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Transcript of IGCSE MATH-
1. The first item the function must satisfy is that it must have either one or two real variables. If another variable is present, it must be a known variable or constant. For example, the function C=2*pi*r is a linear function because only the C and the r are real variables, with the pi being a constant.
2. The second item is that none of the variables can have an exponent or power to them. They cannot be squared, cubed or anything else. All variables must be in the numerator.
3. The third item is that the function must graph to a straight line. Any kind of a curve disqualifies the function.
What Do Linear Functions Look Like Graphed?
Can you guess what they would like when graphed? Yes, they all will have some kind of straight line. The line could be going up and down, left and right, or slanted. But the line is always straight. It doesn't matter where on the graph the function is plotted as long as the line comes out straight.
IGCSE INTERNATIONAL MATH-
Its real life relationship and integration
with other fields of knowledge
Basic activities and techniques integrated with various fields of knowledge used to teach math.
Set is nothing but a collection of distinct things having some common property among them.
Examples of sets:
The objects that make up a set (also known as the elements or members of a set) can be anything: numbers, people, letters of the alphabet, other sets, and so on.
Whole numbers are subset of real numbers.....
Vowels are subset of English alphabets
What is a Subset?
A subset is a set
contained in another set
Principal way of showing sets diagrammatically
A number is a mathematical object used to count, measure, and label.
Natural numbers-All the counting numbers.
Whole numbers-All the counting numbers and zero.
Integers- Zero, natural numbers and their additive inverse numbers.
Rational numbers-A rational number is a number that can be written as a ratio. That means it can be written as a fraction, in which both the numerator (the number on top) and the denominator (the number on the bottom) are whole numbers.
Irrational numbers-An irrational number is any real number that cannot be expressed as a ratio of integers. Irrational numbers cannot be represented as terminating or repeating decimals.
Real numbers-Real numbers are numbers that can be found on the number line. This includes both the rational and irrational numbers.
Percentages are used to express how large or small one quantity is relative to another quantity. A percentage is a number or ratio expressed as a fraction of 100.For example, 45% (read as "forty-five percent") is equal to 45/100, or 0.45. A related system which expresses a number as a fraction of 1,000 uses the terms "per mil" and "millage"
A Half can be written...
As a percentage: 50%
As a decimal: 0.5
As a fraction: 1/2
Money is Not Free to Borrow
People can always find a use for money, so it costs to borrow money.
Speed,distance and time
Integrating with art work and paper presentation of speed,distance and time relationship
Integrating speed,distance and time with PE activity which gives clear understanding of how the three are interdependent
Algebra (from Arabic al-jebr meaning "reunion of broken parts"
Algebra is the language through which we describe patterns. Think of it as a shorthand, of sorts. As opposed to having to do something over and over again, algebra gives you a simple way to express that repetitive process.
Best way to teach this rule and get students to have a hold over it we need to make them practice and test themselves.
In mathematical terms, if x is the length of the side of the field, m is the amount of crop you can grow on a square field of side length 1, and c is the amount of crop that you can grow, then
c= m x^2
If we now do a bit of origami, taking a sheet of A1 paper and then folding it in half (along its longest side), we get A2 paper. Folding it in half again gives A3, and again gives A4 etc. However, the paper is designed so that the proportions of each of the A sizes is the same - that is, each piece of paper has the same shape.
A paper sizes We can pose the question of what proportion this is. Start with a piece of paper with sides x and y with x the longest side. Now divide this in two to give another piece of paper with sides y and x/2 with now y being the longest side. This is illustrated to the right.
The proportions of the first piece of paper are x/y and those of the second are y/(x/2) or 2y/x. We want these two proportions to be equal. This means that
Conic sections come into our story because each of them is described by a quadratic equation. In particular, if (x,y) represents a point on each curve, then a quadratic equation links x and y. We have:
The circle: x^2 + y^2 = 1;
The ellipse: a x^2 + b y^2 = 1;
The hyperbola:ax^2 - by^2 = 1;
The parabola: a x^2 = y
Graphing Linear equations
You plot some selected points for x and then calculate the y value and then plot the points on the graph and then draw a straight line through the points.
When x = 0, y = 0
When x = 10, y = 10
Plot (x,y) = (0,0) on the graph.
Plot (x,y) = (10,10) on the graph.
Draw a straight line between them.
The point (x,y) is x units to the right or left of x = 0, and y units up or down from y = 0.
If x is negative, you move to the left.
If y is negative, you move down.
If x is positive, you move to the right.
If y is positive, you move up.
Graphing quadratic equations
Graphing y = x ^2
We have already discovered
how to graph linear functions.
But what does the graph of
y = x^ 2 look like? To find the
answer, make a data table:
Just for Fun
It's springtime and Irene wants to fill her swimming pool. She doesn't want to stand there all day, but she doesn't want to waste water over the edge of the pool, either. She sees that it takes 25 minutes to raise the pool level by 4 inches. She needs to fill the pool to a depth of 4 feet; she has 44 more inches to go. She figures out her linear equation: 44 inches * (25 minutes/4 inches) is 275 minutes, so she knows she has four hours and 35 minutes more to wait.
What is a Sequence?
A Sequence is a list of things (usually numbers) that are in order.
An auditorium has 20 seats on the first row, 24 seats on the second row, 28 seats on the third row, and so on and has 30 rows of seats. How many seats are in the theater?
To know if it’s arithmetic or geometric, look at the pattern in the problem. There are 20 seats on the first row, 24 on the second row, and 28 on the third row. Each row has four more seats than the one before it. Since we are adding four to each row, this is an arithmetic sequence of numbers that we will be adding up.
The formula for an arithmetic sequence is
The formula for an arithmetic series is
The formula for a geometric series
After knee surgery, your trainer tells you to return to your jogging program slowly. He suggests jogging for 12 minutes each day for the first week. Each week thereafter, he suggests that you increase that time by 6 minutes per day. How many weeks will it be before you are up to jogging 60 minutes per day?
Relations and Functions
A relation is simply a set or collection of ordered pairs. Nothing really special about it. An ordered pair, commonly known as a point, has two components which are the x and y coordinates.
This is an example of an ordered pair.
ordered pair (5,-2) where 5 is the x-coordinate, and -2 is the y-coordinate
As long as the numbers come in pairs, then that becomes a relation. If you can write a bunch of points (ordered pairs) then you already know how a relation looks like. For instance, here we have a relation that has five ordered pairs. Writing this in set notation using curly braces,
On the other hand, function is actually a "special" kind of relation because it follows an extra rule. Just like a relation, a function is also a set of ordered pairs; however, every x-value must be associated to only one y-value.
Recognition of the following function types from the shape of their graphs:
Trains are just one example of things that can be used with linear functions. For instance, to see when two trains travelling at constant rates towards each other meet is a simple linear function.
Source: Boundless. “Applications of Linear Functions and Slope.” Boundless Algebra. Boundless, 27 Jun. 2014. Retrieved 09 Apr. 2015 from https://www.boundless.com/algebra/textbooks/boundless-algebra-textbook/graphs-functions-and-models-2/functions-an-introduction-17/applications-of-linear-functions-and-slope-103-5009/
What Is a Quadratic Function?
A quadratic function is a function or mathematical expression of degree two. Quadratic functions must have two as their highest power. It cannot be lower or higher. If it's lower or higher, it is no longer a quadratic.
How Do They Look Graphed?
Quadratic functions have a certain characteristic that makes them easy to spot
when graphed. They will always graph a certain way. Try graphing the function x^2 by setting up a t-chart with -2, -1, 0, 1, 2 to see what you get. Most likely, you will get something like this.
Quadratics will always graph into some form of parabola.
Refer to the following link for clear understanding of cubic functions.
Absolute Value function
Lets see some examples of linear motion given below:
Parade of the soldiers
Car moving at constant speed
A bullet targeted from the pistol
A man swimming in the straight lane
Train moving in a straight track
Object dropped from a certain height.
Some examples of quadratic motion
1) modeling curves for thing like Roads , teapot spouts,
bends in rivers , electrical wave patterns, roller coaster rides
2) fluids patterns like Air & water flow in curved pipe
In the case of exponential, however, you will be dealing with functions such as g(x) = 2x, where the base is the fixed number, and the power is the variable.
1) Population: Write an equation to model future growth.
2)Half-Life:Find the constant of proportionality for radium-226.
3)Bacteria Growth:A certain strain of bacteria that is growing on your kitchen counter doubles every 5 minutes.
The graph of y = gets closer to the x-axis as the value of x increases, but it never meets the x- axis.
Absolute Value means ...... how far a number is from zero:
"6" is 6 away from zero, and "−6" is also 6 away from zero.
Let's start with the basic sine function, f(t) = sin(t). This function has an amplitude of 1 because the graph goes one unit up and one unit down from the midline of the graph. This function has a period of 2π because the sine wave repeats every 2π units. The graph looks like this:
Properties of Circle
Angles of triangles and polygons
Angles on a straight line
Angles at a point
Angles formed with parallel lines
Interior and Exterior angles
Similarity and congruence--Through online games
Area and volume of similar shapes
The students have to make a pig by following these directions:
1. Choose black, pink, or gray paper.
2. Cut 7 similar circles for the head, body, eyes, snout, and nostrils.
3. Cut 2 congruent triangles for ears.
4. Cut 4 congruent rectangles for legs.
5. Glue all pieces together to assemble one pig.
6. Twirl and attach a pipe cleaner for the tail.
7. Write about your pig.
Angle in a semi-circle
Angle between a tangent and radius
Angle at the center of a circle
Angles in same segment
Angles in opposite segments
Equal chords and perpendicular bisectors
Tangent from external point
Every thing in the world which is man-made is constructed on the basis of shapes that are known to us. Students must have an insight that construction of all things around them are in certain shapes. And shapes are nothing but lines joined at specific
Observe the different shapes.........
The geometry of precision shooting is perhaps the most interesting.
At medium to short distances, gravity is the biggest factor affecting a bullet’s trajectory. Because once the bullet is free of the barrel, gravity begins to pull it toward the ground.
So to hit a target at a long distance, you have to shoot at a “theoretical” point above the target. Let’s use an easier example. A football game. The quarterback throws the ball to a receiver 30 yards away. If he aimed directly at the receiver, the ball would hit the ground before arriving. Without thinking about it, he aims 20 feet above the receiver’s head, but because gravity pulls on the ball once it leaves his hand, it ends up in the receiver’s grasp. The same principal applies to shooting, the bullet’s arc through the air is just much longer and flatter. Also, when a quarterback throws the ball, he’s not looking at the theoretical spot above the receiver’s head, he’s focused on the receiver.
Children should be able to:
Be aware of the space around them.
Develop the skills of using geometric instruments.
Know the importance of neatness and accuracy in mathematics.
Present mathematical data in the form of a picture or diagram.
The United Nations Food and Agriculture Organization still recommends using a (3, 4, 5) triangle to set out right angles in land survey work.
Read more : http://www.ehow.com/info_8247514_real-life-uses-pythagorean-theorem.html
See more at: http://www.ams.org/samplings/feature-column/fcarc-surveying-one#sthash.TTiqhLiP.dpuf
Architecture and Construction
The most obvious application of the Pythagorean Theorem is in the world of architecture and building construction.
Triangulation is a method used for pinpointing a location when two reference points are known. When triangulation is used with a 90-degree angle, the Pythagorean Theorem is used to determine the location.
Crime Scene Investigation
Forensic investigators use the Pythagorean Theorem to determine bullet trajectory. Bullet trajectory shows the path the bullet took before impact. This trajectory tells police the area from which the bullet originated.
Mid point and distance between two points
Gradient of a line
Parallel and perpendicular lines
Equation of a line
X2 - x1 = 18-5
y2 - y1 = 17-3
A ( 5 , 3 ) , B ( 18, 17 )
A ( x1 , y1 ) B ( x2 , y2 )
AB2 = (18 - 5)2 + (17 - 3)2
Using Pythagoras’ Theorem,
AB2 = 132 + 142
17 – 3 = 14 units
18 – 5 = 13 units
Distance between two points.
Introduction of Cartesian coordinate system
influence in mathematics is equally apparent; the Cartesian coordinate system — allowing reference to a point in space as a set of numbers, and allowing algebraic equations to be expressed as geometric shapes in a two-dimensional coordinate system (and conversely, shapes to be described as equations) — was named after him. He is credited as the father of analytical geometry, the bridge between algebra and geometry
We can say that coordinates are those quantities which are used to express the Position of a point in space.
Plotting of points and reading graphs
Coordinate Picture Graphing
Coordinate pictures are a way of helping to reinforce plotting skills with a game of connect-the-dots. Each series of points connects to form a line. The collection of lines reveals a picture.
HOW MUCH LEFT OR RIGHT?
HOW MUCH UP OR DOWN?
Since slope is a measure of the angle of a line from the horizontal, and since parallel lines must have the same angle, then parallel lines have the same slope — and lines with the same slope are parallel.
The other "opposite" thing with perpendicular slopes is that their values are reciprocals; that is, you take the one slope value, and flip it upside down.
We can have some manipulative or hands on activity as shown which gives clear understanding of slope of parallel and perpendicular lines.
Teaching point slope form- a Jingle
Teaching point slope form- full body style
Probability can not be taught on board.
Students need an hands on experience to learn probability of Independent events, combined events and exclusive events.
Some of the activities which can be applied are demonstrated in the following video:
Game of Pig: 2-dice version: Students should be familiar with the one-die version of Pig before playing the 2-dice version. Tossing a one on either die means that the player loses all points collected in that round, if he/she has not stopped before the one is thrown. Any player who is still playing when snake-eyes (double ones) are thrown, loses all points collected thus far in the whole game!
Area,Circumference,surface area and volume of
Area, Surface area, Volume of
Area, Surface area, perimeter, volume
A great activity to teach students why the volume of cone is 1/3rd of a cube?
Teaching about the volume of sphere and cylinder.
Kids tend to dislike word problems for good reason, but with this book – word problems are more engaging! It’s done in a way that students get excited to solve a mystery… they just have to use math to solve it!
Order of Operations Task Cards with QR Codes
Online games/puzzles and quiz
Animated video lessons
Word problem books
Project presentation/paper presentation
Integration with PE activities
Using students body
Science is application of mathematics
Best example is current
and voltage wave
Always integrating math in science whenever possible. For example in deriving formulas, graphing observations, measuring, data collection etc.
Connecting science and math through various topics like Ratio and Proportion, functions, speed, distance, acceleration, percentage etc.
Right and non-right angle triangle
Area of triangle
Clear understanding of right angled and non right angled triangles.
Students need to clearly recognize different types of triangles that are existing. The pattern should be clear in their minds. Then and then students can link triangles to real world scenario.
Teaching triangles phases:
Right angled triangles
Methods to teach triangles:
ICT integration -videos and quiz
Art integration-Craft and drawing
Teaching right triangle:
Solving real life situations using these properties.
Non right triangles
("triangle" and "measure")
is a branch of mathematics that studies relationships involving lengths and angles of triangles.
Trigonometry—a subject whose rules are generally based on right triangles—can still be used to solve a non-right triangle. You need different tools, though.
Identifying the triangle.
Analyzing the given triangle.
Deciding which property is applicable to find unknown parameters.
Similarly analyzing given data and constructing a required triangle.
Linking Real life situations and experiences inside classroom mathematics.
Right from the child's birth the child is surrounded by maths. It only needs a guideline and an inquiring attitude to start recognizing it.
Maths is almost in every facet of life.
Maths is the language of science and engineering.