Newbattle Community High School Maths Department Course Planner (CfE): ALGEBRA TOPICS. Notes on approaches and activities for learning

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1 INDEX (if viewing this file in MS Word, control+click takes you straight to that section) Index... 1 Substitution... Using Formulae... 5 Linear Graphical Relationships... 8 Simplification 1 (Like Terms and Algebraic Fractions)... 1 Simplification (Surds and Indices) Brackets and Factorising Equations and Inequalities... Non-Linear Graphical Relationships... 6 Creating Expressions and Equations... 9 A pentagon class in the Broad General Education should cover all the pentagon content. A pentagon class in the Senior Phase is likely to miss some of the content out. Teachers should consult the course plan and ask the Principal Teacher for advice as required. At dodecagon level, RfH indicates optional content that is not required at National 5, but is advised for pupils intending to attempt one year Higher

2 SUBSTITUTION 3 rd /4 th level CfE Substitution outcomes: I can collect like algebraic terms, simplify expressions and evaluate using substitution. MTH 3-14a Cross Curricular Links (whole school numeracy record): required regularly in science By the end of the topic, pupils should be able to: Notes on approaches and activities for learning SUBSTITUTION (TRIANGLE COURSE) SUBSTITUTE POSITIVE INTEGERS INTO BASIC ALGEBRAIC EXPRESSIONS Prerequisites from Number/Algebra courses and other topics: triangle multiply Evaluate expressions (adding, taking away, multiplying and order of operations involving these) with positive whole number (answers always 0) a + 7, a, a + 3, 5a + b, 3a + b + c, ab, 4 + a, 4a b Dynamic Worksheet: Algebra-04 Worksheet Secondary J17 SUBSTITUTION (PENTAGON COURSE) SUBSTITUTE POSITIVE INTEGERS INTO MORE COMPLEX ALGEBRAIC EXPRESSIONS Prerequisites from Number/Algebra courses and other topics: pentagon multiply, pentagon surds and indices Evaluate expressions involving brackets, fractions, BODMAS, basic squares and square roots - substituting (and answers) positive whole numbers ONLY TEACHING APPROACH: class experiment with values of n to discover that (n+3) and n+3 give different answers; and similarly to discover that 3(x+) and 3x+6 always give the same answer a b 3( a b c), 1, a, a 3, 4 x, 3 x 4 a b c, ( a b) 5, a b, 4 c d, a² (a is a small integer), a where a is a square number Worksheet Secondary J18 TEACHING APPROACH: emphasise "invisible brackets" when evaluating fractions and square roots a b ( a b), x 1 ( x 1) c c - Page -

3 By the end of the topic, pupils should be able to: Newbattle Community High School Maths Department Notes on approaches and activities for learning SUBSTITUTION (OCTAGON COURSE) SUBSTITUTE INTEGERS AND DECIMALS INTO COMPLEX ALGEBRAIC EXPRESSIONS Prerequisites from Number/Algebra courses and other topics: no specific requirements MATHS AND LIFESKILLS ROUTE Evaluate all expressions previously encountered by 5 + a, a + 3b, 3a + b + c, ab, 4b, Dynamic Worksheet: Algebra-04 substituting decimals and (for octagon maths classes) integers a b 1 a 3b, 3(a + b + c),, a, a 3, 4 x, 3 x, 4 ( a b ) 5 a b c, a b, 4 c d, a b (physics), a b c (physics) Evaluate expressions involving squares, cubes, roots TEACHING APPROACH: emphasise "invisible brackets" when evaluating fractions and square roots a², 3b², x² ± 4y², a³, 5b³, xy², (xy)², x²y, 4x, 4 x, b² 4ac a b a b c d ( c d) 3b 3 ( b ) ( ), 4 ( b ac b 4 ac ) Worksheet Secondary J18, J19 - Page 3 -

4 By the end of the topic, pupils should be able to: Newbattle Community High School Maths Department Notes on approaches and activities for learning SUBSTITUTION (DECAGON COURSE) EVALUATE EXPRESSIONS AND FUNCTIONS Prerequisites from Number/Algebra courses and other topics: octagon maths surds and indices Introduce function notation, basic examples f(x) = x² + 4. What is f(3)? f(8)? f(y)? g(x) = x. What is g(3)? g(1)? g(b)? (basic positive integers only) h(x) = 7x + 9. What is h(1)? h( 5)? h(p)? Evaluate quadratic expressions and functions What is x² + 5x + 14 when x =, x = 1,...? f(x) = x² + 5x What is f(), f( 1),? What is 3x² + 5x + 14 when x =, x = 1,...? What is (x 3)² 4 when x =, x = 1,...? g(x) = (x 7)(x + 5). What is g(4)? Dynamic Worksheet: Quadratics- 04, 05 Evaluate expressions involving surds What is 6 x when x = 9? f(x) = 5 x. What is f(36)? What is 3 x when x = 8? SUBSTITUTION (DODECAGON COURSE) BECOME READY FOR HIGHER Prerequisites from Number/Algebra courses and other topics: decagon surds and indices Interpret fractional powers as roots (numerical examples) , , , 5, Interpret negative powers as fractions (numerical examples) 1 3 Evaluate expressions and functions involving negative and fractional powers What is a when a = 3? f(x) = x 4/3. What is f(8)? (RfH) Evaluate all previous expressions and functions by substituting in fractions What is x² + 5x + 14 when x = ½? f(x) = x². What is f(¼)? - Page 4 -

5 USING FORMULAE 3 rd /4 th level CfE Using Formulae outcomes: I can create and evaluate a simple formula representing information contained in a diagram, problem or statement. MTH 3-15b Cross Curricular Links (whole school numeracy record): required very regularly in science, especially physics By the end of the topic, pupils should be able to: Notes on approaches and activities for learning USING FORMULAE (TRIANGLE COURSE) USE SIMPLE FORMULAE IN CONTEXT Prerequisites from Number/Algebra courses and other topics: triangle substitution, triangle multiply Use formulae that model familiar context in words and algebra and function machines: Addition Subtraction Multiplying two numbers Area = Length Breadth, Cost = No. Items Price, A=LB, P=a+b+c, D = ST, D = R (from physics:) V=IR, W=QV, p=mv, Q=It Extend to formulae that include: Division linear operations (i.e. ax+b) multiplying more than two numbers using the fact that multiplying comes before adding and/or taking away Cost = minutes price per minute plus line rental, Distance = Speed Time, V=length breadth height (from Physics Acc3/Int 1 formula sheet in words:) Weight = 10 mass, Voltage gain = output voltage/input voltage, current = power / voltage, S = D/T, (request from Physics and Engineering to use small letters i.e. not S D T ) BH n d C = mp + r, y = mx + c, A, A, r, (from physics:) t F P, E = V + Ir, Emphasise the use of / in formula meaning divide A Dynamic Worksheet: SC01 (F=ma) - Page 5 -

6 By the end of the topic, pupils should be able to: Newbattle Community High School Maths Department Notes on approaches and activities for learning USING FORMULAE (PENTAGON COURSE) DEVELOP KNOWLEDGE OF FORMULAE INCLUDING BODMAS Prerequisites from Number/Algebra courses and other topics: pentagon multiply, pentagon substitution Use formulae that model real-life situations in Perimeter of Rectangle P=A + B or P=(A+B), words and algebra including: A = ½BH, A = L², L A unitary fractions ½ ( v u) (physics:) a [with visible and/or invisible brackets], E=mgh squaring, square rooting t powers BODMAS invisible brackets - Page 6 - RESOURCES (in addition to topic SmartBoard files) USING FORMULAE (OCTAGON COURSE) EXTEND KNOWLEDGE OF FORMULAE Prerequisites from Number/Algebra courses and other topics: pentagon integers, octagon substitution, pentagon equations MATHS AND LIFESKILLS ROUTE 5 5M 9 9C Continue to use a wide range of formulae, K M or K, F C 3 or F 3, r h V including: substituting decimal and (for octagon 4 r V, A = πr², A=4πr², V= πr²h, c a b, y=½x+8 maths classes) integer values dealing with invisible brackets Develop the "change side do the opposite" method (learnt for solving equations at pentagon level), into a strategy for changing the subject of a simple formula Hence or otherwise, develop a strategy for going backwards in a real life context 3 (from physics:) s = ut + ½at², s = ½(u+v)t, E = ½mv², PV/T = constant, V=rt(PR), V=rt(E/mR), P = I²R, V k R P, I, E = ½CV², V s R d R R V 1 d Change the subject of s to d or t, (request from Physics that we t use small letters rather than capitals) Change the subject of C = d to d, y = a ± b to a or b. From Physics: P=IV, V=IR, F=ma, E = mgh, P = I²R (to R only) Extend as appropriate to class. The volume of a cuboid is 450cm³. If the length is 10cm and the breadth is 4 5cm, what is the height? The circumference of a circle is 60cm. What is the diameter? Using v = u + at, find t when u = 0, v = 30, a = 750 Active Worksheet Website: Worksheet Secondary J34

7 By the end of the topic, pupils should be able to: Notes on approaches and activities for learning USING FORMULAE (DECAGON COURSE) DEVELOP FORMULA SKILLS NEEDED FOR NATIONAL 5 Prerequisites from Number/Algebra courses and other topics: octagon formulae Continue to use a wide range of formulae, including those that will be encountered in the volume topic Change the subject of: linear formulae (of the form y=mx+c) formulae that involve only multiplying [e.g. of the form V = abcd] Hence or otherwise, develop a strategy for going backwards in a real life context Substitute confidently into all formulae encountered in the National 5 course, including dealing with invisible brackets Examples: V= (πr²h)/3, V= (4πr³)/3, A=πr², V= πr²h, (from physics:) v² = u² + as, 1/R T = 1/R 1 + 1/R, 1/R T = 1/R 1 + 1/R + 1/R 3 Change the subject of: y = mx + c to x, V = r²h to h V 3V 1 V 3 r hto h [giving answer as h rather than h ] 1 3 r r If y = 4x 5, and y = 3, what is x? The volume of a cylinder is 000cm³ and the radius is 3cm, what is h? 1 Using s ( u v ) t, find v when a = 0 1, t = 30, u = 0 This is a key skill in the final exam USING FORMULAE (DODECAGON COURSE) USE ALL FORMULAE REQUIRED FOR NATIONAL 5 CONFIDENTLY sine rule, cosine rule, cosine rule for angles, gradient formula (with integers), volume formulae (with decimals), quadratic formula (with integers), equation of a parabola RESOURCES (in addition to topic SmartBoard files on server) Active Worksheet Website: Worksheet Secondary J34 Change the subject of more complicated formulae that may involve squaring, square rooting, brackets (visible or invisible) and fractions Change the subject of: 1 V= r²h to r, A ha ( b) to b or h, mn K to m, p q s to s. p 1 1 From Physics: s ( u v) t, s ut at b A d to c, c Hence or otherwise, develop a strategy for going backwards in a real life context The volume of a cuboid is 450cm³. If the length is 10cm and the breadth is 4 5cm, what is the height? The circumference of a circle is 60cm. What is the diameter? Physics (H): find a when s = 0 6, v = 30, u = 0 using v² = u² + as - Page 7 -

8 LINEAR GRAPHICAL RELATIONSHIPS 3 rd /4 th level CfE Linear Graphical Relationships outcomes: Having investigated the pattern of the coordinate points lying on a horizontal or vertical line, I can describe the pattern using a simple equation. MTH 4-13c I have discussed ways to describe the slope of a line, can interpret the definition of gradient and can use it to make relevant calculations, interpreting my answer for the context of the problem. MTH 4-13b I can use a given formula to generate points lying on a straight line, plot them to create a graphical representation then use this to answer related questions. MTH 4-13d Cross Curricular Links (whole school numeracy record): none have been identified yet (will be added later as required) By the end of the topic, pupils should be able to: No content at triangle No content at pentagon Notes on approaches and activities for learning RESOURCES (in addition to topic SmartBoard files on server) LINEAR GRAPHICAL RELATIONSHIPS (OCTAGON COURSE) BE ABLE TO DRAW A GRAPH FROM AN EQUATION USING A TABLE OF VALUES, AND THEN TO DISCOVER THE LINK BETWEEN THE EQUATION, GRADIENT AND Y-INTERCEPT (INTEGERS ONLY) Prerequisites from Number/Algebra courses and other topics: no specific requirements MATHS ROUTE Follow simple rules in words to identify and plot points, and to draw a line "the x coordinate is double the y coordinate", "the y coordinate is " Draw lines of the form x = a or y = b (horizontal/vertical) without needing a table of values Use tables of values to draw lines of the form y = mx + c on a four-quadrant grid Understand the concept of gradient as how far you go up for every one you go along Identify the gradient of a straight line from a diagram V using m (answers whole numbers only) H Draw x =, y = 4 without a table Draw y = x + 7, y = 3 x, y = ½x + 3, y = 3x + with a table Gradient of as "along 1 up " Gradient of 3 as "along 1, down 3" Gradient of zero is horizontal Identify gradient of y = x + 5 from its graph. Identify gradient of y = 3 x from its graph. Identify gradient of line joining (1, 1) and (4, 10) from a diagram Dynamic Worksheet: StraightLine-0 Past Intermediate 1 exam questions - Page 8 -

9 Identify the gradient and y-intercept of a given straight line from its graph Compulsory practical task: Investigation using Desmos in ipads to discover connection between gradient, y-intercept and equation of a straight line leading to y = mx + c Write down the gradient and y-intercept of a straight line when told its equation in the form y = mx + c (m, c integers) LIFESKILLS ROUTE No content Pupil shown graph of y = 4x + 5 on Autograph and will be able to identify its gradient and y intercept The equation of a line is y = 4x + 5. What is its gradient? The equation of a line is y = 3 x. Where does it go through the y axis? Autograph Autograph - Page 9 -

10 By the end of the topic, pupils should be able to: Newbattle Community High School Maths Department Notes on approaches and activities for learning LINEAR GRAPHICAL RELATIONSHIPS (DECAGON COURSE) USE Y = MX + C FOR ALL LINES INCLUDING THOSE WITH A FRACTIONAL GRADIENT Prerequisites from Number/Algebra courses and other topics: octagon linear graphical relationships Extend the concept of gradient to fractional and undefined gradients gradient of ½ as "along 1 up ½" leading to "along, up 1" gradient of ¾ as "along 4, down 3" gradient of zero is horizontal vertical line has undefined gradient RESOURCES (in addition to topic SmartBoard files on server) Understand and use gradient formula y y m x x 1 1 Find the gradient of the line joining (, 1) and (3, 4). Identify gradient of a line drawn on squared paper by choosing suitable points. Identify gradient of a line drawn using a non standard scale. Dynamic Worksheet: StraightLine-01 Draw a straight line (in the form y = mx + c) from its equation on squared paper without requiring a table of values Sketch a straight line from its equation on plain whiteboard, correctly annotating the sketch Draw y = 4x 5, y = 3 + ½x, y = 5 x, y = ¾x +, y = 5, x = without needing a table of values. Sketch y = x 5, sketch y = 5, sketch y = 5x, sketch a possible graph for y = ax + b given that a > 0 and b = 0 etc. Identify the equation of any straight line from a diagram Any equation of the form y = mx + c Worksheet Secondary K31a, K31c Laminated card group work tasks Use formula to complete a coordinate point on a given straight line given either the x or y coordinate, including x and y intercepts The point ( 3, a ) lies on the straight line y = 5x 3. Find a. Find the x intercept of y = x 4. The point ( b, 1 ) lies on the line y = 7 x. Find b Use y b m( x a) to create and simplify the equation of a given straight line (m is an integer) Identify the equation from a diagram showing the point (4, ) marked and a gradient of 3 Dynamic Worksheet: StraightLine-03 Worksheet Secondary K31b - Page 10 -

11 By the end of the topic, pupils should be able to: Newbattle Community High School Maths Department Notes on approaches and activities for learning LINEAR GRAPHICAL RELATIONSHIPS (DODECAGON COURSE) SOLVE PROBLEMS USING THE EQUATION OF A STRAIGHT LINE Prerequisites from Number/Algebra courses and other topics: no specific requirements Use y b m( x a) to create and simplify the equation of a given straight line (m may be a fraction) Use y = mx + c to identify the gradient and y-intercept of any straight line in non-routine questions (e.g. those where the equation requires rearranging first) Teaching point: pupils should be taught not to use y b m( x a) as default but to develop the ability to be selective in their approach. Identify the equation from a diagram showing the point (4, ) marked and a gradient of ¾ A line has equation y + 3x = 1. What is its gradient? A line has equation 3y + 6x = 1, what is its gradient? RESOURCES (in addition to topic SmartBoard files on server) Dynamic Worksheet: StraightLine- 03 Worksheet Secondary K31b Understand the term point of intersection and how it can be found graphically. Identify the point of intersection of two lines expressed in the form y = mx + c algebraically. The strategy of equating, substitution and simultaneous equations should be explored. Use formulae to complete a coordinate point on a given straight line given either the x or y coordinate, including x and y intercepts Find the point of intersection of y = x + 1 and y = 3 x by drawing both graphs on the same axes from first principles and identifying the point of intersection Find the point of intersection of y = x + 1 and y = 3 x by first forming and solving the equation x + 1 = 3 x. Find the coordinates of the point when x = 3 on the line y + 3x = 6. What are the coordinates of the point that the line y + 3x = 6 crosses the x axis? - Page 11 -

12 SIMPLIFICATION 1 (LIKE TERMS AND ALGEBRAIC FRACTIONS) 3 rd /4 th level CfE Simplification 1 (Like terms and Algebraic Fractions) outcomes: I can collect like algebraic terms, simplify expressions and evaluate using substitution. MTH 3-14a Cross Curricular Links (whole school numeracy record): none have been identified yet (will be added later as required) By the end of the topic, pupils should be able to: Notes on approaches and activities for learning SIMPLIFICATION 1 (LIKE TERMS AND ALGEBRAIC FRACTIONS) (TRIANGLE COURSE) INTRODUCE THE IDEA THAT EXPRESSIONS CAN BE EQUIVALENT Prerequisites from Number/Algebra courses and other topics: no specific requirements Briefly discuss basic equivalences x+x+x+x always gives same answer as 4x, so x+x+x+x = 4x. Simplify y + y + y. Simplify basic expressions involving one letter and no constants. Simplify a + a. Simplify 3b + b. Simplify m 7. Simplify 7a a. Dynamic Worksheet: Algebra-03 SIMPLIFICATION 1 (LIKE TERMS AND ALGEBRAIC FRACTIONS) (PENTAGON COURSE) SIMPLIFY BASIC EXPRESSIONS Prerequisites from Number/Algebra courses and other topics: no specific requirements Be aware of common equivalences: add/multiply, unitary fractions x = 1x, x+3 = 3 + x, ab = ba, 1 x x 4 4 Collect like terms in expressions involving letters and constants a + a + a, a a + 3, a + 3a + a, a + + 4a, 5 + 3a + 3, a + b + a + b, a b + 3, 6a 4a, 5x + y x + 3y Dynamic Worksheet: Algebra-01 Worksheet Secondary J8 Simplify expressions involving multiplication of constants, and up to two letters (no powers except squaring) p q=pq, a b =ab, a 3 b = 6ab, a a, 4a a, 3y y, 5a 3a b Dynamic Worksheet: Algebra-0 Worksheet Secondary J31 - Page 1 -

13 By the end of the topic, pupils should be able to: Newbattle Community High School Maths Department Notes on approaches and activities for learning SIMPLIFICATION 1 (LIKE TERMS AND ALGEBRAIC FRACTIONS) (OCTAGON COURSE) COLLECT LIKE TERMS INCLUDING INTEGERS AND SQUARED TERMS Prerequisites from Number/Algebra courses and other topics: no specific requirements MATHS ROUTE Be aware of all equivalences from pentagon and how they are x = 1x, x 3 = 3 + x, x + ( y) = x y adapted for negative coefficients Be aware of common equivalences with fractions and division x x 3 3 Collect like terms in longer expressions involving letters and b + 3a 4a + 3b, 5 + 3a + 3 b constants (coefficients in answer may be negative) Dynamic Worksheet: Algebra-01 Worksheet Secondary J8 Simplify expressions involving multiplication of constants, and letters (powers up to 3 may be involved) Collect like terms involving a², b², ab etc. (basic examples) LIFESKILLS ROUTE No content b b b, 4 a a, 3 a 4 a, 3a 5a a a² + a², x² + x² + x, a² + b²+ a² + b², xy + 5xy, a 3a a a Dynamic Worksheet: Algebra-0 Worksheet Secondary J31 Dynamic Worksheet: Algebra-01 Worksheet Secondary J8 - Page 13 -

14 By the end of the topic, pupils should be able to: Newbattle Community High School Maths Department Notes on approaches and activities for learning SIMPLIFICATION 1 (LIKE TERMS AND ALGEBRAIC FRACTIONS) (DECAGON COURSE) COLLECT LIKE TERMS INCLUDING CUBED TERMS Prerequisites from Number/Algebra courses and other topics: octagon fractions Simplify all examples from octagon, plus ones involving x², x³ etc. Simplify algebraic fractions b + 3a 4a + 3b, 5 + 3a + 3 b, b b b, 4 a a, 3 a 4 a, 3a 5a a a² + a², x² + x² + x, a² + b²+ a² + b², xy + 5xy, a 3a a a x Simplify x, x x, ab c abc, 5 10a, 4 ab b, 4 3 ( x )( x 4) ( x 3) 6xyz,,, 3 ( x )( x 4) ( x 3) 9x y a 6a 3 Dynamic Worksheet: Algebra-01, 0 Worksheet Secondary J8, J31 Realise when an algebraic fraction cannot be simplified. Teaching approach: Get class good at recognising when to simplify, when to try factorising, and when to do nothing. Emphasise the idea of "invisible brackets" when dealing with fractions, and how when they exist you can only cancel the bracket not the other terms. pq rs, qs a 5 ab etc. Apply the four operations to algebraic fractions Express as a fraction in its simplest form: 5 4 x x, 4 3 m 3 1 3,, a,, a b 4 m b ab b From Physics: a b c, R R R, 1 3 x z, 4 a b y y 6b 8c, a c, a 4b b b y x - Page 14 -

15 By the end of the topic, pupils should be able to: Newbattle Community High School Maths Department Notes on approaches and activities for learning SIMPLIFICATION 1 (LIKE TERMS AND ALGEBRAIC FRACTIONS) (DODECAGON COURSE) BE ABLE TO USE AND SIMPLIFY INDICES AND ALGEBRAIC FRACTIONS FLUENTLY, INCLUDING NEGATIVE AND FRACTIONAL POWERS Prerequisites from Number/Algebra courses and other topics: no specific requirements Simplify algebraic fractions where either or both lines have to x 3 be factorised Simplify x 9, 3x 15 x 7x 10,, ( x 5) x 6x 8 x 10y 1 Realise when an algebraic cannot be simplified pq rs qs, a 5 etc. get class good at ab recognising when to simplify, when to try factorising, and when to do nothing. Emphasise the idea of "invisible brackets" when dealing with fractions, and how when they exist you can only cancel the bracket not the other terms Continue to practice four operations with algebraic fractions, including trickier examples Express as a fraction in its simplest form: 3 4, x 1 x x x 5, 3( x 1) zw y x 1 etc. - Page 15 -

16 SIMPLIFICATION (SURDS AND INDICES) 3 rd /4 th level CfE Simplification (Indices) outcomes: None Cross Curricular Links (whole school numeracy record): negative indices are used in science, especially physics By the end of the topic, pupils should be able to: Notes on approaches and activities for learning No content at triangle No content at pentagon No content at octagon SIMPLIFICATION (SURDS AND INDICES) (DECAGON COURSE) KNOW AND USE SURDS, NEGATIVE AND FRACTIONAL INDICES AND RULES OF INDICES Prerequisites from Number/Algebra courses and other topics: decagon surds and indices Revise numerical skills involving negative indices from Evaluate 5, Evaluate 4 3 Number course RESOURCES (in addition to topic SmartBoard files on server) Revise numerical skills involving basic fractional indices from Number course From Physics: Discuss use of indices (negative and positive) in units a b a b Apply the two rules x x x a b a b, x x x. Answers or questions may include simple negative powers , , 8 4 Physics and Engineering units: Understand why ms -1 is an alternative way of writing m/s Understand why ms - is an alternative way of writing m/s² Other equivalence used in Physics might include m³ (km), m -3 (mm), m -6 (micrometres), m -9 (nanometres), khz = 10³Hz, MHz = 10 6 Hz y y y, 6 y, y 8 y, 10b a³ 5a², 5b, a a 4 a 3 Apply the two rules with basic fractional indices (powers such as 1, 1, 3, 5 ) 1 1 y y, y 4y y, y, 1 4y 3y n Note: answers should be of the form ay : there is no need to reinterpret the fractional power as a surd at this level. - Page 16 -

17 By the end of the topic, pupils should be able to: Notes on approaches and activities for learning SIMPLIFICATION (SURDS AND INDICES) (DODECAGON COURSE) SIMPLIFY MORE COMPLEX EXPRESSIONS INVOLVING SURDS AND INDICES Prerequisites from Number/Algebra courses and other topics: decagon surds and indices Revise decagon NUNP surds and indices. Evaluate and simplify surds, involving a mixture of adding, taking away and multiplication Simplify 63 8, 00 3 RESOURCES (in addition to topic SmartBoard files on server) Simplify surd expressions involving division 50, Rationalise the denominator Rationalise the denominator and simplify Change a negative and/or fractional index into fraction or root form and vice versa 50, 5 10, Rationalise: 5 3 Rationalise: a, 5x, x, Note: complex conjugate denominators are not required 3 x, 4 x, 1 3x, 1 a etc. Revise the two rules of indices learnt at decagon, plus a b ab the third rule ( x ) x a³ 5a², Confidently manipulate expressions involving negative and fractional powers x 1 3 x, b 5b, a a 1a , 4 6, ( m ), ( a ), a a a x x, x 4x, 4a 5 a 0, 10b 4 5b, 3 4 ( y ) m, 1 4 ( a ) ( ) Use squared expressions in trigonometry Simplify simple trig expressions by multiplying and dividing Understand the equivalences sin A cos A 1 and sin A tan A cos A Simplify expressions using sin A cos A 1 and sin A tan A cos A Investigate the meaning of the terms sin A, cos A and the difference between these and cos(x ) and sin(x ) etc. cos A sina sina, cosa cosa cosa, sina sin²a, cos A, 3sin A sin A Using Autograph show the equivalent graphs. Change the subject to get sin A and cos A. See Intermediate /National 5 Past Papers for examples - Page 17 - Autograph

18 BRACKETS AND FACTORISING 3 rd /4 th level CfE Brackets and Factorising outcomes: Having explored the distributive law in practical contexts, I can simplify, I can find the factors of algebraic terms, use my understanding to identify multiply and evaluate simple algebraic terms involving a bracket. MTH 4- common factors and apply this to factorise expressions. MTH 4-14b 14a Cross Curricular Links (whole school numeracy record): none have been identified yet (will be added later as required) By the end of the topic, pupils should be able to: Notes on approaches and activities for learning No content at triangle No content at pentagon BRACKETS AND FACTORISING (OCTAGON COURSE) MULTIPLY OUT BRACKETS AND FACTORISE WHERE COMMON FACTOR IS A POSITIVE WHOLE NUMBER Prerequisites from Number/Algebra courses and other topics: no specific requirements MATHS ROUTE AND LIFESKILLS ROUTE (Maths route: aiming for mastery. Lifeskills route: aiming for an experience only, if time) Understand meaning of multiplying brackets Discover with numbers that (x+3) and x+6 always give the same answer Multiply brackets where multiplier is positive whole number (x + 3), 4(3x 4), 5(1 x) Worksheet Secondary J3 Multiply brackets and simplify (x + 3) + 5, 4(3x 4) + x, 5(1 + x) + 3( + x) Dynamic Worksheet: Algebra-05 Worksheet Secondary J3 Factorise using common factor that is a positive whole number x+4, 3 + 9x, x, 9y 1x Dynamic Worksheet: Algebra-06 Worksheet Secondary J33 - Page 18 -

19 Notes on approaches and activities By the end of the topic, pupils should be able to: for learning BRACKETS AND FACTORISING (DECAGON COURSE) BE ABLE TO WORK WITH BASIC DOUBLE BRACKETS Prerequisites from Number/Algebra courses and other topics: octagon brackets and factorising Multiply any single bracket where multiplier may be a negative number or letter x(x + 3), 4(3x 4), 5(1 x + y) Dynamic Worksheet: Algebra-05 Be aware that pupils who came through the Lifeskills route may have limited (or no) experience of working with single brackets Multiply brackets and simplify x(x + 3) + x, 5(1 + x) 3( + x) Worksheet Secondary J3 Factorise using common factor ab + a, x² + 5x, 5xy + 10yz, 10ab + 5bc + 15bd, 5x³ + x² Worksheet Secondary J33 Understand meaning of double brackets Discover with numbers that (x+)(x+3) and x²+5x+6 (and/or x²+x+3x+6) always give the same answer Multiply double brackets and simplify (a + 7)(a + 3), (x 5)(x + 3), (a )(a ), (x + 3)(x + 5), (a + 7)², (x )² Worksheet Secondary J6 Use department method to factorise trinomials where all coefficients are positive and a > 1 Note: DEPARTMENT POLICY agreed 3/09/16 is to use the method of factorising ax² + bx +c by looking for factors of ac that add to give b [and then factorising each half of the expression. Other methods can be mentioned but this is the core method that must be taught for consistency. Teach a > 1 first to establish the method, and only AFTER simplify this to the case where a = 1 Factorise a² + 9a + 4, 3x² + 5x + Example Dynamic Worksheet: Quadratics-01 Worksheet Secondary J7 3x² + 14x + 8. Need two numbers that multiply to give 4 and add to give 14 (i.e. 1 and ) Either Or 3x² + 14x + 8 3x² + 14x + 8 = 3x² + 1x + x + 8 = 3x² + x + 1x + 8 = 3x(x + 4) + (x + 4) = x(3x + ) + 4(3x + ) = (3x + )(x + 4) = (3x + )(x + 4) With more able pupils this method can even be extended relatively easily to questions such as 6m² + 11mn + 3n² see dodecagon course for more examples. - Page 19 -

20 Use department method to factorise any trinomial (where a = 1 and b or c may be negative Understand meaning of difference of two squares Factorise a² 8a + 15, x² + x Note: care needs to be taken when the term in the third position is negative (e.g. x² + x 5x 10 would be more of a problem, x² 5x + x 10 works better. Discover with numbers that (x + 1)(x 1) and x² 1 always give the same answer and other similar examples Dynamic Worksheet: Quadratics-01 Worksheet Secondary J7 Factorise expressions using a difference of two squares Factorise x² y², a² 4, 1 p², 4x² 5y², 64a² 11, 100a² p² Complete the square where a = 1 and b is even Complete the square for x² + 8x + 13, Complete the square for y² 6y + 1 Dynamic Worksheet: Quadratics-06 - Page 0 -

21 Notes on approaches and activities By the end of the topic, pupils should be able to: for learning BRACKETS AND FACTORISING (DODECAGON COURSE) BE ABLE TO WORK WITH MORE COMPLEX DOUBLE BRACKETS Prerequisites from Number/Algebra courses and other topics: no specific requirements Use department method to factorise any trinomial (where 'a' is Factorise a² 8a + 15, 3x² + x prime and b or c may be negative) Note: the method of factorising ax² + bx +c by looking Note: care needs to be taken when the term in the third for factors of ac that add to give b [and then factorising position is negative (e.g. x² + x 5x 10 would be more each half of the expression (the Hall-Leighton of a problem, x² 5x + x 10 works better. method)] has given some success in the past. Dynamic Worksheet: Quadratics- 01, 07 Worksheet Secondary J7 Revise difference of two squares Factorise 4x² 5y², 64a² 11, 100a² p², Multiply more complex brackets and simplify (4a + 7)², (x 3)², 5x + (6x )(x + 3), (x + )(x² + 5x + 3), (x + 1)(x² 3x ), (sinx + 3)(sinx + 1) Fully factorise more complex expressions where a common factor has to be taken out first Factorise fully: x² + 10x + 1, 50x² 7y², x² 18 Complete the square where a = 1 and b is odd Complete the square for x² + 7x + 13, Complete the square for y² 5y + 1 Dynamic Worksheet: Quadratics-06 Opportunity for deep learning: Use department method to factorise more complex expressions Possible examples should include: (mixture of letters) e.g. a² + ab + b², (more complex negatives) e.g. 1 x x², 9 9x x² More able classes should also experience: (non prime coefficient of x²) e.g. 6x² + 11x + 3, (other powers of x) e.g. 3x 4 + 5x² (trigonometric expressions) sin²x + 5sinx Page 1 -

22 EQUATIONS AND INEQUALITIES 3 rd /4 th level CfE Equations and Inequalities outcomes: Having discussed ways to express problems or statements using Having discussed the benefits of using mathematics to model real-life mathematical language, I can construct, and use appropriate methods to situations, I can construct and solve inequalities and an extended range of solve, a range of simple equations. MTH 3-15a equations. MTH 4-15a Cross Curricular Links (whole school numeracy record): used in science, especially physics By the end of the topic, pupils should be able to: Notes on approaches and activities for learning EQUATIONS AND INEQUALITIES (TRIANGLE COURSE) DEVELOP AN INTUITIVE UNDERSTANDING OF THE SYMBOLS AND METHODS REQUIRED FOR FORMAL SOLVING EQUATIONS IN PENTAGON Prerequisites from Number/Algebra courses and other topics: no specific requirements Work with number sentences where a number is replaced by a symbol, non-trivial examples only (i.e. not ones that can be guessed: not 3 +? = 7) - using a calculator so that focus is on operations and layout rather than calculation 78 + = 503, 3 = 07 Dynamic Worksheet: Equations-06 Develop understanding of symbols =, <, > Identify numbers by interpreting an inequality expressed using symbols (including,, ) Write the correct sign in the space: 43 7, x > 4 what could x be? y 3, what could y be? Dynamic Worksheet: Equations-03 EQUATIONS AND INEQUALITIES (PENTAGON COURSE) DEVELOP "CHANGE SIDE CHANGE OPERATION" TO SOLVE SIMPLE LINEAR EQUATIONS AND INEQUATIONS WITH POSITIVE SOLUTIONS Prerequisites from Number/Algebra courses and other topics: no specific requirements Solve two-step equations with positive integers as solutions using change side change operation, using a calculator and focussing on layout x + 45 = 11, 7x 3 = 00 Dynamic Worksheet: Equations-0 Worksheet Secondary J14 Solve one-step or two-step equations that do not have whole number solutions, giving answer as a fraction in its simplest form Discuss values that will satisfy a simple one or two-step inequation (positive integer solutions) leading to the use of formal methods will not be assessed 3x =, 4 = x, x 4 = 5, 3x 6 = x > 4, what could x be? y 3 5, what could y be? Solve x > 4, y 3 5, 7x 3 > 18 - Page - Dynamic Worksheet: Equations-07

23 By the end of the topic, pupils should be able to: Notes on approaches and activities for learning EQUATIONS AND INEQUALITIES (OCTAGON COURSE) SOLVE ANY LINEAR EQUATIONS AND INEQUATIONS Prerequisites from Number/Algebra courses and other topics: no specific requirements MATHS ROUTE AND LIFESKILLS ROUTE Revise two-step equations and inequations from pentagon including a few examples of equations containing brackets x + 5 > 11, 7x 3 < 11, 3(x + ) = 1 Deep learning opportunity: Class must also experience numerous examples written in nonstandard form such as 4 = x 6 Dynamic Worksheet: Equations-0 Worksheet Secondary J14 Solve equations or inequations with letters on both sides, or containing brackets Solve one-step equations involving dividing and/or fractions, x + 3 = x + 10, 5x > 3x + 4, 3x 1 = 3 x, 3x + > x + 6 Dynamic Worksheet: Equations-08 Worksheet Secondary J14 x 0 1, 4 3 x 13, a Dynamic Worksheet: Equations Solve equations of the form x² = a, knowing that there is a positive and negative solution x 16 x 4 a 5 x 5 In preparation for trigonometry unit solve equations involving sin, cos and tan, emphasising that these follow the same rules of equations as those already encountered x sin 30, cos x = 0 7 Practice selecting the correct strategy by encountering a mixture of different types of equation Worksheets on server: d OCTAGON WS Mixed Equations Hard 1 DGW d OCTAGON WS Mixed Equations Hard DGW - Page 3 -

24 By the end of the topic, pupils should be able to: Newbattle Community High School Maths Department Notes on approaches and activities for learning EQUATIONS AND INEQUALITIES (DECAGON COURSE) SOLVE LINEAR, QUADRATIC, TRIGONOMETRIC AND SIMULTANEOUS EQUATIONS Prerequisites from Number/Algebra courses and other topics: octagon equations, decagon brackets and factorising Be aware that multiplying or dividing an inequality by a x negative reverses the sign 3x > 1, 6, 3 5x x + 4 Worksheet Secondary J14 Solve simple quadratic equations by factorising using double brackets. (x + )(x 3) = 0, x 3x + = 0, Dynamic Worksheet: Quadratics-0 Solve quadratic equations using the quadratic formula Find the roots of x² + 5x = 0, x + 3x = 0, 3x 5x + 1 = 0 Worksheet Secondary J4 Dynamic Worksheet: Quadratics-03 Know that b² 4ac is called the discriminant, and that this has to be positive or zero. If it is negative, there are no are no solutions Calculate the discriminant of a quadratic equation, and use this to determine the nature of the roots ( two real and distinct roots (not two roots ), one real and repeated root or no real roots ) Introduce pupils to the CAST diagram Explain why x² + x + 3 = 0 has no roots What is the discriminant of x² + 3x 10 = 0? Determine the nature of the roots of the equation 3a² + a + 8 = 0 Draw 00 on a CAST diagram. What are the related angles in the other quadrants? Solve simple trigonometric equations solve sinx = 0 3 for 0 x < 360 solve cosx = 0 1 for 0 x < 360, solve sinx = 1 for 0 x < 360. Dynamic Worksheet: Equations-09 Solve trigonometric equations that require rearranging Teaching approach: emphasise how these solutions can be represented both on a graph and with CAST. Know the term system of equations Solve a system of simultaneous equations in two variables with positive coefficients only by cross multiplying to get the first coefficients the same and then subtracting Create and solve simultaneous equations in context solve sinx + 3 = 4 for 0 x < 360 solve 7cosx + 6 = for 0 x < 360 Solve the system of equations: 4x + y = 17, 3x + y = 1. Three pizzas and four kebabs cost Two pizzas and five kebabs cost What is the cost of.? - Page 4 - Dynamic Worksheet: Equations-09 Dynamic Worksheet: Equations-10 Worksheet Secondary J1

25 By the end of the topic, pupils should be able to: Newbattle Community High School Maths Department Notes on approaches and activities for learning EQUATIONS AND INEQUALITIES (DODECAGON COURSE) SOLVE ANY LINEAR, QUADRATIC, SIMULTANEOUS OR TRIGONOMETRIC EQUATION Prerequisites from Number/Algebra courses and other topics: no specific requirements Solve any linear inequality or inequation All previous examples, including brackets and/or negative coefficients. See past credit/national 5 exam papers. Solve any linear trigonometric equation using CAST diagram solve 7cosx + 6 = for 0 x < 360 Rearrange and then solve any quadratic equation that can be factorised Use a condition on the discriminant to find the nature of roots x + 3x = 0, x + 6x + 4 = 0, x + x = 15, x x = 0, 4x 100 = 0, x x 6 = x 6. x(x 5) = 7 e.g. x² + px + 5 = 0 has two real and distinct roots. Find the values of p. Dynamic Worksheet: Quadratics-0 Choose correct strategy and solve any quadratic equation Factorising and/or quadratic formula Solve simultaneous equations in two variables where some coefficients may be negative, including examples in context. Use default method first (cross multiply by first coefficient and subtract) then explore other ways of solving as appropriate (e.g. adding the equations) Solve the system of equations 4x + y = 6, 3x y = 1. Emphasise the term system of equations so that pupils know in an exam that this means to use the simultaneous equations technique. Dynamic Worksheet: Equations-10 Worksheet Secondary J1 - Page 5 -

26 NON-LINEAR GRAPHICAL RELATIONSHIPS 3 rd /4 th level CfE Non-Linear Graphical Relationships outcomes: None Cross Curricular Links (whole school numeracy record): none have been identified yet (will be added later as required) By the end of the topic, pupils should be able to: Notes on approaches and activities for learning No content at triangle No content at pentagon No content at octagon Using a table of values at first, then Autograph as necessary, teacher leads class through sketching basic parabolas, and introducing the concept of a parabola Emphasise turning points, axis of symmetry, (if appropriate roots). NON-LINEAR GRAPHICAL RELATIONSHIPS (DECAGON COURSE) SKETCH AND IDENTIFY SIMPLE PARABOLAS AND TRIG GRAPHS y = x², then y = x², then y = ±x² ±, then y = ±(x ± 3)², then y = ±(x ± 1)² ± 4. Key concept: the original graph of y = x² has been translated. Group work matching graphs and equations Identify the equation (of the form y = kx²) of a graph given one coordinate point by substituting for x and y From the equations of the form y (or f(x) ) = (x b) + c or y= (x b)² + c, state:: co-ordinates and nature of turning point equation of axis of symmetry From the equations sketch the graphs of a parabola of the form y (or f(x) ) = (x b) + c or y= (x b)² + c Graphs should include: clearly annotated coordinates of turning point y-intercept Identify the equation of a parabola when shown the graph (the formula y = ±(x b)² + c would always be given) Identify the solution/roots of a simple quadratic equation of the form x + bx + c = 0 from a graph. Identify equation of the graph of y = kx given the parabola and the point on (1,4). The point (5, 75) lies on the graph of y=kx². What is k? y = (x + 3) 5, f(x) = (x 6)² +. State the coordinates and nature of the turning point. State the equation of the axis of symmetry. Sketch y = (x 4), f(x) = (x + 3) + 5, y = (x 5) 3, y = x², f(x) = x² "happy" parabola shown with turning point (, 3). Pupil has to write down the equation. Using the graph write down solutions of x x 6 = 0. - Page 6 -

27 From the equations sketch the graphs of a parabola of the form y (or f(x) ) = (x a)(x b) Graphs should include: clearly annotated coordinates of turning point y-intercept roots Draw y (or f(x) ) = sinx, y = cosx and y = tanx using table of values and know the features of each graph, and the vocabulary period, amplitude and frequency. Sketch the graphs of y (or f(x) ) = asinbx, y = acosbx and y = tanbx; Key concept: talking about transformations of the graph horizontally and/or vertically. Identify the equations of graphs with equations of the form y (or f(x) ) = asinbx, y = acosbx and y = tanbx from a sketch Sketch the graph of f(x) = (x + )(x 4) showing turning point, y intercept and roots. To avoid unnecessary complications, coordinates of Turning Points should be chosen to be whole numbers only. (This takes some pre planning however) Using Autograph discuss the TPs, x intercepts, periodicity, amplitude. What are the values of x when sin is negative? Sketch the graph of: y = 3cosx, f(x) = cosx, y = tanx, y = sin3x, f(x) = cos3x, y = sin4x, annotating the x intercepts and TPs. - Page 7 -

28 By the end of the topic, pupils should be able to: Notes on approaches and activities for learning NON-LINEAR GRAPHICAL RELATIONSHIPS (DODECAGON COURSE) BECOME READY FOR HIGHER WHEN WORKING WITH GRAPHS Prerequisites from Number/Algebra courses and other topics: dodecagon brackets and factorising Show whether or not a particular point lies on a particular graph Does the point (4, 5) lie on the curve y = x² x + 6? (parabola, trig, straight line) RESOURCES (in addition to topic SmartBoard files on server) Given an x or y value, complete the coordinates of a point on a given straight line, trig graph or curve - in words or from a diagram; including x and y-intercepts By first factorising to get the roots, use symmetry identify the coordinates of the TP of a graph of the form y = x + bx + c. From the equations sketch the graphs of a parabola of the form y (or f(x) ) = x + bx + c (for equations that can be factorised) Graphs should include roots, turning point, y-intercept The point (, c) lies on y = x² + 5. What is c? The point (45, b) lies on the graph y = sin x. What is b? (also repeat a few straight line examples as well) Simple examples such as: Find the coordinates of TP of f(x) = x + 7x + 6, or y = 0 + x x². Sketch the graph y = x + x 15 = 0, annotating the TP, x and y intercepts. Find the turning point of y = x + bx + c by completing the square (a) Complete the square for f(x) = x + 6x + 8 (b) Hence state the turning point of the graph of f(x) = x² + 6x + 8 Given certain co-ordinates, use symmetry properties to calculate other co-ordinates in a diagram involving one or more parabolas. Brief general overview of transformations of graphs. (Function notation will not be required in this context, although could be used if teacher feels class will cope) Sketch graphs of the form y=acos(x ± b), f(x) = asin(x ± b), y=tan(x ± b), f(x) =acos(x) ± c, y=asin(x) ± c, y=tan(x) ± c and identify the period. Key concept: talk about translations of the graph, and link to parabola work e.g. Turning Point and one root shown. Calculate co ordinate of other root. e.g. Three adjoining parabolas shown. Root and turning point of first graph shown. Find equation of third graph. e.g. Graph in context of a projectile. Two roots shown. Calculate maximum point. "if you add a number at the end, the graph moves up", "if you multiply by a number in front, the graph is stretched up" Revise previous sketching trig. graphs examples. Sketch the graphs of y = sin(x + 30), y = cos(x 45), y = sin x + 1, y = cos x etc. Int /National 5 and Credit Past papers - Page 8 -

29 CREATING EXPRESSIONS AND EQUATIONS 3 rd /4 th level CfE Creating Formulae outcomes: Having explored number sequences, I can establish the set of numbers generated by a given rule and determine a rule for a given sequence, expressing it using appropriate notation. MTH 3-13a I can create and evaluate a simple formula representing information contained in a diagram, problem or statement. MTH 3-15b Having explored how real-life situations can be modelled by number patterns, I can establish a number sequence to represent a physical or pictorial pattern, determine a general formula to describe the sequence, then use it to make evaluations and solve related problems. MTH 4-13a Cross Curricular Links (whole school numeracy record): none have been identified yet (will be added later as required) By the end of the topic, pupils should be able to: Notes on approaches and activities for learning CREATING EXPRESSIONS AND EQUATIONS (TRIANGLE COURSE) USE TABLES OF VALUES TO IDENTIFY AND USE RULES CONNECTING TWO VARIABLES Prerequisites from Number/Algebra courses and other topics: triangle formulae Describe the rule from a table of values in context (e.g. Number of Legs = 4 Number of Chairs, tables/chairs) (one operation - add, take away or multiply): first then L = 4C in words, then as a formula CREATING EXPRESSIONS AND EQUATIONS (PENTAGON COURSE) CONSTRUCT MORE DIFFICULT FORMULAE IN CONTEXT AND FROM TABLES OF VALUES Prerequisites from Number/Algebra courses and other topics: no specific requirements Extend ability to describe the rule from a table of values (two operations, dividing, squaring): first in words, then as a formula Cost of Taxi Ride = Miles + 3 (then F = M + 3); Area of Square = Length of side² (then A = L²) Create equations from expressions and solve If x + 45 has an answer of 93, what is the value of x? If 100 is equal to 4a + 0, what is the value of a? If 3y 7 is less than, what are the possible values of y? RESOURCES (in addition to topic SmartBoard files on server) - Page 9 -

30 By the end of the topic, pupils should be able to: Notes on approaches and activities for learning CREATING EXPRESSIONS AND EQUATIONS (OCTAGON COURSE) CONSTRUCT AND USE MORE COMPLEX FORMULAE Prerequisites from Number/Algebra courses and other topics: octagon perimeter/area/volume MATHS ROUTE Describe the rule from a table of values, and hence create a formula in a real-life context. Use formula to work out output value when input value is known (in context) In context: see Standard Grade General past papers and National 4 Mathematics AVU RESOURCES (in addition to topic SmartBoard files on server) Active Worksheet Website: Worksheet Secondary K1, K, K3 (not in context) Use formula to create and solve an equation to calculate the input value when the output value is known (in context) Be aware of the distinction between an expression and a formula Create expressions in a real-life context either from sentences or diagrams Brief discussion A piece of wood is 50cm long. x centimetres is cut off the end. How long is the piece of wood? A garden is 5 metres long is extended with a path that is y metres long. How long is the garden now? Andy is twice as old as Ben. If Ben is x years old, how old is Andy? Create an expression or formula to describe geometric situations Volumes of cuboids, area of triangle, composite shapes, circles, where at least one length is expressed as a letter Use a created expression to create and solve an equation Pupil has just created an expression in a real life context (5x + 0, say). If they are now told the answer is 55, they should then be able to write 5x + 0 = 55 and solve. LIFESKILLS ROUTE No content - Page 30 -

31 CREATING EXPRESSIONS AND EQUATIONS (DECAGON COURSE) DEVELOP A BASIC UNDERSTANDING OF REAL-LIFE APPLICATIONS OF QUADRATIC EXPRESSIONS Prerequisites from Number/Algebra courses and other topics: decagon brackets and factorising Express the area of a rectangle, or volume of a cuboid as a quadratic expression Use a created quadratic expression to create and solve a equation Construct a linear equation with letters on both sides from information given in a real-life context and solve Recap all decagon content (plus simultaneous equations) to create equations and solve problems a rectangle has length x + and breadth x 5, write an expression for its area. A cuboid has dimensions, y and y + 3, write an expression for its volume The area of the rectangle is. How long is x? See textbooks for examples. CREATING EXPRESSIONS AND EQUATIONS (DODECAGON COURSE) BE ABLE TO SOLVE PROBLEMS IN REAL LIFE SITUATIONS Intermediate exam style questions (e.g. area of garden) Look at more difficult questions involving creating an equation in geometric situations e.g. surface area of cuboid, area under a straight line create appropriate questions along a similar style to a Higher optimisation part (a) - Page 31 -