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Mathscape 8 Teaching Program

Stage 4

MATHSCAPE 8Term Topic (Chapter in Mathscape 8) Time

4 1. Algebra (2) 2 weeks / 8 hrs

2. Data representation (4) 2 weeks / 8 hrs

3. Data analysis and probability (12) 2 weeks / 8 hrs

1 4. Pythagoras (3) 2 weeks / 8 hrs

5. Equations, inequations and formulae (8) 2 weeks / 8 hrs

6. Angles and geometrical figures (5) 2 weeks / 8 hrs

7. Geometrical constructions (6) 1 week / 4 hrs

8. Ratio and rates (9) 2 weeks / 8 hrs

2 9. Percentages (1) 2 weeks / 8 hrs

10. Congruence and similarity (13) 2 weeks / 8 hrs

11. Linear relationships (11) 2 weeks / 8 hrs

3 12. Circles and cylinders (10) 2 weeks / 8 hrs

13. Area and volume (7) 3 weeks / 12 hrs

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


Chapter 1. PercentagesSubstrandsFractions, Decimals and Percentages

Text ReferenceMathscape 8Chapter 1: Percentages (pages 1 – 41)

CD ReferencePercentages

Duration2 weeks / 8 hours

Key IdeasPerforms operations with fractions, decimals and mixed numerals.

OutcomesNS4.3 (page 63): Operates with fractions, decimals, percentages, ratios and rates.

Working MathematicallyStudents learn to choose the appropriate equivalent form for mental computation eg 10% of $40 is of $40 (Applying Strategies) question the reasonableness of statements in the media that quote fractions, decimals or percentages

eg ‘the number of children in the average family is 2.3’ (Questioning) interpret a calculator display in formulating a solution to a problem, by appropriately rounding a decimal (Communicating, Applying Strategies) recognise equivalences when calculating

eg multiplication by 1.05 will increase a number/quantity by 5%, multiplication by 0.87 will decrease a number/quantity by 13% (Applying Strategies) solve a variety of real-life problems involving fractions, decimals and percentages (Applying Strategies) use a number of strategies to solve unfamiliar problems, including:

- using a table - looking for patterns - simplifying the problem - drawing a diagram- working backwards- guess and refine

(Applying Strategies, Communicating) interpret media and sport reports involving percentages (Communicating) evaluate best buys and special offers eg discounts (Applying Strategies)



Knowledge and SkillsStudents learn aboutFractions, Decimals and Percentages converting fractions to decimals (terminating and recurring) and percentages converting terminating decimals to fractions and percentages converting percentages to fractions and decimals calculating fractions, decimals and percentages of quantities increasing and decreasing a quantity by a given percentage interpreting and calculating percentages greater than 100% eg an increase from 6

to 18 is an increase of 200%; 150% of $2 is $3 expressing profit and/or loss as a percentage of cost price or selling price ordering fractions, decimals and percentages expressing one quantity as a fraction or a percentage of another eg 15 minutes is

or 25% of an hour

Teaching, Learning and Assessment TRY THIS:Archery winner (page 13): Problem SolvingPure Juice (page 20): Problem SolvingPercentage Rebound (page 27): Practical FOCUS ON WORKING MATHEMATICALLY: Australia’s Indigenous

population(page 35)The web site at the Australian Bureau of Statistics is http://www.abs.gov.au You can find further information starting at the home page http://www.abs.gov.au/ and going to Themes, then choosing People, then choosing Indigenous. EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING

(page 37) CHAPTER REVIEW (page 39) A collection of problems to revise the

chapter.

TechnologyPercentages: speadsheet activity that converts fractions to decimals to percentages and displays result in a pie graph.



Chapter 2. AlgebraSubstrandAlgebraic Techniques

Text ReferenceMathscape 8Chapter 2: Algebra (pages 42 – 77)

CD ReferenceSimplifyExpand Factorise


Key IdeasUses the algebraic symbol system to simplify, expand and factorise simple algebraic expressions.Substitute into algebraic expressions.Apply the index laws to simplify algebraic expressions (positive integral indices only).

OutcomesPAS4.3 (page 85): Uses the algebraic symbol system to simplify, expand and factorise simple algebraic expressionsPAS5.1.1 (page 87): Applies the index laws to simplify algebraic expressions.

Working MathematicallyStudents learn to

check expansions and factorisations by performing the reverse process (Reasoning) interpret statements involving algebraic symbols in other contexts eg creating and formatting spreadsheets (Communicating) explain why a particular algebraic expansion or factorisation is incorrect (Reasoning, Communicating) determine whether a particular pattern can be described using algebraic symbols (Applying Strategies, Communicating) verify the index laws using a calculator eg 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 eg explain why is incorrect (Communicating, Reasoning) examine and discuss the difference between expressions such as

and by substituting values for a (Reasoning, Applying Strategies, Communicating)



Knowledge and SkillsStudents learn about

recognising like terms and adding and subtracting like terms to simplify algebraic expressions eg

recognising the role of grouping symbols and the different meanings of expressions, such as

and simplifying algebraic expressions that involve multiplication and division

eg

simplifying expressions that involve simple algebraicfractions

eg

expanding algebraic expressions by removing grouping symbols (the distributive property)

eg

factorising a single term eg factorising algebraic expressions by finding a common factor

eg

distinguishing between algebraic expressions where letters are used as variables, and equations, where letters are used as unknowns

substituting into algebraic expressions generating a number pattern from an algebraic expression

Teaching, Learning and Assessment TRY THIS:Dot Pentagons (page 45): Number PatternMagic Square (page 48): Find the numbers in the magic square using algebra.Guess my rule (page 54): Partner Activity FOCUS ON WORKING MATHEMATICALLY: Algebra as atool for

exploring patterns (page 72)The idea of this activity is to allow students the opportunity to work mathematically keeping track of their thinking (see Let’s Communicate p74). Teachers should pitch the activity to the ability of their class to make it enjoyable. Collaborative small groups are encouraged.The Islamic magic square on page 72 expresses the number 66 in every direction. The grid is formed by the letters of the word Allah. See Clifford A, Pickover Wonder of Numbers, Oxford University press, 2000, page 233. The web site http://www.godprovenas1.com/Supplement/Allah-Letters.html explains how it works.This point of contact is a good one to explore the contributions of Islamic scholars to mathematics. For example Muhammad Musa al-Khwarizmi was born sometime before 800 A.D. in an area not far from Baghdad and lived at least until 847. He wrote his Al-jabr wa'l muqabala (from which our modern word "algebra" comes) while working as a scholar in Baghdad. The English word "algorithm" derives from the Latin form of al-Khwarizmi's name. See for example http://www.peak.org/~jeremy/calculators/alKwarizmi.htmlMethod 1: a counting strategyBy observation the smallest integer required is 42 and the next is 112. To preserve the conditions you need to continue to add 70. This gives 42, 112, 182, … until you reach the largest 882. By counting there are 13 numbers altogether in the sequence.A spreadsheet could be used to generate the sequence 42, 112, 182, …. 882. The technology folder on the CD accompanying Mathscape 7, chapter 5 Patterns and pronumerals will be useful.Method 2: a reasoning strategySome students may reason from a table of values using the difference method to get the formula like Tn = 70n –38 where n =1, 2, …,13. Others may reason without a table to the equivalent formula 42 + 70(n-1) where n = 1,2,3,…13. Some may guess 42 + 70n where n = 0,1,2,3 … 12.


http://www.peak.org/~jeremy/calculators/alKwarizmi.html

http://www.godprovenas1.com/Supplement/Allah-Letters.html


eg 1 2 3 4 5 6 10 1004 5 6 _ _ _ _ _

replacing written statements describing patterns with equations written in algebraic symbols

eg ‘you add five to the first number to get the second number’ could be replaced with ‘ ’

translating from everyday language to algebraic language and from algebraic language to everyday language

using the index laws previously established for numbers to develop the index laws in algebraic form

eg

establishing that using the index lawseg and

simplifying algebraic expressions that include index notationeg

Reasoning from writing the numbers in the form 14k, the values for k must end in 3 or 8. This leads to 3,8,13,18,23,28,33,38,43,48,53,58 and 63 (by calculator 68 is too big) giving the sequence 42, 112, 182, …882. We can see immediately that there are 13 numbers in all.Extension activity: the most able students should compare the strengths and weaknesses of the different methods used. To answer the question as a multiple choice item it is likely that an arithmetic method using a calculator will be the quickest route. EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING

(page 73) CHAPTER REVIEW (page 76): A collection of problems to revise the

chapter.

TechnologySimplify: this program will attempt to collect like terms and simplify the entered expression according to the algebraic rules provided in the Algebra chapter.Expand: this program will expand a given algebraic expression.Factorise: this program will factorise a given algebraic expression.



Chapter 3. Pythagoras’ TheoremSubstrand Perimeter and Area

Text ReferenceMathscape 8Chapter 3: Pythagoras’ Theorem(pages 78 – 104)

CD ReferencePythagoras TheoremCrow Flying

Duration 2 weeks / 8 hours

Key IdeasApply Pythagoras’ theorem.

OutcomesMS4.1 (page 124): Uses formulae and Pythagoras’ Theorem in calculating the perimeter and area of circles and figures composed of rectangles and triangles.


describe the relationship between the sides of a right-angled triangle (Communicating) use Pythagoras’ theorem to solve practical problems involving right-angled triangles (Applying Strategies) apply Pythagoras’ theorem to solve problems involving perimeter and area (Applying Strategies) identify the perpendicular height of triangles and parallelograms in different orientations (Communicating)


identifying the hypotenuse as the longest side in any right-angled triangle and also as the side opposite the right angle

establishing the relationship between the lengths of the sides of a right-angled triangle in practical ways, including the dissection of areas

using Pythagoras’ theorem to find the length of sides in right-angled triangles solving problems involving Pythagoras’ theorem, giving an exact answer as a

surd (eg ) and approximating the answer using an approximation of the square root

writing answers to a specified or sensible level of accuracy, using the ‘approximately equals’ sign

identifying a Pythagorean triad as a set of three numbers such that the sum of the squares of the first two equals the square of the third

using the converse of Pythagoras’ theorem to establish whether a triangle has a right angle.

Teaching, Learning and Assessment TRY THIS:How long is a tile? (p84): Problem SolvingDemonstrating Pythagoras’ Theorem (p94): Practical proof of Pythagoras’ Theorem FOCUS ON WORKING MATHEMATICALLY: Secret societies,

mathematics and magic (p99)The history of maths web site at St Andrews University, Scotland http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Pythagoras.html is a good one for detail on the life of Pythagoras and the society he formed.Try the web link http://www.cut-the-knot.com/pythagoras/ for excellent visual proofs of the theorem.Check out http://www.aboutscotland.com/harmony/prop.html if you want to see the mathematics of musical notes.

A key point about the activity is that Pythagoras’ theorem for squares enables


http://www.aboutscotland.com/harmony/prop.html

http://www.cut-the-knot.com/pythagoras/

http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Pythagoras.html

http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Pythagoras.html


the calculation of the length of the hypotenuse. This is what led to the discovery of irrational numbers and its importance cannot be emphasised enough. Other shapes may work but the square is the most important. EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING

(p101) CHAPTER REVIEW (p103): A collection of problems to revise the

chapter.

TechnologyPythagoras Theorem: Students use the worksheet and the program to discover how to use and prove Pythagoras’ theorem.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.



Chapter 4. Data representationSubstrandData Analysis and Evaluation

Text ReferenceMathscape 8Chapter 4: Data representation(pages 105 – 174)

CD ReferenceSports Statistics


Key IdeasDraw, read and interpret graphs (line, sector, travel, step, conversion, divided bar, dot plots and stem-and-leaf plots), tables and charts. Distinguish between types of variables used in graphs. Identify misrepresentation of data in graphs. Construct frequency tables. Draw frequency histograms and polygons.

OutcomesDS4.1 (page 114): Constructs, reads and interprets graphs, tables, charts and statistical information.DS4.2 (page 115): Collects statistical data using either a census or a sample, and analyses data using measures of location and range.


choose appropriate forms to display data (Communicating) write a story which matches a given travel graph (Communicating) read and comprehend a variety of data displays used in the media and in other school subject areas (Communicating) interpret back-to-back stem-and-leaf plots when comparing data sets (Communicating) analyse graphical displays to recognise features that may cause a misleading interpretation eg displaced zero, irregular scales (Communicating, Reasoning) compare the strengths and weaknesses of different forms of data display (Reasoning, Communicating) interpret data displayed in a spreadsheet (Communicating) identify when a line graph is appropriate (Communicating) interpret the findings displayed in a graph eg the graph shows that the heights of all children in the class are between 140 cm and 175 cm and that most are in

the group 151–155 cm (Communicating) generate questions from information displayed in graphs (Questioning)




drawing and interpreting graphs of the following types:- sector graphs- conversion graphs - divided bar graphs- line graphs - step graphs

choosing appropriate scales on the horizontal and vertical axes when drawing graphs

drawing and interpreting travel graphs, recognising concepts such as change of speed and change of direction

using line graphs for continuous data only reading and interpreting tables, charts and graphs recognising data as quantitative (either discrete or continuous) or categorical using a tally to organise data into a frequency distribution table (class

intervals to be given for grouped data) drawing frequency histograms and polygons drawing and using dot plots drawing and using stem-and-leaf plots using the terms ‘cluster’ and ‘outlier’ when describing data making predictions from a scatter diagram or graph using spreadsheets to tabulate and graph data analysing categorical data eg a survey of car colours

Teaching, Learning and Assessment TRY THIS:Runners (page 135): Problem SolvingThe Top 40 (page 149): Investigate patterns in the Top 40 chartsLet’s Jump (page 161): Class data generating exercise. FOCUS ON WORKING MATHEMATICALLY: Championship Tennis

(page 166)The key point about the activity is that a tennis draw can be thought of as a representation of data. See http://www.ausopen.org/pages/default.aspx?id=15 for access to the Australian Open. The recent Rugby World Cup held in Sydney in 2003 http://www.rugby2003.com.au/ is a contrasting example of a grouped draw which teachers could use. Local competitions in which students play create interest and enables a connection to the real world of the student.Sporting draws can have a big impact on who reaches the finals. The reasoning behind them can be used effectively to promote the outcomes of working mathematically. EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING

(page 167) CHAPTER REVIEW (page 169) A collection of problems to revise the

chapter.

TechnologySports Statistics: a spreadsheet of a shot put competition which could be used for other competitions.


http://www.rugby2003.com.au/

http://www.ausopen.org/pages/default.aspx?id=15

http://www.ausopen.org/pages/default.aspx?id=15


Chapter 5. Angles and geometric figuresSubstrands AnglesProperties of Geometrical Figures

Text ReferenceMathscape 8Chapter 5: Angles and geometric figures (pages 175 – 216)

CD ReferenceAngle PairsPolygon AnglesQuadrilateralsExterior Angle


Key IdeasConstruct parallel and perpendicular lines and determine associated angle properties. Complete simple numerical exercises based on geometrical properties.

OutcomesSGS4.2 (page 153): Identifies and names angles formed by the intersection of straight lines, including those related to transversals on sets of parallel lines, and make use of the relationships between them.SGS4.3 (page 154) Classifies, constructs and determines the properties of triangles and quadrilaterals

Working MathematicallyStudents learn to use dynamic geometry software to investigate angle relationships (Applying Strategies, Reasoning) recognise that a given triangle may belong to more than one class (Reasoning) recognise that the longest side of a triangle is always opposite the largest angle (Applying Strategies, Reasoning) recognise and explain why two sides of a triangle must together be longer than the third side (Applying Strategies, Reasoning) recognise special types of triangles and quadrilaterals embedded in composite figures or drawn in various orientations (Communicating) apply geometrical facts, properties and relationships to solve numerical problems such as finding unknown sides and angles in diagrams (Applying

Strategies) justify their solutions to problems by giving reasons using their own words (Reasoning)

Knowledge and SkillsStudents learn aboutAngles at a Point

identifying and naming adjacent angles (two angles with a common vertex and a common arm), vertically opposite angles, straight angles and angles of complete revolution, embedded in a diagram

Teaching, Learning and Assessment TRY THIS:Rotating Pencil (page 180): PracticalAngles (page 184): PracticalQuadrilateral Diagram (page 209): Students create a diagram to show the properties of quadrilaterals



using the words ‘complementary’ and ‘supplementary’ for angles adding to 90º and 180º respectively, and the terms ‘complement’ and ‘supplement’

establishing and using the equality of vertically opposite angles

Angles Associated with Transversals identifying and naming a pair of parallel lines and a transversal using common symbols for ‘is parallel to’ ( ) and ‘is perpendicular to’ ( ) using the common conventions to indicate parallel lines on diagrams identifying, naming and measuring the alternate angle pairs, the

corresponding angle pairs and the co-interior angle pairs for two lines cut by a transversal

recognising the equal and supplementary angles formed when a pair of parallel lines are cut by a transversal

using angle properties to identify parallel lines using angle relationships to find unknown angles in diagrams

Triangles using a parallel line construction, to prove that the interior angle sum of a

triangle is 180º

Quadrilaterals establishing that the angle sum of a quadrilateral is 360º investigating the properties of special quadrilaterals (trapeziums, kites,

parallelograms, rectangles, squares and rhombuses) by using symmetry, paper folding, measurement and/or applying geometrical reasoning Properties to be considered include :

opposite sides parallelopposite sides equaladjacent sides perpendicularopposite angles equaldiagonals equal in lengthdiagonals bisect each otherdiagonals bisect each other at right angles diagonals bisect the angles of the quadrilateral

classifying special quadrilaterals on the basis of their properties

FOCUS ON WORKING MATHEMATICALLY: Logos (page 210)The questions refer to the Mitsubishi logo which teachers should bring a copy of to class. In the first printing run the ABC logo was printed by mistake. Teachers will find the material on logos in Mathscape 7 Exercise 10.7 pages 397- 400 useful.Geometer’s sketchpad or Cabri geometry can be used effectively to design logos. Students could be given a competition to design a logo for their maths class, a school club or a sporting team.There are many companies which specialise in the design of logos. Check out the interesting logos which IBM have used over the years at http://www-1.ibm.com/ibm/history/exhibits/logo/logo_8.html and search the net for others. Note: there is an underscore '_' between 'logo' and '8'. EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING


chapter.

Technology


http://www-1.ibm.com/ibm/history/exhibits/logo/logo_8.html

http://www-1.ibm.com/ibm/history/exhibits/logo/logo_8.html


Angle Pairs: interactive geometry to discover cointerior, corresponding and alternate angles.Polygon Angles: this file contains a number of interactive geometric diagrams that focus on the angle sum of polygons.Quadrilaterals: students use the program and worksheet to investigate quadrilateralsExterior Angles: 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.



Chapter 6. Geometric constructionsSubstrands Properties of Geometrical Figures

Text ReferenceMathscape 8Chapter 6: Geometric constructions (pages 217 – 242)

CD ReferenceGeometric ConstructionsEuler Line

Duration 1 week / 4 hours

Key IdeasConstruct parallel and perpendicular lines and determine associated angle properties. Classify, construct and determine properties of triangles and quadrilaterals.

OutcomesSGS4.2 (page 153): Identifies and names angles formed by the intersection of straight lines, including those related to transversals on sets of parallel lines, and make use of the relationships between them.SGS4.3 (page154) Classifies, constructs and determines the properties of triangles and quadrilaterals

Working MathematicallyStudents learn to construct a pair of perpendicular lines using a ruler and a protractor, a ruler and a set square, or a ruler and a pair of compasses (Applying Strategies) bisect an angle by applying geometrical properties eg constructing a rhombus (Applying Strategies) bisect an interval by applying geometrical properties eg constructing a rhombus (Applying Strategies) draw a perpendicular to a line from a point on the line by applying geometrical properties eg constructing an isosceles triangle (Applying Strategies) draw a perpendicular to a line from a point off the line by applying geometrical properties eg constructing a rhombus (Applying Strategies) use ruler and compasses to construct angles of 60º and 120º by applying geometrical properties eg constructing an equilateral triangle (Applying Strategies) explain that a circle consists of all points that are a given distance from the centre and how this relates to the use of a pair of compasses (Communicating,

Reasoning) use dynamic geometry software to investigate the properties of geometrical figures (Applying Strategies, Reasoning)

Knowledge and SkillsStudents learn aboutTriangles

constructing various types of triangles using geometrical instruments, given different information eg the lengths of all sides, two sides and the included angle, and two angles and one side

Teaching, Learning and Assessment TRY THIS:Triangle and Rhombus Construction (page 226): PracticalOrthocentre and Incentre: (page 235): Practical. Good for extension.FOCUS ON WORKING MATHEMATICALLY: Finding south in the night sky (page 235) This activity enables teachers to link mathematics across cultures. The book mentioned on page 238 of Mathscape 8 by Raymond Haynes on the



Quadrilaterals constructing various types of quadrilaterals

observations of stars by Aboriginal Australian people will interest many students. This is an important part of the work of the International Study Group on Ethnomathematics, see http://www.rpi.edu/~eglash/isgem.htm for details.The web site at NASA http://istp.gsfc.nasa.gov/stargaze/ is an excellent source of linked topics about stars and space.The Thinkquest Internet Challenge web pages are an excellent source of information about Australian astronomy and finding south in the night sky. Go to http://library.thinkquest.org/C005462/astronomy.html For specific information about Aboriginal astronomy go to http://library.thinkquest.org/C005462/index2.html EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING


chapter.

TechnologyGeometric Constructions: the program is designed to strengthen students understanding of the underlying geometry of the constructions they have studied in chapter 6 of Mathscape 8. Run the program and then use it to answer the questions. Students will need a pair of compasses to complete some of the questions on the worksheet and the textbook open at chapter 6, section 6.4, on page 226.Euler Line: the Euler line of a triangle is a line that passes through three special points of a triangle. Investigative exercise.


http://library.thinkquest.org/C005462/index2.html

http://library.thinkquest.org/C005462/astronomy.html

http://istp.gsfc.nasa.gov/stargaze/

http://www.rpi.edu/~eglash/isgem.htm


Chapter 7. Area and volumeSubstrands Perimeter and AreaSurface Area and Volume

Text ReferenceMathscape 8Chapter 7: Area and Volume(pages 243 – 284)

CD ReferenceMeasuring Plane ShapesSolid Measurements


Key IdeasDevelop formulae and use it to find the area and perimeter of triangles, rectangles and parallelograms.Find the areas of simple composite figures.Apply Pythagoras’ theorem.Find the surface area of rectangular and triangular prisms.Find the volume of right prisms and cylinders.Convert between metric units of volume.

OutcomesMS4.1 (page 124): Uses formulae and Pythagoras’ Theorem in calculating the perimeter and area of circles and figures composed of rectangles and triangles.MS4.2 (page 131): Calculates surface area of rectangular and triangular prisms and volume of right prisms and cylinders.

Working MathematicallyStudents learn to identify the perpendicular height of triangles and parallelograms in different orientations (Communicating) find the dimensions of a square given its perimeter, and of a rectangle given its perimeter and one side length (Applying Strategies) solve problems relating to perimeter, area and circumference (Applying Strategies) compare rectangles with the same area and ask questions related to their perimeter such as whether they have the same perimeter

(Questioning, Applying Strategies, Reasoning) compare various shapes with the same perimeter and ask questions related to their area such as whether they have the same area (Questioning) explain the relationship that multiplying, dividing, squaring and factoring have with the areas of squares and rectangles with integer solve problems involving surface area of rectangular and triangular prisms (Applying Strategies) solve problems involving volume and capacity of right prisms and cylinders (Applying Strategies) recognise, giving examples, that prisms with the same volume may have different surface areas, and prisms with the same surface area may have different

volumes (Reasoning, Applying Strategies)



Knowledge and SkillsStudents learn aboutSurface Area of Prisms

identifying the surface area and edge lengths of rectangular and triangular prisms

finding the surface area of rectangular and triangular prisms by practical means eg from a net

calculating the surface area of rectangular and triangular prisms

Volume of Prisms converting between units of volume 1 cm3 = 1000 mm3, 1L = 1000 mL = 1000 cm3, 1 m3 = 1000 L = 1 kL using the kilolitre as a unit in measuring large volumes constructing and drawing various prisms from a given cross-sectional diagram identifying and drawing the cross-section of a prism developing the formula for volume of prisms by considering the number and

volume of layers of identical shape calculating the volume of a prism given its perpendicular height and the area

of its cross-section calculating the volume of prisms with cross-sections that are rectangular and

triangular calculating the volume of prisms with cross-sections that are simple

composite figures that may be dissected into rectangles and triangles Volume of Cylinders developing and using the formula to find the volume of cylinders (r is the

length of the radius of the base and h is the perpendicular height)

Teaching, Learning and Assessment TRY THIS:Square area (page 250): Problem SolvingBiggest area (page 256): Practical problemA packing problem (page 263): Challenging problemVolume through liquid displacement (p272): Experiment FOCUS ON WORKING MATHEMATICALLY: Torrential rain in

Sydney (page 278)The issue of water conservation is very topical. Go to the Sydney Water site http://www.sydneywater.com.au/ for ideas to use in class. The idea of actually measuring dripping water over a period of 30 mins during a lesson is highly recommended. The extension activity to find the approximate volume of a single drop works well and is highly motivating. If you type in “Rain gauge” into a search engine you will get lots of sites to help you make one. Try http://sln.fi.edu/weather/todo/r-gauge.html to get started. EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING


chapter.

TechnologyMeasuring Plane Shapes: this file contains hyperlinks to a number of interactive geometric diagrams.Solid Measurements: the activities and questions make use of the Solid Measure.html file which includes a number of interactive diagrams of geometric solids and their nets (the two-dimensional figure that can be ‘folded’ to create the three-dimensional figure).


http://sln.fi.edu/weather/todo/r-gauge.html

http://www.sydneywater.com.au/


Chapter 8. Equations, inequations and formulaeSubstrandAlgebraic Techniques

Text ReferenceMathscape 8Chapter 8: Equations, inequations and formulae (pages 285 – 326)

CD ReferenceYacht Formula


Key IdeasSolve linear equations and word problems using algebra.Solve simple inequalities

OutcomesPAS4.4 (page 86): Uses algebraic techniques to solve linear equations and simple inequalities.


compare and contrast different methods to solve a range of linear equations (Reasoning) create equations to solve a variety of problems, clearly stating the meaning of introduced letters as ‘the number of …’, and verify solutions

(Applying Strategies, Reasoning) use algebraic techniques as a tool for problem solving (Applying Strategies) construct formulae for finding areas of common geometric figures eg area of a triangle (Applying Strategies) determine equations that have a given solution eg find equations that have the solution x = 5 (Applying Strategies) substitute into formulae used in other strands of the syllabus or in other key learning areas and interpret the solutions (Applying Strategies, Communicating)

eg

describe the process of solving simple inequalities and justifying solutions (Communicating, Reasoning)

Knowledge and SkillsStudents learn about solving simple linear equations using concrete materials, such as the balance

model or cups and counters, stressing the notion of doing the same thing to both sides of an equation

Teaching, Learning and Assessment TRY THIS:Car colours (page 297): Problem SolvingBraking Distance (page 302): Use of the formula for the braking distance of a car



solving linear equations using strategies such as guess, check and improve, and backtracking (reverse flow charts)

solving equations using algebraic methods that involve up to and including three steps in the solution process and have solutions that are not necessarily whole numbers

eg

checking solutions to equations by substituting translating a word problem into an equation, solving the equation and translating

the solution into an answer to the problem solving equations arising from substitution into formulae

eg given P = 2l + 2b and P = 20, l = 6, solve for b finding a range of values that satisfy an inequality using strategies such as ‘guess

and check’ solving simple inequalities such as

representing solutions to simple inequalities on the number line

FOCUS ON WORKING MATHEMATICALLY: What is a 12m yacht? (page 321)

This activity should be used in conjunction with the interactive Cabri geometry program What is a 12m yacht? on the Mathscape 8 CD. It enables the student to vary the dimensions L, D, H and S in the formula and see the immediate effects. There is an additional worksheet for teachers in the same folder. See the NZ America’s Cup site at http://www.americascup.co.nz/index.html for interesting information on the history of the race and http://www.acmuseum.com/portfolio1999_2000.htm for photos of the different designs of yachts taking part. Note: there is an underscore '_' between '1999' and '2000'. Keen sailors might be interested in sail design at http://www.quantumsails.com/ EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING


chapter.

TechnologyYacht Formula: the ctivities and questions make use of the Yacht Formula.html program. It has two parts. First, there is a picture of the yacht so that students can see what the variables in the formula measure. The software enables students to see the visual effects changing the length, width, height or sail area has on the formula. Second, there is a spreadsheet to enables students to rapidly calculate the effects of changing the measurements. Mathscape 8 textbook needs to be opened at chapter 8 page 321.


http://www.quantumsails.com/

http://www.acmuseum.com/portfolio1999_2000.htm

http://www.americascup.co.nz/index.html


Chapter 9. Ratio and ratesSubstrandFractions, Decimals and Percentages

Text ReferenceMathscape 8Chapter 9: Ratios and Rates(pages 327 – 362)

CD ReferenceRatiosRates


Key IdeasUse ratios and rates to solve problems.

OutcomesNS4.3 (p 63): Operates with fractions, decimals, percentages, ratios and rates.

Working MathematicallyStudents learn to interpret descriptions of products that involve fractions, decimals, percentages or ratios eg on labels of packages (Communicating) solve a variety of real-life problems involving ratios eg scales on maps, mixes for fuels or concrete, gear ratios (Applying Strategies) solve a variety of real-life problems involving rates eg batting and bowling strike rates, telephone rates, speed, fuel consumption (Applying Strategies)

Knowledge and SkillsStudents learn aboutRatio and Rates

using ratio to compare quantities of the same type writing ratios in various forms

eg , 4:6, 4 to 6

simplifying ratios eg 4:6 = 2:3, :2 = 1:4, 0.3:1 = 3:10 applying the unitary method to ratio problems dividing a quantity in a given ratio interpreting and calculating ratios that involve more than two numbers calculating speed given distance and time calculating rates from given information

eg 150 kilometres travelled in 2 hours

Teaching, Learning and Assessment TRY THIS:Cartoons (page 331): Enlarge a cartoon image using a scale factor of 2:1Scale factor and areas (page 333): InvestigationBiscuits (page 342): Increase the quantities in a recipe using conversionsHeart rates (page 344): Students guess and find heart rate in beats per minuteHow far away is lightning? (page 351): Investigation FOCUS ON WORKING MATHEMATICALLY: The comparative size of

a nuclear submarine (page 357)A good web site for finding out about Kursk is the CNN web site http://www5.cnn.com/SPECIALS/2000/submarine/ for details and pictures. Click on resources at the bottom on the page.There are two excellent worksheets on ratios and rates available on the Mathscape 8 CD. These are based on spreadsheets which you will find in the Ratios and Rates technology folders.To read more about and practice ratios check out http://www.math.com/school/subject1/lessons/S1U2L1GL.html#sm2


http://www.math.com/school/subject1/lessons/S1U2L1GL.html#sm2

http://www5.cnn.com/SPECIALS/2000/submarine/


EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING (page 358)

CHAPTER REVIEW (page 360): A collection of problems to revise the chapter.

TechnologyRatios: Ratios are simplified in this interactive program.Rates: Rates are converted in this interactive program.



Chapter 10. Circles and cylindersSubstrands Perimeter and AreaSurface Area and VolumeProperties of Geometrical Figures

Text ReferenceMathscape 8Chapter 10: Circles and cylinders (pages 363 – 397)

CD ReferenceCircle PartsCircle MeasuringCylinders


Key IdeasInvestigate and find the area and circumference of circles.Find the volume of right prisms and cylinders.

OutcomesMS4.1 (page 124): Uses formulae and Pythagoras’ Theorem in calculating the perimeter and area of circles and figures composed of rectangles and triangles.MS4.2 (page 131): Calculates surface area of rectangular and triangular prisms and volume of right prisms and cylinders.SGS4.3 (page 154): Classifies, constructs, and determines the properties of triangles and quadrilaterals.

Working MathematicallyStudents learn to use mental strategies to estimate the circumference of circles, using an approximate value of eg 3 (Applying Strategies) find the area and perimeter of quadrants and semi-circles (Applying Strategies) find radii of circles given their circumference or area (Applying Strategies) solve problems involving , giving an exact answer in terms of and an approximate answer using , 3.14 or a calculator’s approximation for

(Applying Strategies) compare the perimeter of a regular hexagon inscribed in a circle with the circle’s circumference to demonstrate that > 3 (Reasoning) solve problems involving volume and capacity of right prisms and cylinders (Applying Strategies)

Knowledge and SkillsStudents learn aboutCircumferences and Areas of Circles demonstrating by practical means that the ratio of the circumference to the

diameter of a circle is constant eg by measuring and comparing the diameter and circumference of cylinders

defining the number π as the ratio of the circumference to the diameter of any

Teaching, Learning and Assessment TRY THIS:How Eratosthenes measured the circumference of the Earth (page 369): ProofRoping logs (page 374): Problem solvingStained panes (page 379): Problem solving

FOCUS ON WORKING MATHEMATICALLY: International one-day



circle developing, from the definition of π, formulae to calculate the circumference of

circles in terms of the radius r or diameter d or

developing by dissection and using the formula to calculate the area of circles

developing and using the formula to find the volume of cylinders (r is the length of the radius of the base and h is the perpendicular height)

identifying and naming parts of the circle and related lines, including arc, tangent and chord

investigating the line symmetries and the rotational symmetry of circles and of diagrams involving circles, such as a sector and a circle with a chord or tangent

cricket (page 392)To check rules for the inner circle and restrictions on fielding positions go to the International Cricket Council site http://www.cricket.org/link_to_database/NATIONAL/ICC/RULES/ and click on one day internationals. Note: there is an underscore '_' between 'link' and 'to' and 'to' and 'database'. Women’s cricket is very strong in Australia. The home page of Women’s Cricket in Australia http://www.ausport.gov.au/wca/ deserves a visit. Australia’s loss to NZ in the World Cup 2000 final was a close and wonderful match. See match report at http://www.ausport.gov.au/wca/wcawc2000.htmThe home page for Cricket Australia is a great source of statistical information which teachers can use to design working mathematically activities. Go to http://www-aus.cricket.org/link_to_database/NATIONAL/AUS/ EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING


chapter.

TechnologyCircle Parts, Circle Measuring, Cylinders: a set of Cabri Geometry interactive worksheets that are used for students to explore the parts and use of circles.


http://www-aus.cricket.org/link_to_database/NATIONAL/AUS/

http://www.ausport.gov.au/wca/wcawc2000.htm

http://www.ausport.gov.au/wca/

http://www.cricket.org/link_to_database/NATIONAL/ICC/RULES/


Chapter 11. Linear relationshipsSubstrandLinear Relationships

Text ReferenceMathscape 8Chapter 11: Linear Relationships (pages 398 – 428)

CD ReferenceRobot Lander


Key IdeasInterpret the number plane and locate ordered pairs.Graph and interpret linear relationships created from simple number patterns and equations.

OutcomesPAS4.5 (page 96): Graphs and interprets linear relationships on the number plane.

Working MathematicallyStudents learn to relate the location of points on a number plane to maps, plans, street directories and theatre seating and note the different recording conventions eg 15 E

(Communicating, Reflecting) compare similarities and differences between sets of linear relationships (Reasoning)

eg

sort and classify equations of linear relationships into groups to demonstrate similarities and differences (Reasoning) question whether a particular equation will have a similar graph to another equation and graph the line to check (Questioning, Applying Strategies, Reasoning) recognise and explain that not all patterns form a linear relationship (Reasoning) determine and explain differences between equations that represent linear relationships and those that represent non-linear relationships

(Applying Strategies, Reasoning) explain the significance of the point of intersection of two lines in relation to it being a solution of each equation (Applying Strategies, Reasoning) question if the graphs of all linear relationships that have a negative x term will decrease (Questioning) reason and explain which term affects the slope of a graph, making it either increasing or decreasing (Reasoning, Communicating) use a graphics calculator and spreadsheet software to graph and compare a range of linear relationships (Applying Strategies, Communicating)



Knowledge and SkillsStudents learn about interpreting the number plane formed from the intersection of a horizontal x -axis

and vertical y -axis and recognising similarities and differences between points located in each of the four quadrants

identifying the point of intersection of the two axes as the origin, having coordinates (0,0)

reading, plotting and naming ordered pairs on the number plane including those with values that are not whole numbers

graphing points on the number plane from a table of values, using an appropriate scale

extending the line joining a set of points to show that there is an infinite number of ordered pairs that satisfy a given linear relationship

interpreting the meaning of the continuous line joining the points that satisfy a given number pattern

reading values from the graph of a linear relationship to demonstrate that there are many points on the line

deriving a rule for a set of points that has been graphed on a number plane by forming a table of values or otherwise

forming a table of values for a linear relationship by substituting a set of appropriate values for either of the letters and graphing the number pairs on the number plane eg given y = 3x + 1, forming a table of values using x = 0, 1 and 2 and then graphing the number pairs on a number plane with appropriate scale

graphing more than one line on the same set of axes and comparing the graphs to determine similarities and differences eg parallel, passing through the same point

graphing two intersecting lines on the same set of axes and reading off the point of intersection

Teaching, Learning and Assessment FOCUS ON WORKING MATHEMATICALLY: Dreams, imagination

and mathematical ideas (page 423)The purpose of this activity is to strengthen the use of students’ visual imagery in understanding mathematical ideas. With reference to the quote from Einstein (Let’s Communicate page 425) most teachers will agree that in school mathematics both knowledge and imagination are important. The activity lends itself to working in pairs or small groups.Examples in the extension exercise have been provided for discussion. Teachers will find the working mathematically activity with newspapers, Mathscape 7 page 156, useful for the drawing in example 3.Teachers may like use the “shapely thinking” exercise in chapter 12 of the year 7 text page 477. This exercise focuses on visualising conic sections. See also notes for teachers on CD to accompany Mathscape 7.The web site at St Andrews is a good general reference for getting information about particular mathematicians. You will find Descartes at http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Descartes.html. There is also a poster which can be down loaded (scroll down to the end of the article) and cross references to topics.The novel by Mark Haddon The curious incident of the dog in the night-time, David Fickling books, Oxford, 2003 is highly recommended reading for teachers and students. Fifteen year old Christopher has Asperger’s syndrome, a form of autism. He has a rich mathematical imagination and a photographic memory. He understands mathematics but cannot understand human beings. EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING


chapter.

TechnologyRobot Lander: the Robot Lander project is a game that lets users practise using coordinates. The main activity of the game involves converting between absolute and relative coordinates. The object of the game is to direct a robot that has been sent to another planet of our solar system (including our moon) in search of dilithium crystals.


http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Descartes.html

http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Descartes.html


Chapter 12. Data analysis and probabilitySubstrands Data Analysis and EvaluationProbability

Text ReferenceMathscape 8Chapter 12: Data analysis and probability (pages 429 – 490)

CD ReferenceData AnalysisWeighted Dice


Key IdeasUse sampling and census. Make predictions from samples and diagrams. Analyse data using mean, mode, median and range. Determine the probability of simple events. Solve simple probability problems. Recognise complementary events.

OutcomesDS4.2 (page 115): Collects statistical data using either a census or a sample, and analyses data using measures of location and range.NS4.4 (page 74): Solves probability problems involving simple events.




work in a group to design and conduct an investigationeg decide on an issue

decide whether to use a census or samplechoose appropriate methods of presenting questions (yes/no, tick a box, a scale of 1 to 5, open-ended, etc)analyse and present the datadraw conclusions (Questioning, Reasoning, Applying Strategies, Communicating)

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)

detect bias in the selection of a sample (Applying Strategies) consider the size of the sample when making predictions about the population (Applying Strategies) compare two sets of data by finding the mean, mode and/or median, and range of both sets (Applying Strategies) recognise that summary statistics may vary from sample to sample (Reasoning) draw conclusions based on the analysis of data (eg a survey of the school canteen food) using the mean, mode and/or median, and range (Applying

Strategies, Reasoning) interpret media reports and advertising that quote various statistics eg media ratings (Communicating) question when it is more appropriate to use the mode or median, rather than the mean, when analysing data (Questioning) solve simple probability problems arising in games (Applying Strategies) use language associated with chance events appropriately (Communicating) evaluate media statements involving probability (Applying Strategies, Communicating) interpret and use probabilities expressed as percentages or decimals (Applying Strategies, Communicating) explain the meaning of a probability of 0, 0.5 and 1 in a given situation (Communicating, Reasoning)


formulating key questions to generate data for a problem of interest refining key questions after a trial recognising the differences between a census and a sample

Teaching, Learning and Assessment TRY THIS:Sampling bottle (page 450): Estimation practicalMr and Mrs Average (page 463): Students investigate – What is an average family?Mean versus median (page 470): Students discover which one is appropriate to



finding measures of location (mean, mode, median) for small sets of data using a scientific or graphics calculator to determine the mean of a set of

scores using measures of location (mean, mode, median) and the range to analyse

data that is displayed in a frequency distribution table, stem-and-leaf plot, or dot plot

collecting data using a random process eg numbers from a page in a phone book, or from a random number function on a calculator

making predictions from a sample that may apply to the whole population making predictions from a scatter diagram or graph using spreadsheets to tabulate and graph data analysing categorical data eg a survey of car colours listing all possible outcomes of a simple event using the term ‘sample space’ to denote all possible outcomes eg for tossing a

fair die, the sample space is 1, 2, 3, 4, 5, 6 assigning probabilities to simple events by reasoning about equally likely

outcomes eg the probability of a 5 resulting from the throw of a fair die is

expressing the probability of a particular outcome as a fraction between 0 and 1

assigning a probability of zero to events that are impossible and a probability of one to events that are certain

recognising that the sum of the probabilities of all possible outcomes of a simple event is 1

identifying the complement of an event eg ‘The complement of drawing a red card from a deck of cards is drawing a black card.’

finding the probability of a complementary event

useI win, you lose (page 480): Dice problem FOCUS ON WORKING MATHEMATICALLY: Health risks of smoking

(page 483)The source of the data for the learning activity Self reported behaviours of secondary students, NSW 1999, Statistical Bulletin – tobacco is available from the Cancer Council NSW ([email protected]) or online at the NSW Health web site http://www.health.nsw.gov.au/public-health/health-promotion/tobacco/facts/index.html. Scroll down to the subheading Perceived percentage of the population prevalence and click on Self reported behaviours of secondary students NSW 1999 tobacco Of all groups in the community, Aboriginal people are most at risk from smoking related diseases. See http://www.abs.gov.au/ausstatsThe Australian Institute of Health and Welfare www.aihw.gov.au produces regular statistics on a wide range of drug and alcohol use in Australia. EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING


chapter.

TechnologyData Analysis: the activities and questions make use of the Class Analysis and Data Analysis spreadsheets. The intention is to provide students with an opportunity to analyse data with the help of a spreadsheet. Mathscape 8 is needed for reference.Weighted Dice: this Excel spreadsheet simulates the throwing 100 times of a weighted die, and graphs the outcomes.


http://www.aihw.gov.au/

http://www.health.nsw.gov.au/public-health/health-promotion/tobacco/facts/index.html

http://www.health.nsw.gov.au/public-health/health-promotion/tobacco/facts/index.html

mailto:[email protected]


Chapter 13. Congruence and similaritySubstrandProperties of Geometrical Figures

Text ReferenceMathscape 8Chapter 13: Congruence and similarity (pages 491 – 534)

CD ReferenceTransformations


Key IdeasIdentify congruent figuresInvestigate similar figures and interpret and construct scale drawings.

OutcomesSGS4.4 (page 156): Identifies congruent and similar two dimensional figures stating the relevant conditions.

Working MathematicallyStudents learn to recognise congruent figures in tessellations, art and design work (Reflecting) interpret and use scales in photographs, plans and drawings found in the media and/or other learning areas (Applying Strategies, Communicating) enlarge diagrams such as cartoons and pictures(Applying Strategies) apply similarity to finding lengths in the environment where it is impractical to measure directly eg heights of trees, buildings (Applying Strategies, Reasoning) apply geometrical facts, properties and relationships to solve problems such as finding unknown sides and angles in diagrams (Applying Strategies, Reasoning) justify their solutions to problems by giving reasons using their own words (Reasoning, Communicating) recognise that area, length of matching sides and angle sizes are preserved in congruent figures (Reasoning) recognise that shape, angle size and the ratio of matching sides are preserved in similar figures (Reasoning) recognise that similar and congruent figures are used in specific designs, architecture and art work eg works by Escher, Vasarely and Mondrian; or landscaping

in European formal gardens (Reflecting) find examples of similar and congruent figures embedded in designs from many cultures and historical periods (Reflecting) use dynamic geometry software to investigate the properties of geometrical figures (Applying Strategies, Reasoning)

Knowledge and SkillsStudents learn aboutCongruence identifying congruent figures by superimposing them through a

combination of rotations, reflections and translations matching sides and angles of two congruent polygons naming the vertices in matching order when using the symbol in a

Teaching, Learning and Assessment TRY THIS:Escher tessellations (page 496): Students create a tessellating tileHouses (page 501): Problem solvingSimilar Rectangles (page 516): Problem solving FOCUS ON WORKING MATHEMATICALLY: Geometry in art:

The work of Piet Mondrian (page 528)



congruence statement drawing congruent figures using geometrical instruments determining the condition for two circles to be congruent (equal radii)

Similarity using the term ‘similar’ for any two figures that have the same shape but

not necessarily the same size matching the sides and angles of similar figures naming the vertices in matching order when using the symbol lll in a

similarity statement determining that shape, angle size and the ratio of matching sides are

preserved in similar figures determining the scale factor for a pair of similar polygons determining the scale factor for a pair of circles calculating dimensions of similar figures using the enlargement or

reduction factor choosing an appropriate scale in order to enlarge or reduce a diagram constructing scale drawings drawing similar figures using geometrical instruments

There is an excellent interactive web link at http://www.stephen.com/mondrimat/ where you can paint in the style of Mondrian.There is an excellent access at http://www.artchive.com/ftp_site.htm where you can view the work of many artists. Note: there is an underscore '_' between 'ftp' and 'site'. Click on Mondrian and then VIEW IMAGE LISTYou can view the lay out of many gardens including the Chateau at Villandry in France at http://www.a-castle-for-rent.com/castles/gardens.htm EXTENSION ACTIVITIES, LET’S COMMUNICATE,

REFLECTING (page 529) CHAPTER REVIEW (page 532): A collection of problems to revise

the chapter.

TechnologyTransformations: the activities and questions make use of the interactive Cabri Geometry program in the Transformations.html page.


http://www.a-castle-for-rent.com/castles/gardens.htm

http://www.artchive.com/ftp_site.htm

http://www.stephen.com/mondrimat/

mathscape 7 teaching program - macmillan · web viewpythagoras’ theorem substrand...

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