logging in or signing up mohamed Attia ( teaching mathematics) mohamedattia12000 Download Post to : URL : Related Presentations : Share Add to Flag Embed Email Send to Blogs and Networks Add to Channel Uploaded from authorPOINT lite Insert YouTube videos in PowerPont slides with aS Desktop Copy embed code: (To copy code, click on the text box) Embed: URL: Thumbnail: WordPress Embed Customize Embed The presentation is successfully added In Your Favorites. Views: 40 Category: Education License: All Rights Reserved Like it (0) Dislike it (0) Added: February 10, 2012 This Presentation is Public Favorites: 0 Presentation Description mathematics for grade 9 Comments Posting comment... Premium member Presentation Transcript The international school of Egypt : The international school of Egypt D.NERMIEN ISMAIL LANGUGE SCHOOL MATHS STAFF 2011-2012RELATIONS & FUNCTIONS: RELATIONS & FUNCTIONS CARTESIAN PRODUCT, DOMAIN , RANGE , CODOMAIN , ARROW DIAGRAM & GRAPHS. MATHS STAFF 2011-2012RELATION (CARTESIAN PRODUCT): RELATION (CARTESIAN PRODUCT) We will learn how to link pairs of objects from two sets and then introduce relations b/w the two objects in the pair. AXB = { ( a,b ): a ε A, b ε B} for non-empty sets A,B otherwise AXB= φ , this is the set of all ordered pairs of elements from A and B. n(AXB)=n(A) Xn (B) AXBXC = {( a,b,c ):a ε A,b ε B,c ε C} is called set of all ordered triplets for non-empty sets. n(AXBXC) =n(A) Xn (B) Xn (C) AX(BXC)=(AXB)XC=AXBXC. AXB,BXA are not equal (not commutative). MATHS STAFF 2011-2012SOME BASIC POINTS: SOME BASIC POINTS Let P,Q be two sets then a relation R from P to Q is a subset of PXQ. Example: If A={ a,b,c,d },B={ p,q,r,s },then which of the following are relations ( i ) R1={( a,p ),( b,r ),( c,s )} (ii) R2 ={( q,b ),( c,s ),( d,r )} Solution: R1is a relation because R1is the subset of AXB but R2 is the not the relation of AXB because ( q,b ) ε R2 but not in AXB.(R2 is not a subset of AXB) Total number of relations that can be defined from a set A to B is the number of possible subsets of AXB. If n(A)=P , n(B)=q , then total number of relations is 2 pq OR total number of subsets of AXB. Example: If R is relation on a finite set having n elements, then the number of relations on A is Answer 2 nxn Inverse Function is a relation from B to A defined by { ( b,a ) : ( a,b ) ε R} MATHS STAFF 2011-2012PowerPoint Presentation: Relations and Functions Relations A relation is a set of inputs and outputs, often written as ordered pairs (input, output). We can also represent a relation as a mapping diagram or a graph. For example, the relation can be represented as: Relations A relation is a set of inputs and outputs, often written as ordered pairs (input, output). We can also represent a relation as a mapping diagram or a graph. For example, the relation can be represented as: {(2, –3), (4, 6), (3, –1), (6, 6), (2, 3)} domain : {2, 3, 4, 6} range: {–3, –1, 3, 6} State the domain and range of the following relation. Is the relation a function? they gave me two points with the same x -value: (2, –3) and (2, 3). Since x = 2 gives me two possible destinations, then this relation is not a function. * If output has one more unmapped element say 5 then output becomes codomain then Range is the subset of codomain Range ={-2,1,2,4} and Codomain ={-2,1,2.4,5} MATHS STAFF 2011-2012PowerPoint Presentation: State the domain and range of the following relation. Is the relation a function? {(–3, 5), (–2, 5), (–1, 5), (0, 5), (1, 5), (2, 5)} domain : {–3, –2, –1, 0, 1, 2} range: {5} this relation is indeed a function. ARROW DIGRAMS are given below : State in each case ,whether it is a function or not ? MATHS STAFF 2011-2012PowerPoint Presentation: QUESTION BANK WITH HINTS OF RELATIONS AND FUNCTIONS ↓ MATHS STAFF 2011-2012PowerPoint Presentation: Q.1 The relation R is defined on the set N as R={ (x,x+5) : x <4 ; x,yε N} Write R in roster form .Write its domain & range. Q.2 If AXB={(p,q),(p,r),(m,q),(m,r)},find A&B. Q.3 If A={x,y,z} and B={1,2}.Find the number of relations from A to B. Q.4 Examine the relation :R={{(2,1),(3,1),(4,1)} and state whether it is a function or not? Q.5 Let A={1,2,3,4,5,6,7,8,9,10}.Define a relation R from A to A by R={(x,y):2x – y=0,where x,y εA}. Write down its domain ,co-domain & range. Q.6 Let a relation R1 on the set of real numbers be defined as (a,b)ε R1↔ 1+ab>0 for all a,bεR. Show that R1 is reflexive & symmetric but not transitive . Q.7 If a={1,2,3,4,6}.Let R be the relation on A defined by {(a,b):a,bεA, b is divisible by a} Write R in roster form, domain & range of R. Q.8 If R is a relation from set A={11,12,13} to set B={8,10,12} defined by y=x-3, then write R -1 . Q.9 Let R be the relation on the set N of natural numbers defined by R={(a,b):a+3b=12,a,bεN} Find R, domain of R & range of R. Q.10 If A={1,2,3},B={1,4,6,9} and R is a relation from A to B defined by ‘x is greater than y’.Find the range of R. MATHS STAFF 2011-2012PowerPoint Presentation: Ans.6 Reflexive (a,a)εR1 [1+a²>0] & symmetric also as ab=ba. (1,1/2)εR1,(1/2,-1)ε R1 . Ans.8 {8,11),(10.13)} Ans.9 R={(9,1),(6,2),3,3)} Ans 10 {1} Ans.7 {(1,1),(1,2)……(1,6),(2,4)….(4,4),(6,6),(3,3),(3,6)} , domain=range ={1,2,3,4,6} Ans.5 R={(1,2),(2,4),(3,6),(4,8),(5,10)} , Domain= {1,2,3,4,5} range ={2,4,6,8,10} ,co-domain = A Ans .3 2 2x3 =2 6 Ans.1 R={(1,6),(2,7),(3,8)} domain={1,2,3},range={6,7,8} SOLUTIONS: MATHS STAFF 2011-2012PowerPoint Presentation: Q.1 Find the domain and range of the function f defined by f(x)= Q.2 Let f(x)=x2 and g(x)=2x+1 be two real functions .Find (f – g)(x),(fg)(x),(f/g)(x) Q.3 If f={(1,1),(2,3),(o,-1),(-1,-3)} be a linear function from Z into Z . Find f(x). Q.4 If f(x) = x 2 , find . Q.6 If f(x)= x 2 – 1/x 2 , then find the value of : f(x) +f(1/x) Q.7 Let A= {-2,-1,0.1,2} and f:A→ Z given by f(x) =x 2 – 2x – 3,find range of f and also pre-image of 6,-3,5. Q.8 Find the domain and range of real valued function f(x)= Q.5 Let f(x)= find f(-1) , f(3). Q.9 Y=f(x)= , then show that x=f(y). Q.10 If f(x) = , show that f[f(x)]. MATHS STAFF 2011-2012PowerPoint Presentation: Ans.1 f(x) is defined real function if 9-x 2 >0 ,so domain=(-3,3) and x 2 =(9 - 1/y 2 ) ,x is defined when 9 – 1/y 2 ≥0,so range=[1/3,∞) Ans.2 (f – g)(x)=x 2 -2x-1, (fg)(x)=2x 3 +x 2 and (f/g)(x)= ,x≠-1/2. Ans.3 Let f(x)=ax+b be a linear function. If (0,-1),(1,1)εf then we Can get a=2,b=-1 so linear function becomes f(x)=2x-1 Ans.4 = 2.2 Ans.5 f(-1)=2,f(3)=10. Ans.7 Range of f={0,5,-3,-4} and no pre-image of 6 ;0,2 are the pre-images of -3;-2 is the pre-image of 5. Ans.8 It is defined when x ≠4 otherwise it becomes 0/0 form(indeterminate form) Domain(f)=R-{4}, range (f)={-1} as f(x) =-1 [x-4=-(4-x)] Ans.6 0 SOLUTIONS : MATHS STAFF 2011-2012Polynomial Functions:: Finding where the function is positive and negative Polynomial Functions: MATHS STAFF 2011-2012Main Goal:: Main Goal: Learning how to make and use sign charts correctly and efficiently. Interval Test # F(x) sign MATHS STAFF 2011-2012Step One – Find the Zeros: Example: f(x) = (x-1)(x+2) (x-1)(x+2) = 0 (x-1) = 0 (x+2) = 0 and x = 1 and x = -2 Step One – Find the Zeros MATHS STAFF 2011-2012Step Two – Arrange Zeros on a Number Line: Step Two – Arrange Zeros on a Number Line -2 1 - infinity infinity MATHS STAFF 2011-2012Step Three – Choose Your Test Numbers: Step Three – Choose Your Test Numbers -2 1 0 3 5 infinity -infinity *When choosing, make sure you pick one from each interval.* MATHS STAFF 2011-2012Step Four – Create Your Chart: Step Four – Create Your Chart Interval (-infinity, -2) (-2, 1) (1, infinity) Test # -5 0 3 f(x) -18 -2 10 Sign - - + MATHS STAFF 2011-2012Tips For Making Your Chart: Tips For Making Your Chart Interval (-infinity, -2) (-2, 1) (1, infinity) Test # f(x) Sign Use your zeros! You already chose these endpoints when you made your number line. MATHS STAFF 2011-2012Tips For Making Your Chart (cont.): Tips For Making Your Chart (cont.) Interval Test # -5 0 3 f(x) -18 -2 10 Sign To find f(x), plug the test # into your original equation- f(x) = (x-1)(x+2) MATHS STAFF 2011-2012Tips For Making Your Chart (cont.): Tips For Making Your Chart (cont.) Interval Test # f(x) -18 -2 10 Sign - - + To decide the sign, look at the sign of f(x) MATHS STAFF 2011-2012Example 2:: Example 2: Step One - Factor f(x) = (x-1)(x+6) MATHS STAFF 2011-2012Example 2 (cont.): Example 2 (cont.) Step Two - Find the zeros (x-1) = 0 (x+6) = 0 x = 1 x = -6 MATHS STAFF 2011-2012Example 2 (cont.): Example 2 (cont.) Step Three – Arrange zeros on a number line -infinity infinity -6 1 MATHS STAFF 2011-2012Example 2 (cont.): Example 2 (cont.) -6 1 0 3 -7 infinity -infinity *Any number from between this interval. Do not have to use these numbers.* Step Four – Choose Your Test Numbers MATHS STAFF 2011-2012Example 2 (Cont.): Example 2 (Cont.) Interval (-infinity, -6) (-6, 1) (1, infinity) Test # -7 0 3 f(x) 8 -6 18 Sign + - + Step Five – Create Your Chart MATHS STAFF 2011-2012Practice Problems: Practice Problems MATHS STAFF 2011-2012Quadratic Formula: Quadratic Formula MATHS STAFF 2011-2012Graphing Quadratic Functions: Graphing Quadratic Functions y = ax 2 + bx + c MATHS STAFF 2011-2012PowerPoint Presentation: All the slides in this presentation are timed. You do not need to click the mouse or press any keys on the keyboard for the presentation on each slide to continue. However, in order to make sure the presentation does not go too quickly, you will need to click the mouse or press a key on the keyboard to advance to the next slide. You will know when the slide is finished when you see a small icon in the bottom left corner of the slide. Click the mouse button to advance the slide when you see this icon. MATHS STAFF 2011-2012Quadratic Functions: Quadratic Functions The graph of a quadratic function is a parabola . A parabola can open up or down. If the parabola opens up, the lowest point is called the vertex. If the parabola opens down, the vertex is the highest point. NOTE: if the parabola opened left or right it would not be a function! y x Vertex Vertex MATHS STAFF 2011-2012Standard Form: y = ax 2 + bx + c The parabola will open down when the a value is negative. The parabola will open up when the a value is positive. Standard Form y x The standard form of a quadratic function is a > 0 a < 0 MATHS STAFF 2011-2012Line of Symmetry: y x Line of Symmetry Line of Symmetry Parabolas have a symmetric property to them. If we drew a line down the middle of the parabola, we could fold the parabola in half. We call this line the line of symmetry . The line of symmetry ALWAYS passes through the vertex. Or, if we graphed one side of the parabola, we could “fold” (or REFLECT ) it over, the line of symmetry to graph the other side. MATHS STAFF 2011-2012Finding the Line of Symmetry: Find the line of symmetry of y = 3 x 2 – 18 x + 7 Finding the Line of Symmetry When a quadratic function is in standard form The equation of the line of symmetry is y = ax 2 + bx + c , For example… Using the formula… This is best read as … the opposite of b divided by the quantity of 2 times a . Thus, the line of symmetry is x = 3. MATHS STAFF 2011-2012Finding the Vertex: Finding the Vertex We know the line of symmetry always goes through the vertex. Thus, the line of symmetry gives us the x – coordinate of the vertex. To find the y – coordinate of the vertex, we need to plug the x – value into the original equation. STEP 1: Find the line of symmetry STEP 2: Plug the x – value into the original equation to find the y value. y = –2 x 2 + 8 x –3 y = –2(2) 2 + 8(2) –3 y = –2(4)+ 8(2) –3 y = –8+ 16 –3 y = 5 Therefore, the vertex is (2 , 5) MATHS STAFF 2011-2012A Quadratic Function in Standard Form: A Quadratic Function in Standard Form The standard form of a quadratic function is given by y = ax 2 + bx + c There are 3 steps to graphing a parabola in standard form. STEP 1 : Find the line of symmetry STEP 2 : Find the vertex STEP 3 : Find two other points and reflect them across the line of symmetry. Then connect the five points with a smooth curve. Plug in the line of symmetry ( x – value) to obtain the y – value of the vertex. MAKE A TABLE using x – values close to the line of symmetry. USE the equation MATHS STAFF 2011-2012PowerPoint Presentation: STEP 1 : Find the line of symmetry Let's Graph ONE! Try … y = 2 x 2 – 4 x – 1 A Quadratic Function in Standard Form y x Thus the line of symmetry is x = 1 MATHS STAFF 2011-2012PowerPoint Presentation: Let's Graph ONE! Try … y = 2 x 2 – 4 x – 1 STEP 2 : Find the vertex A Quadratic Function in Standard Form y x Thus the vertex is (1 ,–3). Since the x – value of the vertex is given by the line of symmetry, we need to plug in x = 1 to find the y – value of the vertex. MATHS STAFF 2011-2012PowerPoint Presentation: 5 –1 Let's Graph ONE! Try … y = 2 x 2 – 4 x – 1 STEP 3 : Find two other points and reflect them across the line of symmetry. Then connect the five points with a smooth curve. A Quadratic Function in Standard Form y x 3 2 y x MATHS STAFF 2011-2012Solving Systems of equations: Solving Systems of equations 1)Graphing 2)Substitution 3)Elimination MATHS STAFF 2011-2012Graphing: Graphing Parallel Lines have no solution. That is,they share no common points. Intersecting lines will share exactly one point, (x,y) MATHS STAFF 2011-2012Substitution: Substitution STEP 1 At least one of the equations must be solved for a variable. x -y = 1 X+y = -9 First solve one equation for either x or y: X = 1+y STEP 2 Now substitute this variable into the other Equation: X + y = -9 (1+y) +y =-9 Solve: 1 + 2y = -9 2y = -10 Y = -5 Now go back to step 1. X = 1 +y or x = 1 + -5 ( x,y ) = (-4,-5) MATHS STAFF 2011-2012Elimination:add/subtract/multiply: Elimination:add/subtract/multiply 4x + y =23 3x – y = 12 Notice the y variables are opposites. This is what you always want to have. Add the two equations together. When you add the variables with opposite coefficients, it will be eliminated. The result is: 7x = 35 X = 5 To find the eliminated variable, substitute 3x – y = 12 3(5) - y =12 Y = 3 (5,3) Does it work if you test it?? MATHS STAFF 2011-2012Coefficients are not opposites: Multiply first: Coefficients are not opposites: Multiply first 4x+3y = -2 4x +2y = 3 There are no opposites, so multiply eq #2 by neg 1 4x + 3y = -2 -4x-2y = -3 Now add! Y = -5 then go find x!! 5x-2y = -10 3x + 6y = 66 Mult eq#1 by 3!! Can you see why? 15x -6y = -30 3x + 6y = 66 then add!! 12x = 36 X = 3 go find y!! Final answers are always ( x,y ) MATHS STAFF 2011-2012Exceptions: Exceptions Parallel lines: no solution 3x-y = -2 3x –y = 0 Mult eqn #2 by - 1 3x – y = -2 -3x + y = 0 If you add , you get: 0 = -2 This is NEVER true so the lines NEVER cross, thus no solution. Same line: infinite solutions 2x –y = 3 4x – 2y = 6 Mult eqn #1 by -2 -4x+2y = -6 4x -2y = 6 If you add, you get: 0 = 0 This is ALWAYS true, so these are actually the same line, thus INFINITE solutions MATHS STAFF 2011-2012Algebraic Fractions: Algebraic Fractions Using 4 Rules MATHS STAFF 2011-2012PowerPoint Presentation: In this lesson we will look at examples of simplifying Algebraic Fractions using the “4 Rules of Fractions.” + - × ÷ Menu MATHS STAFF 2011-2012Addition: Addition Simplify + MATHS STAFF 2011-2012Addition: Addition Simplify + MATHS STAFF 2011-2012Subtraction: Subtraction Simplify - Can you see an alternative, quicker method here? MATHS STAFF 2011-2012Subtraction: Subtraction Simplify 4 - MATHS STAFF 2011-2012Multiplication: Multiplication Simplify × MATHS STAFF 2011-2012Multiplication: Multiplication Simplify × Factorising (2x-6) 3 MATHS STAFF 2011-2012Division: Division Simplify ÷ 2 2 MATHS STAFF 2011-2012Equations: Equations Once you know how to simplify algebraic fractions, you Can solve equations containing them For example MATHS STAFF 2011-2012PowerPoint Presentation: Solve MATHS STAFF 2011-2012PowerPoint Presentation: Solve MATHS STAFF 2011-2012PowerPoint Presentation: Adding with Common Denominators Adding with Different Denominators Multiplying Dividing Fractions MATHS STAFF 2011-2012PowerPoint Presentation: If the base (denominator) is the same … Add straight across the top (numerators) Add/Subtract Common Denominators MATHS STAFF 2011-2012PowerPoint Presentation: Can it be simplified? MATHS STAFF 2011-2012PowerPoint Presentation: 3 MATHS STAFF 2011-2012PowerPoint Presentation: Add the numerators Simplify MATHS STAFF 2011-2012PowerPoint Presentation: Make the bases the same (choose a common denominator) Add/Subtract Different Denominators MATHS STAFF 2011-2012PowerPoint Presentation: Multiply top & bottom by the SAME number. (changes the look not the fraction) 30 MATHS STAFF 2011-2012PowerPoint Presentation: Now the base (denominator) is the same … Add straight across the top (numerators). MATHS STAFF 2011-2012PowerPoint Presentation: Can it be simplified? MATHS STAFF 2011-2012PowerPoint Presentation: Make denominators the same Simplify if you can! Multiply top and bottom by same number Add the numerators MATHS STAFF 2011-2012PowerPoint Presentation: Multiply straight across the top (numerators) Multiply Multiply straight across the bottom (denominators) MATHS STAFF 2011-2012PowerPoint Presentation: Can it be simplified? MATHS STAFF 2011-2012PowerPoint Presentation: 2 MATHS STAFF 2011-2012PowerPoint Presentation: Multiply numerators , multiply denominators Simplify MATHS STAFF 2011-2012PowerPoint Presentation: Divide Flip the second fraction upside down (invert) MATHS STAFF 2011-2012PowerPoint Presentation: Multiply straight across the top (numerators) Multiply straight across the bottom (denominators) MATHS STAFF 2011-2012PowerPoint Presentation: Can it be simplified? MATHS STAFF 2011-2012PowerPoint Presentation: Flip the second fraction Simplify if you can! Multiply numerators , multiply denominators MATHS STAFF 2011-2012PowerPoint Presentation: Algebraic Fractions Adding/Subtracting Amsco Math A, Chapter 19 MATHS STAFF 2011-2012PowerPoint Presentation: Combining with Like Denominators Example: Combine the numerators over the denominator The Answer! Simplify MATHS STAFF 2011-2012PowerPoint Presentation: Combining fractions with Different Denominators Example: Multiply by 1 to get the same denominator Distribute Combine the numerators over the denominator MATHS STAFF 2011-2012PowerPoint Presentation: Combining fractions with Different Denominators Example: Factor the numerator Combine like terms in the numerator Cancel The Answer MATHS STAFF 2011-2012Probability : Probability MATHS STAFF 2011-2012What Is Probability, Hun?: What Is Probability, Hun? Probability is the way of expressing knowledge of belief that an event will occur on chance. Usually used in mathematics. Did You Know? Probability originated from the Latin word meaning approval . MATHS STAFF 2011-2012Things You Will Need To Know: Things You Will Need To Know Experiment: Is a situation involving chance or probability that leads to results called outcomes. Outcome: Is the results of a single trial of an experiment . Event: One or more outcomes in an experiment. MATHS STAFF 2011-2012Experimenting !: Experimenting ! A spinner has 4 equal sides colored: yellow, blue, green, and red. Suppose you spin that spinner and it lands on red. What are the chances of this event having that outcome? The Chances of landing on red are 1 in 4 or ¼. These chances are so because the spinner is divided into 4 and it landed on 1 of the equal sides. MATHS STAFF 2011-2012Experimenting !: Experimenting ! A dice has 6 sides, numbered 1-6. Suppose the dice is rolled and lands on a 2. What are the chances that number was the outcome of this event? The chances of the dice landing on a two are 1 in 6 or 1/6. This is possible to occur because it is a 6 sided die that could only land on 1 of the 6 sides. MATHS STAFF 2011-2012PowerPoint Presentation: Horse Racing Horse Racing consists of wagering money on which horse will finish first, second, or third. It is known respectively as betting to win, betting to place, and betting to show. Horse racing odds at a track are usually displayed on a large board, called the tote board. This board represents the horse racing odds in terms of the amount of money you will receive from betting. For example, if the horse racing odds are given as 3:1 on a particular horse, this means that if you wager $1 for the horse to win, and it comes in first place, you will receive $3. Many tracks have a minimum bet of $2, so the tote board also often includes a number showing how much money you would receive on a $2 bet . MATHS STAFF 2011-2012PowerPoint Presentation: MATHS STAFF 2011-2012Questions: : Questions: 1. Which of the following is an experiment? A) Tossing a coin. B) Rolling a single 6-sided die. C) Choosing a marble from a jar. D) All of the above. 2. Which of the following is an outcome? A) Rolling a pair of dice. B) Landing on red. C) Choosing 2 marbles from a jar. D) None of the above. MATHS STAFF 2011-2012Answers:: Answers: Question 1- D) Question 2- B) MATHS STAFF 2011-2012Circle Theorems: Circle Theorems MATHS STAFF 2011-2012A Circle features…….: A Circle features……. Circumference … the distance around the Circle… … its PERIMETER Diameter … the distance across the circle, passing through the centre of the circle Radius … the distance from the centre of the circle to any point on the circumference MATHS STAFF 2011-2012A Circle features…….: A Circle features……. … a line joining two points on the circumference. … chord divides circle into two segments … part of the circumference of a circle Chord Tangent Major Segment Minor Segment ARC … a line which touches the circumference at one point only From Italian tangere , to touch MATHS STAFF 2011-2012Properties of circles: Properties of circles When angles, triangles and quadrilaterals are constructed in a circle, the angles have certain properties We are going to look at 4 such properties before trying out some questions together MATHS STAFF 2011-2012An ANGLE on a chord: An ANGLE on a chord An angle that ‘sits’ on a chord does not change as the APEX moves around the circumference … as long as it stays in the same segment We say “Angles subtended by a chord in the same segment are equal” Alternatively “Angles subtended by an arc in the same segment are equal” From now on, we will only consider the CHORD, not the ARC MATHS STAFF 2011-2012Typical examples: Typical examples Find angles a and b Imagine the Chord Angle b = 28º Imagine the Chord Angle a = 44º Very often, the exam tries to confuse you by drawing in the chords YOU have to see the Angles on the same chord for yourself MATHS STAFF 2011-2012Angle at the centre: Angle at the centre Consider the two angles which stand on this same chord Chord What do you notice about the angle at the circumference? It is half the angle at the centre We say “ If two angles stand on the same chord, then the angle at the centre is twice the angle at the circumference ” A MATHS STAFF 2011-2012Angle at the centre: Angle at the centre We say “ If two angles stand on the same chord, then the angle at the centre is twice the angle at the circumference ” It’s still true when we move The apex, A, around the circumference A As long as it stays in the same segment 136° 272° Of course, the reflex angle at the centre is twice the angle at circumference too!! MATHS STAFF 2011-2012Angle at Centre: Angle at Centre A Special Case When the angle stands on the diameter, what is the size of angle a? a a The diameter is a straight line so the angle at the centre is 180° Angle a = 90° We say “The angle in a semi-circle is a Right Angle” MATHS STAFF 2011-2012Inscribed Angles: Inscribed Angles MATHS STAFF 2011-2012PowerPoint Presentation: Inscribed angle – An angle where the vertex is on the circle and its arms are chords in the circle. Inscribed angle MATHS STAFF 2011-2012PowerPoint Presentation: Intercepted arc – the part of the circle inside the arms of the angle Intercepted arc MATHS STAFF 2011-2012Lets go to the internet…: Lets go to the internet… Circle Fun MATHS STAFF 2011-2012PowerPoint Presentation: Theorem – An inscribed angle is half of the intercepted arc. MATHS STAFF 2011-2012What is the measure of x?: What is the measure of x? 50° 60 40° 20° 10° 10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 MATHS STAFF 2011-2012What is the measure of y?: What is the measure of y? 50° 60° 40° 20° 10° 10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 MATHS STAFF 2011-2012PowerPoint Presentation: Corollary 1 – If two inscribed angles intercept the same arc, then the angles are congruent. 1 2 MATHS STAFF 2011-2012Lets go to the internet…: Lets go to the internet… Circle Fun MATHS STAFF 2011-2012PowerPoint Presentation: Corollary 2 – An angle inscribed in a semicircle is a right angle. MATHS STAFF 2011-2012What is x?: What is x? 30° 40° 70° 90° 110° 140° 150° 10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 MATHS STAFF 2011-2012What is y?: What is y? 30° 40° 70° 90° 110° 140° 150° 10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 MATHS STAFF 2011-2012What is z?: What is z? 30° 40° 70° 90° 110° 140° 150° 10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 MATHS STAFF 2011-2012Lets go to the internet…: Lets go to the internet… Circle Fun MATHS STAFF 2011-2012PowerPoint Presentation: Theorem – The measure of an formed by a chord and a tangent is equal to ½ the intercepted arc . 1 MATHS STAFF 2011-2012A Cyclic Quadrilateral: A Cyclic Quadrilateral …is a Quadrilateral whose vertices lie on the circumference of a circle Opposite angles in a Cyclic Quadrilateral Add up to 180° They are supplementary We say “Opposite angles in a cyclic quadrilateral add up to 180°” MATHS STAFF 2011-2012Questions: Questions MATHS STAFF 2011-2012PowerPoint Presentation: MATHS STAFF 2011-2012PowerPoint Presentation: MATHS STAFF 2011-2012PowerPoint Presentation: MATHS STAFF 2011-2012PowerPoint Presentation: Could you define a rule for this situation? MATHS STAFF 2011-2012Tangents: Tangents When a tangent to a circle is drawn, the angles inside & outside the circle have several properties. MATHS STAFF 2011-20121. Tangent & Radius: 1. Tangent & Radius A tangent is perpendicular to the radius of a circle MATHS STAFF 2011-20122. Two tangents from a point outside circle: 2. Two tangents from a point outside circle PA = PB Tangents are equal PO bisects angle APB g g <PAO = <PBO = 90° 90° 90° <APO = <BPO AO = BO (Radii) The two Triangles APO and BPO are Congruent MATHS STAFF 2011-2012 You do not have the permission to view this presentation. In order to view it, please contact the author of the presentation.
mohamed Attia ( teaching mathematics) mohamedattia12000 Download Post to : URL : Related Presentations : Share Add to Flag Embed Email Send to Blogs and Networks Add to Channel Uploaded from authorPOINT lite Insert YouTube videos in PowerPont slides with aS Desktop Copy embed code: (To copy code, click on the text box) Embed: URL: Thumbnail: WordPress Embed Customize Embed The presentation is successfully added In Your Favorites. Views: 40 Category: Education License: All Rights Reserved Like it (0) Dislike it (0) Added: February 10, 2012 This Presentation is Public Favorites: 0 Presentation Description mathematics for grade 9 Comments Posting comment... Premium member Presentation Transcript The international school of Egypt : The international school of Egypt D.NERMIEN ISMAIL LANGUGE SCHOOL MATHS STAFF 2011-2012RELATIONS & FUNCTIONS: RELATIONS & FUNCTIONS CARTESIAN PRODUCT, DOMAIN , RANGE , CODOMAIN , ARROW DIAGRAM & GRAPHS. MATHS STAFF 2011-2012RELATION (CARTESIAN PRODUCT): RELATION (CARTESIAN PRODUCT) We will learn how to link pairs of objects from two sets and then introduce relations b/w the two objects in the pair. AXB = { ( a,b ): a ε A, b ε B} for non-empty sets A,B otherwise AXB= φ , this is the set of all ordered pairs of elements from A and B. n(AXB)=n(A) Xn (B) AXBXC = {( a,b,c ):a ε A,b ε B,c ε C} is called set of all ordered triplets for non-empty sets. n(AXBXC) =n(A) Xn (B) Xn (C) AX(BXC)=(AXB)XC=AXBXC. AXB,BXA are not equal (not commutative). MATHS STAFF 2011-2012SOME BASIC POINTS: SOME BASIC POINTS Let P,Q be two sets then a relation R from P to Q is a subset of PXQ. Example: If A={ a,b,c,d },B={ p,q,r,s },then which of the following are relations ( i ) R1={( a,p ),( b,r ),( c,s )} (ii) R2 ={( q,b ),( c,s ),( d,r )} Solution: R1is a relation because R1is the subset of AXB but R2 is the not the relation of AXB because ( q,b ) ε R2 but not in AXB.(R2 is not a subset of AXB) Total number of relations that can be defined from a set A to B is the number of possible subsets of AXB. If n(A)=P , n(B)=q , then total number of relations is 2 pq OR total number of subsets of AXB. Example: If R is relation on a finite set having n elements, then the number of relations on A is Answer 2 nxn Inverse Function is a relation from B to A defined by { ( b,a ) : ( a,b ) ε R} MATHS STAFF 2011-2012PowerPoint Presentation: Relations and Functions Relations A relation is a set of inputs and outputs, often written as ordered pairs (input, output). We can also represent a relation as a mapping diagram or a graph. For example, the relation can be represented as: Relations A relation is a set of inputs and outputs, often written as ordered pairs (input, output). We can also represent a relation as a mapping diagram or a graph. For example, the relation can be represented as: {(2, –3), (4, 6), (3, –1), (6, 6), (2, 3)} domain : {2, 3, 4, 6} range: {–3, –1, 3, 6} State the domain and range of the following relation. Is the relation a function? they gave me two points with the same x -value: (2, –3) and (2, 3). Since x = 2 gives me two possible destinations, then this relation is not a function. * If output has one more unmapped element say 5 then output becomes codomain then Range is the subset of codomain Range ={-2,1,2,4} and Codomain ={-2,1,2.4,5} MATHS STAFF 2011-2012PowerPoint Presentation: State the domain and range of the following relation. Is the relation a function? {(–3, 5), (–2, 5), (–1, 5), (0, 5), (1, 5), (2, 5)} domain : {–3, –2, –1, 0, 1, 2} range: {5} this relation is indeed a function. ARROW DIGRAMS are given below : State in each case ,whether it is a function or not ? MATHS STAFF 2011-2012PowerPoint Presentation: QUESTION BANK WITH HINTS OF RELATIONS AND FUNCTIONS ↓ MATHS STAFF 2011-2012PowerPoint Presentation: Q.1 The relation R is defined on the set N as R={ (x,x+5) : x <4 ; x,yε N} Write R in roster form .Write its domain & range. Q.2 If AXB={(p,q),(p,r),(m,q),(m,r)},find A&B. Q.3 If A={x,y,z} and B={1,2}.Find the number of relations from A to B. Q.4 Examine the relation :R={{(2,1),(3,1),(4,1)} and state whether it is a function or not? Q.5 Let A={1,2,3,4,5,6,7,8,9,10}.Define a relation R from A to A by R={(x,y):2x – y=0,where x,y εA}. Write down its domain ,co-domain & range. Q.6 Let a relation R1 on the set of real numbers be defined as (a,b)ε R1↔ 1+ab>0 for all a,bεR. Show that R1 is reflexive & symmetric but not transitive . Q.7 If a={1,2,3,4,6}.Let R be the relation on A defined by {(a,b):a,bεA, b is divisible by a} Write R in roster form, domain & range of R. Q.8 If R is a relation from set A={11,12,13} to set B={8,10,12} defined by y=x-3, then write R -1 . Q.9 Let R be the relation on the set N of natural numbers defined by R={(a,b):a+3b=12,a,bεN} Find R, domain of R & range of R. Q.10 If A={1,2,3},B={1,4,6,9} and R is a relation from A to B defined by ‘x is greater than y’.Find the range of R. MATHS STAFF 2011-2012PowerPoint Presentation: Ans.6 Reflexive (a,a)εR1 [1+a²>0] & symmetric also as ab=ba. (1,1/2)εR1,(1/2,-1)ε R1 . Ans.8 {8,11),(10.13)} Ans.9 R={(9,1),(6,2),3,3)} Ans 10 {1} Ans.7 {(1,1),(1,2)……(1,6),(2,4)….(4,4),(6,6),(3,3),(3,6)} , domain=range ={1,2,3,4,6} Ans.5 R={(1,2),(2,4),(3,6),(4,8),(5,10)} , Domain= {1,2,3,4,5} range ={2,4,6,8,10} ,co-domain = A Ans .3 2 2x3 =2 6 Ans.1 R={(1,6),(2,7),(3,8)} domain={1,2,3},range={6,7,8} SOLUTIONS: MATHS STAFF 2011-2012PowerPoint Presentation: Q.1 Find the domain and range of the function f defined by f(x)= Q.2 Let f(x)=x2 and g(x)=2x+1 be two real functions .Find (f – g)(x),(fg)(x),(f/g)(x) Q.3 If f={(1,1),(2,3),(o,-1),(-1,-3)} be a linear function from Z into Z . Find f(x). Q.4 If f(x) = x 2 , find . Q.6 If f(x)= x 2 – 1/x 2 , then find the value of : f(x) +f(1/x) Q.7 Let A= {-2,-1,0.1,2} and f:A→ Z given by f(x) =x 2 – 2x – 3,find range of f and also pre-image of 6,-3,5. Q.8 Find the domain and range of real valued function f(x)= Q.5 Let f(x)= find f(-1) , f(3). Q.9 Y=f(x)= , then show that x=f(y). Q.10 If f(x) = , show that f[f(x)]. MATHS STAFF 2011-2012PowerPoint Presentation: Ans.1 f(x) is defined real function if 9-x 2 >0 ,so domain=(-3,3) and x 2 =(9 - 1/y 2 ) ,x is defined when 9 – 1/y 2 ≥0,so range=[1/3,∞) Ans.2 (f – g)(x)=x 2 -2x-1, (fg)(x)=2x 3 +x 2 and (f/g)(x)= ,x≠-1/2. Ans.3 Let f(x)=ax+b be a linear function. If (0,-1),(1,1)εf then we Can get a=2,b=-1 so linear function becomes f(x)=2x-1 Ans.4 = 2.2 Ans.5 f(-1)=2,f(3)=10. Ans.7 Range of f={0,5,-3,-4} and no pre-image of 6 ;0,2 are the pre-images of -3;-2 is the pre-image of 5. Ans.8 It is defined when x ≠4 otherwise it becomes 0/0 form(indeterminate form) Domain(f)=R-{4}, range (f)={-1} as f(x) =-1 [x-4=-(4-x)] Ans.6 0 SOLUTIONS : MATHS STAFF 2011-2012Polynomial Functions:: Finding where the function is positive and negative Polynomial Functions: MATHS STAFF 2011-2012Main Goal:: Main Goal: Learning how to make and use sign charts correctly and efficiently. Interval Test # F(x) sign MATHS STAFF 2011-2012Step One – Find the Zeros: Example: f(x) = (x-1)(x+2) (x-1)(x+2) = 0 (x-1) = 0 (x+2) = 0 and x = 1 and x = -2 Step One – Find the Zeros MATHS STAFF 2011-2012Step Two – Arrange Zeros on a Number Line: Step Two – Arrange Zeros on a Number Line -2 1 - infinity infinity MATHS STAFF 2011-2012Step Three – Choose Your Test Numbers: Step Three – Choose Your Test Numbers -2 1 0 3 5 infinity -infinity *When choosing, make sure you pick one from each interval.* MATHS STAFF 2011-2012Step Four – Create Your Chart: Step Four – Create Your Chart Interval (-infinity, -2) (-2, 1) (1, infinity) Test # -5 0 3 f(x) -18 -2 10 Sign - - + MATHS STAFF 2011-2012Tips For Making Your Chart: Tips For Making Your Chart Interval (-infinity, -2) (-2, 1) (1, infinity) Test # f(x) Sign Use your zeros! You already chose these endpoints when you made your number line. MATHS STAFF 2011-2012Tips For Making Your Chart (cont.): Tips For Making Your Chart (cont.) Interval Test # -5 0 3 f(x) -18 -2 10 Sign To find f(x), plug the test # into your original equation- f(x) = (x-1)(x+2) MATHS STAFF 2011-2012Tips For Making Your Chart (cont.): Tips For Making Your Chart (cont.) Interval Test # f(x) -18 -2 10 Sign - - + To decide the sign, look at the sign of f(x) MATHS STAFF 2011-2012Example 2:: Example 2: Step One - Factor f(x) = (x-1)(x+6) MATHS STAFF 2011-2012Example 2 (cont.): Example 2 (cont.) Step Two - Find the zeros (x-1) = 0 (x+6) = 0 x = 1 x = -6 MATHS STAFF 2011-2012Example 2 (cont.): Example 2 (cont.) Step Three – Arrange zeros on a number line -infinity infinity -6 1 MATHS STAFF 2011-2012Example 2 (cont.): Example 2 (cont.) -6 1 0 3 -7 infinity -infinity *Any number from between this interval. Do not have to use these numbers.* Step Four – Choose Your Test Numbers MATHS STAFF 2011-2012Example 2 (Cont.): Example 2 (Cont.) Interval (-infinity, -6) (-6, 1) (1, infinity) Test # -7 0 3 f(x) 8 -6 18 Sign + - + Step Five – Create Your Chart MATHS STAFF 2011-2012Practice Problems: Practice Problems MATHS STAFF 2011-2012Quadratic Formula: Quadratic Formula MATHS STAFF 2011-2012Graphing Quadratic Functions: Graphing Quadratic Functions y = ax 2 + bx + c MATHS STAFF 2011-2012PowerPoint Presentation: All the slides in this presentation are timed. You do not need to click the mouse or press any keys on the keyboard for the presentation on each slide to continue. However, in order to make sure the presentation does not go too quickly, you will need to click the mouse or press a key on the keyboard to advance to the next slide. You will know when the slide is finished when you see a small icon in the bottom left corner of the slide. Click the mouse button to advance the slide when you see this icon. MATHS STAFF 2011-2012Quadratic Functions: Quadratic Functions The graph of a quadratic function is a parabola . A parabola can open up or down. If the parabola opens up, the lowest point is called the vertex. If the parabola opens down, the vertex is the highest point. NOTE: if the parabola opened left or right it would not be a function! y x Vertex Vertex MATHS STAFF 2011-2012Standard Form: y = ax 2 + bx + c The parabola will open down when the a value is negative. The parabola will open up when the a value is positive. Standard Form y x The standard form of a quadratic function is a > 0 a < 0 MATHS STAFF 2011-2012Line of Symmetry: y x Line of Symmetry Line of Symmetry Parabolas have a symmetric property to them. If we drew a line down the middle of the parabola, we could fold the parabola in half. We call this line the line of symmetry . The line of symmetry ALWAYS passes through the vertex. Or, if we graphed one side of the parabola, we could “fold” (or REFLECT ) it over, the line of symmetry to graph the other side. MATHS STAFF 2011-2012Finding the Line of Symmetry: Find the line of symmetry of y = 3 x 2 – 18 x + 7 Finding the Line of Symmetry When a quadratic function is in standard form The equation of the line of symmetry is y = ax 2 + bx + c , For example… Using the formula… This is best read as … the opposite of b divided by the quantity of 2 times a . Thus, the line of symmetry is x = 3. MATHS STAFF 2011-2012Finding the Vertex: Finding the Vertex We know the line of symmetry always goes through the vertex. Thus, the line of symmetry gives us the x – coordinate of the vertex. To find the y – coordinate of the vertex, we need to plug the x – value into the original equation. STEP 1: Find the line of symmetry STEP 2: Plug the x – value into the original equation to find the y value. y = –2 x 2 + 8 x –3 y = –2(2) 2 + 8(2) –3 y = –2(4)+ 8(2) –3 y = –8+ 16 –3 y = 5 Therefore, the vertex is (2 , 5) MATHS STAFF 2011-2012A Quadratic Function in Standard Form: A Quadratic Function in Standard Form The standard form of a quadratic function is given by y = ax 2 + bx + c There are 3 steps to graphing a parabola in standard form. STEP 1 : Find the line of symmetry STEP 2 : Find the vertex STEP 3 : Find two other points and reflect them across the line of symmetry. Then connect the five points with a smooth curve. Plug in the line of symmetry ( x – value) to obtain the y – value of the vertex. MAKE A TABLE using x – values close to the line of symmetry. USE the equation MATHS STAFF 2011-2012PowerPoint Presentation: STEP 1 : Find the line of symmetry Let's Graph ONE! Try … y = 2 x 2 – 4 x – 1 A Quadratic Function in Standard Form y x Thus the line of symmetry is x = 1 MATHS STAFF 2011-2012PowerPoint Presentation: Let's Graph ONE! Try … y = 2 x 2 – 4 x – 1 STEP 2 : Find the vertex A Quadratic Function in Standard Form y x Thus the vertex is (1 ,–3). Since the x – value of the vertex is given by the line of symmetry, we need to plug in x = 1 to find the y – value of the vertex. MATHS STAFF 2011-2012PowerPoint Presentation: 5 –1 Let's Graph ONE! Try … y = 2 x 2 – 4 x – 1 STEP 3 : Find two other points and reflect them across the line of symmetry. Then connect the five points with a smooth curve. A Quadratic Function in Standard Form y x 3 2 y x MATHS STAFF 2011-2012Solving Systems of equations: Solving Systems of equations 1)Graphing 2)Substitution 3)Elimination MATHS STAFF 2011-2012Graphing: Graphing Parallel Lines have no solution. That is,they share no common points. Intersecting lines will share exactly one point, (x,y) MATHS STAFF 2011-2012Substitution: Substitution STEP 1 At least one of the equations must be solved for a variable. x -y = 1 X+y = -9 First solve one equation for either x or y: X = 1+y STEP 2 Now substitute this variable into the other Equation: X + y = -9 (1+y) +y =-9 Solve: 1 + 2y = -9 2y = -10 Y = -5 Now go back to step 1. X = 1 +y or x = 1 + -5 ( x,y ) = (-4,-5) MATHS STAFF 2011-2012Elimination:add/subtract/multiply: Elimination:add/subtract/multiply 4x + y =23 3x – y = 12 Notice the y variables are opposites. This is what you always want to have. Add the two equations together. When you add the variables with opposite coefficients, it will be eliminated. The result is: 7x = 35 X = 5 To find the eliminated variable, substitute 3x – y = 12 3(5) - y =12 Y = 3 (5,3) Does it work if you test it?? MATHS STAFF 2011-2012Coefficients are not opposites: Multiply first: Coefficients are not opposites: Multiply first 4x+3y = -2 4x +2y = 3 There are no opposites, so multiply eq #2 by neg 1 4x + 3y = -2 -4x-2y = -3 Now add! Y = -5 then go find x!! 5x-2y = -10 3x + 6y = 66 Mult eq#1 by 3!! Can you see why? 15x -6y = -30 3x + 6y = 66 then add!! 12x = 36 X = 3 go find y!! Final answers are always ( x,y ) MATHS STAFF 2011-2012Exceptions: Exceptions Parallel lines: no solution 3x-y = -2 3x –y = 0 Mult eqn #2 by - 1 3x – y = -2 -3x + y = 0 If you add , you get: 0 = -2 This is NEVER true so the lines NEVER cross, thus no solution. Same line: infinite solutions 2x –y = 3 4x – 2y = 6 Mult eqn #1 by -2 -4x+2y = -6 4x -2y = 6 If you add, you get: 0 = 0 This is ALWAYS true, so these are actually the same line, thus INFINITE solutions MATHS STAFF 2011-2012Algebraic Fractions: Algebraic Fractions Using 4 Rules MATHS STAFF 2011-2012PowerPoint Presentation: In this lesson we will look at examples of simplifying Algebraic Fractions using the “4 Rules of Fractions.” + - × ÷ Menu MATHS STAFF 2011-2012Addition: Addition Simplify + MATHS STAFF 2011-2012Addition: Addition Simplify + MATHS STAFF 2011-2012Subtraction: Subtraction Simplify - Can you see an alternative, quicker method here? MATHS STAFF 2011-2012Subtraction: Subtraction Simplify 4 - MATHS STAFF 2011-2012Multiplication: Multiplication Simplify × MATHS STAFF 2011-2012Multiplication: Multiplication Simplify × Factorising (2x-6) 3 MATHS STAFF 2011-2012Division: Division Simplify ÷ 2 2 MATHS STAFF 2011-2012Equations: Equations Once you know how to simplify algebraic fractions, you Can solve equations containing them For example MATHS STAFF 2011-2012PowerPoint Presentation: Solve MATHS STAFF 2011-2012PowerPoint Presentation: Solve MATHS STAFF 2011-2012PowerPoint Presentation: Adding with Common Denominators Adding with Different Denominators Multiplying Dividing Fractions MATHS STAFF 2011-2012PowerPoint Presentation: If the base (denominator) is the same … Add straight across the top (numerators) Add/Subtract Common Denominators MATHS STAFF 2011-2012PowerPoint Presentation: Can it be simplified? MATHS STAFF 2011-2012PowerPoint Presentation: 3 MATHS STAFF 2011-2012PowerPoint Presentation: Add the numerators Simplify MATHS STAFF 2011-2012PowerPoint Presentation: Make the bases the same (choose a common denominator) Add/Subtract Different Denominators MATHS STAFF 2011-2012PowerPoint Presentation: Multiply top & bottom by the SAME number. (changes the look not the fraction) 30 MATHS STAFF 2011-2012PowerPoint Presentation: Now the base (denominator) is the same … Add straight across the top (numerators). MATHS STAFF 2011-2012PowerPoint Presentation: Can it be simplified? MATHS STAFF 2011-2012PowerPoint Presentation: Make denominators the same Simplify if you can! Multiply top and bottom by same number Add the numerators MATHS STAFF 2011-2012PowerPoint Presentation: Multiply straight across the top (numerators) Multiply Multiply straight across the bottom (denominators) MATHS STAFF 2011-2012PowerPoint Presentation: Can it be simplified? MATHS STAFF 2011-2012PowerPoint Presentation: 2 MATHS STAFF 2011-2012PowerPoint Presentation: Multiply numerators , multiply denominators Simplify MATHS STAFF 2011-2012PowerPoint Presentation: Divide Flip the second fraction upside down (invert) MATHS STAFF 2011-2012PowerPoint Presentation: Multiply straight across the top (numerators) Multiply straight across the bottom (denominators) MATHS STAFF 2011-2012PowerPoint Presentation: Can it be simplified? MATHS STAFF 2011-2012PowerPoint Presentation: Flip the second fraction Simplify if you can! Multiply numerators , multiply denominators MATHS STAFF 2011-2012PowerPoint Presentation: Algebraic Fractions Adding/Subtracting Amsco Math A, Chapter 19 MATHS STAFF 2011-2012PowerPoint Presentation: Combining with Like Denominators Example: Combine the numerators over the denominator The Answer! Simplify MATHS STAFF 2011-2012PowerPoint Presentation: Combining fractions with Different Denominators Example: Multiply by 1 to get the same denominator Distribute Combine the numerators over the denominator MATHS STAFF 2011-2012PowerPoint Presentation: Combining fractions with Different Denominators Example: Factor the numerator Combine like terms in the numerator Cancel The Answer MATHS STAFF 2011-2012Probability : Probability MATHS STAFF 2011-2012What Is Probability, Hun?: What Is Probability, Hun? Probability is the way of expressing knowledge of belief that an event will occur on chance. Usually used in mathematics. Did You Know? Probability originated from the Latin word meaning approval . MATHS STAFF 2011-2012Things You Will Need To Know: Things You Will Need To Know Experiment: Is a situation involving chance or probability that leads to results called outcomes. Outcome: Is the results of a single trial of an experiment . Event: One or more outcomes in an experiment. MATHS STAFF 2011-2012Experimenting !: Experimenting ! A spinner has 4 equal sides colored: yellow, blue, green, and red. Suppose you spin that spinner and it lands on red. What are the chances of this event having that outcome? The Chances of landing on red are 1 in 4 or ¼. These chances are so because the spinner is divided into 4 and it landed on 1 of the equal sides. MATHS STAFF 2011-2012Experimenting !: Experimenting ! A dice has 6 sides, numbered 1-6. Suppose the dice is rolled and lands on a 2. What are the chances that number was the outcome of this event? The chances of the dice landing on a two are 1 in 6 or 1/6. This is possible to occur because it is a 6 sided die that could only land on 1 of the 6 sides. MATHS STAFF 2011-2012PowerPoint Presentation: Horse Racing Horse Racing consists of wagering money on which horse will finish first, second, or third. It is known respectively as betting to win, betting to place, and betting to show. Horse racing odds at a track are usually displayed on a large board, called the tote board. This board represents the horse racing odds in terms of the amount of money you will receive from betting. For example, if the horse racing odds are given as 3:1 on a particular horse, this means that if you wager $1 for the horse to win, and it comes in first place, you will receive $3. Many tracks have a minimum bet of $2, so the tote board also often includes a number showing how much money you would receive on a $2 bet . MATHS STAFF 2011-2012PowerPoint Presentation: MATHS STAFF 2011-2012Questions: : Questions: 1. Which of the following is an experiment? A) Tossing a coin. B) Rolling a single 6-sided die. C) Choosing a marble from a jar. D) All of the above. 2. Which of the following is an outcome? A) Rolling a pair of dice. B) Landing on red. C) Choosing 2 marbles from a jar. D) None of the above. MATHS STAFF 2011-2012Answers:: Answers: Question 1- D) Question 2- B) MATHS STAFF 2011-2012Circle Theorems: Circle Theorems MATHS STAFF 2011-2012A Circle features…….: A Circle features……. Circumference … the distance around the Circle… … its PERIMETER Diameter … the distance across the circle, passing through the centre of the circle Radius … the distance from the centre of the circle to any point on the circumference MATHS STAFF 2011-2012A Circle features…….: A Circle features……. … a line joining two points on the circumference. … chord divides circle into two segments … part of the circumference of a circle Chord Tangent Major Segment Minor Segment ARC … a line which touches the circumference at one point only From Italian tangere , to touch MATHS STAFF 2011-2012Properties of circles: Properties of circles When angles, triangles and quadrilaterals are constructed in a circle, the angles have certain properties We are going to look at 4 such properties before trying out some questions together MATHS STAFF 2011-2012An ANGLE on a chord: An ANGLE on a chord An angle that ‘sits’ on a chord does not change as the APEX moves around the circumference … as long as it stays in the same segment We say “Angles subtended by a chord in the same segment are equal” Alternatively “Angles subtended by an arc in the same segment are equal” From now on, we will only consider the CHORD, not the ARC MATHS STAFF 2011-2012Typical examples: Typical examples Find angles a and b Imagine the Chord Angle b = 28º Imagine the Chord Angle a = 44º Very often, the exam tries to confuse you by drawing in the chords YOU have to see the Angles on the same chord for yourself MATHS STAFF 2011-2012Angle at the centre: Angle at the centre Consider the two angles which stand on this same chord Chord What do you notice about the angle at the circumference? It is half the angle at the centre We say “ If two angles stand on the same chord, then the angle at the centre is twice the angle at the circumference ” A MATHS STAFF 2011-2012Angle at the centre: Angle at the centre We say “ If two angles stand on the same chord, then the angle at the centre is twice the angle at the circumference ” It’s still true when we move The apex, A, around the circumference A As long as it stays in the same segment 136° 272° Of course, the reflex angle at the centre is twice the angle at circumference too!! MATHS STAFF 2011-2012Angle at Centre: Angle at Centre A Special Case When the angle stands on the diameter, what is the size of angle a? a a The diameter is a straight line so the angle at the centre is 180° Angle a = 90° We say “The angle in a semi-circle is a Right Angle” MATHS STAFF 2011-2012Inscribed Angles: Inscribed Angles MATHS STAFF 2011-2012PowerPoint Presentation: Inscribed angle – An angle where the vertex is on the circle and its arms are chords in the circle. Inscribed angle MATHS STAFF 2011-2012PowerPoint Presentation: Intercepted arc – the part of the circle inside the arms of the angle Intercepted arc MATHS STAFF 2011-2012Lets go to the internet…: Lets go to the internet… Circle Fun MATHS STAFF 2011-2012PowerPoint Presentation: Theorem – An inscribed angle is half of the intercepted arc. MATHS STAFF 2011-2012What is the measure of x?: What is the measure of x? 50° 60 40° 20° 10° 10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 MATHS STAFF 2011-2012What is the measure of y?: What is the measure of y? 50° 60° 40° 20° 10° 10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 MATHS STAFF 2011-2012PowerPoint Presentation: Corollary 1 – If two inscribed angles intercept the same arc, then the angles are congruent. 1 2 MATHS STAFF 2011-2012Lets go to the internet…: Lets go to the internet… Circle Fun MATHS STAFF 2011-2012PowerPoint Presentation: Corollary 2 – An angle inscribed in a semicircle is a right angle. MATHS STAFF 2011-2012What is x?: What is x? 30° 40° 70° 90° 110° 140° 150° 10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 MATHS STAFF 2011-2012What is y?: What is y? 30° 40° 70° 90° 110° 140° 150° 10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 MATHS STAFF 2011-2012What is z?: What is z? 30° 40° 70° 90° 110° 140° 150° 10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 MATHS STAFF 2011-2012Lets go to the internet…: Lets go to the internet… Circle Fun MATHS STAFF 2011-2012PowerPoint Presentation: Theorem – The measure of an formed by a chord and a tangent is equal to ½ the intercepted arc . 1 MATHS STAFF 2011-2012A Cyclic Quadrilateral: A Cyclic Quadrilateral …is a Quadrilateral whose vertices lie on the circumference of a circle Opposite angles in a Cyclic Quadrilateral Add up to 180° They are supplementary We say “Opposite angles in a cyclic quadrilateral add up to 180°” MATHS STAFF 2011-2012Questions: Questions MATHS STAFF 2011-2012PowerPoint Presentation: MATHS STAFF 2011-2012PowerPoint Presentation: MATHS STAFF 2011-2012PowerPoint Presentation: MATHS STAFF 2011-2012PowerPoint Presentation: Could you define a rule for this situation? MATHS STAFF 2011-2012Tangents: Tangents When a tangent to a circle is drawn, the angles inside & outside the circle have several properties. MATHS STAFF 2011-20121. Tangent & Radius: 1. Tangent & Radius A tangent is perpendicular to the radius of a circle MATHS STAFF 2011-20122. Two tangents from a point outside circle: 2. Two tangents from a point outside circle PA = PB Tangents are equal PO bisects angle APB g g <PAO = <PBO = 90° 90° 90° <APO = <BPO AO = BO (Radii) The two Triangles APO and BPO are Congruent MATHS STAFF 2011-2012