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[1] Estimation & Measurement MNU112M, MNU113M, MNU217M, MNU218M, MNU219M, MNU315M [2] Rounding MNU301A [3] Addition, Subtraction, Multiplication & Division MNU103C, MNU203C, MNU303C, MTH403C, MTH205C [4] Scientific Notation / Standard Form MTH405G [5] Fractions MNU105H, MNU208H, MNU308H, MTH309H, MTH310H, MTH407H [6] Percentages MNU208H, MNU308H, MNU406H, MNU209H [7] Proportion MNU311J, MNU408J [8] Time MNU214L, MNU216L, MNU314L, MNU413L [9] Algebraic Expressions, Equations & Formulae MTH222R, MTH320R, MTH321R [10] Co-ordinates MNU206D, MTH230U, MTH429U, MTH430U [11] Data & Analysis MNU124W, MNU232W, MNU233W, MNU325W, MNU432W [1] Estimation and Measurement Estimate / measure height and length in mm, cm, m and angle sizes in degrees eg Estimate / measure weight in g and kg, area in cm2, m2 and hectares and volume / capacity in cm3, m3, ml and l eg length of pencil ≈ 10cm width of desk ≈ 0.5m diameter of 1p coin ≈ 15mm bag of sugar ≈ 1kg area of window ≈ 4m2 volume of drinks can ≈ 300ml Learn equivalences 10mm 1000mm 1cm3 1000cm3 10000m2 = = = = = 1cm 100cm 1ml 1000ml 1 hectare = 1m = 1l MNU112M, MNU113M, MNU217M, MNU218M, MNU219M, MNU315M [2] Rounding Round to the nearest whole number, 10 or 100 eg Round to any number of decimal places or significant figures eg 74 to the nearest 10 ≈ 70 347.5 to the nearest whole number ≈ 348 7.51 ≈ 7.5 (1dp) 3.14159 ≈ 3.142 (3dp) ≈ 3.14 (3sp) 0.00231 ≈ 0.002 (1sf) When the next number is a 5 always round up Always round your final answer to the same level of accuracy as your starting values Never round as you go along – just at the end Watch out for necessary rounding eg. If 90 children and 4 teachers go on a trip, how many 40-seater coaches would be needed? 94 40 = 2.35 coaches which has to be rounded up or some people will be left behind ! MNU301A [3] Addition, Subtraction, Multiplication & Division Subtract using decomposition (as a written method) eg 271 - 38 2 33 Do not borrow and pay back Calculate using alternative mental methods when appropriate eg 478 – 99 = 379 by subtracting 100 then adding 1 eg 1+2+3+ ………+8+9+10 = 11 x 5 = 55 Use correct order of operations Remember BODMAS (or BOMDAS) Brackets Order Divide Multiply Add Subtract eg 18 – 7 x 2 = 18 – 14 =4 eg (14 + 12) (6 – 4) = 26 2 = 13 MNU103C, MNU203C, MNU303C, MTH403C, MTH205C [4] Scientific Notation or Standard Form Write large or small numbers in standard form and vice versa eg 24500000 = 2.45 x 107 0.000988 = 9.88 x 10-4 Write as a x 10n where a is between 1 and 10 Use a calculator to carry out calculations involving standard form eg (3.2 x 106) x (1.7 x 10-2) = 3.2 E 6 x 1.7 E (-) 2 = 54400 = 5.44 x 104 To avoid confusion do not use 10x on calculator Different calculators have different displays - learn how yours works MTH405G [5] Fractions Find simple fractions of a quantity eg 1 of 70 5 2 of 120 3 = 70 5 = 120 3 x 2 = 14 = 80 Divide by the denominator, multiply by the numerator Use equivalence of widely used fractions and decimals x5 eg 3 = 0.3 10 3 15 = = 0.15 20 100 x5 Add and subtract fractions eg 3 1 4 +1 2 5 eg 2 1 3 4 =3 5 8 +1 10 10 = 8 3 12 12 =4 13 10 = 5 12 =5 3 10 Common denominator for adding and subtracting Never use decimals in a fraction question MNU105H, MNU208H, MNU308H, MTH309H, MTH310H, MTH407H Multiply fractions eg 1 1 5 x 4 7 2 1 7 x4 3 8 = 5 5 x 4 7 = 7 39 x 3 8 = 25 28 = 91 8 = 11 3 8 Cancel numerators and denominators first if possible to simplify figures Always write final answer as a mixed number Always give your answer in its simplest form Never cancel two numbers on the top / or bottom Never use a common denominator when multiplying Divide fractions eg 2 3 5 4 6 = 11 63 x 42 5 = 33 10 =3 3 10 To divide, invert second fraction and multiply Don’t use a calculator for calculations involving fractions [6] Percentages Find 50%, 33⅓%, 10% and 1% without a calculator and use addition to find other amounts eg 2 40 = = 40% 5 100 23% of £300 = 0.23 x £300 = £69 Express fractions as percentages using a calculator eg A caravan was bought for £3000 and sold for £3250. What was the profit as a percentage of the cost price ? Profit = £3250 - £3000 = £250 Profit % = 250 = 0.0835 = 8.3% 3000 Carry out calculations involving percentage increase and decrease eg 15% of £360 = 10% of £360 + 5% of £360 = £36 + £18 = £54 Find percentages with a calculator eg eg Express some fractions as a percentage without a calculator eg 50% of £240 = ½ of £240 = £120 Increase £350 by 15% 1.15 x £350 = £402.50 Always change percentages to decimals when using a calculator Never use the percentage button on your calculator MNU208H, MNU308H, MNU406H, MNU209H [7] Proportion Use the unitary method (ie. find the value of one first, then multiply by the required value) eg Direct If 5 bananas cost 80p, what do 8 bananas cost ? Bananas 5 1 3 Cost 80p 80 5 = 16p 16p x 3 = 48p 3 bananas cost 48p eg Inverse The journey time at 60km/h is 30 minutes, so what is the journey time at 50km/h ? Speed 60 1 50 Time 30 mins 30 x 60 = 1800 mins 1800 50 = 36 mins The journey time at 50km/h is 36 minutes Always communicate answer Don’t round until the last stage MNU311J, MNU408J [8] Time Convert between 12 and 24 hour clock eg 2327 = 11.27pm Do not write 2327pm Calculate duration in hours and minutes by counting up to the next hour then on to the required time, including pm → am times eg 20mins 10.40pm 15mins 11.00pm 4.00am 5hrs = 5hrs 35mins Remember the cross-eyed frog ! Never use subtraction to find time intervals Change minutes to hours and hours to minutes 4.15am eg 27 mins = 27 60hrs = 0.45hrs eg 0.2hrs = 0.2 x 60mins = 12mins Use the link between time, speed and distance to carry out related calculations Speed = 42 km/h T Distance = 800km D S 800 = 42 = 0.0476 x 60 = 3mins = 19.0476….hrs = 19hrs 3mins MNU214L, MNU216L, MNU314L, MNU413L [9] Algebraic Expressions, Equations and Formulae Solving Equations By balancing / using the flag method Performing the same operation to each side of the equation Doing ‘undo’ operations eg undo ‘+’ with ‘-‘ Using statements like “multiply both sides by …” eg 2x + 3 = 9 -3 -3 2x = 6 2 2 x = 3 One equal sign per line, written underneath each other Work down the page Write the letter ‘x’ differently from a multiplication sign Never change side, change sign Do not write ‘nonsense’ statements, such as 2x = 6 = 3 Formulae Write down the formula first Substitute clearly Simplify the expression Communicate answer fully eg The length of a string smm for the weight wg is given by the formula s = 16 + 3w Find (i) s when w = 3g (ii) w when s = 20.5mm (i) s = 16 + 3w = 16 + 3 x 3 = 16 + 9 = 25 length of string is 25mm (ii) s = 16 + 3w 20.5 = 16 + 3w -16 -16 4.5 = 3w 3w = 4.5 3 3 w = 1.5 weight of string is 1.5g Always show all steps in working Always substitute first, then re-arrange as necessary to solve the equation MTH222R, MTH320R, MTH321R [10] Co-ordinates Cartesian Co-ordinates are pairs of numbers separated by a comma and enclosed in brackets. Each pair of numbers gives the position of a point relative to an origin O, eg. (3, 4) is 3 units to right along the x-axis and 4 units in the positive y-direction and (-3, -2) is 3 to the left and 3 down. The points are marked where the lines cross, and not in the spaces. The order matters in that (3, 4) is not in the same place as (4, 3). (Remember : along the corridor then up (or down) the stairs !) MNU206D, MTH230U, MTH429U, MTH430U [11] Data and Analysis Use a pencil and ruler Give the graph a title Label lines Label the frequency up the side Label on lines, not on spaces Bar Graph Construct and interpret bar graphs Quantities of Litter eg Make sure each bar has equal width Label each bar in its centre Line Graphs Construct and interpret line graphs The distance a gas travels over time has been recorded in the table below Time (s) Distance (cm) 0 0 5 15 10 30 15 45 20 60 25 75 30 90 Distance travelled by a gas over time Plot points neatly using a cross or dot If the lower point of a graph has been missed out, use a jagged line to show this eg Scatter Graph Construct and interpret scatter graphs Draw a line of best fit when there is a correlation Pie Charts Construct pie charts involving simple fractions, decimals or percentages eg 30% of pupils travel to school by bus 10% by car, 55% walk and 5% cycle Bus Car Walk Cycle 30% of 360 = 108 10% of 360 = 36 55% of 360 = 198 5% of 360 = 18 360 Check Construct pie charts of raw data eg 20 pupils were asked “what is your favourite subject ?” Replies were Maths 5, English 6, Science 7, Art 2 Maths English Science Art 5 20 x 360 = 90 6 20 x 360 = 108 7 20 x 360 = 126 2 20 x 360 = 36 360 Check MNU124W, MNU232W, MNU233W, MNU325W, MNU432W Elements Informing the Numeracy Framework for Action Numeracy across the curriculum Teacher knowledge of mathematics learning and inclusive teaching Improved student outcomes in mathematics and numeracy capabilities Understanding numeracy Numeracy leadership © ‘Framework for Action 2007-2010’ Queensland Government