(Sec 14 or Grade 710)
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COPYRIGHT (from 2004)
I am in the process of recategorising and updating the resources.Some of the resources have already been shifted to other pages. At the end of the process, this page will be no more. To access these resources, please go to my Home Page by clicking the link on the top right of this page.
Feel free to amend
and use the materials in this Webpage. However you cannot publish, sell or claim
ownership of these materials. If you make so many major amendments until the
materials are very different from the originals in this Webpage, you can claim
them as your own but you must acknowledge the source. The source is not
necessary me but other teachers who have contributed their works.
This Webpage is divided into 5 main sections:
Section I: Worksheets
and GSP Templates for Students (some worksheets have answers at the back)
A. Arithmetic

Multiplication of Negative Numbers: How do you teach your
students that 4 x 3 = 12 and (4) x (3) = 12? One way is by looking at
patterns and then inferring by induction (see worksheet).



What has mathematics got to do
with rabbits or flowers? Fibonacci sequence
occurs in real life. But there is some controversy regarding flowers and
Fibonacci numbers: so you need to read the PPT file (13MB) for more
details before you give the worksheet to your students.



What has mathematics got to do
with ancient architecture or nautilus shells? This worksheet allows your
students to investigate what Perfect Rectangle
and Golden Ratio are. This topic is
also related to the Fibonacci sequence, so it may be better to let your
students do the worksheet in A2 first.



There is a
relationship between A3 and A4 papers that involves the
square root of 2. You can use this
worksheet when teaching irrational numbers. (For algebraic version, see
B3 below.)



Which natural numbers can be
expressed as a sum of consecutive natural numbers? This worksheet allows
students to investigate the properties of polite
numbers. Prerequisite knowledge includes natural numbers,
number patterns and multiples. Students can also use the Excel template
provided to help in their investigation.



How do you teach the
simple interest formula I = PART / 100?
It may be common sense to us but some students just don't understand why.
This worksheet guides the students to infer the formula by common sense
(at their level)
but please read the notes for teachers first because you need to guide
them properly. It also compares
compound interest with simple interest
in the second part of the worksheet where the students have to make a
decision.



How do you use
mathematicallyrich games to arouse students' interest in mathematics? An
example is to use the Tower of Hanoi.
The worksheet guides students to investigate the minimum number of moves
required to move a certain number of discs from one pole to another. It
involves logical thinking and recognizing number patterns. An interactive Tower of Hanoi is available at
www.mazeworks.com/hanoi.


B. Algebra

How do you teach students to
expand algebraic expressions of the
form a(b+c)?
There are various methods but one of these is to induce the expansion rule
by looking at
patterns (see worksheet).



How do you teach students to
factorise quadratic expressions? One
way is to use the crossmethod but why does it work? This worksheet allows
your students to explore the concept of factorising quadratic expressions
using algebra tiles.



There is a
relationship between A3 and A4 papers that involves the square root of 2. You can use this
worksheet when teaching irrational numbers or
basic algebra because Q1218 require basic algebra.
(For nonalgebraic version, see A4 above.)



What is the use of
standard form? It is used to represent
very big or very small numbers. But where do you encounter such numbers in
real life? You can use this worksheet for consolidation, after teaching
how to convert numbers from ordinary notation to standard form and vice
versa. It contains reallife examples from astronomy and biology.



How do you
teach your students Law 6 and Law 7 of Indices
(namely a^{0} = 1 if
a¹0;
a^{n} = 1/a^{n}
if a¹0)?
One way is to induce the laws by looking at patterns (see worksheet).



How do you
teach your students the Product and Quotient Laws
of Logarithms?
One way is to induce the laws by looking at numerical examples (see worksheet).



One application
of logarithms is the use of the Richter scale in
measuring the magnitude of an earthquake, e.g. the earthquake in Indonesia
on 26 Dec 2004 that sent off giant tidal waves and killed at least 159 000
people measured 9.0 on the Richter scale. What does this magnitude mean
and why do we use a logarithmic scale to measure the magnitudes of
earthquakes? This worksheet will guide students to answer these questions.


C. Geometry

How do you
guide students to explore and discover the four symmetric properties of
circles, the four angle properties of circles (Singapore E. Maths
syllabus) and the four circle theorems (Singapore A. Maths syllabus)? Please see attached GSP
template for all the circle properties (students need to have the GSP software).
Because the A. Maths syllabus includes Midpoint and Intercept Theorems,
these are also included in this template.



An alternative
to discover the first symmetric property of
circles is to draw circles and explore: please see attached
worksheet (this
worksheet contains a reallife application).


D. Mensuration

How do you
guide students to explore and discover the formulae for calculating
arc length and sector area? Please see
attached GSP template (students need to have the GSP software)
and worksheet.



Students usually do not
understand fully the implications of the formulae involving the area and
volume of similar figures and solids.
This worksheet allows the students to investigate the implications of
these formulae on why a giant cannot exist and why a spider is helpless
when covered with water.


E. Pythagoras Theorem

Although generating Pythagorean
Triples is not in the OLevel syllabus, it came out as a problemsolving
question in OLevel Nov 2004 Elementary Mathematics Paper 2. This
worksheet allows your students to generate
Pythagorean Triples using another method. You can use it as an
enrichment.



Many students view Pythagoras'
Theorem as the relationship between certain numbers but they fail to see
it as a property of areas as well. So this worksheet allows your students
to discover the Generalised Pythagoras' Theorem.
Again this is not in the OLevel syllabus but you can use it for
enrichment.


F. Trigonometry

How do you
teach Sine Rule? State the rule and prove it? Many students have problems
with deductive proofs. An alternative is to guide them to discover sine
rule. There are two versions: without IT and with the help of IT. For the
IT worksheet, a GSP template is also attached. But the students need to
have the GSP software.



How do you
teach Cosine Rule? State the rule and prove it? Many students have problems
with deductive proofs. An alternative is to guide them to discover cosine
rule. There are two versions: without IT and with the help of IT. For the
IT worksheet, a GSP template is also attached. But the students need to
have the GSP software.



How do you
teach students the signs of trigonometric ratios
of angles in the four quadrants? You can use the interactive GSP template
to illustrate. You need the GSP software.



How do you help
students to visualise how the
sine curve, cosine curve
and tangent curve is generated from a
unit circle? You can use the interactive GSP template to demonstrate.
You need the GSP software.



This is a
simple worksheet to guide students to explore the effect of
a on the
sine curve
y =
a sin x where a >
0. It does not involve the use of IT.


Section II: Readytouse Resources for Teaching
(including songs and videos)
A. Arithmetic

For enrichment on prime numbers,
you can teach your students to generate primes using the formula by
Mersenne (see PPT). The largest known prime number (found on 23 Aug 2008)
is the 45^{th} Mersenne Prime and it contains 12 978 189 digits
(or 433 newspaper pages)! To impress your students how big this number is,
download the zip file on the right, unzip it and open it with Notepad. For
new update, visit
www.mersenne.org.


B. Algebra

To give a
reallife example of a
parabolic curve which is described by
a quadratic equation, you can show this video clip of the motion of a
projectile for different angles of projection.


C. Geometry

You can use an interactive kaleidoscope
to illustrate the effect of multiple reflections.
Use Internet Explorer to open after unzipping.



If your students have problems visualising the
nets of polyhedra such
as prisms, pyramids etc., Poly is the shareware for you. E.g. you can open
up a cube to show its nets. What you can download from this site is only
an evaluation version. For more details on the shareware and where you can
purchase it, please visit
www.peda.com/poly.


E. Matrices

What is a matrix? It is a rectangular array of numbers and it can used
to store information. To relate this
to real life, you can link it to the movie The Matrix where the matrix is used to store genetic
information (if
you include characters as well as numbers). There are two versions: the
original 2D version and the new 3D version.


Section III: Free Computer Software
A. Graphs

If you have problems drawing
highquality graphs in Word or POT,
Gnuplot is the freeware for you. This graphing software is
for teachers to draw and print highquality graphs, including ease of drawing blank grids and
frequency cumulative curves (by fitting curve to a set of data). It is
not for students to explore graphs because
it is not that userfriendly


B. Geometry

If you want to combine math and arts, Tess is a shareware that allows
you and your students to create colourful art pieces that involve
tessellations. What you can download from this site is only an evaluation
version. For more details on the shareware and where you can purchase it,
please visit www.peda.com/tess.


Section IV: Resources to Help You Prepare Your Own Teaching
Materials
A. Arithmetic

When preparing a worksheet on prime numbers (e.g. which years in this
decade are primes?), it may be useful to refer to a list of big prime
numbers. Download the zip file on the first 1000
primes, unzip and open with WordPad (rightclick on file,
select Open With and choose WordPad). If you use the default Notepad to
open, the alignment of the primes will be out.


B. Graphs and Geometry

Having trouble drawing highquality
blank graph papers? You can choose from the many samples from this Word document
and copy and paste them into your worksheets. Alternatively, the following
Website allows you to create your own graph papers,
number lines, nets,
shapes, etc.):
http://illuminations.nctm.org/ActivityDetail.aspx?ID=205


C. Probability

Pictures of Playing Cards for you to copy and paste when preparing
your worksheet.


Section V: Worksheets Designed by Teachers During My Workshops
Disclaimer:
Some of these worksheets are excellent while others need to be modified. You
need to sieve through yourself.
A. Active
Mathematics Teaching

In these workshops conducted by me, teachers are taught how to design
worksheets that promote active student learning. The worksheets may or may
not use IT to guide students to explore mathematical concepts. There are
two series of workshops: Aug 2005 and Mar 2006. You can download the
teachers' worksheets using the buttons on the right.


B. Engaged
Learning in Mathematics

In these workshops conducted by me, teachers are taught how to engage
students in their minds by guiding them to discover mathematical concepts
using nonIT and/or IT
worksheets, and how to engage students in their hearts by telling
inspiring stories about mathematicians, playing mathematicallyrich games
and using video clips and songs. There are
three series of workshops: Aug 2006, Mar 2007, Feb 2008, Mar 2009 and Mar 2010. You can download the
teachers' works using the buttons on the right.


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If you have any queries, please email
Joseph Yeo. More to come...