Taking away works in exactly the same way as adding. The only difference is that the final answer has a take away sign in place of the add sign.

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$Example Take away, giving your answer as a single fraction in its simplest form: 4a b, x 0 5 x 4a b 5 x 5x 4ax 5b 5x (smile). (kiss) When a fraction has more than one term on the top or bottom (e.g. x rather than just x or ), you need to introduce brackets (i.$

1 Taking away works in exactly the same way as adding. The only difference is that the final answer has a take away sign in place of the add sign. Example Take away, giving your answer as a single fraction in its simplest form: 4a b, x 0 5 x 4a b 5 x 5x 4ax 5b 5x (smile). (kiss) When a fraction has more than one term on the top or bottom (e.g. x rather than just x or ), you need to introduce brackets (i.e. x becomes ( x ) ). You then perform kiss and smile, thinking of the bracket as a single object. Example brackets Add, giving your answer as a single fraction in its simplest form: x, x, x 4 x x4 x x x x4 ( x) ( x4) (smile) ( x)( x4) xx ( 4) ( x) = (kiss) ( x)( x4) 4 6 ( x)( x4) x x x (multiplying out brackets on top line) x x6 (simplifying top line) ( x)( x4) We didn t multiply out the bottom line because the bottom line was already in its simplest possible form. However we could have multiplied it out if we chose to. licensed to: Boclair Academy Page 5

2 Gradient The Meaning of Gradient The gradient of a line is its steepness. It measures how much the line goes up (or down) for every one unit that you move along. The basic definition of gradient is: up vertical gradient along horizontal. A positive gradient (e.g.,, 5 ) means the line slopes upwards. A negative gradient (e.g. 7 5,, ) means the line slopes downwards. 7 A gradient of means along, up. A gradient of means along, down A gradient of 4 means along, up, [more easily thought of as along 4, up ] 4 If a line is horizontal, it has a gradient of zero. If a line is vertical, we say its gradient is undefined. We usually use the letter m to mean gradient. Two lines are parallel if they have the same gradient. Calculating the Gradient Formula Gradient between two points ( x, y ) and ( x, y ) : y y m x x Example from coordinates Find the gradient between the points (, 5) and (, 4) Step label the coordinates: (, 5) (, 4) x y x y Step put into the formula: licensed to: Boclair Academy Page 6

3 y y m x x 4 5 m Answer: Example from a diagram Calculate the gradient of this straight line Step identify any two nice coordinates on the line: (0, ) (, 4) Step label the coordinates: (0, ) (, 4) x y x y Step put into the formula: y y m x x 4 m 0 Answer: m m licensed to: Boclair Academy Page 7

Rounding to Significant Figures Example Example Perimeter, Area and Volume 446.

$00567 Rounded to significant figure is 0.006 Rounded to significant figures is 0.0057 Arc Length and Sector Area An arc in a circle is a fraction of its circumference.$

4 Rounding to Significant Figures Example Example Perimeter, Area and Volume Rounded to significant figure is 400 Rounded to significant figures is 450 Rounded to significant figures is 447 Rounded to 4 significant figures is Rounded to significant figure is Rounded to significant figures is Arc Length and Sector Area An arc in a circle is a fraction of its circumference. A sector of a circle is a fraction of its area. If you divide a circle into two bits, you get two sectors - a bigger one (major) and a smaller one (minor). In the diagram on the right: The smaller blue sector OAB is the minor sector, with the minor arc AB. The larger pink sector OAB is called the major sector, with the longer major arc AB. The key idea in these questions is to identify the fraction of the circle that is in the question. This depends on the angle at the centre of the circle. This fraction is always Angle 60. Formula. These formula are not given on the National 5 Mathematics exam paper. Angle Arc length in a circle: Arc length d 60 Angle Sector area of a circle: Sector Area r 60 You are allowed to use.4 instead of in calculations. Example Arc Length Find the length of the (major) arc RS in this sector of a circle Radius is 8m so diameter is 6m. 5 Arc length d m ( d.p.) licensed to: Boclair Academy Page 8

Note: units for arc length are just normal units (i.e. not squared or cubic units). Example Sector area Calculate the area of (minor) sector AOB in this diagram. 5 A r 60.7488957... 60 5.75m ( d.p.) Note: units for sector area must always be squared units.

Throughout this topic remember that: all volume questions must have answered in cubic units (e.g. m³, cm³, inches³) you should always state your unrounded answer before rounding (see page 6) Formula.

V Ah Volume of a Pyramid: Volume Area of Base Height Example Pyramid The diagram shows a pyramid with height 7cm and a square base with sides of length cm. Calculate the volume of the pyramid.

5 Note: units for arc length are just normal units (i.e. not squared or cubic units). Example Sector area Calculate the area of (minor) sector AOB in this diagram. 5 A r m ( d.p.) Note: units for sector area must always be squared units. Volumes of Solids You should know from National 4 how to calculate the volume of a prism. At National 5 level, you also need to be able to calculate the volume of a pyramid. Throughout this topic remember that: all volume questions must have answered in cubic units (e.g. m³, cm³, inches³) you should always state your unrounded answer before rounding (see page 6) Formula. This formula is not given on the National 5 Mathematics exam paper. V Ah Volume of a Prism: Volume Area of cross section Height Formula. This formula is given on the National 5 Mathematics exam paper. V Ah Volume of a Pyramid: Volume Area of Base Height Example Pyramid The diagram shows a pyramid with height 7cm and a square base with sides of length cm. Calculate the volume of the pyramid. The area of a square is given by the formula A L, so the area of the base of this pyramid is ² = 44cm² Therefore the volume of the whole pyramid is V Ah cm licensed to: Boclair Academy Page 9

Special cases of prisms and pyramids are when the cross-sectional area of the prism is a circle (in which case you have a cylinder) or when the base of a pyramid is a circle (giving a cone).

Volume of a Cylinder: V r h Example cylinder Work out the volume of this cylinder. Round your answer to significant figures. Diameter is 0cm so radius is 5cm V r h 5 0 ( or 5 5 0 ) 570.7967.

6 Special cases of prisms and pyramids are when the cross-sectional area of the prism is a circle (in which case you have a cylinder) or when the base of a pyramid is a circle (giving a cone). In these cases, we can adapt the earlier formulae to give us a quicker formula: Formula. This formula is not given on the National 5 Mathematics exam paper. Volume of a Cylinder: V r h Example cylinder Work out the volume of this cylinder. Round your answer to significant figures. Diameter is 0cm so radius is 5cm V r h 5 0 ( or ) cm ( s.f.) Formula. This formula is given on the National 5 Mathematics exam paper. Volume of a Cone: V r h In the formula for the volume of a cone, the height h refers to the perpendicular height (the one that goes straight up) and not any sloping heights. Example cone Calculate the volume of this cone. Round your answer to significant figures. Diameter is 0cm so radius is 5cm V r h 5 40 ( or 5 40 ) cm ( s.f.) You are also expected to know how to calculate the volume of a sphere Formula. This formula is given on the National 5 Mathematics exam paper. 4 Volume of a Sphere: V r Example 4 sphere Calculate the volume of this sphere. Round your answer to significant figure. Radius is 5cm licensed to: Boclair Academy Page 0

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licensed to: St Andrews Academy Page 2 National 5 Mathematics Revision Notes Last updated January 014 Use this booklet to practise working independently like you will have to in the exam. Get in the habit of turning to this booklet to refresh