MATHS Week 8 Geometry

Multiplication Challenge Multiplication Challenge! You have 5 minutes to fill in as many as you can – good luck!

Answers

Assessment/Target Please stick your target into your folder For those that have got your paper back today – please write a target and hand the sheet into me so I can put your target onto pro-portal

What did we do before half term?

Have you done your directed study? Please have this out so we can go through it

5 9 x2 + 4 6 -2 x3 - 2 13 37 x2 + 7 ALGEBRA REVIEW X a 5m n 2p 4 20m b 3) Multiplying expressions Complete the table below 5) Function machines Find the output for each function machine Find the inputs for the function machine 1) Collecting like terms Simplify these expressions (a) 3m + 5n + 2n + 12m (b) 4p + 6 + p - 2 (c) 8a + b + 5b - a (d) 4y + 3w + y – 7w (e) g – 4h + 8g + 6h (*f) 3ab + 2a + 5ab – 5b X a 5m n 2p 4 20m b 3h 6hp 3p 5 9 x2 + 4 6 -2 x3 - 2 13 37 x2 + 7 4) Expand brackets Expand 3(2y + 1) Expand 5(3m - 4) (*c) Expand 4(2w + 3) + 2(3w + 9) 2) Substitution a = 3, b = 5, c = -2, d = 10 (a) 4b (b) 12d (c) 3c (d) 5a + 2 (e) 3b - 11 (f) 3c + 1 (g) 2c - 6 (h) ab (i) 5² (*j) 4b – 6 2 Objective Mastered I can collect like terms    I can substitute positive values into expressions I can substitute negative values into expressions I can multiply an expression by an integer I can multiply two algebraic expressions I can expand brackets I can use a function machine to find an output when given an input I can use a function machine to find an input when given an output

ALGEBRA REVIEW 4) Formulae Substitution The formula t = v – u a a = 3, b = 5, c = -2, d = 10 (a) 3b – d (b) ac + b (c) b(4a – 3) (d) a(2b + c) 4 4) Formulae The formula t = v – u a is used to calculate time (t). If v = 21, u = 6 and a = 3, calculate what t will be? If v = 14, u = 2 and t = 3, calculate what a is? Expand brackets Expand 3(4y – 5) Expand m(3m + 5) Expand 4(2w – 3) + 2(3w + 9) Expand 5(2y + 1) – 3(3y – 2) Expand (p + 3)(p + 5) Shape algebra Calculate the perimeter and area of this rectangle 3 4n - 1 Objective Mastered I can expand single brackets    I can expand separate brackets I can expand double brackets I can factorise expressions into single brackets I can substitute into expressions I can substitute into formulae I can solve algebra problems in shapes 2) Factorise Factorise 6b – 10 Factorise a² + 3a Factorise 20a – 15ab

What are we going to do this week? Introducing Geometry Use correct terminology Properties of shapes Rotational Symmetry Angles Interior and Exterior angles in polygons

Terminology You have a list of words each Can you explain what they are? You have 5 minutes to fill in as many as you can All these words will be covered in this weeks lessons – your task is to complete ALL explanations by the end of the lesson 2

Think about What is special about…. parallel lines are always the same distance apart perpendicular lines meet at 90

Triangles Think about What is special about …. have 3 sides and 3 angles sum of the angles = 180 Think about What is special about …. a right-angled triangle an obtuse-angled triangle an acute-angled triangle has one angle of 90 has one angle greater than 90 has three angles all less than 90 Think about Why is it not possible for a triangle to have more than one right angle?

Triangles Think about What is special about…. an equilateral triangle an isosceles triangle a scalene triangle has 3 equal sides and 3 equal angles. has 2 equal sides and 2 equal angles. has 3 different sides and 3 different angles. Think about What is each angle in an equilateral triangle?

Labelling angles and shapes 1. What do the little lines mean on the sides of the triangles in the previous slide? 2. How do we refer to the angles in this triangle? 3. … and in this rectangle (notice the letters at the vertices are in order)?

Polygons Think about the names of these polygons…. A quadrilateral has 4 sides. Sum of the angles = 360 A pentagon has 5 sides. A hexagon has 6 sides. An octagon has 8 sides

Quadrilaterals Think about … What is special about these quadrilaterals? Opposite sides equal All sides equal Opposite sides parallel Opposite sides parallel Rectangle All angles are 90⁰ Square All angles are 90⁰ (again – talk about the labelling on these shapes)

Quadrilaterals Think about … What is special about these quadrilaterals? Rhombus Opposite angles equal Opposite sides parallel Opposite sides equal Parallelogram Opposite angles equal Opposite sides parallel All sides equal

Quadrilaterals Think about What is special about these quadrilaterals? Kite Trapezium Two pairs of equal sides One pair of sides parallel One pair of equal angles

Activity - Card match Can you match the picture with the shape and its definition?

Solutions

Solutions

Regular polygons Think about … What does ‘regular’ mean? Think about What is another name for a regular triangle? What is another name for a regular quadrilateral?

Lines of symmetry - Regular Polygons Equilateral Triangle Square Regular Pentagon Regular polygons have lines of symmetry equal to the number of sides/angles that they possess. Regular Regular Hexagon Regular Octagon

Rotational Symmetry

Rotational Symmetry The order of rotational symmetry that an object has is the number of times that it fits on to itself during a full rotation of 360 degrees. Order 1

Rotational Symmetry The order of rotational symmetry that an object has is the number of times that it fits on to itself during a full rotation of 360 degrees. 2 1 Order 1 Order 2

Rotational Symmetry The order of rotational symmetry that an object has is the number of times that it fits on to itself during a full rotation of 360 degrees. 2 1 Order 1 Order 2 3 2 1 Order 3

Rotational Symmetry The order of rotational symmetry that an object has is the number of times that it fits on to itself during a full rotation of 360 degrees. 2 1 Order 1 Order 2 3 3 4 2 1 2 1 Order 4 Order 3

Rotational Symmetry Have a look at the following shapes…

order 4 Eg1. A square It fits on itself 4 times We say that a square has…

order 1 Eg2. A heart shape It fits on itself only once We say that a heart has…

Rotational Symmetry The order of rotational symmetry of a shape is determined by how many times the shape fits onto itself during a 360° turn. Every shape has an order of rotational symmetry, even if it is order 1. Have a look at the following shapes…

What is the order of rotational symmetry? answer

What is the order of rotational symmetry? answer

What is the order of rotational symmetry? answer

What is the order of rotational symmetry? answer

Draw an angle on a piece of paper Measure it (secretly) Now pass it to your partner – do they agree with your measurement? Now draw an angle of 120o – pass it to your partner to measure. Is it accurate enough? (Within 1o?)

Angles on a Straight Line

Q – What is angle a?

Q – What is angle b?

Angles around a point

Q – What is angle c?

Triangles

Angles in a triangle Always add up to 180°

Angles in a Triangle

Q – What is angle o?

Angles within Parallel Lines

Alternate Angles

Corresponding Angles

Vertically Opposite Angles

Co-Interior Angles (Supplementary)

What are angles m and n?

Q – What is angle x?

Q – What is angle x?

Q – What is angle e?

Q – What is angle z? (bit trickier)

Q – What is angle z? (bit trickier)

Q – What are angles g and h? (Slightly trickier)

Q – What are angles a and b?

Activity – Dinky King!

Answers

How can you PROVE that the angles in a triangle add up to 180°?

How can you PROVE that the angles in a quadrilateral add up to 360°?

Look at your answers to Dinky King – they also prove that the angles of a triangle add up to 180°

Draw any quadrilateral. Draw a diagonal on your quadrilateral. b Draw any quadrilateral. Draw a diagonal on your quadrilateral. Label the angles in one of your triangles a, b, c The sum of the angles in the first triangle is 180° so: a + b + c = 180 Label the angles in the other triangle d, e, f The sum of the angles in the second triangle is 180° so: d + e + f = 180 a f c d

= the sum of the angles in ANY quadrilateral = 360° b + e b = a a f f c c + d d So the angles in your quadrilateral are: a, b + e, f, and c + d So therefore: the sum of the angles in a quadrilateral = a + b + e + f + c + d Which rearranged: the sum of the angles in a quadrilateral = a + b + c + d + e + f Substituting a + b + c = 180 and d + e + f = 180 (from 4 and 6) the sum of the angles in a quadrilateral = 180 + 180 the sum of the angles in ANY quadrilateral = 360°

Interior and Exterior angles of polygons This is Mr Red!

Polygon properties Mr Red is going to run around the outside of the polygon. The black arrow shows the way he is facing as he runs. Version 2.0 Copyright © AQA and its licensors. All rights reserved.

Polygon properties Watch the arrow as Mr Red runs around the polygon. Version 2.0 Copyright © AQA and its licensors. All rights reserved.

We will repeat the run but this time watch the arrow carefully. Polygon properties What happened to the arrow as Mr Red ran around the polygon? We will repeat the run but this time watch the arrow carefully. Version 2.0 Copyright © AQA and its licensors. All rights reserved.

The arrow turned through 360°. Polygon properties The arrow turned through 360°. This is the case for running around the outside of any polygon. It does not matter how many sides it has. You will always end up facing the way you started having made a whole turn of 360°. Version 2.0 Copyright © AQA and its licensors. All rights reserved.

Let us look closely at a corner The blue arrow shows the angle turned through as we pass a corner. It is outside the polygon so is called an EXTERIOR angle.

We know that if we go around all the corners of the polygon we will turn through all the exterior angles. We have also seen that we turn through 3600. This shows us that all the exterior angles together of any polygon must add up to 3600.

The sum of the exterior angles of any polygon = 3600

What is a REGULAR polygon? All the sides are equal in length. All the angles are equal in size.

For a regular polygon All the exterior angles are the same. All the exterior angles add up to 360. To find an exterior angle we divide 360 by the number of angles (or sides since it is the same).

What is the exterior angle of a regular dodecagon? Regular so all angles the same Dodecagon 12 sides so 12 exterior angles Share 360 equally between the 12 All of them add up to 360

What is the exterior angle of a regular dodecagon? 3600 ÷ 12 = 300

The INTERIOR angle of a polygon is the angle INSIDE the shape Exterior angle INTERIOR angle

INTERIOR angle Exterior angle We can see from the diagram that the exterior angle and the interior angle at any corner of a polygon together make a straight line or 1800 Exterior angle INTERIOR angle

To find the INTERIOR angle of a regular polygon First find the exterior angle by dividing 360 by the number of angles (sides). Take your answer for the exterior angle from 180 to get the interior angle.

What is the interior angle of a regular 18 sided polygon? Exterior angle is 360 ÷ 18 = 20 So each INTERIOR angle is 180 – 20= 1600

Copy and complete this table for Regular Polygons Number of sides Name Exterior Angle Interior Angle 3 Triangle 1200 4 5 6 8 9 10

Complete your table to show the interior angles of Regular Polygons Number of sides Name Exterior Angle Interior Angle 3 Triangle 1200 600 4 Square 900 5 Pentagon 720 1080 6 Hexagon 8 Octagon 450 1350 9 Nonagon 400 1400 10 Decagon 360 1440

What do the angles of an irregular polygon add up to? Here is an irregular pentagon First we put a dot inside the polygon (somewhere near the middle). First we put a dot inside the polygon (somewhere near the middle) Then we join every corner to the dot with a straight line. Then we join every corner to the dot with a straight line

Each triangle has angles that add up to 1800 We have produced 5 triangles. Each side of the polygon produces a triangle when we join its ends to the dot. 5 sides so 5 triangles Each triangle has angles that add up to 1800 180 180 180 180 180 So all the angles inside the shape add up to 5 x 180.

All the angles add up to 5 x 180 = 900 But this total includes the angles around the point in the middle of the polygon. All the angles add up to 5 x 180 = 900 They are not anything to do with the angles of the polygon so we must take them away from the 900. 180 180 180 Angles round a point total 360. 180 180 So the interior angles must add up to (5x180) – 360 = 5400

To calculate the sum of the interior angles of an irregular polygon Multiply the number of sides by 180 (each side produces a triangle if you join it to a point in the middle). Take away 360 (the angles around the point in the middle which are not interior angles of the polygon). Answer is the sum of the interior angles of the polygon.

What do the angles of an irregular 13 sided figure add up to? 13 x 180 = 2340 2340 – 360 = 1980 The angle sum of a 13 sided shape is 19800

What about a polygon with n sides? What is the exterior angle of a regular polygon with n sides? 360 ÷ n What is the interior angle of a regular polygon with n sides? 180 – (360 ÷ n) What is the angle sum of an irregular n sided polygon? 180xn – 360 (180n-360)

Terminology Recap

Directed Study

Complete the AQA Angles worksheet