1 ESO - UNIT 11 - GRAPHS AND TABLES. PROBABILITY.

1. Unit 11 June
1. CARTESIAN COORDINATES.
Using Cartesian Coordinates we mark a point on a graph by how far along and
how far up it is:
The point (12,5) is 12 units along, and 5 units up.
The left-right (horizontal) direction is commonly called X. Is also called The X-
Coordinate Axis or Abscissa Axis.
The up-down (vertical) direction is commonly called Y. Is also called the Y-
Coordinate Axis or Ordinate Axis.
Put them together on a graph:
Axel Cotón Gutiérrez Mathematics 1º ESO 11.1

2. Unit 11 June
Where they cross over is the "(𝟎𝟎, 𝟎𝟎)" point, called Origin of Coordinates; you
measure everything from there.
It is like we put two Number Lines together, one going left-right, and the other
going down-up.
Axel Cotón Gutiérrez Mathematics 1º ESO 11.2

3. Unit 11 June
As 𝒙𝒙 increases, the point moves further right. When 𝒙𝒙 decreases, the point
moves further to the left. As 𝒚𝒚 increases, the point moves further up. When 𝒚𝒚
decreases, the point moves further down. The coordinates are always written in a
certain order:
• the Horizontal distance first,
• then the Vertical distance.
This is called an "Ordered Pair" (a pair of numbers in a special order)
And usually the numbers are separated by a comma, and parentheses are put
around the whole thing like this:
( 𝟒𝟒, 𝟓𝟓)
Just like with the Number Line, you can also have Negative Values. Negative:
start at zero and head in the opposite direction:
• Negative 𝒙𝒙 goes to the left.
• Negative 𝒚𝒚 goes down.
MATH VOCABULARY: Cartesian Coordinates, Graph, Axis, Origin of Coordinates,
Abscissa, Ordinate, Ordered Pair.
Axel Cotón Gutiérrez Mathematics 1º ESO 11.3

4. Unit 11 June
2. GRAPHS.
In everyday life, many quantities depend on one or more changing Variables.
For example:
• Plant growth depends on sunlight and rainfall.
• Distance travelled depends on speed and time taken.
• Test marks depend on attitude, listening in lectures and doing tutorials (among
many other variables!!).
Graphs can be used to quickly get an idea of how one quantity varies as
another quantity changes.
This graph shows Dan’s journey on his bike. We can see we use points to see where
the graphs changes.
Axel Cotón Gutiérrez Mathematics 1º ESO 11.4

5. Unit 11 June
A Function is a relation between two variables such that for every value of the
first, there is only one corresponding value of the second. We say that the second
variable is a function of the first variable.
The first variable is the Independent Variable (usually 𝒙𝒙), and the second
variable is the Dependent Variable (usually 𝒚𝒚). A good way of presenting a Function is
by Graphical Representation. Graphs give us a visual picture of the function.
Normally, the values of the Independent Variable (generally the 𝒙𝒙 − 𝒗𝒗𝒗𝒗𝒗𝒗 𝒗𝒗𝒗𝒗𝒗𝒗)
are placed on the Horizontal Axis, while the values of the Dependent Variable
(generally the 𝒚𝒚 − 𝒗𝒗𝒗𝒗𝒗𝒗 𝒗𝒗𝒗𝒗𝒗𝒗) are placed on the Vertical Axis.
The following graph shows the present people (in thousands) in a shopping centre
during a day.
The function increases in the intervals (9, 12) and (14, 18).
Axel Cotón Gutiérrez Mathematics 1º ESO 11.5

6. Unit 11 June
The function decreases in the intervals (12, 14) and (18, 24).
The point (12, 3) is a local maximum and the point (14, 2) is a local minimum.
At 09:00 the function has the global minimum and at 18:00 the function has the global
maximum (5000 people).
MATH VOCABULARY: Function, Independent Variable, Dependent Variable, Graphical
Representation, Interval, Maximum, Minimum.
3. STATISTICS.
Statistics is the science of making effective use of data relating to groups of
individuals or experiments. It deals with all aspects of this, including not only the
collection, organization, presentation, analysis and interpretation of data, but also the
planning of the collection of them, in terms of the design of surveys and experiments.
Imagine you have to make a study of the leisure activities in a city. The
Population of this study will be the group of inhabitants of that city. A part of this
group is called Sample. Therefore, the Population is the whole group of units that you
are going to study and a Sample is a subset of the population. Each unit of the
population is called Individual.
The owner of a screw factory wants to make a quality control. He picks up 1 out of
every 100 produced screws and then he analyses them.
The population is the total number of the screws of the factory.
The sample is 1% of the population.
The individuals are each one of the screws.
Statistical Variables are things that we measure, control or manipulate in
research. They can be numbers, words, measurements, observations ... Statistical
Variables can be Qualitative or Quantitative.
Axel Cotón Gutiérrez Mathematics 1º ESO 11.6

7. Unit 11 June
• Qualitative Variable is descriptive information (describes something).
Examples: eyes color, car brands, leisure activities …
• Quantitative Variable is numerical information (numbers). Examples: height,
age, number of televisions in each house …
You can collect data using a Data-Collection Sheet. This one is a tally chart.
Data can also be shown in a Frequency Table:
The number of televisions in each house of my street is shown in the frequency table.
The number of houses in my street are:
Axel Cotón Gutiérrez Mathematics 1º ESO 11.7

8. Unit 11 June
1 + 5 + 12 + 9 + 1 = 28 houses
The number of TVs are:
0 + 5 + 24 + 27 + 4 = 60 TVs
The Relative Frequency is a percentage of the Frequency:
𝑹𝑹𝑹𝑹 =
𝑭𝑭𝑭𝑭𝑭𝑭𝑭𝑭𝒖𝒖𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆
𝑻𝑻𝑻𝑻𝑻𝑻𝑻𝑻𝑻𝑻 𝑵𝑵𝑵𝑵𝑵𝑵𝑵𝑵𝑵𝑵𝑵𝑵 𝒐𝒐𝒐𝒐 𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰 𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰
MATH VOCABULARY: Statistics, Population, Sample, Individual, Statistical Variables,
Qualitative, Quantitative, Data-Collection Sheet, Frequency Table, Relative Frequency.
4. STATISTICS GRAPHS.
You can use a variety of Diagrams and Charts to display data.
A Bar Graph (also called Bar Chart) is a graphical display of data using bars of
different heights.
Axel Cotón Gutiérrez Mathematics 1º ESO 11.8

9. Unit 11 June
It is a really good way to show Relative Sizes: we can see which types of movie
are most liked, and which are least liked, at a glance. We can use bar graphs to show
the relative sizes of many things, such as what type of car people have, how many
customers a shop has on different days and so on.
A Histogram is a graphical display of data using bars of different heights.
It is similar to a Bar Chart, but a histogram groups numbers into Ranges.
Pie Chart: a special chart that uses "pie slices" to show relative sizes of data.
Axel Cotón Gutiérrez Mathematics 1º ESO 11.9

10. Unit 11 June
It is a really good way to show Relative Sizes: it is easy to see which movie
types are most liked, and which are least liked, at a glance. To calculate each sector we
use the Relative Frequency to find the angle of the sector.
We can use a Frequency Polygon to display grouped continuous data:
MATH VOCABULARY: Chart, Bar Graph, Histogram, Ranges, Pie Chart, Frequency
Polygon.
Axel Cotón Gutiérrez Mathematics 1º ESO 11.10

11. Unit 11 June
5. MEASURES OF CENTRAL TENDENCY.
Measures of Central Tendency –Mean, Mode, and Median- can help you
capture, with a single number, what is typical or representative of the data.
• The Mean of a set of data is the total of all the values divided by the number of
values, that is, the average value of all the data in the set. The mean is denoted
by 𝒙𝒙�.
• The Mode is the value that occurs most often. (The mode is sometimes called
the modal value).
• The Median is the middle value when the data is arranged in order, that is, the
value that has exactly half the data above it and half bellow it.
Calculate the mean, mode and median of 8, 3, 8, 7, 5
𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒 = 𝒙𝒙� =
8 + 3 + 8 + 7 + 5
5
=
31
5
= 6.2
𝑀𝑀𝑜𝑜𝑜𝑜𝑜𝑜 = 8, 𝑎𝑎𝑎𝑎 8 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜
For the median, first arrange the numbers in numerical order
3 5 7 8 8
The one in the Middle is 7 → 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 = 7
MATH VOCABULARY: Measures of Central Tendency, Mean, Mode, Median.
Axel Cotón Gutiérrez Mathematics 1º ESO 11.11

12. Unit 11 June
6. PROBABILITY.
Random experiments are those that can’t be predicted with total certainty.
Tossing a coin, rolling a dice, choosing a card from a deck, choosing a marble from a
bag, …
The set of all possible Outcomes (results) of a random experiment is called the
Sample Space (𝑺𝑺). The basic outcomes of an experiment are also known as Elementary
Events. An Event is a set of outcomes, that is, a subset of the Sample Space.
Sample Space: 𝑆𝑆 = {1, 2, 3, 4, 5, 6}
{1}, {2}, {3}, {4}, {5} and {6} are elementary events
𝐴𝐴 = {5, 6} is an event that has two elementary events.
• Sure Event: event that always occurs, whatever the result of the experiment is.
The sample space 𝑺𝑺 is a sure event.
• Impossible Event: event that never occurs, whatever the result of the
experiment is. The empty set ∅ is an impossible event.
We can calculate the Theoretical Probability when an event is fair or unbiased.
Theoretical probability is based on Equally Likely Outcomes
Axel Cotón Gutiérrez Mathematics 1º ESO 11.12

13. Unit 11 June
𝑃𝑃[𝑅𝑅𝑅𝑅𝑑𝑑 𝑏𝑏𝑎𝑎𝑙𝑙𝑙𝑙 𝑖𝑖𝑛𝑛 𝐴𝐴 𝑏𝑏𝑜𝑜𝑜𝑜] =
4
7
= 0.57
𝑃𝑃[𝐺𝐺𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑏𝑏𝑎𝑎𝑙𝑙𝑙𝑙 𝑖𝑖𝑛𝑛 𝐵𝐵 𝑏𝑏𝑜𝑜𝑜𝑜] =
4
7
= 0.57
MATH VOCABULARY: Random Experiments, Sample Space, Elementary Events, Sure
Event, Impossible Event, Theoretical Probability.
Axel Cotón Gutiérrez Mathematics 1º ESO 11.13