Mathematics General Stage 6 Syllabus

HSC Mathematics General 1 Course Support Material

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Contents Support material for HSC Mathematics General 1 Strands and Focus Studies Strand: Financial Mathematics.......................................................................................................... 4

Strand: Data and Statistics................................................................................................................ 7

Strand: Measurement........................................................................................................................ 9

Strand: Probability........................................................................................................................... 10

 

Strand: Algebra and Modelling........................................................................................................ 11

 

Focus Study: Mathematics and Design........................................................................................... 12

 

Focus Study: Mathematics and Household Finance....................................................................... 16

 

Focus Study: Mathematics and the Human Body ........................................................................... 18

 

Focus Study: Mathematics and Personal Resource Usage............................................................ 23

 

 

 

 

HSC Mathematics General 1 Course Support Material

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Strand: Financial Mathematics

FM4CEC Credit and borrowing The following online resources could be used to support the teaching and learning of this topic.

Credit card calculator (calculate the total amount that will be paid if only the minimum repayment is made each month)

https://www.moneysmart.gov.au/tools-and-resources/calculators-and-tools/credit-card-calculator

Credit card fees and charges explained www.stgeorge.com.au/assets/stg/downloads/accounts_and_cards/sgb_cc_fee_charges_0612.pdf

Credit card interest explained (includes exemplar statements covering a three-month period)

www.nab.com.au/vgnmedia/downld/Facts_about_credit_card_interest_40014A0308.pdf

Credit card interest and fees explained (includes credit card statement explained)

www.commbank.com.au/personal/apply-online/download-printed-forms/ADB3181-a-question-of-interest.pdf

Credit card interest-free periods explained www.creditcardfinder.com.au/what-does-55-days-interest-free-really-mean.html

Credit card jargon buster www.commbank.com.au/personal/credit-cards/credit-card-jargon-buster/

Credit card statements explained www.creditcardfinder.com.au/understanding-credit-card-statement-features.html www.anz.com.au/personal/credit-cards/calculators-tools/how-to-read-your-statement/ www.citibank.com.au/global_docs/statement_demo/ http://learn.nab.com.au/how-to-read-your-credit-card-statement/

Credit cards (includes videos) https://www.moneysmart.gov.au/borrowing-and-credit/credit-cards

Credit cards and store cards facts sheet https://www.moneysmart.gov.au/media/283208/cfs-credit-cards-and-store-cards.pdf

For the teacher

Credit cards • The conditions and calculations of interest, fees, balances and payments for credit card

accounts vary depending on the type of card and the issuer.

• By law, all Australian credit card issuers are required to provide those who apply for a credit card with a ‘key facts sheet’ containing information on the:

– minimum repayment (including how it will be calculated)

– interest rate that applies to purchases and to cash advances

– interest rate that applies to balance transfers (and for how long)

– promotional interest rate (if any)

– length of the interest-free period (if any)

– annual and late payment fees (if any).

Search the websites of credit card issuers to locate the key facts sheets for particular credit cards.

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Terminology used in credit card statements

Term Meaning Notes annual fee or monthly fee

A fee charged by the issuer for maintenance of the account.

Annual fees are the most common fees on credit card accounts. They usually range from $0 to $300, but they can be higher.

available credit The total amount of money available within the credit limit for purchases/cash advances.

balance transfer The act of transferring the balance from an existing credit card account to a different credit card account.

Balances transferred to a new card often attract a significantly lower interest rate for an introductory period. After the introductory period expires, any remaining balance usually attracts the standard interest rate for purchases, although in some cases it attracts the interest rate for cash advances.

cash advance Cash withdrawn from a credit card account.

Transactions considered to be cash advances by most credit card issuers include: • withdrawing cash at an ATM or at

a branch • ‘taking cash out’ when making

a purchase at a store • using a credit card to gamble, either

online or at a casino • using a credit card to buy foreign

currency. cash advance fee A fee charged by the issuer

when the cardholder takes out a ‘cash advance’.

Cash advance fees are usually the greater of a specified: • percentage of the cash advance

(usually between 1% and 3%) or

• amount between $2 and $5. closing balance The amount owing at the end of the

particular statement cycle.

credit limit The maximum amount of money that the issuer will allow the cardholder to spend using the credit card account.

interest-free period or interest-free days

The maximum number of days for which a transaction will not incur interest charges, provided that the previous ‘closing balance’ is paid in full by the ‘payment due date’. The actual number of interest-free days applicable to a particular transaction depends on when the date of the purchase occurs in the statement cycle.

Any interest-free period usually applies to purchases only (not cash advances) and is typically between 40 and 55 days. A transaction on the first day of the statement cycle will have the maximum number of interest-free days. For an account with a 55-day interest-free period on purchases and a 30-day statement cycle, a purchase on day 10 of the statement cycle will have 45 interest-free days (ie the remaining 20 days of the statement cycle plus the 25 days to make the full payment).

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Term Meaning Notes interest rate The annual interest rate or the daily

interest rate that applies for a specified type of transaction.

The interest rate that applies to cash advances is usually higher than the interest rate that applies to purchases.

late payment fee A penalty fee charged by the issuer when the cardholder does not pay at least the ‘minimum repayment’ by the ‘payment due date’.

Late payment fees are typically between $10 and $50.

minimum repayment, minimum amount due or minimum payment due

The amount the issuer requires the cardholder to pay by the due date.

The minimum repayment is usually the greater of a specified: • percentage of the cash advance

(usually between 2% and 5%) or

• amount between $10 and $30. opening balance or prior month balance

The amount owing at the start of the particular statement period (ie the closing balance on the previous statement).

overdue amount or outstanding balance

The amount overdue if the minimum repayment for the previous statement period was not made.

payment due date The date by which at least the minimum amount due is to be paid.

statement cycle or statement period

The period of time that the particular statement covers.

The statement period is usually 30 days and may be expressed in terms of a ‘statement start date’ and a ‘statement end date’.

transactions or transaction details

An itemised list of all transactions, including purchases, cash advances, interest, fees and repayments.

The transaction list usually includes details of all purchases, cash advances, payments and credits during the statement period, including the date, description and amount of each transaction.

• The terms ‘outstanding balance’ and ‘outstanding amount’ (where the adjective ‘outstanding’ is used to mean ‘unpaid’) may need to be explained explicitly, as some students may be unfamiliar with this meaning of ‘outstanding’.

• Questions involving calculations for credit card accounts should be carefully constructed to ensure that students have sufficient information to perform the calculations.

• In the HSC Mathematics General 1 course: – simple interest is to be assumed for all interest calculations, with interest calculated on

the daily outstanding account balance for each transaction and applied at the end of the statement period

– the same interest rate is to be assumed to apply for all transaction types – the daily interest rate is to be found by dividing the annual percentage interest rate by 365 – students are expected to carry out calculations, including calculations of simple interest for

one billing cycle, for credit card statements on which the ‘opening balance’ is $0 and also for credit card statements on which the ‘opening balance’ is not $0

– students are not expected to carry out calculations involving balance transfers, although this could be considered for some students if appropriate.

Teachers should refer to the ‘Considerations’ section of FM4CEC on page 133 of the Mathematics General Stage 6 Syllabus for further information about expectations regarding credit card calculations in the HSC Mathematics General 1 course.

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Strand: Data and Statistics

DS4CEC Distributions The following online resource could be used to support the teaching and learning of this topic.

Applets to create and/or investigate statistical displays

www.shodor.org/interactivate/activities/ (select ‘Statistics’ for a list of applicable applets)

DS5CEC Interpreting sets of data The following online resources could be used to support the teaching and learning of this topic.

Scootle resources – enter the following reference numbers into the Scootle search function: • L3513 Box plot/histogram: select

the histogram option – automatically generates a histogram and calculates mean and median as data values are entered

• L5905 Graph investigator: reaction time • L10338 Graph investigator: reaction

time (ESL) • L5906 Graph investigator: hand

preference • L10339 Graph investigator: hand

preference (ESL) • L5912 Stem-and-leaf plots: an

introduction – focuses on back-to-back stem-and-leaf plots

• M009386 Back-to-back stem-and-leaf plot of sex vs concentration time – uses CensusAtSchool data

www.scootle.edu.au/ec/p/home

DS6CEC Working with statistics The following online resources could be used to support the teaching and learning of this topic.

Reversal Paradox http://repository.upenn.edu/cgi/viewcontent.cgi?article=1014&context=wharton_research_scholars

Simpson’s Paradox http://en.wikipedia.org/wiki/Simpson%27s_paradox#cite_note-12

Refer also to page 5 of the Preliminary Mathematics General course support material for links to resources relating to Data and Statistics.

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For the teacher The Reversal Paradox is also known as Simpson’s Paradox, the Amalgamation Paradox or the Yule–Simpson Effect. The Reversal Paradox applies where the statistical relationship between two variables is reversed in situations where, rather than considering the relevant data in subgroups, the data of the subgroups is combined and viewed as a whole.

Reversal Paradox example: Kidney stone treaThe following example of the Reversal Paradox is adapted from Wikipedia and is based on a real-life study of the success of two different kidney stone treatments (Treatment A and Treatment B).

tment study

The table below shows the success rates and the numbers of treatments for cases involving small kidney stones and for cases involving large kidney stones.

Treatment A Treatment B

Small stones Group 1

93% (81/87) Group 2

87% (234/270)

Large stones Group 3

73% (192/263) Group 4

69% (55/80)

Both 78% (273/350) 83% (289/350)

The paradoxical conclusion is that Treatment A is more effective when used on small stones and also when used on large stones, yet Treatment B is more effective when considering both sizes at the same time. In this example, the ‘lurking variable’ or ‘confounding variable’ of the stone size was not known to be important until it was decided to include its effects.

Determining which treatment is better requires considering the inequality between two fractions (successes/total) in the table. The reversal of the inequality between the fractions (Simpson's Paradox) becomes evident because two effects occur together:

1. The sizes of the groups, which are combined when the lurking variable is ignored, are very different. Doctors tend to give the severe cases (large stones) the better treatment (Treatment A), and the milder cases (small stones) the inferior treatment (Treatment B). Therefore, the totals are dominated by Group 2 and Group 3, and not by the two much smaller groups, Group 1 and Group 4.

2. The lurking variable has a large effect on the ratios, ie the success rate is more strongly influenced by the severity of the case than by the choice of treatment. Therefore, the group of patients with large stones using Treatment A (Group 3) does worse than the group with small stones, even if the latter was using the inferior Treatment B (Group 2).

Source: Adapted from Wikipedia, ‘Simpson’s Paradox’, http://en.wikipedia.org/wiki/Simpson%27s_paradox, citing CR Charig, DR Webb, SR Payne and JE Wickham (1986) ‘Comparison of treatment of renal calculi by open surgery, percutaneous nephrolithotomy, and extracorporeal shockwave lithotripsy’, Br Med J (Clin Res Ed) 292:879.

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Strand: Measurement

MM4CEC Further applications of area and volume The following online resources could be used to support the teaching and learning of this topic.

Nets – interactive www.learner.org/interactives/geometry/area_surface.html

Nets of prisms, pyramids, cylinders and cones (create and print)

http://illuminations.nctm.org/ActivityDetail.aspx?ID=205 (select the ‘Nets’ tab)

Solids to nets – interactives (made using GeoGebra)

http://mrskrummel.com/apps/Geometry/ch11_SurfaceArea.html

Refer also to page 9 of the Preliminary Mathematics General course support material for links to resources relating to Measurement.

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Strand: Probability

PB2CEC Multistage events and applications of probability The following online resources could be used to support the teaching and learning of this topic.

Dice rolling simulator (two dice) with histogram comparing experimental results to theoretical results

https://www.math.duke.edu//education/postcalc/probability/dice/index.html

Probability simulations (coin, spinner and others)

www.mathsonline.co.uk/nonmembers/resource/prob/index.html

Probability tree diagram generator http://kera.name/treediag/

Vehicle registration plates of the world http://en.wikipedia.org/wiki/Vehicle_registration_plate

See note (1) below

Refer also to page 10 of the Preliminary Mathematics General course support material for links to resources relating to Probability.

Note (1) Go to ‘License plates by country or territory’ and select a country to obtain specific information

about that country’s vehicle registration plates.

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Strand: Algebra and Modelling The following online resource could be used to support the teaching and learning of this Strand.

Algebraic modelling activities www.thefutureschannel.com/algebra_real_world.php

Refer also to page 11 of the Preliminary Mathematics General course support material for links to resources relating to Algebra and Modelling.

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Focus Study: Mathematics and Design

FSDe1CEC Scale drawings and house plans The following online resources could be used to support the teaching and learning of this topic.

Building plan examples and free software www.edrawsoft.com/Building-Plan-Examples.php

Drawing and scale diagram tool (for floor plans, for example)

www.gliffy.com/

Floor plan examples and free software www.edrawsoft.com/floorplan.php

Floor plan symbols explained www.floorplansfirst.com/symbols.cfm

Floor plans of properties for sale www.realestate.com.au/buy www.domain.com.au/

From Concept to Construction (video about mathematics used by engineers)

www.mathscareers.org.uk/viewitem.cfm?cit_id=383234

Graph paper generator (includes a variety of styles)

http://incompetech.com/graphpaper/

Graph paper generator (includes isometric paper)

http://illuminations.nctm.org/ActivityDetail.aspx?ID=205 (select ‘Graph Paper’)

Home plan examples and free software www.edrawsoft.com/homeplan.php

Homestyler (drag-and-drop floor plan design tool)

www.homestyler.com/designer

Math by Design (interactive design of local amenities)

http://mathbydesign.thinkport.org/ (select either ‘Flossville Town Park’ or Windjammer Environmental Center’)

Math in Action (videos) http://mathbydesign.thinkport.org/educator_resources/mathinaction.aspx

Sample floor plan with features explained www.homedesigndirectory.com.au/articles/plans/LargeFloorPlan2.html

Sample house plan with dimensions (opens in Microsoft Word)

https://www.dlsweb.rmit.edu.au/toolbox/electrotech/toolbox1204/resources/04diagrams/02architectural/03floor_plan.htm

Sample house plans with dimensions www.maddisonconstructions.com.au/newhome_example_plans.html

Teacher package: ‘Maths and art’ http://plus.maths.org/content/teacher-package-maths-and-art#design

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FSDe2CEC Design The following online resources could be used to support the teaching and learning of this topic.

Escher gallery of designs www.mcescher.com/Gallery/gallery.htm

GeoGebra (freeware) www.geogebra.org/cms/

Golden ratio (includes tessellations) www.miqel.com/fractals_math_patterns/visual-math-phi-golden.html

Golden ratio and beauty in art http://library.thinkquest.org/trio/TTQ05063/phibeauty3.htm

Golden ratio calculator http://goldenratiocalculator.com/

Golden ratio explained http://en.wikipedia.org/wiki/Golden_ratio www.goldennumber.net/ www.phiday.org/phi-golden-ratio/

Golden ratio – The Divine Proportion: Phi (video)

www.youtube.com/watch?v=3sERcM9o25g

Golden Ratio − Universal Principles of Design (video)

www.youtube.com/watch?v=jCUJzNsKGW4

Islamic tiling pattern (photograph, with inquiry questions)

www.aamt.edu.au/digital-resources/R11271/index.html

Logos designed using the golden ratio www.banskt.com/blog/golden-ratio-in-logo-designs/

Pyramid composed of glass tiles (photograph, with inquiry questions)

www.aamt.edu.au/digital-resources/R11197/index.html

Tessellating garden design (photograph, with inquiry questions)

www.aamt.edu.au/digital-resources/R11276/index.html

Tessellation creation (animation) http://library.thinkquest.org/16661/escher.html

Tessellation creators http://illuminations.nctm.org/ActivityDetail.aspx?ID=205 (select ‘Tessellations’ tab) www.shodor.org/interactivate/activities/Tessellate/

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For the student

Designs: golden or not? (1) Research the ‘golden ratio’ (also known as the ‘golden proportion’, the number ‘phi’

and various other names) online (eg www.youtube.com/watch?v=3sERcM9o25g, http://en.wikipedia.org/wiki/Golden_ratio, www.goldennumber.net/).

(2) What is the value of the golden ratio, correct to three decimal places?

(3) Consider your own bank card, or the example shown here. The same dimensions are used for credit cards, loyalty cards, driver licences, and so on.

Are the dimensions (length and width) of the bank card in the golden ratio?

CC-BY-SA-3.0-MIGRATED. Released under the GNU Free Documentation Licence.

© BMW AG RE. 97. L. 1491.

(4) Consider the logo of the BMW car company. The design includes an ‘outer’ circle and an ‘inner’ circle.

Measure the radius of each circle and determine whether or not the circles were constructed using the golden ratio.

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© BP Australia Pty Ltd, reproduced with permission.

(5) Consider the logo of the BP oil and gas company. The design includes concentric circles (circles with the same centre). These concentric circles and two axes have been drawn on the image at left.

Measure the diameter of each of the circles and determine whether or not the golden ratio has been used in the design of the logo.

(6) Consider the logo of the iCloud service

operated by Apple. It is composed of four circles and a base line. The four circles have been drawn on the image at right in two pairs, one pink and one blue. A red rectangle has been drawn around the cloud to show the length and width of the cloud.

Has the golden ratio been used in the iCloud logo? If so, describe where it has been applied.

© Apple Inc., registered in the US and other countries.

(7) Find a logo or other commonly used symbol with dimensions that are in the golden ratio. Record a picture of your logo or symbol. Label the picture with the dimensions that you have measured that are in the golden ratio. Write down the calculations that you did to determine that the dimensions are in the golden ratio.

(8) Design a logo for a new car company called ‘Movus’ using simple shapes and the golden ratio correct to one decimal place. You may use appropriate technology, or pencil and paper. Grid paper may also be helpful.

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Focus Study: Mathematics and Household Finance

FSHo1CEC Accommodation costs: buying and renting The following online resources could be used to support the teaching and learning of this topic.

Buying a home https://www.moneysmart.gov.au/tools-and-resources/life-events/buying-a-home

Buying a home – costs www.anz.com.au/personal/home-loans/getting-started/costs-buying-home/

Buying into a strata scheme www.fairtrading.nsw.gov.au/Tenants_and_home_owners/Buying_property/Buying_into_a_strata_scheme.html

Buying property (includes three videos) www.fairtrading.nsw.gov.au/Tenants_and_home_owners/Buying_property.html

Home loans https://www.moneysmart.gov.au/borrowing-and-credit/home-loans

Leaving home costs (includes renting, sharing, setting up utilities, etc)

https://www.moneysmart.gov.au/tools-and-resources/information-for/under-25s/leaving-home

Mortgage calculator https://www.moneysmart.gov.au/tools-and-resources/calculators-and-tools/mortgage-calculator

Mortgage calculator (shows amount owing over time in a graph or table)

www.infochoice.com.au/calculators/home-loan-calculator/ (select ‘Yearly Breakdown’ tab for table)

Property buying cost calculators (including additional costs)

www.yourmortgage.com.au/calculators/homecost/ http://service.commbank.com.au/personal/home-loans/calculators/additional-cost-calculator.aspx www.ingdirect.com.au/home_loans/calculators/property_buying_cost_calculator.htm

Renting a property www.fairtrading.nsw.gov.au/Tenants_and_home_owners/Renting_a_home.html

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FSHo2CEC Costs of running a household, maintenance, and repairs The following online resources could be used to support the teaching and learning of this topic.

Budget planner https://www.moneysmart.gov.au/tools-and-resources/calculators-and-tools/budget-planner (select either online or Excel version)

Do-it-yourself (DIY) calculators (including for paint and flooring)

www.bunnings.com.au/learn-how-to-diy_online-diy-tools.aspx

Drawing and scale diagram online tool (for floor plans, for example)

www.gliffy.com/

Energy bills explained www.agl.com.au/home/billing-and-payments/Pages/agl-bill-explainers.aspx www.energyaustralia.com.au/residential/account-tools/bills-payments/understanding-your-bill/different-types-of-bill www.integral.com.au/wps/wcm/connect/IE/NSW/NSW+Homepage/yourAccountNav/Your+bill+explained/ www.originenergy.com.au/1374/Understanding-your-bill www.redenergy.com.au/docs/BillExplainerQuarterlyBill.pdf

Energy retailers in NSW www.myenergyoffers.nsw.gov.au/useful-information/energy-retailers.aspx

Floor plan examples and free software www.edrawsoft.com/floorplan.php

Graph paper generator (includes a variety of styles)

http://incompetech.com/graphpaper/

Household budget facts sheet https://www.moneysmart.gov.au/media/344193/household-budgets.pdf

How to read your energy bills and track your consumption

www.transport.wa.gov.au/mediaFiles/AT_LS_P_read_bills_track_consumption.pdf

Insurance https://www.moneysmart.gov.au/managing-my-money/insurance

Kitchen planning and designing www.mitre10.com.au/Kitchens/Plan-And-Design-Your-Kitchen/ www.kitchenplanneronline.com/

Phone bills explained http://optus.custhelp.com/app/answers/detail/a_id/291 www.telstra.com.au/telstrabills/index.htm

Sample energy bill (with links to bill examples from South Australia)

www.sa.gov.au/subject/Water,+energy+and+environment/Energy/Energy+efficiency/Understanding+your+energy+use/Understand+your+energy+bills

Spending habits of Australians by state (infographic)

https://www.moneysmart.gov.au/managing-my-money/budgeting/spending/australian-spending-habits

Spending money https://www.moneysmart.gov.au/managing-my-money/budgeting/spending

Strata levies www.strataman.com.au/levies.html

Strata levies listed in advertisements of properties for sale

www.realestate.com.au/buy www.domain.com.au/

‘Your money’ brochure (includes scenario and sample budget)

https://www.moneysmart.gov.au/media/142419/your-money.pdf

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Focus Study: Mathematics and the Human Body

FSHu1CEC Blood The following online resources could be used to support the teaching and learning of this topic.

Blood pressure http://en.wikipedia.org/wiki/Blood_pressure

Blood pressure chart www.bloodpressureuk.org/BloodPressureandyou/Thebasics/Bloodpressurechart

Blood safety and availability (includes world map of blood donation rates)

www.who.int/mediacentre/factsheets/fs279/en/index.html

Blood type http://en.wikipedia.org/wiki/Blood_type

Blood types (includes blood compatability two-way table)

www.donateblood.com.au/about-blood/types

Heart rate http://en.wikipedia.org/wiki/Heart_rate

Heart rate and exercise (activities)

www.bioedonline.org/resources/files/tso_hc_07_s.pdf www.k8science.org/resources/files/PowerPlay_s/03_P-Pre_s.pdf http://illuminations.nctm.org/LessonDetail.aspx?ID=U74

Inheritance patterns of blood types

www.transfusion.com.au/blood_basics/blood_groups/inheritance_patterns

FSHu2CEC Body measurements The following online resources could be used to support the teaching and learning of this topic.

Biometric data http://biostat.mc.vanderbilt.edu/twiki/bin/view/Main/DataSets?CGISESSID=10713f6d891653ddcbb7ddbdd9cffb79

Body measurements www.who.int/growthref/en/ See note (1) below

CensusAtSchool random sampler www.cas.abs.gov.au/cgi-local/cassampler.pl

Guessing correlations http://istics.net/stat/correlations/

Least-squares regression line (applet)

http://illuminations.nctm.org/LessonDetail.aspx?ID=L491#applet

Linear regression and body measurements (PowerPoint)

www.google.com.au/url?sa=t&rct=j&q=&esrc=s&source=web&cd=3&ved=0CC8QFjAC&url=http%3A%2F%2Fwww.cs.sunysb.edu%2F~mueller%2Fteaching%2FvolumeGraphicsSeminar%2Fch18Reg.ppt&ei=cM-iUKK6LqeUiAemx4CgAg&usg=AFQjCNGmnxCkExONDkCVBCkDNlt0Y76wxw&sig2=8waim-kJSlp289ofMJ1aPg

Regression by eye (applet) www.ruf.rice.edu/~lane/stat_sim/reg_by_eye/index.html

Scatterplot, correlation and line of best fit creator

http://nlvm.usu.edu/en/nav/frames_asid_144_g_4_t_5.html?open=activities&from=category_g_4_t_5.html

Note (1) The World Health Organization (WHO) website provides data on height, weight and body

mass index (BMI) by age.

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For the teacher

Anthropometry (from the Greek anthropos, meaning ‘man’, and metron, meaning ‘measure’) is the measurement of human individuals. Many historical units of length were based on body measurements. The ancient Egyptians used the ‘cubit’ and the ‘palm’ to measure length. A cubit is the distance from the elbow to the tip of the middle finger when extended. A palm is the distance across the knuckles at the base of the fingers when all four fingers are extended. The ancient Egyptians equated one cubit to seven palms.

Students could investigate relationships between any or all of the following body measurements:

• cubit – the distance from the elbow to the tip of the middle finger when extended

• fathom – the distance from fingertip to fingertip when the arms are outstretched (armspan)

• foot – the length of the foot from the back of the heel to the tip of the longest toe

• handspan – the distance from the tip of the outspread little finger to the tip of the outspread thumb

• height – the distance from the floor to the top of the head when standing with bare feet flat on the ground

• inch – the width of the thumb

• navel (‘belly button’) height – the distance of the ‘belly button’ from the floor when standing with bare feet flat on the ground

• pace – the distance covered in one step (obtained by dividing a distance walked, eg 10 metres, by the number of ‘paces’ required to cover that distance)

• palm – the distance across the knuckles at the base of the fingers when all four fingers are extended.

The Metropolitan Museum of Art, Rogers Fund, 1930 (30.4.137) Image © The Metropolitan Museum of Art. Source: www.metmuseum.org.

Teachers should be aware of the following:

• Taking body measurements may be a sensitive issue for some students. Teachers should adjust activities accordingly.

• The ‘trendline’ created on charts in Microsoft Excel is the least-squares line of best fit.

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For the student

What relationships can be found between various body measurements? Many historical units of length were based on body measurements, eg the ancient Egyptians used the ‘cubit’ and the ‘palm’ to measure length. Some body measurements to investigate include:

cubit – the distance from the elbow to the tip of the middle finger when extended fathom – the distance from fingertip to fingertip when the arms are outstretched (armspan)

foot – the length of the foot from the back of the heel to the tip of the longest toe handspan – the distance from the tip of the outspread little finger to the tip of the outspread thumb

height – the distance from the floor to the top of the head when standing with bare feet flat on the ground

inch – the width of the thumb navel (‘belly button’) height – the distance of the ‘belly button’ from the floor when standing with bare feet flat

on the ground pace – the distance covered in one step (obtained by dividing a distance walked,

eg 10 metres, by the number of ‘paces’ required to cover that distance) palm – the distance across the knuckles at the base of the fingers when all four fingers

are extended fathom handspan inch palm

cubit

In the following activities, pairs of body measurements will be investigated to determine:

• whether or not there is a correlation between a selected pair of body measurements

• the strength of the correlation, if one exists

• an algebraic model for the relationship between a selected pair of body measurements (ie a line of best fit).

(1) Work in small groups using a tape measure to obtain the body measurements listed in the table below for each person in the group. Record all measurements in the table in centimetres to the nearest centimetre, with the exception of the inch and palm, which should be measured in centimetres to the nearest millimetre.

Body measurement (cm) Student A Student B Student C Student D

Sex (please circle M or F) M / F M / F M / F M / F

Cubit

Fathom (armspan)

Foot length

Handspan

Height

Inch (to nearest 0.1 cm)

Navel (‘belly button’) height

Palm (to nearest 0.1 cm)

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(2) Enter the data for each student into a spreadsheet (see below) so that all students in the class have access to the body measurements of each member of the class. If students of other classes are also participating in this investigation, combine the sets of data into one spreadsheet. It is preferable to have as many data values as possible.

Spreadsheet for use by students Spreadsheet for use by teachers/students (includes hidden formulae) Spreadsheet for use by teachers (includes formulae displayed)

(3) Use the appropriate spreadsheet function to calculate the mean of each set of body measurements.

Note: In Microsoft Excel, the mean function is =AVERAGE(array).

(4) Use the appropriate spreadsheet function to calculate the sample standard deviation of each set of body measurements.

Note: In Microsoft Excel, the sample standard deviation function is =STDEV(array). The population standard deviation is =STDEVP(array).

(5) Use the appropriate spreadsheet function to calculate the correlation coefficient for each pair of body measurements. Correlation coefficients can be arranged in a table for easy reference (see below).

Note: In Microsoft Excel, the correlation function is =CORREL(array1,array2). Try entering the formula in each of the cells in the column for ‘Cubit’ using ‘absolute cell reference’ (by pressing F4 after typing the cell reference or by manually entering $ symbols), then use ‘fill across’ to save time, eg the formulae in cells C38, C39 and C40 above would be:

=CORREL($C$3:$C$32,C3:C32) and =CORREL($D$3:$D$32,C3:C32) and =CORREL($E$3:$E$32,C3:C32) respectively.

(a) Are any values in the table identical? Why? Describe any patterns that you notice in the table.

(b) Which pair of body measurements is most strongly correlated?

(c) Which pair of body measurements is least strongly correlated?

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(6) Use the chart function of the spreadsheet application to create scatterplots that compare each pair of body measurements, eg cubit with handspan, foot length with palm size, height with foot length. Ensure that all scatterplots are labelled appropriately (heading and axes).

Note: To select cells in columns that are not adjacent, hold down the CTRL key while selecting cells.

(7) Match the appropriate correlation coefficient to each scatterplot.

Describe the relationship between the spread of data points on the scatterplot and the value of the correlation coefficient for the pair of body measurements that is:

(a) most strongly correlated

(b) least strongly correlated.

(8) For each pair of body measurements that has a correlation coefficient greater than 0.5, use the appropriate spreadsheet function to construct and display the equation of the least-squares line of best fit on the scatterplot.

Note: The line of best fit may be known as the ‘trendline (linear)’ or similar. Non-linear trendlines are beyond the scope of this course.

(9) Use the equation of the appropriate line of best fit to calculate expected values for particular body measurements given another body measurement, eg calculate the expected handspan for a person of height 154 cm.

(10) Extension: The golden ratio

Research the ‘golden ratio’. What is it? Where does it occur in the natural environment? Where has it been used in the built environment? Where has it been used in the arts? Where is it found in the ‘ideal’ human body?

These online resources may be helpful:

• Pythagoras: How to Measure Beauty − The Human Face (video, BBC Worldwide), www.youtube.com/watch?v=mVVroi8q0Y0

• Golden Ratio in Human Body (Golden Mean in Mankind) (video), www.youtube.com/watch?v=085KSyQVb-U

• the golden ratio, http://en.wikipedia.org/wiki/Golden_ratio

• the golden ratio in the human body, http://merlib.org/node/1377.

FSHu3CEC Lung capacity The following online resources could be used to support the teaching and learning of this topic.

Lung volumes http://en.wikipedia.org/wiki/Lung_volumes

Measuring lung capacity (includes the use of balloons)

www.biologycorner.com/worksheets/lungcapacity.html

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Focus Study: Mathematics and Personal Resource Usage

FSPe1CEC Water usage and collection The following online resources could be used to support the teaching and learning of this topic.

‘Let’s have a school rainwater tank!’ (student activity)

www.rouswater.nsw.gov.au/cmst/rw010/res.asp?id=745

Rainfall data (Hunter Region catchments) www.hunterwater.com.au/Water-and-Sewer/Water-Supply/Rainfall-Data.aspx

Rainfall in a catchment calculator www.calctool.org/CALC/other/default/rainfall

Rainfall tables www.bom.gov.au/jsp/watl/rainfall/pme.jsp

Water (Australian Government) www.environment.gov.au/water/index.html

Water (NSW Government) www.nsw.gov.au/water

Water availability and usage (includes data for international locations)

www.publish.csiro.au/?act=view_file&file_id=9780643103283_Chapter_1.pdf

Water education recources www.environment.gov.au/water/education/index.html

Water harvesting calculations http://oasisdesign.net/water/rainharvesting/drylandsbook/Appendix3Calculations.pdf

Water saving ideas www.waterforgood.sa.gov.au/using-water/saving-water/

Water suppliers in NSW www.ewon.com.au/index.cfm/suppliers/suppliers-in-nsw/water-suppliers/

Water tanks www.watertankfactory.com.au/Water-Tank-Range.php http://tankworld.com.au/

Water usage and conservation www.abs.gov.au/AUSSTATS/abs@.nsf/DetailsPage/4602.0.55.003Mar 2010?OpenDocument www.abs.gov.au/ausstats/abs@.nsf/mf/4602.0.55.003

Water usage calculator (generate a personal household water usage report)

www.sawater.com.au/interactivehouse/

Water usage statistical indicators (state and territory)

www.abs.gov.au/ausstats/abs@.nsf/Lookup/by+Subject/1367.0~2011~Main+Features~Water+Use~6.37

For the teacher Find information about your local water supplier. It is also worth exploring suppliers from other areas, as they may have more data and information that you could use with your students.

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FSPe2CEC Electricity The following online resources could be used to support the teaching and learning of this topic.

Energy ratings for different appliances http://reg.energyrating.gov.au/comparator/product_types/

Energy Savings Scheme (ESS) www.ess.nsw.gov.au/Home

Energy tools and calculators www.aglsmarterliving.com.au/energy-efficiency-advice/energy-tools/ www.redenergy.com.au/billbenchmark/

Energy usage www.yourhome.gov.au/technical/fs61.html

kW and kWh explained www.energylens.com/articles/kw-and-kwh

Reading electricity meters www.actewagl.com.au/Help-and-advice/How-to-read-your-meters.aspx

Sample energy rating labels www.energyrating.gov.au/programs/e3-program/energy-rating-labelling/obtain/

Tips on buying energy-efficient appliances www.livinggreener.gov.au/energy/energy-efficient-appliances

Understanding demand and consumption www.think-energy.net/KWvsKWH.htm

Your power usage www.savepower.nsw.gov.au/get-the-facts/your-power-usage.aspx

FSPe3CEC Sustainability and energy-efficient housing The following online resources could be used to support the teaching and learning of this topic.

BASIX – Building Sustainability Index www.basix.nsw.gov.au/information/index.jsp

Building an energy-efficient home www.environment.nsw.gov.au/energy/home.htm

Designing an energy-efficient home www.sa.gov.au/subject/Water,+energy+and+environment/Energy/Energy+efficiency/Home+energy+efficiency/Designing+an+energy+efficient+home

Ecological footprint (measure your impact on the environment)

www.epa.vic.gov.au/ecologicalfootprint/globalfootprint/index.asp

Energy-efficient home renovator www.synergy.net.au/at_home/energy_efficient_renovator.xhtml

Energy saving tips interactive www.originenergy.com.au/calculator

Home energy audits (includes heating and cooling, lighting, and water heating)

www.sa.gov.au/subject/Water,+energy+and+environment/Energy/Energy+efficiency/Understanding+your+energy+use/Check+your+home+energy+use+-+home+energy+audit/Do+your+own+home+energy+audit

Interactive energy-efficient house www.sa.gov.au/subject/Water,+energy+and+environment/Energy/Energy+efficiency/Home+energy+efficiency/Interactive+energy+efficient+house

Water tanks and BASIX package www.rainwatertanksdirect.com.au/

Your Home Technical Manual (includes information about sustainable design and practices)

www.yourhome.gov.au/technical/index.html (select ‘Material Use’, ‘Energy Use’, ‘Water Use’, etc)

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For the student (FSPe1CEC and FSPe3CEC)

How much could you save on your water bill with a rainwater tank? (1) Use an online calculator (eg www.sawater.com.au/interactivehouse/) to estimate and record

the number of litres of water used by your household in one year.

Note: You may need the assistance of other household members to complete this part of the activity.

(2) Research rainfall data for your region to obtain and record the mean annual rainfall.

(3) Use a real estate website to find a suitable house to represent your household in your region. Determine the ‘footprint’ of the ‘drip line’ of the house using the dimensions shown on the floor plan of the house, ie the roof area available for water harvesting.

Note: The pitch/slope of the roof does not matter, as it is the footprint of the drip line of the roof that determines how much water can be harvested.

(4) Roofs typically lose 5–20% of the rainwater that falls on them. The reasons for this include the type of roof surface, light rain events that may be completely absorbed by the roof surface, and water that remains on the roof after a rain event (and is subsequently lost through evaporation). What is a conservative estimate of the percentage of rain that you can expect to collect from the roof of the selected house (ie a conservative estimate of the ‘runoff coefficient’)?

(5) The net annual volume of rainwater that can be collected from a roof can be calculated using the formula V = A × R × C, where V is the volume of water collected in cubic metres, A is the area of the collection surface in square metres, R is the mean annual rainfall expressed in metres, and C is the runoff coefficient expressed as a decimal. Use the formula to calculate a conservative net annual volume of water that can be collected from the roof of your selected house.

(6) Determine whether or not the estimated net annual volume of rainwater to be collected from the roof of your selected house is enough to meet your household’s annual water usage. How much extra water is needed, or how much excess water is left over?

(7) Research and calculate the cost of water for your annual household usage if you do not use any collected rainwater. On average, how much could you save on water bills in one year by using rainwater collected from the roof of your selected house?

(8) Using online sources, research rainwater tanks suitable for household use and select a tank with a capacity large enough to hold your estimated annual collected rainwater.

(a) Record the manufacturer’s name and URL, and the name, dimensions, capacity and cost of the tank as stated on the website.

(b) Draw a diagram of the tank and clearly label it with the stated dimensions.

(c) Calculate the volume of the tank using the stated dimensions and find its capacity in litres based on these calculations.

(d) Compare the capacity that you calculated for the tank with the capacity stated by the manufacturer and explain why the calculated capacity might be different from that stated.

(e) Determine the number of years required to completely recoup the cost of buying the tank through savings made on your annual water bill if water pricing remains unchanged (include recouping installation costs, if these can be estimated).