12.5 Drawing a Locus

A locus is a set of points that follow a given rule. The plural of locus is loci. You need a ruler and compass to draw a locus accurately

There are five basic locus theorems (rules). 

 Each theorem will be explained in detail in the following sections under this topic.  Even though the theorems sound confusing, the concepts are easy to understand.

When attempting to solve a locus problem, there are certain steps that should be followed:

reference: http://www.regentsprep.org/Regents/math/geometry/GL1/What.htm

12.4 Enlarging Shapes

When you enlarge a shape, all the lengths of the sides of the shape increase in the same proportion. This is called the scale factor. All the angles in the shape stay the same size.

When you describe an enlargement you must give:

• The scale factor of the enlargement.
• The position of the center of the enlargement

The picture above shows you how an enlarged shape looks like. This one is a triangle. Unlike tessellating shapes, you can enlarge any shapes to any size. 

The opposite of enlargement is reduction.

12.3 Transforming Shapes

You can use a combination of reflections, translations and rotations to transform a shape. You can also describe the transformation that maps an object onto its image.

To describe a reflection you must give: 

• the equation of the mirror line

To describe a translation you must give: 

• the column vector
  
  

To describe a rotation you must give: 

• the centre of rotation

•  the number of degrees of rotation (or fraction of whole turn)

• the direction of the reaction (clockwise or anticlockwise)

12.2 Solving trasnformations problem

 you may want to copy the grid on your book or do the questions on grid paper! :D

12.1 Tessellating Shapes

A tessellation is a pattern made of identical shapes. You can make your own tessellations by fitting copies of a shape together, without gaps or overlaps. 

You say that the shape tessellates, or is a tessellating shape. 

Here are some examples of shapes that tessellate with themselves. 

Here are some examples of shapes that do not tessellate with themselves. There are gaps between the shapes. 

When you make a tessellation you can move the shape by translating, rotating or reflecting it.
For example, here are some of the ways you can tessellate a rectangle. 

Many tesselations are made by repeating a shape and using half-turn rotations of the same shape.

For example, a normal triangle and a half-turn rotation of the same triangle fit together perfectly to make a tessellation like this:

In any tessellations, the sum of the angles at the point where the vertices of the shape meet is 360°.

Look closely the tessellation below:

90° + 90° + 90° + 90° = 360°

Worked Example 12.1

a. Show that the triangle below will tessellate by drawing a tessellation on squared paper.

b. Explain why a regular pentagon will not tessellate. 

Answer:

11.3 Percentage changes

In a value, some increase and decrease may happen. These are best counted using percentages. You can use percentages to describe a change in a quantity. It could be an increase or a decrease. A percentage change is always calculated as a percentage of initial value. Percentage is a simple way to determine an increase and/or decrease

The initial value is 100%. It is important to choose the correct value to be 100%.

Worked example 11.3:
In May 800 people visited a museum. In June 900 people visited. In July, the number was 800 again.
Work out:
a. The percentage increase from May to June.
b. The percentage decrease from June to July.

a. 100% =  800                        [the initial value in may]
    The increase is 100.            [900-800]
    The percentage increase is:
    100 / 800 x 100 = 12.5%      [the fraction 100/800 simplifies to 1/8]

b. 100% = 900                         [the initial value is 900 this time]
    The decrease is 100            [a decrease from 900 to 800]
    The percentage decrease is:
    100 / 900 x 100 = 11.1%      [the fraction 100/900 simplifies to 1/9]
*the percentages are NOT the same.

Here are more examples about percentage changes:

Question 1:
The price of a car was $20 000. In sale, the price decreased by 4%. After the sale it increased by 4%.
Ahmad:"The price after the sale is $20 000 again"
a. What mistake has Ahmad made?
answer:
We must first solve the question before going to Ahmad's mistake.
Decrease: 
= $20 000 - (4/100 x 20 000)
= $20 000 - $800
= $19 200
Increase:
= $19 200+ (4/100 x 19 200)
= $19 200 + $768
= $19 968
Ahmad is incorrect because he answered $20 000 instead of $19 968

b. What is the correct price after the sale?
From what I have counted above, the correct price is $19 968

Question 2:
One week the height of a plant increased from 30 cm to 35 cm.
a. Work out the percentage increase.
answer:
35 - 30 = 5
5/30 x 100 = 17% (rounded off to the nearest whole number)
The increase percentage is 17%

The following week the height increase by 12%.
b. Work out the new height.
answer:
12/100 x 35 = 4.2 cm
4.2 cm + 35 cm
= 39.2 cm


Here is a video about this subchapter.

11.4 Practical Examples

Here are some real-life example uses of percentages. This time, it's more about the transaction that happens among us in our daily life.

• If you buy something and sell it, the differences between the two prices is a profit or a loss.

          It is given as a percentage of the buying price.
          If you buy something for $20 and sell it for $15 you make a loss of $5 or 25%

• When you buy something you may be offered a discount.

          This is a reduction in the price. It is usually given as a percentage.
          If the price is normally $20 and you get a 10% discount, you only pay 18%.

• If a bank helps you to buy an item, you may have to pay back more than you borrow. This is the interest that the bank charges. 

          It is given as a percentage of the cost.
          If a car costs $20 000 and the rate of the interest is 3$, you will pay $20 600.

• If you buy something the price may include a tax. This is called a purchase tax. When you earn money you may have to pay tax on what you earn. This is called income tax.

Let's take a look and see if we can solve the question below!

Worked Example:
A man earns $45 000 in a year.
He can earn $16 000 without paying any tax. He pays 24% tax on anything above $16 000.

a. Work out how much tax he pays.
answer:
$45 000 - $16 000 = 29 000    [This is his taxable income. He pays tax on this amount.
24% of 29 000 = 6 960            [That is 0.24 x 29 000]
He pays $6 960

b. What percentage of his income does he pay in income tax?
answer:
6960 / 45 000 x 100
= 15.5%                                     [45 000 = 100%. This answer is rounded to one decimal place]


Another questions? That'll do!

Question 1:
A restaurant must add 15% tax to the price of a meal.

a. Here are some bill totals before tax is added. Work out the bill after tax is added.
    i. $42.20 ii. $19.50
 answer:
i. $42.20 + (15 / 100 x 42.20)
   = $48. 53
ii. $19.50 + (15 / 100 x 19.50)
   = $22.43

Question 2:

A man invests $4500 in a bank. The bank pays 8% interest.
a. Work out the interest, in dollars.
answer: 
8 / 100 x $4500
= $360
b. Work out the total.
answer:
$4500 + $360
= $4860
Got it? If you still don't get this subchapter please kindly contact the math teacher.

As this is the end of the chapter, I hope Allah eases you in learning this chapter and I hope the posts I made about these are helpful to you.

The video below also explain about this subchapter!! :D

11.2 Comparing different quantities

On daily basis, there are times when you are required to compare different quantities, like which price is cheaper, which bands have the most fans or anything that includes comparing. In this post I am going to show you how to compare different quantities :)

You will often need to compare groups that are different sizes.
Suppose that, in one school, 85 students took an exam and 59 passed. 
In another school, 237 took an exam and 147 passed.

Which school did better? It is hard to say because each school had a different number of students.

The worked example below shows you how to use percentages to help to answer questions like this.


Worked example 11.2:
In school A, 85 students took a mathematics exam and 59 passed.

In school B, 237 took a mathematics exam and 147 passed. 

Which school had a better pass rate?

59 out of 85 = 59 ÷ 85 = 69% [59 ÷ 85 = 0.694... = 69% to the nearest whole number.]

147 out of 237 = 147 ÷ 237 = 62% [147 ÷ 237 = 0.620... = 62% to the nearest whole number] 

The pass rate in school A is better by 7 percentage points. Why? Because the percentage of students who passed the exam in school A is more than the percentage of students who passed the exam in school B. The difference between 62% and 69% is given in "percentage points".

Understand? No? Ok then let's try with another example :-)

Question 1:
There were 270 people in a cinema. There were 168 women and 102 men.
There were 152 people in a theater. There are 78 women and 74 men.

a. Work out the percentage of women in each venue
answer: 
cinema   = 168/270 x 100
             = 62% (rounded off to the nearest whole number)
theater  = 78 /152 x 100
             = 51% (rounded off to the nearest whole number)
from that, we know that there are more women in cinema than there are in theaters.
b. Work out the percentage of men in each venue
answer:
cinema   = 102/270 x 100
             = 28% (rounded off to the nearest whole number)
theater  = 74 /152 x 100
             = 49% (rounded off to the nearest whole number)
from that, we know that there are more women in cinema than there are in theaters.

 Question 2:
This table below shows the results of a survey in a factory.

a. What percentage of men are smokers?
answer: 
12 / 76 x 100  
= 16% (rounded off to the nearest whole number)
b. Compare the percentages of men and women who are non-smokers.
answer: 
men (nonsmokers)     = 64 / 76 x 100
                               = 84% (rounded off to the nearest whole number)
women (nonsmokers) = 32 / 41 x 100
                               = 78% (rounded off to the nearest whole number)
From the data we have, we know that more men are nonsmokers compared to women.

To practice more about this subchapter, I suggest this interactive educational game for you.
http://www.mathplayground.com/balloon_invaders_percent.html
 Enjoy xx

11.1 Using mental methods

Some percentages are easy to find because they are simple fractions like. For example, 50% are easy to find because its just 50/100 and its easily dividable.  Some percentages aren't very easy to find because they aren't simple fractions. For example, 21% is not very easy to find because its not a simple fraction(21/100). There are examples of these on the first page of this unit.

You can use the easy ones to work out more complicated percentages.

You can often do this quite easily. You do not always need a calculator.

Have you ever feel so upset because your favorite stuff are on sale and you don't know how to count those discounts without calculators? No worries!

In this unit I will introduce you a method that can make your life a lot easier. It's called a mental method. This method will help you find the value of something that has complicated percentage.

Hint: 

If you know 10%, you can find any multiple of 10%.

Worked example 11.1:

There are 4600 people in a stadium. 58% are males. How many is that?

100% = 4600

58%   = 50% + 10% - 2% [these are all easy percentage to find]

50%   = 2300 [50% = 1/2]

10%   = 460 [1/10 is easy. Just divide by 10]

1%     = 46 [divide 10% by 10 to find 1%]

58%   = 2300 + 460 - (2 x 46) = 2668 [Do this sum on your head or on paper to practice                                                             your mental methods skill]

*You could have found 50% + 5% + 3%. Is that easier? Show your working with these steps and leave a comment below telling me which one do you think is easier! :)

 Now that I have explained the whole thing about this sub-chapter, I think we should try to answer some questions with this method, shall we? :D

Question 1:

Use the fact above to find:
a. 52% of $78 = ?
answer:
52% of $78 is twice the amount of 26% of 78.
so, we can just count it like this: 
52% of $78 = 2(26% of 78)
                 = 2 x 20.28
                 = $40. 52
b. 13% of 78 kg = ?
answer: 
13% of 78a kg is half the amount of 26% of 78.
so, we can just count it like this: 
13% of 78 kg = (26% of 78) / 2
                 = 20.28 / 2
                 = 10.14 kg
Question 2:
150% of 62 = ?
answer:
150%    = 100% + 50%
           = 62 + (62 / 2)
           = 93
Question 3:
0.5% of 7000 = ?
answer:
0.5%    = 1% / 2
           = (1% of 7000) / 2
           = 70 / 2
           = 35

Here is a game to help your understanding about percentages:
http://www.bgfl.org/bgfl/custom/resources_ftp/client_ftp/ks2/maths/percentages/index.htm
Hope it'll help :)

10.2 Calculating Statistics

Now you can work out several different statistical measures.
In a real situation, you need to decide which one to use.
If you want to measure how spread out a set of measurement is, the range is the most useful statistic.
If you want to find a representative measurement, you need an average. Should it be the mode, the median or the mean? That depends on the particular situation.

Here is a summary to help you which average to choose.

• Choose the mode if you want to know which is the most commonly occurring number.
• The median is the middle value, when the data values are put in order. Half the numbers are greater than the median and half the numbers are less than the median.
• The mean depends on every value. If you change one number you can change the mean.

Worked example 10.2

Here are the ages, in years, of the players in a football team. Work out the average age. Give a reason for your choice of average.

• There are three modes(18, 20, 21). Each has a frequency of 2. Therefore, the mode is not a good choice.
• They are much older and will distort the value. In fact the mean is 22.1 and nine people are younger than this, only two are older. The mean will be affected by the two oldest people.
• The median is 20. Five players are younger than the median and five are older. Therefore the median is the best average to use in this case.

Another example..
The number of hours in a week that teenagers spent on the computer are recorded below. Determine the mean, the mode and the median and decide which average is the best to use in this case.

• Mean: (10+11+12+13+14+15+16+16+16+16+16+16) ÷ 12 = 14.3

clearly mean is not the best average to use in this case since more than half of the teenagers whose data are

recorded above spent more than 14.3 hours a week on the computer.

13.2 hours on the computer in a week.

• Mode: The mode of this data is 16 which is the best average to use.

Because most of the teenagers there spent 16 hours a week on the computer. 

• Median: the middle value of the data above is (15 + 16) ÷ 2 = 15.5 

This could be the a good average to use but mode is the best average to use in this case.

-pristina-

Math: Processing and Presenting Data [10.1]

   10.1 Calculating Statistics

You can use statistics to summarise sets of data.

You can also use them to compare different sets of data.

You should already be able to calculate three different averages: the mode, the median, the mean.

Remember that the range is not an average. It measures how spread out a set of values or numbers is. 

For a large set of data, it is not practical to list every number separately. Instead, you can record the data in a frequency table. 

The mode is the most common value or number.

The median is the middle value, when they are listed in order.

The mean is the sum of all the values divided by the number of values.

The range is the largest value minus the smallest.

A frequency table is any table that records how often (frequently) data values occur.

WORKED EXAMPLE 10.1

Number of Beads	25	30	35	40	45	50
Frequency	34	48	61	30	15	12

The table shows the number of beads on 200 necklaces.

a. Find the mode.

- the mode is the number with the highest frequency. The number of beads of with the highest frequency in this table is 35. Therefore the mode is 35.

b. Find the mean.

- (25 x 34 + 30 x 48 + 35 x 61 + 40 x 30 + 45 x 15 + 50 x 12) ÷ the sum of all the frequencies. The sum of frequencies is: 34 + 48 + 61 + 30 + 15 + 12 = 200. Now that we know sum of frequencies, we can find the mean.

6900 ÷ 200 = 34.5

This is a reasonable answer because it is near the middle of all the possible number of beads.

c. Find the range.

The first step to find the range is to determine the largest and the smallest number of beads. In the table above, the largest number of beads is 50 and the smallest number of beads is 25. So, that'll be:
50 - 25 = 25.
the range is 25.

Another example..
Students spin coins until they get heads. They record their attempts in the table below.

Students	Attempts
Caspar	1
Louise	3
Zoe	7
Alfie	6
Marcus	3
Joe	7
Jim	1
Bethany	9
Alexa	9
Anthony	3
Carly	2
Jack	2
Finn	4
Michelle	3
Tanya	8

a. Find the Mean: To find the mean, we must first count the total amount of attempts which is 68. And we have to count the number of students which is 15. We must then divide 68 by 15.
68 ÷ 15 = 4.5 
So the mean is 4.5

b. Find the Median:
 Median is the middle value when they are listed in orders. So we have to list the number of attempts in order first. 
1,1,2,2,3,3,3,3,4,6,7,7,8,9,9

We have found that the middle value is 3, so the median is 3.

c. Find the Mode:
Mode is the most common value or number. From the data above we know that the most common number is 3. So the mode is 3.

d. Find the Range:
Range is the biggest value subtracted by the smallest value. In this case the biggest value is 9 and the smallest value is 1.
9 - 1 = 8
So the range is 8.

-pristina-