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N5 N5 N5 N5 N5 N5 National Five N5 N5 N5 N5 N5 N5 Relationships Teachers’ Notes N5 N5 N5 N5 N5 N5 N5 1.1 Applying algebraic skills to linear equations A1.1 Function Notation A1.1a Introduction to f(x) notation. Build on the pupils’ previous knowledge of number machines. Teejay Int 2 Book 2 P40 Ex 4.1 Q 1 to 6 A1.1b The common sets and basic set builder notation could be introduced here. This can be linked to inequalities and reinforced as lesson starters. Graph drawing packages could be used to draw graphs of functions by letting y = f(x). Teejay Int 2 Book 2 P40 Ex 4.1 Q 7 to 9 B1.1 Straight Line Straight Line is covered earlier in the course. This section should only be a quick revision. It should also be reinforced through lesson starters. B1.1a I can calculate the gradient of a straight line given two points which lie on the line: National 5 Bk Page 41 Ex 6.2 By this point pupils should be familiar with finding the gradient simply as a process of substitution into a formula. B1.1b I have investigated the gradients of horizontal and vertical lines and can state their equations. Departmental PowerPoint Horizontal lines can be explained as all the points where y = k and similarly for vertical lines. The PowerPoint ‚BHStraightLineHorizontalAndVertical‛ is available. B1.1c I have worked with others to investigate the effect of m and c in the equation Investigation, Board Examples and Graphing Software Graphing software is available to illustrate what happens to the graph of a straight line as m and c are varied. Pupils should attempt to predict graphs by using a Cartesian show me board. This will be revision of S3 work. B1.1d I can write down the equation of a straight line given the gradient and yintercept and vice versa. Natioanl 5 Bk B1.1e Ex 9.1 I can rearrange the equation of a line into the format National 5 Bk B1.1f Page 62 Page 65 Ex 9.3 I can find the equation of a straight line given the gradient and one point using National 5 Bk . Page 63 Ex 9.2 C1.1 Linear Equations and Inequalities This topic has been well covered during the BGE. It should be reinforced by lesson starters very, very regularly. C1.1a I can solve equations that contain brackets. National 5 Bk P72 Ex 10.4 National 5 Bk P73 Ex 10.5 C1.1b I can solve equations which contain fractions. National 5 Bk C1.1c P74 Ex 10.6 I can solve inequalities and understand through investigation that when then etc. National 5 Bk P77 Ex 10.8 Natioanl 5 Bk P78 Ex 10.9 C1.1d I can use Equations to make mathematical models. National 5 Bk P75 Ex 10.7 D1.1 Simultaneous Equations Simultaneous Equations are covered earlier in the course. This section should only be a quick revision. It should also be reinforced through lesson starters. D1.1a I can solve simultaneous equations (2 equations and 2 unknowns) graphically. National 5 Bk P 82 Ex 11.1 D1.1b I can solve simultaneous equations (2 equations and 2 unknowns) by elimination. National 5 Bk D1.1c P85 Ex 11.3 Nation al 5 BK P86 Ex 11.4 I can solve simultaneous equations (2 equations and 2 unknowns) by substitution. National 5 Bk P85 Ex 11.5 D1.1h I can model real life problems with simultaneous equations and solve them. National 5 Bk P88 Ex 11.6 E1.1 Changing the subject Pupils will have been solving equations using inverse operations for a number of years by this point. Changing the subject should initially be introduced as a series of inverse operations on the new subject. e.g. y = 2x + 3 – x is multiplied by two then 3 is added. The inverse of this is subtracting 3 followed by division by two. Apply this to both sides and hey presto!! There are then 3 special cases where some manipulation prior to finding the inverse operations is required a) New subject is part of a negative term. b) New subject is part of a denominator. c) New subject appears in more than one term. Departmental PowerPoint is available E1.1a I can substitute into Expressions and Formulae National 5 Bk E1.1b P92 Ex 12.1 I can Change the subject of formulae Int 2/C2 P22 Ex 2.5 Q1 – 9 Int 2/C2 P24 Ex 2.6 Q1(a – d), 2 – 8 National 5 Bk P 94 Ex 12.2 National 5 Bk P 95 Ex 12.3 Nationa 5 Bk P95 Ex 12.4 Revision and sub skill assessment Relationships 1.1 N5 1.2 Applying algebraic skills to graphs of quadratic relationships A1.2 Quadratic functions and Graphs Pupils will be familiar with the term PARABOLA from the flying car experiment on projectiles. The National 5 textbook avoids the notation f(x) but this should be covered by the teacher. It should also be made clear that the function is f(x) and the graph is y = f(x). A1.2a I can make a table of values for a function (f(x)) and draw the graph of y = f(x). National 5 Bk P99 Ex 13.1 A1.2b I can recognise, from its graph, y = x2 and any related graph of the form y = ax2 + b. A1.2c National 5 Bk P101 Ex 13.2 Int 2/C2 P96 Ex 9.3 Q1 to Q5 I have revised the procedure known as Completing the Square. National 5 Bk P30 Ex 4.8 Q3 to Q7 A1.2d I can identify the turning point (and line of symmetry) of a quadratic when the quadratic is given in the form and determine the nature of this turning point. I can also determine the equation of a quadratic in the form from its graph. National 5 Bk P103 Ex 13.3 Int 2/C2 P94 Ex 9.1 Q1 to Q6 Int 2/C2 P95 Ex 9.2 Q1 to Q4 Revision and sub skill assessment Relationships 1.2 N5 1.3 Applying algebraic skills to quadratic equations A1.3 Quadratic Equations (Graphically) Graph drawing packages used with show me boards allow pupils to experience a large number of examples of finding the points where the graph cuts the x axis and making the connection between this and the solution to the equation. Teaching Idea – Have 20 or so graphs of quadratic functions around the classroom or corridor. Pupils are given some quadratics equations and have to go on a gallery walk to find the solution. They will first have to rearrange the equations into the correct form before searching for the correct graph. Include examples which do not have a solution. A1.3a I know the meaning of the term ‘roots of a quadratic equation’ and solve quadratic equations graphically. National 5 Bk P106 Ex 14.1 Int 2/C2 P60 Ex 6.2 Q1 to Q3 B1.3 Quadratic Equations (by factorisation) Revision of factorisation should be happening regularly as lesson starters for a number of weeks prior to starting this topic. B1.3a B1.3b I have revised how to factorise National 5 Bk P29 Ex 4.6 and Ex 4.7 Int 2/C2 : P61 : Ex 6.3 : Q1 to Q4 I can solve a Quadratic Equation by factorisation. National 5 Bk P108 Ex 14.2 Int 2/C2 : P62 : Ex 6.4 : Q1 to Q13 C1.3 Quadratic Formula and the Discriminant Pupils should ask ‚does the quadratic equation have real roots‛ by calculating the Discriminant. The Discriminant can then be substituted into the quadratic formula if it is >=0. It is therefore important to give pupils the opportunity to experience quadratic equations with no real roots. A graph drawing package could be used to illustrate the three conditions graphically while checking answers. This is well suited to pupils working in active pairs with a show me board between them. This will reinforce and deepen our pupils’ understanding of the discriminant. Negative – No real roots. Can’t solve the equation (for now!) Zero – Repeated roots, equal roots producing one solution to the equation. Positive – Two real and distinct roots producing two solutions to the equation. C1.3a I know how to determine nature of the roots of a quadratic equation using the discriminant. National 5 Bk C1.3b C1.3c P110 Ex 14.4 Q1 and Q2 I can solve a quadratic equation using the Quadratic Formula National 5 Bk P109 Ex 14.3 National 5 Bk P110 Ex 14,4 Q10 Int 2/C2 P98 Ex 9.4 Q1 to 10 I can identify problems where the discriminant can be used as a strategy to solve a problem. National 5 Bk P110 Ex 14.4 Q3 to Q9 D1.3 Problem Solving Using Quadratic Equations D1.3a I can find a Quadratic Equation which models a real life situation and by solving the Quadratics Equation I can find and communicate the solution to a problem. National 5 Bk P112 Ex 14.5 Revision and sub skill assessment Relationships 1.3 N5 1.4 Applying geometric skills to lengths, angles and similarity A1.4 Pythagoras (converse and 3D) Converse has previously been covered in the BGE. 3D Pythagoras is new at this point and could be perfect for some model building. Build a skeleton model. Use Pythagoras to calculate the length of a space diagonal. Cut out a length and check that it fits. What types of 3D shape could be used. Are there packs available commercially? Could we just use straws? A1.4a I have revised the use of Pythagoras Theorem including the Converse of Pythagoras. National 5 Bk P115 Ex 15.1 A1.4b I can use Pythagoras to solve problems in 3 Dimensions. National 5 Bk P118 Ex 15.3 B1.4 Properties of Shape (Triangles, Quadrilaterals, Polygons, Angles in a Circle) Angles In The Circle has been covered in the BGE. This could be new having this type of plane geometry so prominent at this level. Development required. Some form of extension to the quadrilateral project might be good only for regular polygons. B1.4a I have spent time revising properties of 3D shape National 5 Bk P123 Ex 16.1 Q3 to Q6 National 5 Bk P125 Ex 16.2 Q5 to Q7 National 5 Bk P127 Ex 16.3 B1.4b I have revised using angle facts in a circle. National 5 Bk P130 Ex 16.5 C1.4 Similarity C1.4a I know how to check if two shapes are Similar and can use the fact that two shapes are Similar to calculate unknown sides. C1.4b National 5 Bk P135 Ex 17.1 Int 2 C2 P48 Ex 5.1 I can show when triangles are Similar and can use Similarity in Triangles to find the lengths of unknown sides. C1.4c National 5 Bk P137 Ex 17.2 Int 2 C2 P50 Ex 5.2 Int 2 C2 P52 Ex 5.3 I know the relationships between Length Scale Factor, Area Scale Factor and Volume Scale Factor and can apply Scale Factor to solve problems. National 5 Bk P140 Ex 17.3 Int 2 C2 P54 Ex 5.4 Int 2 C2 P56 Ex 5.5 Revision and sub skill assessment Applications 1.4 N5 1.5 Applying trigonometric skills to graphs and identities A1.5 Related Angles in the range 0 - 360˚ Pupils should use graphs of sin, cos and tanx to identify related angles which satisfy sin, cos, tan x = ±k. Pupils should then be guided towards an understanding that every angle 0 - 360 has a related angle in each quadrant (with the exception of the border angles 0, 90, 180, 270, 360) and that sometimes it is positive and sometimes it is negative. This may seem a bit back to front but it lets pupils see where they are going before introducing the new definition of sinx, cosx and tanx as ratio of coordinates produced by a rotating arm rotating counter clockwise by an angle x. This cognitive conflict caused by rejecting the nice opp/hyp etc can cause some distress, particularly to the fixed mindset pupils, as they will not like finding out that something they though was easy can in fact be quite challenging. Care must be taken !! A1.5a I understand the new definitions of , and Insert Required A1.5b Given an angle θ where where , I can state a related base angle A and all of its related angles (both positive and negative) in the other three quadrants. Insert Required National 5 Bk P149 Ex 18.2 B1.5 Trig Graphs (basic, amplitude, vertical translation, horizontal translation (phase angle), multiple angles This topic will build upon the translation work in the blue course and the quadratic graph section of this unit. Pupils will create their own graphs of , and . Then they will investigate the input transformations. i.e. Given there are two transformations taking place. Both affect the graph in the x-direction. is first translated by –b in the x-direction according to the transformation matix . The new period is found using the formula . Finally they will investigate the output transformations. i.e. Given there are two transformations taking place. Both affect the graph in the y-direction. The range of values of y is changed from amplitude changes from 1 to p. to so the is translated by q in the y-direction according to the transformation matix . Pupils could be asked to match a list of Trig Graphs to the Equations using a gallery walk similar to that in section A1.3 of this unit. B1.5a I have used a table of values to plot the graphs of . , and Class activity B1.5b I know the meaning of the terms Transformation; Translation; Period and Amplitude. Class activity B1.5c Sketch and identify trigonometric functions which involve transformations to the basic graphs of , and . National 5 Bk P146 Ex 18.1 Int 2/C2 P75 Ex 7.4 Q1 to Q6 Int 2/C2 P77 Ex 7.5 Q1 to Q5 Int 2/C2 P82 Ex 7.7 Q1 to Q2 Int 2/C2 P83 RR Q1 to Q6 C1.5 Trig Equations in the range 0 - 360˚ Pupils were introduced to the concept of a base angle in the first quadrant and related angles in the other three. This will be developed into a formal procedure for solving trig equations as below. this is the base angle in the first quadrant From the diagram below we can see that Cosine is negative in the 2nd and 3rd quadrant Sine 180 - A All A 180 + A Tan 360 - A Cos Leading to our two solutions Solution 1 C1.5a Solution 2 I can solve Trig Equations Algebraically National 5 Bk P150 Ex 18.3 Int 2/C2 : P116 Ex 11.1 Q1 - 11 D1.5 Trig Identities D1.5a I can use the Trig Identities and National 5 Bk P152 Ex 18.4 Int 2/C2 P120 Ex 11.3 Revision and sub skill assessment Applications 1.5 . Q1 to Q4, Q6(a – e), Q7(a, b)