Note the equivalence between the laws of indices and the laws of logarithms. Use logarithm tables for the purposes of calculation. It can only give a positive Teacher: Chart demonstrating laws of logarithms; answer. So, that is why we say that in logax, logarithm tables as used by Examinations Boards. This can be other variable. Give them enough examples to practise this. The second expression cannot really be simplified without using log tables or a scientific calculator.
Solve quadratic equations using factorisation, completing the square, the quadratic formula and graphical methods. Transform word problems into linear equations that may be solved algebraically. Transform word problems into quadratic equations that may be solved algebraically.
Rewrite the equation in the standard form of drawing curves wire or twine. Teacher: Graph boards or if an overhead projector 2. Write the values of a, b and c into the actually factorise quadratic trinomials. If they brackets. Use a calculator to work out the answer. Use short sentences when you write a problem for them to solve. Ask them to write down their steps. Areas of difficulty and common mistakes 3. Sometimes the problem can also be made factorisation, students do not realise that right-hand easier by representing it in table form side of the equation must always be equal to 0.
Make sure are asked specifically to complete the square to that they do by helping those with problems find the root s of a quadratic equation. Teach individually. Solve for x: tend to use completion of the square. Calculate the length of a circular arc. Calculate the area of sectors and segments of a circle. Calculate the outcomes when adding or subtracting compatible areas and volumes.
Solve problems involving change of shape of quantities such as liquids. Use linear, area and volume factors d, d 2, d 3 when comparing the dimensions of shapes that are geometrically similar. You can and the angle between the two sides.
For example: to km, they do not know whether they must Length ratio of two similar shapes is in the divide by 1 or multiply by 1 This problem can be You find the base of a prism if you imagine that avoided if they use a table like the example above you cut through it like you would cut bread and or if they convert the units before working out form congruent slices.
Then the base is parallel the area or volume. If, for example, the disk is 5 mm an error is and correct it or you can see where thick and has a diameter of 8 cm, either the 8 cm they went wrong and correct a wrong thinking has to be converted to mm or the 5 mm has to process. Supplementary worked examples 1. The table on the next page gives information about enlargements of the parallelogram shown here. Copy the table in your book and complete it.
This table shows the volume of the cuboid shown here. Copy and complete the table. Solution 1. Calculate the simple and compound interest on a given principal at a given rate over a given period of time. Calculate the depreciation as an item loses value over a given period of time. Calculate how prices are affected by inflation over time. Calculate the dividend arising from investing in stocks and shares. Calculate the financial benefits in purchasing an annuity. Calculate the income tax levied on given incomes.
Determine the value-added tax VAT paid on certain goods and services. Linear depreciation more or less like simple compound increase works: interest. Reducing-balance depreciation where the Let us say that Mr Okoro pays 24 per year depreciation is also a depreciation on the to save for his retirement. It can be compared He pays this amount at the end of each year for to compound interest. The formula is: 30 years. See number 18 or Exercise 5a. The second amount paid at the end of the 2nd year Let us say that somebody pays instalments of would yield interest for 1 year: 10 1.
The 20 includes the interest and we annuity, it means that it is money that will be want to know what the actual amount per received in the future. It usually is a way to save month without interest is, because we want to for retirement. Steven starts saving at the end of the first month. Answer: 1,, A dividend is always an amount per share. Now this amount is the already increased amount.
The taxable amount must always be broken down into the tax bands and the tax for each band worked out and added to give the total income tax. Teaching and learning materials The sin-ratio contains a y-coordinate and is, Students: Textbook, exercise book, writing therefore, positive in the second quadrant.
Teacher: Graph boards, aids for drawing curves So, all the angles in the second quadrant can be wire or twine. They answer, the tangent-ratio is positive in the third are the cos and tan ratios.
They are quadrant in which the radius of the circle falls. Determine: consists of separate parts, and we say that this a tan Determine: — Where the graph intersects the y-axis the a tan Emphasise that they equation. Then translate that graph 1 unit upwards, or first 0. So, let them draw the graphs in steps.
Distinguish between great circles and small circles on a spherical surface. Recall and use the definitions of lines of longitude including the Greenwich Meridian and latitude including the equator on the surface of the Earth.
Calculate the distance between two points on a great circle meridian and equator. Calculate the distance between two points on a parallel latitude. Compare great-circle and small-circle routes on the surface of the Earth. Organise data and information in a rectangular array, or matrix of rows and columns. Multiply matrices by a scalar. Transpose a matrix. Teaching and learning materials If we multiply two matrices A and B by each Students: Textbook, exercise book and writing other, we multiply each entry of the first row materials.
That gives us the first multiplication; examples of matrices from entry of the product of the two matrices. That again gives us the Teaching notes second entry of the product of the two matrices. The number of columns vertical. This means that: This is similar to writing down the coordinates of a point.
We then write down the vertical component y-coordinate. This rule is especially useful the entries of the matrix over the main diagonal. This will only work for square matrices, however. So: Step 3: Now, take the matrix —53 4 —1 and replace. Step 6: Now, take the matrix —53 4 —1 and replace. So: —53 — Let them use arrows as illustrated above and also let them write out their work.
Let them practise the method. Recall and use various forms of the equations of lines that are parallel to the Cartesian axes. Recall and use various expressions for the gradient of a straight line. Find the intercepts that a line makes with the x- and y-axis and use them to sketch the line. Find the distance between two points on a Cartesian plane. Find the coordinates of the midpoint of a straight line joining two points of the Cartesian plane.
Recall and use the conditions for two lines to be a parallel and b perpendicular. Calculate the angle between two lines on the Cartesian plane. So, division by 0 is undefined. Tell them to check their work 2 and to remember how the gradient is obtained. Tell them that, if we want to find the x-coordinate in the middle of two other e y x-coordinates, then we always add the two 4, 1 x x-coordinates and divide the answer by 2.
The same principle applies for finding the y-coordinate of the midpoint of a line segment. So, we show the gradient 2 and the fact that the line passes through the origin. So, we show the gradient and the fact that the line passes through the x origin. Note the direction of the line because of the —2 fact that the gradient is positive.
You must show that the graph is parallel to the x-axis, because it is a sketch and facts are only facts if they are shown on a sketch. Find the gradient functions of a simple quadratic curve and hence find the gradient of the curve at any point. Use the chain rule, and the product and quotient rules to differentiate complex algebraic expressions. The derivative of a h constant is equal to 0. Then you add the two products. From this you subtract the derivative of the part underneath the line multiplied by the part above the line.
This whole expression is divided by the square of the part underneath the line. Determine the equation of a line, or curve given its gradient function. Use the inverse of differentiation to find the general solutions of simple differential equations. Apply the rules of integration to integrate algebraic expressions. Calculate the arbitrary constant, given sufficient relevant information. Apply integral calculus to solve problems relating to: coordinate geometry.
Teaching and learning materials 2. Teacher: Charts showing standard integrals; In words: For a power of a variable say x add 1 calculus-related computer instructional materials to the exponent of x and divide the power of where available.
In words: The integral of a constant is that constant times x, and then add C, the arbitrary constant of integration. The integration integral: constant, C, is not necessary. This is also called the definite integral with respect to x. You will have to explain this very carefully. Recall and use the method of calculating the standard deviation of a set of discrete data. Interpret the variation or spread of a data set in terms of its standard deviation. Calculate the mean deviation of a set of discrete data.
Select and use a working mean to simplify calculation. Calculate the variance of a given data set. Calculate the standard deviation of a set of grouped data.
Teaching and learning materials Step 2 Add all the values you got in Step 1. Students: Textbook, exercise book, writing Step 3 Add all the frequencies. Glossary of terms Step 3 Square all the differences d 2 in Step 2 to remove the negative of some values. Standard deviation shows the extent of the Step 4 Multiply f by the square of each of the variation or spread of the distribution. Students do not know which inequality sign to Below is a summary of some of the words used: At least means at the minimum or not less 10 than or bigger than or equal.
Not more than means less than or equal. Up to means at the most or the maximum x or less than or equal. Not less than means it must be more than or The opposite happens, if one wants to equal.
Making y the objective function as a family of parallel lines. They do not understand that to maximise Now the y-intercept, which is part of the cost, they have to move the line as far as possible up must be as low as possible while still touching along the y-axis, while still keeping the same at least one point that is part of the feasible gradient.
So, part of the line must at least still touch 10 one of the points of the lines that make up the boundaries of the region that contains the possible values of x and y. This region is called the feasible region. Now emphasise that, if you have or , the , 1 line is solid, because the point are included. This method will only work when the inequality is written in the form where only y is the subject of the formula. The arrow shows that the numbers 5 go on. Use the language of probability, including applications to set language, to describe and evaluate events involving chance.
Define and use experimental probabilities to estimate problems involving chance. Define and use theoretical probabilities to calculate problems involving chance.
Illustrate probability spaces by means of outcome tables, tree diagram and Venn diagrams, and use them to solve probability problems. Teaching and learning materials Areas of difficulty Students: Textbook, exercise book, graph paper, Words like at most and at least could give coins, and a simple die made from a hexagonal problems. Explain as follows: pencil or ballpoint pen. Say you are choosing 3 cards from a pack of Teacher: Class sets of coins, dice, counters, playing cards and you want to know what the probability cards, graph board.
Then choosing at most 3 spades means that Glossary of terms you could choose 3 spades, 2 spades, 1 spade or Outcome is a possible result of an experiment. Each possible outcome of a particular Choosing at least 2 spades means that you can experiment is unique. Only one outcome will choose 2 spades or 3 spades. If a choice is made and the object is not put Certain event is something that is certain to back, students tend to forget that the total happen and its probability is always equal to 1.
The quantity of the other kinds of objects Impossible event is something that will never stays the same. For example, For example, you have 3 black and 4 white the probability that the sun rises in the West in balls in a bag. London is 0. If you take out one black ball and you do not Theoretical probability is probability based on logical put it back, there are 2 black balls left and the total number of balls now is 6.
Disjoint events are also mutually exclusive events. Define and apply the properties of a tangent of a circle. Teaching and learning materials A A tangent Students: Textbook, exercise book. Teacher: Posters, cardboard models, chalkboard O P O P instruments especially a protractor and a compass , tangent computer instructional materials where available.
Tangent of a circle is a line that intersects the x circle in only one place. We can also say that it B tangent D B D has only one point in common with the circle. To help them you could give them summaries If they still cannot recognise this, let them put of the theorems and suggestions of the reasons their forefinger and thumb on the chord that they can give.
Below are suggestions: shares the point of contact with the tangent and follow the two lines at its ends to the angle on the circumference of the circle. Define and distinguish between displacement, equivalent and position vectors. Calculate the magnitude and direction of a vector. Find the sum and difference of vectors. Define a null zero vector. Multiply a vector by a scalar. Resolve a vector into components. Solve vector algebra problems.
Teaching and learning materials Areas of difficulty Students: Textbook, graph exercise book, ruler and Students must remember that, if they represent a pencil. A negative x component is a horizontal displacement to the left. Components of vectors are the scalar multiples of y i and j. The symbol for A 3 equivalent vectors is. The reason is that if an area is calculated, y two lengths are used and if each length is, for 6 example, 2 times longer, the corresponding B area is 2 2 bigger.
Present and interpret grouped data in frequency tables, class intervals, histogram and frequency polygons. Calculate, illustrate and interpret the mean, median and mode of grouped data. Construct cumulative frequency tables. Draw cumulative frequency curves and use them to deduce quartiles and medians.
Teaching and learning materials If the scale of the cumulative frequency curve Students: Writing materials, exercise book, is too small, accurate readings of the median, textbook, graph paper, a soft wire that can be used first quartile, third quartile, and so on cannot be to draw the curve of the cumulative frequency taken.
Students should make the scale for this curves and a ruler. Teacher: Graph chalkboard; computer with Make sure that students know that for the ogive appropriate software if available; copies of graph or cumulative frequency curve, they plot the paper on transparencies, transparency pen and an upper class boundaries on the horizontal axis overhead projector, if available.
Areas of difficulty The accuracy of the readings from the ogive also depends on how accurate the curve is drawn. Students make the mistake of not starting and Students can use a soft wire and bend it so that ending a frequency polygon on the horizontal it passes through all the plotted points.
They can axis. Teach them that they always start with the then trace the curve of the wire on the graph class midpoint of the class before the first class paper to draw an accurate curve. Explain what is meant by geometrical transformations, congruencies and enlargements. Define and apply the properties of translation, reflections and rotation in a plane. Use reflection and rotation to determine properties of plane shapes. Determine and use the scale factor of an enlargement.
Construct enlargements of given shapes. Locate the centre of an enlargement. Solve problems involving combined transformations. Teaching and learning materials Areas of difficulty Students: Textbook, graph exercise book, ruler and When you want to explain how the angle of pencil.
Join P, the corresponding point of the image Glossary of terms to K, the centre of rotation. Congruencies in transformation geometry Measure the angle PKA. Translation, reflection and rotation from A to P is anti-clockwise.
It is, therefore, the length 7 of an imaginary straight path; and it is usually 6 K 2, 6 not the actual path travelled by P. A displacement vector represents the length and direction of 5 this imaginary straight path. If the displacement R. Usually it can be seen from the form of the They will then see that A' is the point figure what the images of the different points 2,2. The same process could be repeated are otherwise it must be given because to find B' and C': different images would result in different A general rule can, therefore, be enlargements as can be seen from 2.
It states that if any point Exercise 15f. If the scale factor is 3 and This rule also applies for a rotation. Supplementary worked examples Students then could take the same triangle 1. When a figure is rotated around the origin, again and measure an angle of 90 clockwise specific rules can be deduced. Students should around the origin and follow the same find these rules out for themselves. The same process could be repeated to find B' and C': 4 A general rule can, therefore, be deduced.
It states that if any point is A 2, 2 2 A 2, 2 rotated clockwise through 90around the origin, then x,y x,y. When the origin is the centre of enlargement, clockwise rotation of around the special rules can also be deduced.
If students look at the coordinates of The same process could be repeated to corresponding points, the coordinates of the find B' and C': enlarged triangle are twice the coordinates of The general rule can, therefore, be the original triangle. So, in general, if the sides of the deduced. It states that if any point is rotated through clockwise or enlarged triangle are k times longer anti-clockwise around the origin, then than those of the original triangle, then x,y x,y.
If on the other hand, the enlargement factor is negative, the image is on the other side of the origin, the centre of enlargement. Calculate the gradient of a straight line. Determine the equation of a straight line from given data. Draw the tangent to a curve at a given point. Use the tangent to find an approximate value for the gradient of the curve at a given point. Areas of difficulty If they do not have plastic curves, let them Students find the concept of gradient difficult bend a piece of soft wire in such a way that the and often divide the change in the x-values wire passes through all the points they plotted by the change in y-values.
Emphasise that the on the graph paper. Then let them trace the curve of the wire with The reason why the gradient of a line parallel to a pencil. This is when a gradient is positive and when it is because the x-value stays unchanged: negative. Division by 0 is, therefore, undefined. Example 5 could also be done like this: 1.
From the point go you take first, as long as you take the approximately 4 units down and 3 units to corresponding x-value also first. Downwards is negative:. To the left is negative:. Use factorisation to simplify algebraic fractions by reducing them to their lowest terms. Multiply and divide algebraic fractions, reducing them to their lowest terms. Add and subtract algebraic fractions to give a single algebraic fraction. Use substitution of numerical values or algebraic terms to simplify given algebraic fractions.
Solve equations that contain algebraic fractions. Understand what is meant by an undefined fraction. Determine the values that make a fraction undefined. Teaching and learning materials When multiplying and dividing algebraic Students: Textbook, exercise book and writing fractions, their numerators and denominators materials.
For these algebraic fractions, the factor form Areas of difficulty of their numerators and denominators is the simplest form for these specific purposes. This other form is Although we do not write numerical fractions simpler or easier to use in certain situations. When multiplying and dividing algebraic So this expression is simpler for the purpose fractions, students tend to also cancel terms of substitution. In this chapter, the numerator instead of first factorising the numerators and denominator of algebraic fractions must and denominators of fractions.
Before you do be written in factor form before the fraction Examples 4 to 6, you could also do this example can be written in its lowest terms. This can lead to multiplying all the terms of the numerator mathematical errors. Then they should Students tend to treat the addition and substitute the root in the right-hand side of subtraction of algebraic fractions the same as the equation and work out the answer. To test whether this it is a completely different matter.
Have two sides left- Are added by putting hand side and right- 11 is a root of the equation. Remove the fraction to 1. Determine the rule that generates a sequence of terms, extending the sequence as required.
Find an algebraic expression for the nth term of a simple sequence. Define an arithmetic progression in terms of its common difference, d, and first term, a. Define a geometric progression in terms of its common ratio, r, and first term, a.
Distinguish between a sequence and a series. Calculate the sum of the first n terms of an arithmetic series. Calculate the sum of the first n terms of a geometric series.
Calculate the sum to infinity of a geometric series. Apply sequences and series to numerical and real-life problems.
Teaching and learning materials where there is a constant increase or decrease Students: Textbook, exercise book and writing from one term to the next term. If the gradient is equal to 2, the y-values Teacher: Textbook, chalkboard and chalk. This diagram shows this: Areas of difficulty y. Teach them that the first formula can only be used if they know the last 2 term of the series.
The second formula is used when the last term is not known. The gradient for a straight line is constant. So this means that we actually have the same condition here as with an arithmetic sequence,. If we 2. Write down the common difference, d. Complete the following for the infinite series 3.
When he becomes so small that we can leave it out was still in school, his teacher one day tried to 4. Complete the following for the infinite series keep the class busy by telling them to add all the in number 2: a 1 rn natural numbers from 1 to In Umrissen, Erfunden Und Gestochen.
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