between the names of shapes and the formulas used to calculate their surface area and volume when comparing two dimensional objects to three dimensional objects? Prerequisites (Prior knowledge) Students need to know the formulas for surface area and area on two dimensional objects so they can fully understand the three dimensional formulas. Stage 1 – Desired Results Content Area Standard(s) (include complete standard, not just standard #)
This unit addresses the following NYS Performance Indicators:

G.G.12 Know and apply that the volume of a prism is the product of the area of the base and the altitude G.G.13 Apply the properties of a regular pyramid, including: - lateral edges are congruent - lateral faces are congruent isosceles triangles - volume of a pyramid equals one-third the product of the area of the base and the altitude G.G.14 Apply the properties of a cylinder, including: - bases are congruent - volume equals the product of the area of the base and the altitude - lateral area of a right circular cylinder equals the product of an altitude and the circumference of the base G.G.15 Apply the properties of a right circular cone, including: - lateral area equals one-half the product of the slant height and the circumference of its base - volume is one-third the product of the area of its base and its altitude G.G.16 Apply the properties of a sphere, including: - the intersection of a plane and a sphere is a circle - a great circle is the largest circle that can be drawn on a sphere - two planes equidistant from the center of the sphere and intersecting the sphere do so in congruent circles - surface area is 4 r 2 - volume is

4 3 r 3

© Gradel & Jabot 2009 (adapted from Jabot, Maheady, Rey 2005 (adapted from UbD, Wiggins & McTighe))

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Intended Learning Outcome (Should define what students will know and be able to do and at what
level of mastery they should be able to do it.) Students will know… Student will be able to…

How to apply the Pythagorean Theorem to three dimensional objects Know the names of all three dimensional objects talked about in this unit. How to use the three dimensional formulas associated with surface area and volume. How to use a blogging website.

Understand what the difference is between lateral and surface area. Recall their two dimensional formulas well and be able to apply them to the three dimensional shapes. Have a better understanding of the reference sheet given to them on the Geometry Regents. Know how to show proper work and labels for answering surface area and volume questions. Think of real world examples to apply to mathematics and blog about it. Use their Google account and blogging information to see the class blog anytime.

Stage 2 – Assessment Evidence
Students will demonstrate their learning/understanding in the following way(s):

Teacher-Created Assessments Pre-test: See Attached Pre-test Post-test: See attached Quiz Blogging: The blogging assignment is an assignment to measure student performance on how they relate the current three dimensional objects lesson to the world outside of school. (Performance Assessments:)
I will assess student performance by:

Seeing how well students do on questions after each sub-topic Asking students to volunteer answers to questions in front of the class Walk around the room and see how students are doing on their own. Check to see if students are showing and labeling their work. Checking on the blog site to see if students were able to do the assignment and blog it successfully (Other Assessments: Peer, Self)
Students can assess themselves by:

Checking their work with answers on the projector screen. Checking their work for the proper steps for each problem. Making sure they labeled their answer with units. Logging onto the blog to see other student answers to the blogging question.
© Gradel & Jabot 2009 (adapted from Jabot, Maheady, Rey 2005 (adapted from UbD, Wiggins & McTighe))

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(Assessment Adaptations) Depending on how long it takes to do various planned out assessment, you could always assess students quickly during class by seeing how they answer questions when called upon rather than wait for a quiz or test. Having the blogging as an assignment, this can be used to assess how the class is doing relating mathematics to the world outside of school.

Stage 3 – Learning Plan Learning Activities
Instructional Strategies/Learning Activities: e.g., demonstration, discussion, small groups, role play, etc. On the first day when introducing, as I show the class the various objects we will mainly be having a small discussion on names of these objects and where they have seen or heard of them before. When getting into showing students some formulas, I will be doing some questions with them demonstrating how I would like the students to show and label their work. When students are working on questions, I will let them work with a partner when answering some of the questions. This is mainly for a check and balances system so students are bouncing ideas back and forth off each other to try and solve the problems. Going to the computer lab to have students do the blogging assignment on their blog.

http://bernsteingeomety.blogspot.com/
Introducing the lesson: (capturing students’ attention, activating students’ prior knowledge).

The introduction to this lesson is showing the students the various shapes we will be working with. On the projector, showing the students the transparent three dimensional shapes will get the students interested. I will also pass around the objects for the students to personally handle so they get an idea of what these figures like and maybe why the formulas are what they are for certain objects.
Instructional Sequence: (representing the content: teaching/learning activities, connecting to students’ prior knowledge, etc.)

Teacher activity (The teacher is doing….) Day One 1) Handing out the Pre-test reminding the students it will not be for a grade and to try their best. 2) Showing the transparent three dimensional objects on the projector for the students to see. Asking them questions like:  What two dimensional object does this look like?  What is the name of this three
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Student activity (The student is doing…) Day One 1) Putting their name on the pre-test and completing to the best of their ability. 2) Looking at the projector to see the three dimensional objects. Answering what other two dimensional objects they look like and what they thing the object is named. 3) Looking at the three dimensional objects and passing them along to their fellow

© Gradel & Jabot 2009 (adapted from Jabot, Maheady, Rey 2005 (adapted from UbD, Wiggins & McTighe))

dimensional object? 3) Passing out three dimensional objects for students to hold. 4) Assign students their homework. Day Two 5) Go over the homework and refresh student’s minds with the material from the day before. 6) Hand out the guided notes on lateral & surface area and go thru the formulas with the students. 7) Go thru some of the first few problems so students can see how they should answer the problem and how much work they should be showing. This is a good place to emphasize on students how important it is to label their work with the proper units.

classmates. 4) Write down their homework assignment for the night.

Day Two 5) Correct their homework and remind them what they worked on the previous day. 6) Following along on their guided notes learning how certain formulas apply to various three dimensional objects. 7) Writing down all the steps the teacher is doing for solving the first few examples. Writing down the units that the teacher is showing them. Write a small reminder on the page to always label with units!

8) Working on some problems on their own. Once a few are done and the 8) Have students work on the Lateral & teacher says you can work in pairs, find Surface Area questions. Have them do a a partner and continue with the few on their own then allow them to work questions. in pairs with each other. 9) Correct their work as the teacher goes 9) Go over the questions the students over it. Some students are being called worked on. Call up some students to on for volunteering to go to the board to write on the board how they solved put up their work for some of the some questions. questions. 10) End the class and assign students their homework. Day Three 11) Go over the homework and refresh student’s minds with the material from the day before. 12) Hand out the guided notes on volume and go thru the formulas with the students. 13) Go thru some of the first few problems
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10) Copy down the assignment for homework. Day Three 11) Correct their homework and remind them what they worked on the previous day. 12) Following along on their guided notes learning how certain formulas apply to various three dimensional objects. 13) Writing down all the steps the teacher is

© Gradel & Jabot 2009 (adapted from Jabot, Maheady, Rey 2005 (adapted from UbD, Wiggins & McTighe))

so students can see how they should answer the problem and how much work they should be showing. This is a good place to emphasize on students how important it is to label their work with the proper units. 14) Have students work on the volume questions. Have them do a few on their own then allow them to work in pairs with each other. 15) Go over the questions the students worked on. Call up some students to write on the board how they solved some questions. 16) End the class and assign students their homework.

doing for solving the first few examples. Writing down the units that the teacher is showing them. Write a small reminder on the page to always label with units!

14) Working on some problems on their own. Once a few are done and the teacher says you can work in pairs, find a partner and continue with the questions. 15) Correct their work as the teacher goes over it. Some students are being called on for volunteering to go to the board to put up their work for some of the questions. 16) Copy down the assignment for homework.

Day Four 17) Explain the blogging assignment to the students and what they are expected to do.

Day Four 17) Listen and ask questions on the blogging assignment so everyone understands what they need to do for the current assignment. 18) Walk over to the computer lab 19) Begin thinking/searching the web for real life explains where mathematics is used. Day Five 20) Asking questions on lateral and surface and/or volume and make sure they know what is expected of them. 21) Taking the quiz.

18) Take students over to the computer lab. 19) Monitor students to make sure they are on tasking working on the blogging assignment. Day Five 20) Answer any questions the class has on lateral and surface area and/or volume and go thru them. 21) Hand out the end of the unit quiz.

Adaptations to the Instructional Sequence to Differentiate: Depending on how the sequence goes, if time is running low make sure you get at least some examples done with the class so they know what is expected of them. Homework can always be shortened to allow for students to work on questions not gotten to in class.
© Gradel & Jabot 2009 (adapted from Jabot, Maheady, Rey 2005 (adapted from UbD, Wiggins & McTighe))

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Discussion and Assessment of Learning: (Pointing out to students how what they are learning is related to the driving question; assessing students’ learning as a result of the lesson)

Remind students that learning how to solve surface area and volume questions of three dimensional objects is better preparing them for the real work. Solving these kinds of questions is more likely to have when compared to two dimensional objects because we live in a three dimensional world.
Closure: Extensions for early finishers:

These students can move onto their homework questions for the day. Students can also try to come up with more real life examples where these three dimensional objects are used and seen. They can also post a new blog of cool math websites that are interactive with calculating surface area and volume.
Alternate strategies for struggling students or those who learn differently

Have these students hold the objects they are working with so they can perhaps visually see where they are applying the formula they are working with .
Procedures: (already established procedures to be used and procedures to be taught for this lesson)
Instructional Protocol/Itinerary:

Begin with assessing student knowledge with a short pre-test Start the unit off with identifying all the shapes they will be working with throughout the unit – Engage the students with hand held examples of these objects Hand out the notes on lateral & surface area and show the students the formulas that go along with them. Have them practice using the formulas to compute these areas Hand out the notes on volume and have the students practice computing volume with the formulas Review all of the formulas with the objects and on the last day of the unit assess their knowledge with a test. Explain to the students the importance of recognizing mathematics in the real work. Explain the blogging assignment to the students.

© Gradel & Jabot 2009 (adapted from Jabot, Maheady, Rey 2005 (adapted from UbD, Wiggins & McTighe))

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Lesson Development Resources Technology Tools and Materials:
(classroom set-up, preparations, resources, etc.)
This lesson requires the following materials:

Overhead projector Hand-held transparent objects Guided notes packet Pre/post tests Calculator Notify/sign-up the class for a day at a computer lab Computer lab – computer for each student Students need to create or have a Google account (Contact Information) For questions, I can be contacted at: Steven Bernstein
bern4990@fredonia.edu

Blog Site: http://bernsteingeomety.blogspot.com/ Class Web Page: http://sites.google.com/site/mrbernsteinsgeometryclass/

© Gradel & Jabot 2009 (adapted from Jabot, Maheady, Rey 2005 (adapted from UbD, Wiggins & McTighe))

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Name: _________________________ Geometry

Date: _______________ Blogging Assignment

Directions: Today we will be working in the computer lab. It is your assignment to post a blog post. See below of what you will be posting.

Post an example of how any of the three dimensional objects we learned about is used in the world outside of school. Your post must follow Post 1 directions or Post 2 directions.
Post 1: You are posted a location or name of a three dimensional object that we learned about. This is simply just seeing the shape of this figure. It can be outside or a picture on the web. What is it representing?

Example of Post 1: Ice cream cone.

Note: Having a picture of your findings makes the blog post look nice! ------------------------------------------------------------------------Post 2: Talking about a profession (Job) that uses these three dimensional objects all the time. Make sure you explain how it relates to that profession!

Example of Post 2: Architect. This profession calls upon these objects all the time when designing various models. Being able to calculate surface area and volume of figures is something an Architect does every day to make sure they are maximizing the space of their creation or drawing.

Be sure to sign the post with your name before posting it to our blog! Ask me for help if you are having trouble coming up with ideas. Blog your post as a comment under the directions. (like the teacher example)

Visit: http://sites.google.com/site/mrbernsteinsgeometryclass/ to get to the class page. From here you will see a link to our blogging page where you will put your post. http://bernsteingeomety.blogspot.com/

© Gradel & Jabot 2009 (adapted from Jabot, Maheady, Rey 2005 (adapted from UbD, Wiggins & McTighe))

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© Gradel & Jabot 2009 (adapted from Jabot, Maheady, Rey 2005 (adapted from UbD, Wiggins & McTighe))

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© Gradel & Jabot 2009 (adapted from Jabot, Maheady, Rey 2005 (adapted from UbD, Wiggins & McTighe))

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I decided to use blogging as my web 2.0 technology. I decided to incorporate this into a unit plan on lateral & surface area and volume. Students always complaining how the things they learn they will never use again. I decided to take that head on and incorporate some technology to show students that there are real life applications. Not only am I having them use technology, but I’m also having them blog about real world examples outside of school on how three dimensional objects are seen and used. This really ties the lesson plans together and tackles two common problems with your average lesson: Not incorporating technology Not incorporating applications or examples to the world out of school.

© Gradel & Jabot 2009 (adapted from Jabot, Maheady, Rey 2005 (adapted from UbD, Wiggins & McTighe))

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