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Mathematics Year 7 Parents’ Guidebook The Mathematics Department Ysgol Gyfun Gwynllyw Mathematics Year 7 – Parents’ Guidebook This booklet is intended to be a useful tool as your child undertakes their year 7 mathematics work this year. In the booklet you’ll find a description of the topics studied, appropriate examples and a list of useful vocabulary. These are the topics studied per term, as a rule: Autumn Term Special Numbers Number patterns Rounding Symmetry Place value Polygons Algebra Displaying data Spring Term Fractions and Percentages Area and Perimeter Further Algebra Angles Probability Solids Summer Term Plotting points Graph-reading Averages Drawing to scale Throughout the year, it’s important that pupils have the appropriate equipment. You child will be given a work book during the initial lessons and it is expected that these be covered and kept tidily at all times. The equipment definitely needed is: Pens, pencils, rubber and sharpener. Protractor and compass. Scientific calculator Special Numbers: This topic allows pupils to become more familiar with certain types of numbers. There will be a need to use some of these types, whereas recognising others will be sufficient. Odd numbers: 1, 3, 5, 7, 9 and all other numbers which finish with these digits. Even numbers: 2, 4, 6, 8 and all other numbers which finish with these digits. * Every number which finishes with 0 is also even, e.g. 10, 340, 1870. Square number: Cube number: Prime number: A number gotten by multiplying a number with itself: 1, 4, 9, 16, 25, ... 1x1=1 6 x 6 = 36 2x2=4 7 x 7 = 49 3x3=9 8 x 8 = 64 4 x 4 = 16 9 x 9 = 81 5 x 5 = 25 10 x 10 = 100 A number multiplied with itself three times: 1, 8, 27, 64, 125, ... 1x1x1=1 6 x 6 x 6 = 216 2x2x2=8 7 x 7 x 7 = 343 3 x 3 x 3 = 27 8 x 8 x 8 = 512 4 x 4 x 4 = 64 9 x 9 x 9 = 729 5 x 5 x 5 = 125 10 x 10 x 10 = 1000 A number that can only be divided by 1 and the number itself to have a whole-number answer: 2, 3, 5, 7, 11, 13, 17, 19, 23, 39, 31, 37,... Multiples: Sequences of number which follow the multiplication tables: Multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, ... Multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, ... Factors: Numbers that divide a larger number to give a whole-number answer: The factors of 6 are 1, 2, 3, 6 because 6 divided by each of these give a whole-number answer. *4 is not a factor of 6 because 6 ÷ 4 = 1.5, which is not whole. Common factor: A number which is a factor to 2 or more numbers. 7 is a common factor of 14 and 21 Highest Common Factor: The largest number in a set of common factors Triangular numbers Number machines: A number is input into a machine, changed and an output gained: Indices: Rounding: OUTPUT +5 INPUT 3 +5 8 12 +5 17 -1 +5 4 Numbers noted using powers: 2 x 2 x 2 x 2 = 24 5 x 5 x 5 = 53 c x c = c2 d x d x d x d x d = d5 Numbers noted as a close number, e.g. to the nearest 10/100/whole number... 6.8 is noted as 7 to the nearest whole number. 13 is noted as 10 to the nearest 10. 267 is noted as 300 to the nearest 100. BUT A number that ending in 5 is rounded up every time: 3.5 → 4 15 → 20 150 → 200 BODMAS: The order to follow when calculating answers: BRACKETS 7 12 4 8 4 3 OF POWERS 72 4 8 4 3 DIVISION 49 4 8 4 3 MULTIPLICATION 49 4 2 3 ADDITION SUBTRACTION Symmetry: 49 8 3 57 3 54 Reflection: Note a mirror-line where the first side is a perfect mirror image of the second side. Rotation: How many tines does an object fit onto it’s original shape when rotating around a full circle. Rotational symmetry order: 4 Rotational symmetry order: 8 Turns: Measure of turning in a circle: ½ turn clockwise ¼ turn clockwise Decimals: ¼ turn anticlockwise Use numbers with decimal points: Add and subtract in the usual way, but ensuring that the numbers and decimal points are in columns: 12 .3 + 17 .7 4 .6 + 16 .9 13 .2 18 .6 24 .1 7 .5 - 18 .7 11 .1 5 .4 - 30 .9 When multiplying decimals there are a number of valid methods, but this is easiest to begin: 2.5 x 4 25 x 4 = 100 ÷ 10 2.5 x 4 = 10 Place value: Put numbers in the correct columns: 7063.208, 612.059 Thousands Hundreds Tens Units Tenths Hundredths Thousandths 1/10 1/100 1/1000 7 0 6 3 2 0 8 0 6 1 2 0 5 9 Polygon: Enclosed shape where each side is a straight line: Number of Name of sides polygon 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon 9 Nonagon 10 Decagon Quadrilaterals: Square Rectangle Rhombus Trapezium Parallelogram Kite Arrowhead When discussing polygons, many unfamiliar words are used: Algebra: Parallel Lines in the same direction that will never meet Vertex A corner of a 2D or 3D shape Diagonal A line drawn from one vertex to another Convex A shape that goes outward Concave A shape where some side come back into the shape Use of letters to represent numbers. There are definite techniques for using algebra, and it’s important to remember how to multiply: Collect terms: 2 x a = 2a 6 x b = 6b c x d = cd a x a = a2 This is the methods of bringing similar terms together to simplify expressions: a + a + a = 3a 2b + 3b + b = 6b 5a – 2a + a = 4a Substitution: Put numbers is place of letters and work out values, so if a=2, b= 3 and c=5: a+c=2+5=7 2b = 2 x b = 2 x 3 = 6 5a – 3b = (5 x a) – (3 x b) = (5 x 2) – (3 x 3) = 10 – 9 = 1 Graphs: Pictogram → Pictures represent values → There must be a key telling how many each picture represents Bar chart → Draw axes with the quantity on the vertical axis Pie chart Fractions: → There are 360o in a circle, so ½ is 180o, ¼ is 90o... The numerator say how many pieces are used and the denominator tells the total amount of pieces. In this circle, there are 3 equal parts, 2 coloured, so the fraction of the circle coloured is Improper fraction: Mixed number: 2 3 This is when the numerator is more than the denominator, e.g. A number and a fraction together, e.g. 6 23 9 15 There are specific methods for converting an improper (top-heavy) fraction into a mixed number, and vice versa: Improper → Mixed Mixed → Improper 1. How many times does the 1. Multiply the large denominator fit into the number which the numerator? This is the number denominator and add before the fraction. the numerator. This 2. How much is left over? This is the new numerator. 3. The denominator doesn’t numerator. 2. The denominator doesn’t change. change. 16 1 3 5 5 Equivalent fractions: is the new 19 5 2 7 7 These are different fractions with the same value: 1 2 3 11 37 2 4 6 22 74 All of these are equal to 1/2. 2 6 8 22 36 5 15 20 55 90 All of these are equal to 3/5. 21 8 Simplifying fractions: In order to simplify fractions we need to find factors of the denominator and numerator, and then divide both by that number: 3 is a factor of 6 and 9 so if 6 and 9 were divided by 3, the answers would be 2 and 3, so Percentages: 6 2 . 9 3 Converting between decimals, fractions and simple percentages. Decimal Fraction Percentage 0.01 1 100 1% 0.05 5 1 100 20 5% 0.1 10 1 10 10% 0.15 15 3 15% 0.2 20 0.25 25 1 100 4 25% 0.3 30 3 10 30% 0.4 40 2 40% 0.5 50 0.6 60 0.7 70 0.75 75 3 100 4 0.8 80 0.9 90 1 100 100 100 100 100 100 100 100 100 100 100 20 1 1 3 7 100 5 2 5 10 4 9 5 5 10 1 20% 50% 60% 70% 75% 80% 90% 100% Area: The amount of space inside a 2D shape, and there are formulas to use in each case Perimeter Shape Formula Square length x width Rectangle length x width Parallelogram base x height Triangle ½ x base x height Trapezium ½ x (a + b) x u Rhombus base x height The total length of the sides of shapes, calculated by adding the length of all the sides together. Equations: Calculate the value of a letter so that the equation is valid: 2 x 1 11 2 x 10 x5 Take away 1 from both sides Divide both sides by 2 After getting an answer, it can be checked by putting it into the equation: (2 x 5) + 1 = 11 Trial and improvement: Choose a initial value and improve it according to the answer What is the value of x so that 2x 7 29 x 2x 7 10 27 Too small 12 31 Too large 11 29 Solids: 3D objects Sphere Cube Cuboid Cone Cylinder Square base pyramid (Pyramid) Triangular base pyramid (Tetrahedron) Triangular prism Hexagonal prism Nets: The method of drawing shapes in 2D where they can be folded to make the necessary shape: Net of a triangular prism Probability: Describing the chance an event will happen, in words and numbers: Numbers can be used to correspond with the words: Angles: 0 Impossible 0.5 Even chance 1 Certain The method of noting the space between two lines, and there are specific names for the different angles: Acute angle Angle between 0o and 90o Right angle Angle equal to 90o Obtuse angle Angle between 90o and 180o Straight angle Angle equal to 180o Reflex angle Angle between 180o and 360o A triangle’s internal angles add to give 180o. A quadrilateral’s internal angles add to give 360o. In isosceles triangles, two angles are equal and one different. Opposite angles are equal. Measure: This topic studies the length of objects and how they can be measured. There is some specific vocabulary in measure: Imperial – older units such as inches, feet, yards, miles Metric – newer units such as centimetre (cm), millimetre (mm), meter (m), kilometre (km) Use is also made of additional older units: Span – the distance from the tip of the thumb to the small finger Hand-breadth – the distance from one side of the hand to the other Cubit – the distance from the elbow to the tip of the middle finger Span Hand-breadth Cubit We also needs to remember some important ratios: Metric units Imperial units 1cm 10mm 1 foot 12 inches 1m 100cm 1 yard 3 feet 1km 1000m 1 mile 1760 yards Imperial ↔ metric units Average: 1 inch 2.5cm 1 yard 0.9m 1 mile 1.6km This is a general word for 4 specific types of average: mean, mode, median and range (although this is not strictly a type of average). Mean: Add all the numbers, divide with the amount of numbers Mode: The most popular number (can be more than one mode) Range: The difference between the largest and smallest number Median: Arrange the numbers largest → smallest and choose the number in the middle BUT if there are two numbers in the middle (as in the example below) add the two numbers together and halve the answer Calculate the mean, mode, median and range of 8, 6, 9, 4, 9, 6 Mode: 6 and 9 appear twice, so the mode is 6 and 9. Range: 9 – 4 = 5. Mean: (8 + 6 + 9 + 4 + 9 + 6) ÷ 6 = 42 ÷ 6 = 7 Median: 4, 6, 6, 8, 9, 9 As 6 and 8 are the middle numbers, the median is (6 + 8) ÷ 2 = 14 ÷ 2 = 7. It’s possible to have different answers for the 4 averages. Co-ordinates: The method of noting the location of points on graphs. We go horizontally first, then vertically: The points’ co-ordinates are: A (-3, 4) B (0, 5) C (3, 0) D (0, 3) H (5, 0) I (4, -3) P( -2, 3) S (5, -2) T (2, -5) W (3, -2) Y (-4, -1) The horizontal axis is called the x axis, and the vertical axis the y axis. Vocabulary addition adio inch modfedd rectangle petryal acute angle angle ongl lem index indecs reflection adlewyrchiad ongl isosceles isosgeles reflex angle ongl atblyg area arwynebedd mean cymedr rotation cylchdro average cyfartaledd median canolrif rotation troad BODMAS CORLAT mile milltir rounding talgrynnu brackets cromfachau rhif cymysg simplify symleiddio circle cylch mixed number mode modd square sgwâr collect casglu multiple lluosrif rhif sgwâr cube number decimal rhif ciwb multiply lluosi square number square root degolyn net rhwyd substitution amnewid decimals degolion ongl aflem subtract tynnu divide rhannu obtuse angle odd number odrif symmetry cymesuredd equation hafaliad percentage canran to square sgwario pendrwm ailisradd equilateral hafalochrog percentages canrannau top-heavy equivalent cywerth perimeter perimedr even number factor eilrif rhif cysefin ffactor prime number probability trial and cynnig a improvement gwella triangle triongl foot troedfedd quadrilateral pedrochr fraction ffracsiwn range tebygolrwydd amrediad triangular number whole number yard rhif trionglog rhif cyfan llathen