TITLE: Area of a Parallelogram

TASK DEVELOPER: Muhammad Shahbaz Ali

CONTENT AREA AND GRADE: 11th Grade Basic Geometry

SCOPE AND SEQUENCE: Section 8.3 Basic Geometry Text

TARGET TEACHING DATE: March 26, 2007

SCHOOL: John F. Kennedy High School

STANDARDS:

GEOMETRY AND MEASUREMENT - GRADE 9-12

STANDARD 4.2 GEOMETRY AND MEASUREMENT: All students will develop spatial sense and the ability to use geometric properties, relationships, and measurement to model, describe and analyze phenomena

Strand A. Geometric Properties: Building upon knowledge and skills gained in preceding grades, by the end of Grade 12, students will:

Apply the properties of geometric shapes.

  • Parallel lines.
  • Minimal conditions for a shape to be a special quadrilateral
  • Self-similarity


Strand B. Transforming Shapes: Building upon knowledge and skills gained in preceding grades, by the end of Grade 12, students will:

Determine whether two or more given shapes can be used to generate a tessellation.

  • Patterns in areas and perimeters of self-similar figures

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PERFORMANCES:
  1. The students will construct geometric figures.
  2. Students will construct a rectangle given length and width and calculate the area.
  3. Using same dimensions students will construct a parallelogram and calculate area.

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SETTING:

Real World Setting: Farming

You are a farmer. You are faced with finding the area to find the cost of grading of a field which is a parallelogram shape.You must express the base of the parallelogram that could be any one of the sides and height is the length of the altitude. Once you have completed this job, you will find area by multiplying base to the height. Then, you will find out the cost of grading.

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SMARTSKILLS:

Level I: Acquiring Data - Data students will acquire in this standards-based task:

  • Vocabulary: Students will learn the definition of base and altitude.
  • Skills: Students will use pencil and ruler to construct a parallelogram and find length of its base and the height.
  • Concepts: Students will use the basic concept of length and width of the rectangle and finding the area of rectangle.
  • Processes: Students will follow a formula to find out the area.

Level II: Visualizing Information - Data from Level I that are visualized as information in this standards-based task: Teacher will draw the rectangle on the board and then convert its sides to the bases of the parallelogram.

  • Organizing: Students will mention the length of the base and the height of the altitude and compare them with the length and width of the rectangle with the same dimension of the parallelogram.

Level III: Applying Knowledge - Visualized information from Level II that is applied knowledge in this standards-based task:

  • Making decisions: Students will use same dimension to find out the area of rectangle and triangle.
  • Solving problems: Students will find area by measuring sides and using correct formula. 
  • Creating solutions: Students will use the solution to find out the cost of ploughing, seeding etc.

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PREFERENCES:

Student Involvement - The students will complete the task:

  • individually
  • as a cooperative group
  • as an action team
  • in a project group
  • through independent study

Instruction - Activities will be organized and delivered:

  • by differentiating the activities or strategies to offer appropriate ways for students to learn
    in a student service learning setting in the local community

Special Education Accommodations - Students with special needs will require the following electronic devices:

  • Calculator

Use of Resources - The school will provide:

  • Calculator and Text Book
  • Classroom time to complete the task

Use of Resources - The students will provide homework time.

Customer for Student Work - The student will present their work as evidence of task completion to teachers.

Assessment of Student Work - The following people will be involved in assessing student work generated to complete the task:

  • The student's teacher
  • The student's self evaluation

Assessment of Student Work - The following forms of assessment will be used to determine progress and results:

  • Performance assessment
  • Multiple choice test items

Reporting Results - The assessment results will be reported:

  • as a score point on a rubric, rule, or key
  • as the total points for a checklist
  • as a letter grade

Timeline - The estimated time needed to plan, teach, and score this task is one class period of 80 minutes.

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ACTIVITIES:

Activity 1: Review of yesterday's lesson and new definitions (Estimated time : 5 minutes)

  1. Ask students to draw a rectangle and mention the length and width.
  2. Find out the area of rectangle by using length and width as :

Area = length . width

A = l . w

Materials: notebook, pencil, and ruler. Students will produced yesterday's homework, provide the area of rectangle, and write the new definitions in their notebooks.

Activity 2: Construction of a parallelogram (Estimated time: 15 minutes)

The teacher will direct the students through this process, and will draw each step of the construction on the board while the students will work at their seats in a group of two.

  1. Use the ruler and pencil to draw a rectangle.
  2. Write the dimensions.
  3. Find out the area by using formula.
  4. Use the same dimension to draw a parallelogram.

Materials: Pencil, notebook, and ruler. Students will use these tools to draw the required construction.The students will be able to visually observe the difference of angles between rectangle and parallelogram. The students will be able to visually observe the sides of parallelogram which are parallel to the opposite sides.

Activity 3: Synthesis and analysis of data ( Estimated time 20 minutes)

Teacher will draw a rectangle on the board with mentioning length and the width. Teacher will also calculate the area of rectangle by using formula

Teacher will ask each student to draw the rectangle of same dimension and find out the area.

Then  teacher will draw a pair of parallel lines which are opposite but of same dimension of rectangle by mentioning the concept of parallel lines. Students will be asked to do the same by turning the sides of rectangle to the opposite parallel sides but same dimension of which they have already for the rectangle. At this point teacher will review the base and the altitude. On new figure, which turns from rectangle to parallelogram, teacher will write down the new names of sides as the base and the altitude.At this point, teacher will explain about the base of a parallelogram that it could be any one of the sides and the height is the length of the altitude. Furthermore teacher will explain that the height or altitude of a parallelogram is always perpendicular to the base of the parallelogram.

Students will be asked to draw at least four rectangles of different dimensions and calculate their areas by using formula, and turn these  rectangles into parallelograms.They will use ruler to draw each rectangle and then parallelogram.

At this point, students will be asked to mention the base and the height of each parallelogram.

Activity 4:Find out the area by using formula ( Estimated time: 20 minutes)

 The teacher will write down the formula to find out the area of parallelogram:

Area of parallelogram = base . height/ length of the altitude

A = b . h

Teacher will find out the area of parallelogram by substituting the value of base and height. Teacher will mention that the the both areas i,e area of the rectangle and parallelogram are the same.

Students will be asked to do the same with their rectangles on the papers by converting lengths and widths to the bases and and the heights.

Students will be asked to use the formula of parallelogram, i.e., A = b . h to find out the areas of the parallelograms which will be the same as of rectangles.

Materials: notebook, pencil, and ruler.

Activity 5: Discuss students answers and more practice to find unknown base, or height if area is given (Estimated time: 20 minutes)

Teacher will make sure that each student or each group of two students calculated area of each parallelogram and is the same of the converted rectangle.

Teacher will now ask students to write down the formula and substitute the calculated area, and height or base to find out  the base or height.

Teacher will do the example on board from "Pacemaker Geometry Book" , pages 216-217, and involve students to mention base and height and then finding area of parallelograms. Students will be asked to do more practice from the book (page 127). Teacher will check the students work and if someone needs help, teacher would explain on the board, and will mention how common mistakes could be rectified. 

Materials: notebooks, pencil, ruler, chalk

Source:Beech, M & McClain, K. (2003)

Pacemaker Geometry

Pearson Learning Group

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BENCHMARKING:

Student Performance One: Construction

Assessment Benchmarking Example: Students will construct parallelogram and mention its base and height.

Student Performance Two: Calculation.

Real World Benchmarking Example: Students will calculate the area of parallelogram by using base and height and can calculate the cost of grounding etc.

Student Performance Three: Verify

Real World Benchmarking Example: Students will verify that the area of rectangle and area of parallelogram is same if they use the same dimensions.

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SCORING:

New Jersey High School Proficiency Test (HSPT)

Holistic Scoring Guide for Mathematics Open-Ended Items (Generic Rubric)
3-Point Response

The response shows complete understanding of the problem's essential mathematical concepts. The student executes procedures completely and gives relevant responses to all parts of the task. The response contains few minor errors, if any. The response contains a clear, effective explanation detailing how the problem was solved so that the reader does not need to infer how and why decisions were made.

2-Point Response

The response shows nearly complete understanding of the problem's essential mathematical concepts. The student executes nearly all procedures and gives relevant responses to most parts of the task. The response may have minor errors. The explanation detailing how the problem was solved may not be clear, causing the reader to make some inferences.

1-Point Response

The response shows limited understanding of the problem's essential mathematical concepts. The response and procedures may be incomplete and/or may contain major errors. An incomplete explanation of how the problem was solved may contribute to questions as to how and why decisions were made.

0-Point ResponseThe response shows insufficient understanding of the problem's essential mathematical concepts. Their may be no explanation of the solution or the reader may not be able to understand the explanation. The reader may not be able to understand how and why decisions were made.
Source: New Jersey State Department of Education

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RESULTS:

Analyze: I will examine the data to look for trends, contributing factors, and implications of student performance over a series of assessments of the same learning standard.

  • Trends: I will look for improvement relative to previous knowledge which included area of square and rectangle.

Reflect: As I relate my students' results with my lesson activities, I noticed that...

Having the students perform a construction helped them to have a better retention of geometric concepts because they had a chance to see, draw, and measure shapes rather than just calculations and substitutions in the formulas. Students performed well on multiple choices, i.e. 80%, but their scores on open ended questions was 68%. The scoring difference between these type of questions was due to lack of showing the complete process in the open ended questions. I found that they can find the correct answer in a multiple choice format, because the process is not needed.

This connects to previous and subsequent lessons in the chapter on area and perimeter. The students are becoming familiar with the basic concepts, construction, and substitution the dimensions in given formulas.

Action Plan: I will complete the following Task Builder Figure 8 Strategy Action Plan to prepare for my next standards-based task.

1. Plan - My next standards-based task will focus on:

  • Title: Area of triangles
  • Content Area: Basic Geometry
  • Learning Standard(s): NJCCCS 4.2
  • Intent: Use the basic concept of base and the height from the parallelogram to find out the area of the triangle.

5. Team or Grade Level Portfolio and School Web Site - I will insert the standards-based instruction or assessment task, results, samples of student work, and summary into my Team or Grade Level Portfolio and upload them to my School's Instructional Web Site on the following dates:

  • Target date for Team or Grade Level Portfolio:
  • Target date for School Instructional Web Site: May 17

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