Order of Topics. A-Level
I'm seeing a massive disparity between schools in their approach to delivering the A-Level Spec.
Today, I learned that a very experienced Head of Maths had chosen to do Integration by substitution, before finishing differentiation at y13.
Also today, a different school 2nd in dept chose to differentiate sin(x) 'from 1st principles' before even discussing addition/dbl angle formulae.
So, if you are, or were to be, employed as Head of Maths in a school, what would be your preferred order?
Here's mine:
Surds & Indices [2 lessons]
Coord Geometry, not circles [2 lessons]
Quads [8 lessons]
Function Transforms & sketching [2 lessons]
Circles [2 lessons]
Diff from 1st principles [2 lessons]
Diff of polynomials [3 lessons]
Binomial expansion, factorials, Pascal's triangle [4 lessons]
APs [2 lessons]
Trig : Radians, formulae for sectors, Sine & Cosine rules & "half absinC"
[3 lessons]
Trig : graphs & transforms, simple identities & solving eqns [4 lessons]
Integration [3 lessons]
Logs & exponentials [4 lessons]
GPs [3 lessons]
Binomial Expansion (-ve & fractional indices) [2 lessons]
Partial Fractions [2 lessons]
Other Sequences & recurrence relationships, Sigma notation [2 lessons]
Trig: sec, cosec & cot: graphs & identities [2 lessons]
Trig: addition & dbl angle formulae [3 lessons]
Trig: Harmonic formulae {Rcos(theta minus alpha)} [2 lessons]
Diff : sinx & cosx proofs from '1st principles' using small angle approximations & also using a squeeze [1 lesson]
Diff : Chain, Product & Quotient rules [3 lessons]
Diff : generate diff results for tan, sec, cosec & cot [1 lesson]
Diff: Implicit & Rates [2 lessons]
Iteration : Newton-Raphson [1 lesson]
Functions: Transformations involving stretch & translate in same & both directions, NB transforms of trig graphs especially to explain/discuss Rcos(theta minus alpha) [2 lessons]
Functions: Modulus, graph & transform & solve eqn [2 lessons]
Functions: mappings, domain & range, combinations, inverses, solving [3 lessons]
Integration : All 11 sub-topics [8 lessons]
Parametrics : cartesian equivalents, diff, integrations [3 lessons]
Vectors : 2D, 3D, parallel, perpendicular, angle between, vector algebra, magnitude, dot & cross products & their relationship to tan(y/x) and cosine rule, line equations, area formulae incl. matrix result for parallelogram, simple eqns of plane & normals [6 lessons]
Total : 90 lessons.
ONLY THEN would I start Mech & Stats. 32- 40 lessons (16 -20 each)
Plenty of space in a 72-week course in which to fit in topic tests, end of year test (in July, not before) & 2 mocks in y13 (BEFORE Christmas, and again after Feb half term). Also space left to do some extra things, like a trip, a few investigations, a bit of history of maths incl some notable historic maths figures/characters (not so much Newton! he has a bad rep!), but Euler, Eratosthenes, Euclid, Andrew Wiles, Pascal, Toricelli & his relationship to Galileo & Mersenne, Gauss, Bernouilli (most of the family! & their relationship to Eurler), Sophie Germain, Descartes, Hilbert.
Today, I learned that a very experienced Head of Maths had chosen to do Integration by substitution, before finishing differentiation at y13.
Also today, a different school 2nd in dept chose to differentiate sin(x) 'from 1st principles' before even discussing addition/dbl angle formulae.
So, if you are, or were to be, employed as Head of Maths in a school, what would be your preferred order?
Here's mine:
Surds & Indices [2 lessons]
Coord Geometry, not circles [2 lessons]
Quads [8 lessons]
Function Transforms & sketching [2 lessons]
Circles [2 lessons]
Diff from 1st principles [2 lessons]
Diff of polynomials [3 lessons]
Binomial expansion, factorials, Pascal's triangle [4 lessons]
APs [2 lessons]
Trig : Radians, formulae for sectors, Sine & Cosine rules & "half absinC"
[3 lessons]
Trig : graphs & transforms, simple identities & solving eqns [4 lessons]
Integration [3 lessons]
Logs & exponentials [4 lessons]
GPs [3 lessons]
Binomial Expansion (-ve & fractional indices) [2 lessons]
Partial Fractions [2 lessons]
Other Sequences & recurrence relationships, Sigma notation [2 lessons]
Trig: sec, cosec & cot: graphs & identities [2 lessons]
Trig: addition & dbl angle formulae [3 lessons]
Trig: Harmonic formulae {Rcos(theta minus alpha)} [2 lessons]
Diff : sinx & cosx proofs from '1st principles' using small angle approximations & also using a squeeze [1 lesson]
Diff : Chain, Product & Quotient rules [3 lessons]
Diff : generate diff results for tan, sec, cosec & cot [1 lesson]
Diff: Implicit & Rates [2 lessons]
Iteration : Newton-Raphson [1 lesson]
Functions: Transformations involving stretch & translate in same & both directions, NB transforms of trig graphs especially to explain/discuss Rcos(theta minus alpha) [2 lessons]
Functions: Modulus, graph & transform & solve eqn [2 lessons]
Functions: mappings, domain & range, combinations, inverses, solving [3 lessons]
Integration : All 11 sub-topics [8 lessons]
Parametrics : cartesian equivalents, diff, integrations [3 lessons]
Vectors : 2D, 3D, parallel, perpendicular, angle between, vector algebra, magnitude, dot & cross products & their relationship to tan(y/x) and cosine rule, line equations, area formulae incl. matrix result for parallelogram, simple eqns of plane & normals [6 lessons]
Total : 90 lessons.
ONLY THEN would I start Mech & Stats. 32- 40 lessons (16 -20 each)
Plenty of space in a 72-week course in which to fit in topic tests, end of year test (in July, not before) & 2 mocks in y13 (BEFORE Christmas, and again after Feb half term). Also space left to do some extra things, like a trip, a few investigations, a bit of history of maths incl some notable historic maths figures/characters (not so much Newton! he has a bad rep!), but Euler, Eratosthenes, Euclid, Andrew Wiles, Pascal, Toricelli & his relationship to Galileo & Mersenne, Gauss, Bernouilli (most of the family! & their relationship to Eurler), Sophie Germain, Descartes, Hilbert.
(edited 5 years ago)
First 5/6 are the same we did but we’ve done besides vectors too and some stats
(edited 5 years ago)
Original post by Y12_FurtherMaths
First 5/6 are the same we did but we’ve done besides vectors too and some stats
Cool. But according to my schedule, above, those 5 topics should be 18 lessons or so.
By now, you've had 30+ maths lessons, and so 12 lessons or so on some vectors & some stats?
I can see why there's some call to do some vectors & some stats before end of this year, together with some mech : so that the dept might use AS papers from June 2017 (& possibly June 2018) for your end of year exams. (this is a bit lazy as far as your dept is concerned - only my opinion!)
So, can you answer the following 3 questions:
1. Given 6(root8)^n / (12 x 4^(2n-1)) = 2^p
Find p in terms of n
2. Given y = 2x + k , k>0, is a tangent to the circle which is described by the eqn x^2 - kx + y^2 - 4y = 3k
a) find k, as a simplified surd
b) using your value of k, find
i) the length of the tangent from (0,k) to the circle
ii) the radius of the circle
3. A curve, C, is defined by the equation y = 3x^0.5 - 2 / x^3.5
P and Q are points on the curve where x=4, and x=12, respectively.
a) show that the gradient of the line PQ is an approximation to the tangent to C when x=8
b) explain how you might find a better approximation to the gradient of the curve at x=8
c) hence, or otherwise, find the gradient of the curve at x=8, correct to 3sig.fig.
Original post by begbie68
Cool. But according to my schedule, above, those 5 topics should be 18 lessons or so.
By now, you've had 30+ maths lessons, and so 12 lessons or so on some vectors & some stats?
I can see why there's some call to do some vectors & some stats before end of this year, together with some mech : so that the dept might use AS papers from June 2017 (& possibly June 2018) for your end of year exams. (this is a bit lazy as far as your dept is concerned - only my opinion!)
So, can you answer the following 3 questions:
1. Given 6(root8)^n / (12 x 4^(2n-1)) = 2^p
Find p in terms of n
2. Given y = 2x + k , k>0, is a tangent to the circle which is described by the eqn x^2 - kx + y^2 - 4y = 3k
a) find k, as a simplified surd
b) using your value of k, find
i) the length of the tangent from (0,k) to the circle
ii) the radius of the circle
3. A curve, C, is defined by the equation y = 3x^0.5 - 2 / x^3.5
P and Q are points on the curve where x=4, and x=12, respectively.
a) show that the gradient of the line PQ is an approximation to the tangent to C when x=8
b) explain how you might find a better approximation to the gradient of the curve at x=8
c) hence, or otherwise, find the gradient of the curve at x=8, correct to 3sig.fig.
By now, you've had 30+ maths lessons, and so 12 lessons or so on some vectors & some stats?
I can see why there's some call to do some vectors & some stats before end of this year, together with some mech : so that the dept might use AS papers from June 2017 (& possibly June 2018) for your end of year exams. (this is a bit lazy as far as your dept is concerned - only my opinion!)
So, can you answer the following 3 questions:
1. Given 6(root8)^n / (12 x 4^(2n-1)) = 2^p
Find p in terms of n
2. Given y = 2x + k , k>0, is a tangent to the circle which is described by the eqn x^2 - kx + y^2 - 4y = 3k
a) find k, as a simplified surd
b) using your value of k, find
i) the length of the tangent from (0,k) to the circle
ii) the radius of the circle
3. A curve, C, is defined by the equation y = 3x^0.5 - 2 / x^3.5
P and Q are points on the curve where x=4, and x=12, respectively.
a) show that the gradient of the line PQ is an approximation to the tangent to C when x=8
b) explain how you might find a better approximation to the gradient of the curve at x=8
c) hence, or otherwise, find the gradient of the curve at x=8, correct to 3sig.fig.
1) P=(-5n-6)/2
2)a) K=(46+2root705)/11. I used a calculator.
Want to get my value of k checked please before i go further.
Original post by Y12_FurtherMaths
1) P=(-5n-6)/2
2)a) K=(46+2root705)/11. I used a calculator.
Want to get my value of k checked please before i go further.
2)a) K=(46+2root705)/11. I used a calculator.
Want to get my value of k checked please before i go further.
1. p is not correct.
2. k IS correct
nice one!Original post by begbie68
1. p is not correct.
2. k IS correct nice one!
2. k IS correct
nice one!I don’t see where I’ve gone wrong then unless I’ve copied the question out incorrectly. Let me post my working. Did you mean the sixth root instead of 6root? Sorry my working is all over the place https://imgur.com/a/GMmCAds
(edited 5 years ago)
Original post by Y12_FurtherMaths
I don’t see where I’ve gone wrong then unless I’ve copied the question out incorrectly. Let me post my working. Did you mean the sixth root instead of 6root? https://imgur.com/a/GMmCAds
no. i mean what it says:
6 x (root8)^n
or
6 x 8^(n/2) , if that helps
Original post by begbie68
no. i mean what it says:
6 x (root8)^n
or
6 x 8^(n/2) , if that helps
6 x (root8)^n
or
6 x 8^(n/2) , if that helps
Can you check my working please and identify where I’ve made an error?
Original post by Y12_FurtherMaths
Can you check my working please and identify where I’ve made an error?
When you take the denominator up (final step)
2n - 1
becomes
-2n + 1
They're exponents of 4, so double to match the 2
Original post by mqb2766
When you take the denominator up (final step)
2n - 1
becomes
-2n + 1
They're exponents of 4, so double to match the 2
2n - 1
becomes
-2n + 1
They're exponents of 4, so double to match the 2
Silly me forgetting brackets. Thanks! Will answer the rest later
Original post by begbie68
Cool. But according to my schedule, above, those 5 topics should be 18 lessons or so.
By now, you've had 30+ maths lessons, and so 12 lessons or so on some vectors & some stats?
I can see why there's some call to do some vectors & some stats before end of this year, together with some mech : so that the dept might use AS papers from June 2017 (& possibly June 2018) for your end of year exams. (this is a bit lazy as far as your dept is concerned - only my opinion!)
So, can you answer the following 3 questions:
1. Given 6(root8)^n / (12 x 4^(2n-1)) = 2^p
Find p in terms of n
2. Given y = 2x + k , k>0, is a tangent to the circle which is described by the eqn x^2 - kx + y^2 - 4y = 3k
a) find k, as a simplified surd
b) using your value of k, find
i) the length of the tangent from (0,k) to the circle
ii) the radius of the circle
3. A curve, C, is defined by the equation y = 3x^0.5 - 2 / x^3.5
P and Q are points on the curve where x=4, and x=12, respectively.
a) show that the gradient of the line PQ is an approximation to the tangent to C when x=8
b) explain how you might find a better approximation to the gradient of the curve at x=8
c) hence, or otherwise, find the gradient of the curve at x=8, correct to 3sig.fig.
By now, you've had 30+ maths lessons, and so 12 lessons or so on some vectors & some stats?
I can see why there's some call to do some vectors & some stats before end of this year, together with some mech : so that the dept might use AS papers from June 2017 (& possibly June 2018) for your end of year exams. (this is a bit lazy as far as your dept is concerned - only my opinion!)
So, can you answer the following 3 questions:
1. Given 6(root8)^n / (12 x 4^(2n-1)) = 2^p
Find p in terms of n
2. Given y = 2x + k , k>0, is a tangent to the circle which is described by the eqn x^2 - kx + y^2 - 4y = 3k
a) find k, as a simplified surd
b) using your value of k, find
i) the length of the tangent from (0,k) to the circle
ii) the radius of the circle
3. A curve, C, is defined by the equation y = 3x^0.5 - 2 / x^3.5
P and Q are points on the curve where x=4, and x=12, respectively.
a) show that the gradient of the line PQ is an approximation to the tangent to C when x=8
b) explain how you might find a better approximation to the gradient of the curve at x=8
c) hence, or otherwise, find the gradient of the curve at x=8, correct to 3sig.fig.
2)b)i) 18.1(3sf) Might be off slightly due to rounding. ii) 7.16(3sf) again might be off due to rounding.
3)a) I calculated the gradient and got 0.5314 but not sure how to show that its an approximation.
b) i'd make the x coordinates closer to 8.
c) -2?
These questions are a lot harder than what ive been used to in class tbh, did you write them yourself?
Original post by Y12_FurtherMaths
2)b)i) 18.1(3sf) Might be off slightly due to rounding. ii) 7.16(3sf) again might be off due to rounding.
3)a) I calculated the gradient and got 0.5314 but not sure how to show that its an approximation.
b) i'd make the x coordinates closer to 8.
c) -2?
These questions are a lot harder than what ive been used to in class tbh, did you write them yourself?
3)a) I calculated the gradient and got 0.5314 but not sure how to show that its an approximation.
b) i'd make the x coordinates closer to 8.
c) -2?
These questions are a lot harder than what ive been used to in class tbh, did you write them yourself?
Cool. Top marks.
Q1 & 3 are slight variations to GCSE questions from past IGCSE papers. In fact Q1 was pretty much the same as a question from last year's IGCSE paper.
Variations of Q3 have often been used by OCR/MEI in the old C1/C2 exams.
Q2 is a variant of a specimen-type question used by all the boards, but I made up the coefficients without checking them... my bad. My choices made it super-(if not hyper-)technical in terms of surds.
My reasoning is that if you can answer those, then you can pretty much answer anything that might be thrown at you in a Paper1 (on those topics). If not, then hopefully you've gained a good deal in the experience. (If only to recognise the importance of brackets!).
your query on 3a is a good one. You are taught to recognise [f(x+h)-f(x)] / h, but there is an equivalent which is often better (when/if it can be used) : [f(x+h)-f(x-h)] / 2h ... think about the graph and take an interval above x, and the same interval below x in your standard 'from first principles' diff.
I'm impressed by your persistence in getting correct answers!
Original post by begbie68
Cool. Top marks.
Q1 & 3 are slight variations to GCSE questions from past IGCSE papers. In fact Q1 was pretty much the same as a question from last year's IGCSE paper.
Variations of Q3 have often been used by OCR/MEI in the old C1/C2 exams.
Q2 is a variant of a specimen-type question used by all the boards, but I made up the coefficients without checking them... my bad. My choices made it super-(if not hyper-)technical in terms of surds.
My reasoning is that if you can answer those, then you can pretty much answer anything that might be thrown at you in a Paper1 (on those topics). If not, then hopefully you've gained a good deal in the experience. (If only to recognise the importance of brackets!).
your query on 3a is a good one. You are taught to recognise [f(x+h)-f(x)] / h, but there is an equivalent which is often better (when/if it can be used) : [f(x+h)-f(x-h)] / 2h ... think about the graph and take an interval above x, and the same interval below x in your standard 'from first principles' diff.
I'm impressed by your persistence in getting correct answers!
Q1 & 3 are slight variations to GCSE questions from past IGCSE papers. In fact Q1 was pretty much the same as a question from last year's IGCSE paper.
Variations of Q3 have often been used by OCR/MEI in the old C1/C2 exams.
Q2 is a variant of a specimen-type question used by all the boards, but I made up the coefficients without checking them... my bad. My choices made it super-(if not hyper-)technical in terms of surds.
My reasoning is that if you can answer those, then you can pretty much answer anything that might be thrown at you in a Paper1 (on those topics). If not, then hopefully you've gained a good deal in the experience. (If only to recognise the importance of brackets!).
your query on 3a is a good one. You are taught to recognise [f(x+h)-f(x)] / h, but there is an equivalent which is often better (when/if it can be used) : [f(x+h)-f(x-h)] / 2h ... think about the graph and take an interval above x, and the same interval below x in your standard 'from first principles' diff.
I'm impressed by your persistence in getting correct answers!
Hmm not too sure really. Is there more chance that the gradient will be closer to being parallel with that of the required gradient?
Original post by Y12_FurtherMaths
Hmm not too sure really. Is there more chance that the gradient will be closer to being parallel with that of the required gradient?
and you've just let yourself down!! you had been doing sooo well!!!
come on ... if the gradient of the approximation is close to the required gradient (to begin with!!!) we've almost won right at the start ... but we STILL make h tend to zero
Original post by begbie68
and you've just let yourself down!! you had been doing sooo well!!!
come on ... if the gradient of the approximation is close to the required gradient (to begin with!!!) we've almost won right at the start ... but we STILL make h tend to zero
come on ... if the gradient of the approximation is close to the required gradient (to begin with!!!) we've almost won right at the start ... but we STILL make h tend to zero
Ahh haha. Thanks for the questions! Definitely a challenge.
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