Practical Numeracy - Steps To Success And Worked Examples

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S a i n t A n d r e w โ€™s R C S e c o n d a r ySchoolPutting Young People FirstSTEM NUMERACY SKILLSSteps to SuccessAndWorked Examples1

CONTENTSIntroductionPage 4S1Types of Data:Page 6Unit conversions and prefixesPage 7AveragesPage 8RangesPage9Creating PercentagesPage 10Creating tablesPage 11Using Information from a tablePage 12Bar graphsPage 13Line graphsPage 14S2Pie chartsPage 15Multiple Graphs (Batch Data)Page 16Grid References 4 figurePage 17Grid References 6 figurePage 18Interpreting Graphs: ExtrapolationPage 19Interpreting Graphs: Finding ValuesPage 20Accuracy and Precision.Page 21Counting Significant figuresPage 22Solving an equation (substitution)Page 232

S3Changing to Scientific notationPage 24Changing from Scientific notationPage 25PrefixesPage 26Equations with significant figuresPage 27Solving equations (changing the subject)Page 28Simplifying ratiosPage 31Using ratiosPage 32Interpreting Graphs: GradientsPage 34Interpreting Graphs: EquationPage 35Interpreting Graphs: Drawing ConclusionsPage 36AppendixPage 433

STEM Numeracy Skills: Worked examples andSteps to SuccessSTEM is the term used to describe Science, Technology, Engineering and Mathsbased subjects.The STEM subjects offered in St Andrews are MathsScience (Biology, Chemistry, Physics, Lab Science and HealthSector)ICT (Accounts, Administration, Business Management andComputing)GeographyTechnical (Design and Manufacturing, Graphics and Practical Craft)All of the subjects above use numeracy skills and maths skills to solve problems.Practical Numeracy is a course run from S1-S3. The Practical Numeracy coursewill help to develop the numeracy skills you will use in your practical STEMsubjects.The numeracy skills you will use in Practical Numeracy are the same skills youwill be using in all your other STEM subjects. These are called transferableskills.The 4transferable skills you will learn are:To understand different types of data.To change to and from scientific notationTo use the correct number of significant figuresTo change between standard unitsTo display and use information in tablesTo calculate averagesTo use a rangeTo calculate a percentageTo use ratios correctlyTo solve and rearrange equationsTo draw bar graphsTo draw line graphsTo draw a pie chartTo make conclusions from a graph.To use grid references

This booklet will take you through each of the skills above and detail the steps tosuccess in using these skills. It will also give you examples of them being used.Numeracy skills in the BGE (S1-S3)Most of the numeracy skills listed above will be met at all levels of the Senior Phase(S4-S6). They will also be met across the different STEM subject areas in the BGE(S1-S3).The table below shows the Numeracy Skills used,the subjects they are used in andthe year pupils will first meet the skillNumeracy SkillData itconversionsAverages andrangeConstructingand usingtablesPlotting bargraphsPlotting linegraphsUsinginformation ona graphConclusion froma graphPie ChartsRatiosPercentagesUsing equationsBatch dataGridreferencesPrecision S2S3S3S1S2S2S3S3S2S2S1S1S3S1S2S3

Types of dataSteps to success:1. Identify which types of datacannot be measured in numbers.This is qualitative data.2. Identify why types of data canbe assigned a numerical value.This is quantitative data.3. If the quantitative data set canonly have 1 specific value it issaid to be discrete.4. If the quantitative data set canonly have all range of numericalvalues it is said to be continuous.Example:Hair colour, height, hand span, number of freckles, gender, and blood group are allways of describing characteristics of human beings.Separate these characteristics into qualitative quantities and quantitative quantities.You must say if the quantitative information is continuous or discrete.Answer:Qualitative:Hair colour, gender, blood groupQuantitative:height (continuous), hand span (continuous), number of freckles (discrete)6

Example:Unit conversionsConvert the following table measurementsSteps to Success:into mm.1. Identify the prefix that you wishto change to. This is the letterbefore the standard units you willhave been taught, e.g. mV theprefix is m. You will mostly use k,c or m prefixes.2. Identify the prefix of the unityou currently have.3. Work along the flow chart below. 100 100 1000Length: 1.5mDepth: 70cmAnswerFor lengthFor DepthStep 1 and 2.Length: m to mmDepth: cm to mmStep 3:1.2m x 100 120cm70 cm x 100 700mm120 cm x 100 1200mmm X 1007c X 100standard unit kX 1000

Averages (mean)Steps to Success:1. Add all of the values up.2. Divide the total number by howmany values you have.3. Record the mean with units.Example:Chloe runs is timed for running 50 m. She repeats this 7 times and records theinformation in the table below.50 mTrialTime (s)112214311410512613Calculate the average (mean) time taken for the trials.Answer:๐ด๐‘ฃ๐‘’๐‘Ÿ๐‘Ž๐‘”๐‘’ ๐ด๐‘ฃ๐‘’๐‘Ÿ๐‘Ž๐‘”๐‘’ ๐‘†๐‘ข๐‘š๐‘๐‘ข๐‘š๐‘๐‘’๐‘Ÿ ๐‘œ๐‘“ ๐‘ฃ๐‘Ž๐‘™๐‘ข๐‘’๐‘ (12 14 11 10 12 13 12)7๐ด๐‘ฃ๐‘’๐‘Ÿ๐‘Ž๐‘”๐‘’ 847๐ด๐‘ฃ๐‘’๐‘Ÿ๐‘Ž๐‘”๐‘’ 128712

RangesSteps to Success:1. Find the biggest number and thesmallest number2. Subtract the smallest numberfrom the biggest number.3. The answer is the range of theset of values.Example:Calculate the range of values in time below.TrialTime (s)112214311410512613712Answer:Step 1:Biggest number 14s Smallest number 10sStep 2:Range 14 -10 4Step 3:Range 4 sNote: In Physics and Geography a range is considered to be between 2 values i.e. in theexample above, the range would be from 10s to 14sSubject Notes:Physics: A range is considered tobe between two values. In theexample given the range would befrom 10s to 14s.9

Creating PercentagesSteps to Success:1. Identify the whole (originalamount) and the part (new value)in the same units2. Divide the part by the whole (thiscreates a fraction)3. Multiply the fraction by 100. Useโ€œ%โ€ as units.Example :A power station uses 348 MJ of chemical energy to generate 194MJ of electricalenergy and 154MJ of waste energy. Calculate the percentage of chemical energychanged to electrical.Answer:Step 1:Energy in (Whole) 348MJ, Energy out (Part) 194MJStep 2:Fraction isStep 3% 194348348X 100% 55.7 %10194

Creating tablesExample:The population of Scotland reached its highest in 2014. It isSteps to Success:estimated to be 5347600. This is a rise of 19,000 people since themiddle of 2013. This is much larger than in the 1800โ€™s. In 1800 it1. Find out how many columns youneedfromlookingatwas estimated to be 1.6 million. In 1820 there was 2 million and 2.5theinformation the question wantsmillion in 1840. This was due to people moving from the Highlandsyou to display.and Ireland during the potato famine. By 1860 the population was 3million. By 1920 the population reached 5 million. It has increased2. Find out how many rows you needand decreased around this mark since.by counting how many matchingsets of data you have and add anextra row for your headings.3. Draw outline of table using aruler.4. Mark your headings in the top ofthese columns (remember units).5. Fill in table, with the matchingsets of data going in the samerow.11Construct a table to show the population in Scotland between18002014. You must include 5 pieces of information.Answer:(Year and population 2 columns). (6 years with 6 matching populations 6 sets ofdata so 7 rows)Year180018201840186019002014Population (million)1.62.02.53.05.05.0

Using information from aExample:tableYou have been asked to design a table for a client. The client has specified that theSteps to Success:1. Write down what you are askedto find from the rows andcolumns2. Move along the row headings untilyou find the heading asked for.3. Move along the column headingsuntil you find the heading askedfor.4. Find where this column and rowmeet. This is the table value yourequire5. Carry out any furthercalculations as asked using thisvalue.table calculate the minimum height of the table from the ground for a 50th percentiletable must have a 100mm clearance above the knees. Using the information from theuser.No.Dimension5th percentile50th percentile889768526406942819567444758Sitting height330Eye height, sitting529Knee height, sitting678Popliteal heightAll sizes in mm95thpercentile995869609481Answer:50th percentile and knee height.No.Dimension5th percentile50th percentile758330Sitting heightEye height, sitting88976894281995thpercentile995869529Knee height, sitting526567609678Popliteal height406444481thFrom the table the 50 percentile user has a Knee height, sitting of 567mm.Table height knee height 100mm clearance 567 100 667mm12

Bar GraphSteps to Success:1. Label your axis (with units).2. Mark your scale (each divisionshould be worth the sameamount).3. Plot your bars. First bar shouldnot touch axis.4. Give it a title.Example:An experiment is carried out to measure how much rainfall occurs in the first 6months if the year. The results rerecorded in the table belowMonthRain fall(cm)Jan20Feb18March19April15May12Plot a graph of these results.Answer:Rainfall each month.2520Geography: In geography you mustdraw your bars touching.Rain fall )cm)Subject Notes:15105013JanFebMarchAprilMonthMayJunJun11

Line GraphSteps to Success:1. Label your axis (with units).2. Mark your scale (each divisionshould be worth the sameamount).3. Plot your points.4. Draw the best fit line/connectpoints (subject dependant).5. Give it a title.Example:An experiment is carried out to measure how the length of a spring changesdepending on the mass attached to it. The results are shown below.Plot a graph of these 04.27.08.110.212.014.916.1Answer:Mass vs Length of SpringSubject Notes:18In practical subjects the term16โ€œline graphโ€ is used when talking14and scatterplots.12Length (cm)about line graphs, curved graphsDepending on the subject you maybe asked to draw a best fit line orconnect the points. As a generalrule draw a best fit line in Physicsand Chemistry, connect the pointsin Geography and Biology.141086.420012Mass (kg)34

Pie ChartSteps to Success:1. Find how many total parts youhave by adding up each type ofdata you have2. Find the number degrees 1 partis worth by dividing 360 by thenumber of parts.3. Find the number of degrees foreach different type of data bymultiplying the value by youranswer to step 2.4. Draw a full circle5. Using a protractor divide yourcircle into the correct number ofdegrees for each data type6. Label each part with thepercentage or value.7. You can include a key if you wishExample:Display the land use survey results below on a pie chartLand useNo. of ep1:Total number of buildings: 3 5 2 10Step 2:1 building is worth 360/10 36 Step 3:Commercial 3 x 36 108 Residential 5 x 36 180 Services 2 x 36 72 Step 4-7:Land use of differenttypes of properties2(20%)5(50%)153(30%)Commercial

Data)Steps to Success:1. Label your axis (with units).2. Mark your scale.3. Plot your points for the initialreading.4. Draw the best fit line/connectpoints (subject dependant).5. Plot your points for the next dataset using a different colour ofpen or different mark (e.g. cross,dots, square) and mark a best fitline/join points (subjectdependant).6. Repeat step 5 for remaining datasets.7. Give it a title.Example:Draw a graph to detail the information below.Time (min)Glucose6346281000048121620Answer:Colorimeter Reading (Units)Maltose63615635100Colorimeter Reading against6050Colorimeter Reading(Units) Glucose40Colorimeter Reading(Units) Maltose30Colorimeter Reading(Units) Lactose20100016Lactose63636363636370Colorimeter Reading (units)Multiple Graphs (Batch1020Time (min)30

Grid References (4 fig)Steps to Success:Example:4 figure:1. Find the symbol on the map.2. Move along to bottom of the map(Eastings) and find the bottomshown.left corner of the box with theAnswer:symbol in it.4 figure gridline3. Write this down as a two digitnumber.4. Move up the side of the map(Northings) and find the bottomleft corner of the box it thesymbol in it.5. Write this down as a two digitnumberafterthepreviousnumber.6. Combine the numbers to get yourgrid reference.17Find the grid reference for the Museum in the mapStep 2 and 3:92Step 4 and 5:93Step 6:9293

Grid References (6 fig)6 figure:1. Find the symbol on the map.2. Move along to bottom of the map(Eastings) and find the bottomleft corner of the box with thesymbol in it. Write this down as atwo digit number.3. Split the box that the symbol ininto 10 imaginary sections (called0-9) along the Eastings, and upthe Northings.The middle of the line is 5.After the line is 6. Before theline is 4. On the line is 0.4. Identify where the symbol fallson the imaginary line. Write thisnumber after your first twodigits.5. Repeat for the Northings line.6. Combine these numbers to getthe grid reference.18Example:Find the gridline for the Museum in the map shownAnswer:6 figure โ€“Step 2 and 3 : 92Step 4 and 5: 3Step 6: 93 and 8Step 7: 923 938

Interpreting Graphs:ExtrapolationExample:The graph shows howVoltage varies withcurrent across aSteps to Success:1. Line up your ruler up with a bestfit line.2. Drag the line back with yourfinger or a pencil until it meetsthe axis. Where the line meetsthe axis is your answer.resistor.Using the informationfrom the graph find thevoltage supply (where itcuts the potentialdifference axis).Answer:Line cutspotentialdifference axisat 4 boxes above4.0.4 boxes 4 x 0.2 0.8 so;4.8 V19

Interpreting Graphs:Finding Values1. Calculate how much each box isworth in both scales. (e.g. takethe value at 10 boxes and divideby 10. This details how much eachsmall box is worth).2. Move a ruler along the x axis torequired value.3. Move up the y axis at this pointuntil we meet the line.Read from the y-axis the valueExample:The graph showshow Voltage varieswith current acrossa resistor.Using theinformation fromthe graph find thepotential differenceat a current of 1.0A.Answer: 2.8 V20

Accuracy and Precision.Steps to Success:1. Calculate the maximum value ofthe range (x).2. Calculate the minimum value ofthe range (y).3. Write range is between x and y.4. If the range is very small thenthe result is precise.5. If accepted value is between xand y then result is accurate.21Example:The value of g is measured to be 9.9 0.04 Nkg-1. Theaccepted value of g is 9.82 Nkg-1. Is the studentโ€™s value ofg accurate?Answer:Student value of g 9.9 0.04 Nkg-1Step 1:Studentโ€™s maximum g 9.9 0.04 9.94 Nkg-1Step 2:Studentโ€™s minimum g 9.9-0.04 9.86 Nkg-1Step3:Range of students value for g 9.86 to 9.94 Nkg-1Step 4 and 5:Students result is not accurate as accepted value of g is notwithin the range of the experimental result.

Counting significantfiguresSteps to Success:1. Find the first zero from the lefthand side. This is your firstsignificant figure2. Including this digit count allnumbers that follow. This is thenumber of significant figures.3. In practical subjects we do notcount trailing zeros as significantunless they are finished with โ€œ.0โ€or come after a decimal point.22Example:How many significant figures are in the numbers below?a.b.c.d.e.124 mm10.0 mm100 mm0.015 mm0.010 mmAnswer:a.b.c.d.e.331 (in practical subjects), 3 (in maths)22

Solving an equation(substitution)Example:A resistor of 2 ฮฉ is measured to have a current of 2A passing through it. Calculatethe voltage across the resistorSteps to Success:1. Identify and write down all givenvalues in the question.2. Select equation to be used usingthe symbols given.3. Substitute values.4. Solve with correct units.232A2ฮฉAnswer:Step 1:I 2A, R 2 ฮฉ V ?Step 2:V IRStep 3:V 2 X 2Step 4:V 4V

Changing to scientificnotationExample:Change the numbers below into scientific notationa. 1230000 mb. 0.000123ASteps to Success:1. Move the first non-zero numberto immediately before yourdecimal point and drop all trailingzeros. This gives you yourcoefficient.2. Count how many places you havemoved your first significantfigure. This will give you yourpower.3. If moving numbers right then thepower will be positive. If movingdecimal point left the power willbe negative.4. Write in form โ€œ coefficient x 10power โ€œ.Answer:a. Step 1:Step 2 3Step 41230000 becomes 1.23Number of places the number has moved 6 right1.23 x 10 6 mb. Step 1:Step 2 3Step 40.000123 becomes 1.23Number of places the number has moved 4 left1.23 x 10 -4 ASubject Notes: In Science it is perfectly acceptable to have 123 x 10-94as an answer. In factthis is encouraged when using prefixes (e.g. 670 nm 670x10 m). We will look at prefixeslater.24

Changing from scientificnotationSteps to Success:1. Identify if your power is positiveor negativeExample:Change the numbers below into scientific notationa. 1.23 x 10 6 mb. 1.23 x 10-4 AAnswer:For positive powers2. If it is positive then add zerosuntil the amount of numbersafter the decimal is the same asthe power.3. Drop the decimal pointFor negative powers2. If it is negative then add thisnumber of zeros before.3. Drop the old decimal point andplace new decimal point betweenthe first two zeros.25a. Step1:Step2:Step 3: 6 (6 numbers after decimal)1.2300001230000 mb. Step1:Step 2Step3-4 (4 zeros before first digit)00001.230.000123 A

PrefixesSteps to SuccessExample:Convert the following values into SIunits.a. 12MWb. 0.3mAc. 2.3km1. Identify the prefix. This is the letter before the standard units you will havebeen taught.2. Using the prefix table shown, either substitute the prefix with the scientificnotation value OR multiply the value given by the decimal number given.3. If you are now attempting to add a prefix then divide the number given by thedecimal number in the table.4. Finish with the SI unit.Answer:a. 12MWStep 1: Prefix is MStep 2: M is x 106 or x 1 000 000So 12MW 12 x 106 W or 12 000000 Wb. 0.3 X 10 -3 A or 0.003Ac. 2.3 X 103 m or 2300 mbolTGMkcmฮผnpScientific notationx 10 12x 10 9x 10 6x 10 3x 10 -2x 10 -3x 10 -6x 10 -9x 10-12Decimal number1 000 000 000 0001 000 000 0001 000 0001 0000.010.00 10.00 000 10.00 000 0010.00 000 000 000 1

Equations with SignificantExample:figuresA resistor of 2500 ฮฉ is measured to have a current of 0.00169A passing through it.Steps to Success:1. Identify and write down all givenvalues in the question.2. Highlight the value that has thelowest number of significantfigures using the rules for apractical subject. This is howmany significant figures youranswer should be to.3. Solve question following thesteps you normally would forsolving an equation.4. Round the answer to the numberof significant figures identifiedin step 2.Calculate the voltage across the resistorAnswer:Step 1 2:I 0.00169A (3 sig fig), R 2500 ฮฉ (2 sig fig) V ?Step 3:V IRV 0.00169 X 2500V 4.225 (4 sig fig)Step 4V 4.2 V (2 sig fig).SubjectNotes: Your answer is allowed to be within a range of significant figures. It can be 1significant figure less than step 2 and 2 significant figures more than step 2.In Maths R 2500ฮฉ would have 4 significant figures but in practical subjects we would say ithas 2 because it has been measured. You will be instructed how many significant figures toround to in Maths.27

Solving equations(changing the subject)Steps to Success:1. Identify and write down all givenvalues in the question.2. Select equation to be used usingthe symbols given.3. Substitute values.4. Rearrange by balancing.5. Solve with units (applying sig. fig.rule).Example 1:100cm3 of gas is expanded to 200cm3 at constanttemperature. Its initial pressure was measuredto be 100kPa. Calculate its final pressure at itsnew volume.Answer:Step 1V1 100cm3, V 200cm3, P1 100kPa, P2 ?Step 2P1V1 P2V2Step 3100 x 100 P2 X 200Step 4Divide both sides by 200 givesP2 Step 528100 ๐‘‹100200P2 50 kPa

ORExample 1:Steps to Success:1. Identify and write down all givenvalues

Practical Numeracy is a course run from S1-S3. The Practical Numeracy course will help to develop the numeracy skills you will use in your practical STEM subjects. The numeracy skills you will use in Practical Numeracy are the same skills you will be using in all your other STEM subjects. These are called transferable skills.

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