- Have your pens (incl spares), pencil, calculator (check battery), geometry set ready well before the exam time.
- Manage your time. 2½ hours for exam = 25 mins per question. Exam starts at 9:30, you should be well into your 2nd question by 10am.
- Write clearly. Number all questions and parts of questions. Don't waste time rewriting the question. Highlight your answer clearly.
- Keep all parts of each question together. It is a good idea to keep questions in order. If you have to come back to re-attempt a question later in the exam, it will be easier to find it.
- Read questions carefully, do your work and then read question again to make sure you have done what was asked and to make sure you format your answer properly (correct to 2 decimal places, as a fraction, in surd form, in km/h etc.).
- Attempt all questions. Where there is a part (i), (ii), (iii) etc., don't assume that if you cannot finish part (i) that you cannot attempt part (ii), (iii).
- If you are running out of time remember how to get attempt marks - “any correct substitution”.
- Some students find it reassuring to write all the formulae you have memorised down as soon as you are told that you may start.
- Remember, for every blunder (-3 marks) you lose ½%, so be careful! Check your answers. Sanity check answers to “real-world” type questions (speed of truck = 640km/h, boy's height = 3.7m), check roots of equations and verify algebra using substitution if you have time.
- Construction marks are part of your answer. “Using your graph” means what it says. Show how you have used your graph.
Wednesday, June 4, 2008
Best of luck in the exams!
... and because you shouldn't need luck to succeed, here again are my top 10 tips for maximising your grade
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.
Monday, April 28, 2008
Wrapping up geometry
We covered everything I planned in class today - we will repeat for the other half of the class on Tuesday.
Work for our next proper maths class is from the 2005 and 2006 papers.
In both papers, answer Q2 (c) and the complete question 3 and 4.
As it is a really short week maths-wise and there is a long weekend approaching, we will touch on functions and graphs and statistics so that you'll be able to do some useful work on these topics.
Work for our next proper maths class is from the 2005 and 2006 papers.
In both papers, answer Q2 (c) and the complete question 3 and 4.
As it is a really short week maths-wise and there is a long weekend approaching, we will touch on functions and graphs and statistics so that you'll be able to do some useful work on these topics.
Sunday, April 27, 2008
Planning notes
As alternative halves of the class will be out Monday/Tuesday - we will have a repeating class, covering the deductions of the circle theorem we did on Friday, the last circle theorem and an elegant proof of Pythagoras.
We will have only 1 proper maths class with everyone present this week, so we must finish the geometry.
According to the plan, we should be doing functions + graphs this week, so I will try to get this started during our Thursday class so that you can work through this topic over the long weekend.
We made our original plan at the end of February, let me know if you want to change the order of revision.
We will have only 1 proper maths class with everyone present this week, so we must finish the geometry.
According to the plan, we should be doing functions + graphs this week, so I will try to get this started during our Thursday class so that you can work through this topic over the long weekend.
We made our original plan at the end of February, let me know if you want to change the order of revision.
Thursday, April 24, 2008
Geometry up as far as circle theorems
Today we went over the idea of a proof of a geometric problem as opposed to proof of a theorem.
Insofar as possible, keep the proof of problems as formal as the proofs of theorems:
You should know the first 6 theorems by now. It is important to remember the order so that you know which previous theorems you can use in the course of the theorem you are asked to prove.
Tomorrow we will look at the constructions and start proving the first of the circle theorems.
Insofar as possible, keep the proof of problems as formal as the proofs of theorems:
- drawing a diagram
- stating what is to be proven in terms of your diagram
- doing a construction or a relabelling if needed
- justifying each step along the way.
You should know the first 6 theorems by now. It is important to remember the order so that you know which previous theorems you can use in the course of the theorem you are asked to prove.
Tomorrow we will look at the constructions and start proving the first of the circle theorems.
Wednesday, April 23, 2008
Geometry
Today we looked at the various transformations of the plane (translations, central and axial symmetry and the new one: rotations).
You need to know how to rotate a point or a line or a shape through a given angle and how to identify how many orders of rotational symmetry a shape has.
If you have a point a, its image under a given transformation will be called a'.
Homework is to look at some shapes and work out what their order of rotational symmetry is and to answer a question about rotations on the coordinated plane. It is important the the rotation is through 90degrees.
If you have are rotating about the point a(1,2) and you are rotating the point b(4,0) you need to find a point b' which is the same distance from a as the point b is, where ab is perpendicular (90 degrees) to ab'. See if you can work out a quick way of doing this, without using slope and distance formula.
We also started looking at an overview of the theorems which we will be covering.
Don't forget to bring you geometry set to each class from now on.
You need to know how to rotate a point or a line or a shape through a given angle and how to identify how many orders of rotational symmetry a shape has.
If you have a point a, its image under a given transformation will be called a'.
Homework is to look at some shapes and work out what their order of rotational symmetry is and to answer a question about rotations on the coordinated plane. It is important the the rotation is through 90degrees.
If you have are rotating about the point a(1,2) and you are rotating the point b(4,0) you need to find a point b' which is the same distance from a as the point b is, where ab is perpendicular (90 degrees) to ab'. See if you can work out a quick way of doing this, without using slope and distance formula.
We also started looking at an overview of the theorems which we will be covering.
Don't forget to bring you geometry set to each class from now on.
Monday, April 21, 2008
Arithmetic continued
Today we went over the topics of compound interest and income tax.
On compound interest we looked at how you could use the compound interest formula to work out the principal, given the amount of interest earned over a given number of years.
E.g. what principal will earn 315.35 interest at 5% over 3 years?
P +315.25 = P(1.05)³
P +315.25 = P(1.157625)
P(1.157625) - P = 315.25
Factor LHS:
P(1.157625 -1) = 315.25
P = 315.25/0.157625 = 2000
Income Tax questions are straightforward once you know the terms used and lay your work out clearly. Practice the questions where you have to work backwards to find the original Gross Income.
Also, watch out for older editions of the textbook. You do not need to know about Tax Free Allowance - just standard rate cut-off and tax credits.
Tomorrow we will get started on geometry.
On compound interest we looked at how you could use the compound interest formula to work out the principal, given the amount of interest earned over a given number of years.
E.g. what principal will earn 315.35 interest at 5% over 3 years?
P +315.25 = P(1.05)³
P +315.25 = P(1.157625)
P(1.157625) - P = 315.25
Factor LHS:
P(1.157625 -1) = 315.25
P = 315.25/0.157625 = 2000
Income Tax questions are straightforward once you know the terms used and lay your work out clearly. Practice the questions where you have to work backwards to find the original Gross Income.
Also, watch out for older editions of the textbook. You do not need to know about Tax Free Allowance - just standard rate cut-off and tax credits.
Tomorrow we will get started on geometry.
Sunday, April 20, 2008
Arithmetic
It is important not to overlook the arithmetic section of the course. The questions here can be subtly tricky.
These are the main pitfalls we went through in class:
Ratios:
If there are fractions, multiply the entire ratio by the Lowest Common Denominator before you start to work out the individual shares. So 1/2 : 1/4 : 1 becomes 2:1:4 when you multiply across by 4.
Distance Speed and Time:
Be careful subtracting 24hour clock questions - we talked about the different ways of doing this - stick with whatever works best for you, but double check your answer. Make sure that you don't treat 3h20mins as 3.2 hours. Work in fractions.
Finding average speed over a 2-part journey, don't add the 2 speeds and divide by 2, DO divide the total journey time by the total time taken.
Percentages + Tax
If you are going to use the % button on your calculator, make sure you know how to use it properly. If you want to find 19% of a number you can divide by 100 and multiply by 19 or just multiply by 0.19. To add 19% on in one step, just multiply by 1.19. You don't really need the percentage button at all.
For questions such as "a bill including VAT at 12% comes to €106.40, what was the bill before tax was added" you need to write down
106.4 = 112%
then divide accross by 112 to find 1% then multiply by 100 to find 100%.
The compound interest formula is useful for working out the original principle after n years of the same rate of interest being applied. One of the homework questions requires you to use it. Set up an equation using the formula and solve it.
We will try to finish off Arithemtic in the next class.
These are the main pitfalls we went through in class:
Ratios:
If there are fractions, multiply the entire ratio by the Lowest Common Denominator before you start to work out the individual shares. So 1/2 : 1/4 : 1 becomes 2:1:4 when you multiply across by 4.
Distance Speed and Time:
Be careful subtracting 24hour clock questions - we talked about the different ways of doing this - stick with whatever works best for you, but double check your answer. Make sure that you don't treat 3h20mins as 3.2 hours. Work in fractions.
Finding average speed over a 2-part journey, don't add the 2 speeds and divide by 2, DO divide the total journey time by the total time taken.
Percentages + Tax
If you are going to use the % button on your calculator, make sure you know how to use it properly. If you want to find 19% of a number you can divide by 100 and multiply by 19 or just multiply by 0.19. To add 19% on in one step, just multiply by 1.19. You don't really need the percentage button at all.
For questions such as "a bill including VAT at 12% comes to €106.40, what was the bill before tax was added" you need to write down
106.4 = 112%
then divide accross by 112 to find 1% then multiply by 100 to find 100%.
The compound interest formula is useful for working out the original principle after n years of the same rate of interest being applied. One of the homework questions requires you to use it. Set up an equation using the formula and solve it.
We will try to finish off Arithemtic in the next class.
Tuesday, April 15, 2008
Algebra test results
I just finished correcting today's test.
A couple of points before we move on to the next topic.
On the question about the buffet meal, everyone managed to get the two expressions representing the old price per person (160/x) and the new price per person (160/(x+4)).
The tricky part is putting the equation together.
To do this you need to say that the old price minus the new price is 2 euro (as they tell you that the new price is cheaper).
(160/x) - (160/(x+4)) = 2
And solve.
[You could also work it out by saying the new price + 2 euro = old price]
A few people made worrying mistakes on BEMDAS.
In ....
With the double inequality, you can solve it all in one go but it is easier to follow if you split it into two inequalities and solve them separately.
Most of you managed the long division nicely.
In the equation with algebraic fractions many of you made it very complicated, looking for a large common multiple ... instead of just multiplying both sides of the equation by the smallest number that would make all the fractions disappear.
Finally, many of you managed to express 3/(2x-1) - 2/(3x+1) as a single fraction, but completely missed the point of verifying your answer by setting x = 1.
If you just sub x=1 into your answer you are not verifying (i.e. proving) anything.
The point is that if you replace x with 1 in the above expression you get 3/(2-1) - 2/(3+1) = 3 - 1/2 = 2.5
and if you replace x with 1 in the answer you got for the single fraction
(should be (5x + 5)/(2x-1)(3x+1), you'll get (5+5)/(2-1)(3+1) = 10/4 = 2.5
You verify that you haven't changed the value of the expression by subbing x=1 into the original version as you were given and your new improved version of the expression and getting the same answer in both.
Note that you can do this verification to check that you haven't made a mistake anytime you are asked to simplify an expression, write as a single fraction etc.
Tomorrow we will start on arithmetic.
A couple of points before we move on to the next topic.
On the question about the buffet meal, everyone managed to get the two expressions representing the old price per person (160/x) and the new price per person (160/(x+4)).
The tricky part is putting the equation together.
To do this you need to say that the old price minus the new price is 2 euro (as they tell you that the new price is cheaper).
(160/x) - (160/(x+4)) = 2
And solve.
[You could also work it out by saying the new price + 2 euro = old price]
A few people made worrying mistakes on BEMDAS.
In ....
3(-1)²
it is the -1 that gets squared, not the 3, so the answer is 3(1) = 3With the double inequality, you can solve it all in one go but it is easier to follow if you split it into two inequalities and solve them separately.
Most of you managed the long division nicely.
In the equation with algebraic fractions many of you made it very complicated, looking for a large common multiple ... instead of just multiplying both sides of the equation by the smallest number that would make all the fractions disappear.
Finally, many of you managed to express 3/(2x-1) - 2/(3x+1) as a single fraction, but completely missed the point of verifying your answer by setting x = 1.
If you just sub x=1 into your answer you are not verifying (i.e. proving) anything.
The point is that if you replace x with 1 in the above expression you get 3/(2-1) - 2/(3+1) = 3 - 1/2 = 2.5
and if you replace x with 1 in the answer you got for the single fraction
(should be (5x + 5)/(2x-1)(3x+1), you'll get (5+5)/(2-1)(3+1) = 10/4 = 2.5
You verify that you haven't changed the value of the expression by subbing x=1 into the original version as you were given and your new improved version of the expression and getting the same answer in both.
Note that you can do this verification to check that you haven't made a mistake anytime you are asked to simplify an expression, write as a single fraction etc.
Tomorrow we will start on arithmetic.
Thursday, April 10, 2008
Using algebra to solve problems
We focussed today on the type of questions where an unknown is parcelled up in two different ways and a relationship between them is given.
There is no doubt about it - these problems can be tricky. Partly because you have to parse through a lot of text, and partly because the equation that you end up with can be awkward.
Some good examples are in the 2006, 2005 and 2004 papers.
In the 2006 q, you make 2 expressions
540/x gives us the number of days the first lot of hay will last.
300/x+1 gives us the number of days the second lot will last
The sum of these two expressions is 90 days.
When you build your equation and simplify it you get
After you have factored, don't forget to discard the negative root and write your answer as x = 9 bales of hay and to work out the last part of the problem ... how many days it took to complete the first set of 540 bales. In other words, evaluate the first expression with x = 9 and write down the answer as 60 days.
There is no doubt about it - these problems can be tricky. Partly because you have to parse through a lot of text, and partly because the equation that you end up with can be awkward.
Some good examples are in the 2006, 2005 and 2004 papers.
In the 2006 q, you make 2 expressions
540/x gives us the number of days the first lot of hay will last.
300/x+1 gives us the number of days the second lot will last
The sum of these two expressions is 90 days.
When you build your equation and simplify it you get
| 3x² -25x - 18 = 0 |
Tuesday, April 8, 2008
Long Division in Algebra
The main things to remember are :
DMSB - the divide, multiply, subtract, bring-down repeated sequence is the same as long division.
If you have something like x³ + 2x²- 12, you have to add in the missing term to make it x³ + 2x² +0x - 12 before you start dividing.
The expressions will always divide in equally. You should not get a remainder.
Practice as many of these as you can.
DMSB - the divide, multiply, subtract, bring-down repeated sequence is the same as long division.
If you have something like x³ + 2x²- 12, you have to add in the missing term to make it x³ + 2x² +0x - 12 before you start dividing.
The expressions will always divide in equally. You should not get a remainder.
Practice as many of these as you can.
Thursday, April 3, 2008
Quadratic Equations
Quadratic equations are equations that involve a squared term.
If
If
x² = 4
then x could be +4 or -4.
So quadratic equations usually have 2 possible solutions.
We solve quadratics by factoring or by using the quadratic formula.
The more you practice factoring of quadratic trinomials, the more easily the answer will jump off the page for you when you do them.
The quadratic formula also takes practice, plus you need to commit it to memory. Practice writing it out regularly and always check in the book to ensure that you have got it exactly right before you use it.
For homework you are doing a selection of quadratics.
- a straightforward equation that has a coefficient in front of the x²
- one that will only take the ax² + bx + c = 0 format after you have done some manipulation
- one that requires the use of the formula
- one that needs a second round of equation solving after you have worked out possible values for x.
Tuesday, April 1, 2008
Algebra
We have just started revising Algebra.
We started with simultaneous equations. These are really important as they test a lot of mathematical skills and they can come up in co-ordinate geometry (intersection of 2 lines) also.
We will look at some worded problems which lead to simultaneous equations in the next class.
The next topic we'll cover is called changing the focus of en equation.
E.g. a = 2(b+c) Express b in terms of a and c.
We started with simultaneous equations. These are really important as they test a lot of mathematical skills and they can come up in co-ordinate geometry (intersection of 2 lines) also.
We will look at some worded problems which lead to simultaneous equations in the next class.
The next topic we'll cover is called changing the focus of en equation.
E.g. a = 2(b+c) Express b in terms of a and c.
Sunday, March 30, 2008
Surds, Indices and Index Notation
In the last classes we looked at these topics and I gave you all a handout of all the related questions from previous Junior Cert papers.
If you can do all of these you will have no problems with this topic in your exam.
In Monday's class we will go over these questions and get ready spend the remainder of this week and all of next week on algebra.
If you can do all of these you will have no problems with this topic in your exam.
In Monday's class we will go over these questions and get ready spend the remainder of this week and all of next week on algebra.
Tuesday, March 11, 2008
Co-ordinate geometry
This topic is examined on Paper 2 in question 2.
You need to apply the concepts of translations and symmetry on the co-ordinated plane.
You also need to remember all the formulae and be able to apply them.
This is an overview of how the connections between the different concepts of midpoint, distance, slope and equation of a line.
You need to apply the concepts of translations and symmetry on the co-ordinated plane.
You also need to remember all the formulae and be able to apply them.
This is an overview of how the connections between the different concepts of midpoint, distance, slope and equation of a line.
Sunday, March 2, 2008
Sets
We looked at Set notation, basics of Venn diagrams etc in class. If you are unsure of any of this, go back over the Sets chapter in book 1, as book 2 really just glosses over the foundations.
The big difference with Higher Level Sets is that you have to handle unknowns.
Take a class where 8 study Latin and 10 study Business. If you don't know how many study both, you put :
x in the L intersection B region of the diagram
(8 - x) in the L\B region
(10 - x) in the B\L region.
Then if you are told there are 14 students altogether you can create the equation:
(8 - x ) + x + (10 - x) = 14
and solve for x
18 - x = 14
x = 4 .... so 4 students study both Latin and Business.
You don't need the brackets in the above example, but it is a good idea to always use them as there are times you need to work out expressions like in q 13 on page 195
For the region of the Venn diagram representing W only the x - 2 is worked out from:
16 - (10 - x) - x - (8 - x)
= 16 - 10 + x - x - 8 + x
= -2 + x or x -2
Homework was p189: q11, q13
The big difference with Higher Level Sets is that you have to handle unknowns.
Take a class where 8 study Latin and 10 study Business. If you don't know how many study both, you put :
x in the L intersection B region of the diagram
(8 - x) in the L\B region
(10 - x) in the B\L region.
Then if you are told there are 14 students altogether you can create the equation:
(8 - x ) + x + (10 - x) = 14
and solve for x
18 - x = 14
x = 4 .... so 4 students study both Latin and Business.
You don't need the brackets in the above example, but it is a good idea to always use them as there are times you need to work out expressions like in q 13 on page 195
For the region of the Venn diagram representing W only the x - 2 is worked out from:
16 - (10 - x) - x - (8 - x)
= 16 - 10 + x - x - 8 + x
= -2 + x or x -2
Homework was p189: q11, q13
Wednesday, February 27, 2008
Schedule of revision
Today we decided on the order in which we'll study the maths course in preparation for the Junior Cert Exam.
We decided to start with Sets since it has been a while since we studied them in detail.
Here is the full running order:
I haven't allocated time for reviewing the mocks, since I don't know when these will arrive.
There will undoubtedly be changes to the above schedule, but we will try to keep to this or a revised schedule insofar as possible.
Let me know if you'd like me to make any changes.
We decided to start with Sets since it has been a while since we studied them in detail.
Here is the full running order:
| Week starting | Topics to be covered | |
| Week 1 | 03/Mar | 2 days sets / 3 days co-ordinate geometry |
| Week 2 | 10/Mar | 3 days co-ordinate geometry / 2 days surds, indices, index notation |
| 17/Mar | ||
| 24/Mar | ||
| Week 3 | 31/Mar | 1 day surds, indices, index notation / 4 days algebra |
| Week 4 | 07/Apr | 1 day algebra / 4 days using algebra to solve problems |
| Week 5 | 14/Apr | 5 days arithmetic |
| Week 6 | 21/Apr | 5 days geometry and theorems |
| Week 7 | 28/Apr | 4 days functions + graphs |
| Week 8 | 05/May | 3 days trigonometry / 1 day statistics |
| Week 9 | 12/May | 2 days statistics / 3 days perimeter area + vol |
| Week 10 | 19/May | Revision |
| Week 11 | 26/May | Revision |
I haven't allocated time for reviewing the mocks, since I don't know when these will arrive.
There will undoubtedly be changes to the above schedule, but we will try to keep to this or a revised schedule insofar as possible.
Let me know if you'd like me to make any changes.
Thursday, February 7, 2008
Trigonometry wrap-up
Today we covered the Trig question that was done for homework and took a look at the marking scheme for this question.
We took a glance through a full paper and answered a few questions.
The reciprocal of a value = 1 / the value.
Eg Reciprocal of x is 1/x
For fractions, getting the reciprocal is the same as inverting them.
E.g. reciprocal of 1/2 is 1/(1/2) = 2
E.g. reciprocal of 2/3 is 1/(2/3) = 1 x 3/2 = 3/2
Friday's class will be a general q+a class.
We took a glance through a full paper and answered a few questions.
The reciprocal of a value = 1 / the value.
Eg Reciprocal of x is 1/x
For fractions, getting the reciprocal is the same as inverting them.
E.g. reciprocal of 1/2 is 1/(1/2) = 2
E.g. reciprocal of 2/3 is 1/(2/3) = 1 x 3/2 = 3/2
Friday's class will be a general q+a class.
Wednesday, February 6, 2008
Algebra wrap-up
We looked at algebra today, specifically at question 3 and 4 on this paper.
Q3
Part a: remember the laws of indices, but check your result on your calculator.
Part b: the first two are straightforward factoring questions. Part iii can be solved by multiplying it out or by observing that it is something (2x-1) squared minus something else (x-1) squared and so can be treated as one big difference of 2 squares.
Part c: find an expression for one gram of powder, then for one gram of powder during the promotion. Then set up an equation which states that the difference between these is = 1.
Q4
Part c: note the key difference between an expression and an equation. To write both expressions as a single fraction, take each fraction and multiply above and below the line by the same thing, in such a way that you get a common denominator in the two fractions. Once you get to part ii you have an equation, so you can now multiply both sides of the equation by the denominator of the LHS to get rid of all the fractions. The a±√b format tells you you'll need to use the quadratic formula to find the roots of the equation - don't waste time looking for factors.
Key points from today:
Tomorrow we'll do Trigonometry. It is 2005 Paper 2 Q 5 for homework this evening.
Q3
Part a: remember the laws of indices, but check your result on your calculator.
Part b: the first two are straightforward factoring questions. Part iii can be solved by multiplying it out or by observing that it is something (2x-1) squared minus something else (x-1) squared and so can be treated as one big difference of 2 squares.
Part c: find an expression for one gram of powder, then for one gram of powder during the promotion. Then set up an equation which states that the difference between these is = 1.
Q4
Part c: note the key difference between an expression and an equation. To write both expressions as a single fraction, take each fraction and multiply above and below the line by the same thing, in such a way that you get a common denominator in the two fractions. Once you get to part ii you have an equation, so you can now multiply both sides of the equation by the denominator of the LHS to get rid of all the fractions. The a±√b format tells you you'll need to use the quadratic formula to find the roots of the equation - don't waste time looking for factors.
Key points from today:
- An expression squared, like (2x-1)² is (2x-1) multiplied by itself i.e. (2x-1)(2x-1).
- A quick way to do this is to square the first term, square the second term and twice their product. If in doubt, do it out the long way.
- x is an element of Q gives you a hint that the result is going to be a fraction (rational number)
Tomorrow we'll do Trigonometry. It is 2005 Paper 2 Q 5 for homework this evening.
Tuesday, February 5, 2008
Introduction
Welcome to the Junior Cert Maths Blog - designed to assist the maths learning for my 3rd year students.
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