Mathematics

1. Mathematics Geometry and Spatial Sense

2. BIG IDEAS: • Shapes of different dimensions and their properties can be described mathematically • There are always many representations of a given shape • New shapes can be created by either combining or dissecting existing shapes • Shapes can be located in space and relocated by using mathematical processes

4. Geometry and Spatial Sense: Grade 8 *sorting quadrilaterals by geometric properties involving diagonals , *constructing circles; *investigating relationships among similar shapes; *determining and applying angle relationships for parallel and intersecting lines; *relating the numbers of faces, edges, and vertices of a polyhedron; *determining and applying the Pythagorean relationship geometrically; *plotting the image of a point on the coordinate plane after applying a transformation

5. The Big Ideas For Geometry 1. Shapes of different dimensions and their properties can be described mathematically

6. The Big Ideas For Geometry 2. There are always many representations of a given shape.

7. The Big Ideas For Geometry 3. New shapes can be created by either combining or dissecting existing shapes.

8. The Big Ideas For Geometry 4. Shapes can be located in space and relocated by using mathematical processes

9. 3D- 3 Dimensional five Platonic solids tetrahedron, cube, octahedron, dodecahedron, icosahedron

10. Polyhedrons determine, through investigation using concrete materials, the relationship between the numbers of faces, edges, and vertices of a polyhedron

11. Overall expectations • demonstrate an understanding of the geometric properties of quadrilaterals and circles and the applications of geometric properties in the real world; • develop geometric relationships involving lines, triangles, and polyhedra, and solve problems involving lines and triangles • represent transformations using the Cartesian coordinate plane, and make connections between transformations and the real world.

12. circle • circle. The points on a plane that are all the same distance from a centre. degree. A unit for measuring angles. For example, one full revolution measures 360º. diameter. A line segment that joins two points on a circle and passes through the centre.

13. construct a circle, given its centre and radius, or its centre and a point on the circle, or three points on the circle

14. Quadrilaterals • sort and classify quadrilaterals by geometric properties, including those based on diagonals, through investigation using a variety of methods

15. Lines • bisector. A line that divides a segment or an angle into two equal parts. A line that divides another line in half and intersects that line at a 90º angle is called a perpendicular Bisector.

16. vertex. The common endpoint of the two line segments or rays of an angle.

17. Lines • parallel lines

18. diagonal • diagonal. A line segment joining two vertices of a polygon that are not next to each other (i.e., that are not joined by one side). • diagonal in a rectangle • diagonal in a pentagon

19. Angles • angle. A shape formed by two rays or two line segments with a common endpoint. • degree. A unit for measuring angles. For example, one full revolution measures 360º. • acute angle. An angle whose measure is between 0º and 90º.

20. Alternate Angles alternate angles. • Two angles on opposite sides of a transversal when it crosses two lines. • The angles are equal when the lines are parallel. • The angles form one of these patterns:

21. solve angle-relationship problems involving Triangles interior angles complementary angles supplementary angles opposite angles parallel lines and transversals alternate angles corresponding angles)

22. Right Angles • solve problems involving right triangles geometrically, using the Pythagorean relationship;

23. Pythagoras' Theorem • Years ago, a man named Pythagoras found an amazing fact about triangles: • If the triangle had a right angle (90°) ... • ... and you made a square on each of the three sides, then ... • ... the biggest square had the exact same area as the other two squares put together!

24. 5x5=25 c2 3x3=9 a2 It is called "Pythagoras' Theorem" and can be written in one short equation: a2 + b2 = c2 4x4=16 b2

25. Definition The longest side of the triangle is called the "hypotenuse", so the formal definition is: In a right angled triangle:the square of the hypotenuse is equal to the sum of the squares of the other two sides. • Example: A "3,4,5" triangle has a right angle in it.

26. Why Is This Useful? If we know the lengths of two sides of a right angled triangle, we can find the length of the third side. (But remember it only works on right angled triangles!) How Do I Use it? Write it down as an equation:

27. Now you can use algebra to find any missing value, as in the following examples: Example: Solve this triangle.

28. Example: What is the diagonal distance across a square of size 1?

29. Triangles determine, through investigation using a variety of tools and strategies ,the angle relationships for intersecting lines and for parallel lines and transversals, and the sum of the angles of a triangle

30. Pythagorean relationship • determine the Pythagorean relationship, through investigation using a variety of tools

31. Transformations

32. Reflection. • reflection. A transformation that flips a shape over an axis to form a congruent shape. A reflection image is the mirror image that results from a reflection. Also called flip.

33. Rotation rotation. A transformation that turns a shape about a fixed point to form a congruent shape. A rotation image is the result of a rotation. Also called turn.

34. Translation • Translation. A transformation that moves every point on a shape the same distance, in the same direction, to form a congruent shape. A translation image is the result of a translation. Also called slide.

35. Rotational Symmetry. • rotational symmetry. A geometric property of a shape whose position coincides with its original position after a rotation of less than 360º about its centre. For example, the position of a square coincides with its square has rotational symmetry.

36. Cartesian coordinate grid.

37. graph the image of a point, or set of points, on the Cartesian coordinate plane after applying a transformation to the original point(s) (i.e., translation; reflection in the x-axis, the y-axis, or the angle bisector of the axes that passes through the first and third quadrants; rotation of 90°, 180°, or 270° about the origin);

38. Geometric Relationships • determine, through investigation using a variety of tools (e.g., dynamic geometry software, concrete materials, geoboard), relationships among area, perimeter, corresponding side lengths, and corresponding angles of similar shapes

39. The Five Platonic Solids construct the five Platonic solids [i.e., tetrahedron, cube, octahedron, dodecahedron, icosahedron], and compare the sum of the numbers of faces and vertices to the number of edges for each solid.).

40. a tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids. an octahedron (plural: octahedra) is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.

41. a dodecahedron ('twelve' 'base', 'seat' or 'face') is any polyhedron with twelve flat faces, but usually a regular dodecahedron is meant: a Platonic solid. It is composed of 12 regular pentagonal faces, with three meeting at each vertex, an icosahedron (twenty ) is a regularpolyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids.

42. Octahedron net tetrahedron net Icosahedron net

44. volume. The amount of space occupied by an object; measured in cubic units, such as cubic centimetres.

45. dynamic geometry software. dynamic geometry software. Computer software that allows the user to explore and analyse geometric properties and relationships through dynamic dragging and animations. Uses of the software include plotting points and making graphs on a coordinate system; measuring line segments and angles; constructing and transforming two-dimensional shapes; and creating two-dimensional representations of three-dimensional objects. An example of the software is The Geometer’s Sketchpad.