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 mathematically-rich 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.

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 cross-method 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 Q12-18 require basic algebra. (For non-algebraic 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 real-life examples from astronomy and biology.

How do you teach your students Law 6 and Law 7 of Indices (namely a⁰ = 1 if a¹0; a^-n = 1/aⁿ 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.

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 real-life application).

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.

Although generating Pythagorean Triples is not in the O-Level syllabus, it came out as a problem-solving question in O-Level 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 O-Level syllabus but you can use it for enrichment.

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.

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.

To give a real-life 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.

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.

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.

If you have problems drawing high-quality graphs in Word or POT, Gnuplot is the freeware for you. This graphing software is for teachers to draw and print high-quality 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 user-friendly

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.

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 (right-click on file, select Open With and choose WordPad). If you use the default Notepad to open, the alignment of the primes will be out.

Having trouble drawing high-quality 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

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

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.

In these workshops conducted by me, teachers are taught how to engage students in their minds by guiding them to discover mathematical concepts using non-IT and/or IT worksheets, and how to engage students in their hearts by telling inspiring stories about mathematicians, playing mathematically-rich 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.