Ad. Maths Test

Scatter Graphs: Used to compare 2 sets of data at the same time. Used to find correlation. Plot dots, draw a line of best fit to get the same amount on each side, going through dots if necessary. You can find whether there is a positive or negative correlation. The closer to the line the stronger the correlation.

Line Graphs: Tracks dada that changes constantly over periods of time. Lines go from dot to dot according to the info. Y Axis is usually what is being measured, and X Axis is usually the time period. A straight line graph uses an equation to solve what x/y equals. Example: Y= 2x + 3. 2 is gradient, c is intercept. Intercept shows where it crosses the y axis. To find the equation with the graph, plot a right angled triangle on any 2 points and work out the rise and the run. Divide rise by run. This is the gradient. Then find the intercept using the y axis. Graph is in the + format. We can also use this to solve simultaneous equations graphically.

Pie Chart: To draw a Pie Chart you must multiply the fraction the slice is of the whole number (Eg 13/100) by 360. This will get the angle the slice will be. To read a pie chart you must find the angle and see what fraction it is of the whole circle. (Eg. A 180 degree angle is half of the circle, therefore that slice is 1/2 of the total data)

Probability: Probability can be written as a fraction, decimal or percentage. The probabilities must add up to 1.0. OR Rule: Add. AND/BOTH Rule: Multiply, but only if the events are independent of each other. Eg 1: Find the probability of rolling a 1 or 2. Add probability (1/6 add 1/6) to get 2/6.   Eg 2: Find the probability of rolling a 1 and 2 in two tries. Multiply probability (1/6 x 1/6) to get 1/36. Drawing 2 way tables. This finds every possible outcome. Easy to draw.

Mean, Mode, Median and Range: In a Frequency Table: Mean is fx divided by f. Mode is the largest f. Median is the x that contains f/2. Range is largest x minus smallest x.                             

In a Grouped Frequency Table: Mean is fa divided by f (a is the middle value of x) Mode is the group with the largest f. Median is the group which has f/2. Range is max value over min value.

Ratio: A ratio is used to compare two or mre related quantities. x:y is read as the ratio of x to y. Always give ratio in its simplest form. Convert both terms to the same unit before dividing. To share a quantity in a given ratio, add the total parts in the ratio, work out what one part of the ratio is and work out what the other parts are worth.                                                                                               

Product of Prime Factors: Divide the number by the smallest numbers you can each time. Also remember to express the product in indices. (E.g. 64 = 2, 2, 2, 2, 2, 2 = 26)

Highest Common Factor (HCF): The HCF is the greatest number that is a factor of all the given numbers. Write down any common prime number between all the given numbers, picking the smallest indices. Multiply these together. Note: One is always a Common Factor.

Lowest Common Multiple (LCM): The LCM is the smallest number that is a multiple of all the given numbers. Write down every prime number between all the given numbers. picking the largest indices. Multiply these together. 

Rules of Indices:  When multiplying, add all the indices. When dividing, subtract the indices, When raising one power to the next ((1x)3) multiply the indices. Anything to the power 0 is 1. Anything to the power 1 is itself. Anything to the power -1 is reciprocal of that number (5-1 = 5/5 = 1/25)

Proportion: When we use proportion to compare part of something to the whole. Eg. There are 6 boys and 7 girls. The proportion of boys is 6/13. Two men can paint a tower in a week. It takes one man two weeks to paint the same tower.

Sequences: A sequence is a list of numbers which has a relationship between them. To find the nth term we find the relationship (eg +1) and the 0^th term (eg n-relationship) We then make the equation relationshipn 0^th term. (Eg 1n-1)

Expanding Brackets: We multiply what’s in front of the bracket by each thing inside the bracket. (Eg. 2(3a-4) = 6a-8)

Polygons: Triangle – 3 sides. Quadrilateral- 4 sides. Pentagon – 5 sides. Hexagon- 6 sides. Septagon/Heptagon- 7 sides. Octagon- 8 sides. Nonagon- 9 sides. Decagon- 10 sides. Hendecagon – 11 sides. Dodecagon- 12 sides. Regular polygons have equal sides and equal angles. An interior angle and an exterior angle adds up to 180 degrees in a regular polygon. (N of sides – 2) makes triangles. 180 (n-2) is the sum of interior angles. The sum of exterior angles is 360 degrees so each exterior angle is 360/n of sides.

Substitution: Substitution is replacing letters with numbers.(Format: Let x be 1 and y be 2. What is 2x – y?)

Equations: The aim of solving an equation is to get ‘one letter = ‘Number’ You remove the brackets, collect the letter terms on the side with most letters, collect the number terms on the other side, simplify and then solve. 

Solving inequalities: < and > symbols. The hungry crocodile always eats the largest number.       < with a _ under it means greater than or equals to. We can show inequalities in a number line. When there is an open circle, that number is not included. If it is filled then the number is included. We solve inequalities the same way we solve equations.

Pythagoras’s Theorum: h2 = a2 + b2. H is the side opposite the right angle. We can use this equation to find the size of h, a/b, or to find whether a triangle is right angled.

Speed, Distance and Time: The formula triangle has D at the top and S and T at the bottom. This means S = D/T. T = D/S and D = S x T. We write out all the information we receive and then we use one of the equations to find the missing information. Sometimes there are distance time graphs. The x axis is time and the y axis is distance. The steeper the slope, the faster the speed.

Data: Discrete Data is often found by counting. Continuous data is found by measuring. When a survey is carried out we collect raw data.  We sort the raw data by putting it in a tally chart or frequency table.

Tally Chart: A tally chart shows the categories on the left side and then the frequency on the right. The frequency is shown in tallies, which are little slashes. Every 5 tallies you put a line through the previous 4 tallies, so frequency of 9 would look like. llllllll. 

Bar Chart: A bar chart uses bars to represent each category. The height of each bar represents the frequency. Frequency goes on the vertical axis. The graph should have a title and all axes should be labelled. All bars should be equal in width. If the data is discrete there is a space between the bars. The graph is as large as possible.

Bar Line Chart: In bar line charts you use lines insted of bars to represent each category. The height of each line represents the frequency of each category.

Line Graph: Line graphs are useful for displaying changing data. X axis has the time period and Y axis is whats being measured. Lines connecting points give estimates of what is being measured.

Scatter Graph: Used to compare two sets of data at the same time. It is used to show the correlation between two sets of data. Positive correlation is going up diagonally. Negative is going down diagonally. No correlation has no pattern.

Pie Charts: The slices are the categories. Find the angle of each marked space to get the angle of each slice and the amount of data in each slice. We can change tables to pie charts and vice versa. 

Perimeter, Area and Volume: Find the perimeter by simply measuring each side. Area of regular rectangles and squares is length x bredth. Area of trapezium is half the sum of the parallel sides x the perpendicular height. Area of triangle is bredth x height divided by 2.Area of parallelogram is bredth x height. Area of kite is bredth x height divided by 2. The circumference of a circle is 2x pi r. Area of a circle is pi x radius squared. Radius is from the middle of the circle to the outside in a straight line. Volume of a cylinder is area of circle x length. Volume of a prism is area of trapezium x length.

Special Numbers

Prime Numbers: A prime number is a number which is divisable by only two numbers/has 2 factors, 1 and itself. The first prime numbers are 2,3,5 and 7. 

Factors: These are whole numbers which divide exactly into another number. Factors of 20 are 1,2,4,5,10,20.

Multiples: The multiples are just the numbers in the multiplication chart. Multiples of 8 are 8,16,24,32. The 11th multiple would be 8 x 11 = 88.

Square Numbers: A square number is the result of a number to the power of 2. The first square numbers are 1,3,9,16 and 25.

Cube Numbers: A cube number is the result of a number to the power of 2. The first cube numbers are 1,8,27,64 and 125.

Triangular Numbers: This is a well-known pattern. The sequence is n(n+1) / 2

BIDMAS: We do things in this order: Brackets Indices Division Multiplication Addition Subtraction This means in an equation like 4+6x6/3, we don't go left to right. We do Division first, then multiplication, then addition. This means we are actually doing 4+(6x(6/3)), which results in 16 as our result, instead of 20. In a question with only addition or subtraction in them, you should ignore BIDMAS and do it in the left to right order.

Flow Charts: Flow Charts are a diagrammatic representation of a set of instructions which must be followed. Flow charts are made up of different boxes, which each have different functions. A rectangular box contains instructions. A box with curved edges signifies the start of the end of a flow chart. A square box turned 45 degrees is a decision box. Boxes are seperated with lines, and an arrow points which way to move.

Expressions: Expressions are equations that don't have an equals sign. For example, 5x is an expression, but 5=x is an equation. We can easily form expressions out of problems. We then simplify by collecting like terms to get the smallest numbers. Finally we solve the equation. Example: Joe has y cookies. Jack takes 2 cookies away. John gives 3 cookies back. How many cookies does Joe have now? y - 2 + 3. Simplify to y + 1.

Rounding Numbers: Numbers can be rounded in two different ways. They are through decimal places (d,p) and significant figures. 

Decimal Places: Every number after a decimal point is one decimal place. To round something to one decimal place, you remove all the numbers except the first decimal place. The last number you are asked for is called the last required digit. The last required digit is affected by the decider, which is the number after the LRD. If the decider is 5 or more, you add one to the last required digit. If it is less than 5, you do nothing.

Significant Figures: The first non-zero number is the first significant figure. Any number after the first significant figure is also a significant figure, even if it is a zero. All the other rules apply.

Negative Numbers: When adding and subtracting with negative numbers, we use the number line. The number line has negative numbers to the left and positive to the right. 0 or the original number is usually in the middle of the number line. When adding, we move to the right. When subtracting, we move to the left. 

Estimation: Estimation is a good way of checking answers. To estimate round the numbers to easy numbers with 1/2 significant figures. Then solve the equation using an  estimate equal sign ≈.

Co-ordinates: When reading co-ordinates read across first then up or down. (0,0) is called the origin. X axis is across. Y axis is up. Co-ordinates are always written with brackets and a comma in between the 2 numbers. Eg (9,-2) means 9 across and then two down. The first number refers to the x co-ordinates, whilst the second refers to the y co-ordinates. 

Line Symmetry: Line symmetry is when a line can be draw through the shape, and one half mirrors the other half. We do this by taking each corner of the shape seperately and measuring how far it is from the mirror line. Then we form a point. We do the same at the other side of the mirror line and form a point. Doing this for every corner will tell us if there is line symmetry. 

Rotational Symmetry: The order of rotational symmetry is the number of times a shape  looks the same when it is turned 360 degrees. The centre is the point about which the shape should be rotated. All shapes have rotational symmetry of at least one. Squares have rotational symmetry of 4, and rectangles 2. 

Fractions: There are two types of fractions: Proper Fractions and Improper Fractions. Proper Fractions have a larger denominator (The bottom number) Improper Fractions have a larger numerator (The top number) Mixed numbers have a whole number and a fraction. E.g. 1 1/2. Equivalent Fractions are fractions that are equal to each other. E.g. 1/2, 7/14 and 37/74.

Lowest Common Denominator: the smallest number into which all of the denominators will divide. To simplify a fraction you divide both the numerator and the denominator by the same number until we get the smallest possible number. To see if which of two fractions are largest change them so they have the same denominator.

Adding/Subtracting Fractions: Convert all mixed numbers into improper fractions. Write both fractions so they have the same denominator. And or subtract the numerators.

Multiplying Fractions: Convert mixed numbers into improper fractions. Multiply the numerators. Multiply the denominators. If the answer is an improper fraction write it as a mixed number.

Dividing Fractions: Convert mixed numbers into improper fractions. Turn the fraction which follows the division sign upside down. Change the division sign to a multiplication sign. Follow the rules of multiplication.

Changing Decimals: If you are multiplying a decimal by a multiple of ten, move the decimal point to the right. If you're dividing by a multiple of ten, move it to the left. 

Ordering Decimals: Using the hungry crocodiles, you can order decimals. Make sure to pay attention to whether the decimals are positive or negative. Also check the decider number to see which is larger. We can also order decimals in ascending order, using commas in between. Eg 1. 78.15 } 78.149. Eg 2. 0.003, 0.02, 0.1. 

Adding and Subtracting Decimals: We line up the decimal places above each other in a long addition or long subtraction equation so we don't make a mistake in the equation.

Dividing Decimals: The first thing we do is make a long division equation, and putting the decimal place in first. If we are dividing by a decimal we rewrite the sum so we are dibinding by a whole number. 

Finding a percentage of a given amount: First, find 1% of the amount. Then multiply the result by the percentage you are asked to find.

Increasing/Decreasing amounts by a given percentage: Find the increase or decrease by finding 1% and then multiplying by the given percentage. Add or subract this to the starting amount.

Finding numbers as percentages of other numbers: Write the given amount as a fraction of the other number. Multiply the fraction by 100.

Finding percentage increases/decreases: A pecentage increase is the actual increase divided by the initial value multiplied by 100. A percentage decrease is the actual decrease divided by the initial value multiplied by 100.

Fraction into decimal: To change a fraction into a decimal, divide the numerator by the denominator.

Decimal into fraction: To change a decimal into a fraction, use place value (E.g. 0.5 is 5 tenths)

Fraction into percentage: To change a fraction into a percentage change the fraction into a decimal and then multiply by 100

Percentage into fraction: To change a percentage into a fraction put the percentage over 100 and simplify if possible.

Changing from Decimal into percentage: To change a decimal into a percentage multiply by 100

Changinf from Percentage into decimal: To change a percentage into a decimal divide by 100.

Angle Properties: An amount of turn is called an angle. To describe an angle we use a measurement called the degree. The lines that form an angle are called the arms of the angle. The point where the angles meet is called the vertex of the angle. An acute angle is between 0 and 90 degrees. A right angle is 90 degrees. An obtuse angle is between 90 and 180 degrees. A straight line angle is 180 degrees. A reflex angle is between 180 and 360 degrees. 

Estimating Angle Size: When estimating angle size we compare it to a right angle. Eg. About half of a right angle. We can also estimate certain angles. Angles around a point add up to 180, and angles on a straight line add up to 180. Vertically opposite sides are equal.

Angles and Triangles: In an equilateral triangle all angles are 60. In an isosceles triangle opposite sides are equal. In a scalene triangle no sides are equal. In a right angle triangle the right angle is 90 and the other two add up to 90. An isosceles right angle has the other two sides at 45 degrees.

Other Shapes and Angles: Corresponding angles (an  F shape) are equal. The alternate angles in a Z shape are equal. Interior angles in a C shape add up to 180. 

Constructing Triangles

Given 1 side and 2 angles: Eg. Draw triangle ABC. AB is 7cm, A is 35 degrees and B is 40 degrees. Draw a rough sketch of ABC. Draw the line out accurately. Use a protractor to make the two angles. Extend both lines until the cross. Where they cross is C.

Given the length of 3 sides: Draw a rough sketch. Draw the first line. Set the compass to the next line using a ruler abd draw a wide arc. Set the compass to the length of the final kline using a ruler and draw another arc. Where the arcs meet is the angle.

Given 2 sides and the angle between the two sides: Sketch. Draw the sides whose lengths we know first. Use a protractor to make the angle. Then use a compass to make an arc and join the arc and angle together.

Converting between Units

Length: 10mm is 1cm. 100cm is 1m. 1000m is 1km. 1 inch is 2.54cm. 12 inches are 1 foot. 1 foot is 30.48cm.

Mass: 1000mg is 1 gram. 1000 grams are 1kg. 100kg is 1 tonne. 1 ounce is 28 grams.1 pound is 453 grams.

Volume: 1ml is 1cm cubed. 1 litre is 1000ml. 1000 litres is 1 metre cubed.

Parallelogram: Opposite sides are parallel and equal. Opposite angles are equal. Adjacent angles add up to 180. If a parallelogram has 1 right angle all angles are right and it is a rectangle.

Rectangle: Opposite sides are parallel and equal. All angles are 90 degrees. A rectangle is a right angled parallelogram.

Rhombus: All sides are equal. Opposite angles are equal. Diagonals are at 90 degrees

Square: All sides are equal and parallel. All angles are equal (90 degrees)

Trapezium: Bases are parallel. Nothing is equal.

Right Angled Triangle: Has one angle of 90 degrees. Can be scalene or isosceles.

Scalene Triangle: Nothing is equal.

Isosceles Triangle: 2 equal sides and 2 equal angles.

Equilateral Triangle: All sides equal. All angles 60 degrees. 

Transformations

Translations: This is moving a shape right, left, up or down a grid. It is shown in a displacement vector. The top number is how far right it moves. If the number is negative move left. The bottom number his how high up it moves. If the number is negative move left. We translate by manually moving each corner by the vector and then joining the corners together.

Reflections: This is mirroring a shape so it is backwards. We need to get each corner manually, measure the distance from the mirror line and then replicate it. 

Rotations: This is rotating a shape. We need to find the point in between the original and the rotated shape. This is the centre point. We then rotate every point using a compass to find the degrees it has been moved. If nothing has changed it has rotated either 0 or 360.

Enlargement: This is making a shape larger or smaller. If we double the size the scale factor is 2. If we half the size the scale factor is -2. A centre of enlargement is the specific point from which enlargement is being measured. We multiply the distance from the centre point by the scale factor.