Maths - National 5 Index

Online Mathematic Lessons for National 5 Students in Scotland

• Surds
• Simplifying expressions using the laws of Indices
• Expansion of brackets
• Factorising expressions
• Completing the square
• Algebraic Fractions
• Arcs and Sectors
• Volume of solids

Surds: A surd is an expression that includes a square root, cube root or other root symbol. Surds are used to write irrational numbers precisely – because the decimals of irrational numbers do not terminate or recur, they cannot be written exactly in decimal form.

Simplifying surds: surds can be simplified if the number in the surd has a square number as a factor. Some general rules that help are demonstrated within the lesson.

Rationalising the denominator: a fraction whose denominator is a surd can be simplified by making the denominator rational. This process is called rationalising the denominator. Rationalising an expression means getting rid of any surds from the bottom (denominator) of fractions.

Simplifying expressions using the laws of indices: indices show where a number has been multiplied by itself, eg squared or cubed, or to show roots of numbers, for example, the square root. Some terms with indices can be simplified using the laws of indices.

Multiplying and dividing indices: if the two terms have the same base (in this case x) and are to be multiplied together their indices are added. In general: x^m \times x^n = x^{m+n}. If the two terms have the same base (in this case x) and are to be divided their indices are subtracted. In general: \dfrac{x^m}{x^n}=x^{m-n}

More rules of indices:

Laws of indices provide us with rules for simplifying calculations or expressions involving powers of the same base. An index is a small number that tells us how many times a term has been multiplied by itself. You’ll learn what the laws of indices are and how we can use them. You’ll learn how to multiply indices, divide indices, use brackets and indices, how to raise values to the power of 0 and to the power of 1, as well as fractional and negative indices.

Calculations using scientific notation: Scientific Notation (also called Standard Form) is a special way of writing numbers . It can also make calculations easier to use big and small values. This lesson will give you a brief review of scientific notation. We will then look at how to use scientific notation.

Expansion of brackets: expanding brackets is the process of multiplying each term in the bracket by the expression outside the bracket. To remove brackets, multiply the term on the outside of the bracket with each term inside the brackets. This process to remove brackets is also known as the distributive law.

Multiplying out single brackets: to expand the brackets, you need to multiply the part outside the brackets by every part inside the brackets. This lesson will demonstrate how it is done.

Multiplying out double brackets: when expanding double brackets, every term in the first bracket has to be multiplied by every term in the second bracket. It is helpful to always multiply the terms in order so none are forgotten. One common method used is FOIL: First, Outside, Inside, Last.

Multiplying out triple brackets: When expanding triple brackets, we just multiply the first two brackets together, then multiply the result with the final bracket.

Factorising expressions: factorising is the reverse process of expanding brackets. A factorised answer will always contain a set of brackets. To factorise an expression fully, take out the highest common factor (HCF) of all the terms.

Common factors: a common factor is a number that can be divided into two different numbers, without leaving a remainder.  Often numbers can share more than one common factor. “Factors” are numbers we can multiply together to get another number.

Difference of two squares: when we are subtracting a squared term from another squared term we can use the difference of two squares method. One squared thing subtracted from another squared thing. This means they are straightforward to factorise.

Trinomials: the definition of a trinomial is a math equation that has three terms which are connected by plus or minus notations. An example of trinomial is 6x squared + 3x + 5.

Completing the square: some quadratics cannot be factorised. An alternative method to solve a quadratic equation is to complete the square. This lesson explains what completing the square means.

Algebraic fractions: algebraic expressions in fraction form are rational. Methods of adding, subtracting, multiplying and dividing fractions plus expanding and factorising can be used to simplify rational expressions.

Simplifying algebraic fractions: Simplifying rational expressions or algebraic fractions works in the same way as simplifying normal fractions. A common factor must be found and divided throughout

Adding and subtracting: to be able to begin to add or subtract a fraction the denominators must be the same, so the two parts that have been split, when added or subtracted will equal the same size. When fractions have the same denominator it is simple as the equal parts are the same size. This is why we have to find an equivalent fraction when the denominators aren’t the same so we get the same size overall.

Multiplying and dividing: to divide two fractions, multiply the first fraction by the reciprocal of the second fraction. This means simply that the divide sign is swapped for a multiply sign, and the second fraction is flipped upside down. This lesson will demonstrate how.

Arcs and sectors: a circular sector, also known as circle sector or disk sector, is the portion of a disk enclosed by two radii and an arc, where the smaller area is known as the minor sector and the larger being the major sector.

Calculating the arc length: arc length is the distance between two points along a section of a curve. Length of an Arc = θ × r, where θ is in radian. Length of an Arc = θ × (π/180) × r, where θ is in degree.

Calculating the sector area: the formula for sector area is simple – multiply the central angle by the radius squared, and divide by 2: Sector Area = r² * α / 2

Volume of solids: the volume of a solid is the measure of how much space an object takes up. It is measured by the number of unit cubes it takes to fill up the solid.

Volume of prisms including cylinder: to find the volume of a prism (it doesn’t matter if it is rectangular or triangular) we multiply the area of the base, called the base area B, by the height h. A cylinder is a tube and is composed of two parallel congruent circles and a rectangle which base is the circumference of the circle. The lesson will demonstrate more on volume of prisms including cylinder.

Volume of cone, sphere and pyramid:  complete lesson on calculating the volume and surface area of pyramids, cones and spheres. We will examine how to find the surface area of a cylinder and develop the formulae for the volume and surface area of a pyramid, a cone and a sphere. These solids differ from prisms in that they do not have uniform cross sections.

Equation of a straight line: in this lesson we find the equation of a straight line, when we are given some information about the line. The general equation. of a straight line is y = m x + c , where m is the gradient and ( 0 , c ) the coordinates of the y-intercept.

Find the gradient of y-intercept of a line from its equation: the general equation of a straight line is y = mx + c, where m is the gradient, and y = c is the value where the line cuts the y-axis. This number c is called the intercept on the y-axis. The equation of a straight line with gradient m and intercept c on the y-axis is y = mx + c.

Find the equation of a straight line when give two points: practise estimating the equation of a line of best fit through data points in a scatter plot. Then, use the equation to make a prediction. The ‘line of best fit’ goes roughly through the middle of all the scatter points on a graph. The closer the points are to the line of best fit.

Find the equation of a line of best fit and make predictions: practice estimating the equation of a line of best fit through data points in a scatter plot. Then, use the equation to make a prediction. The ‘line of best fit’ goes roughly through the middle of all the scatter points on a graph. The closer the points are to the line of best fit.

Equations and inequations: working with linear equations and inequations. As with other equations and inequations, when working with algebraic equations and inequations it is key to change operation when you change side. When solving inequations, the inequality sign can mostly be treated the same way as an equals sign. This lesson will demonstrate this with easy-to-understand examples and illustrations

Solving equations: NO COPY!!!!

Solving inequations: When solving inequations, the inequality sign can mostly be treated the same way as an equals sign. This lesson will demonstrate this with easy-to-understand examples and illustrations.

Simultaneous equations: simultaneous equations require algebraic skills to find the values of letters within two or more equations. They are called simultaneous equations because the equations are solved at the same time. Equations that have more than one unknown can have an infinite number of solutions. Simultaneous equations can be solved algebraically or graphically. Knowledge of plotting linear and quadratic graphs is needed to solve equations graphically.

Solve equations graphically: To find solutions from graphs, look for the point where the two graphs cross one another. This is the solution point.

Solving simultaneous equations algebraically (basic): Solving simultaneous equations graphically is the process that allows us to solve two or more algebraic equations that share variables. To find solutions from graphs, look for the point where the two graphs cross one another. This is the solution point.

Solving simultaneous equations algebraically (harder): simultaneous equations that contain a quadratic and equation can also be solved graphically. As with solving algebraically, there will usually be two pairs of solutions.

Solving equations in context: this lesson will demonstrate solving equations in context. Equations are often written in the context of a geometric situation.

Change the subject of a formula: When changing the subject of a formula, we rearrange the formula so that we have a different subject. In other words, if you move a term from one side of the equals sign to the other, change the operation to do the opposite. (The opposite of an operation is called its inverse).

Quadratic graphs: a quadratic graph is produced when you have an equation of the form y = a x 2 + b x + c , where ‘and’ can be zero, but cannot be zero. All quadratic graphs have a line of symmetry. Positive quadratic graphs are U-shaped and have a turning point at the bottom of the curve.

Equation of a quadratic from its graph (y=kx2): a Maths lesson that helps pupils to obtain the quadratic equation in the form y = kx^2 using a coordinate

Sketching a quadratic: to sketch a quadratic function you must first determine the roots, nature and coordinates of the turning point and the y-intercept.

Quadratic graphs (in complete square form): the steps to sketch a quadratic equation by completing the square are: Equation should be in y = a(x – h) 2 + k form. Determine shape and turning point of the curve. If a < 0, it is a maximum curve .Find the x-intercepts and y-intercept. Plot the graph.

Quadratic equations: in algebra, a quadratic equation is any equation that can be rearranged in standard form as ax^{2}+bx+c=0 where x represents an unknown, and a, b, and c represent known numbers, where a ≠ 0.

Solving quadratic equations using factorising. Learn how to solve quadratic equations like (x-1)(x+3)=0 and how to use factorisation to solve other forms of equations. Quadratics are algebraic expressions that include the term, x^2, in the general form, · ax^2 + bx + c.

Using the quadratic formula: learn to use quadratic equations and to use factorisation to solve other forms of equations. Quadratics are algebraic expressions that include the term, x^2, in the general form, · ax^2 + bx + c.

Using discriminant to find the nature of roots: to determine the nature of roots of quadratic equations (in the form ax^2 + bx +c=0) , we need to caclulate the discriminant, which is b^2 – 4 a c. When discriminant is greater than zero, the roots are unequal and real. When discriminant is equal to zero, the roots are equal and real.

Converse of Pythagoras: the converse of the Pythagorean Theorem states that if the square of the third side of a triangle is equivalent to the sum of its two shorter sides, then it must be a right triangle. In other words, the converse of the Pythagorean Theorem is the same Pythagorean Theorem but flipped.

Properties of shapes: properties of shape is the identification and comparison of a shape’s attributes. For example, how many sides or vertices (corners) does a shape have? 2D (two dimensional) shapes are flat, e.g. square or circle. 3D (three dimensional) shapes can have more than one edge and face (the side of a shape), e.g. cube.

Properties of circles: the circles are said to be congruent if they have equal radii. The diameter of a circle is the longest chord of a circle. Equal chords of a circle subtend equal angles at the centre. The radius drawn perpendicular to the chord bisects the chord. Circles having different radius are similar.

Solving circle problems using Pythagoras Theorem: circles have different angle properties described by different circle theorems. Circle theorems are used in geometric proofs and to calculate angles. Students learn to solve problems involving properties of circles, the Pythagorean Theorem, special right triangles, and area.

Similar shapes: when a shape is enlarged, the image is similar to the original shape. It is the same shape but a different size.

Scale factor: enlargement of a shape or quantity is determined by its scale factor. When describing an enlargement you must state by how much the shape has been enlarged. This is called the scale factor. For example, a scale factor of 2 means that the new shape is double the size of the original shape. When a scale factor is a fraction the shape decreases in size, but we still call this an enlargement. So a scale factor of ¼ means that the new shape is 4 times smaller than the original. The position of a shape is determined by where the centre of enlargement is located. The centre of enlargement can either be within the shape or outside it.

Area scale factor: scale factors can be used to compare lengths and areas. Scale factors are calculated differently for area. This table shows that if a shape’s lengths are increased by a scale factor of 2, the surface area will be increased by a scale factor of 4. To calculate the area scale factor, square the length scale factor.

Volume scale factor: scale factors can be used to compare lengths and areas. Scale factors are calculated differently for area. This table shows that if a shape’s lengths are increased by a scale factor of 2, the surface area will be increased by a scale factor of 4. To calculate the area scale factor, square the length scale factor.

Trig graphs and equations: Trigonometric graphs can be sketched when you know the amplitude, period, phase and maximum and minimum turning points.

Graph of y = asinbx + c: the sine and cosine graphs are very similar as they both: have the same curve only shifted along the x-axis have an amplitude (half the distance between the maximum and minimum values) of 1 have a period (size of one wave) of 360˚.

Graph of y = acosbx + c: the sine and cosine graphs are very similar as they both: have the same curve only shifted along the x-axis have an amplitude (half the distance between the maximum and minimum values) of 1 have a period (size of one wave) of 360˚.

Trig equations using CAST: the CAST diagram helps us to remember the signs of the trigonometric functions in each of the quadrants. The CAST diagram is also called the Quadrant Rule or the ASTC diagram. In the first quadrant, the values are all positive. In the second quadrant, only the values for sin are positive. In the third quadrant, only the values for tan are positive. In the fourth quadrant, only the values for cos are positive.

Trig identities: they are sine, cosine, tangent, cosecant, secant, and cotangent. All these trigonometric ratios are defined using the sides of the right triangle, such as an adjacent side, opposite side, and hypotenuse side. All the fundamental trigonometric identities are derived from the six trigonometric ratios.

Trigonometry: a branch of mathematics that studies relationships between side lengths and angles of triangles. The three basic functions in trigonometry are sine, cosine and tangent.

Using Trigonometry to calculate the area of a triangle: trigonometry can be used to find the area of a triangle when you know the length of two sides and the included angle. To be able to calculate the area of a triangle, you need to know two sides and the included angle.

Using the sine rule to find the side and angle: the sine rule. The trigonometric ratios sine, cosine and tangent are used to calculate angles and sides in right angled triangles. Trigonometry can be used to find the area of a triangle when you know the length of two sides and the included angle. To be able to calculate the area of a triangle, you need to know two sides and the included angle.

Finding the cosine rule for length of side and angle: to use the Cosine Rule you will need to know at least one pair of a side with its opposite angle. Trigonometry can be used to find the area of a triangle when you know the length of two sides and the included angle. To be able to calculate the area of a triangle, you need to know two sides and the included angle.

Vectors:  vector describes a movement from one point to another. A vector quantity has both direction and magnitude (size). A vector is an element of a vector space. For many specific vector spaces, the vectors have received specific names

Adding and subtracting vectors using direct line segments: vectors can be represented graphically using a directed line segment. However, unlike directed line segments, vectors do not have a unique starting or ending. A vector is a directed line segment symbolised by putting an arrow at the end of the line segment in the required direction.

Using vectors to find coordinates from a 3d object:  a vector describes a movement from one point to another. 3D vectors exist in the xyz plane. The 3D coordinates for any point have three values.

Adding and subtracting vectors using components: the angle that the vector makes with the horizontal can be used to calculate. It is often simpler to add or subtract vectors by using their components. To add or subtract two vectors, add or subtract the corresponding components.

Finding the magnitude of a vector: the magnitude of a vector is its size. It can be calculated from the square root of the total of the squares of the individual vector components.

Percentages: in mathematics, a percentage is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, “%”, although the abbreviations “pct.”, “pct” and sometimes “pc” are also used. A percentage is a dimensionless number; it has no unit of measurement.

Solving problems using reverse percentages: using reverse percentages is a way of working backwards on a percentages problem in order to find the original amount. In order to do this, we: either add/subtract the percentage given in the problem from 100% to determine what percentage we have. Find 1% by dividing by percentage found in previous step. Find 100% (original amount) by multiplying your answer in step 2 by 100.

Calculating appreciation and depreciation: appreciation, depreciation and compound interest can all be calculated using the multiplier method. Appreciation refers to when the value of something increases over time. The value of a house usually increases with time. Therefore its value is said to appreciate. Depreciation refers to when the value of something goes down over time. The value of a car is usually decreases in value with time. Therefore its value is said to depreciate.

Fractions: a fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. Fractions lesson includes: Classify Fractions, Equivalent Fractions, Simplify, Compare and Order. Convert Fractions to Mixed Numbers. Students learn how to add fractions with different denominators. Number: Interactive Lessons · Dividing Fractions and Mixed Numbers.

Adding and subtracting fractions: common denominators. It’s easy to add and subtract fractions when the numbers on the bottom are the same. These are called the denominators. With equivalent fractions, however, sometimes the denominators are different. You use equivalent fractions to make them the same.

Multiplying and dividing fractions: to divide two fractions, multiply the first fraction by the reciprocal of the second fraction. This means simply that the divide sign is swapped for a multiply sign, and the second fraction is flipped upside down.

Statistics: the discipline that concerns the collection, organisation, analysis, interpretation, and presentation of data.

Finding the mean, standard deviation and comparing results: data sets can be compared using averages, box plots, the interquartile range and standard deviation. Standard deviation is an important measure of spread or dispersion. It tells us how far, on average, the results are from the mean. Therefore if the standard deviation is small, this tells us that the results are close to the mean, whereas if the standard deviation is large, the results are more spread out.

Finding the median, interquartile ranges and comparing results: we know that for a set of ordered numbers, the median  is the middle number which divides the data into two halves. Similarly, the lower quartile divides the bottom half of the data into two halves, and the upper quartile also divides the upper half of the data into two halves.

Diagram of interquartile range: the interquartile range is the difference between the upper quartile and lower quartile.