Monday, May 26, 2008
Paper 2 - Coordinate Geometry
This is a 1-page overview of what you need to know for this question.
See also previous posts:
The connections ("stepping stones") between the different aspects of coordinate geometry - here
See also previous posts:
The connections ("stepping stones") between the different aspects of coordinate geometry - here
Paper 1 - Number systems and number theory
This topic covers
There is some information on Sets - here
- HCF and LCM
- Primes and prime factors
- Number systems (what it means to be x ∈ N (naturals) , Z (integers), R (reals), Q (rationals))
- Sets
There is some information on Sets - here
Paper 1 - Arithmetic
This question incorporates everything in the orange book chapters 8 to 13 inclusive and chapter 11 of the purple book.
This is a 1-page overview of this topic.
I have a few older blog entries related to this topic.
An overview of arithmetic - here
A look at compound interest, especially where the interest and rate is given, but not the principal - here
Paper 1 - Algebra
This is an overview of the algebra course for Junior Cert.
The most important thing to remember is the difference between
These previous blog entries are also related to Algebra.
Long division in Algebra - here
Quadratic equations - here
Using Algebra to solve problems - here
Some notes on an algebra test we did, with common pitfalls - here
The most important thing to remember is the difference between- algebra where you are just factoring, simplifying or evaluating an expression
- equations, where you are solving and finding the value(s) of unknowns.
These previous blog entries are also related to Algebra.
Long division in Algebra - here
Quadratic equations - here
Using Algebra to solve problems - here
Some notes on an algebra test we did, with common pitfalls - here
Wednesday, May 21, 2008
Feedback on Paper 2 Mini-mock
Overall the results on paper 2 were lower than for paper 1 ... what is it about co-ordinate geometry?
Going through the questions one by one :
The trigonometry question required you to use the table (that I wrote up on the board) that is in the maths tables book. There you see that the angle whose tan = 1 is the angle π/4 i.e. 45º
Then you go to your CAST circle and count 45º as your reference angle in each of the 2 quadrants where Tans are negative.
The geometry part a) question was well answered in general - just check that you are looking at the correct angles e.g. |<opq|
The co-ordinate geometry question went a little haywire for some. In part (i) you were told that the point (-1,h) was on the line 3x - 4y + 7 = 0.
This means that 3(-1) - 4(h) + 7 must be = 0 so write it as
3(-1) - 4(h) + 7 = 0
and solve. Then you get the full co-ordinates of this point - (-1, 1).
Part (ii) involved simultaneous equations. Note that a simultaneous equations don't necessarily always work out evenly, but simultaneous equations in co-ordinate geometry almost always will.
Part (iii) said to show the 3 points and two lines on a coordinate plane. If you are asked to do this, make sure that you show all the points and lines you are asked for. Some students joined p and q, forming a triangle. This wasn't part of the question. When you are drawing a coordinated plane, use a pencil and make sure that the scales on the X- and Y-axes are even and equal.
Part (iv) required you to find the slopes of each line and prove that their product was -1.
The theorem was poorly answered. Nobody got full marks. Many didn't even attempt it. Don't leave it blank in your exam. You get attempt marks (6 out of 20) for just demonstrating that you get the gist of it. As I said when we studied this theorem before the mocks, the trickiest part is figuring out which facts you are allowed to assume (in this case it is that a line which is parallel to one side of a triangle divides the other two sides in the same ratio) and facts which you have to establish (that the line you constructed is parallel, and you do this by proving SAS congruence).
No matter how elusive the subtleties of a theorem may be, there is no excuse for not writing down, Given: To prove: Construction: and Proof: along with a diagram.
The question on volume of a cone was reasonably well attempted - but there was one glaring recurring error.
The 2nd cone had height = 2x and radius = 1.5x
So the total volume was
1/3 π r² h
= 1/3 π(2x)²(1.5x)
= 1/3 π4x²(1.5x)
= 1/3 π6x³
= 2x³π
Even in the middle of the perimeter area and volume question, you need to know your algebra.
Finally the statistics question was pretty well answered. Use a pencil for the graph, make the scales even and mark the points of your graph carefully - some students are getting a bit sloppy here, so don't get into bad habits.
We will go over these and other Paper 2 issues tomorrow. Line up any questions you may have for our last 2 classes.
Going through the questions one by one :
The trigonometry question required you to use the table (that I wrote up on the board) that is in the maths tables book. There you see that the angle whose tan = 1 is the angle π/4 i.e. 45º
Then you go to your CAST circle and count 45º as your reference angle in each of the 2 quadrants where Tans are negative.
The geometry part a) question was well answered in general - just check that you are looking at the correct angles e.g. |<opq|
The co-ordinate geometry question went a little haywire for some. In part (i) you were told that the point (-1,h) was on the line 3x - 4y + 7 = 0.
This means that 3(-1) - 4(h) + 7 must be = 0 so write it as
3(-1) - 4(h) + 7 = 0
and solve. Then you get the full co-ordinates of this point - (-1, 1).
Part (ii) involved simultaneous equations. Note that a simultaneous equations don't necessarily always work out evenly, but simultaneous equations in co-ordinate geometry almost always will.
Part (iii) said to show the 3 points and two lines on a coordinate plane. If you are asked to do this, make sure that you show all the points and lines you are asked for. Some students joined p and q, forming a triangle. This wasn't part of the question. When you are drawing a coordinated plane, use a pencil and make sure that the scales on the X- and Y-axes are even and equal.
Part (iv) required you to find the slopes of each line and prove that their product was -1.
The theorem was poorly answered. Nobody got full marks. Many didn't even attempt it. Don't leave it blank in your exam. You get attempt marks (6 out of 20) for just demonstrating that you get the gist of it. As I said when we studied this theorem before the mocks, the trickiest part is figuring out which facts you are allowed to assume (in this case it is that a line which is parallel to one side of a triangle divides the other two sides in the same ratio) and facts which you have to establish (that the line you constructed is parallel, and you do this by proving SAS congruence).
No matter how elusive the subtleties of a theorem may be, there is no excuse for not writing down, Given: To prove: Construction: and Proof: along with a diagram.
The question on volume of a cone was reasonably well attempted - but there was one glaring recurring error.
The 2nd cone had height = 2x and radius = 1.5x
So the total volume was
1/3 π r² h
= 1/3 π(2x)²(1.5x)
= 1/3 π4x²(1.5x)
= 1/3 π6x³
= 2x³π
Even in the middle of the perimeter area and volume question, you need to know your algebra.
Finally the statistics question was pretty well answered. Use a pencil for the graph, make the scales even and mark the points of your graph carefully - some students are getting a bit sloppy here, so don't get into bad habits.
We will go over these and other Paper 2 issues tomorrow. Line up any questions you may have for our last 2 classes.
Monday, May 19, 2008
Feedback on Paper 1 Mini-mock
I have just finished correcting this.
Most students handled the long division very well.
However most students failed to write 44100 as a product of its prime factors. Look back in the early chapters of the yellow book for how to do this.
Prime numbers are 2,3,5,7,11 etc
Start dividing 44100 by primes,
2 ⌊44100
2 ⌊22050
3 ⌊11025
3 ⌊3675
etc
Then list all the factors as 2x2x3x3x5.... etc
The scientific notation was well done.
The tax question could have been better laid out in a few cases. Make sure that you don't just write down a jumble of numbers. Also, check what you are asked for: in this case it was net income, not net tax.
The factoring was well done but a few of you couldn't figure out how to factor
6x² - 7x - 24
and resorted to the quadratic formula.
This is an acceptable "cheat" as long as you realise that
a) you were given an expression to factor not an equation to solve and
b) you take the "roots" of the "equation" and convert them back into factors.
For example:, you should have ended up with x = -1.5 as one of your "roots".
Working on this:
x = -1.5
2x = -3
2x + 3 = 0
So (2x + 3) is one of the factors you are looking for.
Do the same for the other "root" and write down your answer as if you hadn't used the quadratic formula (covering your tracks)
6x² - 7x - 24
= (2x + 3)(3x - 8)
On the Sets questions, lots of you lost marks because you didn't show enough work.
If you are asked (A\B) ∪ (A ∩ C)
then write down the solution set of (A\B) first, then the solution set of (A∩C), then the final union of both sets.
The solving for y using a Venn diagram question was reasonably well answered. A few of you left out the 4 people who hadn't visited any of the countries when calculating y.
Another common error was to not answer the last part of the question (how many people visited 1 country only), throwing away 5 marks.
On the functions question, lots of you answered using trial and error. After all those years of practicing solving quadratic trinomials by factoring, this was an opportunity to apply that know-how.
The quadratic function was of the form x² + 2x - 8 and you had to find the two values of x for which the output of the function = 0.
In other words, solve:
x² + 2x - 8 = 0
(x + 4)(x -2)=0
x = -4, 2
(This kind of questions goes to the very core of understanding functions. The fancier those calculators get, the more you will see questions like this on the exam!)
BTW, we are not finished yet. Lots of you lost marks because you didn't pay attention to the detail of what you were asked for: the co-ordinates of a and b. If you don't write down (-4,0) and (2,0) you lose marks.
For finding the intersection with the y-axis at c, you need to recognise that the input into the function at this point is x = 0.
Sub in x=0 and solve.
f(x) or y = 0² + 2(0) - 8 = -8
So the co-ordinates of c = (0,-8).
Finally, the last part of the question (which several students left out) asked for the range of values for which f(x) ≤ 0
As you were told x ∈ R, it isn't enough to write down: -4,-3,-2 .... etc.
You have to write it as -4 ≤ x ≤ 2
Overall, the results were pretty good, given the rush. Make sure that you know where you went wrong. We will go over paper 1 issues tomorrow and do a paper 2 mini-mock on Wednesday.
Most students handled the long division very well.
However most students failed to write 44100 as a product of its prime factors. Look back in the early chapters of the yellow book for how to do this.
Prime numbers are 2,3,5,7,11 etc
Start dividing 44100 by primes,
2 ⌊44100
2 ⌊22050
3 ⌊11025
3 ⌊3675
etc
Then list all the factors as 2x2x3x3x5.... etc
The scientific notation was well done.
The tax question could have been better laid out in a few cases. Make sure that you don't just write down a jumble of numbers. Also, check what you are asked for: in this case it was net income, not net tax.
The factoring was well done but a few of you couldn't figure out how to factor
6x² - 7x - 24
and resorted to the quadratic formula.
This is an acceptable "cheat" as long as you realise that
a) you were given an expression to factor not an equation to solve and
b) you take the "roots" of the "equation" and convert them back into factors.
For example:, you should have ended up with x = -1.5 as one of your "roots".
Working on this:
x = -1.5
2x = -3
2x + 3 = 0
So (2x + 3) is one of the factors you are looking for.
Do the same for the other "root" and write down your answer as if you hadn't used the quadratic formula (covering your tracks)
6x² - 7x - 24
= (2x + 3)(3x - 8)
On the Sets questions, lots of you lost marks because you didn't show enough work.
If you are asked (A\B) ∪ (A ∩ C)
then write down the solution set of (A\B) first, then the solution set of (A∩C), then the final union of both sets.
The solving for y using a Venn diagram question was reasonably well answered. A few of you left out the 4 people who hadn't visited any of the countries when calculating y.
Another common error was to not answer the last part of the question (how many people visited 1 country only), throwing away 5 marks.
On the functions question, lots of you answered using trial and error. After all those years of practicing solving quadratic trinomials by factoring, this was an opportunity to apply that know-how.
The quadratic function was of the form x² + 2x - 8 and you had to find the two values of x for which the output of the function = 0.
In other words, solve:
x² + 2x - 8 = 0
(x + 4)(x -2)=0
x = -4, 2
(This kind of questions goes to the very core of understanding functions. The fancier those calculators get, the more you will see questions like this on the exam!)
BTW, we are not finished yet. Lots of you lost marks because you didn't pay attention to the detail of what you were asked for: the co-ordinates of a and b. If you don't write down (-4,0) and (2,0) you lose marks.
For finding the intersection with the y-axis at c, you need to recognise that the input into the function at this point is x = 0.
Sub in x=0 and solve.
f(x) or y = 0² + 2(0) - 8 = -8
So the co-ordinates of c = (0,-8).
Finally, the last part of the question (which several students left out) asked for the range of values for which f(x) ≤ 0
As you were told x ∈ R, it isn't enough to write down: -4,-3,-2 .... etc.
You have to write it as -4 ≤ x ≤ 2
Overall, the results were pretty good, given the rush. Make sure that you know where you went wrong. We will go over paper 1 issues tomorrow and do a paper 2 mini-mock on Wednesday.
Sunday, May 18, 2008
Paper 1 Mini-Mock on Monday
Don't forget, we will have a mini-mock exam with a selection of 2 part (a)s, 2 part (b)s and 2 part (c)s from Paper 1 questions.
Thursday, May 15, 2008
Perimeter Area and Volume
Looking at the 2006 question:
a) application of the formula for volume of cylinder. Opportunity to lose 3 marks by not spotting that the diameter is given rather than the radius!
b) note that the circumference of a wheel is the same as the distance it travels in one revolution. (think of an opisometer or trundle-wheel.)
c) note that you should answer the whole question without setting π = 3.14 or 22/7. In part (i) your answer will be in terms of π (e.g. 32πm²) In part (ii) the πs will cancel.
This is a one-page overview of the main points you need to know for this topic.
a) application of the formula for volume of cylinder. Opportunity to lose 3 marks by not spotting that the diameter is given rather than the radius!
b) note that the circumference of a wheel is the same as the distance it travels in one revolution. (think of an opisometer or trundle-wheel.)
c) note that you should answer the whole question without setting π = 3.14 or 22/7. In part (i) your answer will be in terms of π (e.g. 32πm²) In part (ii) the πs will cancel.
This is a one-page overview of the main points you need to know for this topic.
Trigonometry wrap-up
We covered all the basics of trigonometry in our last class.
The 2006 trigonometry question shouldn't pose any major difficulty.
For part a) draw a rough diagram and identify which side is which before you draw it exactly.
For part b) remember that you need SAS to get the area of the triangle, so you need to find a missing angle first.
For part c) part (i) is straightforward sine rule (after finding an angle to complete an angle/opposite side pair). For part (ii) there are a few ways you could do this either using area of a triangle or using the basic ratios for right-angled triangles (the shortest line from f to the opposite bank of the river is perpendicular to the river bank).
The 2006 trigonometry question shouldn't pose any major difficulty.
For part a) draw a rough diagram and identify which side is which before you draw it exactly.
For part b) remember that you need SAS to get the area of the triangle, so you need to find a missing angle first.
For part c) part (i) is straightforward sine rule (after finding an angle to complete an angle/opposite side pair). For part (ii) there are a few ways you could do this either using area of a triangle or using the basic ratios for right-angled triangles (the shortest line from f to the opposite bank of the river is perpendicular to the river bank).
Tuesday, May 13, 2008
Trigonometry overview
I have put together a 1-page overview of Trigonometry which you may find useful.
We will go through this in tomorrow's class.
After Wednesday, we will have a class on Friday, and hopefully, 5 classes next week.
Think of any topics you would like to go over before each class.
We will go through this in tomorrow's class.
After Wednesday, we will have a class on Friday, and hopefully, 5 classes next week.
Think of any topics you would like to go over before each class.
Tuesday, May 6, 2008
Revised revision schedule
We need to change the order of revision a little and the time spent on the next few topics.
We will complete Statistics and Functions and Graphs on Wednesday/Thursday this Week.
Then we will go on to do Trigonometry - starting on Friday.
Next week, we will probably miss class on Monday and definitely miss class on Tuesday.
We will try to complete Trig on Wednesday and then spend Thursday and Friday on perimeter area and volume, finishing last bits and pieces (including the final theorem) On Monday/Tuesday of the following week.
That will leave us a few days and our block class during the last week in May to tie up any loose ends.
I also want to fit in a Paper 1 and Paper 2 mini-mock as soon as possible. These will be 2 40 minute exams, where you tackle 2 part (a)s, 2 part (b)s and 2 part (c)s from assorted questions on the paper.
We will complete Statistics and Functions and Graphs on Wednesday/Thursday this Week.
Then we will go on to do Trigonometry - starting on Friday.
Next week, we will probably miss class on Monday and definitely miss class on Tuesday.
We will try to complete Trig on Wednesday and then spend Thursday and Friday on perimeter area and volume, finishing last bits and pieces (including the final theorem) On Monday/Tuesday of the following week.
That will leave us a few days and our block class during the last week in May to tie up any loose ends.
I also want to fit in a Paper 1 and Paper 2 mini-mock as soon as possible. These will be 2 40 minute exams, where you tackle 2 part (a)s, 2 part (b)s and 2 part (c)s from assorted questions on the paper.
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