PERIMETER AND AREA Grade Level: 6 th 10-Day Unit Plan Technology & Manipulatives Used: Graphing Calculator Geometers Sketchpad One-inch square tiles Magnetic polygons TI-Connect Tangrams Geoboards Jaclyn Hutchinson & Nicole Kwiatkowski
Table of Contents Objectives and Standards......3 Resources......7 Overview of Unit......8 Materials and Equipment......10 Day One: Perimeter......12 Day Two: Area......19 Day Three: Area continued......24 Day Four: Constant Area, changing perimeter..29 Day Five: Constant perimeter, changing area...37 Day Six: Constant perimeter, changing area.....42 Day Seven: Area of Parallelograms......47 Day Eight: Area of Triangles........56 Day Nine: Making Parallelograms from Triangles 64 Day Ten: Area of Trapezoids......69 2
Objectives and Standards Students will be able to: Differentiate between area and perimeter. Identify that different shapes can have the same perimeter or area. Construct diagrams and tables to represent data. Derive formulas for finding the area of geometric shapes including rectangles, parallelograms, triangles, trapezoids, and irregular figures. Calculate the area of various geometric shapes given their dimensions. Discover relationships between the formulas for finding area of different shapes. Standards: NCTM Algebra Represent, analyze, and generalize a variety of patterns with tables, graphs, words, and when possible, symbolic rules Geometry Understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects Draw geometric objects with specified properties, such as side lengths or angle measures Use visual tools to represent and solve problems Recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life. Measurement Select and apply techniques and tools to accurately find area Develop and use formulas to determine area of triangles, parallelograms, and trapezoids Data Analysis and Probability Select, create, and use appropriate graphical representations of data, including histograms, box plots, and scatter plots Problem Solving Build mathematical knowledge through problem solving Solve problems that arise in mathematics and in other contexts. 3
Apply and adapt a variety of appropriate strategies to solve problems Reasoning and Proof Examine patterns and structures to detect regularities Formulate generalizations and conjectures about observed regularities Communications Organize and consolidate their mathematical thinking through communication Communicate their mathematical thinking coherently and clearly to peers, teachers, and others Analyze and evaluate the mathematical thinking and strategies of others Use language of mathematics to express mathematical ideas precisely Connections Recognize and use connections among mathematical ideas Recognize and apply mathematics in contexts outside of mathematics Representation Create and use representations to organize, record, and communicate mathematical ideas Use representations to model and interpret physical, social, and mathematical phenomena NYS Problem Solving Strand 6.PS.6: Translate from a picture/diagram to a numeric expression 6.PS.8: Select an appropriate representation of a problem 6.PS.10: Work in collaboration with others to solve problems 6.PS.11: Translate from a picture/diagram to a number or symbolic expression 6.PS.12: Use trial and error and the process of elimination to solve problems 6.PS.13: Model problems with pictures/diagrams or physical objects 6.PS.14: Analyze problems by observing patterns 6.PS.15: Make organized lists or charts to solve numerical problems 4
6.PS.21: Explain the methods and reasoning behind the problem solving strategies used Reasoning and Proof Strand 6.RP.5: Justify general claims or conjectures, using manipulatives, models, expressions, and mathematical relationships Communication Strand 6.CM.4: Share organized mathematical ideas through the manipulation of objects, numerical tables, drawings, pictures, charts, graphs, tables, diagrams, models, and symbols in written and verbal form 6.CM.8: Consider strategies used and solutions found by others in relation to their own work 6.CM.10: Use appropriate vocabulary when describing objects, relationships, mathematical solutions, and rationale Connections Strand 6.CN.4: Understand multiple representations and how they are related 6.CN.6: Recognize and provide examples of the presence of mathematics in their daily lives 6.CN.7: Apply mathematics to problem situations that develop outside of mathematics Representation Strand 6.R.1: Use physical objects, drawings, charts, tables, graphs, symbols, equations, or objects created using technology as representations 6.R.5: Use representations to explore problem situations 6.R.6: Investigate relationships between different representations and their impact on a given problem 6.R.7: Use mathematics to show and understand physical phenomena (e.g., determine the perimeter of a bulletin board) Geometry Strand 6.G.2: Determine the area of triangles and quadrilaterals (squares, rectangles, rhombi, and trapezoids) and develop formulas 6.G.3: Use a variety of strategies to find the area of regular and irregular polygons 6.G.11: Calculate the area of basic polygons drawn on a coordinate plane (rectangles and shapes composed of rectangles having sides with integer lengths) Statistics and Probability Strand 5
6.S.2: Record data in a frequency table 6.S.4: Determine and justify the most appropriate graph to display a given set of data (pictograph, bar graph, line graph, histogram, or circle graph) 6.S.7: Read and interpret graphs 6
Resources Internet 1. The Core Curriculum Companions for the New York State Mathematics Resource Guide http://www.emsc.nysed.gov/ciai/mst/math-companion.pdf 2. Strong Museum: National Museum of Play http://www.strongmuseum.org 3. Illuminations http://illuminations.nctm.org Text 1. Malloy, Carol E. Perimeter and Area Through the Van Hiele Model. National Council of Teachers of Mathematics: Mathematics Teaching in the Middle School 5 (1999): 87-90 2. Lappan, Fey, Fitzgerald, Friel, and Phillips. Covering and Surrounding: Two-Dimensional Measurement. Connected Mathematics Series. Glenview, Illinois: Prentice Hall 7
Overview of Unit Day One: Perimeter Students will use one-inch square tiles to investigate perimeter. Through group exploration, students will determine how you can increase or decrease perimeter by adding square tiles to a figure. Students will be able to identify that different shapes can have the same perimeter. Day Two: Area Students will read a story that introduces students to manipulatives called tangrams. Students will then use the tangrams to investigate the meaning of area. In groups, students will work to construct two figures from the story. Students will become familiar with geometric shapes and see how they can combine to form other shapes. Day Three: Area continued Students will continue to use tangrams to explore area. Individually, students will be asked to complete an activity of drawing rectangles with the same area. Their partner according to the rubric on the overhead will grade this activity. Students will develop and understanding that different shapes can have the same area. Students will also have a chance to obtain bonus points by completing a fun activity with tangrams. Day Four: Constant Area, Changing Perimeter Students will explore that perimeters of rectangles can vary even when their area is constant. Students will use geoboards to construct rectangles with a given area. As a class, students will develop a chart that demonstrates that the perimeters vary. Students will then compete a similar activity individually that adds another component of cost. Day Five: Constant Perimeter, Changing Area Students will complete a similar activity as in day four; only today students will discover that areas of rectangles with a fixed perimeter can vary. Students will use their geoboards to investigate this concept. Students will also construct diagrams and tables that allow them to explore maximum and minimum questions relating to the areas for rectangles of fixed perimeter. Day Six: Constant Perimeter, Changing Area Students will work on the same problem as the previous day except this time they will use the graphing calculator to arrive at their solution. Students will work in groups to complete a scatter plot by following the instructional worksheet. They will also analyze tables to determine the values needed for lists. Students will see how you can use a scatterplot and the trace function to maximize area. 8
Day Seven: Area of Parallelograms This lesson is designed to allow for students to deepen their understanding of area by finding areas of parallelograms. Students will use their geoboards to construct parallelograms and then count and estimate to find the area. Students will develop a formula for finding the area of a parallelogram, and will discover that they can relate it to something they can already do, which is finding the area of a rectangle. Day Eight: Area of Triangles Students will use again use their geoboards to complete a similar activity as yesterday. Students will construct triangles and then count and estimate to find the area. Through self-discovery students will develop a formula for finding the area of a triangle. Day Nine: Making Parallelograms from Triangles Students will cut out triangles in this activity and arrange them to form different shapes. This is designed to help students see that a triangle is half of a parallelogram. Students will discover that two congruent triangles can always be arranged to make a parallelogram, as well as that a parallelogram can always be cut to form two congruent triangles. Day Ten: Area of Trapezoids Students will look at a rectangle and triangle in this lesson to see if they can find the similarities and differences compared to a trapezoid. Using their geoboards to construct trapezoids and then count and estimate to find the area. Students will develop a formula for finding the area of a trapezoid. Students will discover that a trapezoid can be cut and arranged to form a rectangle. 9
Materials and Equipment Day One: Perimeter Set of one-inch square tiles for each group, Perimeter Investigation Worksheet, Grid Paper, Perimeter Homework Worksheet, Overhead Projector, Transparency of Anticipatory Set Day Two: Area Grandfather Tang s Story, Grid Paper, Set of tangrams for each student, Overhead Projector, Transparency of homework solutions, Transparency of rubric for assignment, Tangram investigation worksheet Day Three: Area continued Overhead Projector, Tangram Investigation Worksheet, Overhead of solution to Tangram Worksheet, Transparency of On Your Own Task, Fun with Tangram Worksheet, Solutions to Fun with Tangram Worksheet Day Four: Constant Area, Changing Perimeter Geoboards and rubber bands, Grid Paper, Transparency of solution to anticipatory set, Storm Shelter Worksheet, Transparency of On Your Own Task, Sandbox Worksheet Day Five: Constant Perimeter, Changing Area Geoboards and rubber bands, Grid Paper, Overhead Projector, Transparency of Perimeter = 12 chart, Fido s Fence: Day One Worksheet Day Six: Constant Perimeter, Changing Area Overhead Projector, Overhead graphing calculator, Graphing calculator for each student, Fido s Fence: Day Two Worksheet, TI Instructional Worksheet Day Seven: Area of Parallelograms Calculator, Graph Paper, Geoboards and rubber bands, Geoboard Grid Paper, Area of Parallelogram Worksheet, Parallelogram homework Worksheet, Overhead Projector, Transparency of anticipatory set, Magnetic parallelograms/rectangles Day Eight: Area of Triangles Calculator, Grid Paper, Geoboards and rubber bands, Geoboard Grid Paper, Area of Triangles Worksheet, Overhead Projector, Transparency of anticipatory set, Homework Worksheet 10
Day Nine: Making Parallelograms from Triangles Calculator, Grid Paper, Scissors, Straight Edge, Making Connections Worksheet, Magnetic Triangles/Parallelograms, Overhead Projector, Transparency of anticipatory set, Transparency of On Your Own Task Day Ten: Area of Trapezoids Calculator, Grid Paper, Geoboards and rubber bands, Geoboard Grid Paper, Magetic Polygons, Area of Trapezoid Worksheet, Overhead Projector, Transparency of anticipatory set 11
Day 1: Perimeter Objectives: 6 th grade students will expand their knowledge of perimeter and methods of calculating perimeter. Students will be able to identify that different shapes can have the same perimeter. Standards: NCTM: Algebra, Geometry, Problem Solving, Reasoning and Proof, Communications, Representations, Connections NYS: 6.PS.10, 6.PS.12, 6.PS.16, 6.PS.21, 6.CN.4, 6.R.1, 6.R.6, 6.R.7 Materials: Set of one-inch square tiles for each group Perimeter Investigation Worksheet Grid Paper Perimeter Homework Worksheet Overhead Projector Transparency of anticipatory set Opening Activity: I will have the following figures on the overhead with directions for the students to Calculate the space surrounding each of the following figures. This will act as a review of perimeter because they were introduced to perimeter in 5 th grade. 8 7 6 5 4 3 2 1 x-10-8 -6-4 -2 2 4 6 8 10-1 -2-3 -4-5 -6-7 Ask Students: o What is the space surrounding each figure and how did you calculate it. Square: 14 inches, Triangle: 16 inches, Parallelogram: 8 inches 12
Main Activity: Investigation: In your groups use the one-inch square tiles to answer the guided questions. Throughout this activity you will be further investigating the properties of perimeter. To Begin: Set-up 6 one-inch square tiles as done in the diagram below: Tell Groups: o Use one-inch square tiles to answer the guided questions on the worksheet provided. Individual Assessment: o I will make sure groups are on task and call on random individuals in each group to explain to me the how that individuals group arrived at their answer. Closing: As a class we will discuss the groups responses to the guided questions for the one-inch square investigation. Group Assessment: o I will call on a random group to give their responses to the guided questions. Homework: Students will complete the Perimeter worksheet. 13
Name PERIMETER INVESTIGATION 1. What is the perimeter of the given object? 2. Where would I place a tile to increase the perimeter by zero? By one? By 2? By 3? Draw a sketch to explain responses. 3. What is the fewest number of tiles that can be added to increase the perimeter to 16 units? Draw a sketch of this figure. 4. What is the greatest number of tiles that you can add to increase the perimeter to 16 units? Draw a sketch of this figure. 5. Is there a way to add a tile to decrease the perimeter? 14
PERIMETER INVESTIGATION Answer Key 1. What is the perimeter of the given object? 17 units 2. Where would I place a tile to increase the perimeter by zero? By one? By 2? By 3? Draw a sketch to explain responses. By zero - in either of the indented corners. By one - not possible. By two - on any of the square on the side of the figure. By three anywhere except the ends or the indented corners. 3. What is the fewest number of tiles that can be added to increase the perimeter to 16 units? Draw a sketch of this figure. Two tiles, placed on the ends. 4. What is the greatest number of tiles that you can add to increase the perimeter to 16 units? Draw a sketch of this figure. Four tiles. Two in indented corners and two anywhere else. 5. Is there a way to add a tile to decrease the perimeter? No. 15
Name Grid Paper 16
Name Perimeter Worksheet a. Find the perimeter of Mr. Ruffino s garden? Show work. 8cm 14cm 15cm 4cm b. Find another polygon that is non-congruent to Mr. Ruffino s garden, but has the same perimeter. Draw the figure in the space below. Show work. (Use a straight edge). 17
Perimeter Worksheet Answer Key a. Find the perimeter of Mr. Ruffino s garden? Show work. 8cm 8 + 15 + 14 + 4 = 41cm 14cm 15cm 4cm b. Find another polygon that is non-congruent to Mr. Ruffino s garden, but has the same perimeter. Draw the figure in the space below. Show work. (Use a straight edge). An example of a 3 response: Answers will vary 7 cm 17 cm P = 17 cm + 5 cm + 7 cm + 12 cm P = 41cm 5 cm This response is complete and correct. It also demonstrates a thorough understanding of the mathematical concepts. 12 cm An example of a 2 response: 4.5 cm 11 cm P = 11 + 11 + 4.5 + 4.5 P = 31 cm 11 cm 4.5 cm This student has an incorrect solution, but applies a mathematically appropriate process. He/she forgot to carry the tens place in her addition in question #1. An example of a 1 response: 13 cm 17 cm 21 cm This student has an incorrect response for question #1 and no work is shown. For question #2 they have the correct response but didn t provide the required work. 18
Day 2: Area Objectives: Students will become familiar with geometric shapes and how they can combine them to form other shapes. Students will use tangrams to investigate the meaning of area. Standards: NCTM: Algebra, Measurement, Geometry, Problem Solving, Reasoning and Proof, Communications, Representations, Connection NYS: 6.PS.10, 6.PS.12, 6.PS.13, 6.PS.16, 6.PS.21, 6.CM.4, 6.CM.8, 6.CM.10, 6.CN.4, 6.R.1, 6.R.5, 6.R.6, 6.R.7, 6.G.2, 6.G.3, 6.G.11 Materials: Grandfather Tang s Story Grid paper Set of tangrams for each student Overhead Projector Transparency of homework solutions Transparency of rubric for assignment Tangram Investigation Worksheet Homework Check: Students will check their answer to question #1 with the solution on the overhead. For question #2 students will give themselves a score of 1, 2, or 3 according to a given rubric. Then I will collect the assignment. Opening Activity: I will have my students read Grandfather Tang s Story independently. Tell Students: o The tangrams that Grandfather Tang uses in the story are actual ancient Chinese puzzles that are still used today. Each group has two sets of tangrams in which you will be using to complete the following investigation. 19
Main Activity: Investigation: After Reading Grandfather Tang s Story Tia and Marquez were playing with 5 of the seven-tangram pieces that Grandfather Tang used. They removed the two big triangles. Tia made a rectangle and Marquez made a square. In your groups reconstruct Tia s rectangle and Marquez s square. Tell Students: o In each group of four, one pair should make Tia s rectangle and the other pair should make Marquez s square. Then you should trace each figure on the grid paper provided and complete the Tangram Worksheet #1-7. Marquez s Square Tia s Rectangle Individual Assessment: o I will make sure groups are on task and call on random individuals in each group to explain to me the how that individuals group arrived at their answer. Closing: As a class we will discuss the groups responses to questions #1-4 on the tangram investigation worksheet. Group Assessment: o I will call on a random group to give their responses to the guided questions. Homework: NONE!!! GROUP MEMBERS 20
Tangram Investigation Worksheet 1. What are the similarities in the two shapes? 2. What are the differences in the two shapes? 3. How do I measure the space surrounding each object? 4. What is the perimeter of each figure? Tia: Marquez: 5. How do I measure the space each shape covers? 6. What is the area of each figure? Tia: Marquez: 7. How do we know the areas are the same? 8. Make a rule for finding the area of squares and rectangles. 21
Tangram Investigation Worksheet Answer Key 1. What are the similarities in the two shapes? They use all the shapes. 2. What are the differences in the two shapes? One is a square, and the other one is a rectangle. 3. How do I measure the space surrounding each object? Count up all the squares around the figure. 4. What is the perimeter of each figure? Tia: 40 cm Marquez: 32 cm 5. How do I measure the space each shape covers? Count up the squares inside each figure. 6. What is the area of each figure? Tia: 64 cm² Marquez: 64 cm² 7. How do we know the areas are the same? Because they have the same number of squares inside of them. 8. Make a rule for finding the area of squares and rectangles. Answers will vary. 22
Name GRID PAPER 23
Day 3: Area Continued Objectives: Students will be able to discover different shapes can have the same area. Students will develop a formula for finding the area of squares and rectangles. Materials: Overhead projector Tangram Investigation Worksheet Overhead of solution to Tangram Worksheet Transparency of On Your Own Task Fun with Tangram Worksheet Solutions to Fun with Tangram Worksheet Standards: NCTM: Algebra, Geometry, Problem Solving, Reasoning and Proof, Communications, Representations, Connection NYS: 6.PS.8, 6.PS.10, 6.PS.13, 6.CM.4, 6.CM.10, 6.CN.4, 6.R.1, 6.R.5, 6.R.6, 6.R.7, 6.G.2, 6.G.11 Opening Activity: Have students recall the tangram investigation from the previous day. We will briefly review how the groups found the space surrounding each of the figures and their exact dimensions, as well as the area covering each figure and their exact dimensions. Main Activity: Instruct Students: o Have students get into their groups to finish the Tangram Investigation worksheet from yesterday. Discussion: As a class we will discuss the groups responses to questions #8 on the tangram investigation worksheet. Group Assessment: o I will call on a random group to give their responses to that specific question. On Their Own: Students will be asked to complete the following task individually: 24
Draw all the rectangles that have an area = 14 square units on the grid paper provided. Show all the possible perimeters that each of the rectangles can have. 14 12 10 8 6 4 2-15 -10-5 5 10 Informal Assessment: o As the students are working I will be spot-checking the students work to make sure they have a good understanding of the concept and task at hand. Tell Students: o That once they have completed the assignment to switch papers with one of their group members and compare their responses to the questions that I will have on the overhead (4 rectangles). I will instruct the students to write the score (out of four) on the top and I will spot-check the scores before I collect them. In the Time Remaining: Inform Students: o Have students attempt a mini investigation of shapes with the same area. They are to use their tangrams to create the eight given shapes on the fun with tangrams worksheet and trace their solutions. o I will let students know at the end of class that the groups that manage to get all eight shapes correct will receive bonus points on their next assignment. Homework: Finish fun with tangrams worksheet for homework (optional). 25
Name GRID PAPER 26
Fun with Tangrams Directions: Use all 7-tangram pieces to create the figures below. 1. CHAIR 2. SHIRT 3. BOOT 4. ROOSTER 5. KANGAROO 6. HORSE 6. CANDLE 8. SAILBOAT 27
Answer Key 28
Day 4: Constant Area, Changing Perimeter Objectives: Students will understand how the perimeters of rectangles can vary considerably even when the area is held constant. They will construct diagrams and tables to organize data for them to explore maximum and minimum questions relating to the perimeter for rectangles of fixed area. Materials: Geoboards and rubber bands Grid paper Transparency of solution to anticipatory set Storm Shelter Worksheet Transparency of On Your Own Task Sandbox Worksheet Standards: NCTM: Algebra, Geometry, Problem Solving, Reasoning and Proof, Communications, Representations, Connection NYS: 6.PS.10, 6.PS.13, 6.PS.14, 6.PS.15, 6.PS.16, 6.RP.5, 6.CM.4, 6.CM.10, 6.CN.4, 6.N.6, 6.N.7, 6.R.1, 6.R.5, 6.R.7, 6.G.2, 6.G.3, 6.G.11 Homework Check: The answers to last night s homework will be placed on the overhead. Students will be allowed five minutes to check their answers and discuss with their neighbor any problems they had. I will be walking around the room to see if each student made a conscious effort to complete the assignment. Opening Activity: Instruct Students: o Build a rectangle with an area of 12 square units using a geoboard. Transfer your figures on the grid paper. As a class we will construct a chart of the rectangles the students built by listing the dimensions (length and width) of the rectangle, as well as the area and perimeter. LENGTH WIDTH AREA PERIMETER 1 unit 12 units 12 square units 26 units 2 units 6 units 12 square units 16 units 3 units 4 units 12 square units 14 units 4 units 3 units 12 square units 14 units 6 units 2 units 12 square units 16 units 29
12 units 1 unit 12 square units 26 units Ask Students: o How do we know that we have found all the rectangles that can be made using 12 square units? Explain task to Students: Because we have used all the possible factor pairs as lengths and widths of the rectangle. Investigation: Building Storm Shelters BACKGROUND: From March 12-14, 1993, a fierce winter storm hit the eastern United States form Florida to Maine. Thousands of people were stranded in the snow, far from shelter. A group of 24 Michigan students, who had been hiking in the Smoky Mountains of Tennessee, were among those stranded. To prepare for this kind of emergency, parks often provide shelters at points along major hiking trails. Since shelters are only for emergency use, they are designed to be simple and inexpensive buildings that are easily maintained. INVESTIGATION: The rangers in Great Smoky Mountains National Park want to build several inexpensive storm shelters. The shelters must have 24 square meters of floor space. Suppose that the walls are made of sections one meter wide and cost $125. Tell Students: o Use your geoboards to create the different rectangle shapes. As you complete the investigation worksheet, record your findings in a table similar to the one we made in the opening activity, but you will need to add a column for cost. Informal Assessment: o Make sure students are recording the rectangles onto the grid paper that they made on their geoboards, as well as completing the chart. Closure: We will discuss the worksheet as a class. Group Assessment: o I will ask random groups to explain to the class how they arrived at their responses. 30
On their Own: Have student complete the following problem individually: If the park ranger wanted to build storm shelters with 20 square meters of floor space instead of 24 square meters, which design would be the least expensive to build? LENGTH WIDTH AREA PERIMETER COST OF WALLS 1 meter 20 meters 20 square meters 42 meters $5250 2 meters 10 meters 20 square meters 24 meters $3000 4 meters 5 meters 20 square meters 18 meters $2250 5 meters 4 meters 20 square meters 18 meters $2250 10 meters 2 meters 20 square meters 24 meters $3000 20 meters 1 meters 20 square meters 42 meters $5250 The 4 x 5 (or 5 x 4) shelter would be the least expensive to build. Individual Assessment o The activity above will act as an individual assessment. I will collect and grade each student s paper. Homework: Students will complete the Sandbox worksheet. NAME 31
BUILDING A STORM SHELTER INVESTIGATION: The rangers in Great Smoky Mountains National Park want to build several inexpensive storm shelters. The shelters must have 24 square meters of floor space. Suppose that the walls are made of sections one meter wide and cost $125. 1. Use your geoboards to experiment with different rectangular shapes. Sketch each possible floor plan on grid paper. Record your group s data in a table with these column headings: LENGTH WIDTH AREA PERIMETER COST OF WALLS 2. Based on the cost of the wall sections, which design would be the least expensive to build? Describe what that shelter looks like. 3. Which shelter plan has the most expensive set of wall sections? Describe what that shelter would look like. Answer Key 32
BUILDING A STORM SHELTER INVESTIGATION: The rangers in Great Smoky Mountains National Park want to build several inexpensive storm shelters. The shelters must have 24 square meters of floor space. Suppose that the walls are made of sections one meter wide and cost $125. 1. Use your geoboards to experiment with different rectangular shapes. Sketch each possible floor plan on grid paper. Record your group s data in a table with these column headings: LENGTH WIDTH AREA PERIMETER COST OF WALLS LENGTH WIDTH AREA PERIMETER COST OF WALLS 1 meter 24 meters 24 square meters 50 meters $6250 2 meters 12 meters 24 square meters 28 meters $3500 3 meters 8 meters 24 square meters 22 meters $2750 4 meters 6 meters 24 square meters 20 meters $2500 6 meters 4 meters 24 square meters 20 meters $2500 8 meters 3 meters 24 square meters 22 meters $2700 12 meters 2 meters 24 square meters 28 meters $3500 24 meters 1 meters 24 square meters 50 meters $6250 2. Based on the cost of the wall sections, which design would be the least expensive to build? Describe what that shelter looks like. The 4 x 6 (or 6 x 4) shelter is the least expensive to build. The floor plan is the most square-like of the possibilities. The shelter would have the most open and the fewest wall sections. 3. Which shelter plan has the most expensive set of wall sections? Describe what that shelter would look like. The 1 x 24 (or 24 x 1) shelter is the most expensive to build. The floor plan is long and skinny, with the least open space and the most wall sections. 33
Name GRID PAPER 34
NAME Sandbox Worksheet Directions: Complete the following problem. Show your work and give a brief explanation on how you arrived at your solution. PROBLEM: Alyssa is designing a rectangular sandbox. The bottom is to cover 16 square feet. What shape will require the least amount of material for the sides of the sandbox? Explain how you arrived at your solution. 35
NAME Answer Key Sandbox Worksheet Directions: Complete the following problem. Show your work and give a brief explanation on how you arrived at your solution. PROBLEM: Alyssa is designing a rectangular sandbox. The bottom is to cover 16 square feet. What shape will require the least amount of material for the sides of the sandbox? Explain how you arrived at your solution. LENGTH WIDTH AREA PERIMETER 1 feet 16 feet 16 square meters 34 meters 2 meters 8 meters 16 square meters 20 meters 4 meters 4 meters 16 square meters 16 meters 8 meters 2 meters 16 square meters 20 meters 16 meters 1 meters 16 square meters 24 meters The 4 x 4 sandbox would require the least amount of material for the sides because it has the smallest perimeter compared to the other lengths and widths 36
Day 5: Constant Perimeter, Changing Area Objectives: Students will discover that areas of rectangles with a fixed perimeter can vary considerably. Students will construct diagrams and tables to organize data for them to explore maximum and minimum questions relating to the areas for rectangles of fixed perimeter. Standards: NCTM: Algebra, Geometry, Problem Solving, Reasoning and Proof, Communications, Representations, Connection NYS: 6.PS.8, 6.PS.10, 6.PS.13, 6.PS.14, 6.PS.15, 6.PS.16, 6.CM.4, 6.CM.10, 6.CN.4, 6.CN.6, 6.CN.7, 6.R.1, 6.R.5, 6.R.6, 6.R.7, 6.G.2, 6.G.3, 6.G.11 Materials: Geoboards and rubber bands Grid paper Overhead Projector Transparency of Perimeter = 12 chart Fido s Fence: Day One Worksheet Homework Check: Pop homework quiz. Students will transfer their explanations and solutions for their maximum area sandbox problem onto a separate sheet of paper. Opening Activity: Tell Students: o Yesterday we looked at finding the least expensive perimeter for a given area. Today we are going to look at finding the largest area for a given perimeter. o Suppose we make a rectangle with a perimeter of 12. What are the possible rectangles that could be made? Work with your groups and use geoboards to find all the possible rectangles. Record the answers on grid paper. Then find the area of each rectangle. As a class we will compile a list of data in the form of a chart. LENGTH WIDTH PERIMETER AREA 1 unit 5 units 12 units 5 units 2 units 4 units 12 units 8 units 3 units 3 units 12 units 9 units 37
Present the following problem to the class: Fido, our mutt, keeps running away and we need to fence him in. The fencing costs $3 a foot and we have only $72 to spend. How much fencing material can we buy? Fido is a very hyper dog. We need to build the largest possible rectangular pen so he had plenty of room to run. What do the dimensions of the pen need to be? Informal Assessment: o As the groups are working on the assigned tasks I will ensure the students are properly constructing their rectangles and filling in their charts. In addition, I will check that the students are accurately computing the area of each rectangle. Closure: We will discuss the worksheet as a class. Group Assessment: o I will ask random groups to explain to the class how they arrived at their responses. Name 38
FIDO S FENCE: DAY ONE 1. How many feet of fencing are we able to buy? 2. Sketch the possible rectangles on your grid paper. 3. Compile all the data into a chart with the following headings: LENGTH WIDTH PERIMETER AREA 4. What is the maximum area of the Fido s pen? What are the dimensions of the pen that yield the maximum area? 39
Answer Key FIDO S FENCE: DAY ONE 1. How many feet of fencing are we able to buy? 72/3 = 24. We are able to buy 24 feet of fencing 2. Sketch the possible rectangles on your grid paper. 3. Compile all the data into a chart with the following headings: LENGTH WIDTH PERIMETER AREA LENGTH WIDTH PERIMETER AREA 1 foot 11 feet 24 feet 11 square feet 2 feet 10 feet 24 feet 20 square feet 3 feet 9 feet 24 feet 27 square feet 4 feet 8 feet 24 feet 32 square feet 5 feet 7 feet 24 feet 35 square feet 6 feet 6 feet 24 feet 36 square feet 7 feet 5 feet 24 feet 35 square feet 8 feet 4 feet 24 feet 32 square feet 9 feet 3 feet 24 feet 27 square feet 10 feet 2 feet 24 feet 20 square feet 11 feet 1 foot 24 feet 11 square feet 4. What is the maximum area of the Fido s pen? What are the dimensions of the pen that yield the maximum area? The max area is 36 square feet. The length and the width would both be 6 feet. 40
Name GRID PAPER 41
Day 6: Constant Perimeter, Changing Area Objectives: Students will reproduce steps of solving a maximization problem using the TI-84. Students will analyze tables to determine values for specific lists. Students will be able to describe the aspects and values of the scatter plot. Students will describe in their own words the process of using the TI-84 to solve the maximization problem. Standards: NCTM: Algebra, Geometry, Data Analysis and Probability, Problem Solving, Reasoning and Proof, Communications, Representations, Connection NYS: 6.PS.10, 6.PS.11, 6.PS.15, 6.PS.16, 6.CM.4, 6.CM.10, 6.CN.6, 6.CN.7, 6.R.1, 6.R.5, 6.R.7, 6.G.3, 6.S.2, 6.S.4, 6.S.7 Materials: Overhead projector Overhead graphing calculator Graphing calculator for each student Fido s Fence: Day Two Worksheet TI Instructional Worksheet Homework Check: Pop homework quiz. (I don t think students would plan on having a pop homework quiz 2 days in a row). Students will transfer their perimeter charts from the previous evening s homework onto a separate sheet of paper. Opening Activity: Pick-up a graphing calculator, TI-Instructional Worksheet, and Fido s Fence: Day One Worksheet. Also, get your Fido s Fence: Day One worksheet. Tell Students: o Today we are going to work on the same problem as we did the previous day only today we are going to use the calculator to help us arrive at a solution. Main Activity: We will complete the investigation of Fido s Fence as a class. 42
NOTE: The class will be sitting in their cooperative learning groups. We will go through the worksheets as a class. To complete the task and answer the questions on the worksheet the students will do so in their groups. Then we will come back together as a class to discuss their responses. Informal Assessment: o As the groups are working on the assigned tasks I will make sure the students are entering information into their calculator correctly and that they are filling out their worksheets properly. Group Assessment: o When we come back together as a class, (at a couple different times during the lesson) I will ask random groups to explain to the class how they arrived at their responses. Closure: As a class, we will discuss how we would go about completing the following task: o Imagine your friend has asked you how you found the maximum area for Fido s Pen. Write her directions on how you went about finding the maximum area. Homework: Write the steps to find the maximum area of a rectangle with the area 30. Then find the dimensions of the rectangle with the maximum area. 43
TI-INSTRUCTIONAL WORKSHEET 1. Press STAT 1:EDIT and press ENTER. 2. Enter the whole numbers from 1 to 11(the largest possible length) into L1 on your calculator by typing each number and pressing ENTER until all the lengths have been entered. 3. You should have noticed that the number of fencing pieces remaining for the width of the pen can be found by subtracting twice the number used for a length from L2 fencing pieces available. We want to store in L2 the widths that correspond to the lengths in L1. 4. Press to move to the second list. Press to move to the top so that L2 is highlighted. 5. Enter 12 - L1 as the definition for L2. Your display should look like the example on the right. 6. Press ENTER and the column of possible lengths should appear in L2. 7. Press the to move to the third list and then press to move to the top so that L3 is highlighted. 8. Enter 2(L1) + 2(12 L1) as the definition for L3. Your entire list should be equal to 24. 9. Press the to move to the fourth list and then press to move to the top so that L4 is highlighted. 10. Press 2nd [L1] x 2nd [L2] ENTER to enter the expression for L4 that you found in question #2 on Day Two worksheet. 11. Answer question #4 on the Day Two worksheet. 12. A scatterplot is often used to present a visual display of the relationship between two sets of paired data like the length and area measurements. Your calculator can produce a scatterplot display of the numbers in its lists. 13. Press 2nd [STAT PLOT] ENTER. Edit the window so that yours looks like the one at the right. To make a selection move the blinking cursor on top of the desired location and then press ENTER. 44
14. Press Y= and clear any equations in any of the lines. Press GRAPH to view a scatterplot of the pen areas in relation to the pen lengths. 15. We are using comparing L1 (length) to L4 (area). This way when we find the greatest area we are able to look down and see what length is associated with it. Then we can find the width if we know the length and thus we have the dimension of the figure that yields the greatest area. 16. Press ZOOM 9:ZoomStat to get a picture of all of your data on your calculator screen 17. You can view the coordinates of each plotted point by pressing TRACE followed by the left and right arrow keys. 18. Answer questions #6, 7, and 8 on the Day Two worksheet. 45
FIDO S FENCE: DAY TWO 1. We are going to use our data from our table and enter each column into a different list. You will use L1 to store the possible lengths and then calculate values for the corresponding areas. Once you calculate the values, you will store the lengths in L2, the perimeters in L3, and the areas in L4. 2. If L1 = length, then what would the formulas be for the following? a) L2 = width = 24-2L1 = 12 L1 b) L3 = perimeter = 2(L1) + 2(12 L1) c) L4 = area = L1 x L2 3. Go to #1 on the Instructional Guide. 4. Examine the fourth list to find the dimensions of the area of the pen that has the largest possible area. Complete the following sentence to provide a solution to the original question: A rectangle with a length of 6 feet and a width of 6 feet gives the largest possible pen for Fido of 36_ square feet. 5. Return to the #13 on the Instructional worksheet. 6. You used TRACE to move through the data points in the scatterplot. Which point corresponds to the maximum area? What sets it apart from the other points on the plot? 7. What is the length and width for the maximum area? 8. Sketch a diagram of Fido s pen. Label the dimensions. 6 ft A = 36 sq. ft 6 ft 46
Day 7: Area of Parallelograms Objectives: Students will be able to discover relationships between rectangles and parallelograms Students will develop an understanding of the area of parallelograms through self-discovery. Students will be able to calculate the area of parallelograms using the formula they developed given the dimensions of the shape, as well as determine the dimensions of given the area. Students will discover that a parallelogram can be cut to form a rectangle and the area remains constant. Standards: NCTM: Algebra, Geometry, Measurement, Problem Solving, Reasoning and Proof, Communications, Connections, Representation NYS: 6.PS.10, 6.PS.13, 6.PS.14, 6.PS.15, 6.PS.16, 6.CM.4 6.CN.4 6.R.1, 6.R.5, 6.R.6, 6.G.2, 6.G.11 Materials: Calculator Graph Paper Geoboards and rubber bands Geoboard Grid Paper Area of parallelogram worksheet Parallelogram homework worksheet Overhead projector Transparency of anticipatory set Magnetic parallelograms/rectangles Homework Check: Students will be allowed five minutes to discuss their homework in groups of four. I will be walking around the room to check to see if the students made a conscious effort to complete the assignment and to make sure students are on task. I will only answer questions if each group member has the same question, which is, everyone agrees that they do not understand something. Opening Activity: I will have the following figures on the overhead with directions for the students to find the area of the given rectangle and find the area of the parallelogram using graph paper. 47
4 2 5 10 15-2 -4 Ask Students: o What is the area of the rectangle and how did you arrive at your answer? Students will estimate the area by counting the number of squares in both the rectangle and parallelogram and find that the areas both equal 52 square units. Tell Students: o We talked about finding the areas of rectangular figures. Today we will find shortcuts for finding the area of special figures called parallelograms. Main Activity: Part I: o With their cooperative learning groups and using their geoboards, students will be asked to create five parallelograms with different areas. One side of the parallelogram must be parallel to a side of the geoboard. o Students will record their parallelograms on geoboard grid paper, as well as find and record the area of each parallelogram on their parallelograms worksheet. Informal Assessment: o I will observe groups to ensure they have created the parallelograms on their geoboards correctly. In addition I will make sure groups are correctly determining the area of their parallelograms. Part II: o When students have finished recording the area of each parallelogram, they will be asked to find the length of the side that 48
is parallel to a side of the geoboard for each parallelogram. They are to label it as the base. They will then find the length of the base of each parallelogram and record that amount on their worksheet as well. o In addition, they will be asked to determine the height of the parallelogram from the base to the top or highest point of the parallelogram and label that amount as the height. Students will find and record the height of each parallelogram on their parallelogram worksheet. o If any student questions why we measure the height of the highest point and not the slant I will draw and explain to students that when you go to the doctor they always measure your height. Do you ever stand like this: No! They measure your height standing straight up and from your lowest point (feet) to your highest point (head). Like this: o In their cooperative learning groups, students will answer the guided learning questions #1-4 on their parallelogram worksheets. Informal Assessment: o I will walk around the room and make sure students are correctly labeling their figures and see if there are any misconceptions on how to determine the area of a parallelogram. Closing: We will come back together as a class to discuss the student s findings during the activity. We will talk about the student s responses to the questions on their worksheets. 49
Group Assessment: o I will call on a random group member to represent their group s responses to particular questions from the worksheet and to explain how they arrived at their response. I will show the students why their formulas are true using the square and parallelogram magnets as follows: Homework: Students will complete the finding area of parallelograms handout. 50
Name Area of Parallelograms Worksheet Geoboard Paper 51
Name Area of Parallelograms Worksheet Record your data in the table below: Figure Base Height Area 1 2 3 4 5 Answer the following: 1.) What relationship can you find between the three measurements of base, height and area? 2.) Can you predict the data in one column if you know the data in the other two? 3.) Explain how you could find the area of a parallelogram if you know its base and height. 4.) Write a formula using A for area, B for base, and H for height that shows the relationship between these three measurements. 52
Answer Key Record your data in the table below: Figure Base Height Area 1 2 3 4 5 Answers Will Vary Answer the following: 1.) What relationship can you find between the three measurements of base, height and area? The area always is equal to the base multiplied by the height 2.) Can you predict the data in one column if you know the data in the other two? Yes 3.) Explain how you could find the area of a parallelogram if you know its base and height. If you knew the base and the height, you could multiply them together to find the area 4.) Write a formula using A for area, B for base, and H for height that shows the relationship between these three measurements. A = b*h 53
Finding Area of Parallelograms Homework Name Date Use the Formula A = b*h to find the area of each parallelogram. Don t forget to show all of your work! 1. 2. 3 cm 4 cm 10 m 15 m 6 cm 5 m 3. 24 in 26 in 9.3 in 4. Find the base of a parallelogram with a height of 16 meters and an area of 32 m 2. 5. Draw a parallelogram with the area of 30 square meters and label its dimensions. 54
Answer Key 1. 2. 3 cm 4 cm 10 m 15 m 6 cm A = b*h A = b*h A = 6*3 A =5*10 A = 18 cm 2 A = 50 m 2 5 m 3. A = b*h 24 in A = 9.3*24 A = 223.2 in 2 26 in 9.3 in 4. Find the base of a parallelogram with a height of 16 meters and an area of 32 m 2. A = b*h 32 = b*16 32/16 = b 2 m = b 5. Draw a parallelogram with the area of 30 square meters and label its dimensions. 5 m OR OR 3 m 6 m 15 m 2 m 10 m 55
Day 8: Area of Triangles Objectives: Students will develop an understanding of the area of triangles through self-discovery and employing strategies. Students will discover patterns and relationships between triangles to determine a formula to calculate the area of triangles. Students will be able to calculate areas of triangles given relevant side lengths as well as determine the dimensions of a triangle given the area. Standards: NCTM: Algebra, Geometry, Measurement, Problem Solving, Reasoning and Proof, Communications, Connections, Representation NYS: 6.PS.10, 6.PS.13, 6.PS.14, 6.PS.15, 6.PS.16, 6.CM.4 6.CN.4 6.R.1, 6.R.5, 6.R.6, 6.G.2, 6.G.11 Materials: Calculator Grid paper Geoboards and rubber bands Geoboard Grid Paper Area of Triangles Worksheet Overhead projector Transparency of anticipatory set Homework worksheet Homework Check: I will choose students to present one of their answers on the board from last night s homework so that every problem is displayed on the board. Students will check their answers while each student is presenting his or her answer. Homework will be collected and checked. Opening Activity: I will have the following figure on the overhead with directions for the students to find the area of the given triangle using grid paper: 4 2-5 5 56
Ask Students: o What did you determine the area to be? What strategies did you use to find the area? Students will estimate the area by counting the number of squares and find that the area equals 6 square units Tell Students o We are going to complete a task similar to the one we did yesterday to develop a formula for finding the area of a triangle. Main Activity: Part I: o With their cooperative learning groups and using their geoboards, students will be asked to create five triangles with different areas. One side of the triangle must be parallel to a side of the geoboard. o Students will record their triangles on geoboard grid paper, as well as find and record the area of each triangle on their triangle worksheet. Informal Assessment: o I will observe groups to ensure they have created the triangles on their geoboards correctly. In addition I will make sure groups are correctly determining the area of their triangles. Part II: o When students have finished recording the area of each triangle, they will be asked to find the length of the side of each triangle that is parallel to a side of the geoboard. They are to label this as the base. They will then find the length of the base of each triangle and record this on their triangle worksheet. o In addition, they will be asked to determine the height of the triangle from the base to the top or highest point of the triangle and label that amount as the height. Students will find and record the height of each triangle on their triangle worksheet. o In their cooperative learning groups, students will be asked to answer questions #1-4 on their triangle worksheet. Informal Assessment: 57
o I will walk around the room and make sure students are correctly labeling their figures and see if there are any misconceptions on how to determine the area of their triangles. Closing: We will come back together as a class to discuss the student s findings during the activity. We will discuss the group s responses to the questions on their worksheets. Group Assessment: Homework: o I will call on a random group member to represent their group s responses to particular questions from the worksheet and to explain how they arrived at their response. Students will complete the finding area of triangles handout. 58