The concept of decimals are presented in grid format to enhance the understanding of decimals. The subtle difference between 0.3 and 0.30 is made approachable to younger pupils of primary school age. The idea of money is used to explaining that. For example, while the former represents 3 ten pence coins, the latter represents 30 one pence coins. In more advanced way, one may explain it in the following way: While in 0.3, one is not quite sure whether the digit 3 is the result of rounding up or down, whereas in 0.30, one will have the confidence that it is not. Another way of looking at it is, that, if these figures were the results of measurements, then, 0.30 is more accurately measured than 0.3

Unit circle is very useful to find the values of sine, cosine and tangent of angles such as 0, 90º, 180º, 270º or 360º degrees without a calculator. One can also use equilateral triangle to find the values of sine, cosine and tangent of 30º 60º . The sine, cosine and tangent of 45º can be found using isosceles right angle triangle. Proof of some trigonometric identities are also provided. Examples on how to use the unit circle and exercises on its applications and solutions to the exercises are also included.

A line segment is divided into to parts. The division of the segment is carried out in different ratios. The line segment need not be horizontal or vertical. Three coordinate, P(a,b), Q(x, y) and R(c, d) that are on a line segment are related in such a way that PQ:QR = m:n After introduction, three different examples are provided. Given P(a, b) and R(c, d) and PQ:QR, the coordinates of Q(x, y) is be found. All three coordinates are given and the ratio of PQ:QR is sought. The intermediate coordinate, Q(x, y), the ratio PQ:QR and one other coordinate is given to find the third coordinate. This qualifies for full lesson. Additional exercises with answers provided.

The aim is to establish, experimentally, the value of ‘pi’. The relationship between the circumference and the diameter of a circle is to be investigated. Pupils are encouraged to measure the circumference and diameter of several round objects. They record their measures in the table provided. A plot of circumference against diameter is to be done and gradient (slope) to be measured , which is the approximate value of pi. The instructions how to carry out the experiment is given in one of the sheets. There is also small exercise on other worksheet. Points of discussion: Is the mean better summary value than individual measures? Discussion on measurement errors To try to find the mean of all the means of whole class

Material suitable for a complete lesson in sequences are provided in two worksheets. One deals with a differentiated lesson and the other, the more involved one, is to find the nth term of a triangular sequence. The steps arriving at the nth term of the triangular sequence are given, but the pupils are expected to answer as to why each step is true, until one arrives at the last step. Because of the inductive nature of the process, it is good practice to check if the generalised expression of the sequence works for some other situations of the sequence number.

A bundle of theorems are included here. They are: angle in a semicircle, angle in a major and minor sector, cyclic quadrilateral, angle subtended by an arc, tangent to circle property and alternate segment theorem. These materials are good for discussion and exercises can always be found in text books. The focus here is on concepts. Also, there are applications of perpendicular and angle bisectors as applied to the sides and angles of triangles to achieve circumscribed and inscribed circles. This can enhance and perfect the skills required to carry out perpendicular and angle bisectors.

Interactive puzzle that is engaging. It gives pupils the opportunity to solve a problem that requires flexible thinking. The problem is simple. Numbers form 1 to 9 is used, but each number is used only once. The goal to achieve is the sum of each row and column, and diagonally, should be 15. There are four different ways to achieve that. Note: for 3 x3 magic square 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45 45 /3 = 15 ( 3 rows or columns share the 45) For 5 x 5 magic square the sum of each row, column or diagonal is 65. For 7 x 7 the sum is 175. The rule: Enter 1 in the middle of the first row. Go through right corner (small arrow line through the corner will help). Once outside the boundary of the main square fill the next number at end of that row or column. Keep on going through right corner and if you find empty square fill the next number there, but if you find it is already occupied, fill the next number in the square immediately below. Exception: since there there is no row or column after filling top right corner square, fill the next number in the square immediately under it. The general formula for the row sum is: Last x (Last + 1) /( 2*R) where R = Number of rows Example, for 3 x 3 magic square (9 x 10 )/(2 x 3) =15 for 5 x 5 magic square (25 x 26)/(2 x 5) = 65 for 7 x 7 magic square (49 x 50)/(2 x 7) = 175 Hint: One could use the 5 x 5 and 7 x7 magic squares to practice mental arithmetic, using the idea of number bonds.

This visual proof of Pythagoras theorem is an adaptation of an excellent animation by Alan Kitching. Search “YouTube” under the title “Pythagoras in 60 seconds”, it is great fun to see.

Interesting collections of the basic concepts of fractions, decimals and percentages are provided. The equivalence of the three types of the fractions are explained in terms of diagrams. It is hoped children at KS2 will benefit from it. Very visual presentation of the subject.

We believe multiplications, presented in grid format, is very good way to teach multiplications, as long as pupils know how to partition numbers. We also believe that it is an excellent alternative to the algorithmic (long division method) to teaching divisions by way of successive subtractions, as long as the pupils know how to multiply by 10, 100 and 1000. It is hoped, this exercise should build pupils confidence on how to operate on numbers flexibly.

The material in this worksheet is suitable for children of primary school age. It enhances the understanding, through picture, the concept of units, tens and hundreds. A unit is represented by a little square, the tens is represented by a strip of ten units and hundreds is represented by a sheet of one hundred unit squares. One can think of hundred also as being made of ten strips. It is hoped the place values concept is strengthened by the representations of these figures. The teacher may choose to discuss the what we mean by a numbers and numerals at this stage. For example, 235 represents one number but three numerals are used to express it.

Areas of three types of triangles (an acute, obtuse and right angled), a rectangle, a parallelogram and two trapezium types are to be visualised and through experimentation, understanding is enhanced. For example, if the right angled triangle formed with a base length of three square units on the left of the parallelogram is cut and pasted on the blank space on the right side of the parallelogram, one gets a rectangle equivalent to the rectangle on left of the parallelogram. Now, half of the area of the rectangle is easy to comprehend to be the area of triangle (right triangle). Since the area of the rectangle and that of the parallelogram are equal, one can infer that half of the parallelogram is that of an obtuse triangle of the same area as that of right triangle of the rectangle. Similarly, one can appreciate the area of a trapezium is also the sum of two triangles. The pupils are left to investigate, given the dimensions suggested in the worksheet to arrive at the formulas of the areas of the rectangle, parallelogram, triangles (the three types) and that of the trapeziums.