Term 4 Week 3

Coordinate Geometry continued...

We have hopefully reached an understanding that any line can be defined by an equation of this form:

y = mx + b

where 'm' is the gradient of the line, and 'b' is the place the line crosses the y axis - ie the y intercept.

By understanding 'm' we can compare lines with each other. We can say that if two lines have the same 'm' value that they must be parallel. Furthermore we find that if we want to check if two lines are perpendicular we can multiply their 'm' values and if the result is -1 then we can conclude that they are.


Coordinate Geometry Assignment 2 is here

Coordinate Geometry Assignment 3 is here

Term 3 Week 9

The new topic is Coordinate Geometry

This is where pictures can be described by numbers.

To do this we use a two dimensional space called the 'number plane'. This is an extension of the number line into two dimensions: left/right (the 'x' axis) and up/down (the 'y' axis)

To specify a 'point' on this number plane we use a pair of numbers like this: (3,5), where the first number '3' is how far left/right it is, and the second number '5' is how far up/down it is. The point (3,5) would be in the top right hand side of the number plane 3 across and 5 up.

By putting multiple points on we can build up lines, and then we can make any shape or picture that we want. So we could draw a black and white Mona Lisa simply be specifying a whole series of points.

This system was worked out by a French dude name Renee Descartes about 400 years ago, and is also known as the 'Cartesian' coordinate system. Most computer games use this system to calculate and display objects on the screen.

The Coordinate Geometry Assignment 1 is here

Term 3 Week 8

Greetings maths beings!

Quite a long break there, what with trips to NZ and a couple of weeks to recover from the flu!

We have just finished a week of Indices, and there are only two or three more days before the exam. This means the exam will be either on Friday or Tuesday.

SO, keep up the practice!

Indices 3 Assignment is here

General Revision Assignment 3 is here

Term 3 Week 3

More Indices

The new shortcuts this week are:

A power divided by a power:

a^m / aⁿ = a<sup>m-n

example</sup>
2⁵ / 2² = 2^5-2= 2³

A power to a Power:

(a^m)ⁿ = a^{m x n
example:}
(5²)⁴ = 5^2x4 = 5⁸

Powers of products and quotients

(ab)ⁿ = aⁿb^n
example:
(2b)³ = 2³b³

(a/b)ⁿ = aⁿ/b^n
example:
(3/7)⁵ = 3⁵/7⁵

Class 9 is on camp for the next two weeks - enjoy, and leave the maths until you get back!

Here is the Indices 2 Assignment

Term 3 Week 2

Our new topic is Indices.

An Indice or Index is a bit of mathematics speak for writing down how many times you want to multiply something together:

10³ = 10 x 10 x 10 = 1000

We say in words 'Ten to the power of three' and this means we multiply three 10s together. The '3' is the Indice or Index or Power which tells us how many 10's we want to multiply. The '10' is the 'Base' which is 'raised' to the Indice.

It means we can work with very large and very small numbers in a simple and easy way.

For instance if we wanted to work out a problem that involved the number 'one million billion' we could write 1000000000000000 down and work with that, or we could simply write 10¹⁵

There are some excellent short cuts that can be taken with Indices. If we had to multiply two numbers like 10² and 10³ we notice this:

10² x 10³ = (10 x 10) x (10 x 10 x 10) = 10 x 10 x 10 x 10 x 10 = 10⁵

And we realise that we can use a more general rule when multiplying anything with the same 'base':

a^m x aⁿ = a^m+n

Which means that we can simply add up the indices to find the answer:

9⁵ x 9³ = 9^{5 + 3} = 9⁸

How easy is that!

More good shortcuts with Indices next week.

The Indices Assignment 1 is here

Term 3 Week 1

Welcome back for term 3!

Apologies for the delay, my home computer has been down this week.

We are still just finishing perimeter area and volume - test on Wednesday.

General Revision Assignment 2 is here

Term 2 Week 9

The final week of Perimeter Area and Volume

This week we consolidate this topic and will have an end of topic test Wednesday of week 10.

Volume. We will get our heads around the idea that we can calculate the volume of any prism by multiplying the area of the base times the vertical height.

So if we had a triangular prism (looking at the end you see a triangle) all we have to do is multiply the area of that triangle (1/2 base x height) by the length of the prism.

This is true for any prism - ie any shape that is 'extruded' like pasta. It does not matter how tricky the end looks if we can calculate the area of it then we can work out the volume of the whole shape.

The Perimeter, Area and Volume Assignment 3 is here

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Term 2 Week 8

This image is computer generated - which required the calculation of many thousands of areas of colour to render - the computer uses the same rules for area that we learn.

This week we move from calculating Perimeter to calculating area of shapes, and the the surface area of 3d shapes.

The secret is to know how to calculate the area of the standard shapes:

• Squares, rectangles, Parallelograms and Rhombus all have Area = base x vertical height

• Triangles have Area = 1/2 base x vertical height

• Kites and Rhombus have Area = 1/2 diagonal1 x diagonal2 (rhombus area can be calculated either way)

• Circles have Area = Pi x r², Part circles (sector) have Area = angle of sector / 360 x Pi x r²

The area of any shape that is a combination of the above shapes can be calculated by adding up (or subtracting as required) the area of the simpler shapes within it.

Calculating the surface are of 3d shapes is the same - just work out each basic shape and add up the areas. The only trick is to be sure that all of the sides of the object are included.

Perimeter, Area and Volume Assignment 2 is here

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Term 2 Week 7

The volume of Jupiter is more than that of all the other planets combined, but still only a small fraction of the volume of the sun.

This week we start the topic: Perimeter, Area and Volume. Students will already know how to work out the perimeter and volume of many shapes. This year we will extend that knowledge so students will be able to be able to add shapes or volumes together to make more complex shapes or volumes.

We will revisit Pythagoras so that lengths of right angle triangles can be computed. We will also do some calculations with circles and parts of circles (like pizza slices)

An example problem would be to work out the perimeter of a 20cm radius pizza slice - like you were an adventurous ant doing a walk around the edge of one. The pizza is divided equally into six parts.

To answer the question we need to add up the lines around the edge of the slice - there are two straight lines and a curved line. The straight ones are easy - they are just the radius of the circle - 2 x 20cm, but the round part is a little harder. We can find it by finding a fraction of the whole circle. - in this case 1/6th of a circle. So we multiply the circumference of the whole circle (Pi x diameter) by the fraction of a circle (1/6) . This gives us 3.14 x 40cm x 1/6 = 125.6/6 cm = ~ 31 cm. So the total edge of the slice is: 2 x 20 cm + 31 cm = 71 cm.

The Perimeter, Area and Volume assignment is here

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Term 2 Week 6

We finalise the work with investigative geometry this week. The test will most likely be on Friday, so now is a most auspicious time to revise.

This geometry has been about solving problems using 'well known' geometric facts. Students should be able to use these and name them.

In particular students will need to know these: Supplementary angles, Vertically Opposite angles, Angles at a point, Angle sum of triangle, Angle sum of quadrilateral, Angle sum of polygon formula, Exterior angle of polygon formula, Corresponding angles, Alternate angles, and Co-Interior angles.

How to get top results? Re-read the chapter, Do the assignments on time, Do some extra examples from the text, after that relax and enjoy!

Investigative Geometry Assignment 3 is here

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Term 2 Week 5

Geometry can get very complex - a 2d representation of a 4 dimensional 'polygon' ('polychora')

This week we are exploring the nature of angles in many sided figures (polygons)

By dividing 4 sided figures (quadrilaterals) into two triangles we understand that the internal angles add up to the same as that of two triangles - ie 360 degrees.

By doing the process for 5 sided figures we get 3 triangles - ie 540 degrees

We quickly discover that there is a pattern here - the internal angles of a shape with any number of sides always add up to 180 x (number of sides - 2) degrees.

So for example we can now work out that all 20 sided figures have internal angle sum of 180 x (20 - 2) degrees - ie 3240 degrees.

After getting familiar with solving these problems we will have a topic test early next week.

Investigative Geometry Assignment 2 is here
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Term 2 Week 4

God using geometry to create the world

The Hammer group is having an exam on Wednesday to complete their work with Products and Factors.

We will all start a new topic this week: Investigative Geometry.

Geometry is everywhere! Every building, every machine, In chemical compounds, In the arrangement of our bones - just about any structure we can think of has a geometry of some sort.

Being able to identify geometric figures, and being able to work out the relations of lines and angles within them is an important life skill. Particularly for anyone who wishes to work in trades as diverse as construction, engineering or space flight.

Investigative Geometry Assignment 1 is here
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Term 2 Week 3

A very large parabolic mirror used to concentrate the sun's energy - the optimal shape of the mirror is defined by a quadratic equation.

Greetings Al Jabr Lovers!

The Test for the topic Products and Factors is set for Wed week 4 ( 16/05/07)

Make sure to review the text and do as many examples as you can bear. They will get easier.

This Algebra topic is probably the most challenging maths topic that you will do this year. It lays a very good foundation for advanced maths in class 9, 10 and beyond.

This week we are covering the strategies to choose when factorising different types of expressions.

Based on the number of terms in the expression we can choose what to do:
If 2 terms (eg x² - 9) take out any common factors, and check to see if it is a difference of squares.

x² - 9 // yes this is a 'difference of squares'
= (x - 3)(x + 3)

If 4 terms (eg ab + ac + db + dc) take out any common factors, and then group in pairs, and factorise.

ab + ac + db + dc
= (ab + ac) + (db + dc) //this is the 'group in pairs' step
= a(b + c) + d(b + c) //this is the first factorise step - do each pair seperately
= (a + d)(b + c) // finally factorise the whole thing

If 3 terms (eg x² + 5x + 4) take out any common factors, and then use the 'ac' and 'b' values to work out how to split the 'b' term. (a is the coeficient of the x squared part, b is the coeficient of the x term, and c is the constant part at the end) When two values are found that multiply to equal ac and also add to equal b then these values are used to 'split' the b term. The new expression can then be grouped in pairs as per the 4 term expression above. In this example the values 1,4 are the correct ones: 1 x 4 = 4 = ac, and 1 + 4 = 5 = b.

x² + 5x + 4
= x² + 1x + 4x + 4 // split the 'b' term with the values found by using 'ac' and 'b'
= x(x + 1) + 4(x + 1) // group in pairs and factorise
= (x + 4)(x + 1) // finally fully factorise

This stuff is NOT for the faint of heart - but you will become a way better mathemetician and logical thinker if you persevere.

The Johnny Bushelle group is doing a test on Exploring Numbers this Thursday.

The Products and Factors Assignment 3 is here
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Term 2 Week 2

graph of a quadratic equation - very useful for calculating how things fall

This weeks algebra delves more deeply into the mystery of factorising binomials and trinomials.

Last week we covered how to factorise by grouping in pairs.

This week students should be able to recognise quadratic trinomials of the form ax² + bx + c (where a, b and c are numbers)

This week we are continuing to look for ways to factorise expressions.

One strategy is to look for any expressions that match the pattern for 'difference of squares' For example:

a² - 4

we recognise as a 'difference of squares', and so can easily factorise it to:

(a + 2)(a - 2)

However expressions like this are more tricky, and will require a lot more patience and practice:

5k² -12k + 4

To do this one we have to find two numbers that multiply to equal 20 (a times c, or 5 times 4) and also add up to -12 (b). We work out with trial and error that -2 & -10 multiply to equal 20 and add up to -12. Now we can 'split' the middle term and apply the strategy 'group in pairs' that we learned last week.

= 5k² -10k -2k + 4
= 5k(k - 2) -2(k - 2)
=(5k - 2)(k -2)

There, fully factorised! Wasn't that easy! Actually this is one of the most challenging Algebraic things students will do all year. If you are finding this hard you are not alone! Make sure you get one of the maths teachers to help you at lunch or outside of class.

Good luck!

The Products and Factors Assignment 2 is here
(note c9 assignments are now given and due on Wednesdays)
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Term 2 Week 1

A promising start to the development of Nuclear Fusion - people who develop their maths abilities do have the option to become mad scientists...

Welcome back to maths in second term!

The Bushelle group is doing more work with Exploring Number.

The Hammer group is doing more advanced Algebra - Products and Factors.

For example - what if you had to multiply two expressions together?

(a + 1) x (b + 2)

Because we are scrupulously fair we must multiply everything in the first expression by everything in the second one so we end up with:

= a x b + a x 2 + 1 x b + 1 x 2
= ab + 2a + b + 2

As always we can test this by putting in some actual numbers - say a = 5 and b = 7

so (a + 1)(b +2) = (5 + 1)(7 + 2) = (6)(9) = 54
and ab + 2a + b + 2 = 5 x 7 + 2 x 5 + 7 + 2 = 35 + 10 + 7 + 2 = 54

Which gives us some confidence that our multiplication of algebra expressions works!

The Products and Factors Assignment 1 is here

Week 10

fractal broccoli

Congratulations to all those who did well in the recent Algebra topic test!

We are ready now to move on to new topics. John's group is doing more work with number calculations. The Hammer group is spending a little time with some more advanced Algebra - doing binomial expansions, and then hopefully factorising polynomials.

How to do a binomial expansion? It is just multiplying one 'expression' with another 'expression':

(a + 2) x (b + 3)

The trick is to multiply everything in the first expression by everything in the second (don't leave any bit out)

= a x b + a x 3 + 2 x b + 2 x 3
= ab + 3a + 2b + 6

Don't get too stressed about this - we will do lots of practice, and as a bonus there will be no assignment over the holidays, so relax your brain and enjoy!

Week 9

The Trapezium Cluster of stars in the Orion Nebula about 1,500 light years away from us. Light from the European 'Dark Ages' will be just reaching these stars from us now. Fortunately much mathematics was preserved through that period in history by the Irish and then the Arabs.

We have completed the Algebra exam - well done everyone, and what a relief!

We are now ready to start the next topics. John's class will be moving on to 'Investigative Geometry', and Josh the Hammer's group will continue with a more advanced Algebra topic: 'Products and Factors'.

Some students will be changing class depending on their needs and preferences.

This weeks (Hammer) maths assignment is here, and is due on Wed April 4th

Week 8

This week we finish this Algebra unit. Wednesday and Thursday will be revision, and on Friday we will conclude with a Topic test.

The following Tuesday we will re-shuffle the class to ensure that students needs are met, and then we will start on the next topic.

Click here for the latest (Hammer) assignment - due Tue 27th March.
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Week 7

Algebra continues in Week 7, following a relatively non mathematical week at Nimboida.

This week we cover operations on fractions - adding, subtracting, multiplying and dividing - all with the added twist that some of the numbers are variables - ie unknown.

Don't despair - the same rules for ordinary fractions still apply - just a little more care is required, and lots of practice!

example:

a/3 + a/4 (one third of 'a' plus one quarter of 'a')
= 4a/12 + 3a/12 (before adding we must make the denominators the same)
= (4a + 3a)/12
= 7a/12 (which is the same as 7/12ths of a)

Click here for the latest (Hammer) maths assignment which is due Tue March 20th.
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Week 5

Drawing of a Mobius strip by MC Escher

Students are forging ahead with Al-Jabr this week! We are planning to squeeze in some work with 'grouping symbols' (ie brackets) and learn how to e-x-p-a-n-d an al-jabr expression:

Example: expand 2 x (m + n) - Solution: simple! everything inside the brackets is multiplied by 2, so we get 2 x m + 2 x n = 2m + 2n

And then we will reverse the process, and become confident in taking out common parts of expressions (aka 'factorisation')

Example: factorise 24 + 6q Solution: find something that divides into both parts - in this case '6' - and take it outside some brackets. This leaves two 'factors' : 6 x (4 + p) - which is the reverse of how we expanded an expression above.

Click here for the next Hammer Maths Assignment
which is due Tue March 13th (first Tuesday after Nimboyda)

Week 4

4000 year old Babylonian Inscription Representing the Square Root of 2

Following the successful completion of the 'Exploring Numbers' test, the class is now divided into two groups. John has one group in the Class 9 room, and Josh has the other in the Library Mezanine/Computer Room2/Class 8F (depending on the day)

Algebra (or Al Jabr as I often refer to it) is our topic for the next 4 weeks - we will be brushing up the skills already learned in class 7 and 8, and moving boldly forward into some new teritory.

Algebra is something people seem to love or hate with a passion. It is an extremely powerful tool which allows humanity to solve many difficult problems - like predicting how much our polution is affecting global temperatures. Being competent with Algebra is essential for many subjects in science and economics. Even better, algebra helps people to think creatively when they solve problems.

And the good news is that it is not as hard as it looks! We follow exactly the same rules as for 'normal' addition, subtraction, multiplication and division. Furthermore when we use a letter to represent a number it is not instantly complicated. The number is still simply an amount or quantity like any other number, we just have to allow that we havn't counted it up yet...

Our goal, of course, is to take students on a successful journey to confidence with this important subject.

Josh the Hammer's maths group (aka 'Hammer Maths') has new homework - click here to see it. This is due on Tuesday February 27 (ie next Tuesday). Email Josh if you have any questions or need assistance with this homework. (see the first post below for the address)

Number Asignments

Assignment 1 - Due Thu 01.02.07

Welcome to Class 9 Maths Middles in 2007

Click here for course information

contact john: johnb AT shearwater DOT nsw DOT edu DOT au
contact josh: joshs AT shearwater DOT nsw DOT edu DOT au

Click here for the Shearwater website