Unit Plan CANDURO! 1 of! 9. Lesson #2 - Geometry. Lesson Plan: Principles of Mathematics, Grade 9, Academic (MPM1D)

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1 Unit Plan CANDURO! 1 of! 9 Lesson #2 - Geometry Lesson Plan: Principles of Mathematics, Grade 9, Academic (MPM1D) Purpose Context Standard Lesson Outcome(s) The purpose of this lesson is for students to derive the formula needed to calculate the surface area of a pyramid (using the definition of pyramid as provided on dictionary.com). Specifically, students will review concepts of measurement, congruency, and of area for 2-dimensional figures (including composite figures), as well as the process of generalizing an algebraic expression. Students will also learn to derive an equation using a 3- dimensional model of a pyramid and will apply this knowledge in calculating the amount of material needed to make modifications to the exterior of such a shape. This lesson would occur near the middle of a unit on Geometry and Measurement and would follow a unit on Linear Relations. Students would have already been introduced to concepts and procedures, such as: determining the maximum area of a rectangle given a perimeter, determining the minimum perimeter of a rectangle given an area, and solving problems involving the areas and perimeters of composite two-dimensional shapes. Students would have also explored the significance of optimal area and the geometric representation of the Pythagorean Theorem, and would have previously developed the formulas for the volume of a pyramid, a cone, and a sphere. Following this lesson would be several lessons focused on using previously derived formulas to solve for surface area and volume of various 3-dimensional shapes (including composite shapes). From THE ONTARIO CURRICULUM, GRADES 9 AND 10: MATHEMATICS, Measurement and Geometry, MPM1D (p.37): Determine, through investigation, the relationship for calculating the surface area of a pyramid. By the end of this lesson, students will be able to: use a 3-dimensional model of a pyramid or the net of a pyramid to derive the equation necessary to calculate the surface area for that pyramid, generalize the above equation using algebraic terms, use such knowledge to determine the amount of material needed to make exterior modifications to a pyramid.

2 Unit Plan CANDURO! 2 of! 9 The assessment activity for this lesson will be as follows: Assessment Activity Students will have previously measured the dimensions of the net they had created during the performance-based task, and would have worked to derive a formula for calculating the surface area of the resulting pyramid. Students must use the formula derived to calculate the surface area of the pyramid they have created and must use such calculations to determine how many materials will be needed to cover the pyramid in paper mâché and to paint it. Students will only be given the exact amount of material which they have calculated that they will need. The calculations of surface area will be marked for correctness and completeness. They will also be marked based on how well the material covers the pyramid. Students will be placed in groups of 4 and will be asked to choose one of the following polygons: triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon or decagon. **Note, if there are more groups than polygon choices, multiple groups will work on the same polygon. ***Students will be provided with a formula sheet with equations for the surface area of all aforementioned shapes. Performance- Based Task Students will be asked to create a net for a pyramid which has the chosen polygon as the base. The net must satisfy the definition for pyramid provided by dictionary.com. Any polygon used to create the net must have equal side lengths. Students will be responsible to measure the dimensions of the pyramid which the net would form and to derive the equation needed to calculate the total surface area of the pyramid. **Note: Students will be given the formula for area of the polygon which they initially chose to work with, as well as the formula for area of a triangle. The goal of this activity is for students to realize that finding the surface area entails finding the sum of the areas of each face of a shape.

3 Unit Plan CANDURO! 3 of! 9 Procedures Formative Assessment Before: -Students will be invited to participate in a discussion about the area of a 2-dimensional figure. To facilitate this discussion, I will draw several shapes on the board and will call several students up to the board to fill in the area of the shapes. I will then have students work in groups of 4 to develop a working definition for area. This definition will be posted on the board, so that it is easily visible by all. -Students will then be asked to contemplate the idea of surface. They will be asked to provide examples of surfaces, with the hope that students will begin to describe 3-dimensional objects. -A brief period of direct instruction will follow, where the idea of the surface area of an object will be clearly defined and added below the definition for area. During: -Students will be given a formula sheet, on which the formulas for surface area of multiple regular polygons would be provided. -Students will be placed in random groups of 4 and will be given the choice of the regular polygon they wish to work with. They will then be required to work in groups to construct a net for a pyramid having their chosen polygon as the base. -Students will be required to measure the dimensions of the pyramid that their net would form, and to derive the equation necessary to compute the surface area of the pyramid. -The intention of the groups was to promote rich discussion and to aid with students who are struggling. If a student continues to struggle, I will hold a brief conference with those students and would use simple 3-dimensional shapes to help them realize that surface area simply involves taking the sum of the areas of all the sides of an object. -Students who have reached a conclusion very quickly will be given several complex objects and asked to calculate the surface area. -Students will be required to write an individual reflection on the process used to determine the equation for surface area for a pyramid. They will be encouraged to express the struggles they faced, and what concepts or ideas aided in clarifying any elements of confusion. -The assessment project will be explained, and materials will be provided for students to build pyramids, based on their previously constructed nets. After (the following class period): -Groups will work together to write a reflection of whether the materials were sufficient in covering the exterior of their pyramid. If not, the group must work together to resolve errors in the calculations used. Groups may work with other groups to proof work or to provide a set of fresh eyes. The teacher will hold a conference with each student as they are working to derive the formula needed to calculate surface area, to ensure they are headed in the correct direction. Students will then submit the formula prior to obtaining materials for their pyramid to ensure they are using the correct formula for the activity. Students will not be permitted to continue with the assessment activity if their work for the performance-based task is not correct.

4 Unit Plan CANDURO! 4 of! 9 Materials Room Setup Standards for Mathematical Practice Reflection Paper and pencil, Formula sheet, Rulers, Calculators, Papier Mâché, Paint of various colours, Paintbrushes. Desks will be arranged in clusters of four to promote a collaborative environment. As well, this arrangement is suitable given that students will be working in groups of four to complete their performance-based task and their assessment project. Students will be permitted to move their desks to a more isolated area, in order to write their individual reflections. (MP1): Despite that students are familiar with concepts as area and surface, they are required to derive a formula to find surface area of a more complex 3-dimensional shape. They are required to persevere in finding the correct solution to the problem given. (MP5): Students must use the tools given (formula sheet and measurement tools) to derive an equation and to solve for surface area. (MP6): Students must be precise in their measurements and calculations in order to obtain the correct amount of material needed to cover the exterior of the pyramid. After completing this lesson plan, I few things that I could improve upon include: -Using a simple shape, such as a cube, to initiate a class discussion on surface area, and have the class derive a formula for surface area of a cube together, -Holding conferences with students after they have derived the formula for surface area, -Eliminating the triangle from the possible polygons to be used as a base, as this results in an easier task, -Have students switch shapes after they have derived a formula, in order to introduce more choice to the lesson.

5 Unit Plan CANDURO! 5 of! 9 Lesson #3 - Geometry Lesson Plan: Principles of Mathematics, Grade 9, Academic (MPM1D) Purpose Context Standard Lesson Outcome(s) The purpose of this lesson is for students to become familiar with equations for surface area and volume of various shapes, and to become comfortable in applying such equations in solving for surface area and volume of real-world objects. Specifically, students will learn how to utilize the formulas provided for various geometric shapes, in order to calculate the surface area and volume of prisms, pyramids, cylinders, cones and spheres. Students will also learn to calculate volume and surface area for composite figures (composed of the aforementioned shapes). Students will use such calculations to compare the volume and surface area of different shapes, and relate such information to real-life situations. This lesson is meant to familiarize students with equations for surface area and volume, and would thus occur near the middle of a unit on Measurement and Geometry and would follow a unit on Linear Relations. Students will have had practice with determining the maximum area of a rectangle given a perimeter, determining the minimum perimeter of a rectangle given an area, and solving problems involving the areas and perimeters of composite twodimensional shape. Students would have also previously explored the significance of optimal area and the geometric representation of the Pythagorean Theorem. This lesson has much emphasis on real-world applications, and is aimed to strengthen students abilities to manipulate and combine equations to solve for a complex composite figure. Following this lesson will be lessons emphasizing the application of geometric relationships. Specifically, students will explore the properties and relationships of the interior and exterior angles of triangles, quadrilaterals, and other polygons and will be invited to make conjectures about geometric relationships. From THE ONTARIO CURRICULUM, GRADES 9 AND 10: MATHEMATICS, Measurement and Geometry, MPM1D (p.37): Solve problems involving the surface areas and volumes of prisms, pyramids, cylinders, cones, and spheres, including composite figures (Sample problem: Break-bit Cereal is sold in a singleserving size, in a box in the shape of a rectangular prism of dimensions 5 cm by 4 cm by 10 cm. The manufacturer also sells the cereal in a larger size; in a box with dimensions double those of the smaller box. Compare the surface areas and the volumes of the two boxes, and explain the implications of your answers). By the end of this lesson, students will be able to: solve problems involving the surface areas and volumes of prisms, pyramids, cylinders, cones, and spheres, including composite figures, use such knowledge to design a building and to calculate the surface area and volume of the building.

6 Unit Plan CANDURO! 6 of! 9 Assessment Activity The assessment for this lesson is as follows: Students are required to design a building using paper and pencil, utilizing all of the following shapes: prisms, pyramids, cylinders, cones and spheres. Students will then be responsible to measure the dimensions of their building designs and to calculate the corresponding surface area and volume of the buildings. Refer to the image below for the rubric of this assessment activity: Performance- Based Task Students will be asked the following question: Given the choice of a cone, cylinder or rectangular prismshaped container, which would you choose to eat popcorn from (with the intended answer being the largest one)?. Students will be provided with real containers in the aforementioned shapes and will be placed in pairs in order to measure the dimensions of their chosen container. The groups will then be responsible to calculate the surface area and volume of the chosen container using the formula sheet given. Students will then discuss which container had the largest/smallest volume and thus which would be the most suitable choice for a hungry customer. **Note: If done in an actual class, I would provide popcorn. ***Refer to Handout #1 below for the formula sheet given.

7 Unit Plan CANDURO! 7 of! 9 Procedures Before: -The lesson will begin with a brief power point reviewing the concepts of surface area and volume (in two and three dimensions). The power point will also include a visual representation of various geometric shapes, accompanied by the formulas for surface area and volume. -Given that concepts, such as surface area and volume, are review, this portion of instruction will be indirect and will involve much student interaction. Students will be invited to discuss what exactly the terms surface area and volume mean and a working definition of such terms will evolve from this discussion. -Students will be called upon, and asked to draw one of the following shapes on the board: prism (of any kind), pyramid, cylinder, cone, and sphere. Students will then be asked to provide the formulas (using the formula sheet given) needed to calculate surface area and volume for each of the shapes. -The teacher will demonstrate how to calculate the surface area and volume of one of the shapes drawn on the board, using dimensions made up by students. Student input will be expected during this demonstration. During: -Students will work with a partner to complete the performance-based task (refer to Performance-Based Task portion of lesson plan for details). -The purpose of placing students in pairs for this activity is to aid those students who are struggling. If a student continues to struggle, I will hold a brief conference with that student in which I will demonstrate how to calculate the surface area and volume of various simple geometric shapes. -Students who have completed the activity early will be given a 3-dimensional composite figure and asked to calculate volume and surface area for that figure. -After the pairs of students have completed calculations, a classdiscussion will be initiated, during which calculations will be shared and compared. Students will then be responsible to write a reflection on the activity, being careful to make note of which container would be best to eat popcorn from. -Instructions for the assessment activity will be given. After (the following class period): -Students will present their assessment projects to the class, and will walk through the calculations that were necessary to measure surface area and volume for their building. -Students will be invited to positively critique the work of others, and will be encouraged to work through any disagreements regarding calculations.

8 Unit Plan CANDURO! 8 of! 9 Formative Assessment Materials Room Setup Standards for Mathematical Practice Reflection The performance-based task will act as the formative assessment for this lesson. Students will be required to record their work for the calculations of surface area and volume for the shape that they chose. This work will be handed in at the end of the activity and will be graded for completeness and correctness. Students must also provide a reflection comparing their surface area and volume calculations to that of other groups, and must make note of which shape would be most beneficial to consumers, and which would be most beneficial for the business selling the product. Containers in the following shapes: cone, cylinder and rectangular prism, Formula sheet, Rulers, Pencils and paper, Calculators, Popcorn. Desks will be arranged individually in rows. For the performance-based task, students will be permitted to move desks in groups of two, given that the task requires students to work with a partner. For the assessment activity, students will be working individually, but will be allowed to share ideas with those around them, and so clusters of three to four will be permitted. (MP1): Students must design a building and learn to break such a complex shape up into manageable sections in order to complete calculations for volume and surface area. (MP3): Students are asked to critique the buildings of others during the presentation portion of the lesson. (MP4): Students are asked to use known shapes to build complex structures. (MP6): Students must attend to precision in measuring shapes and in making calculations in order to arrive at a correct solution. After completing this lesson plan, a few things I have noticed which could be improved upon include: -More shapes should have been used for the performance-based task, to give students more of a variety and to better satisfy the standard for the lesson, -The calculations needed for a composite 3-D figure should have been demonstrated to the class as a whole, given that a large portion of their grade relies on such calculations.

9 Unit Plan CANDURO! 9 of! 9 Handout #1 - Formula Sheet Note: The above handout was given to me while I was a student in Grade 9 Math at St. Mary s College, in Sault Ste. Marie Ontario.