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Mathscape 9 Teaching Program

Stage 5

MATHSCAPE 9

Term Chapter Time

1 1. Rational numbers 2 weeks / 8 hrs

2. Algebra 2 weeks / 8 hrs

3. Consumer arithmetic 2 weeks / 8 hrs

4. Equations, inequations and formulae 2 weeks / 8 hrs

2 5. Measurement 3 weeks / 12 hrs

6. Data representation and analysis 2 weeks / 8 hrs

7. Probability 1 week / 4 hrs

8. Indices 3 weeks / 12 hrs

4 9. Geometry 2 weeks / 8 hrs

10. The linear function 2 weeks / 8 hrs

11. Trigonometry 3 weeks / 12 hrs

12. Co-ordinate geometry 2 weeks / 8 hrs

Published by Macmillan Education Australia. © Macmillan Education Australia 2004.


Chapter 1. Rational numbersSubstrandRational numbers

Text referencesMathscape 9Chapter 1. Rational Numbers(pages 1–25)

CD referenceSignificant figuresRecurring decimalsRates

Duration2 weeks / 8 hours

Key ideasRound numbers to a specified number of significant figures.Express recurring decimals as fractions.Convert rates from one set of units to another.

OutcomesNS5.2.1 (page 67): Rounds decimals to a specified number of significant figures, expresses recurring decimals in fraction form and converts rates from one set of units to another.

Working mathematicallyStudents learn to recognise that calculators show approximations to recurring decimals e.g. displayed as 0.666667 (Communicating)

justify that (Reasoning)

decide on an appropriate level of accuracy for results of calculations (Applying Strategies) assess the effect of truncating or rounding during calculations on the accuracy of the results (Reasoning) appreciate the importance of the number of significant figures in a given measurement (Communicating) use an appropriate level of accuracy for a given situation or problem solution (Applying Strategies) solve problems involving rates (Applying Strategies)

Knowledge and skillsStudents learn about identifying significant figures rounding numbers to a specified number of significant figures using the language of estimation appropriately, including:

rounding approximate level of accuracy

using symbols for approximation e.g.

Teaching, learning and assessment TRY THIS

Fermi Problem (page 10): Estimation problem solvingDesert Walk (page 15): Problem solvingPassing Trains (page 20): Travel graph problem

FOCUS ON WORKING MATHEMATICALLYArt, Magic Squares and Mathematics (page 20): If you would like to learn how to make a magic square start with John Webb's article in the June 2000 journal of nrich, the mathematics enrichment page of the Millenium Mathematics Project based at the University of Cambridge http://nrich.maths.org/mathsf/journalf/jun00/art2/ There are many sites


http://nrich.maths.org/mathsf/journalf/jun00/art2/


determining the effect of truncating or rounding during calculations on the accuracy of the results

writing recurring decimals in fraction form using calculator and non-calculator methods e.g. , ,

converting rates from one set of units to another e.g. km/h to m/s, interest rate of 6% per annum is 0.5% per month

which will provide instructions but this is a good one to begin. From January 2004 the nrich web home page can be found at http://nrich.maths.org/public/viewer.php?obj_id=1376 and the home page of the project at http://mmp.maths.org/index.html The web page http://www-history.mcs.st-and.ac.uk/history/Mathematicians/Durer.html will get you straight to Albrecht Durer. You can scroll through the text to get a look at his engraving Melancholia which is highlighted in blue. From here you can go to the main index and look for "magic squares" under topics, or check out Leonhard Euler under mathematicians.

EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING (page 23)

CHAPTER REVIEW (page 24) a collection of problems to revise the chapter.

TechnologySignificant Figures: this spreadsheet in designed to round off a given number to a desired number of significant figures. To be used with the text.Recurring Decimals: this spreadsheet converts recurring decimals to fractions.Rates: this spreadsheet deals with rates and ratios in units.


http://www-history.mcs.st-and.ac.uk/history/Mathematicians/Durer.html

http://www-history.mcs.st-and.ac.uk/history/Mathematicians/Durer.html

http://mmp.maths.org/index.html

http://nrich.maths.org/public/viewer.php?obj_id=1376


Chapter 2. AlgebraSubstrandAlgebraic techniques

Text referencesMathscape 9Chapter 2. Algebra(pages 26–66)

CD referenceSimplify (with fractional indices)ExpandRailway tickets


Key ideasSimplify, expand and factorise algebraic expressions including those involving fractions or with negative and/or fractional indices.

OutcomesPAS5.2.1 (page 88): Simplifies, expands and factorises algebraic expressions involving fractions and negative and fractional indices.

Working mathematicallyStudents learn to describe relationships between the algebraic symbol system and number properties (Reflecting, Communicating) link algebra with generalised arithmetic

e.g. use the distributive property of multiplication over addition to determine that (Reflecting) determine and justify whether a simplified expression is correct by substituting numbers for pronumerals (Applying Strategies, Reasoning) generate a variety of equivalent expressions that represent a particular situation or problem (Applying Strategies) check expansions and factorisations by performing the reverse process (Reasoning) interpret statements involving algebraic symbols in other contexts e.g. spreadsheets (Communicating) explain why an algebraic expansion or factorisation is incorrect e.g. Why is the following incorrect? (Reasoning,

Communicating)

Knowledge and skillsStudents learn about simplifying algebraic expressions involving fractions, such as

expanding, by removing grouping symbols, and collecting like terms where possible, algebraic expressions such as


Flags (page 35): Algebraic problem solvingOverhanging the overhang (page 42): PracticalRailway Tickets (page 58): Complete a table and find a rule

FOCUS ON WORKING MATHEMATICALLYParty Magic (page 59): Teachers may wish to down load the Party Magic with Algebra worksheet in the technology folder for chapter 2 Algebra. This worksheet explores the algebraic structure of the games using technology. The web link http://atschool.eduweb.co.uk/ufa10/tricks.htm at


http://atschool.eduweb.co.uk/ufa10/tricks.htm


factorising, by determining common factors, algebraic expressions such as

Birmingham in England has great resources for students and teachers.The web page http://www.umassmed.edu/bsrc/tricks.cfm has good links and lots of activites to show that maths really can be fun. For the addicted to fun and games check out Martin Gardner's books at http://thinks.com/books/gardner.htm



TechnologySimplify (with fractional indices): algebraic program that simplifies algebraic terms. To be used with the worksheets. Also to be used with the Focus on Working mathematically section. Expand: this program will expand a given algebraic expression.Railway Tickets: worksheet to use with the “Try This” problem on page 58.


http://thinks.com/books/gardner.htm

http://www.umassmed.edu/bsrc/tricks.cfm


Chapter 3. Consumer arithmeticSubstrandConsumer arithmetic

Text referencesMathscape 9Chapter 3. Consumer Arithmetic(pages 67–107)

CD referenceMoney


Key ideasSolve simple consumer problems including those involving earning and spending money.Calculate simple interest and find compound interest using a calculator and tables of values.Use compound interest formula.Solve consumer arithmetic problems involving compound interest, depreciation and successive discounts.

OutcomesNS5.1.2 (page 70): Solves consumer arithmetic problems involving earning and spending money.NS5.2.2 (page 71): Solves Consumer arithmetic problems involving compound interest, depreciation, and successive discounts.

Working mathematicallyStudents learn to read and interpret pay slips from part-time jobs when questioning the details of their own employment (Questioning, Communicating) prepare a budget for a given income, considering such expenses as rent, food, transport etc

(Applying Strategies) interpret the different ways of indicating wages or salary in newspaper ‘positions vacant’ advertisements e.g. $20K (Communicating) compare employment conditions for different careers where information is gathered from a variety of mediums including the Internet

e.g. employment rates, payment (Applying Strategies) explain why, for example, a discount of 10% following a discount of 15% is not the same as a discount of 25% (Applying Strategies, Communicating,

Reasoning)

Knowledge and skillsStudents learn about calculating earnings for various time periods from different sources, including:

- wage- salary- commission- piecework- overtime


Sue’s Boutique (page 72): Problem SolvingTelephone Charges (page 92): Problem SolvingProgressive Discounting (page 98): Investigation

FOCUS ON WORKING MATHEMATICALLYSydney Market prices in 1831 (page 102): The purpose of the learning activities is for students to think about the cost of living in Australia today



- bonuses- holiday loadings- interest on investments

calculating income earned in casual and part-time jobs, considering agreed rates and special rates for Sundays and public holidays

calculating weekly, fortnightly, monthly and yearly incomes calculating net earnings considering deductions such as taxation and

superannuation calculating a ‘best buy’ calculating the result of successive discounts

using market prices in 1831 as a starting point. As an extension students are given opportunity to explore inflation and how the consumer price index (CPI) is calculated. An invitation to a member of the Economics staff to your class could be stimulating. Teachers should note that the further apart the years being compared, the less valid it is to use the relative prices of goods in those years to measure the standard of living. This point is well made in the article by Nell Ingalls published on the web site http://www.sls.lib.il.us/reference/por/features/98/money.html. This is a useful source of information on the value of money. A good summary of how the CPI is calculated in Australia can be found at http://www.aph.gov.au/library/pubs/mesi/features/cpi.htm



TechnologyMoney: series of worksheets to use with spreadsheets to explore Commission, Net Income, Piece Work, Salaries, Wages and a Weekly Budget.


http://www.aph.gov.au/library/pubs/mesi/features/cpi.htm

http://www.sls.lib.il.us/reference/por/features/98/money.html


Chapter 4. Equations, inequations and formulaeSubstrandAlgebraic techniques

Text referencesMathscape 9Chapter 4: Equations, inequations and formulae (pages 108–40)

CD referenceEvaluatingFloodlighting


Key ideasSolve linear and simple quadratic equations of the form Solve linear inequalities

OutcomesPAS5.2.2 (page 90): Solves linear and simple quadratic equations, solves linear inequalities and solves simultaneous equations using graphical and analytical methods.



Working mathematicallyStudents learn to compare and contrast different methods of solving linear equations and justify a choice for a particular case (Applying Strategies, Reasoning) use a number of strategies to solve unfamiliar problems, including:

- using a table- drawing a diagram- looking for patterns- working backwards- simplifying the problem and- trial and error (Applying Strategies, Communicating)

solve non-routine problems using algebraic methods (Communicating, Applying Strategies) explain why a particular value could not be a solution to an equation (Applying Strategies, Communicating, Reasoning) create equations to solve a variety of problems and check solutions (Communicating, Applying Strategies, Reasoning) write formulae for spreadsheets (Applying Strategies, Communicating) solve and interpret solutions to equations arising from substitution into formulae used in other strands of the syllabus and in other subjects. Formulae such as the

following could be used:

(Applying Strategies, Communicating, Reflecting) explain why quadratic equations could be expected to have two solutions (Communicating, Reasoning) justify a range of solutions to an inequality (Applying Strategies, Communicating, Reasoning)

Knowledge and skillsStudents learn aboutLinear and Quadratic Equations solving linear equations such as


A Prince and a King (page 129): Two Ancient ProblemsArm Strength (page 132): Formulae InvestigationFloodlighting by formula (page 136): Formulae Investigation



solving word problems that result in equations exploring the number of solutions that satisfy simple quadratic equations of the

form solving simple quadratic equations of the form solving equations arising from substitution into formulae

Linear Inequalities solving inequalities such as

FOCUS ON WORKING MATHEMATICALLYBushfires (page 137): Teachers may wish to use a spreadsheet to evaluate F given C and vice versa. Go to the Evaluating the subject.xls spreadsheet in the technology folder for Chapter 4. There is also a useful worksheet. Extension students could discuss whether F = 9C/5 + 32 is a formula or an equation and what constitutes the difference -- see page 136 on Floodlighting for example. For newspaper reports of the fires try the Sydney Morning herald web site http://www.smh.com.au/. If you type 'Sydney bushfires' into a search engine you will get a range of options.http://www.gi.alaska.edu/ScienceForum/ASF13/1317.html will give you a short account how the two men Daniel Fahrenheit and Anders Celcius constructed their scales. This will be very useful link with the study of science.



TechnologyEvaluating: students analyse a spreadsheet and then design their own.Floodlighting: activity to complement the “Try This” problem on page 136.


http://www.gi.alaska.edu/ScienceForum/ASF13/1317.html

http://www.smh.com.au/


Chapter 5. MeasurementSubstrandAlgebraic techniques

Text referencesMathscape 9Chapter 5:.Measurement(pages 141–205)

CD referencePerigalMeasuring plane shapesCircle measuring


Key ideasDevelop formulae and use to find the area of rhombuses, trapeziums and kites.Find the area and perimeter of simple composite figures consisting of two shapes including quadrants and semicircles. Find area and perimeter of more complex composite figures.

OutcomesMS5.1.1 (page 126): Use formulae to calculate the area of quadrilaterals and find areas and perimeters of simple composite figures.MS5.2.1 (page 127): Find areas and perimeters of composite figures.

Working mathematicallyStudents learn to identify the perpendicular height of a trapezium in different orientations (Communicating) select and use the appropriate formula to calculate the area of a quadrilateral (Applying Strategies) dissect composite shapes into simpler shapes (Applying Strategies) solve practical problems involving area of quadrilaterals and simple composite figures (Applying Strategies) solve problems involving perimeter and area of composite shapes (Applying Strategies) calculate the area of an annulus (Applying Strategies) apply formulae and properties of geometrical shapes to find perimeters and areas e.g. find the perimeter of a rhombus given the lengths of the diagonals

(Applying Strategies) identify different possible dissections for a given composite figure and select an appropriate dissection to facilitate calculation of the area

(Applying Strategies, Reasoning)

Knowledge and skillsStudents learn about developing and using formulae to find the area of quadrilaterals:

- for a kite or rhombus, Area where x and y are the lengths of the diagonals;

- for a trapezium, Area where h is the perpendicular height and a and b the lengths of the parallel sides

calculating the area of simple composite figures consisting of two shapes


Bags of potatoes (page 147): Problem SolvingOverseas Call (page 154): Problem SolvingPythagorean Proof by Perigal (page 160): ProofThe box and the wall (page 163): Problem SolvingCommand Module (page 174): Investigation of Apollo 11The area of a circle (page 185): Archimedes methodArea (page 195): Challenge Problem

FOCUS ON WORKING MATHEMATICALLY



including quadrants and semicircles calculating the perimeter of simple composite figures consisting of two shapes

including quadrants and semicircles calculating the area and perimeter of sectors calculating the perimeter and area of composite figures by dissection into

triangles, special quadrilaterals, semicircles and sectors

The Melbourne Cup (page 198): These activities focus on units of measurement linked to the Melbourne cup. The history of the cup provides insight into the dramatic changes in the Australian way of life since the race began in 1861. Students also explore the origin of the words used to describe the units. The web page http://www.unc.edu/~rowlett/units/ is a dictionary of unusual units you will find fascinating. If you type in 'Melbourne cup' into a search engine, you will have lots of choice. Try the VRC web page http://home.vicnet.net.au/~basiced3/cup/history.html The web page http://www.equine-world.co.uk/about_horses/height.htm will show you nice diagram on the way the heights of horses are measured.



TechnologyPerigal: Cabri Geometry interactive worksheet on the Pythagorean proof by Perigal.Measuring Plane Shapes: this file contains hyperlinks to a number of interactive geometric diagrams.Circle Measuring: a set of Cabri Geometry interactive worksheets that are used for students to explore the parts and use of circles.


http://www.equine-world.co.uk/about_horses/height.htm

http://home.vicnet.net.au/~basiced3/cup/history.html

http://www.unc.edu/~rowlett/units/


Chapter 6. Data representation and analysisSubstrandData representation and analysis

Text referencesMathscape 9Chapter 6. Data Representation and Analysis (pages 206–53)

CD referenceData analysisCumulative analysis


Key ideasConstruct frequency tables for grouped data.Find mean and modal class for grouped data.Determine cumulative frequency.Find median using a cumulative frequency table or polygon

OutcomesDS5.1.1 (page 116): Groups data to aid analysis and constructs frequency and cumulative frequency tables and graphs.

Working mathematicallyStudents learn to construct frequency tables and graphs from data obtained from different sources (e.g. the Internet) and discuss ethical issues that may arise from the data

(Applying Strategies, Communicating, Reflecting) read and interpret information from a cumulative frequency table or graph (Communicating) compare the effects of different ways of grouping the same data (Reasoning) use spreadsheets, databases, statistics packages, or other technology, to analyse collected data, present graphical displays, and discuss ethical issues that may

arise from the data (Applying Strategies, Communicating, Reflecting)

Knowledge and skillsStudents learn about constructing a cumulative frequency table for ungrouped data constructing a cumulative frequency histogram and polygon (ogive) using a cumulative frequency polygon to find the median grouping data into class intervals constructing a frequency table for grouped data constructing a histogram for grouped data finding the mean using the class centre finding the modal class


The English Language (page 232): InvestigationEarthquakes (page 246): Can we predict the number of Earthquakes there will be in a year?

FOCUS ON WORKING MATHEMATICALLYWorld Health (page 246): This investigation provides an opportunity for students to analyse two indicators of world public health and to apply their skills in Working mathematically. The objective is to show how statistical evidence can play a role in arguing a case for the development of programs to support global health. There is an excellent opportunity for class discussion about the sort of data governments need in order to make sensible policy decisions for global health.



A good international web site is http://www.globalhealth.gov/worldhealthstatistics.shtml The frequently asked questions page at http://www.globalhealth.gov/faq.shtml provides useful background information for teachers



TechnologyData Analysis: students Analyse data with the help of a spreadsheet.Cumulative Analysis: students use the spreadsheet to calculate the median using the cumulative frequency


http://www.globalhealth.gov/faq.shtml

http://www.globalhealth.gov/worldhealthstatistics.shtml


Chapter 7. ProbabilitySubstrandProbability

Text referenceMathscape 9Chapter 8. Probability (pages 254–80)

CD referenceProbabilityCraps simulationWeighted dice

Duration1 week / 4 hours

Key ideasDetermine relative frequencies to estimate probabilities.Determine theoretical probabilities.

OutcomesNS5.1.3 (page 75): Determines relative frequencies and theoretical probabilities.

Working mathematicallyStudents learn to recognise and explain differences between relative frequency and theoretical probability in a simple experiment (Communicating, Reasoning) apply relative frequency to predict future experimental outcomes (Applying Strategies) design a device to produce a specified relative frequency e.g. a four-coloured circular spinner (Applying Strategies) recognise that probability estimates become more stable as the number of trials increases (Reasoning) recognise randomness in chance situations (Communicating) apply the formula for calculating probabilities to problems related to card, dice and other games (Applying Strategies)

Knowledge and skillsStudents learn about repeating an experiment a number of times to determine the relative frequency of

an event estimating the probability of an event from experimental data using relative

frequencies expressing the probability of an event A given a finite number of equally likely

outcomes as

=

where n is the total number of outcomes in the sample space using the formula to calculate probabilities for simple events simulating probability experiments using random number generators


Two-Up (page 265): ExperimentThe game of Craps (page 270): SimulationWinning Chances (page 274): Problem Solving

FOCUS ON WORKING MATHEMATICALLYGetting through traffic lights (page 275): This activity is designed to introduce students to the idea of a simulation. It is designed for all students to enjoy. Teachers should carry out the simulation first using the technology they wish to use in class.The Maths Online web site at http://www.mathsonline.co.uk/nonmembers/resource/prob/ is a great help to teachers looking for lesson plans to simulate real life probability problems. Includes on line flash movies which will draw graphs directly from your input.


http://www.mathsonline.co.uk/nonmembers/resource/prob/


For a good reference text with a CD ROM to simulate probability problems using a graphics calculator try Winter MJ and Carlson RJ (2001) Probability Simulations, Key Curriculum Press, Emeryville, California. Barry Kissane's web page http://wwwstaff.murdoch.edu.au/%7Ekissane/graphicscalcs.htm is invaluable for CASIO users. The Open University Centre for Mathematics Education at http://mcs.open.ac.uk/cme/TIcourses/timain.html has some excellent courses for teachers with little experience in using a TI graphics calculator.



TechnologyProbability: the spreadsheet simulates the drawing of different coloured balls from a bag with replacement.Craps Simulation: this spreadsheet explores the probabilities of winning and losing a game of craps.Weighted Dice: dice simulation spreadsheet.


http://mcs.open.ac.uk/cme/TIcourses/timain.html

http://wwwstaff.murdoch.edu.au/~kissane/graphicscalcs.htm


Chapter 8. IndicesSubstrandsRational numbersAlgebraic techniques

Text referencesMathscape 9Chapter 8. Indices (pages 281–312)

CD referenceSimplify (with fractional indices)Expand


Key ideasDefine and use zero index and negative integral indices.Develop the index laws arithmetically.Use index notation for square and cube roots.Express numbers in scientific notation (positive and negative powers of 10)Apply the index laws to simplify algebraic expressions (positive integral indices only).Simplify, expand and factorise algebraic expressions including those involving fractions or with negative and/or fractional indices.

OutcomesNS5.1.1 (page 65): Applies index laws to simplify and evaluate arithmetic expressions and uses scientific notation to write large and small numbers.PAS5.1.1 (page 87): Applies the index laws to simplify algebraic expressions.PAS5.2.1 (page 88): Simplifies, expands and factorises algebraic expressions involving fractions and negative and fractional indices.

Working mathematicallyStudents learn to solve numerical problems involving indices (Applying Strategies) explain the incorrect use of index laws e.g. why (Communicating, Reasoning)

verify the index laws by using a calculator e.g. to compare the values of , and 5 (Reasoning)

communicate and interpret technical information using scientific notation (Communicating) explain the difference between numerical expressions such as and , particularly with reference to calculator displays (Communicating, Reasoning) solve problems involving scientific notation (Applying Strategies) verify the index laws using a calculator e.g. use a calculator to compare the values of and (Reasoning)

explain why (Applying Strategies, Reasoning, Communicating) link use of indices in Number with use of indices in Algebra (Reflecting) explain why a particular algebraic sentence is incorrect e.g. explain why is incorrect (Communicating, Reasoning)

examine and discuss the difference between expressions such asPublished by Macmillan Education Australia. © Macmillan Education Australia 2004.


and by substituting values for a (Reasoning, Applying Strategies, Communicating) explain why finding the square root of an expression is the same as raising the expression to the power of a half (Communicating, Reasoning) state whether particular equivalences are true or false and give reasons

e.g. Are the following true or false? Why?

(Applying Strategies, Reasoning, Communicating) explain the difference between particular pairs of algebraic expressions, such as and (Reasoning, Communicating)

Knowledge and skillsStudents learn about describing numbers written in index form using terms such as base, power, index,

exponent evaluating numbers expressed as powers of positive whole numbers establishing the meaning of the zero index and negative indices e.g. by patterns

9 3 1

writing reciprocals of powers using negative indices e.g.

translating numbers to index form (integral indices) and vice versa developing index laws arithmetically by expressing each term in expanded form

e.g.

using index laws to simplify expressions using index laws to define fractional indices for square and cube roots


Power Pulse Graphs (page 283): InvestigationSmallest to Largest (page 295): Problem SolvingDigit Patterns (page 300): Investigation

FOCUS ON WORKING MATHEMATICALLYMathematics is at the heart of Science (page 308): The Powers of 10 web site http://www.powersof10.com/ should be explored before starting this Working mathematically activity. There are excellent pictures and ideas for creating absorbing lessons. The learning activities are suitable for students working in pairs. Calculators are recommended. In particular try the patterns section at http://www.powersof10.com/powers/patterns/patterns.html The ABC web site http://www.abc.net.au/science has a wealth of ideas to enable students to see how mathematics lies at the heart of science. The Dr Karl page has a live Q & A opportunity. The class could formulate a question, send it in and listen to the answer on radio or online. There is also a news page which provides great ideas for lesson starters. Teachers are encouraged to liaise with science staff for further information and to invite them to the lesson.


CHAPTER REVIEW (page 311) a collection of problems to revise the Published by Macmillan Education Australia. © Macmillan Education Australia 2004.

http://www.abc.net.au/science

http://www.powersof10.com/powers/patterns/patterns.html

http://www.powersof10.com/


e.g. and , hence

writing square roots and cube roots in index form e.g. recognising the need for a notation to express very large or very small numbers expressing numbers in scientific notation entering and reading scientific notation on a calculator using index laws to make order of magnitude checks for numbers in scientific

notation e.g. converting numbers expressed in scientific notation to decimal form ordering numbers expressed in scientific notation using the index laws previously established for numbers to develop the index laws

in algebraic form

e.g.

establishing that using the index laws e.g. and

simplifying algebraic expressions that include index notatione.g.

applying the index laws to simplify expressions involving pronumerals

establishing that using index laws to assist with the definition of the fractional index for square root

given

chapter.



and

then

using index laws to assist with the definition of the fractional index for cube root using index notation and the index laws to establish that

, , , …

applying the index laws to simplify algebraic expressions such as

TechnologySimplify (with fractional indices): algebraic program that simplifies algebraic terms. To be used with the worksheet. Also to be used with the Focus on Working mathematically section. Worksheet included.Expand: this program will expand a given algebraic expression.



Chapter 9. GeometrySubstrandProperties of geometric figures

Text referencesMathscape 9Chapter 9. Geometry (pages 313–63)

CD referenceN polygonExterior angleEuler lineDuration

2 weeks / 8 hours

Key ideaEstablish sum of exterior angles result and sum of interior angles result for polygons.

OutcomesSGS5.2.1 (page 157): Develops and applies results related to the angle sum of interior and exterior angles for any convex polygon.

Working mathematicallyStudents learn to express in algebraic terms the interior angle sum of a polygon with n sides e.g. (n–2) 180 (Communicating) find the size of the interior and exterior angles of regular polygons with 5,6,7,8, … sides

(Applying Strategies) solve problems using angle sum of polygon results (Applying Strategies)

Knowledge and skillsStudents learn about applying the result for the interior angle sum of a triangle to find, by dissection,

the interior angle sum of polygons with 4,5,6,7,8, … sides defining the exterior angle of a convex polygon establishing that the sum of the exterior angles of any convex polygon is 360 applying angle sum results to find unknown angles


The badge of the Pythagoreans (page 337): Historical ProblemFive Shapes (page 348): Problem SolvingHow many diagonals in a polygon? (page 353): Investigation

FOCUS ON WORKING MATHEMATICALLYA surprising finding (page 354): In this activity we arrive at Pythagoras' theorem from a cutting and pasting activity with hexagons of equal area. It is designed as a fun activity for all students. However there is a deeper mathematical idea which is really for teachers but could be used as an extension. The 4 hexagons drawn on pages 355–356 each tessellate the plane. See http://www.cut-the-knot.org/pythagoras/index.shtml for details -- read proof 16 and 38 of Pythagoras' theorem.


CHAPTER REVIEW (page 357): A collection of problems to revise the chapter.

TechnologyPublished by Macmillan Education Australia. © Macmillan Education Australia 2004.

http://www.cut-the-knot.org/pythagoras/index.shtml


N Polygon: this geometry program draws regular polygons at speed and displays their diagonals. Explores a curious geometrical pattern that would be time consuming if drawn by hand.Exterior Angle: this learning activity makes use of the exterior angle property of a triangle. Students have the opportunity to apply the reasoning to solve a problem in geometry.Euler Line: the Euler line of a triangle is a line that passes through three special points of a triangle. Investigative exercise.



Chapter 10. The linear functionSubstrandCo-ordinate geometry

Text referencesMathscape 9Chapter 10. The Linear Function(pages 364–99)

CD referenceLine equationIntersecting lines

Duration 2 weeks / 8 hours

Key ideasUse a diagram to determine midpoint, length and gradient of an interval joining two points on the number plane.Graph linear and simple non-linear relationships from equations.

OutcomesPAS5.1.2 (page 97): Determines the midpoint, length and gradient of an interval joining two points on the number plane and graphs linear and simple non-linear relationships from equations.

Working mathematicallyStudents learn to explain the meaning of gradient and how it can be found for a line joining two points (Communicating, Applying Strategies) distinguish between positive and negative gradients from a graph (Communicating) describe horizontal and vertical lines in general terms (Communicating) explain why the x -axis has equation y = 0 (Reasoning, Communicating) explain why the y -axis has equation x = 0 (Reasoning, Communicating) determine the difference between equations of lines that have a negative gradient and those that have a positive gradient (Reasoning) use a graphics calculator and spreadsheet software to graph, compare and describe a range of linear and simple non-linear relationships(Applying Strategies,

Communicating) apply ethical considerations when using hardware and software (Reflecting)

Knowledge and skillsStudents learn about

Midpoint, Length and Gradient using the right-angled triangle drawn between two points on the number plane and

the relationship

to find the gradient of the interval joining two points


Size 8 (page 374): Problem SolvingHanging around (page 383): Problem SolvingLatitude and Temperature (page 389): Investigation

FOCUS ON WORKING MATHEMATICALLYPaper Sizes in the printing industry (page 394): The web link http://www.cl.cam.ac.uk/~mgk25/iso-paper.html is a good overview of the length to breadth relationship of A4 to A3. The web link http://www.twics.com/~eds/paper/papersize.html


http://www.twics.com/~eds/paper/papersize.html

http://www.cl.cam.ac.uk/~mgk25/iso-paper.html


determining whether a line has a positive or negative slope by following the line from left to right – if the line goes up it has a positive slope and if it goes down it has a negative slope

finding the gradient of a straight line from the graph by drawing a right-angled triangle after joining two points on the line

Graphs of Relationships constructing tables of values and using coordinates to graph vertical and

horizontal lines such as

identifying the x - and y -intercepts of graphs identifying the x -axis as the line y = 0 identifying the y -axis as the line x = 0 graphing a variety of linear relationships on the number plane by constructing a

table of values and plotting coordinates using an appropriate scale e.g. graph the following:

determining whether a point lies on a line by substituting into the equation of the line

provides you with more information about international paper sizes. Note that the B series is about half way between two A sizes. It is intended as an alternative to the A sizes, and used primarily for books, posters and wall charts. Note that the ratio of length to breadth for the B series is also √2 : 1 and the ratio of their areas 2 : 1.



TechnologyLine Equation: interactive program with accompanying worksheet.Intersecting Lines: interactive program with accompanying worksheet.



Chapter 11. TrigonometrySubstrandTrigonometry

Text referencesMathscape 9Chapter 11. Trigonometry(pages 400–38)

CD referenceSine CosineSOHCAHTOA


Key ideasUse trigonometry to find sides and angles in right-angled triangles.Solve problems involving angles of elevation and angles of depression from diagrams

OutcomesMS5.1.2 (page 139): Applies trigonometry to solve problems (diagrams given) including those involving angles of elevation and depression.

Working mathematicallyStudents learn to label sides of right-angled triangles in different orientations in relation to a given angle (Applying Strategies, Communicating) explain why the ratio of matching sides in similar right-angle triangles is constant for equal angles (Communicating, Reasoning) solve problems in practical situations involving right-angled triangles e.g. finding the pitch of a roof (Applying Strategies) interpret diagrams in questions involving angles of elevation and depression (Communicating) relate the tangent ratio to gradient of a line (Reflecting)

Knowledge and skillsStudents learn aboutTrigonometric Ratios of Acute Angles identifying the hypotenuse, adjacent and opposite sides with respect to a given

angle in a right-angled triangle in any orientation labelling the side lengths of a right-angled triangle in relation to a given angle e.g.

the side c is opposite angle C recognising that the ratio of matching sides in similar right-angled triangles is

constant for equal angles defining the sine, cosine and tangent ratios for angles in right-angled triangles using trigonometric notation e.g. sin A using a calculator to find approximations of the trigonometric ratios of a given

angle measured in degrees using a calculator to find an angle correct to the nearest degree, given one of the

trigonometric ratios of the angleTrigonometry of Right-Angled Triangles


Height to Base Ratio (page 408): InvestigationMake a Hypsometer (page 421): PracticalPilot Instructions (page 432): Problem Solving

FOCUS ON WORKING MATHEMATICALLY: (page 433) Finding your latitude from the sunThis is designed as a fun outdoor activity. Teachers need to prepare well in advance because of the restricted days of the equinox. However the activity could be carried out on a day close to the equinox if it happens to be cloudy. An explanation of the difference can be found in Mathscape 9 Extension page 481. The geometry should be discussed carefully in class before the outdoor lesson.See what a sailor does to determine latitude using an astrolabe at http://www.ruf.rice.edu/~feegi/measure.html A great site to look at navigation in the 15th century is http://www.ruf.rice.edu/~feegi/site_map.html


http://www.ruf.rice.edu/~feegi/site_map.html

http://www.ruf.rice.edu/~feegi/measure.html


selecting and using appropriate trigonometric ratios in right-angled triangles to find unknown sides, including the hypotenuse

selecting and using appropriate trigonometric ratios in right-angled triangles to find unknown angles correct to the nearest degree

identifying angles of elevation and depression solving problems involving angles of elevation and depression when given a

diagram

Read about advances in navigational technology from the Astrolabe to today's Global Positioning System at http://www.canadiangeographic.ca/Magazine/ND01/ findingourway.html



TechnologySine Cosine: explores the range of Trig graphs.SOHCAHTOA: investigation of the tan ratio.


http://www.canadiangeographic.ca/Magazine/ND01/%20findingourway.html

http://www.canadiangeographic.ca/Magazine/ND01/%20findingourway.html


Chapter 12. Co-ordinate geometrySubstrandCo-ordinate geometry

Text referencesMathscape 9Chapter 12. Co-ordinate Geometry (pages 439–70)

CD referenceIntersecting linesCrow flying


Key ideasUse a diagram to determine midpoint, length and gradient of an interval joining two points on the number plane.Graph linear and simple non-linear relationships from equationsUse midpoint, distance and gradient formulae.Apply the gradient/intercept form to interpret and graph straight lines.

OutcomesPAS5.1.2 (page 97): Determines the midpoint, length and gradient of an interval joining two points on the number plane and graphs linear and simple non-linear relationships from equations.PAS5.2.3 (page 99): Uses formulae to find midpoint, distance and gradient and applies the gradient/intercept form to interpret and graph straight lines.

Working mathematicallyStudents learn to describe the meaning of the midpoint of an interval and how it can be found (Communicating) describe how the length of an interval joining two points can be calculated using Pythagoras’ theorem (Communicating, Reasoning) relate the concept of gradient to the tangent ratio in trigonometry for lines with positive gradients (Reflecting) explain the meaning of each of the pronumerals in the formulae for midpoint, distance and gradient (Communicating) use the appropriate formulae to solve problems on the number plane (Applying Strategies) use gradient and distance formulae to determine the type of triangle three points will form or the type of quadrilateral four points will form and justify the

answer (Applying Strategies, Reasoning) explain why the following formulae give the same solutions as those in the left-hand column

and (Reasoning, Communicating)

Knowledge and skillsStudents learn aboutMidpoint, Length and Gradient determining the midpoint of an interval from a diagram graphing two points to form an interval on the number plane and


Car Hire (page 459): Problem SolvingTemperature Rising (page 462): Problem Solving

FOCUS ON WORKING MATHEMATICALLY



forming a right-angled triangle by drawing a vertical side from the higher point and a horizontal side from the lower point

using the right-angled triangle drawn between two points on the number plane and Pythagoras’ theorem to determine the length of the interval joining the two points

Midpoint, Distance and Gradient Formulae using the average concept to establish the formula for the

midpoint, M, of the interval joining two points and

on the number plane

using the formula to find the midpoint of the interval joining two points on the number plane

using Pythagoras’ theorem to establish the formula for the distance, d, between two points and on the number plane

using the formula to find the distance between two points on the number plane

using the relationship

to establish the formula for the gradient, m, of an interval joining two points and on the number plane

using the formula to find the gradient of an interval joining two points on the number plane

Finding the gradient of a ski run (page 463)The resource book Kleeman, G. and Peters A. (2002) Skills in Australian Geography, Cambridge University Press, Cambridge is your best guide for this activity. Try the Social Science department for a copy or your school library.For a good model of calculating gradient from contour maps go to http://academic.brooklyn.cuny.edu/geology/leveson/core/linksa/map_sample_answer2.html. However measurements are calculated in feet which are still used in the USA. A good site written for scouts which looks at gradients, contours and features of ordinance survey maps is http://www.scoutingresources.org.uk/mapping_contour.html Note that the Sun moves from east to west through the northern sky in our (southern) hemisphere. This means the sun will shine on the northern and eastern slopes during the day. Hence the preference for these slopes. Just what we need to enjoy skiing.

EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING (page 466) CHAPTER REVIEW (page 468) a collection of problems to revise the chapter.


http://www.scoutingresources.org.uk/mapping_contour.html

http://academic.brooklyn.cuny.edu/geology/leveson/core/linksa/map_sample_answer2.html

http://academic.brooklyn.cuny.edu/geology/leveson/core/linksa/map_sample_answer2.html


Gradient/Intercept Form rearranging an equation in general form

(ax + by + c = 0) to the gradient/intercept form determining that two lines are parallel if their gradients are equal

TechnologyIntersecting Lines: interactive program with accompanying worksheet.Crow Flying: students use Pythagoras’ Theorem to investigate how much distance they would save if they could fly in a straight line (as the crow flies) across city blocks. students create their own spreadsheet and investigate when the saving is greatest.Midpoint: interactive geometry worksheet.


stage 5 - macmillanfile/mathscape9_teach_prog… · web viewstage 5. mathscape 9. term chapter...

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