Some of you may have done approximation many times but not know its practical use.
Imagine yourself as a sports reporter, you wish to report on the number of people who attended a soccer match. Would it matter to the reader whether 1522 or 1523 people attended the soccer match?
Usually, people would approximate in this case and report that around 1500 attended the match.
For larger numbers, it is the same case. Suppose the population of a country was 4 340 400, usually people would not quote the exact figure. An approximate figure would be enough to give a rough idea, in this case, one can round off 4 340 400 to 4 million. "4 million" is much shorter/convenient to write or say out compared to writing "4 340 400" or saying "four million, three hundred and forty thousand, and four hundred" right?
Friday, June 13, 2008
Sign Change for Multiplication!
Some of you might not have caught algebra well. Maths sometimes is as simple as spotting a pattern.
2(a) = 2a
b(3c) = 3bc
5d(3ek) = 15dek
Spot any pattern?
2(a) = 2 × a = 2a
b(3c) = b × 3c = 3bc
5d(3ek) = 5d × 3ek = 15dek
That was simple enough?
Let's get to what is usually tested:
Q1. 2(a + b) = 2a + 2b
Q2. 3(c + 4d) = 3c + 12d
Let's get to the part where lots of mistakes occur! When you see a negative sign, you have to check for? SIGN CHANGE!
(+)(+) = (+) <-- Do not need to check for sign change because you do not see the negative sign!
(-)(-) = (+)
--------------------
(+)(-) = (-)
(-)(+) = (-)
Q3. 5(f - 2g) = 5f - 10g
Why is it -10g? This is because of +5 × (-2g) = -10g.
Q4. -5(f - 2g) = -5f + 10g
Why is it -5f? This is because of -5 × (+f) = -5f.
Why is it +10g? This is because of -5 × (-2g) = +10g.
Q5. -5(-f - 2g) = 5f + 10g
Why is it -5f? This is because of -5 × (-f) = +5f.
Why is it +10g? This is because of -5 × (-2g) = +10g.
Let's try something longer:
Q6. 3(-2h + 5k - 7m) = -6h + 15k - 21m
Why is it -6h? This is because of 3 × (-2h) = -6h.
Why is it +15k? This is because of 3 × (+5k) = +15k.
Why is it -21m? This is because of 3 × (-7m) = -21m.
Q7. -3(-2h + 5k - 7m) = 6h - 15k + 21m
Why is it -6h? This is because of -3 × (-2h) = +6h.
Why is it +15k? This is because of -3 × (+5k) = -15k.
Why is it -21m? This is because of -3 × (-7m) = +21m.
Q8. -3(-2h + 5ka - 7mn) = 6h - 15ka + 21mn
Why is it -6h? This is because of -3 × (-2h) = +6h.
Why is it +15k? This is because of -3 × (+5ka) = -15ka.
Why is it -21m? This is because of -3 × (-7mn) = +21mn.
Remembered we did index notation?
a × a = a2
Some of you might wonder why not just write a × a = aa?
Well, that seems alright for a × a × a = aaa. However, lots of things practiced in maths is to convenient/help us.
How about?
a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a = aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
Wouldn't it be so so so much easier to write a79?
Some of the most difficult questions that will come out for your level would look like:
Q9. -2b(-3 + 5b - 8ab2) = 6b - 10b2 + 16ab3
Why is it 6b? This is because of -2b × (-3) = +6b.
Why is it -10b2? This is because of -2b × (+5b) = -10b2.
Why is it +16ab3? This is because of -2b × (-8ab2) = 16 × b × a × b × b = 16abbb = +16ab3.
Now, do you understand the topic better and have become more confident at how easy maths can be?
2(a) = 2a
b(3c) = 3bc
5d(3ek) = 15dek
Spot any pattern?
2(a) = 2 × a = 2a
b(3c) = b × 3c = 3bc
5d(3ek) = 5d × 3ek = 15dek
That was simple enough?
Let's get to what is usually tested:
Q1. 2(a + b) = 2a + 2b
Q2. 3(c + 4d) = 3c + 12d
Let's get to the part where lots of mistakes occur! When you see a negative sign, you have to check for? SIGN CHANGE!
(+)(+) = (+) <-- Do not need to check for sign change because you do not see the negative sign!
(-)(-) = (+)
--------------------
(+)(-) = (-)
(-)(+) = (-)
Q3. 5(f - 2g) = 5f - 10g
Why is it -10g? This is because of +5 × (-2g) = -10g.
Q4. -5(f - 2g) = -5f + 10g
Why is it -5f? This is because of -5 × (+f) = -5f.
Why is it +10g? This is because of -5 × (-2g) = +10g.
Q5. -5(-f - 2g) = 5f + 10g
Why is it -5f? This is because of -5 × (-f) = +5f.
Why is it +10g? This is because of -5 × (-2g) = +10g.
Let's try something longer:
Q6. 3(-2h + 5k - 7m) = -6h + 15k - 21m
Why is it -6h? This is because of 3 × (-2h) = -6h.
Why is it +15k? This is because of 3 × (+5k) = +15k.
Why is it -21m? This is because of 3 × (-7m) = -21m.
Q7. -3(-2h + 5k - 7m) = 6h - 15k + 21m
Why is it -6h? This is because of -3 × (-2h) = +6h.
Why is it +15k? This is because of -3 × (+5k) = -15k.
Why is it -21m? This is because of -3 × (-7m) = +21m.
Q8. -3(-2h + 5ka - 7mn) = 6h - 15ka + 21mn
Why is it -6h? This is because of -3 × (-2h) = +6h.
Why is it +15k? This is because of -3 × (+5ka) = -15ka.
Why is it -21m? This is because of -3 × (-7mn) = +21mn.
Remembered we did index notation?
a × a = a2
Some of you might wonder why not just write a × a = aa?
Well, that seems alright for a × a × a = aaa. However, lots of things practiced in maths is to convenient/help us.
How about?
a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a × a = aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
Wouldn't it be so so so much easier to write a79?
Some of the most difficult questions that will come out for your level would look like:
Q9. -2b(-3 + 5b - 8ab2) = 6b - 10b2 + 16ab3
Why is it 6b? This is because of -2b × (-3) = +6b.
Why is it -10b2? This is because of -2b × (+5b) = -10b2.
Why is it +16ab3? This is because of -2b × (-8ab2) = 16 × b × a × b × b = 16abbb = +16ab3.
Now, do you understand the topic better and have become more confident at how easy maths can be?
Wednesday, June 11, 2008
Semester 2
A Sneak Preview of Things to Come in Semester 2
(01) Perpendicular Bisector and Angle
This is an easy topic. When someone asks you to bisect something, it means divide it equally into 2 parts.
As such, perpendicular bisector means divide a straight line into 2 equal parts. For example, a line is 5cm long, when bisected, one part will be 2.5cm. This can be easily done using a ruler. Here, you will learn how to bisect a line using a pair of compasses only (no ruler).
You will be amazed at how using the same pair of compasses, you can also bisect an angle.
(02) Construction of Triangles (3-sided figures)
This is something you learnt in primary school. Using a pair of compasses and a ruler, you are to construct a triangle with given data. The data usually appear as 2 items: length and angle.
(03) Properties of Quadrilaterals (4-sided figures)
Again, some of it you have learnt in primary school. You will get to learn angle properties of quadrilaterals (4-sided figures). Some of the figures include parallelogram, rectangle, square, rhombus, kite and trapezium.
(04) Construction of Quadrilaterals (4-sided figures)
This is something similar to the construction of triangles except now with the same equipment and given data, you are to construct 4-sided figures instead.
(05) Straight Line Graphs
This is an easy to score topic for those who like step-by-step questions (longwinded type). Usually the examiner will give you an equation of a straight line (e.g. y = 2x + 1) and a table of values. The table of values will contain some values of x and y. You are to fill up the empty blanks according to a rule, that is the "equation of a straight line". Then comes the exciting part, according to the table of values, you will plot and draw the straight line. More details when we do this topic...
(06) Ratio, Rate and Proportion
Old friends! By the sound of the topic, you may have guessed it is something you learnt in primary school. Another topic up for grabs!
(07) Solving Simple Inequalities
Something that is ALWAYS confused with equations because of the SIGN.
Equation:
3x = 21
x = 7
Inequality (it is called inequality because it is NOT equal!):
3x > 21
x > 7
Notice the difference?
(08) Number Patterns and Number Sequences
Usually, questions on this topic require you to complete a number pattern or number sequence.
For example,
Number sequence: 3, 5, 7, 9, __, 13.
The number in the missing blank is 11.
However, in secondary school, you also need to learn how to write the general term (sometimes called the nth term) for a number pattern or number sequence.
Using the same example,
nth term = 2n + 1
Why?
This will give a clearer picture:
1st term = 2(1) + 1 = 3
2nd term = 2(2) + 1 = 5
3rd term = 2(3) + 1 = 7
4th term = 2(4) + 1 = 9
5th term = 2(5) + 1 = 11
thus giving the number sequence: 3, 5, 7, 9, 11, 13.
Why do we have to know how to write the nth term formula? This is such that if someone asks you to find the 1280th term, you do not need to waste time writing 3, 5, 7, 9, 11, 13... all the way to the 1280th term.
Instead you can just use the nth term formula:
1280th term = 2(1280) + 1 = 2561
(09) Statistics
Another old friend. Does bar charts and pie charts ring a bell? You will however learn new but very simple skills of how to do up a tally after the collection of data.
(10) Perimeter and Area of 2D Figures
This is an extension of what you learnt in your primary school. By now, you must be an expert in calculating the perimeter and area of some shapes such as squares, rectangles, triangles, circles etc. Here, you will learn to use the formulae to find the area of parallelogram, trapezium and composite figures (meaning made up of 2 or more basic shapes).
One important new thing you will also learn (which you will also do in science. great isn't it? study one item and score for 2 subjects): convert between cm² and m².
(11) Volume and Surface Area of 3D Figures
Again, an extension of what you learnt in primary school. Old stuff you will see: net, volume of cubes, cuboids and cylinders. New stuff: volume and surface area of prisms, convert between cm³ and m³.
The above is not the exact list of what will be covered but a rough idea of what to look out for. To score, there is no substitute other that hard work.
(01) Perpendicular Bisector and Angle
This is an easy topic. When someone asks you to bisect something, it means divide it equally into 2 parts.
As such, perpendicular bisector means divide a straight line into 2 equal parts. For example, a line is 5cm long, when bisected, one part will be 2.5cm. This can be easily done using a ruler. Here, you will learn how to bisect a line using a pair of compasses only (no ruler).
You will be amazed at how using the same pair of compasses, you can also bisect an angle.
(02) Construction of Triangles (3-sided figures)
This is something you learnt in primary school. Using a pair of compasses and a ruler, you are to construct a triangle with given data. The data usually appear as 2 items: length and angle.
(03) Properties of Quadrilaterals (4-sided figures)
Again, some of it you have learnt in primary school. You will get to learn angle properties of quadrilaterals (4-sided figures). Some of the figures include parallelogram, rectangle, square, rhombus, kite and trapezium.
(04) Construction of Quadrilaterals (4-sided figures)
This is something similar to the construction of triangles except now with the same equipment and given data, you are to construct 4-sided figures instead.
(05) Straight Line Graphs
This is an easy to score topic for those who like step-by-step questions (longwinded type). Usually the examiner will give you an equation of a straight line (e.g. y = 2x + 1) and a table of values. The table of values will contain some values of x and y. You are to fill up the empty blanks according to a rule, that is the "equation of a straight line". Then comes the exciting part, according to the table of values, you will plot and draw the straight line. More details when we do this topic...
(06) Ratio, Rate and Proportion
Old friends! By the sound of the topic, you may have guessed it is something you learnt in primary school. Another topic up for grabs!
(07) Solving Simple Inequalities
Something that is ALWAYS confused with equations because of the SIGN.
Equation:
3x = 21
x = 7
Inequality (it is called inequality because it is NOT equal!):
3x > 21
x > 7
Notice the difference?
(08) Number Patterns and Number Sequences
Usually, questions on this topic require you to complete a number pattern or number sequence.
For example,
Number sequence: 3, 5, 7, 9, __, 13.
The number in the missing blank is 11.
However, in secondary school, you also need to learn how to write the general term (sometimes called the nth term) for a number pattern or number sequence.
Using the same example,
nth term = 2n + 1
Why?
This will give a clearer picture:
1st term = 2(1) + 1 = 3
2nd term = 2(2) + 1 = 5
3rd term = 2(3) + 1 = 7
4th term = 2(4) + 1 = 9
5th term = 2(5) + 1 = 11
thus giving the number sequence: 3, 5, 7, 9, 11, 13.
Why do we have to know how to write the nth term formula? This is such that if someone asks you to find the 1280th term, you do not need to waste time writing 3, 5, 7, 9, 11, 13... all the way to the 1280th term.
Instead you can just use the nth term formula:
1280th term = 2(1280) + 1 = 2561
(09) Statistics
Another old friend. Does bar charts and pie charts ring a bell? You will however learn new but very simple skills of how to do up a tally after the collection of data.
(10) Perimeter and Area of 2D Figures
This is an extension of what you learnt in your primary school. By now, you must be an expert in calculating the perimeter and area of some shapes such as squares, rectangles, triangles, circles etc. Here, you will learn to use the formulae to find the area of parallelogram, trapezium and composite figures (meaning made up of 2 or more basic shapes).
One important new thing you will also learn (which you will also do in science. great isn't it? study one item and score for 2 subjects): convert between cm² and m².
(11) Volume and Surface Area of 3D Figures
Again, an extension of what you learnt in primary school. Old stuff you will see: net, volume of cubes, cuboids and cylinders. New stuff: volume and surface area of prisms, convert between cm³ and m³.
The above is not the exact list of what will be covered but a rough idea of what to look out for. To score, there is no substitute other that hard work.