stage 4 mathscape 7 - macmillan publishers · stage 4 mathscape 7 ... the long division interactive...

Mathscape 7 Teaching Program

Published by Macmillan Education Australia copy Macmillan Education Australia 2004

Stage 4

MATHSCAPE 7 Term Chapter Time

1 1 Whole numbers and number systems 3 weeks 12 hrs

2 Number theory 2 weeks 8 hrs

3 Time 2 weeks 8 hrs

2 4 Fractions 3 weeks 12 hrs

5 Number patterns and pronumerals 2 weeks 8 hrs

6 Decimals 3 weeks 12 hrs

3 7 Integers 2 weeks 8 hrs

8 Algebra 2 weeks 8 hrs

9 Angles 2 weeks 8 hrs

10 Properties of geometrical figures 2 weeks 8 hrs

4 11 Measurement length and perimeter 2 weeks 8 hrs

12 Solids 2 weeks 8 hrs

13 Area 2 weeks 8 hrs

14 Sets (Additional CD extension content) 1 week 4 hrs

Mathscape 7 Teaching Program Page 2


Chapter 1 Whole numbers and number systems Substrands Operations with whole numbers Integers

Duration 3 weeks 12 hours

Text reference Mathscape 7 Chapter 1 Whole numbers and number systems (pages 1ndash48)

CD reference Ancient spreadsheets Long multiplication Long division

Key ideas Explore other counting systems Use index notation for positive integral indices Apply mental strategies to aid computation Divide two-or three-digit numbers by a two-digit number Simplify expressions involving grouping symbols and apply order of operations

Outcomes NS 41 (page 56) Recognises the properties of special groups of whole numbers and applies a range of strategies to aid computation NS 42 (page 58) Compares orders and calculates with integers

Working mathematically Students learn to bull discuss the strengths and weaknesses of different number systems (Communicating Reasoning) bull describe and recognise the advantages of the HindundashArabic number system (Communicating Reasoning)

Knowledge and skills Students learn about bull using index notation to express powers of numbers (positive indices only) for

example 8 = 23 bull comparing the HindundashArabic number system with number systems from

different societies past and present bull using an appropriate non-calculator method to divide two- and three-digit

numbers by a two-digit number bull applying a range of mental strategies to aid computation bull a practical understanding of associativity and commutativity for example

2 times 7 times 5 = 7 times (2 times 5) = 70 bull to multiply a number by 12 first multiply by 6 and then double the result bull to multiply a number by 13 first multiply the number by ten and then add 3

times the number

Teaching learning and assessment bull Open ended questions

1 The answer to a multiplication is 75 What is the question 2 List ways of finding the result to 18 times 9 without using a calculator (Communicating Reasoning)

bull Investigations for class discussions and presentations 1 Discuss the need for an order of operations in solving a problem like 9 + 5 times 4 (Communicating Reasoning) 2 Clairersquos calculator is missing the 5 and 7 and the multiplication key Explain how she could show the number 57 on the screen (Communicating Reasoning) 3 Create number sentences for each of the numbers 1 to 10 by using four 2rsquos

and any operations For example 22221 +minus=

22

222 +=



bull to divide by 20 first halve the number and then divide by 10 bull a practical understanding of the distributive law for example to multiply any

number by 9 first multiply by 10 and then subtract the number bull using grouping symbols as an operator bull applying order of operations to simplify expressions

bull TRY THIS Ten pin bowling (page15) students explore the use of whole numbers in scoring a bowling game 3-digit numbers (page 23) students discover a number trick and why it works Add a sign (page 32) use operations to make the number sentence true

bull FOCUS ON WORKING MATHEMATICALLY The Mayan people of Mexico (page 45) teachers should look at the website httpwww-historymcsst-andacukhistory before using the activity in class Click on History Topics Index and then click on Mayan mathematics Students could be given some material to read before the lesson It is important to note that the place value system described in the Focus on working mathematically activity is the one used by the priests and astronomers They used it for describing observations of the stars and making calendar calculations Another good site is the Mayan World Study Center httpwwwmayacalendarcommayacalendarmenuhtml where you will find a Mayan calculator

bull EXTENSION ACTIVITIES LETrsquoS COMMUNICATE REFLECTING (page 47)

bull CHAPTER REVIEW (page 48) a collection of problems to revise the chapter

Technology Ancient spreadsheets interactive worksheet for students to explore how ancient number systems work Use for a presentation to the class or for individual discovery Worksheets included for students to play ldquoConversion Challenge I and IIrdquo in pairs Long multiplication students check their multiplication skills by using the program in interactive or non-interactive mode Use the worksheet to discover patterns in long multiplication Long division the long division interactive program encourages students to practice and explore the mathematics behind long division The associated worksheet also uses the terms dividend divisor quotient and remainder



Chapter 2 Number theory Substrands Operations with whole numbers Fractions decimals and percentages


Text reference Mathscape 7 Chapter 2 Number theory (pages 51ndash80)

CD reference All factors Prime numbers Prime factors LCM and HCF

Key ideas Investigate groups of positive whole numbers Determine and apply tests of divisibility Express a number as a product of its prime factors Find squaresrelated square roots cubesrelated cube roots Finding highest common factors and lowest common multiples

Outcomes NS 41 (page 56) Recognises the properties of special groups of whole numbers and applies a range of strategies to aid computation NS43 (page 63) Operates with fractions decimals percentages ratios and rates

Working mathematically Students learn to bull question whether it is more appropriate to use mental strategies or a calculator to find the square root of a given number (Questioning) bull apply tests of divisibility mentally as an aid to calculation (Applying Strategies) bull verify the various tests of divisibility (Reasoning)

Knowledge and Skills Students learn about bull expressing a number as a product of its prime factors bull using the notation for square root ( ) and cube root ( )3 bull recognising the link between squares and square roots and cubes and cube roots

for example 23 = 8 and 283 = bull exploring through numerical examples that

- ( ) 222 baab = for example (2 times 3)2 = 22 times 32 - baab times= for example 4949 times=times

bull finding square roots and cube roots of numbers expressed as a product of their prime factors

bull finding square roots and cube roots of numbers using a calculator after first


1 Using as many different operations as you like write an expression that equals 8

bull Investigations for class discussions and presentations 1 In pairs students are given a particular number group to investigate (for example Lucas numbers palindromic numbers Kaprekar numbers etc) and prepare a poster showing the number group and interesting features of it This could be extended to students creating their own number type and features of it 2 Find the values of P and Q so that 8P1Q is divisible by 9

bull Games 1 Twenty questions Teacher thinks of a number Students ask yesno questions to guess the number ndash for example lsquoIs it oddrsquo lsquoIs it a Fibonacci



estimating bull identifying special groups of numbers including figurate numbers palindromic

numbers Fibonacci numbers numbers in Pascalrsquos triangle bull determining and applying tests of divisibility

numberrsquo lsquoIs it perfectrsquo etc (Questioning) bull TRY THIS

Palindromes (page 57) students investigate if every whole number can be changed into a palindromic number A trick (page 65) students discover a number trick for odd and even numbers Mathematics of pool tables (page 71) investigation activity

bull FOCUS ON WORKING MATHEMATICALLY Our Beautiful Earth (page 77) Go to the University of St Andrews in Scotland httpwww-historymcsst-andacukhistory Click on History Topics Index and then click on Perfect Numbers Go back to the main menu click on Mathematicians and select Nicomachus for further information Teachers note that the 37 perfect numbers discovered to date are all even No one has yet proved that this might be true in general A good reference book on perfect numbers is Pickover Clifford (2001) The Wonder of Numbers Oxford University Press New York See chapter 94 pages 212ndash215 The sum of the proper divisors of 220 is 284 The sum of the proper divisors of 284 is 220 Such number pairs are called amicable numbers See Pickover (2001) pages 212ndash213 Students can explore this According to Pickover over 1000 amicable numbers have been found



Technology Use Excel to develop number patterns and do divisibility tests All factors a program that outputs all the factors of a number in a given range A PDF file is also included of all the factors of the numbers from 1 to 1000 For explanations of the games lsquoFactor Nimrsquo and lsquoFactor Fishrsquo as well as a list of associated questions for use with the program see lsquoAboutrsquo Prime numbers the program can calculate primes up to 10 000 000 by using the sieve of Eratosthenes Worksheet and lsquoPrime Compositersquo game Prime factors the program allows users to enter a number between 2 and 5002 and outputs the prime factors LCM and HCF program to find LCM and HCF of a given pair of numbers



Chapter 3 Time Substrand Time


Text reference Mathscape 7 Chapter 3 Time (pages 82ndash109)

CD reference Time calculator

Key ideas Perform operations involving time units Use international time zones to compare times Interpret a variety of tables and charts related to time

Outcomes MS43 (page 138) Performs calculations of time that involve mixed units

Working mathematically Students learn to bull plan the most efficient journey to a given destination involving a number of connections and modes of transport (Applying Strategies) bull ask questions about international time relating to everyday life for example whether a particular soccer game can be watched live on television during normal waking

hours (Questioning) bull solve problems involving calculations with mixed time units for exaple lsquoHow old is a person today if heshe was born on 3061989rsquo (Applying Strategies)

Knowledge and skills Students learn about bull adding and subtracting time mentally using bridging strategies for example from

245 to 300 is 15 minutes and from 300 to 500 is 2 hours so the time from 245 until 500 is 15 minutes + 2 hours = 2 hours 15 minutes

bull adding and subtracting time with a calculator using the lsquodegrees minutes secondsrsquo button

bull rounding calculator answers to the nearest minute or hour bull interpreting calculator displays for time calculations for example 225 on a

calculator display for time means 2 41 hours

bull comparing times and calculating time differences between major cities of the world for example lsquoGiven that London is 10 hours behind Sydney what time is it in London when it is 600 pm in Sydneyrsquo

bull interpreting and using tables relating to time for example tide charts sunrisesunset tables standard time zones bus train and airline timetables


1 What were you doing 1 million seconds ago Students guess then calculate (Questioning) 2 Give each student a copy of a different day from the TV guide Students write 5 questions for a different student to answer in the class about time (Questioning Applying Strategies)

bull Games 1 lsquoHow long is a minutersquo Without timing students sit quietly with their eyes closed and stand up when they think a minute has passed 2 The 40 second walk Starting at the back of the classroom students slowly walk to the front of the room so that they touch the front wall on their estimate of 40 seconds lapsed

bull TRY THIS lsquoSee you in portrsquo (page 88)Time problem



Pendulum clocks (page 102) Students investigate the length or pendulum needed for a swing time of 1 second

bull FOCUS ON WORKING MATHEMATICALLY The Calendars of the Mayan people of Mexico (page 106) Start with the St Andrews weblink as for chapter 1 The article on Mayan mathematics contains interesting information about their calendars At the bottom of the article click on The Mayan World Study Center httpwwwmayacalendarcommayacalendarmenuhtml where you will find further information about the Mayan calendars For specific information on how different religious calendars were constructed from observations of the sun and stars go to Calendars from the Sky httpwebexhibitsorgcalendars The answer to the extension activity is 630 AD



Technology Time calculator The program allows users to add and subtract time to and from dates and times and also lengths of time The length of time between dates and times can be calculated Also lengths of time can be converted to other units Worksheet also has research ideas for students For further instructions about using the timecalculator program read the Aboutpdf file



Chapter 4 Fractions Substrand Fractions decimals and percentages


Text reference Mathscape 7 Chapter 4 Fractions (pages 112ndash161)

CD reference Fraction simplifier

Key ideas Perform operations with fractions decimals and mixed numerals

Outcomes NS43 (page 63) Operates with fractions decimals percentages rates and ratios

Working Mathematically Students learn to bull explain multiplication of a fraction by a fraction using a diagram to illustrate the process (Reasoning Communicating) bull explain why division by a fraction is equivalent to multiplication by its reciprocal (Reasoning Communicating) bull recognise and explain incorrect operations with fractions for example explain why 7

341

32 ne+ (Applying Strategies Reasoning Communicating)

bull question the reasonableness of statements in the media that quote fractions (Questioning) bull solve a variety of real-life problems involving fractions (Applying Strategies)

Knowledge and skills Students learn about bull finding equivalent fractions bull reducing a fraction to its lowest equivalent form bull adding and subtracting fractions using written methods bull expressing improper fractions as mixed numerals and vice versa bull adding mixed numerals bull subtracting a fraction from a whole number

for example 31

32

32 2123 =minus+=minus

bull multiplying and dividing fractions and mixed numerals bull determining the effect of multiplying or dividing by a number less than one bull calculating fractions of quantities bull expressing one quantity as a fraction or a percentage of another for example

15 minutes is 41 or 25 of an hour


1 The answer is ⅝ What is the question if you are a) adding two fractions b) subtracting two fractions c) dividing two fractions d) multiplying two fractions (Questioning)

bull Investigations for class discussions and presentations 1 Explain why 7

341


2 Explain why 1 divided by ⅓ = 3 (Applying Strategies Reasoning Communicating)

bull TRY THIS Egyptian fractions (page 151) investigation activity Plus equals times (page 155) 2 times 2 = 2 + 2 Students discover whether a similar result works for fractions

bull FOCUS ON WORKING MATHEMATICALLY Printing Newspapers (page 156) a good site for teachers to view the area and



length to breadth relationship of paper sizes is at httpwwwclcamacuk~mgk25iso-paperhtml The guide to international paper sizes is at httpwwwtwicscom~edspaperpapersizehtml Some teachers may wish to omit the Extension Activity which introduces the term limit and is clearly beyond stage 4 However for those who do the representation of a double page of the SMH as 1 and the subsequent halving

process gives a clear visual meaning to the sum K++++161

81

41

21



Technology Fraction simplifier this spreadsheet will convert improper fractions to mixed fractions and draw a pi graph of a proper fraction This spreadsheet uses the GCD function if you see name appearing anywhere you will need to turn the GCD function on by going to the Tools menu and selecting Add-Ins and put a tick next to Analysis ToolPak



Chapter 5 Number patterns and pronumerals Substrands Algebraic Techniques Number Patterns Duration 2 weeks 8 hours

Text reference Mathscape 7 Chapter 5 Number patterns and pronumerals (pages 162ndash209)

CD reference Patterns Fibonacci sequence

Key ideas Use letters to represent numbers Recognise and use simple equivalent algebraic expressions Create record and describe number patterns using words Use algebraic symbols to translate descriptions of number patterns Represent number pattern relationships as points on a grid

Outcomes PAS41 (page 82) Uses letters to represent numbers and translates between words and algebraic symbols PAS42 (page 83) Creates records analyses and generalises number patterns using words and algebraic symbols in a variety of ways

Working mathematically Students learn to bull generate a variety of equivalent expressions that represent a particular situation or problem (Applying Strategies) bull describe relationships between the algebraic symbol system and number properties (Reflecting Communicating) bull link algebra with generalised arithmetic eg for the commutative property determine that a + b = b + a (Reflecting) bull determine equivalence of algebraic expressions by substituting a given number for the letter (Applying Strategies Reasoning) bull ask questions about how number patterns have been created and how they can be continued (Questioning) bull generate a variety of number patterns that increase or decrease and record them in more than one way (Applying Strategies Communicating) bull model and then record number patterns using diagrams words and algebraic symbols (Communicating) bull check pattern descriptions by substituting further values (Reasoning) bull describe the pattern formed by plotting points from a table and suggest another set of points that might form the same pattern (Communicating Reasoning) bull describe what has been learnt from creating patterns making connections with number facts and number properties (Reflecting) bull play lsquoguess my rulersquo games describing the rule in words and algebraic symbols where appropriate (Applying Strategies Communicating) bull represent and apply patterns and relationships in algebraic forms (Applying Strategies Communicating) bull explain why a particular relationship or rule for a given pattern is better than another (Reasoning Communicating) bull distinguish between graphs that represent an increasing number pattern and those that represent a decreasing number pattern (Communicating) bull determine whether a particular number pattern can be described using algebraic symbols (Applying Strategies Communicating)



Knowledge and skills Students learn about bull using letters to represent numbers and developing the notion that a letter is used to

represent a variable bull using concrete materials such as cups and counters to model

expressions that involve a variable and a variable plus a constant for example 1 +aa

expressions that involve a variable multiplied by a constant for example aa 32 sums and products for example )1(212 ++ aa equivalent expressions such as yyxyyxyyyxx ++=++=++++ )(222 and to assist with simplifying expressions such as

53)32()2()32()2(

+=+++=+++

aaaaa

bull using a process that consists of building a geometric pattern completing a table of values describing the pattern in words and algebraic symbols and representing the relationship on a graph

bull modelling geometric patterns using materials such as matchsticks to form squares for example hellip

bull describing the pattern in a variety of ways that relate to the different methods of building the squares and recording descriptions using words

bull forming and completing a table of values for a geometric pattern for example

Number of squares 1 2 3 4 5 10 100

Number of matchsticks 4 7 10 13 _ _ _

bull representing the values from the table on a number grid and describing the pattern formed by the points on the graph (note ndash the points should not be joined to form a line because values between the points have no meaning)

bull determining a rule in words to describe the pattern from the table ndash this needs to be expressed in function form relating the top-row and bottom-row terms in the table

bull describing the rule in words replacing the varying number by an algebraic symbol bull using algebraic symbols to create an equation that describes the pattern


1 The answer is 4ysup3 What is the question bull Reflecting

1 Write a letter to a friend who has missed to lessons on Algebra and explain the difference between asup2 and 2a (Reasoning Communicating)

bull TRY THIS Diagonals (page 189) discover patterns for the number of squares a diagonal passes through for rectangles of different dimensions Trips (page 205) number pattern problem

bull FOCUS ON WORKING MATHEMATICALLY Seeing is believing (page 207) try httpwwweducation2000comdemodemobotchtmlarithserhtm which is a good site to link the story of Gauss with the problem of adding arithmetic series Teachers can check out number patterns generally at httpwwwlearnerorgteacherslabmathpatterns In the Reflecting activity the emphasis is on exploring series for which the method does not work for example geometric series Students will not know the names of these series but can have a lot of fun exploring ideas





bull creating more than one equation to describe the pattern bull using the rule to calculate the corresponding value for a larger number bull using a process that consists of identifying a number pattern (including decreasing

patterns) completing a table of values describing the pattern in words and algebraic symbols and representing the relationship on a graph

bull completing a table of values for a number pattern for example

a 1 2 3 4 5 10 100 b 4 7 10 13 _ _ _

Technology Patterns allows users to create a sequence of 100 terms based on one two or three stage rules Graphs are also included so users can see the pattern Fibonacci sequence users can modify the Fibonacci sequence by changing the first two numbers The graphs included can show students how quickly the Fibonacci Sequence increases patterns in the Fibonacci sequence are also discussed the accompanying worksheet



Chapter 6 Decimals Substrand Fractions decimals and percentages


Text reference Mathscape 7 Chapter 6 Decimals (pages 214ndash253)

CD reference Decimals

Key ideas Performs operations with fractions decimals and mixed numerals

Outcomes NS43 (page 63) Operates with fractions decimals percentages ratio and rates

Working mathematically Students learn to bull question the reasonableness of statements in the media that quote fractions decimals or percentages for example lsquothe number of children in the average family is 23rsquo

(Questioning) bull interpret a calculator display in formulating a solution to a problem by appropriately rounding a decimal (Communicating Applying Strategies) bull solve a variety of real-life problems involving fractions decimals and percentages (Applying Strategies) bull use a number of strategies to solve unfamiliar problems including

- using a table - looking for patterns - simplifying the problem - drawing a diagram - working backwards (Applying Strategies)

bull guess and refine (Applying Strategies Communicating)

Knowledge and skills Students learn about bull adding subtracting multiplying and dividing decimals (for multiplication and

division limit operators to two-digits) bull determining the effect of multiplying or dividing by a number less than one bull rounding decimals to a given number of places bull using the notation for recurring (repeating) decimals for example

0333 33hellip = 30amp 0345 345 345hellip = 5430 ampamp bull converting fractions to decimals (terminating and recurring) and percentages bull converting terminating decimals to fractions and percentages


1 Write down 12 numbers between 145 and 273 2 Find two numbers that have a product of 06

bull TRY THIS The Dewey decimal system (page 219) students explain how the system is organised Judging Olympic diving (page 236)

bull FOCUS ON WORKING MATHEMATICALLY Olympic decathalon 2000 (page 250) teachers will be rewarded if they



bull calculating fractions decimals and percentages of quantities prepare this well in advance Students can be given material to explore before the Focus on working mathematically activity on the Olympic Decathlon A good website for a general introduction is httpwwwdecathlonusaorghomehtml At the time of writing there was a photo of Chris Huffins for students to view Answers Huffins needed 735 points to the win gold medal He had to run 4min 2929 s to get these points He actually ran 4 min 3871 s so he was 942 s short Huffins went into the 1500 m knowing that his best time to that date was 4 min 4970 s He ran 11 s faster than his personal best Teachers may wish to use this as a starting point for the Lets communicate discussion suggested on page 252 The real issue as far as the mathematics is concerned is the importance of decimals to elite athletes



Technology Decimals three interactive worksheets on place value adding and subtracting decimals Good for teacher instruction for the class



Chapter 7 Integers Substrand Integers


Text reference Mathscape 7 Chapter 7 Integers (pages 256ndash281)

CD reference Number line

Key ideas Perform operations with directed numbers Simplify expressions involving grouping symbols and apply order of operations

Outcomes NS42 (page 58) Compares orders and calculates with integers

Working mathematically Students learn to bull interpret the use of directed numbers in a real world context eg rise and fall of temperature (Communicating) bull construct a directed number sentence to represent a real situation (Communicating) bull apply directed numbers to calculations involving money and temperature (Applying Strategies Reflecting) bull use number lines in applications such as time lines and thermometer scales (Applying Strategies Reflecting) bull verify using a calculator or other means directed number operations eg subtracting a negative number is the same as adding a positive number (Reasoning) bull question whether it is more appropriate to use mental strategies or a calculator when performing operations with integers (Questioning)

Knowledge and skills Students learn about bull recognising the direction and magnitude of an integer bull placing directed numbers on a number line bull ordering directed numbers bull interpreting different meanings (direction or operation) for the + and ndash signs

depending on the context bull adding and subtracting directed numbers bull multiplying and dividing directed numbers bull using grouping symbols as an operator bull applying order of operations to simplify expressions bull keying integers into a calculator using the +ndash key bull using a calculator to perform operations with integers


1 List the many uses of directed numbers in the real world (Communicating) 2 Create a real world directed number problem

bull TRY THIS Temperature (page 262) estimation exercise Multiplication with directed numbers (page 272) modelling directed number multiplication on a number line

bull FOCUS ON WORKING MATHEMATICALLY The loss of the Russian nuclear submarine Kursk (page 276) the intention here was to provide an example of the use of directed numbers in the context of submarines However the size of this nuclear submarine is so great that it was thought teachers could capitalise on a comparison with a bus a jumbo jet and a football field Go to the CNN website



httpwww5cnncomSPECIALS2000submarine for more details and pictures of the Kursk disaster Teachers might explore depth sounders and profiles of the sea-bed to show the close links between mathematics science and technology



Technology Number Line activities and questions that make use of the NumberLine file which includes an interactive number line diagram This file requires the MicroWorlds Web Player plug-in to operate properly See the documentation notes for further instructions about this



Chapter 8 Algebra Substrand Algebraic techniques Duration 2 weeks 8 hours

Text reference Mathscape 7 Chapter 8 Algebra (pages 284ndash308)

CD reference Simplify Expand

Key ideas Translate between words and algebraic symbols and between algebraic symbols and words Recognise and use simple equivalent algebraic expressions Uses the algebraic symbols system to simplify expand and factorise simple algebraic expressions Substitute into algebraic expressions

Outcomes PAS41 (page 82) Uses letters to represent numbers and translates between words and algebraic symbols PAS43 (page 85) Uses the algebraic symbol system to simplify expand and factorise simple algebraic expressions

Working mathematically Students learn to bull generate a variety of equivalent expressions that represent a particular situation or problem (Applying Strategies) bull describe relationships between the algebraic symbol system and number properties (Reflecting Communicating) bull link algebra with generalised arithmetic eg for the commutative property determine that a + b = b + a (Reflecting) bull determine equivalence of algebraic expressions by substituting a given number for the letter (Applying Strategies Reasoning) bull generate a variety of equivalent expressions that represent a particular situation or problem (Applying Strategies) bull determine and justify whether a simplified expression is correct by substituting numbers for letters (Applying Strategies Reasoning) bull interpret statements involving algebraic symbols in other contexts eg creating and formatting spreadsheets (Communicating) bull explain why a particular algebraic expansion or factorisation is incorrect (Reasoning Communicating)

Knowledge and skills Students learn about bull recognising and using equivalent algebraic expressions for example

baba

abbawww

yyyyy

=divide

=times=times

=+++2

4

Teaching learning and assessment bull TRY THIS

Vital capacity (page 292) students use a formula to find out how much air their lungs hold Square magic (page 296) magic square puzzle Number think and back again (page 300) number puzzles

bull FOCUS ON WORKING MATHEMATICALLY Colouring Maps (page 305) having established Eulers formula in the Learning activity students might see if it works with the map of Australia on page 305 Some of the state boundaries are coasts which makes the exercise quite different The sea as a region surrounding the continent also has to be



bull translating between words and algebraic symbols and between algebraic symbols and words

bull recognising like terms and adding and subtracting like terms to simplify algebraic expressions for example nmnmn 3442 +=++

bull recognising the role of grouping symbols and the different meanings of expressions such as 12 +a and ( )12 +a

bull simplifying algebraic expressions that involve multiplication and division for example

aabxa

3234312

timestimesdivide

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

ababaa

xxaa

+=+

minusminus=+minus+=+

2)(

105)2(563)2(3

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

bull substituting into algebraic expressions bull translating from everyday language to algebraic language and from algebraic

language to everyday language

taken into account However it does enable a teacher to ask whether Eulers formula will work if the boundaries are not straight lines Students could be asked about Tasmania ndash should it be included Here are some sites to explore Eulers formula further

httpwwwmathohio-stateedu~fiedorowmath655Eulerhtml this website gives nice clear pictures of the formula applying to three-dimensional solids httpwwwmathucalgaryca~lafcolorful4colorshtml this site has an interesting game to play with the four colour problem httpwwwuwinnipegca~ooellermguthrieFourColorhtml this site will tell you more about Francis Guthrie and the connection of the four colour problem to rare flowers in South Africa



Technology Simplify 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 Teachers may wish to use these programs as an introduction to discovering how the algebraic methods of expanding and simplifying work



Chapter 9 Angles Substrand Angles


Text reference Mathscape 7 Chapter 9 Angles (pages 310ndash356)

CD reference Angles Angle pairs Polygon angles

Key ideas Classify angles and determine angle relationships Construct parallel and perpendicular lines and determine associated angle properties Complete simple numerical exercises based on geometrical properties

Outcomes SGS42 (page 153) Identifies and names angles formed by the intersection of straight lines including those related to transversals on sets of parallel lines and makes use of the relationships between them

Working mathematically Students learn to bull recognise and explain why adjacent angles adding to 90ordm form a right angle (Reasoning) bull recognise and explain why adjacent angles adding to 180ordm form a straight angle (Reasoning) bull recognise and explain why adjacent angles adding to 360ordm form a complete revolution (Reasoning) bull find the unknown angle in a diagram using angle results giving reasons (Applying Strategies Reasoning) bull apply angle results to construct a pair of parallel lines using a ruler and a protractor a ruler and a set square or a ruler and a pair of compasses (Applying Strategies) bull apply angle and parallel line results to determine properties of two-dimensional shapes such as the square rectangle parallelogram rhombus and trapezium (Applying

Strategies Reasoning Reflecting) bull identify parallel and perpendicular lines in the environment (Reasoning Reflecting ) bull 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) bull use dynamic geometry software to investigate angle relationships (Applying Strategies Reasoning)

Knowledge and skills Students learn about Angles at a Point

bull labelling and naming points lines and intervals using capital letters bull labelling the vertex and arms of an angle with capital letters bull labelling and naming angles using angA and angXYZ notation bull using the common conventions to indicate right angles and equal angles on

diagrams


Angular vision (page 327) students measure their widest range of view Leaning towers (page 340) when will the Leaning Tower of Pisa fall over Mirror bounce (page 351) practical

bull FOCUS ON WORKING MATHEMATICALLY The Sunrsquos rays (page 354) this activity makes use of the right-angled isosceles triangle to calculate inaccessible heights Similar triangles have not been introduced at this stage of the course The measurement of the heights of



bull 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

bull using the words lsquocomplementaryrsquo and lsquosupplementaryrsquo for angles adding to 90ordm and 180ordm respectively and the terms lsquocomplementrsquo and lsquosupplementrsquo

bull establishing and using the equality of vertically opposite angles Angles Associated with Transversals

bull identifying and naming a pair of parallel lines and a transversal bull using common symbols for lsquois parallel torsquo ( ) and lsquois perpendicular torsquo ( perp ) bull using the common conventions to indicate parallel lines on diagrams bull 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

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

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

the Pyramids of Egypt using a shadow stick provides an historical connection to early geometry A good website to start exploring ancient Egypt is httpwwweyelidcouk At this site you get a good overall view of the geometry of different pyramids and how they were constructed To see the Great Pyramid of Khufu go to httpwwwguardiansnetegyptgp1htm The site httpwwwcrystalinkscomegypthtml has a video with it and some interesting diagrams of the tombs and their construction The suns rays are an interesting context in which to study angle Not just types of angles but the heating effect of the size of the angle at which the suns rays strike Earth The tilt of the Earth (235deg) means that equal sized rays from the sun have to cover different sized areas depending on the angle at which they arrive This explains the hot tropics around the equator the temperate zones and the cold polar regions Two good websites to explore are James Risers website httpwwwk12trainingcomJamesRiserScienceseasonsseasons2htm and David Sterns From Stargazers to Starships httpwww-istpgsfcnasagovstargazeSintrohtm Scroll down the menu for Astronomy of the Earths motion in space and click on 4 Please note that the diagrams are for the northern hemisphere If you use this site make sure you make the appropriate adjustments to the summer and winter dates



Technology Angles this file includes two interactive geometric diagrams By dragging the lines or points students can observe how the angle size and type changes 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



Chapter 10 Properties of geometrical figures Substrand Properties of geometrical figures Duration 2 weeks 8 hours

Text reference Mathscape 7 Chapter 10 Properties of geometrical figures (pages 362ndash406)

CD reference Plane shapes

Key ideas Classify construct and determine properties of triangles and quadrilaterals Complete simple numerical exercises based on geometrical properties

Outcomes SGS43 (page 154) Classifies constructs and determines the properties of triangles and quadrilaterals

Working mathematically Students learn to bull sketch and label triangles and quadrilaterals from a given verbal description (Communicating) bull describe a sketch in sufficient detail for it to be drawn (Communicating) bull recognise that a given triangle may belong to more than one class (Reasoning) bull recognise that the longest side of a triangle is always opposite the largest angle (Applying Strategies Reasoning) bull recognise and explain why two sides of a triangle must together be longer than the third side (Applying Strategies Reasoning) bull recognise special types of triangles and quadrilaterals embedded in composite figures or drawn in various orientations (Communicating) bull determine if particular triangles and quadrilaterals have line andor rotational symmetry (Applying Strategies) bull apply geometrical facts properties and relationships to solve numerical problems such as finding unknown sides and angles in diagrams (Applying Strategies) use dynamic geometry software to investigate the properties of geometrical figures(Applying Strategies Reasoning)

Knowledge and skills Students learn about Notation bull labelling and naming triangles (eg ABC) and quadrilaterals (eg ABCD) in text and

on diagrams bull using the common conventions to mark equal intervals on diagrams Triangles bull recognising and classifying types of triangles on the basis of their properties

(acute-angled triangles right-angled triangles obtuse-angled triangles scalene triangles isosceles triangles and equilateral triangles)

bull justifying informally by paper folding or cutting and testing by measuring that


Triangle trouble (page 378) can an equilateral triangle be rearranged to form a square Angle sum of a polygon (page 395)

bull FOCUS ON WORKING MATHEMATICALLY Stars (page 403) most websites are too complex for the properties of the pentagram at this stage However there is a colourful fun game at httpwwwkidwizardcomGamesNumberMagicPentagonasp which involves counting triangles in pentagrams and shows the fractal-like repetition of the pentagram as you join diagonals inside This site gives access to a wide



the interior angle sum of a triangle is 180ordm and that any exterior angle equals the sum of the two interior opposite angles

bull proving using a parallel line construction that any exterior angle of a triangle is equal to the sum of the two interior opposite angles

Quadrilaterals bull distinguishing between convex and non-convex quadrilaterals (the diagonals of a

convex quadrilateral lie inside the figure) bull establishing that the angle sum of a quadrilateral is 360ordm bull constructing various types of quadrilaterals bull investigating the properties of special quadrilaterals (trapeziums kites

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

ndash opposite sides parallel ndash opposite sides equal ndash adjacent sides perpendicular ndash opposite angles equal ndash diagonals equal in length ndash diagonals bisect each other ndash diagonals bisect each other at right angles ndash diagonals bisect the angles of the quadrilateral

range of other games to explore In the Extension activity teachers may wish to raise the question as to why for a pentagram the angle sum of the pointed angles is 180deg whereas for six and eight sides it is 360deg Some advanced students may wish to investigate further with more polygons There is also a simple proof based on exterior angles for the sum A + B + C + D + E in the Learning activity Find an exterior angle equal to A + C Then find another for B + D These two angles lie in the small triangle with apex A at the top of the star What do you conclude This proof will be outside the knowledge base for most stage 4 students



Technology Plane shapes the file is Adobe Acrobat Reader format and the shapes can be printed as flash cards mathematical decorations or templates for cutting out cardboard shapes to make the faces for solids Good accompanying worksheet



Chapter 11 Measurement length and perimeter Substrand Perimeter and area


Text reference Mathscape 7 Chapter 11 Measurement length and perimeter (pages 410ndash444)

CD reference Measuring plane shapes

Key ideas Describe the limits of accuracy of measuring instruments Develop formulae and use to find the area and perimeter of triangles rectangles and parallelograms Convert between metric units of length and area

Outcomes MS41 (page 124) Uses formulae and Pythagorasrsquo theorem in calculating perimeter and area of circles and figures composed of rectangles and triangles

Working mathematically Students learn to bull consider the degree of accuracy needed when making measurements in practical situations (Applying Strategies) bull choose appropriate units of measurement based on the required degree of accuracy (Applying Strategies) bull make reasonable estimates for length and area and check by measuring (Applying Strategies) bull select and use appropriate devices to measure lengths and distances (Applying Strategies) bull discuss why measurements are never exact (Communicating Reasoning) bull solve problems relating to perimeter area and circumference (Applying Strategies)

Knowledge and skills Students learn about Length and Perimeter bull estimating lengths and distances using visualisation strategies bull recognising that all measurements are approximate bull describing the limits of accuracy of measuring instruments ( plusmn 05 unit of

measurement) bull interpreting the meaning of the prefixes lsquomillirsquo lsquocentirsquo and lsquokilorsquo bull converting between metric units of length bull finding the perimeter of simple composite figures


Police patrol (page 415) problem solving Small thickness (page 421) how do you measure the thickness of a piece of paper Mobius Strips (page 435) students construct and analyse a Mobius strip

bull FOCUS ON WORKING MATHEMATICALLY Baseball (page 440) the shape of the field in the diagram on page 441 has a simplified outfield Teachers might compare with the photograph on page 440 This is to make the instructions easier to follow The purpose of the Focus on working mathematically activity is to explore measurement especially length and perimeter in the context of baseball Students are asked however to



reproduce a scale drawing of the field Teachers can simplify the task at their discretion The pitchers plate is 10 inches (about 25 cm) above the other bases Students may like to discuss why The website httpwwwbaseball-almanaccomrule1shtml is a good source for teachers who wish to access the rules of baseball However the major league website httpwwwmlbcom will be more than sufficient You will need to scroll down to the bottom of the page to locate baseball basics



Technology Measuring Plane Shapes this file contains hyperlinks to a number of interactive geometric diagrams



Chapter 12 Solids Substrand Properties of solids


Text Reference Mathscape 7 Chapter 12 Solids (pages 447ndash479)

CD Reference Solids

Key ideas Determine properties of three-dimensional objects Investigate Platonic solids Investigate Eulerrsquos relationship for convex polyhedra Make isometric drawings

Outcomes SGS41 (page 147) Describes and sketches three-dimensional solids including polyhedra and classifies them in terms of their properties

Working mathematically Students learn to bull interpret and make models from isometric drawings (Communicating) bull recognise solids with uniform and non-uniform cross-sections (Communicating) bull analyse three-dimensional structures in the environment to explain why they may be particular shapes for example buildings packaging (Reasoning) bull visualise and name a common solid given its net (Communicating) bull recognise whether a diagram is a net of a solid (Communicating) bull identify parallel perpendicular and skew lines in the environment (Communicating Reflecting)

Knowledge and skills Students learn about bull describing solids in terms of their geometric properties

ndash number of faces ndash shape of faces ndash number and type of congruent faces ndash number of vertices ndash number of edges ndash convex or non-convex

bull identifying any pairs of parallel flat faces of a solid bull determining if two straight edges of a solid are intersecting parallel or skew bull determining if a solid has a uniform cross-section

Teaching learning and assessment bull Investigations for class discussions and presentations

1 Students create a scale model of a solid from an isometric drawing or net bull TRY THIS

Painted cube (page 460) problem solving The Soma Puzzle (page 475)

bull FOCUS ON WORKING MATHEMATICALLY Shapely Thinking Cones and Conic Sections (page 477) Training students to form and manipulate mathematical images is an important part of their mathematical development This activity focuses on sections of a cone Teachers may wish to use perspex or wooden models of the conic sections after students have had a go at imagining what they look like Handling these



bull classifying solids on the basis of their properties ndash a polyhedron is a solid whose faces are all flat ndash a prism has a uniform polygonal cross-section ndash a cylinder has a uniform circular cross-section ndash a pyramid has a polygonal base and one further vertex (the apex) ndash a cone has a circular base and an apex

bull All points on the surface of a sphere are a fixed distance from its centre bull identifying right prisms and cylinders and oblique prisms and cylinders bull identifying right pyramids and cones and oblique pyramids and cones bull sketching on isometric grid paper shapes built with cubes bull representing three-dimensional objects in two dimensions from different views bull confirming for various convex polyhedra Eulerrsquos formula F + V = E + 2

relating the number of faces (F) the number of vertices (V) and the number of edges (E)

bull exploring the history of Platonic solids and how to make them bull making models of polyhedra

models is better than just looking at the websites The links with nature are important and addressed in the extension activities Websites abound but check out the five Platonic solids and conic sections at httphometeleportcom~tpgettysplatonicshtml The University of Utah has an interactive site where students can rotate and examine Platonic solids httpwwwmathutahedu~alfeldmathpolyhedrapolyhedrahtml Mathworld has a nice diagram of the conic sections at httpmathworldwolframcomConicSectionhtml



Technology Solids two files are included ndash the Solids file which includes animated diagrams of various solids and the Paper Folding file which includes nets of six interesting polyhedral solids



Chapter 13 Area Substrand Perimeter and Area Duration 2 weeks 8 hours

Text reference Mathscape 7 Chapter 13 Area (pages 401ndash509)


Key ideas Describes the limits of accuracy of measuring instruments Develop formulae and use to find the area and perimeter of triangles rectangles and parallelograms Find the areas of simple composite figures Convert between metric units of length and area


Working mathematically Students learn to bull consider the degree of accuracy needed when making measurements in practical situations (Applying Strategies) bull choose appropriate units of measurement based on the required degree of accuracy (Applying Strategies) bull make reasonable estimates for length and area and check by measuring (Applying Strategies) bull select and use appropriate devices to measure lengths and distances (Applying Strategies) bull discuss why measurements are never exact (Communicating Reasoning) bull find the dimensions of a square given its perimeter and of a rectangle given its perimeter and one side length (Applying Strategies) bull solve problems relating to perimeter area and circumference (Applying Strategies) bull 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) bull compare various shapes with the same perimeter and ask questions related to their area such as whether they have the same area (Questioning) bull explain the relationship that multiplying dividing squaring and factoring have with the areas of squares and rectangles with integer side lengths (Reflecting)

Knowledge and skills Students learn about Areas of squares rectangles triangles and parallelograms bull developing and using formulae for the area of a square and rectangle bull developing (by forming a rectangle) and using the formula for the area of a

triangle bull finding the areas of simple composite figures that may be dissected into rectangles

and triangles


1 Draw a rectangle with an area of 12cmsup2 2 Compare rectangles with the same area Do they have the same perimeter (Questioning Applying Strategies Reasoning)

bull TRY THIS How many people are there in your classroom (page 492) estimation exercise



bull developing the formula by practical means for finding the area of a parallelogram eg by forming a rectangle using cutting and folding techniques

bull converting between metric units of area 1 cm2 = 100 mm2 1 m2 = 1 000 000 mm2 1 ha = 10 000 m2 1 km2 = 1 000 000 m2 = 100 ha

Building blocks (page 498) problem solving bull FOCUS ON WORKING MATHEMATICALLY

Goal The world cup 2002 (page 506) the website httpfifaworldcupyahoocom gives access to all the information students need to answer question 5 of the Extension activities for the World Cup 2002 When you get into the site select English then tournament then scroll down to statistics and select goalkeepers The Reflecting exercise can be done at the same time as this data is considered If teachers need information about the playing field and laws of the game go to httpwwwfifacomengame






Chapter 1 Whole numbers and number systems Substrands Operations with whole numbers Integers


Text reference Mathscape 7 Chapter 1 Whole numbers and number systems (pages 1ndash48)

CD reference Ancient spreadsheets Long multiplication Long division

Key ideas Explore other counting systems Use index notation for positive integral indices Apply mental strategies to aid computation Divide two-or three-digit numbers by a two-digit number Simplify expressions involving grouping symbols and apply order of operations

Outcomes NS 41 (page 56) Recognises the properties of special groups of whole numbers and applies a range of strategies to aid computation NS 42 (page 58) Compares orders and calculates with integers

Working mathematically Students learn to bull discuss the strengths and weaknesses of different number systems (Communicating Reasoning) bull describe and recognise the advantages of the HindundashArabic number system (Communicating Reasoning)

Knowledge and skills Students learn about bull using index notation to express powers of numbers (positive indices only) for

example 8 = 23 bull comparing the HindundashArabic number system with number systems from

different societies past and present bull using an appropriate non-calculator method to divide two- and three-digit

numbers by a two-digit number bull applying a range of mental strategies to aid computation bull a practical understanding of associativity and commutativity for example

2 times 7 times 5 = 7 times (2 times 5) = 70 bull to multiply a number by 12 first multiply by 6 and then double the result bull to multiply a number by 13 first multiply the number by ten and then add 3

times the number


1 The answer to a multiplication is 75 What is the question 2 List ways of finding the result to 18 times 9 without using a calculator (Communicating Reasoning)

bull Investigations for class discussions and presentations 1 Discuss the need for an order of operations in solving a problem like 9 + 5 times 4 (Communicating Reasoning) 2 Clairersquos calculator is missing the 5 and 7 and the multiplication key Explain how she could show the number 57 on the screen (Communicating Reasoning) 3 Create number sentences for each of the numbers 1 to 10 by using four 2rsquos

and any operations For example 22221 +minus=

22

222 +=











































































341




for example 31

32







341









81

41

21


















53)32()2()32()2(

+=+++=+++

aaaaa






















a 1 2 3 4 5 10 100 b 4 7 10 13 _ _ _

























































baba

abbawww

yyyyy

=divide

=times=times

=+++2

4










aabxa

3234312

timestimesdivide


ababaa

xxaa

+=+


2)(

105)2(563)2(3





















diagrams





































































CD Reference Solids
































and triangles























































































341




for example 31

32







341









81

41

21


















53)32()2()32()2(

+=+++=+++

aaaaa






















a 1 2 3 4 5 10 100 b 4 7 10 13 _ _ _

























































baba

abbawww

yyyyy

=divide

=times=times

=+++2

4










aabxa

3234312

timestimesdivide


ababaa

xxaa

+=+


2)(

105)2(563)2(3





















diagrams





































































CD Reference Solids
































and triangles

















81

41

21


















53)32()2()32()2(

+=+++=+++

aaaaa






















a 1 2 3 4 5 10 100 b 4 7 10 13 _ _ _

























































baba

abbawww

yyyyy

=divide

=times=times

=+++2

4










aabxa

3234312

timestimesdivide


ababaa

xxaa

+=+


2)(

105)2(563)2(3





















diagrams





































































CD Reference Solids
































and triangles



























53)32()2()32()2(

+=+++=+++

aaaaa






















a 1 2 3 4 5 10 100 b 4 7 10 13 _ _ _

























































baba

abbawww

yyyyy

=divide

=times=times

=+++2

4










aabxa

3234312

timestimesdivide


ababaa

xxaa

+=+


2)(

105)2(563)2(3





















diagrams





































































CD Reference Solids
































and triangles


















a 1 2 3 4 5 10 100 b 4 7 10 13 _ _ _

























































baba

abbawww

yyyyy

=divide

=times=times

=+++2

4










aabxa

3234312

timestimesdivide


ababaa

xxaa

+=+


2)(

105)2(563)2(3





















diagrams





































































CD Reference Solids
































and triangles




































































baba

abbawww

yyyyy

=divide

=times=times

=+++2

4










aabxa

3234312

timestimesdivide


ababaa

xxaa

+=+


2)(

105)2(563)2(3





















diagrams





































































CD Reference Solids
































and triangles



















aabxa

3234312

timestimesdivide


ababaa

xxaa

+=+


2)(

105)2(563)2(3





















diagrams





































































CD Reference Solids
































and triangles

























diagrams





































































CD Reference Solids
































and triangles














































































CD Reference Solids
































and triangles


































and triangles













stage 4 mathscape 7 - macmillan publishers · stage 4 mathscape 7 ... the long division interactive...

Documents