%Unit: Perimeter & Area% %Grade level: 10- Geometry% %Time span: Five days (45 minute classes) % %Tools used: Geometer s Sketchpad, Geoboards, National Library of Virtual Manipulatives, TI-84 Plus%
This portion of the area and perimeter unit should take five days. Topics include perimeter and area of rectangles and squares, inputting data into the calculator and then analyzing it. Day 1: Review of perimeter. Discussion of perimeter and area, and their relation to real life situations. Discussion of an upcoming journal entry. Development of perimeter formula s. Day 2: A Day in the computer lab with Geometer s Sketchpad. Developing a house layout, and then finding the perimeter and area of the house. Day 3: Introduction to area with geoboards. Development of area formulas. Day 4-5: Flower Science, given perimeter, find all area possibilities. Enter data into calculator, graph the data, and what do you notice? Overall Objectives: Students will be able to determine the difference between area and perimeter. Students will be able to use their knowledge of area and perimeter to handle real life situations. Students will develop general understanding for area and perimeter and they will see where the formulas are derived from. Students will use the TI-84 and its graphing functions to see relations of area. The students will further their knowledge of GSP (Geometer s Sketch Pad) and measure area and perimeter within GSP. Basic introduction of Geoboards (also seen in the National Library of Virtual Manipulatives).
Students will be using their creativity and English skills to write word problems. NCTM Standards: Content: Algebra Geometry Measurement Data analysis Process: Problem solving NYS Standards Key Ideas Measurement Mathematical reasoning Modeling/ Multiple Representation Materials and Resources: Centimeter grid paper (a piece for each student). The Geometer s Sketchpad (available for use by each student). Class set of geoboards (at least enough for each pair to have one). An overhead projector and overhead geoboard (so the class can see fellow classmate s present questions that I may have). Geoboard paper. Flower Science worksheet created by myself. Graph paper for each student. Class set of graphing calculators (TI-84). Class text book: Geometry, Tools for a Changing World; Bass, Hall, Johnson & Wood; 1998; Chapter 5, pages 242-254 Teacher s Guide, Do Bees Build It Best? Interactive Mathematics Program. Dan Fendel and Diane Resek; 1998. Day 2, pages 12-16. National Library of Virtual Manipulatives.
Title: Intro-discussion. Objective: This will be a class mainly consisting of discussion. The students will begin to think about math and relating it to real life situations. A journal entry will also be included in the lesson to encourage the understanding of perimeter vs. area. The writing of word problems will give the students good practice with writing and allow them to become creative when it comes to math. Procedure: After homework is collected and gone over, the students will get into groups (fellow students they are sitting near). They will then listen to me and complete the following activity. Activity: 1) The students will get into groups and they will draw the following on the provided graph paper: A rectangle with length of 5 cm and width 3 cm (P = 16 cm, A = 15 sq. cm). A rectangle with height 2 cm and base 8 cm (P = 20 cm, A = 16 sq. cm). A rectangle with each side measuring 4 cm (P = 16 cm, A = 16 sq. cm). *note: answers will not be provided to the students. At this time, I will have a student from the class designing these rectangles in the National Library of Virtual Manipulatives- Geoboard section. I will have my computer hooked up to a projector for all the students to see. I will be able to use this student s work as the class progresses and questions arise. 2) After a quick review/introduction of perimeter, the students will find the perimeter of each of the rectangles (the students already have previous knowledge of perimeter, so I will not be spending much time on it). 3) Once the students have recalled perimeter and they have answered the above questions, the following discussion questions will be asked and discussed: To find area, record the number of square centimeter in the interior. Find the area of each rectangle. Do the rectangles with the same area; have the same perimeter (No)? Why (Different measurements)? *note: this question will be part of a journal entry that they are going to do for tonight s homework. 4) After these questions are gone over, we will begin to relate this part of math to real life situations. We will discuss how the following sorts of questions relate to math. The questions will consist of: How many posters could be placed around Sally s bedroom? (Perimeter) How many tiles does Martha need to tile her bathroom? (Area) Mrs. Jacobs wants to put a fence around her house, how much fence will she need? (Perimeter) How much space will the math club s poster take up when they hang it in the union? (Area) 5) While still in their groups, the students will think of word problems that only relate to perimeter. This will help the students to better understand perimeter
vs. area. Once the problems are written, I will have some volunteers read their problems aloud. Conclusion: If time permits, we will begin working on actual perimeter examples (like finding perimeter with measurements). By looking at the rectangles we drew at the start of class, we will review how we got the actual perimeter of the rectangles (by adding the sides together). We will begin to develop the formulas to find perimeter, that is, perimeter of a rectangle = 2(b)+2(h) and perimeter of a square= 4(s). Development of these formulas will prepare the students for tomorrow s lesson. Homework: Journal entry (Do the rectangles with the same area; have the same perimeter? Why? Also, page 245 (4-11).
Title: Designer for the Day. Objective: The students will be designing a layout and estimating measurements in relation to height. Then, they will find the area and perimeter of the layout, seeing that math can be handy and helpful. The students will be using yesterday s perimeter formulas and the definition of area that we also went over yesterday. Procedure: For today s class, we will all be heading to the computer lab (or we could just meet in the computer lab). Each student will have their own computer and once they enter the lab, their computer will be locked (by me) until directions are gone over. Activity: 1) Once the students have entered the lab and are settled down, I will begin the lesson. For today s class, we will be finding the perimeter of one story in the student s house (the story that their bedroom is located on) using GSP. 2) Since the students already have so much experience with GSP, no much explanation of the program will be needed (they will have many notes and handouts on the program from previous classes). 3) I will begin by showing the students a floor plan for my studio apartment (the entire demonstration of my apartment can be found in appendix). 4) I expect the student s layout to be in some sort of ratio, for instance, one foot = one centimeter (like my layout is). I want them estimate the size of the room s in their house in relation to their height. For example, let s say that a student is five feet tall, how many of that student (lying down) could fit along a specific wall? 5) I want the perimeter of each room/closet/hallway/etc and the area of each if we have time. The perimeter of each room is something that must be completed using the formulas that we used yesterday. Conclusion: Once the students have completed the project and handed it in, they will be starting their homework until the bell rings. Assignment: Page 245 (12-20 and 22, 23). Also, answer the following question: Why would we need to know the area of your living room or bedroom (so we could put in carpet or tile)?
Title: How would you define area? Objective: After this lesson is complete, students will have the general understanding that area is the number of units that fit within a figure. From these observations, students will be developing formulas based upon their discoveries. Procedure: We shall begin the class by discussing area. I will be asking the students such questions as, How would you define area?, What kinds of things have area?, and having the students distinguish the differences between perimeter, area and volume. I will ask the students to recall the definition of area that we discussed on day 1. After I feel that students generally have an idea of area, we shall begin the activity. Activity: 1) Nailing Down Area: An introduction activity to the geoboard to help students to better understand the meaning of area. 2) Students will get with a partner (the student sitting beside them), and each pair will be given a geoboard (including rubber bands), and geoboard paper (a piece for each student). As a class we will quickly review the rules of the geoboard. 3) I will give the students about five minutes to play with the geoboards. This will allow to students to see how they can form different shapes and such with the rubber bands and the pegs. 4) While the students are playing with the geoboards, I will be putting a few questions on the black board. These questions will consist of: Create a triangle whose area is half of a unit. Create quadrilaterals whose area is exactly 1 unit. Create quadrilaterals/triangles whose area is larger than 1 unit. 5) The students will now begin to work on the questions that I have written on the board. The students will also begin to generate some formulas for the area of quadrilaterals. Once students have generated a formula for quadrilaterals I will then encourage them to use that information to develop a formula for triangles. As a class we will review the student s observations and their formulas they have developed. I will be asking for volunteers to come to the overhead and present their findings. Conclusion: As a closing to this activity, I will be putting a picture of an arbitrary shape on the overhead. I will ask the students to tell me what the closed
shape area would be (I will present the size of the unit to the students as well). This closing part of the activity will allow students to get a better understanding that the unit can very in size depending upon the shape. This will prepare students to do their homework and for the upcoming activity. Homework Assignment: In the text book, page 245-248 (24-33,35-42).
Title: Flower Science Objective: Students will be applying all their knowledge of area and perimeter to a real life situation. From their knowledge of the calculator, they will be inputting their data and then graphing it as well. Procedure: Class will begin by reviewing the formulas that we went over yesterday and the previous days as well. The students will be expected to have a formula page within their notebook and this is where the formula for area (and perimeter) will be written out. We, as a class, will do a quick review of what we discovered yesterday (area formula for rectangles and triangles) and we will be writing them in their notebook, if they are not already in there. Activity: 1) After we have discussed the formulas, I will be handing out the worksheet that the students will be doing in class today (see Appendix for the worksheet and its answers). Graph paper will also be handed out. 2) After a quick overview of the worksheet, the students will begin their work. The students will be asked to complete the entire problem (all parts) and then asked to graph their findings. The directions of the activity are on the actual worksheet. 3) For the rest of the class, the students will just follow the directions on the worksheet until it s complete. There is a lot of data on the worksheet, and I do expect it to take two days. The first day (day 4) the students will be completing the chart. The chart is quite long and should take an entire class period. 4) On day 5, I will be doing a quick review of inputting data into the calculator and then graphing it (the transparency will be used and can be found in the Appendix). Conclusion: At the end of day five, once the chart is complete and the data is graphed, the students will be asked to graph their findings (in more detail) on graph paper for homework. They can begin this once the worksheet is complete. Homework Assignment: Finish the worksheet, if it s not already done, and graph Flower Science in greater detail.
Appendix Day 2 This is my presentation of my apartment. This is what I will be expecting of the students. I m not looking for the students to have tabs or buttons like mine does, but I do expect them to include the measurements (lengths, area and perimeters) of the following things: Bedrooms Bathrooms Closets Entire floor Kitchen, etc. This is the first picture that the students will see. This is the general outline of my apartment. For this picture, 1 cm= 1 foot. To see the actual measurements, just press the button. Show Distance Measurements A D H F E I C
After you have pressed the button, this is what you will see: Hide Distance Measurements A D AD = 17.99 cm DC = 12.04 cm AE = 12.01 cm H FH = 4.52 cm F IH = 6.03 cm E I C EC = 17.99 cm CI = 4.50 cm Now, we will press the tab (at the bottom of the sketch) to see the next page. Here, we have found the perimeter of the apartment: A D AD = 17.97 cm AE = 12.01 cm Perimeter of entire apartment = 59.95 cm DC = 12.01 cm DC = 12.01 cm H FH = 4.52 cm F IH = 6.03 cm E I C EC = 17.97 cm CI = 4.49 cm
If you click on the next tab (page 3), it will show us the perimeter of the bathroom: A D AD = 17.97 cm AE = 12.01 cm Perimeter of entire apartment = 59.95 cm DC = 12.01 cm DC = 12.01 cm H FH = 4.52 cm F IH = 6.03 cm Perimeter of Bathroom = 21.08 cm E I C EC = 17.97 cm CI = 4.49 cm The next tab (page 4) will then show you the area of the entire apartment: A D AD = 17.97 cm AE = 12.01 cm Perimeter of entire apartment = 59.95 cm DC = 12.01 cm DC = 12.01 cm Area of Entire Apartment = 215.80 cm 2 H FH = 4.52 cm F IH = 6.03 cm Perimeter of Bathroom = 21.08 cm E I C EC = 17.97 cm CI = 4.49 cm
Lastly, tab 5 will show you the area of the bathroom: A D AD = 17.97 cm AE = 12.01 cm Perimeter of entire apartment = 59.95 cm DC = 12.01 cm DC = 12.01 cm Area of Entire Apartment = 215.80 cm 2 H FH = 4.52 cm F IH = 6.03 cm Perimeter of Bathroom = 21.08 cm Area of Bathroom = 27.29 cm 2 E I C EC = 17.97 cm CI = 4.49 cm This is the end of my apartment presentation.
Worksheet for Day 4-5, the answers are provided in RED. Appendix Flower Science Name: Jon has 32 yards of fencing. Jon wants to make a garden for his mom for her birthday. He knows that if he doesn t put a fence around the garden, that the deer will eat all of her tomatoes. Jon wants to make a rectangular garden for his mother. What are the dimensions of the garden to result in the maximum area? Draw some possible rectangular gardens and find their area. Dimensions (yards) Students will be asked to generate a chart like the following one: Base Height (16-b=H) Amount of fencing (perimeter) Area (square yards) 15 x 1 1 15 32 yards 15 14 x 2 2 14 32 yards 28 13 x 3 3 13 32 yards 39 12 x 4 4 12 32 yards 48 11 x 5 5 11 32 yards 55 10 x 6 6 10 32 yards 60 9 x 7 7 9 32 yards 63 8 x 8 8 8 32 yards 64 7 x 9 9 7 32 yards 63 6 x 10 10 6 32 yards 60 5 x 11 11 5 32 yards 55 4 x 12 12 4 32 yards 48 3 x 13 13 3 32 yards 39 2 x 14 14 2 32 yards 28 1 x 15 15 1 32 yards 15 After the chart is complete, make a graph of it (in your calculator). Graph the values of the base on the horizontal axis and the values of the area on the vertical axis (see graphing transparency for directions on inputting lists into the calculator and then graphing them).
Answer the following questions: 1. Where does the maximum area occur? The maximum area occurs when the base is 8 and so is the height. The maximum area is 64 square yards. 2. To have the maximum area for his mother s garden, what should his dimensions be? 8 x 8. 3. If you decided to put the garden along the garage, so you would only need to fence three sides, how would this affect your area? It would make my area bigger because I could use the same amount of fencing but I would only need it for three sides instead of four. 4. Jon wants to buy his mother some seeds so she can plant them in her garden as well. One package of seeds costs $1.59 and covers three square yards. How many packets of seeds will he need and how much will it cost him? He will need 22 packets, because three goes into 64 21.33 times, and we round up so he will have enough seed. He will be spending $34.98 on seeds for his mother s garden. We got the price by multiplying 22 times $1.59. For homework: Draw the graph on graph paper in more detail. Be sure to include labels and titles.
Appendix Transparency for Day 5 (Steps on inputting the chart into their calculator and creating a graph of the information they gathered- what the students will actually see on their calculator is provided). Using the TI-84 we begin inputting our data into our calculator by naming our list 1 our base column and our list 2, our area column. Step 1: Go to List and hit ENTER(or hit 1 on the key pad). Step 2: This will bring up a screen to enter your data into list 1, list 2, etc. We will begin by highlighting list 1, hitting clear and then enter. This will clear out any information that is already in list 1. We should do the same for list 2 as well. Step 3: Begin to type your data into list 1 (your base answers will be entered here). Simply hit enter after each entry. Once you have done this, do the same for list 2 but enter the area answers here. Be sure the correct area matches up with right base.
Step 5: Now, choose Stat Plot. Step 6: Choose Plot 1 by hitting enter. Highlight ON, select the desired graph, and a marker of your choice. Step 7: Now, choose Zoom, and highlight 9: ZoomStat. Step 8: Now, simply hit enter, and the graph will appear on your screen.