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Geometry

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EVOLUTION OF GEOMETRY K-7 GRADE8 - 12

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WHY GEOMETRY ? The National Council of Teachers of Mathematics defines geometry as one of the five content strands critical for the middle grades, and states the importance of geometry both for later studies of mathematics as well as for life outside of the mathematics classroom. Existing research has linked students' geometric skills to competence with higher-order mathematics processes, including logical reasoning, application of knowledge in arithmetic and geometry, management of data and procedures, and proportional reasoning. Most high school geometry classes attempt to develop student's ability to reason at higher levels, and these classes are often the only subject in which students are asked to construct proofs based on logic, higher order thinking skills, scientific thinking, and rigor. Because of this, geometry class is often the sole opportunity for students to experience these types of sophisticated thinking and reasoning, both as they apply to mathematics as well as to other domains.

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BIG IDEA Grade K- 7 Properties of two-dimensional shapes and three-dimensional figures Geometric relationships Location and movement Grade Students studying Geometry in high school, further develop analytic and spatial reasoning. They apply what they know about two- dimensional figures to three-dimensional figures in real-world contexts, building spatial visualization skills and deepening their understanding of shape and shape relationships. Connections between transformations of linear and quadratic functions to geometric transformations should be made. Earlier work in linear functions and coordinate graphing leads into coordinate Geometry.

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MISCONCEPTIONS Shape Properties Rectangles are always long Rectangles have two long sides and two short sides Orientation and Rotation of Shapes In the diagram below students may not recognize the second shape as the same square, but instead a diamond or a kite. Perpendicular lines Vertical and horizontal lines are easily recognized as such, and quickly perceived as perpendicular to one another. But the angles between lines oriented differently are not so easily perceived as right angles. In Transformations The mirror line for a reflection does not need to be vertical or horizontal as misconceived. Students can mirror shapes etc. using mirror lines of different slopes. The use of mirrors placed in different positions/orientations demonstrates that mirror lines can be other than vertical or horizontal. Rotations are not always about the origin as one thinks – they can be about ANY point. Practical activities with graph paper and varying centres of rotation can resolve the misconception.

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EXPECTATION BY NCTM Grades 9–12 Expectations: In grades 9–12 all students should– -Analyze properties and determine attributes of two- and three- dimensional objects; -Explore relationships (including congruence and similarity) among classes of two- and three-dimensional geometric objects, make and test conjectures about them, and solve problems involving them. -Establish the validity of geometric conjectures using deduction prove theorems, and critique arguments made by others; -Use trigonometric relationships to determine lengths and angle measures Grades 6–8 Expectations: In grades 6–8 all students should– -Describe sizes, positions, and orientations of shapes under informal transformations such as flips, turns, slides, and scaling; -Examine the congruence, similarity, and line or rotational symmetry of objects using transformations

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HIGH SCHOOL GEOMETRY: WHY IS IT SO DIFFICULT? Lack of proof and proving in earlier school years Lack of understanding of geometry concepts

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GRADE - 8 Measurement C2 draw and construct nets for 3-D objects C3 determine the surface area of right rectangular prisms, right triangular prisms, right cylinders to solve problems. C4 develop and apply formulas for determining the volume of right prisms and right cylinders 3-D Objects and 2-D Shapes C5 draw and interpret top, front, and side views of 3-D objects composed of right rectangular prisms Transformations C6 demonstrate an understanding of tessellation

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The ability to visualize is particularly important in the study of geometry. Visualization “involves thinking in pictures and images, and the ability to perceive, transform and recreate different aspects of the visual-spatial world” (Armstrong 1993, p. 10 )Students need to be able to draw images in their mind of how figures look, and to be able to manipulate these images mentally. In elementary school, geometry is sometimes reduced to learning the names of shapes and measuring a few angles, but they need opportunities to explore geometry: describe shapes using the correct name and description and experience with relationships among shapes, such as two triangles can make a rectangle, leads to an understanding of formulas for finding area of shapes and the concept that shapes that look different can have the same area.(gr.7)

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SURFACE AREA AND VOLUME PRIOR KNOWLEDGE Area Formulas Rectangle: A = l × w Triangle: A = (b × h) ÷ 2 Parallelogram: A = b × h The area is the number of square units needed to cover a surface.

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TESSELLATIONS Tessellations are created using transformations. Three regular polygons and irregular polygons tessellate the plane. Polygons whose interior angles measure 360º or whose interior angles add to a factor of 360º will tessellate the plane.

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GRADE 9 GEOMETRY

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Grade 9 PLO’s C1- Solve problems and justify the solution strategy using circle properties. C2 – Determining the surface area of composite 3-D objects to solve problems. C3 - Demonstrate an understanding of similarity of polygons. C4 - Draw and interpret scale diagrams of 2-D shapes. C5 - Demonstrate an understanding of line and rotation symmetry. C6 – Transformations demonstrate an understanding of tessellation.

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EVOLUTION OF THE TOPIC Grade 6- angles, angle measure, transformations, 3-D and 2-D objects and shapes Grade 7- same as in Grade 6

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SIMILARITIES Corresponding angles are equal. Corresponding sides are enlarged or reduced.

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OCTAGONS And other similar polygons. Scale factor = length on scale diagram / length on original diagram.

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TRANSFORMATION Reflection - object and image - line of reflection Rotation - rotational symmetry -center of rotation - order - angle of rotation

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Circle Properties -Tangent, radius, chord, measure of angles, arc, perpendicular 3-D objects and 2-D shapes. Surface areas

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SURFACE AREA AND VOLUME

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MATH 11

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Apprenticeship and Workplace Mathematics B1. Solve problems that involve two and three right triangles.[CN, PS, T, V] B2. Solve problems that involve scale. [PS, R, T, V] B3. Model and draw 3-D objects and their views. [CN, R, V] B4. Draw and describe exploded views, component parts and scale diagrams of simple 3-D objects. [CN, V]

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Foundation Grade 11 B1. Derive proofs that involve the properties of angles and triangles. [CN, R, V] B2. Solve problems that involve the properties of angles and triangles. [CN, PS, V] B3. Solve problems that involve the cosine law and the sine law, including the ambiguous case. [CN, PS, R ]

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Application of mathematics Shape and space Apprenticeship and workplace Mathematics Geometry Essentials of Mathematics Shape and space Principles of Mathematics Shape and space

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Application of mathematics Shape and space C1 enlarge or reduce a dimensioned object, according to a specified scale. C2 calculate maximum and minimum values, using tolerances, for lengths, areas, and volumes. C3 solve problems involving percentage error when input variables are expressed with percentage errors.

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Apprenticeship and workplace Mathematics Geometry B1. Solve problems by using the sine law and cosine law. B2. Solve problems that involve: triangles quadrilaterals regular polygons. B3. Demonstrate an understanding of transformations on a 2-D shape or a 3-D object.

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Shape and Space Design and Measurement C1 analyse objects shown in “exploded” format C2 draw objects in “exploded” format C3 solve problems involving estimation and cost of materials for building an object when a design is given C4 design an object within a specified budget Essentials of Mathematics

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Principles of Mathematics Shape and Space Transformations B1 describe how vertical and horizontal translations of functions affect graphs and their related equations: − y = f(x − h) − y − k = f(x) B2 describe how compressions and expansions of functions affect graphs and their related equations: − y = af(x) − y = f(kx)

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B3 describe how reflections of functions in both axes and in the line y = x affect graphs and their related equations: − y = f(−x) − y = −f(x) − y = f −1(x) B4 using the graph and/or the equation of f(x), describe and sketch B5 using the graph and/or the equation of f(x), describe and sketch |f(x)| B6 describe and perform single transformations and combinations of transformations on functions and relations

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How to help students to develop understanding of a single geometry concept ? A computer can really help in geometry teaching, since it allows a dynamic, interactive manipulation of a figure. A child can move, rotate, or stretch the figure, and observe what properties stay the same When studying a concept, show correct AND incorrect examples, and in different ways or representations (rotate the pictures upside down etc.!). Students are asked to distinguish between correct and incorrect examples. This will help prevent misconceptions. Ask students to draw correct and incorrect examples of a geometry concept. For example, ask them to draw parallel lines and lines that are not parallel. Tying in with this, ask them to draw a parallelogram and a quadrilateral that is not a parallelogram. Tying in with the previous point, you can ask the students to provide a definition for a concept. This can get them to thinking about which properties in the definition are really necessary and which are not. For example, ask them to define an "equilateral triangle". Allow the students to experiment, investigate, and play with geometrical ideas and figures. For this you could use manipulative, lots of drawing, and computer programs (more on them below). Have each student make his/her own geometry concepts notebook, with examples, non-examples, definitions and other notes or pictures

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References Developing Essential Understanding of Geometry for Teaching Mathematics in Grades 9-12 By Nathalie Sinclair, David Pimm, Melanie Skelin, Rose Mary Zbiek Teaching and learning geometry : issues and methods in mathematical education By Doug French

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