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|MA5.2 – 11MG Area and Surface Area | Mathematics Stage 5.1 /5.2 Year 9 2014 |

|Summary of Sub Strands |Duration: 3.5 weeks |

|MA5.1-1WM, MA5.1-2WM |Start Date: |

|MA5.1-8MG |Completion Date |

|S5.1 Area and Surface Area |Teacher and Class: |

|Unit overview |Outcomes |Big Ideas/Guiding Questions |

|In this unit, students will calculate the area |MA5.1-1WM uses appropriate terminology, diagrams | |

|of composite shapes (including an annulus) and |and symbols in mathematical contexts | |

|the surface area of right prisms, cylinders. |MA5.1-2WM selects and uses appropriate strategies | |

|They will solve problems related to these |to solve problems | |

|concepts using a variety of real life |MA5.1-8MG calculates the areas of composite shapes, | |

|applications. |and the surface areas of rectangular and triangular | |

| |prisms | |

| |MA5.2-1WM selects appropriate notations and | |

| |conventions to communicate mathematical ideas and | |

| |solutions | |

| |MA5.2-2WM interprets mathematical or real-life | |

| |situations, systematically applying appropriate | |

| |strategies to solve problems | |

| |MA5.2-11MG calculates the surface areas of right | |

| |prisms, cylinders and related composite solids | |

| | |Key Words |

| | |Area, composite shapes, triangles, quadrilaterals, quadrants, semicircles, sectors, symmetry, |

| | |annulus, surface area, right prism, rectangular prism, triangular prism, net, Pythagoras’ theorem, |

| | |surface area, cylinder, composite solid |

|Catholic Perspectives |School Free Design |

|MacKillop College Bathurst is a Catholic faith community, dedicated to the education of young women. |This is a free design area for schools to add local additional areas. This could include: |

|The Mathematics teachers undertake to uphold the ethos and teachings of the Catholic church and to |Context if you prefer the unit overview and context to be separate |

|support the liturgical life of the College. They promote in the classroom a sense of compassion, |School focus for learning – eg blooms taxonomy, solo taxonomy, contemporary learning, habits of |

|respect between students and staff and a positive and supportive learning environment. |mind, BLP (building learning power) |

|Numeracy operates within a variety of social contexts. From a Catholic perspective, numeracy must be |Any specific social and emotional learning which could be embedded into the unit eg enhanced group |

|infused with a vision of the innate dignity of all students, as created in the image and likeness of a|work |

|loving, generous and creating God. Teachers of Mathematics have an obligation to not only teach their | |

|students the skills and knowledge to be numerate, but to teach from a Catholic perspective. Teachers | |

|are called to challenge their students to use the skills and knowledge they have acquired to bring | |

|about social change in the world. | |

|Assessment Overview |

|Pretest on Stage 4 ‘Angles and Two-dimensional Shapes’ and ‘Area of Plane Shapes’. |

|Undertake an investigation such as ‘Work out the surface area of a cylinder which will hold three tennis balls in a row’. |

|Ask questions such as ‘Do solids with the same surface area have the same volume or capacity? |

|Estimate the relative area of each of the states from a map of Australia |

|Answer questions like the following: (a) do rectangles with the same perimeter have the same area? (b) do parallelograms with the same perimeter have the same area? |

|Reg |Content |Teaching, learning and assessment |Resources |

| |At the end of this unit the student should be able to: |Review the area of plane shapes using an activity such as a matching exercise (match the shape with the| |

| |Stage 5.1 - Area and Surface Area |area formula), worksheet, smartboard activity or a pretest. | |

| |Calculate the areas of composite shapes (ACMMG216) |Students calculate the area of a simple shape in their environment, for example, the classroom or a |Measuring tapes, rulers, trundle wheels |

| |calculate the areas of composite figures by dissection into |sports field. | |

| |triangles, special quadrilaterals, quadrants, semicircles and |Then introduce the concept of composite shapes. Students brainstorm a method to calculate the area of | |

| |sectors |the shape and then carry it out using a range of measuring instruments. | |

| |identify different possible dissections for a given composite |At this point, discuss the different ways in which the one shape could be dissected. Use an example in | |

| |figure and select an appropriate dissection to facilitate |their environment. | |

| |calculation of the area (Problem Solving) [pic] |Adjustment | |

| | |Shapes can increase in complexity as students become more confident. |Composite shape templates |

| | |Provide students with a range of composite shapes to explore. They can cut the composite shapes into | |

| | |the component shapes and calculate the area. |(utube – Area of a Composite Figure) |

| | |For example, the shape below can be divided into two semicircles and a square as shown by the dotted | |

| | |line. Its area can then be found using the formulas for a circle and a square. | |

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| | |Include composite shapes such as the one below which require a subtraction method. | |

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| |solve a variety of practical problems involving the areas of | | |

| |quadrilaterals and composite shapes |Practical problems could include the following: | |

| | |‘Fence It’ – imagine you had 40 one metre sections of fencing. What is the largest rectangular area of | |

| | |land you could fence off? | (‘Fence it’). |

| | |The local swimming pool (or school pool if possible) needs to be cleaned. Using measuring instruments, | (problems |

| | |calculate the area of wall that needs to be kept clean. |related to area, perimeter, surface area |

| | |Calculate the cost of laying turf on the school grounds. |and volume). |

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| | |Students should have already discussed the derivation of the formula for area of triangle in Stage 4. | |

| |apply properties of geometrical shapes to assist in finding |Briefly discuss and demonstrate with examples. | |

| |areas, eg symmetry (Problem Solving, Reasoning) [pic] |For example, look at the triangle below which has been formed by cutting a rectangle in half. | |

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| | |Students can see that the area of a triangle is equal to half the area of a rectangle of the same base | |

| | |length and height. | |

| | |Trapeziums can be broken into a rectangle and two triangles (equal triangles in the case of an | |

| | |isosceles trapezium below). | |

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| | |Rhombuses and kites can be split into two identical triangles. | |

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| | |Adjustment | |

| | |Some students could carry out a practical investigation. For example, they could investigate car | |

| | |parking spaces. | |

| | |Car parking bays are based on either rectangles or parallelograms as shown: | |

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| | |Students could investigate: | |

| | |What dimensions of each shape would be needed to provide a suitable size for parking a car. (Remember | |

| | |to allow for clearance between adjacent cars). | |

| | |Which arrangement uses the greatest area. | |

| | |What are the advantages and disadvantages of each arrangement. | |

| | |Students could visit carparks to make measurements and to consider other constraints eg. Number of | |

| | |entrances, turning circles of vehicles, etc. | |

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| | |Allow students to discover a method for calculating the area of an annulus independently, ie.by | |

| | |subtracting the area of the smaller circle from the larger circle. | |

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| | |Adjustment | |

| | |The formula for the area of an annulus could be introduced to the more capable students. | |

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| | |Students could investigate the torus (plural: tori) which is a three dimensional representation of a | |

| |calculate the area of an annulus (Problem Solving) |revolving circle, for example, a doughnut. | |

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| | |Check that students can define what a prism is. Describe and/or show how slices can be cut from a prism| |

| | |so that each one has the same size and shape. The cross-sections formed by slicing the prism are | |

| | |congruent. | |

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| | |Students could be given nets of rectangular prisms and triangular prisms. They could name them and then| |

| | |draw the solid that it represents. | |

| |Solve problems involving the surface areas of right prisms |Students could then measure the necessary edges and calculate the surface area. |Practical Surface Area: Design a Toblerone |

| |(ACMMG218) |The solid could then be constructed and compared with the student’s diagram. |Box |

| |identify the edge lengths and the areas making up the ‘surface|Students bring in a variety of solids from their home environment eg. Food packaging |

| |area’ of rectangular and triangular prisms [pic] [pic] | |sign.pdf |

| |visualise and name a right prism, given its net [pic] [pic] |Adjustment | |

| |recognise whether a diagram represents a net of a right prism |Some students can explore the surface area of more complicated solids. For example, a tissue box with | |

| |(Reasoning) [pic] |an ellipse opening removed on the top. Students could research the area formula for an ellipse to use. | |

| |visualise and sketch the nets of right prisms [pic] | | |

| |find the surface areas of rectangular and triangular prisms, |Assessment | |

| |given their net |Explore the relationship between surface area and volume. Areas of investigation could include fat | |

| |calculate the surface areas of rectangular and triangular |content of French fries, the function of the alveoli in the lungs and sizes of organisms. | |

| |prisms |Investigate the surface area of the human body. Is there a formula to estimate this? | |

| | |Check that students can use Pythagoras’ theorem to find the hypotenuse and a short side in right-angled| |

| | |triangles. This was introduced in Stage 4. Then apply to problems related to triangular prisms. | |

| | |Give students a variety of problems to solve, ensuring that they choose one that is based on a | |

| | |rectangular prism and one that is about a triangular prism. Suggestions include the following: | |

| | |Investigate the cost of packaging foods. What shapes are more economical? | |

| | |What area of canvas is needed to make a tent? Students could construct the tent. | |

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| | | |Body surface area calculator |

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| |apply Pythagoras' theorem to assist with finding the surface |Students are given a closed cylinder (such as a small container of Pringles) and explore in groups how | |

| |areas of triangular prisms (Problem Solving) |they would find the surface area. If possible, they could cut the cylinder into parts as in the | |

| | |following diagram: | |

| |solve a variety of practical problems involving the surface | | |

| |areas of rectangular and triangular prisms |[pic] |Surface area and chocolate (includes |

| | | |activities about packaging chocolate). |

| | |Practical problems could include the following: |

| | |Students can bring their own cans in to find the surface area of the label. This could lead to a |-and-chocolate |

| | |discussion about the overlap of labels and the cost of the label. | |

| | |Investigate the cost of using cylinders for packaging as opposed to other shapes. Discuss the reasons | |

| | |for using cylinders as a packing shape for things like tennis balls and Pringles. | |

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| | |Adjustment | |

| | |Some students could find the surface area of cylindrical solids like those below: | |

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| |Stage 5.2 - Area and Surface Area | | |

| |Students: | |Animation showing derivation of surface |

| |Calculate the surface areas of cylinders and solve related | |area of cylinder formula (utube) |

| |problems (ACMMG217) | | |

| |recognise the curved surface of a cylinder as a rectangle and |Discuss a variety of problems where the cylinder might not have closed ends or only one closed end. For| |

| |so calculate the area of the curved surface [pic] |example, cylindrical cups have one closed end and pipes might have both ends open. | |

| |develop and use the formula to find the surface areas of | | |

| |closed right cylinders: | | |

| |[pic] where [pic] is the length of the radius and [pic] is the| | |

| |perpendicular height [pic] | | |

| | |With students, explore a variety of composite shapes. | |

| | |Encourage students to examine the shape and: | |

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| | |Check the number and type of the surfaces | |

| | |Calculate the area of each surface | |

| |solve a variety of practical problems involving the surface |Develop a system for checking that all surfaces have been counted, especially the ones that cannot be | |

| |areas of cylinders, eg find the area of the label on a |seen in the diagram. | |

| |cylindrical can | | |

| | |Adjustment | |

| | |The complexity of the problems can increase as students become confident. Start with basic composite | |

| | |shapes like the shape below: | |

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| |interpret the given conditions of a problem to determine | | |

| |whether a particular cylinder is closed or open (one end only | | |

| |or both ends) (Problem Solving) [pic] | | |

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| |Solve problems involving surface area for a range of prisms, |Then move onto more demanding shapes. | |

| |cylinders and composite solids (ACMMG242) | | |

| |find the surface areas of composite solids involving right | | |

| |prisms and cylinders | | |

| |solve a variety of practical problems related to surface areas| | |

| |of prisms, cylinders and related composite solids, eg compare | | |

| |the amount of packaging material needed for different objects | | |

| |interpret the given conditions of a problem to determine the | | |

| |number of surfaces required in the calculation (Problem | | |

| |Solving) [pic] | | |

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