NUAP Full Course Outline - Newbattle Community High School

17/08/2014
Newbattle Community High School Maths Department
Newbattle Understanding Algebra Progression S1‐S6
August 2014
Course
Aim
Triangle Course
Get a concrete understanding of Develop formal methods using the relationships between positive numbers only
numbers and how letters can be before they move to the next level, pupils will be expected to master:
Substitution
Pentagon Course
substitute positive integers into basic algebraic expressions
* evaluate expressions (adding, taking away, multiplying and order of operations involving these) with positive whole number (answers always ≥ 0)
Octagon Course
Decagon Course
To extend formal methods to National 4 Added Value Unit standard using integers
before they move to the next level, pupils will be expected to master:
before they move to the next level, pupils will be expected to master:
substitute positive integers into more complex algebraic expressions
substitute integers and decimals into complex algebraic expressions
Dodecagon Course
To develop the skills required for To complete the skills required for the National 5 exam and to National 5 ‐ first part of course
become Ready for Higher
before they move to the next level, pupils will be expected before they move to the next level, pupils will be expected to master:
to master: RfH indicates optional content that is not required at National 5, but is advised for pupils attempting one year Higher
substitute into all previous plus quadratic expressions
become Ready for Higher
* evaluate expressions involving brackets, fractions, BODMAS, basic squares and square * evaluate all expressions previously encountered * evaluate quadratic terms
roots ‐ substituting (and answers) positive whole by substituting integers and decimals
numbers ONLY
* introduce function notation
Examples: 5+2a, 2a+3b, 3a+2b +c, 2ab, ‐4b, Examples: what is x² + 5x + 14 when x = 2, x = –1 Examples: 3(a+b+c), (a+b)/2, 2(a +b)+ 5, ½a, Examples: f(x)=x²+5x+4, what is f(2)? f(10)? 2a‐3b, 3(a+b+c), (a+b)/2, 2(a +b)+ 5, ½a, etc? what is 3x² + 5x + 14 when x = 2, x = –1 a/2, ¾x, 3x/4 , (a+b+c)/4, (a + b)/(c+d), a² (a a/2, ¾x, 3x/4, (a+b+c)/4, (a + b)/(c+d), etc? what is (x – 3)² – 4 when x = 2, x = –1 etc? g(x)=x²‐1. What is g(2)? g(‐2)? g(a)?
Examples: a + 7, 2a, 2a + 3, 5a + 2b, 3a + 2b is a small integer), sqrt(a) where a is a square 1/a + 1/b (physics), 1/a + 1/b + 1/c (physics) What is (x – 7)(x + 5) when x = 4?
number + c , 2ab, 4 +2 a, 4a – 2b
Suggested teaching approach: class experiment * evaluate expressions involving surds
with values of x to discover that 2(n+3) and 2n+3 * evaluate expressions involving squares, cubes
give different answers; see that 3(x+2) and 3x+6 always give the same answer
Examples: a², 3b², 2x²± 4y², a³, 5b³, xy², Examples: what is 6/√x when x = 9? What is 3√x (xy)², x²y, √(4x), 4√x, b² ‐ 4ac
when x = 8?
* TEACHING APPROACH: make class aware of * TEACHING APPROACH: make class aware of "invisible brackets" when evaluating fractions and "invisible brackets" when evaluating fractions and square roots
square roots
a  b ( a  b)
a  b (a  b)
b  4 ac  (b  4ac )

,
x  1  ( x  1) e.g. c  d  (c  d ) ,
e.g. 2
2
c
c
2
2
3b  3(b )
Using Formulae
be able to use simple formulae in context
* use formulae that model familiar context in words and algebra and function machines: addition, subtraction, and multiplying two numbers
develop knowledge of formulae including BODMAS
extend knowledge of formulae
* continue to use a wide range of formulae, * use formulae that model real‐life situations in * continue to use a wide range of formulae, words and algebra: including unitary fractions, ×½ including substituting decimal and integer values including those that will be encountered in the volume topic
, squaring, square rooting, powers, BODMAS and and dealing with invisible brackets
invisible brackets
Examples: K = 5/8M or 5M/8, F=9/5C+32 or, F =9C/5 +32, V= (πr²h)/3, V= (4πr³)/, 3, Examples: Perimeter of Rectangle P=2A + 2B or A=πr², A=4πr², V= πr²h, , c = √(a² + b²), Examples: Area = Length × Breadth, Cost = No. P=2(A+B), A = ½BH, A = L², L = √A, (from y=½x+8, (from physics:) s = ut + ½at² , s = Items × Price, A=LB, P=a+b+c, D=ST, D=2R
physics:) a = (v ‐ u ) / t [with visible and/or ½(u+v)t, E=½mv², PV/T = constant, V=rt(PR), (from physics:) V=IR, W=QV, p=mv, Q=It
invisible brackets], E=mgh
V=rt(2E/mR), P = I²R, P = V²/R, I = k/d², E = ½CV², V2 = (R2/(R1+R2))Vs
* extend to formulae that include division, and linear operations (e.g. ax+b), multiplying more than two numbers, using the fact that multiplying comes before adding and/or taking away
Examples: 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 s=d/t,) C=mp+r, y=mx+c, A = LB/2, A=n/t, R=D/2, (from physics:) P = F/A, E = V + Ir, Emphasise the use of / in formula meaning divide
develop formula skills needed for Intermediate 2
* evaluate expressions involving negative and fractional powers
Examples: what is a‐2 when a = 3? What is x4/3 when x = 8?
* (RfH) evaluate all previous expressions by substituting in fractions
Examples: what is x² + 5x + 14 when x = ½? What is 2x² when x = ¼?
use all formulae required for Intermediate 2 confidently
* substitute confidently into all formulae encountered in the intermediate 2 course, including dealing with invisible brackets
Examples sine rule, cosine rule, cosine rule for Examples: V= (πr²h)/3, V= (4πr³)/3, A=πr², angles, gradient formula (with integers), volume V= πr²h, (from physics:) v² = u² + 2as, 1/RT = formulae (with decimals), quadratic formula (with 1/R1 + 1/R2, 1/RT = 1/R1 + 1/R2 + 1/R3 integers), equation of a parabola
* develop the "change side do the opposite" * change the subject of linear formulae (of the method learnt for solving equations at pentagon form y=mx+c) and formulae that involve only level, into a strategy for changing the subject of a multiplying [e.g. of the form V = abcd]
simple formula
* change the subject of more complicated formulae that may involve squaring, square rooting, brackets (visible or invisible) and fractions
Examples: change the subject of S=D/T to D or T, (Request from Physics and Engineering to use Examples: change the subject of y=mx+c to x, small letters s=d/t,) change the subject of C=d V=r²h to h , V=1/3r²h to h [giving answer to d, y=a±b to a or b. From Physics P=IV, V=IR, as h = V / (1/3r²) rather than 3V/r² ] F=ma Extend as appropriate to class. From Physics P=I²R, P=V²/R, E=mgh
Examples: change the subject of V=r²h to r , h=1/2(a+b) to b, h=1/2h(a+b) to h, A = b/c + d to c, K = (m2 n)/p to m, p= √(q + s) to s. From Physics s=1/2(u+v)t, S=ut+1/2at²
* hence or otherwise, develop a strategy for going * hence or otherwise, develop a strategy for going * hence or otherwise, develop a strategy for backwards in a real life context
backwards in a real life context
going backwards in a real life context
Examples: the volume of a cuboid is 450cm³. If Examples: if y = 4x ‐ 5, and y = 23, what is x? the length is 10cm and the breadth is 4.5cm, what The volume of a cylinder is 2000cm³ and the is the height? The circumference of a circle is radius is 3cm, what is h?
60cm. What is the diameter?
Examples: 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?
17/08/2014
Newbattle Understanding Algebra Progression S1‐S6
Linear Graphical Relationships
begin to develop a concept of how a mathematical rule can be drawn as a graph on a four quadrant grid
* understand the meaning of the terms x‐
coordinate and y‐coordinate (on a 4 quadrant grid)
be able to draw a graph from an equation using a table of be able to use y = mx + c for all lines including those with a values, and then to discover the link between the fractional gradient
equation, gradient and y‐intercept ( integers only )
* follow simple rules in words to identify and plot * extend the concept of gradient points, and to draw a line
Examples: "the x‐coordinate is double the y‐
coordinate", "the y‐coordinate is ‐2"
* draw lines of the form x = a or y = b (horizontal/vertical) without needing a table of * follow simple rules in words to identify and plot values
points, and to draw a line
Examples: "give me a coordinate point where the Examples: Draw x = 2, y = –4 without a table
y‐coordinate is 6", "give me a point where the x‐
coordinate is double (or half, treble, 2 bigger than) the y‐coordinate". * use tables of values to draw lines of the form y=mx+c on a four‐quadrant grid, including integers and fractions with a grid
Examples: Draw y = 2x+7, y = 3‐2x, y = ½x+3, y = –3x + 2 with a table (past Intermediate 1 exam questions)
* understand the concept of gradient
solve problems using the equation of a straight line
* use y ‐ b = m ( x ‐ a ) to create the equation of a given straight line (m and c integers only)
Example: gradient as vertical over horizontal; gradient of ½ as "along 1 up ½" leading to "along Examples: identify the equation from a diagram 2, up 1", gradient of ‐¾ as "along 4, down 3", showing the point (4 , 2) marked and a gradient of gradient of zero is horizontal; vertical line has 3
undefined gradient
* use y = mx + c to identify the gradient and y‐
* understand and use gradient formula intercept of any straight line in non routine m = (y2‐y1)/(x2‐x1)
questions (e.g. those where the equation requires rearranging first)
Teaching point: pupils should be taught not to Examples: find the gradient of the line joining (–2, use y‐b=m(x‐a) as default but to develop the 1) and (3, –4). Identify gradient of a line drawn ability to be selective in their approach. Example: on squared paper by choosing suitable points. a line has equation y + 3x = 1. What is its Identify gradient of a line drawn using a non‐
gradient? A line has equation 3y + 6x = 12, what standard scale.
is its gradient?
* be able to draw a straight line (in the form * understand the term point of intersection and y=mx+c) from its equation on squared paper how it can be found graphically.
without requiring a table of values
Example: find the point of intersection of y = x + Examples: draw y = 4x – 5, y = 3 + ½x, y = 5 1 and y = 3 – x by drawing both graphs on the –x, y = ¾x + 2, y = 5, x = –2 without same axes from first principles and identifying the needing a table of values.
point of intersection
* be able to sketch a straight line from its * use algebra to identify the point of intersection equation on plain paper, correctly annotating the of two lines expressed in the form y = mx + c
sketch
The strategy of equating, substitution and Example: gradient as steepness; as how far you Examples: sketch y = 2x – 5, sketch y = 5, simultaneous equations should be explored. go up for every one you go along; gradient of 2 as sketch y = –5x, sketch a possible graph for y = ax Example: Find the point of intersection of y = x + "along 1 up 2", gradient of ‐3 as "along 1, down 1 and y = 3 – x by first forming and solving the + b given that a > 0 and b = 0 etc
3"; gradient of zero is horizontal
equation x + 1 = 3 – x . * use algebra to complete a coordinate point on a * be able to identify the gradient of a straight line * be able to identify the equation of any straight given straight line given either the x or y from a diagram using m = V/H (answers whole line from a diagram numbers only)
coordinate, including x and y intercepts
Examples: identify gradient of y = 2x + 5 from its graph. Identify gradient of y = 3 – x from its Examples: any equation of the form y = mx + c
graph. Identify gradient of line joining (1, 1) and (4, 10) from a diagram
* use algebra to complete a coordinate point on a * be able to identify the gradient and y‐intercept given straight line given either the x or y of a given straight line from its graph
coordinate, including x and y intercepts
Examples: the point ( 3, a ) lies on the straight line Example: pupil shown graph of y = 4x + 5 on y = 5x – 3. Find a. Find the x‐intercept of y = 2x – Autograph and will be able to identify its gradient 4. The point ( b, 1 ) lies on the line y = 7 – x. Find and y‐intercept
b
* discover connection between gradient, y‐
intercept and equation of a straight line leading to y=mx+c
Examples: use Autograph, investigation
* be able to write down the gradient and y‐
intercept of a straight line when told its equation in the form y = mx + c (m, c integers)
Examples: the equation of a line is y = 4x + 5. What is its gradient? The equation of a line is y = 3 – 2x. Where does it go through the y‐axis?
Examples: Find the coordinates of the point when x = 3 on the line 2y + 3x = 6. What is the coordinates of the point that the line 2y + 3x = 6 crosses the x‐axis? 17/08/2014
Newbattle Understanding Algebra Progression S1‐S6
Simplification
introduce the idea that expressions can be equivalent
simplify basic expressions
Simplification Part 1 ‐ * simplify basic expressions involving one letter * be aware of common equivalences: Like terms and add/multiply, unitary fractions
algebraic fractions and no constants.
Example: x+x+x+x always gives same answer as 4x , so x+x+x+x = 4x. Simplify y + y + y. Simplify 2a + a. Simplify 3b + 2b. Simplify m × 7. Simplify 7a – 2a.
x=1x, 2x+3 = 3+2x, ab = ba , ¼ x = x/4
collect like terms including squared terms
be able to use and simplify indices and algebraic fractions fluently, including negative and fractional powers
* be aware of all previous equivalences and how * simplify all previous examples, plus ones they are adapted for negative coefficients
involving x², x³ etc
Examples: –x = –1x, 2x – 3 = –3 + 2x, x + (–y) = x – y
* be aware of common equivalences with fractions and divide
* simplify algebraic fractions
Examples: ⅔x = 2x/3
Examples: simplify x/x², x²/x, ab²c/abc², Examples: simplify (x + 3)/(x² – 9), (3x – 5/10a, (2x + y)/4, 4ab/2b, 2a²/6a³, (x + 2)(x + 15)/(x – 5)², (x² + 7x + 10)/(x² + 6x + 8), (2x + 10y)/12
4)/(x – 2)(x + 4), (x + 3)²/(x + 3)³, 6x4y²z³/9x²y
* collect like terms in longer expressions involving * collect like terms in expressions involving letters letters and constants (answer can be negative)
and constants
Example: a + a + a, a+1+a+3, 2a + 3a + a, 2a + 2 + 4a, 5 + 3a + 3, a+b+a+b, 2a+5+ 3b +3, 6a – Examples: 2b + 3a – 4a + 3b, , 5 + 3a + 3 – 2b
2 – 4a, 5x + 2y – x + 3y
* simplify expressions involving multiplication of * simplify expressions involving multiplication of constants, and up to two letters (no powers constants, and letters (powers up to 3 may be except squaring)
involved)
Simplication part 2 ‐ Laws of indices
collect like terms including cubed terms
* realise when an algebraic fraction cannot be simplified
* simplify algebraic fractions where either or both lines have to be factorised
* realise when an algebraic cannot be simplified
Examples: (pq + rs)/qs, (a + 5)/ab etc ‐ 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
* apply the four operations to algebraic fractions
* continue to practice four operations with algebraic fractions, including trickier examples Examples: express as a fraction in its simplest form: 5/x + 4/x, 4/a + 3/b, m/4 – 3/m, a – Examples: express as a fraction in its simplest Examples: b × b × b, 4 × a × a, 3 × a × 4 × a,
1/b , 3/ab + 2/b, From Physics: 1/a+1/b+1/c, form: 2/(x + 1) + 3/x, 4/(x – 2) – 2/(x + 5), 3a × 5a × a
1/R1+1/R2+1/R3, x/y × z/y, 4a/6b × 2b/8c, 3(x + 1)²/y × zw/(x + 1) etc
Examples: p×q=pq, 2×a×b =2ab, 2×a×3×b=6ab, a/b ÷ c/b, 2a/y ÷ 4b/x
a × a, 4a × a , 3y × 2y, 5a × 3a × 2b
* collect like terms involving a², b², ab etc (basic examples)
Examples: 2a² + a², x² + x² + x, a² + b²+ a² + b², 2xy + 5xy, 2a x 3a ‐ 2a x 2a
* revise negative and fractional powers from * be able to change a negative and/or fractional NUNP at octagon and decagon level (surds and index into fraction or root form and vice versa
indices). Discuss use of indices (negative and positive) in units Examples: see NUNP plan. Examples of units ‐2
5
from Physics and Engineering: m‐3, m‐6, m‐9, m/s = Examples: 5x , 10/a , x^(½), x^(3/2), 3x^(½), a^(‐½) etc
‐2
6
ms‐1, m/s² = ms , kHz = 10³Hz, MHz = 10 Hz * be able to apply the rules xa × xb = xa+b, xa ÷ xb = * be able to manipulate expressions involving xa‐b, (xa)b=xab. Answers or questions may include powers, including negative and fractional powers
simple negative powers
Examples: x^(1/2) × x^(3/2), x‐4 ÷ x‐3, 2x½ ÷ 4x, 4a‐5 Examples: x7 × x8, x8 ÷ x7, x7 ÷ x8, 2a³ × 5a², 10b4 ÷ 5b, (m7)8, (a‐3)4 * use squared expressions in Trig.
¾
× 2a20, 10b4 ÷ 5b, (m3) , (a‐½)4
* Understand the equivalences sin²A + cos²A = 1 and tanA = sinA/cosA
Using Autograph show the equivalent graphs. Investigate the meaning of the terms sin2A, cos2A Change the subject to get sin2A and cos2A.
and the difference between cos(x2) and sin(x2) etc. * Simplify simple trig expressions by multiplying * Simplify expressions using sin²A + cos²A = 1 and and dividing
tanA = sinA/cosA
Examples: sinA × sinA, cosA × cosA × cosA, 2
sinA×sin²A, cos²A/cosA, 3sinA/sin A
See Intermediate 2 Past Papers for Examples
17/08/2014
Newbattle Understanding Algebra Progression S1‐S6
Brackets and Factorising
multiply out brackets and factorise where common factor is a positive whole number
* understand meaning of multiplying brackets
be able to work with basic double brackets
* multiply any single bracket
be able to work with more complex double brackets
* complete the square where a=1 and b is even
Examples: discover with numbers that 2(x+3) and Examples: x(x+3), –4(3x – 4), 5(1–2x+2y)
2x+6 always give the same answer
Examples: complete the square for x² + 8x + 13, complete the square for y² – 6y + 12
* multiply brackets where multiplier is positive whole number
Examples: 2(x + 3), 4(3x – 4), 5(1 – 2x)
* multiply brackets and simplify
Examples: 2(x + 3) + 5, 4(3x – 4) + 2x, 5(1 + 2x) + 3(2 + x)
* factorise using common factor that is a positive whole number
* factorise any trinomial (where 'a' is prime and b or c may be negative)
* multiply brackets and simplify
Examples: x(x+3) + x, 5(1+2x) – 3(2+x)
* factorise using common factor
Examples: ab + a, x² + 5x, 5xy + 10yz, 10ab + 5bc + 15bd, 5x³ + 2x²
* understand meaning of double brackets
Examples: discover with numbers that (x+2)(x+3) Examples: 2x+4, 3 + 9x, 12 + 36x, 9y – and x²+5x+6 (and/or x²+2x+3x+6) always give the Examples: Factorise a² – 8a + 15, 3x² + x – 2
12x
same answer and other similar examples
* multiply double brackets and simplify
* multiply more complex brackets and simplify
Examples: (4a+7)², (2x–3)², 5x + (6x‐2)(x+3), Examples: (a+7)(a+3), (x‐5)(x+3), (a–2)(a–2), (x+2)(x²+5x+3), (2x+1)(x²–3x–2), (sinx + (2x+3)(x+5), (a+7)², (x–2)²
3)(sinx+1)
* factorise trinomials where all coefficients are * factorise more complex expressions using a positive and 'a' is prime
difference of two squares
Examples: Factorise 4x² – 25y², 64a² – 121, Examples: Factorise a² + 8a + 15, 3x² + 5x + 2
100a² – p²
* factorise any trinomial (where 'a' is prime and b * factorise more complex expressions
or c may be negative)
Examples: Factorise a² – 8a + 15, 3x² + x – 2
* understand meaning of difference of two squares
Examples: Factorise a² + 2ab + b², 3a²–24a+45, 9–9x–2x², 6x²+11x+3, 2x²–18, 3x4+5x²‐2 (is explicitly on Int 2 syllabus), sin²x+5sinx+6
METHOD: methods may involve trial and error, drawing a table, or any other method that works
Note: the method of factorising ax² + bx +c by Examples: discover with numbers that (x+1)(x–1) looking for factors of ac that add to give b [and and x²–1 always give the same answer and other then factorising each half of the expression (the similar examples
Hall‐Leighton method)] has given some success with weaker S5 classes in the past.
* factorise expressions using a difference of two squares
Examples: Factorise x² – y², a² – 4, 1 – p²
17/08/2014
Newbattle Understanding Algebra Progression S1‐S6
Equations and Inequalities
develop an intuitive understanding of the symbols and methods required for formal solving equations in pentagon
develop "change side change operation" to solve simple linear equations and inequations with positive solutions
solve linear equations and inequations with positive solutions, including fractions
* work with number sentences where a number is replaced by a symbol, non‐trivial examples only * revise two‐step equations from pentagon and (i.e. not ones that can be guessed ‐ not 23 + ? = * solve two‐step equations with positive integers extend to inequalities
as solutions using change side change operation, 27!) ‐ using a calculator so that focus is on using a calculator and focussing on layout
operations and layout rather than calculation
Examples: 2x + 5 = 11, 7x – 3 = 200
Examples: 2x + 5 > 11, 7x – 3 < 11, 4 = 2x – 6
Examples: 278 + ___ = 503, 23 × ___ = 207
* solve equations or inequations with letters on both sides, or containing brackets (inequalities with * solve one‐step or two‐step equations that do letters on both sides will not be assessed in finishing not have whole number solutions, giving answer exercise but should be included as a natural part of class * develop understanding of symbols =, <, > as a fraction
teaching)
Solve any linear and simple Quadratic Equations
Solve any linear and any Quadratic Equation
* be aware that multiplying or dividing an inequality by a negative reverses the sign
* Solve any linear inequality or inequation
Examples: –3x > 12, x/–3 < 12
All previous examples
* Solve simple Quadratic Equations by factorising * Solve any quadratic equation that can be using double brackets. factorised.
Examples: 2x + 3 = x + 10, 5x –2 > 3x+4, 2 Examples: 43 ___ 27, 2 + 3 ___ 4 ‐ 1 ‐ write Examples: 3x = 2, 4 = 2x, 2x – 4 = 5, 3 – 2x = 3x – 12, 3(x + 2) = 12, 3x + 2 Examples: (x + 2)(x – 3) = 0, x – 3x + 2 = 0, > 2x + 6, 5(x – 3) > 20
the correct sign in the space
3x – 6 = 2 Examples: 2x2 + 3x – 2 = 0, 2x2 + 6x + 4 = 0, x2 + 2x = –15, 2x2 = 4, 2x2 – x = 0, 4x2 – 100 = 0, x2 – x – 6 = x – 6.
* solve one‐step equations involving dividing * Solve Quadratic Equations by using the and/or fractions, From Physics: solving equations quadratic formula
after numbers have been substituted
* Choose correct strategy and solve any quadratic equation
* identify numbers by interpreting an inequality expressed using symbols (including ≥, ≤, ≠)
* give values that will satisfy a simple one‐step inequation (positive integer solutions) formal methods will not be assessed
Examples: x > 4 what could x be? y ≥ 3, what could y be?
Examples: x/4 = 20, 1/3x = 13 From Physics: find v when a = 0.1, t = 30, u = 0.2 using s = Example: Find the roots of x 2 + 3x – 2 = 0, Examples: 2x > 4, what could x be? y – ½(u+v)t, find t when u = 0, v = 30, a = 750 using v 3x2 – 5x + 1 = 0
= u+at
3 ≤ 5, what could y be?
* solve equations of the form x²=a, knowing that * know that b² – 4ac is called the discriminant, there is a positive and negative solution (will not and that this has to be positive or zero. If it is be assessed in finishing exercise)
negative, there are no are no solutions
* substitute and solve using Physics formulae Examples: x² = 16, a² = 25
Examples: From Physics (H): find a when s = 0.6, v = 30, u = 0 using v² = u² + 2as
Example: explain why x² + x + 3 = 0 has no roots
* as part of trigonometry unit solve equations involving sin, cos and tan, emphasising that these * introduce pupils to the CAST diagram
follow the same rules of equations as those already encountered
Examples: x/7 = sin30, cosx = 0.2
Examples: draw 200° on a CAST diagram. What are the related angles in the other quadrants?
* solving simple trigonometric equations Examples: solve sinx = 0.3 for 0 ≤ x < 360, solve cosx = –0.1 for 0 ≤ x < 360, solve sinx = 1 for 0 ≤ x < 360. * solving trigonometric equations that require rearranging
Examples: solve 2sinx + 3 = 4 for 0 ≤ x < 360, solve 7cosx + 6 = 2 for 0 ≤ x < 360; Teaching approach: emphasise how these solutions can be represented on a graph and CAST.
* solving simultaneous equations in two variables * solving simultaneous equations in two variables where some coefficients may be negative. Use with positive coefficients only by cross multiplying default method first (cross multiply by first coefficient and subtract) then explore other ways to get the first coefficients the same and then subtracting
of solving as appropriate (e.g. adding the equations)
Examples: solve 4x + 2y = 17, 3x + y = 12. Three pizzas and four kebabs cost £34.07. Two Examples: solve 4x + 2y = 26, 3x ‐ y = 12. pizzas and five kebabs cost £34.73. What is the cost of…. ?
17/08/2014
Newbattle Understanding Algebra Progression S1‐S6
Non‐Linear Graphical Relationships
Sketch and identify simple parabolas and trig graphs
* 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
Become Ready for Higher when working with graphs
* show whether or not a particular point lies on a particular graph (parabola, trig, straight line)
Examples: y = x², then y = 2x², then y = ±x² ± 2, then y = ±(x ± 3)², then y = ±(x ± 1)² ± 4. Discuss turning points, axis of symmetry, (if appropriate Example: Does the point (4, 5) lie on the curve y = roots). Key concept: the original graph of y = x² x² – 2x + 6? has been translated. Group work matching graphs and equations?
* From the equations sketch the graphs and * given an x or y value, complete the coordinates identify coordinates and nature of turning point of a point on a given straight line, trig graph or and equation of axis of symmetry of a parabola of curve ‐ in words or from a diagram
the form y = (x – b)2 + c or y= –(x – b)² + c Examples: Sketch y = (x – 4)2, y=(x + 3)2 + 5, Example: The point (2, c) lies on the curve with y = –(x – 5)2 – 3, y = x², y = –x², including equation y = x² + 5. What is c? The point (45, lines of symmetry. Identify the coordinates b) lies on the graph y = sin 2x. What is b? (also and nature of the TP of the graph of y = –(x + 3)2 – repeat a few straight line examples as well)
5 * Identify the equation of a parabola when shown * find the turning point of y = x 2 + bx + c by a picture of the graph from the turning point (the completing the square
formula y = ±(x – b)² + c would always be given)
Example: (a) Complete the square for y = x2 + 6x + Examples: "happy" parabola shown with turning 8 (b) Hence state the turning point of the graph point (2, 3). Pupil has to write down the formula.
of y = x² + 6x + 8
* Identify the equation (of the form y = kx²) of a * Find the x and y‐intercepts of a straight line, graph given one coordinate point by substituting parabola, and trig graph
for x and y
Examples: Identify equation of the graph of y = kx2 given the parabola and the point on (1,4). Revise all previous work and extend to y‐
The point (5,75) lies on the graph of y=kx². What intercept.
is k? * By first factorising to get the roots, use * show pupils the connection between the symmetry identify the coordinates of the TP of a equation y = x² + bx + c and y = (x – d)² + e
graph of the form y = x 2 + bx + c . Example: Simple examples such as: Find the 2 Examples: as a lesson starter, multiply out one or twcoordinates of TP of y = x + 7x + 6.
2 * Identify the solution/roots of a simple quadratic * Sketch graphs of the form y = x + bx + c by factorising, finding the x and y‐intercepts and TP equation of the form x 2 + bx + c = 0 from a graph .
using symmetry
Example: Using the graph write down solutions of Example: Sketch the graph y = x2 + 2x – 15 = 0, annotating the TP, x and y‐ intercepts.
x2 – x – 6 = 0. * Know that we can find the roots of a parabola through equating y to 0, factorising then solving * Solve problems associated with parabolas, trig the equation (link to solving quadratic equations graphs or straight lines in context. by factorising from last section)
Examples: Identify the coordinates that the graph y = x2 + 10x + 9 intercepts the x‐axis. * Draw y = sinx, y = cosx and y = tanx using table of values and know the features of each graph, and the vocabulary period, amplitude and frequency.
Using Autograph discuss the TPs, x‐intercepts, periodicity, amplitude. What are the values of x when sin is negative? …
See Int 2 and Credit Past papers for examples
* brief general overview of transformations of graphs. No function notation.
Example: "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"
* Sketch graphs of the form y=cos(x ± a), y=sin(x ± a), y=tan(x ± a), y=cos(x) ± a, y=sin(x) ± a, * Sketch the graphs of y = asinbx, y = acosbx and y y=tan(x) ± a and identify the period. Key concept: = tanbx; talk about translations of the graph, and link to parabola work
17/08/2014
Newbattle Understanding Algebra Progression S1‐S6
Examples: Sketch the graph of: y = 3cosx, y = –cosx, y = tan2x, y = sin3x, y = 2cos3x, Revise previous sketching trig. graphs examples. y = –2sin4x, annotating the x‐intercepts and TPs. Examples: Sketch the graphs of y = sin(x + 30), y = Key concept: talking about transformations of the cos(x – 45) , y = sin x + 1, y = cos x – 2 etc.
graph horizontally and/or vertically.
* be able to find the points of intersection of a * Identify the equations of graphs with equations quadratic and a straight line (easily factorised of the form y = asinbx, y = acosbx and y = tanbx examples) or trig graph and a straight line of the from a sketch
form y=b
Examples: diagram showing y = sinx and y = 0.1; or y = (x – 2)² + 5 (or y = x² – 4x + 9) and y = 14 or y = x² – x and y = x – 1. Find the points of intersection
* be able to sketch trigonometric and quadratic graphs from their equation, understanding which features are important in a sketch and which are not
Examples: sketch y = 3sin2x, sketch y = (x + 3)² + 2
Creating Formulae
use tables of values to identify and use rules connecting two variables
* be able to describe the rule from a table of values in context (e.g. tables/chairs) (one operation ‐ add, take away or multiply): first in words, then as a formula
Examples: Number of Legs = 4 × Number of Chairs, then L = 4C
construct more difficult formulae in context and from tables of values
construct more complex formulae
* create a formula to describe more complex * create a formula in words then algebra in simple geometric situations
geometric situations
develop a basic understanding of real‐life applications of quadratic expressions
* understand how to express the area of a rectangle, or volume of a cuboid as a quadratic expression
Examples: a rectangle has length x + 2 and breadth x – 5, write an expression for its area. A Examples: perimeter, area of rectangle, volumes Examples: composite shapes, circles, where at cuboid has dimensions 2, y and y + 3, write an of cuboids, Area of a triangle ‐ where at least one least one length is expressed as a letter
length is expressed as a letter
expression for its volume
th
* Create a formula to describe the n term of a * Create a formula to describe the nth term of a * extend ability to describe the rule from a table pattern in a table, both in and out of context. Use pattern. Use formula to work out values in either top or bottom row of table. Consider deletion? of values (two operations, dividing, squaring): first formula to work out values in either top or bottom row of table.
DW Aug 14
in words, then as a formula
Examples: Cost of Taxi Ride = Miles × 2 + £3 (then Examples: C = 5n – 3. What is C when n = 58, Examples: See SG Credit past exam questions F = 2M + 3); Area of Square = Length of side² What is n when C = 482? SG General past exam (keep an eye on this area)
(then A = L²)
questions
be able to solve problems in real life situations
* create formulae and solve problems
Examples: Intermediate 2 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
Examples: create appropriate questions along a similar style to a Higher optimisation part (a)

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