The unit plan we have chosen to cover deals with concepts of area.  We assume the students have previous knowledge of geometric concepts and figures.  This knowledge includes basic properties of shapes (particularly rectangles, parallelograms, triangles, and circles), how to calculate the perimeter of each of these shapes, and measurement skills.  In concordance with the Standards and Principles of NCTM, our lesson will develop Geometry and Spatial Sense and their abilities to communicate the language of mathematics.  In addition to this, various assessment tools are built into our lesson plans.
     Developing knowledge of area is important to the middle grade student because it is a foundation for further geometric concepts and real world applications.  Since geometry surrounds middle schoolers in everyday life, it is important for teachers to take advantage of their knowledge of this subject and incorporate it into their classrooms.  Beyond being able to manipulate these figures physically, students should learn to represent their properties mathematically.  For example, it is important to let the students derive formulas from the hands-on activities presented in the lesson plans.  This gives them the chance to learn the formulas in a constructivist environment.
     In compliance with real life applications, it is important that we provide students with the technology to support these applications.  For our technology, we have incorporated the use of spreadsheets to calculate cost based on area.  Students are given a floor plan in which they need to find the cost and amount of flooring needed to cover the entire surface of the floor.  The spreadsheet will enable them to grasp the formulas needed for such calculations and organize their data effectively.  This activity will take place in a computer lab, with one student to a computer.  Microsoft Excel will be used and technical support will be provided by the teacher, as well as other lab personnel.  Our technology will address the constructivist approach we initiated in the classroom, with self-discovery being promoted.  By using this activity, we will meet the Technology Principle presented by NCTM by using the technology in a way that “enhances mathematics learning, supports effective math teaching, and influences what math is taught” (NCTM Standards 2000, online edition).
     The students using this technology is a class of 24 sixth grade students.  This application was appropriate for their ability level as well as the mathematics being introduced.  Excel has many applications beyond manipulating formulas and it is important to introduce this software to students early due to its user-friendly nature.  The strengths of this technology are numerous.  It enables them to have an organized workplace for their data and to investigate formulas based on different shapes and sizes.  One weakness associated with using Microsoft Excel is teaching the students to manipulate the technology.  Since it might be new for them, it is important for the teacher to take this into consideration and remain patient throughout the activity.  Despite this weakness, the popularity of Excel in today’s society makes it a valuable tool for students to learn to use.

Figures for Lesson
Worksheet

I. Goals

• To develop a firm understandings of the underlying concepts of the area of shapes.

• By the end of the lesson, the students will be able to:
• Give examples of where the measurement of area would be used.
• Measure the area of simple shapes by utilizing the space shown inside.
• Measuring the approximate area of a complex shape by use of a grid.

• Each student should receive a transparent grid at a point in class specified in the lesson.
• Each student should also pick up the two worksheets as depicted later.

1. Begin by reading a passage from one of the plays that the students have, are, or will be reading in their literature class.
2. Discuss with the class how the stage may be built out of cubes that are 1m*1m*1m.
3. Present problem as follows:
    You are planning to build a stage for the play that you and your classmates are putting on.  You feel that to make the stage look awesome that each of the squares on top of the stage (from the top of each of the cubes) should be painted a different color.  If your stage looks like that in Figure 1, how many colors would you need?

*Transition – Explain how this can be translated to the area of the stage in that the amount of colors needed is also the amount of cubes, and thus the amount of meters’ squared.

V. Lesson Procedure

4. Define and discuss area so that the students have a firm understanding of area, at least in rectangular shapes.  Have the students give examples of where they would use area in their everyday life.
5. Give examples of odd shapes as shown in Figure 2 and have the students attempt to figure out the area.  Show the students how all they need to do is to count the squares.  Give students some examples and see if they can figure out the area on their own.

*Transition – Ask students how they would measure the area of the state of Illinois if they had a map, and pass out a worksheet with Figure 3 on it.
6. Hand out the grids (shown in figure 4) and show the students how they can approximate the area this way using the squares in Figure 5.  Have the students count how many of each squares (1, ¾, ½, ¼) and have the class settle on a final tally.   Total up the area.   Multiply by 400 because each blocks is 20 mi. x 20 mi. to convert to square miles.  (The area of Illinois is 56,400 sq. mi. and in our example we get about 54,400 sq. mi.)

VI. Closure

7. Have the students reflect on the different ways they learned to measure area and pass out the worksheet (Figure 6) and see if there is questions.

VII. Extension

If time allows, after 3, start a discussion as to why the area of the state of Illinois and the figure we came up with in class are different.  Is our answer wrong then?  Should we disregard our answer?  Let the class engage in a debate/discussion on the topic.  Also, instead of telling the class what the actual area of Illinois, you can have them look the information up on the web to find it.

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Worksheet
 

Goals

• To develop an understanding of the area of rectangles and parallelograms through self-discovery

• The student will develop the relationship between the interior of a surface and its dimensions (specifically for rectangles)
• The student will represent a parallelogram in terms of a rectangle, thus developing a relationship between their areas
• The student will be able to calculate the area of both rectangles and parallelograms given the dimensions of the shapes
• The students will be able to determine the dimensions of a rectangle or parallelogram given the area

• The students will be expected to have a basic understanding of rectangles and parallelograms
• They must have completed the assignment that was given for homework last night

• Depending on the special needs of my students, some accommodations may be needed for this lesson
• Certain students may have a difficult time with the scissors – These students can do this part of the lesson with another student
• Some students with communication problems may have a difficult time conveying their results of the activity with other students – For these students special care must be taken about whom they are paired up with. These students must be paired with other students who are patient and understand with their problem.

• Each student will need a pencil, scissors, and handout that will be provided in class
• Each student will need his or her homework from the previous night
• The teacher will need 24 handouts of parallelograms to distribute to the students

1.  Tell them that we can find the area of rectangles without having to count the number of squares on the interior of the shape
2.  Tell students that by understanding the area of rectangles they can understand the area of other shapes
3.  Remind students of the practical applications for finding area of both parallelograms and rectangles (such helping them decide how much of something they will need to buy)

Lesson Procedure

4.  Tell each student to take our his or her homework
5.  Review several of the problems that were on the worksheet by writing them on the board
6.  Ask students if the see a connection between the dimensions of the rectangles and their areas
7.  Be sure to ask this question in a way that allows the students to think
8.  They should see that multiplying the dimensions gives them the area
9.  After the students are clear about this relationship, instruct them to take out their scissors and give each of them a handout
10.  Have them do the activity on the handout. The directions are clearly marked
11.  When students have completed the activity in the handout, put them in groups of four to discuss their findings
12.  After discussion time, bring the students back together as a large group and discuss their results by writing them on the board. The formula for the area of a parallelogram should have been discovered
13.  Pass out the homework on rectangles and parallelograms for that night

Closure

14.  We started the day by looking at the relationship between the interior of a rectangle and its dimensions. Then, we looked at the relationship between the area of parallelograms and rectangles. How did knowing the area of rectangles help us find the area of parallelograms. Ask if they think this can help us find the area of other shapes as well. (This is a good way to get them interested in tomorrow’s lesson)

Extension

• If time allows, have the students begin working on the handout they were given. Walk around and help them if they have any problems answering the questions. This worksheet should enforce the ideas of today’s lesson

•  Reflect after lesson

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Mini-lab Handout
Homework

I. Goal(s):

• To develop the concepts of area as they relate to triangles

• The student will use manipulatives to discover the relationship between the area of a triangle and the area of a parallelogram
• The student will use manipulatives to discover the relationship between the area of a triangle to the area of a rectangle.
• The student will discover the formula for the area of a triangle

• 1 handout per student
• 1 piece of construction paper per student
• At least 6 pairs of scissors to be passed around the room and shared among students
• 1 ruler per student

1. Sailors have told stories of unusual occurrences in the area known as the Bermuda Triangle.  This imaginary triangle has Melbourne, Florida; Bermuda; and Puerto Rico at its vertices as shown on the board (Copy figure 1 onto the chalkboard).  What is the area enclosed by the Bermuda Triangle?
2. Explain to students that triangles are all around them in their every day lives.  Ask them for some examples.
3.  Tell the students that they will now be completing an activity that will lead them to the discovery of the formula for the area of a triangle.

FIGURE 1

 

V.   Lesson Procedure:

 4.   Handout materials: Select 1 student to handout a piece of construction paper to each student, select one student to handout rulers to the students, select 1 student to hand out pairs of scissors, and select 1 student to hand out the activity directions sheet.
 5.   Read over the directions on the activity sheet.  Answer any questions that the students
have, and clarify that they will be completing the activity individually.  They only  need to think about the 2 questions on the activity sheet.  Remind them that they will be sharing their answers with the class in a discussion format.
6.   Account for any students with disabilities.  Make sure they have enough work space, and assist with cutting and measuring if necessary.  If a teacher’s aid is not available, seat them near a patient, kind student who can assist them with the activity.
7.  Allow time (about 10 minutes) to complete the activity on their sheet, and reflect on the 2 questions provided.
8.  Bring the classes attention back to the front of the room.  Ask for students to share their answers/discoveries to the 4 questions.  Desired answers/discoveries to question #1: The area of one of the triangles is half of the area of a parallelogram
9.  Relate their answers to question #1 to question #2.  Have them share the formulas they came up with for the area of a triangle (AREA = (1/2)*b*h)
10.  Now ask them to attempt to from their triangles into a different parallelogram (a rectangle). Ask them to reflect on the same to questions with regards to this rectangle.
11.  Does our derived formula still work, based on what we know?

VI.  Closure:

12.   Hand out the worksheet on finding the area of a triangle.  Inform them that it will be graded in class the following day.  Answer any questions they  may have about the homework.  Remind them that if they get confused about the formula, to think about a triangle’s relation to a parallelogram.

VII.  Extension:

Go back to the Bermuda example from the beginning of class.  Given the
measurements/lengths of the sides of the triangle provided, what is the area of
The Bermuda Triangle (don’t forget units!).  Have them tell me how to calculate
the area as I record it on the board.  Have 1 student with a calculator do the
calculation, and write this answer down.  This allows them to see an application of the formula and the relevance of the lesson.

VIII.  Reflections:

 Record these following the lesson/class period.

TRY THIS:

• Trace two identical triangles on a sheet of paper using your straight edge (ruler).
• Cut out both triangles.
• Play around with the two triangles.  Try to place the triangles together to form a parallelogram.

QUESTIONS:
1. How does the area of one of the triangles compare to the area of the parallelogram?
2. Write a formula for finding the area of a triangle.
 
 

Homework

I. Goal(s):

• To introduce and explain the area of a circle

• The student will visualize the relationship between the area of a circle and the area of a parallelogram
• The student will calculate the area of a circle given the radius or diameter

• 1 paper plate, divided into 8 equal sections
• 1 pair of scissors

1.  Circles are all around you.  For example, you probably eat pizza on a regular basis.  Let’s say that you would like to figure out what the best deal would be for the amount of pizza you are getting.  To do this, you would want to compare the amount of money charged for a pizza to the area of the pizza.
2.  In order to solve a problem like this, we need to know how to calculate the area of a circle.  So, we’ll move on to finding this formula, and then we’ll come back to our pizza problem.
3.  Can any one tell me the formula for the area of a parallelogram?  (= base * height).  Keep this formula in mind as we proceed with the lesson on the area of a circle.

V.  Lesson Procedure:

4.  Begin the teacher demonstration of finding the area of a circle using a paper plate.
5.  Inform the students that it represents the pizza that we discussed in the beginning of class.
6.  Cut along the drawn lines on the paper plate so that 8 equal “pizza slices” are formed.
7  Arrange the pieces to form a parallelogram.  Do this by taping the pieces to the chalkboard so that all students can easily see the demonstration.
8.  Ask the students: How would you calculate the area of this figure?
9.  Use probing questions to direct the students to the formula for the area of a parallelogram.
10.  Record the area of the parallelogram on the board, and probe the students about how to represent the base and height of this figure.
11.  Revisit the formula for finding the circumference of a circle.  Hopefully this will help them with deriving the formula for the area of a circle.
12.  Follow these steps:

• A = b*h
• A = ((1/2)*c)*r
• A = (1/2)*(2*p*r)*r
• A = p*r*r
• A = p*r²

VI. Closure:

15.  Make sure that students are comfortable with the new material learned.  This can be done by allowing them to come up to my desk for questions during the last few minutes of class.
16.  Revisit the pizza problem that was introduced at the beginning of the class.  Provide them with the following example:
A medium pizza has a radius of 8 inches, and costs $8.
An extra large pizza has a radius of 14 inches, and costs $12.
First figure out how much pizza you get with each size pizza, by calculating its area.
Divide the cost of the pizza by the area you just calculated to get cost per square inch for each pizza.
Which pizza is the better deal?

VII. Extension:

• This will be covered in tonight’s homework, as it involves critical thinking

• Take note of any problems in conceptualizing during the demonstration.  Keep these in mind for future lessons, and reflect after the lesson as well.

Lab Activity
Test Review
Check out Accommodations for this lesson!

I. Goal(s):

• To introduce the importance of using technology effectively in conjunction with mathematical concepts
• To develop understanding about the relationship of math to real-life problems

• The students will use their knowledge of area and the information presented in order to use Microsoft Excel to develop their calculation and manipulation skills.
• The students will learn to enter data into the software as both numbers and formulas
• The students will calculate various estimates of flooring for a specified area of the floor plan each is given.

• 1 handout per student
• Assign each student to a computer in the lab
• An overhead projector of attached to a computer

1. Ask the students if they feel confident in their abilities at calculating and manipulating the different formulas for area
2. Ask them if they would like to know an easier way to find the area
3. Introduce them to the idea of using some computer software and find out how many have used the software before.

V. Lesson Procedure:

4. Lead the students to the lab.  Ask the students who are comfortable with the software to disperse themselves throughout in order to help students who are not familiar with it.
5. Account for students who might have special needs by making sure that they have enough room and are either seated at a computer for the disabled or near the front so it is easier to see, hear, and receive additional help from the teacher or lab attendant.
6. Hand out the activity page to the students.
7. Go over examples provided on worksheet and a give a short (no more than 5 minutes) introduction to using Excel.  Stick to the necessary information that the students will need to complete the activity.
8. Explain the activity to them, paying special attention to the process of using the formula of finding cost accurately.
9. Let the students begin the activity, allowing them to rely on other students if they need help, or to call on the teacher for help.
10. Complete the activity (if time runs out, inform the students that they will have time after the test or during study halls to complete it.)

VI. Closure:

11. Hand out the review sheet for the test.  Reassure them that there will be no new material and that I am sure they will be able to pass with flying colors.
VII. Extension:
Ask the students about there encounters with using the technology and whether they found it informative and easy to use.  Rely on their feedback for future project ideas.

 VIII.   Reflections: (after the lesson)

Today’s activity involves using Microsoft Excel for calculating the cost of carpeting, tiling, etc. a certain section of the floor plan you are choosing.
I want to first show you some examples about using Excel.  Excel works much like a fancy calculator...DON’T BE SCARED.
There are three main types of entries to use in Excel:
1. Words
2. Numbers
3. Formulas
For example, let’s try multiplying 2x3 by using a formula.  Watch the overhead as I do this.
You can use this same technique to calculate the area of a room and to find how much it would cost to lay each different type of flooring over this area.
The following is an approximation of the costs of different types of flooring:
 Carpet:   $10.00 per square foot
 Linoleum: $2.00 per square foot
 Tile:      $5.00 per square foot
************************************************************************************
The equation you will use to calculate cost is:  Total Cost = Area x Cost per sq ft.
************************************************************************************
Your spreadsheet should have the following basic set-up:
 

	A	B	C	D	E	F	G
1	Room	Length- Side 1	Length- Side 2	Area	Total Cost: Carpet	Total Cost: Linoleum	Total Cost: Tile
2
3

Some basic hints to follow:
*Column A should consist of words
*Column B and Column C should be numbers that you enter
*Columns D, E, F, and G should be formulas.  It is up to you to decide what these formulas are and how to enter them.
*All formula cells should begin with an = (equal sign).
*Remember that entering a formula means you need to use other cell names, such as B2, B3, etc. when entering them into the formula.

You should have each room appearing in column A with the rest of the columns filled out for that room.  Print out your page and hand it in.

The test will cover area of the following shapes:
  *Rectangles
  *Parallelograms
  *Triangles
   *Circles

The formulas for the area of these shapes are:

Area of Rectangle = b x h

Area of Parallelogram = b x h

Area of Triangle = ½ x b x h

Area of Circle = p x r²

1.  You should know how to solve for the area given the necessary lengths and you should also be able to determine the lengths given the area.  Remember to include the correct units!!!

2.  You will want to review homework from this week and the mini-labs we did in class.

3.  You will be asked to estimate the area of an irregular shape...you will be given grid squares to help you with this.

4.  The test may include a figure which looks like this:
Have a strategy for finding the area of the total figure.
 
 
 
 
 

GOOD LUCK, I HAVE COMPLETE FAITH IN YOUR ABILITIES AND SO SHOULD YOU!!!

Test

Goals

• To assess the students’ overall knowledge of area related concepts

• To assess the student’s knowledge of how to find the area of a rectangle, parallelogram, circle, and triangle give dimensions
• To assess the students’ knowledge of the dimension of a shape given its area
• To assess the students’ understanding of estimating area of unusual shapes
• To assess the students’ knowledge of how area applies to real life concepts
• To assess the students’ understanding of how the area of rectangles, triangles, circles, and parallelograms relate

• Students need a pencil
• The teacher needs the test which will be prepared ahead of time

1.  The teacher should seat and settle the students
2.  The tests should be distributed to the students
3.  Read the directions on the test and tell them to follow them carefull
4.  The test should take the entire class period

Extensions

• If students finish early tell them to work quietly on some other homework

 

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TEST