Numeracy Acrss The Curriculum Hw tpics invlving numbers are taught within Budehaven Cmmunity Schl Cmpiled by James Grill - 1 -
Cntents Tpic Page Intrductin 3 Basics 4 Estimating 5 Runding 6 Subtractin 7 Multiplicatin 8 Divisin 9 Divisin Methds 10 Fractins 11 C-rdinates 12 Percentages 13 Prprtin 14 Equatins 15 Bar and Frequency Graphs 16 Line Graphs 17 Pie Charts 18 Time Calculatins 19 Using Frmulae 20, 21 Data Analysis 22 Using Indices and Standard Frm 23 Order f Operatins BIDMAS 24-2 -
Intrductin This infrmatin bklet has been prduced t infrm parents and teachers hw and when each tpic is taught, within the Maths Department at the schl. Other departments will use this bklet t make them aware f hw and when tpics are taught in Maths. Teaching f tpics will then be mre unifrm thrughut the schl which shuld make it easier fr pupils t learn. It is hped that the use f the infrmatin in this 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. - 3 -
Basics When pupils cme t secndary schl, they start many different subjects and have a lt f new interests, but it is still imprtant that they practise their basic number wrk, especially times tables t 15 x 15. Every pupil shuld knw their tables and these can be practised at hme. Place value is imprtant. Remember: hundreds tens units Decimal tenths hundredths Pint 3 5 6. 7 5 This number is said as: three hundred and fifty six pint seven five. 3 678 023 This number is said as three millin, six hundred and seventy eight thusand and twenty three. Pupils experience bth metric and imperial weights and measures. Fr example they shuld be aware f their wn height and weight in bth. Opprtunities t use mney and time in a practical situatin will be f value. The better yur child knws the basics, the easier it will be fr him r her t make prgress. - 4 -
Estimating We expect pupils t: Estimate height and length in centimetres (cm) and metres (m). e.g. length f pencil = 10cm width f desk = 2 1 m r 0.5m Knw apprpriate units f measure fr estimating distance, weight and vlume. e.g. bag f sugar = 1kg Knw useful cnversin facts: 1kg 2.2 lbs 1 litre 1.75 pints 8 km 5 miles 2.5 cm 1 inch - 5 -
Runding We expect pupils: t rund any whle number less than 1000 t the nearest 10 r 100 e.g 74 t the nearest 10 is 70; 386 t the nearest 100 is 400 t rund decimals t the nearest whle number e.g 23.54 t the nearest whle number is 24 t rund any number t 1 r 2 decimal places e.g 2.456 t 1 dp is 2.5 2.456 t 2 dp is 2.46 t rund t 1, 2 r 3 significant figures eg 3.14159 t 1 sf is 3 3.14159 t 2 sf is 3.1 3.14159 t 3 sf is 3.14-6 -
Subtractin We d: subtractin using decmpsitin r number line methds; check by additin; prmte alternative mental methds where apprpriate. Decmpsitin Number line Methds Cunting n: 2 6 7 1 1 3 4 9 0 1 0-3 8-7 4 2 3 3 3 2 6 We d nt brrw and pay back T slve 41 27, cunt n frm 27 until yu reach 41. Answer Finding the difference Breaking up the number being subtracted: e.g. T slve 41 27, subtract 20 then subtract 7 Answer - 7 -
Multiplicatin We encurage pupils t have a variety f strategies t multiply, building n the methds they have used in their Primary schl. A: Grid Methd 346 x 9 is apprximately 350 x 10 = 3500 346 x 9 x 300 40 6 9 2700 360 54 = 3114 72 x 38 is apprximately 70 x 40 = 2800 72 x 38 x 70 2 30 2100 60 2160 8 560 16 +576 2736 B. Partitining Shrt multiplicatin: HTU x U 316 x 9 is apprximately 350 x 10 = 3500 346 346 X 9 x 9 300 x 9 2700 leading t 3114 40 x 9 360 6 x 9 54 3114 Lng multiplicatin: TU x TU 72 x 38 is apprximately 70 x 40 = 2800 72 X 38 72 x 30 2160 72 x 8 576 2736 Extend t simple decimals with ne decimal place. Multiply by a single digit, apprximately first. Knw that decimal pints shuld line up under each ther. 4.9 x 3 is apprximately 5 x 3 = 15 4.9 x 3 4.0 x 3 = 12.0 0.9 x 3 = 2.7 14.7-8 -
Divisin We encurage students t realise that divisin is the inverse peratin f multiplicatin and s familiarity with multiplicatin tables is essential. 2 x 5 = 10 5 x 2 = 10 10 5 = 2 Is a set f related facts 10 2 = 5 When we divide we say: Hw many lts f 2 are there in 10? - 9 -
Divisin Methds 144 6 Chunking 144-60 10 x 6 84-60 10 x 6 24-24 4 x 6 0 Bus Stp 6 024 144 The bus stp is the standard methd taught fr divisin. 24 x 6 = 144 s 144 6 = 24 Chunking wrks by repeated subtractin f multiples f 6 148 5 5 029.6 148.0 148-100 20 x 5 48-45 9 x 5 3 0.6 x 5 0 29.6 x 5 = 148 148 5 = 29.6 6 x 5 = 30 0.6 x 5 = 3 Or with a remainder f 3/5 3/5 means 3 5-10 -
Fractins T knw the equivalence f cmmnly used fractins and decimals 3 e.g. = 0.3 10 We expect pupils t calculate simple fractins f amunts 1 1 f 9 = 3 (9 3); f 70 = 14 (70 5) 3 5 We expect pupils t find mre cmplex fractins f amunts 3 f 176 = 132 (176 4 x 3) 4 Essential Fractin skills: 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 Multiply Divide Make the denminatrs the same Multiply the tp and multiply the bttm. Invert the secnd fractin and multiply the tp and bttm. 1 1 2 3 1 2 2 3 2 + X 3 2 6 6 4 5 20 4 5 5 = 6 1 = 10 3 5 15 X 4 2 8 7 = 1 8-11 -
C-rdinates We expect pupils t: 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). We expect pupils t use c-rdinates in all fur quadrants t plt psitins. WORKED EXAMPLE: Plt the fllwing pints: A (5,2), B (7,0), C (0,4), D (-4,2), E (-3,-2) Remember (x) Alng the crridr (y) up the stairs - 12 -
Percentages We expect pupils t: find 50%, 25%, 10% and 1% withut a calculatr and use additin t find ther amunts. e.g. Find 36% f 250 10% is 25 30% is 75 (10% x 3) 5% is 12.50 (10% 2) 1% is 2.50 (10% 10 ) 36% is 90 ( 30% + 5% + 1% ) Express a fractin as a percentage and as a decimal equivalent 2 20 40 40% 0. 4 e.g. 5 50 We expect pupils t: 100 find percentages with a calculatr (e.g 23% f 300 = 0.23 x 300 = 69) recgnise that f means multiply. Slve prblems invlving percentage increase and decrease. e.g. If 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 e.g. Increase 350 by 15% 15% increase f 350 = 1.15 X 350 = 402.50 WE DO NOT use the % buttn n the calculatr because f incnsistencies between mdels - 13 -
Prprtin We expect pupils t: identify direct and inverse prprtin; use the unitary methd (i.e. find the value f ne first then multiply by the required value). Direct Unitary Methd If 5 bananas cst 80 pence, what d 3 bananas cst? b anan as cst (p enc e) 5 80 1 80 5 = 16p 3 16 x 3 = 48p Inverse Unitary Methd If the jurney time at 60 km/h is 30 minutes, 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 - 14 -
Equatins We expect pupils t slve simple equatins by: Balancing r perfrming the same peratin t each side f the equatin. Inverse peratins e.g und +with -, und with +, und x with, und with x 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 This shuld read 3 x 4 = 12 12 2 = 6 6 x 3 = 18 WORKED EXAM PLES: 2 x + 3 = 9 take away 3 frm bth sides 2 x = 6 divide by 2 bth sides x = 3 3 x + 6 = 2 (x 9) 3 x + 6 = 2 x -18 (subtract 6 frm bth sides) 3 x = 2 x 24 (subtract 2 x frm bth sides) x = -24 WE DO NOT change the side, change the sign. - 15 -
Bar and Frequency Graphs A Bar Chart t Shw the Clur f 80 cars in a Car Park A Frequency Diagram t shw the Distributin f Pupils' Heights in Year 11 90 25 80 20 70 Frequency 15 10 Frequency 60 50 40 30 5 20 0 10 Red Si l ver Whi te Gr een Bl ue Bl ack Other Clur 0 120 130 140 150 160 170 180 190 200 Height in cm Discrete Data Cntinuus Data Discrete data can nly take certain values in a given range. Examples: She size : 2, 2.5, 3, 3.5, 4 etc; Hair Clur; Number f Children in a Family; Number f Cars in a Car Park. When Pltting Discrete Data in a Bar Chart make sure there is: A gap between the bars. The Y axis is labelled Frequency (The number f times the data appeared). The X axis is labelled with the categries. The title says what type f graph yu are pltting and what it is shwing. Cntinuus data can take any value in a given range. Height is cntinuus data because it can never be measured precisely as it is always pssible t divide the unit being used by ten. Examples f cntinuus data are: Length; Weight; Mass; Area; Vlume. When pltting cntinuus data we use a frequency diagram make sure there is: N gaps between the bars. The data has been gruped crrectly. Grups intervals are the same size. The Y axis is labelled Frequency (The number f times the data appeared). The X axis is labelled with the categries The title says what type f graph yu pltting and what it is shwing. - 16 -
Line Graphs We expect pupils t: 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); 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. This is called a Brken Axis. WORKED EXAMPLES: In a science experiment, the distance a gas travels ver time has been recrded in the table belw: Time (s) Distance (cm) 0 5 10 15 20 25 30 0 15 30 45 60 75 90 Distance travelled by Gas ver Time 0 5 10 15 20 25 30 Time (s) - 17 -
Pie Charts We expect pupils t: use a pencil; label all the slices r insert a key as required; give the pie chart a title. All pupils shuld: interpret a pie chart. Mre advanced students can: cnstruct pie charts invlving simple fractins r decimals; cnstruct pie charts f data expressed in percentages; cnstruct pie charts f raw data. Wrked Examples 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. 360 = 18º represents 1 pupil 20 Maths 5 x 18=90º English 6 x 18=108º Science 7 x 18=126º Art 2 x 18=36º Favurite subject Science, 7-18 -
Time Calculatins We expect pupils t: cnvert between the 12 and 24 hur clck (23:27 = 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) WOR KED EXAMPLES: Hw lng is it frm 07:55 t 09:48? 07:55 08:00 09:00 0948 ( 5 m i n s ) + ( 1 h r ) + ( 4 8 m i n s ) Ttal time = 1 hr 53 minutes Change 27 minutes int the hurs equivalent. 27 min = 27 60 = 0.45 hurs - 19 -
Using Frmulae Frmulae triangles are used in Science, Design Technlgy as well as Maths. 10 = 5 x 2 2 = 10 5 10 2 5 5 = 10 2 D S T Distance = Speed x Time Speed = Distance Time Time = Distance Speed Frce = Pressure x Area F P A Area = Frce Pressure Pressure = Frce Area Vltage(V) = Current(I) x Resistance (R) I V R Current = Vltage Resistance Resistance = Vltage Current - 20 -
The length f a string S mm fr the weight f W g is given by the frmula: S = 16 + 3W Find S when W = 3 g S = 16 + 3W S = 16 + 3 x 3 S = 16 + 9 S = 25 (write frmula) (replace letters by numbers(this is called substitutin)) (slve the equatin by ding and unding) 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) - 21 -
Data Analysis We expect pupils t: 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. Eg. 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 5, 2, 5, 5, 3 Mean = 5+2+5+5+3 Median = 2, 3, 5, 5, 5 Mde = 5 5 = 4 Median = 5-22 -
Using Indices and Standard Frm All pupils shuld be able t use pwers and square rts. Mre advance pupils slve prblems invlving calculating with pwers, rts and numbers expressed in standard frm, checking fr crrect rder f magnitude. We use standard frm when wrking with very big r very small numbers. We teach that a number in standard frm cnsists f a number between ne and ten multiplied by 10 t sme pwer. Fr example 24,500,000 = 2.45 x 10 7 0.000988 = 9.88 x 10-4 On a calculatr display 2.45 x 10 7 may appear as 2.45 07. Other calculatrs may differ and pupils need t be familiar with their wn. - 23 -
Order f Operatins r BIDMAS BIDMAS 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 this 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 BIDMAS t give the crrect answer: (B)rackets (I)ndices (D)ivisin (M)ultiplicatin (A)dditin (S)ubtractin Accrding t BIDMAS, multiplicatin shuld always be dne befre additin, therefre 17 is the crrect answer. A scientific calculatr applies BIDMAS autmatically and shuld give the answer 17 when yu type in 2 + 3 x 5 <enter>. Indices means a number raised t a pwer such as 2 ² r (-3)³. The pwer is als called the rder leading t an alternative mnemnic BODMAS but bth mean the same thing. Wrked example: Calculate 4 + 70 10 x (1 + 2) 2-1 accrding t the BIDMAS rules. Brackets gives 4 + 70 10 x (3) 2 1 Indices 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 66-24 -