Ratio of objects 3-Oct-15 Calculate ratios from a real life situation Level 6+ D+

STARTER The string of beads contains 6 red and 12 black beads. This can be written as a ratio RED : BLACK 6 : 12 By dividing both numbers by 6, the ratio can be written in a way that tells you that there are twice as many black beads Red : Black 1 : 2 6 ÷ 6 = 1 12 ÷ 6 = 2

2 cm 5 cm This can be done to any ratio. The rectangle has a width of 2 cm and a height of 5 cm so the ratio of width to height is W : H 2 : 5 1 : ÷ 2 = 1 5 ÷ 2 = 2.5 This rectangle is 2.5 times higher than its width 1.8 cm 5.8 cm This rectangle has a width of 1.8 cm and a height of 5.8 cm so the ratio of width to height is W : H 1.8 : : 3.22 (to 2 dp) 1.8 ÷ 1.8 = ÷ 1.8 = 3.22… This rectangle is 3.22 times higher than its width

4 cm 8 cm W : H 4 : 8 1 : 2 8 ÷ 4 = 2 3 cm 12 cm W : H 3 : 12 1 : 4 This rectangle is 2 times higher than its width 12 ÷ 3 = 4 This rectangle is 4 times higher than its width How many times higher than the width?

1.2 cm 6.6 cm W : H 1.2 : : ÷ 1.2 = cm 7.2 cm W : H 0.9 : : 6 This rectangle is 5.5 times higher than its width 7.2 ÷ 0.9 = 8 This rectangle is 8 times higher than its width How many times higher than the width?

2.7 cm 5.2 cm W : H 2.7 : : ÷ 2.7 = 1.925… 1.2 cm 6.5 cm W : H 1.2 : : 5.42 This rectangle is 1.93 (to 2dp) times higher than its width 6.5 ÷ 1.2 = … This rectangle is 5.42 (to 2 dp) times higher than its width How many times higher than the width?

Do You Look Like a Greek God or Goddess? The Greeks believed that the ratio 1 : 1.62 was attractive. The 1.62 is an approximation for something called the Golden Ratio. 12 cm 19.4 cm W : H 12 : : 1.62 This rectangle is 1.62 (to 2dp) times higher than its width

Everyday Objects Object Length Width Ratio Textbook Passport Whiteboard Computer Building Brick Door Frame

Extension What else can you find that will have this same ratio? Go around the room and see what you can find that has a ratio of 1 : ( or as close to it as you can get)

Ratio of objects 3-Oct-15 Be able to construct a spiral Level 6+ D+

Golden Spiral Video

Creating the golden spiral. Construct a 1cm square, It should be about 7cm from the bottom of the page 7cm 1cm

This square represents the square that you have just constructed. Creating the golden spiral 1 : 1 What is the width : length ratio of the square?

Extend the top edge of the square 1cm to the left and draw the arc with centre at the top left-hand corner of the square and radius 1cm. Creating the golden spiral

Extend the bottom edge of the square 1cm to the left and then complete the rectangle. Creating the golden spiral What is the width : length ratio of the rectangle? 2 : 1

Use these diagrams to complete the next steps of the construction. Creating the golden spiral 2 : 3 What is the width : length ratio of the new rectangle?

The last two diagrams show the next stage in the process. Describe it and then construct the new rectangle. Creating the golden spiral 3 : 5 What is the width : length ratio of this new rectangle?

Describe exactly how this process of creating new rectangles works. Creating the golden spiral The longer side of each rectangle is rotated 90° clockwise about a corner to form an extension to the shorter side. Together with the shorter side, this makes the longer side of a new rectangle. The corner used as centre of rotation is the next corner clockwise each time.

Constructing a spiral Continue with this pattern until the spiral no longer fits onto your page. How large does it get? What do you notice about the lengths of the sides?

Method two

Plenary The spiral formed by the quarter-circle arcs is called the golden spiral. Why is this an appropriate name for the spiral? As more rectangles are created, the ratio of width : length of the rectangles approaches 1 : Ø.)

Golden Spiral

Ratio of Gods and Goddesses 3-Oct-15 Calculate ratios from a real life situation to find out if you are a Greek god or goddess? Level 6+ D+ The Greek goddess called ‘Dike’ later became the Roman god with the name of ‘Justice’. Where have you seen her?

Golden Ratio Activity

Video on Golden Ratio Video on Golden Ratio 2

The Greeks also believed that their gods would have features that are in the ratio 1 : Working with a partner, measure and calculate the ratio of your features by following the instructions on the next three slides

Width of head (w) Top of the head to the chin (h) Start by measuring the distance from the top of your head to the chin (h) and then the width of your head (w). Now divide h by w and write down your answer. h ÷ w = (Your Answer) If your answer is near to 1.62, you are the shape of a Greek God/Goddess!

Pupils to where the lips meet (m) Top of the head to the pupils (p) Measure p and m then divide p by mand write down your answer. p ÷ m = Answer If your answer is near to 1.62, you are the shape of a Greek God/Goddess! Write down the two numbers that you have found so far and calculate the mean average

m p h w n f u e Now that you have the idea, measure these features and then divide. Find the mean average of all your answers h ÷ w = p ÷ m = c ÷ s = e ÷ f = u ÷ n = c s

Measurements a = Top-of-head to chin = ………cm b = Top-of-head to pupil = ……… cm c = Pupil to nosetip = ……… cm d = Pupil to lip = ……… cm e = Width of nose = ……… cm f = Outside distance between eyes = ……… cm g = Width of head = ……… cm h = Hairline to pupil = ……… cm i = Nosetip to chin = ……… cm j = Lips to chin = ……… cm k = Length of lips = ……… cm l = Nosetip to lips = ……… cm

Ratios Now, find the following ratios: a/g = ……… cm b/d = ……… cm i/j = ……… cm i/c = ……… cm e/l = ……… cm f/h = ……… cm k/e = ……… cm

Investigation Find pictures of four people that are considered to be attractive. Measure their features on the photograph and see which is the nearest to a Greek God or Goddess. Put the four people into rank order and then do a survey to see if your people’s opinions are the same as the your ratio results. Find out more about the Golden Ratio A student in my class did the Greek Goddess test on a picture of Cheryl Cole and discovered that she fitted the ratio. Try it.