UNIVERSITY HILL SECONDARY SCHOOL

Math 8 Review Package Chapter 8

Math 8 Blk: F

Timmy Harrison

Jan

2013/6/7

This is a unit review package of chapter 8 in Theory and Problems for Mathematics 8 This review Package includes that following elements: Main concepts and skills required to master for this unit What math skills you need to master this unit Example questions Thorough solution for the questions

For Mr. Low

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Chapter 8 Review Summary Timmy, Harrison, Jan Blk:F

Section 8.1 – Isometric Drawings Isometric drawings represent 3D objects in two dimensions. Isometric dot paper is required like the one shown below. The front face of the drawn object is always located at the width. The top view of the isometric drawing is the aerial view looking down at the object

from the top. The right view is what you see at the right of the object. The front view is what you see at the front of the object. Mat plan is the top view that indicates how many cubes there are in each column Learn to draw a mat plan and point of views of an isometric drawing. Learn to draw an isometric drawing based on views or mat plan.

Isometric dot paper front, right and top view of an object

Section 8.2 – Rotating Cubes Horizontal rotation is the rotation about the vertical axis. Vertical rotation is the rotation about the horizontal axis. Clockwise is in the same direction of the hands in a clock. Learn how to rotate an object vertically and horizontally.

Horizontal and vertical axis Section 8.3 -- Tessellations Tessellation is a pattern of identical and repeated shapes that covers a surface

completely without overlapping or creating any gap. Historians have found tessellations as ancient as 4000 BC. A regular tessellation is made up of equal regular polygons. Only three regular polygons tessellate. They are triangles, square and hexagons. This

is because the sum of angles at any point where the regular polygons meet is 360º. The sum of the angles at a point in a tessellation must be 360º

Main Concepts

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Section 8.4 – Transformations Transformations are the movement of geometric figures. There are three types of

transformations: translation, rotation and reflection. Translation is the sliding of the figure in a straight path without turning or rotating it. Rotation is the turning of the figure about a fixed central point. It can be clockwise

or counter clockwise. Reflection is a mirror image of the figure over a line of reflection. Learn to transform a figure with translation, rotation and reflection. Learn to know how a shape has been transformed by looking at the original shape

and the new one. Learn to find the reflection line.

Types of transformations

Section 8.5 – Creating Tessellations Tessellations can be used to create wonderful and amazing works of art. First, a template is required to create a tessellation. Templates are usually created, starting with a regular shape that will tessellate and

then cut off pieces from the one side and move it to the opposite side.

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Chapter 8 Review Summary Timmy, Harrison, Jan Blk:F

Here are all the math skills you might previously knew that are needed in order to master chapter 8: Draw straight lines neatly using utensils such as rulers. Use both isometric dot paper and ordinary dot paper effectively. Understand the direction of clockwise and counterclockwise. Know how a grid paper and the coordinates work. Know what the degrees of angles mean. Know the traits or characteristics of a dice. Know the degree of the interior angles of some simple regular polygons. Able to calculate simple multiplications and divisions accurately. Able to picture 3D objects easily by using your brain. Able to make creative and beautiful art works with drawing utensils.

Required Math Skills

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Chapter 8 Review Package Timmy, Harrison, Jan Blk:F

1. Draw the front, right, and top views of the object below. Also include a mat plan. 2. Draw an isometric drawing based on the mat plan given below.

3 2 1 2 1

3. Based on the views given, draw an isometric drawing.

4. a) Do a horizontal rotation 90ºclockwise to the object below.

b) do a vertical rotation 90ºcounterclockwise to the objet below.

5. What would be the top number and the front number of the dice when it is rotated: a) 180ºon the vertical axis. b) 270ºcounterclockwise on the horizontal axis.

Sample Questions

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6. Which of the following regular polygons can tessellate?

7. Will there be gaps or overlapping when regular pentagons tessellate. Calculate the degree of it. 8. Explain or draw how can2 triangles and 2 decagons tessellate. 9. For the shape on the grid below.

a) translate it 9 units right and 2 units up. b) rotate it 180ºaround the origin. c) Reflect it over the x axis.

10. There is a point on the grid with a coordinate (x,y), describe what kind of transformation occurred for the point to move to: a) (-x,y) b) (-x,-y) c) (x-12,y+3)

11. Do the following questions according to the grid on the right: a) The blue shape is the refection of the black one, find the

reflection line. b) The orange shape is the translated figure of the black one,

describe the translation. 12. Describe the transformations that occurred to the letter “A” in each of the cases below. a) b) c) d)

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Chapter 8 Review Package Timmy Wang, Harrison, Jan Blk:F

1. According to the object, we can picture what we see at the front, right and the top. We know

that the front is always at the width, so the view are: For mat plan, just write how many

cubes are there in each column of cubes when looking down. Write them in the correct box.

2. To draw an object from a mat plan, first draw the top view on the dot paper then add how many

cubes there are in each column according to the mat plan and the top view like indicated in the correct answer.

Step 1 answer: 3 2 2

1 1 3. From the views given in the question, the answer will be.

Also check to see if it is actually correct. 4. In order to rotate an object, first find the horizontal or vertical axis of it according to what

question is asked, and then rotate about the according axis to the direction asked in the question. Vertical rotation is about the horizontal axis and horizontal rotation is about the vertical axis.

a) b)

Solutions for Sample Questions

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5. Note that the two numbers on opposite faces of a typical dice have a sum of 7. a) Front: 5 Top: 6

If the rotation is about the vertical axis, then the top number stays the same. If you rotate it 180ºabout the vertical axis, then number 2 goes to the back leaving 5 at the front because it is the opposite face of number 2 and they add up to 7.

b) Front: 2 Top: 4 270ºcounterclockwise is basically 90ºclockwise, therefore the dice needs to be rotated 90ºabout the horizontal axis where the front number doesn’t change. When the dice is rotated, 6 will go to the place where 3 was, and 3 will go to the bottom. The top is then the opposite face of 3, which is 4.

6. triangle, square, and hexagon.

Remember that angles at a point in tessellations add to 360º. Regular polygons have equal interior angles, so to find if they will tessellate, we just need to know if their interior angles are factors of 360. Cross out those that can’t tessellate: Triangle: interior∠=60º 360÷60=6 Heptagon: interior∠=128.6º 360÷128.6=2.8 Square: interior∠=90º 360÷90=4 Octagon: interior∠=135º 360÷135=2.67 Pentagon: interior∠=108º 360÷108=3.3 Decagon: interior∠=144º 360÷144=2.5 Hexagon: interior∠=120º 360÷120=3 Circle: No interior∠

7. There will be a gap of 36ºwhen pentagons tessellate.

First, we know that an interior angle of a regular pentagon is 108º, now we will find out how many pentagons can fit at one point of the tessellation.

360÷108=3.33333 3 pentagons can fit at one point 108×3=324 the angles add up to 324ºat one point, so there is a gap 360-324=36º the gap is 36º

8. 2 triangles can tessellate with 2 decagons. Note that an interior angle of a decagon is 150ºso the

two shapes can tessellate like this: 360º=150º+150º+ 60º; like the picture shown below.

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9. a) to translate the shape 9 units right and 2 up, just do (x+9,y+2) to all the coordinates of the shape. The translated shape should be the blue one in the grid b) when you rotate a shape 180º, you basically make the coordinates to their opposite numbers, like (-x,-y). However, you can picture the grid rotating and the shape will land on the green one in the fourth quadrant. c) the x-axis is the horizontal one, so when the shape is reflected over the x-axis, the x of the coordinates stays the same, but the y changes to its opposite number, resulting in (x,-y). In this case, the reflected shape is the orange one in the grid.

10. a) As described similarly in 9c, if y stays the same and x changes to its opposite number, the

point is reflected over the y-axis. b) As described in 9b, if x and y of the coordinate change to their opposite numbers, the dot is

rotated 180ºaround the origin. c) We can tell by the coordinate that the point is translated. X is about left or right, -12 means

moved 12 to the left; y is about up or down, 3 means moved 3 up. Therefore the point is translated 12 units left and 3 units up.

11. a) The reflection line is y=1.5. like shown in the figure, corresponding points of the shape all

have the same distance to the line y=1.5 like drawn in the figure. b) For describing the translation, just count or calculate how many steps and to what direction has the shape moved. The answer is 12 units down, and 10 units right.

12. a) rotation (180ºaround the center)

b) reflection (over the line below the first A) c) translation (the A does not change but just move to somewhere else) d) not a transformation (the shape, A changes in size)