Basics When pupils cme t secndary schl they start a lt f different subjects and have a lt f new interests but it is still imprtant that they practise their basic number wrk which may nt be reinfrced as ften as it was in primary schl. Every pupil shuld knw their tables, particularly as they g up the schl. Their six, seven, eight, and nine times tables are very imprtant and can be practised at hme. Primary Schl learning abut place value is ften frgtten and can be reinfrced at hme. Remember hundreds tens units Decimal tenths hundredths Pint 3 5 6. 7 5 Reading and writing large numbers is a cmmn difficulty that yu can help with. 3 678 023 reads three millin, six hundred and seventy eight thusand, and twenty three. Pupils can be made aware at hme f metric and imperial weights and measures and their wn height and weight in bth. They can practise estimating sensibly and the getting the feel f large and small weights heights and distances, and using mney in a practical way. The better yur child knws the basics, the easier it will be fr him r her t make prgress.
Intrductin Recently Strmness Academy set up a wrking grup t see hw tpics invlving numbers are taught in the varius departments within the schl. They cnsulted all the departments and a sample f senir pupils. This infrmatin bklet has been prduced t tell parents and teachers hw each tpic is taught within the Maths Department at the schl. Other departments can use this bklet t make them aware f hw tpics are taught in Maths. Teaching f numeracy tpics will then be mre unifrm thrughut the schl which shuld make it easier fr pupils t learn. It is hped that use f the infrmatin in the bklet will help yu understand the way number tpics are being taught t yur children in the schl, making it easier fr yu t help them with their hmewrk, and as a result imprve their prgress. Index Page Tpic Page Tpic 1 Bar Graphs 10 Pie Charts 2 C-rdinates 11 Prprtin 3 Data Analysis 12 Runding 4 Equatins 13 Scientific Ntatin 5 Estimating 14 Subtractin 6 Fractins 15 Time Calculatins 7 Line Graphs 16 Using Frmulae 8 Order f Operatins 17 Multiplicatin 9 Percentages 18 Divisin This bklet has been prduced by the Strmness Academy, Numeracy acrss the Curriculum wrking grup
Bar Graphs use a pencil give the graph a title label the axes label the bars in the centre f the bar (each bar has an equal width) label the frequency (up the side) n the lines nt n the spaces make sure there are spaces between the bars Cnstruct bar graphs with frequency graduated in single units Cnstruct bar graphs with frequency graduated in multiple units Cnstruct cnstruct bar graphs invlving simple fractins r decimals WORKED EXAMPLES: clur f eyes quantities f litter she size 5 25 12 4 20 10 3 2 15 10 8 6 4 1 5 2 0 brwn blue green 0 paper plastic 0 3.5 4 4.5 5 5.5 1
C-rdinates use a c-rdinate system t lcate a pint n a grid number the grid lines rather than the spaces use the terms acrss/back and up/dwn fr the different directins use a cmma t separate as fllws : 3 acrss 4 up = (3,4) use c-rdinates in all fur quadrants t plt psitins WORKED EXAMPLE: Plt the fllwing pints: M (5,2), A (7,0), T (0,4), H (-4,2), S (-3,-2) y up 5 H (-4,2) x 4 x 3 2 T (0,4) M (5,2) x -5-4 -3-2 -1 1 0-1 A (7,0) x 1 2 3 4 5 6 7 8 x acrss S(-3,-2) x -2-3 2
Data Analysis analyse ungruped data using a tally table and frequency clumn r an rdered list calculate range f a data set. In Maths this is taught as the difference between the highest and lwest values f the data set. ( Range is expressed differently in Bilgy) calculate the mean (average) f a set f data. use a stem and leaf diagram calculate the mean (average) median ( central value f an rdered list) mde (mst cmmn value) f a data set. btain these values frm an ungruped frequency table. Crrelatin in scatter graphs is described in qualitative terms. e.g. The warmer the weather, the less yu spend n heating is negative crrelatin. The mre peple in yur family, the mre yu spend n fd is psitive crrelatin. Prbability is always expressed as a fractin P (event) = number f favurable utcmes ttal number f pssible utcmes WORKED EXAMPLE The results f a survey f the number f pets pupils wned were 3, 3, 4, 4, 4, 5, 6, 6, 7, 8 Mean = ( 3 + 3 + 4 + 4 + 4 + 5 + 6 + 6 + 7 + 8) 10 = 5 Median = the middle = ( 4 + 5) 2 = 4.5 Mde = mst cmmn = 4 Range = highest lwest = 8 3 = 5 3
Equatins slve simple equatins by. Balancing perfrming the same peratin t each side f the equatin ding Und peratins e.g und + with -, und with + und x with, und with x encuraging statements like: add smething t bth sides multiply bth sides by smething We prefer the letter x t be written differently frm a multiplicatin sign ne equals sign per line equals signs beneath each ther we discurage bad frm such as 3 x 4 = 12 2 = 6 x 3 = 18 WORKED EXAMPLES: 2x + 3 = 9 2x = 6 x = 3 3x + 6 = 2 (x 9) 3x + 6 = 2x -18 3x = 2x 24 x = -24 take away 3 frm bth sides divide by 2 bth sides (subtract 6 frm bth sides) (subtract 2x frm bth sides) change the side, change the sign 4
Estimating estimate height and length in cm, m, 1/2m, 1/10m e.g. length f pencil = 10cm width f desk = 1/2m estimate small weights, small areas, small vlumes e.g. bag f sugar = 1kg estimate areas in square metres, lengths in mm and m e.g. area f a blackbard = 4m 2 diameter f 1p = 15mm 5
Fractins calculate simple fractins f 1 r 2 digit numbers e.g 1 1 f 9 = 3 (9 3); f 70 = 14 (70 5) 3 5 calculate simple fractins f up t 4 digit numbers 3 f 176 = 132 (176 4 x 3) 4 use equivalence f widely used fractins and decimals e.g. 10 3 = 0.3 find widely used fractins mentally find fractins f a quantity with a calculatr use equivalence f all fractins, decimals and percentages add, subtract, multiply and divide fractins with and withut a calculatr WORKED EXAMPLES Add and Subtract Multiply Divide Make the denminatrs equal 1 + 1 2 3 = 3 + 2 6 6 = 5 6 Multiply tp and multiply bttm 2 x 3 3 4 = 6 12 = 1 2 Invert the secnd fractin and multiply 3 2 4 5 = 3 x 5 4 2 15 = = 8 1 7 8 6
Line Graphs use a sharpened pencil and a ruler chse an apprpriate scale fr the axes t fit the paper label the axes give the graph a title number the lines nt the spaces plt the pints neatly (using a crss r dt) fit a suitable line if necessary, make use f a jagged line t shw that the lwer part f a graph has been missed ut. WORKED EXAMPLES: The distance a gas travels ver time has been recrded in the table belw: Time (s) 0 5 10 15 20 25 30 Distance (cm) 0 15 30 45 60 75 90 Distance travelled by a gas ver time 100 90 x 80 x Distance (cm) 70 60 50 x x 40 30 x 20 x 10 0 5 10 15 20 25 30 Time (secs) 7
Order f Operatins r Bdmas BODMAS is the mnemnic which we teach in maths t enable pupils t knw exactly the right sequence fr carrying ut mathematical peratins. Scientific calculatrs use a rule t knw which answer t calculate when given a string f numbers t add, subtract, multiply, divide etc. Fr example What d yu think the answer t 2 + 3 x 5 is? Is it (2 + 3) x 5 = 5 x 5 = 25? r 2 + (3 x 5) = 2 + 15 = 17? We use BODMAS t give the crrect answer.: (B)rackets (O)rder (D)ivisin (M)ultiplicatin (A)dditin (S)ubtractin Accrding t BODMAS, multiplicatin shuld always be dne befre additin, therefre 17 is the crrect answer accrding t BODMAS and shuld als be the answer which yur calculatr will give if yu type in 2 + 3 x 5 <enter>. Order means a number raised t a pwer such as 2² r (-3)³. The pwer is als called the index leading t an alternative mnemnic BIDMAS but bth mean the same thing. Wrked example Calculate 4 + 70 10 x (1 + 2) 2-1 accrding t the BODMAS rules. Brackets gives 4 + 70 10 x (3) 2-1 Order gives 4 + 70 10 x 9-1 Divisin gives 4 + 7 x 9-1 Multiplicatin gives 4 + 63-1 Additin gives 67-1 Subtractin gives 66 Answer 66 8
Percentages we expect pupils t find 50%, 25%, 10% and 1% withut a calculatr and use additin t find ther amunts. we expect pupils t find percentages with a calculatr (e.g 23% f 300 = 300 100 x 23 = 69) and t recgnise that f means multiply. we expect pupils t express a fractin as a percentage via the decimal equivalent. Fr example Find36% f 250 10% is 25 30% is 75 (x 3) 5% is 12.50 (10% 2) 1% is 2.50 (10% 10 ) 36% is 90 ( 30% + 5% + 1% ) Express tw fifths as a percentage 2 4 40 = = = 5 10 100 40% Yu buy a car fr 5000 and sell it fr 3500. What is the percentage lss? Lss = 5000 3500 = 1500 1500 15 30 = = = 30% 5000 50 100 Increase 350 by 15% 15% f 350 = 350 100 x 15 = 52.50 ( t find the increase) (then add n fr the new ttal.) 350 + 52.50 = 402.50 use the % buttn n the calculatr because f incnsistencies between mdels 9
Pie Charts use a pencil label all the slices r insert a key as required give the pie chart a title cnstruct pie charts invlving simple fractins r decimals cnstruct pie charts f data expressed in percentages cnstruct pie charts f raw data WORKED EXAMPLES: Basic Mre difficult 30% f pupils travel t schl by bus, 10% by car, 55% walk and 5% cycle. Draw a pie chart f the data. 20 pupils were asked What is yur favurite subject? Replies were Maths 5, English 6, Science 7, Art 2 Draw a pie chart f the data. 10% f 360º =36º 360 20 ( the ttal ) = 18º Bus 30% = 3 x 10% = 108º Maths 5 5 x 18 = 90º Car 10% = 1 x 10% = 36º English 6 6 x 18 = 108º Walk 55% = 5.5 x 10% =198 Science 7 7 x 18 = 126º Cycle 5% = 0.5 x 36% = 18º Art 2 2 x 18 = 36º Transprt t schl Favurite subject cycle 5% bus 30% Art, 2 Maths, 5 walk 55% car 10% Science, 7 English, 6 10
Prprtin identify direct and inverse prprtin recrd apprpriate headings with the unknwn n the right use the unitary methd (i.e. find the value f ne first then multiply by the required value) if runding is required we d nt rund until the last stage WORKED EXAMPLES: A. Direct Unitary Methd If 5 bananas cst 80 pence, then what d 3 bananas cst? bananas cst (pence) 5 80 1 80 5 = 16 3 16 x 3 = 48 B. Inverse Unitary Methd The jurney time at 6 km/h = 30 minutes, s what is the jurney time at 50km/h? Speed (km/h) Time (mins) 60 30 1 30 x 60 = 1800 minutes 50 1800 50 = 36 minutes 11
Runding rund 2 digit whle numbers t the nearest 10 rund 3 digit whle numbers t the nearest 10 rund any number t the nearest whle number, 10 r 100 rund any number t 1 decimal place rund t any number f decimal places r significant figures Nte: We always rund up fr 5 r abve WORKED EXAMPLES: 74 t the nearest 10 70 386 t 390 347.5 t 348 (t nearest whle number); r t 350 (t nearest ten); r t 300 (t nearest hundred) 7.51 (t 1 decimal place) t 7.5 8.96 (t 1 decimal place) t 9.0 3.14159 (t 3 decimal places) t 3.142; r 3.14 (t 2 decimal places); r 3.14 (t 3 significant figures) 12
Scientific ntatin r Standard Frm In mathematics we intrduce scientific ntatin in S1 /S2. It is als taught at the beginning f S3. In maths we teach that a number in scientific ntatin cnsists f a number between ne and ten multiplied by 10 t sme pwer. Fr example 24,500,000 = 2.45 x10 0.000988 = 9.88x10 7 4 Other subjects may apprach this tpic differently. we intrduce the terms: Kil meaning ne thusand Milli meaning ne thusandth. be able t use pwers and square rts. 13
Subtractin subtract using decmpsitin (as a written methd) check by additin we prmte alternative mental methds where apprpriate WORKED EXAMPLES Decmpsitin: 6 3 9 2 7¹1 4 0¹0 3 8 7 4 2 3 3 3 2 6 Cunting n: T slve 41 27, cunt n frm 27 until yu reach 41 Breaking up the number being subtracted: e.g.t slve 41 27, subtract 20 then subtract 7 brrw and pay back 14
Time Calculatins cnvert between the 12 and 24 hur clck (2327 = 11.27pm) calculate duratin in hurs and minutes by cunting up t the next hur then n t the required time cnvert between hurs and minutes (multiply by 60 fr hurs int minutes) WORKED EXAMPLES: Hw lng is it frm 0755 t 0948? 0755 0800 0900 0948 (5 minutes) + (1 hr) + (48 minutes) Ttal time 1 hr 53 minutes Change 27 minutes int hurs equivalent 27 minutes = 27 60 = 0.45 hurs teach time as a subtractin 15
Using Frmulae cnstruct and use simple frmulae by writing dwn the frmula first rewriting the frmula replacing the letters by the apprpriate numbers (substitutin) slving the equatin interpreting the answer and putting the apprpriate units back int cntext WORKED EXAMPLES: The length f a string S millimetres with a weight f W grams per millimetre is given by the frmula: S = 16 + 3W (a) (b) Find S when W = 3 grams S = 16 + 3W (write frmula) S = 16 + 3 x 3 (replace letters by numbers) S = 16 + 9 S = 25 Length f string is 25 mm (interpret result in cntext) Find W when S = 20.5 mm S = 16 + 3 W (write frmula) 20.5 = 16 + 3W (replace letters by numbers) 4.5 = 3W (slve the equatin by ding and unding) 1.5 = W The weight is 1.5 g (interpret result in cntext) Rearrange the frmula befre substitutin (t difficult) State the answer nly. Wrking must be shwn 16
Multiplicatin multiply single numbers using all the tables frm ne t ten withut a calculatr multiply whle numbers by 10, 100, and 1000 withut a calculatr multiply decimal numbers by 10, 100, and 1000 withut a calculatr multiply whle numbers up t fur digits by a single digit whle number multiply decimal number up t tw decimal places by a single digit whle number multiply whle and decimal numbers by multiples f 10 and 20 use a calculatr t multiply any pair f whle numbers up t 3 decimal places. This is extended t multiplying decimal numbers by decimal numbers withut a calculatr WORKED EXAMPLES 2 5 1 2. 3 x 6 x 4 1 5 0 4 9. 2 3 1 16 x 20 = 16 x 2 x 10 = 320, 1.50 x 400 = 1.50 x 100 x 4 = 600 0.7 x 0.3 = 7 x 3 10 10 = 0.21 32.5 x 60 000 = 195 x 10 000 = 1 950 000 17
Divisin divide whle numbers by 10, 100, and 1000 withut a calculatr. divide decimal numbers by 10, 100, and 1000 withut a calculatr. divide whle numbers up t fur digits by a single digit whle number ( whle number answer). divide decimal numbers up t tw decimal places by a single digit whle number (decimal answer). divide simple whle and decimal numbers by multiples f 10 and 20. use a calculatr t divide any pair f whle numbers up t 3 decimal places. This is extended t dividing decimal numbers by decimal numbers withut a calculatr. WORKED EXAMPLES: 8 2 9. 3 5 2 4 1 2 3 2 4 6 5 4 6. 7 5 16 20 = 16 2 10 = 0.8 1.50 50 = 1.50 10 5 = 3p 0.8 0.2 = 8 2 x 10 10 = 4 32.5 500 = 32.5 5 100 = 6.5 100 = 0.065 18