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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