How does changes in variable(s) (dimension) of a quadrilateral, while maintaining the perimeter as a constant, change other attributes of the quadrilateral?
W: “Today we are going to look at the area of rectangles and see how changing their length and width without a change in perimeter can be applied to everyday problems. Area tells us how much space is available within the length and width of a rectangle. Sometimes, a decision about the length and width of a rectangle is base on the amount of area needed. Which of these boxes with the same perimeter would be best for holding these 12 blocks?” Show individual blocks and two boxes. “Which of these rectangles on the board would be able to fit these 4 square shapes?” Point to the rectangles on the board and the smaller square shapes. “These questions can be answered if you know the area of the boxes and rectangles?”
H: “Look at the boxes.” Point out the length and width of both boxes. “The boxes are different shapes but they have the same perimeter. Do you think they have the same area and both hold the 12 blocks? Turn to the student next to you and discuss your thoughts.” (Once the students have time to dialogue with each other, ask them to share their responses and record the yes and no answers on the chart paper. Ask the students to explain their thinking. What are they basing their answer on? “Now look at the rectangles on the board with the same perimeter.” Point out the length and width of both rectangles. “Do you think they have the same area and will both hold the 4 square shapes?” (Once again give the students time to dialogue with each other and ask them to share their responses. Ask the students to explain their thinking. What are they basing their answer on? “Both of these examples require an understanding of the concept of how area relates to perimeter.”
“I will show you how changing the length and width of the boxes with keeping the perimeter the same will change the area and therefore affect how many block the boxes will hold.” Take the two boxes and show how the 12 blocks fit into only one of the boxes. Ask the students whether they predicted correctly and if they though this was the intended outcome. This example shows the focus of the lesson on how the change in width and length changes the area even if the perimeter remains the same. After the students have predicted which of the rectangles on the board will hold the 12 square shapes, I will use tape and show how the squares only fit in one of the shapes.
“As you can see from the two activities, the area of rectangles can be different even if they have the same perimeter.”
E: “Raise your hand if you ever had a vegetable garden? (I will show the students a packet of vegetable seeds) What did you grow? (Students share responses.) “Imagine if each of you had a chance to plan a vegetable garden of you own. You will be able to plan and grow the vegetables you want. The only limitation you will have is that I will supply the fencing material you will need to keep out animals. To be fair I have to give everyone the same amount of fencing material. Does this mean that everyone’s garden has to be the same? You can decide to have different shape gardens or grow different vegetables based on your own interest and preferences. How much you like to work in a garden, how much space in your yard, or how much you like vegetables might be things to consider. We will see today the shape garden you decide on requires an understanding of width and length and the area these dimensions create. Let’s explore how the same amount of fencing material can form different shape gardens. I am going to give each of you 16’ of fence material. Let’s look at how many different gardens of a rectangular shape can be made.” I will provide the students with a worksheet for a garden plan with a perimeter of 16’ (see Garden Plan worksheet in the Materials section). Additionally, I will display this worksheet on the Interactive whiteboard or overhead. We will complete the information on the worksheet. As we go through the worksheet, the answers will be displayed on the interactive whiteboard or overhead. Ask the students the following questions as we complete the worksheet:
“How many garden plans are possible?” ( Four plans)
“What makes each of the garden plans different?” (The length and width of each garden.)
“What is the same about each of the plans?” (The perimeter is the same.)
“What is the area for each garden?” (A = 7sq. ft., A = 12 sq. ft., A = 15sq. ft., A = 16 sq. ft.)
“In which garden can you plant the most vegetables? (A = 16 sq. ft.)
“In which garden can you plant the least vegetables?” (A = 7 sq. ft.)
“What makes the garden plans different?” (Changing the length and width of each garden.)
“What also changes when you change the length and width?” (The area)
“Why does changing the length and width change the area?” (The formula for area of a rectangle is based on the length and width.)
R: “Now you are going to use a set perimeter and change the lengths and widths of the rectangles to see how the area changes.” Give each of the students a geoboard, several rubber bands and refer students to Table I and Table II worksheet (see Table I and Table II worksheet in the Materials section). “Let’s plan a pet enclosure for your dog. This will keep your dog safe even if you are not outside. Your parents have bought 18’ of wire fence. Let’s look at the different rectangular pet enclosures that you make on the geoboard. Each time you form a rectangle on your geoboard. Record the length and width in the table and then calculate the area for each rectangle and record it in the table. Remember, the perimeter for each rectangle must be 18. Can any body tell me the equation for the perimeter of a rectangle?” Record the equation on the board. “Can anyone tell me the equation for area of a rectangle?” Record the equation on the board. Student will use geoboards to represent the different rectangular spaces and complete the table. “Does the geoboard help you determine the area?” Students share responses. (Counting the spaces on the geoboard) “If you complete Table I go on and work on Table II.” After all the students have complete Table I, display the rectangles using the interactive geoboard on the interactive whiteboard. Additionally, the answers for Table II will be reviewed for those students who completed it. "What is the relationship between the shapes of these 5 rectangles and their areas? (Possible answer, the closer the shape gets to a square, the greater the area).
This activity will reinforce the concept of constant perimeter with varied dimensions will result in different areas. Based on student readiness, this activity can be modified to increase complexity.
Progressing: Students complete Table 1.
Proficient: Students complete Tables 1 and 2.
While students are working, observe the strategies students use to complete the task. Use this time to provide support to those who seem to be struggling with the activity.
E: Monitoring student responses during discussion and individual work can be used as informal assessments to guide instruction. Use student geoboards and the interactive geoboard on the Interactive whiteboard to provide feedback to ensure that students stay on track while you are guiding this part of the lesson. “In today’s lesson, we spent time looking at how the area of a rectangle will change when the length and width are changed, but keeping the perimeter the same. Can rectangles with the same perimeter have different areas?” (Yes. The areas will be different because the area is determined by measurements of length and width.)
T: Use the activities and strategies listed below to tailor the lesson to meet the needs of your students during the year.
Routine: A daily question can be used to reinforce skills from a previous lesson and to formatively assess students. Students complete the daily question at the beginning of class. This allows you to pre-assess students’ knowledge of the concept before beginning the lesson.
Extension: Students can apply the concept of constant perimeter to irregular shapes made up of rectangles and squares to determine how area is affected by changing the length and widths.
An extension for Gifted students could be to complete the activity - Animal Cracker Playground: http://illuminations.nctm.org/LessonDetail.aspx?id=L763
Technology Connection: Students can observe the rectangles that can be formed on Table I of the worksheet: http://standards.nctm.org/document/eexamples/chap4/4.2/part2.htm#applet
Students can use interactive websites that allows them to manipulate the dimensions of rectangles to examine the change in its area: http://illuminations.nctm.org/ActivityDetail.aspx?id=46
O: The focus of this lesson is to understand the change in area when the perimeter of a rectangle remains constant and the length and width are changed. Recognizing that rectangles with the same perimeter can have different areas. Use of geoboards and interactive websites helps students visualize the change in shape and resulting change in area. Using mathematical equations help the students understand the possible variations of length, width and area possible with a constant perimeter.
Rectangles with a constant perimeter will not necessarily have a constant area. Though exploration, students should be able to recognize this difference.
On-going formative assessments can be done during small group work, student interaction, and whole-class discussion.
Today's Lesson Summary Question (see Table I and Table II worksheet in the Materials section).