Mathscape 7 Teaching Program
Page 1
Stage 4
MATHSCAPE 7
Term Chapter
1
2
3
4
Time
1. Whole numbers and number systems
3 weeks / 12 hrs
2. Number theory
2 weeks / 8 hrs
3. Time
2 weeks / 8 hrs
4. Fractions
3 weeks / 12 hrs
5. Number patterns and pronumerals
2 weeks / 8 hrs
6. Decimals
3 weeks / 12 hrs
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
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
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Mathscape 7 Teaching Program
Page 2
Chapter 1. Whole numbers and number systems
Text reference
CD reference
Substrands
Mathscape 7
Chapter 1. Whole numbers and number
systems (pages 1–48)
Ancient spreadsheets
Long multiplication
Long division
Operations with whole numbers
Integers
Duration
3 weeks / 12 hours
Key ideas
Outcomes
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.
NS 4.1 (page 56) Recognises the properties of special groups of whole numbers and
applies a range of strategies to aid computation.
NS 4.2 (page 58) Compares, orders and calculates with integers.
Working mathematically
Students learn to
• discuss the strengths and weaknesses of different number systems (Communicating, Reasoning)
• describe and recognise the advantages of the Hindu–Arabic number system (Communicating, Reasoning)
Knowledge and skills
Teaching, learning and assessment
Students learn about
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using index notation to express powers of numbers (positive indices only), for
example 8 = 23
comparing the Hindu–Arabic number system with number systems from
different societies past and present
using an appropriate non-calculator method to divide two- and three-digit
numbers by a two-digit number
applying a range of mental strategies to aid computation
a practical understanding of associativity and commutativity, for example
2 × 7 × 5 = 7 × (2 × 5) = 70
to multiply a number by 12, first multiply by 6 and then double the result
to multiply a number by 13, first multiply the number by ten and then add 3
times the number
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Open ended questions
1. The answer to a multiplication is 75. What is the question?
2. List ways of finding the result to 18 × 9 without using a calculator
(Communicating, Reasoning)
Investigations for class discussions and presentations
1. Discuss the need for an order of operations in solving a problem like
9 + 5 × 4 (Communicating, Reasoning)
2. Claire’s 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 2’s
2
2 2
and any operations. For example 1 = 2 − 2 + , 2 = +
2
2 2
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Mathscape 7 Teaching Program
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to divide by 20, first halve the number and then divide by 10
a practical understanding of the distributive law, for example to multiply any
number by 9 first multiply by 10 and then subtract the number
using grouping symbols as an operator
applying order of operations to simplify expressions
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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.
FOCUS ON WORKING MATHEMATICALLY
The Mayan people of Mexico (page 45): teachers should look at the website
http://www-history.mcs.st-and.ac.uk/history 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,
http://www.mayacalendar.com/mayacalendar/menu.html, where you will
find a Mayan calculator.
EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING
(page 47)
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 “Conversion Challenge I and II” 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.
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Mathscape 7 Teaching Program
Page 4
Chapter 2. Number theory
Text reference
CD reference
Substrands
Mathscape 7
Chapter 2. Number theory (pages 51–80)
All factors
Prime numbers
Prime factors
LCM and HCF
Operations with whole numbers
Fractions, decimals and percentages
Duration
2 weeks / 8 hours
Key ideas
Outcomes
Investigate groups of positive whole numbers.
Determine and apply tests of divisibility.
Express a number as a product of its prime factors.
Find squares/related square roots; cubes/related cube roots.
Finding highest common factors and lowest common multiples.
NS 4.1 (page 56) Recognises the properties of special groups of whole numbers
and applies a range of strategies to aid computation.
NS4.3 (page 63) Operates with fractions, decimals, percentages, ratios and rates.
Working mathematically
Students learn to
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question whether it is more appropriate to use mental strategies or a calculator to find the square root of a given number (Questioning)
apply tests of divisibility mentally as an aid to calculation (Applying Strategies)
verify the various tests of divisibility (Reasoning)
Knowledge and Skills
Teaching, learning and assessment
Students learn about
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expressing a number as a product of its prime factors
using the notation for square root
and cube root
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recognising the link between squares and square roots and cubes and cube roots,
for example 23 = 8 and 3 8 = 2
exploring through numerical examples that:
2
- (ab ) = a 2b 2 , for example (2 × 3)2 = 22 × 32
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( )
( )
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- ab = a × b , for example 9 × 4 = 9 × 4
finding square roots and cube roots of numbers expressed as a product of their
prime factors
finding square roots and cube roots of numbers using a calculator, after first
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Open ended questions
1. Using as many different operations as you like, write an expression that
equals 8.
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
Games
1. Twenty questions: Teacher thinks of a number. Students ask yes/no
questions to guess the number – for example ‘Is it odd?’, ‘Is it a Fibonacci
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Mathscape 7 Teaching Program
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estimating
identifying special groups of numbers including figurate numbers, palindromic
numbers, Fibonacci numbers, numbers in Pascal’s triangle
determining and applying tests of divisibility
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number?’, ‘Is it perfect?’, etc. (Questioning)
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.
FOCUS ON WORKING MATHEMATICALLY
Our Beautiful Earth (page 77): Go to the University of St. Andrews in
Scotland, http://www-history.mcs.st-and.ac.uk/history. 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 212–215.
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 212–213. Students can explore this. According to Pickover, over
1000 amicable numbers have been found.
EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING
(page 78)
CHAPTER REVIEW (page 80) a collection of problems to revise the chapter.
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 ‘Factor Nim’ and ‘Factor Fish’ as well as a list of associated questions for use with the program, see ‘About’.
Prime numbers: the program can calculate primes up to 10 000 000 by using the sieve of Eratosthenes. Worksheet and ‘Prime Composite’ 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.
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Mathscape 7 Teaching Program
Page 6
Chapter 3. Time
Text reference
CD reference
Substrand
Mathscape 7
Chapter 3. Time (pages 82–109)
Time calculator
Time
Duration
2 weeks / 8 hours
Key ideas
Outcomes
Perform operations involving time units.
Use international time zones to compare times.
Interpret a variety of tables and charts related to time.
MS4.3 (page 138) Performs calculations of time that involve mixed units.
Working mathematically
Students learn to
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plan the most efficient journey to a given destination involving a number of connections and modes of transport (Applying Strategies)
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)
solve problems involving calculations with mixed time units, for exaple ‘How old is a person today if he/she was born on 30/6/1989?’ (Applying Strategies)
Knowledge and skills
Teaching, learning and assessment
Students learn about
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adding and subtracting time mentally using bridging strategies, for example from
2:45 to 3:00 is 15 minutes and from 3:00 to 5:00 is 2 hours, so the time from 2:45
until 5:00 is 15 minutes + 2 hours = 2 hours 15 minutes
adding and subtracting time with a calculator using the ‘degrees, minutes,
seconds’ button
rounding calculator answers to the nearest minute or hour
interpreting calculator displays for time calculations, for example 2.25 on a
calculator display for time means 2 14 hours
comparing times and calculating time differences between major cities of the
world, for example ‘Given that London is 10 hours behind Sydney, what time is it
in London when it is 6:00 pm in Sydney?’
interpreting and using tables relating to time, for example tide charts,
sunrise/sunset tables, standard time zones, bus, train and airline timetables
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Open ended questions
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)
Games
1. ‘How long is a minute?’ 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.
TRY THIS
‘See you in port’ (page 88):Time problem.
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Mathscape 7 Teaching Program
Page 7
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Pendulum clocks (page 102): Students investigate the length or pendulum
needed for a swing time of 1 second.
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,
http://www.mayacalendar.com/mayacalendar/menu.html, 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,
http://webexhibits.org/calendars/.
The answer to the extension activity is 630 AD.
EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING
(page 108)
CHAPTER REVIEW (page 109) a collection of problems to revise the
chapter.
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 About.pdf file.
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Mathscape 7 Teaching Program
Page 8
Chapter 4. Fractions
Text reference
CD reference
Substrand
Mathscape 7
Chapter 4. Fractions (pages 112–161)
Fraction simplifier
Fractions, decimals and percentages
Duration
3 weeks / 12 hours
Key ideas
Outcomes
Perform operations with fractions, decimals and mixed numerals.
NS4.3 (page 63) Operates with fractions, decimals, percentages, rates and ratios.
Working Mathematically
Students learn to
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explain multiplication of a fraction by a fraction using a diagram to illustrate the process (Reasoning, Communicating)
explain why division by a fraction is equivalent to multiplication by its reciprocal (Reasoning, Communicating)
recognise and explain incorrect operations with fractions, for example explain why 23 + 14 ≠ 73 (Applying Strategies, Reasoning, Communicating)
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question the reasonableness of statements in the media that quote fractions (Questioning)
solve a variety of real-life problems involving fractions (Applying Strategies)
Knowledge and skills
Teaching, learning and assessment
Students learn about
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finding equivalent fractions
reducing a fraction to its lowest equivalent form
adding and subtracting fractions using written methods
expressing improper fractions as mixed numerals and vice versa
adding mixed numerals
subtracting a fraction from a whole number
for example 3 − 23 =2 + 1 − 23 =2 13
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multiplying and dividing fractions and mixed numerals
determining the effect of multiplying or dividing by a number less than one
calculating fractions of quantities
expressing one quantity as a fraction or a percentage of another, for example
15 minutes is 14 or 25% of an hour
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Open ended questions
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)
Investigations for class discussions and presentations
1. Explain why 23 + 14 ≠ 73 (Applying Strategies, Reasoning, Communicating)
2. Explain why 1 divided by ⅓ = 3 (Applying Strategies, Reasoning,
Communicating)
TRY THIS
Egyptian fractions (page 151): investigation activity.
Plus equals times! (page 155): 2 × 2 = 2 + 2. Students discover whether a
similar result works for fractions.
FOCUS ON WORKING MATHEMATICALLY
Printing Newspapers (page 156): a good site for teachers to view the area and
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Mathscape 7 Teaching Program
Page 9
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length to breadth relationship of paper sizes is at
http://www.cl.cam.ac.uk/~mgk25/iso-paper.html. The guide to international
paper sizes is at http://www.twics.com/~eds/paper/papersize.html. 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
1 1 1 1
process gives a clear visual meaning to the sum + + + + K ?
2 4 8 16
EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING
(page 158)
CHAPTER REVIEW (page 159) a collection of problems to revise the
chapter.
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.
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Mathscape 7 Teaching Program
Page 10
Chapter 5. Number patterns and pronumerals
Text reference
CD reference
Substrands
Mathscape 7
Chapter 5. Number patterns and
pronumerals (pages 162–209)
Patterns
Fibonacci sequence
Algebraic Techniques
Number Patterns
Duration
2 weeks / 8 hours
Key ideas
Outcomes
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.
PAS4.1 (page 82) Uses letters to represent numbers and translates between words
and algebraic symbols.
PAS4.2 (page 83) Creates, records, analyses and generalises number patterns
using words and algebraic symbols in a variety of ways.
Working mathematically
Students learn to
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generate a variety of equivalent expressions that represent a particular situation or problem (Applying Strategies)
describe relationships between the algebraic symbol system and number properties (Reflecting, Communicating)
link algebra with generalised arithmetic eg for the commutative property, determine that a + b = b + a (Reflecting)
determine equivalence of algebraic expressions by substituting a given number for the letter (Applying Strategies, Reasoning)
ask questions about how number patterns have been created and how they can be continued (Questioning)
generate a variety of number patterns that increase or decrease and record them in more than one way (Applying Strategies, Communicating)
model and then record number patterns using diagrams, words and algebraic symbols (Communicating)
check pattern descriptions by substituting further values (Reasoning)
describe the pattern formed by plotting points from a table and suggest another set of points that might form the same pattern (Communicating, Reasoning)
describe what has been learnt from creating patterns, making connections with number facts and number properties (Reflecting)
play ‘guess my rule’ games, describing the rule in words and algebraic symbols where appropriate (Applying Strategies, Communicating)
represent and apply patterns and relationships in algebraic forms (Applying Strategies, Communicating)
explain why a particular relationship or rule for a given pattern is better than another (Reasoning, Communicating)
distinguish between graphs that represent an increasing number pattern and those that represent a decreasing number pattern (Communicating)
determine whether a particular number pattern can be described using algebraic symbols (Applying Strategies, Communicating)
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Mathscape 7 Teaching Program
Page 11
Knowledge and skills
Teaching, learning and assessment
Students learn about
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using letters to represent numbers and developing the notion that a letter is used to
represent a variable
using concrete materials such as cups and counters to model:
expressions that involve a variable and a variable plus a constant, for example
a, a + 1
expressions that involve a variable multiplied by a constant, for example 2a, 3a
sums and products, for example 2a + 1, 2 (a + 1)
equivalent expressions such as x + x + y + y + y = 2 x + 2 y + y = 2 ( x + y ) + y
and to assist with simplifying expressions, such as
(a + 2) + ( 2a + 3) = (a + 2a ) + (2 + 3)
= 3a + 5
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
modelling geometric patterns using materials such as matchsticks to form squares,
for example
,
,
,
,…
describing the pattern in a variety of ways that relate to the different methods of
building the squares, and recording descriptions using words
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
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Open ended questions
1. The answer is 4y³. What is the question?
Reflecting
1. Write a letter to a friend who has missed to lessons on Algebra and explain
the difference between a² and 2a. (Reasoning, Communicating)
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.
FOCUS ON WORKING MATHEMATICALLY
Seeing is believing (page 207): try
http://www.education2000.com/demo/demo/botchtml/arithser.htm, 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
http://www.learner.org/teacherslab/math/patterns.
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.
EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING
(page 208)
CHAPTER REVIEW (page 209) a collection of problems to revise the
chapter.
representing the values from the table on a number grid and describing the pattern
formed by the points on the graph (note – the points should not be joined to form
a line because values between the points have no meaning)
determining a rule in words to describe the pattern from the table – this needs to
be expressed in function form relating the top-row and bottom-row terms in the
table
describing the rule in words, replacing the varying number by an algebraic symbol
using algebraic symbols to create an equation that describes the pattern
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Mathscape 7 Teaching Program
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creating more than one equation to describe the pattern
using the rule to calculate the corresponding value for a larger number
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
completing a table of values for a number pattern, for example
a
1
2
3
4
5
10
100
b
4
7
10
13
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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.
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Mathscape 7 Teaching Program
Page 13
Chapter 6. Decimals
Text reference
CD reference
Substrand
Mathscape 7
Chapter 6. Decimals (pages 214–253)
Decimals
Fractions, decimals and percentages
Duration
3 weeks / 12 hours
Key ideas
Outcomes
Performs operations with fractions, decimals and mixed numerals.
NS4.3 (page 63) Operates with fractions, decimals, percentages, ratio and rates.
Working mathematically
Students learn to
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question the reasonableness of statements in the media that quote fractions, decimals or percentages, for example ‘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)
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 (Applying Strategies)
guess and refine (Applying Strategies, Communicating)
Knowledge and skills
Teaching, learning and assessment
Students learn about
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adding, subtracting, multiplying and dividing decimals (for multiplication and
division, limit operators to two-digits)
determining the effect of multiplying or dividing by a number less than one
rounding decimals to a given number of places
using the notation for recurring (repeating) decimals for example
0.333 33… = 0.3& , 0.345 345 345… = 0.3& 45&
converting fractions to decimals (terminating and recurring) and percentages
converting terminating decimals to fractions and percentages
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Open ended questions
1. Write down 12 numbers between 1.45 and 2.73
2. Find two numbers that have a product of 0.6
TRY THIS
The Dewey decimal system (page 219): students explain how the system is
organised.
Judging Olympic diving (page 236)
FOCUS ON WORKING MATHEMATICALLY
Olympic decathalon 2000 (page 250): teachers will be rewarded if they
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Mathscape 7 Teaching Program
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calculating fractions, decimals and percentages of quantities
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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
http://www.decathlonusa.org/home.html. 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 29.29 s to get these points.
He actually ran 4 min 38.71 s, so he was 9.42 s short. Huffins went into the
1500 m knowing that his best time to that date was 4 min 49.70 s. He ran 11 s
faster than his personal best. Teachers may wish to use this as a starting point
for the Let's communicate discussion suggested on page 252. The real issue as
far as the mathematics is concerned is the importance of decimals to elite
athletes.
EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING
(page 252)
CHAPTER REVIEW (page 253) a collection of problems to revise the
chapter.
Technology
Decimals: three interactive worksheets on place value, adding and subtracting decimals. Good for teacher instruction for the class.
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Mathscape 7 Teaching Program
Page 15
Chapter 7. Integers
Text reference
CD reference
Substrand
Mathscape 7
Chapter 7. Integers (pages 256–281)
Number line
Integers
Duration
2 weeks / 8 hours
Key ideas
Outcomes
Perform operations with directed numbers.
Simplify expressions involving grouping symbols and apply order of operations.
NS4.2 (page 58) Compares, orders and calculates with integers.
Working mathematically
Students learn to
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interpret the use of directed numbers in a real world context eg rise and fall of temperature (Communicating)
construct a directed number sentence to represent a real situation (Communicating)
apply directed numbers to calculations involving money and temperature (Applying Strategies, Reflecting)
use number lines in applications such as time lines and thermometer scales (Applying Strategies, Reflecting)
verify, using a calculator or other means, directed number operations eg subtracting a negative number is the same as adding a positive number (Reasoning)
question whether it is more appropriate to use mental strategies or a calculator when performing operations with integers (Questioning)
Knowledge and skills
Teaching, learning and assessment
Students learn about
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recognising the direction and magnitude of an integer
placing directed numbers on a number line
ordering directed numbers
interpreting different meanings (direction or operation) for the + and – signs
depending on the context
adding and subtracting directed numbers
multiplying and dividing directed numbers
using grouping symbols as an operator
applying order of operations to simplify expressions
keying integers into a calculator using the +/– key
using a calculator to perform operations with integers
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Open ended questions
1. List the many uses of directed numbers in the real world. (Communicating)
2. Create a real world directed number problem.
TRY THIS
Temperature (page 262): estimation exercise.
Multiplication with directed numbers (page 272): modelling directed number
multiplication on a number line.
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
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Mathscape 7 Teaching Program
Page 16
•
•
http://www5.cnn.com/SPECIALS/2000/submarine/ 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.
EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING
(page 280)
CHAPTER REVIEW (page 281) a collection of problems to revise the
chapter.
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.
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Mathscape 7 Teaching Program
Page 17
Chapter 8. Algebra
Text reference
CD reference
Substrand
Mathscape 7
Chapter 8. Algebra (pages 284–308)
Simplify
Expand
Algebraic techniques
Duration
2 weeks / 8 hours
Key ideas
Outcomes
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.
PAS4.1 (page 82) Uses letters to represent numbers and translates between words
and algebraic symbols.
PAS4.3 (page 85) Uses the algebraic symbol system to simplify, expand and
factorise simple algebraic expressions.
Working mathematically
Students learn to
•
•
•
•
•
•
•
•
generate a variety of equivalent expressions that represent a particular situation or problem (Applying Strategies)
describe relationships between the algebraic symbol system and number properties (Reflecting, Communicating)
link algebra with generalised arithmetic eg for the commutative property, determine that a + b = b + a (Reflecting)
determine equivalence of algebraic expressions by substituting a given number for the letter (Applying Strategies, Reasoning)
generate a variety of equivalent expressions that represent a particular situation or problem (Applying Strategies)
determine and justify whether a simplified expression is correct by substituting numbers for letters (Applying Strategies, 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)
Knowledge and skills
Teaching, learning and assessment
Students learn about
•
•
recognising and using equivalent algebraic expressions, for example
y + y + y + y = 4y
w × w = w2
a × b = ab
a
a ÷b =
b
•
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
FOCUS ON WORKING MATHEMATICALLY
Colouring Maps (page 305): having established Euler's 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
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Mathscape 7 Teaching Program
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translating between words and algebraic symbols and between algebraic symbols
and words
recognising like terms and adding and subtracting like terms to simplify algebraic
expressions, for example 2n + 4m + n = 4m + 3n
recognising the role of grouping symbols and the different meanings of
expressions, such as 2a + 1 and 2(a + 1)
simplifying algebraic expressions that involve multiplication and division, for
example
12a ÷ 3
4x × 3
2ab × 3a
•
expanding algebraic expressions by removing grouping symbols (the distributive
property), for example
3(a + 2) = 3a + 6
− 5( x + 2) = −5 x − 10
a (a + b) = a 2 + ab
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taken into account. However, it does enable a teacher to ask whether Euler's
formula will work if the boundaries are not straight lines. Students could be
asked about Tasmania – should it be included? Here are some sites to explore
Euler's formula further:
http://www.math.ohio-state.edu/~fiedorow/math655/Euler.html, this
website gives nice clear pictures of the formula applying to threedimensional solids.
http://www.math.ucalgary.ca/~laf/colorful/4colors.html, this site has an
interesting game to play with the four colour problem.
http://www.uwinnipeg.ca/~ooellerm/guthrie/FourColor.html, this site
will tell you more about Francis Guthrie and the connection of the four
colour problem to rare flowers in South Africa.
EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING
(page 307)
CHAPTER REVIEW (page 308) a collection of problems to revise the
chapter.
distinguishing between algebraic expressions where letters are used as variables,
and equations, where letters are used as unknowns
substituting into algebraic expressions
translating from everyday language to algebraic language and from algebraic
language to everyday language
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.
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Mathscape 7 Teaching Program
Page 19
Chapter 9. Angles
Text reference
CD reference
Substrand
Mathscape 7
Chapter 9. Angles (pages 310–356)
Angles
Angle pairs
Polygon angles
Angles
Duration
2 weeks / 8 hours
Key ideas
Outcomes
Classify angles and determine angle relationships.
Construct parallel and perpendicular lines and determine associated angle properties.
Complete simple numerical exercises based on geometrical properties.
SGS4.2 (page 153) Identifies and names angles formed by the intersection of
straight lines, including those related to transversals on sets of parallel lines, and
makes use of the relationships between them.
Working mathematically
Students learn to
• recognise and explain why adjacent angles adding to 90º form a right angle (Reasoning)
• recognise and explain why adjacent angles adding to 180º form a straight angle (Reasoning)
• recognise and explain why adjacent angles adding to 360º form a complete revolution (Reasoning)
• find the unknown angle in a diagram using angle results, giving reasons (Applying Strategies, Reasoning)
• 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)
• 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)
identify parallel and perpendicular lines in the environment (Reasoning, Reflecting )
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)
use dynamic geometry software to investigate angle relationships (Applying Strategies, Reasoning)
Knowledge and skills
Teaching, learning and assessment
Students learn about
Angles at a Point
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•
labelling and naming points, lines and intervals using capital letters
labelling the vertex and arms of an angle with capital letters
labelling and naming angles using ∠A and ∠XYZ notation
using the common conventions to indicate right angles and equal angles on
diagrams
•
TRY THIS
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
FOCUS ON WORKING MATHEMATICALLY
The Sun’s 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
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Mathscape 7 Teaching Program
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Page 20
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
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
•
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the Pyramids of Egypt using a shadow stick provides an historical connection
to early geometry. A good website to start exploring ancient Egypt is
http://www.eyelid.co.uk/. 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 http://www.guardians.net/egypt/gp1.htm.
The site http://www.crystalinks.com/egypt.html has a video with it and
some interesting diagrams of the tombs and their construction. The sun's 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 sun's rays strike Earth.
The tilt of the Earth (23.5°) 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 Riser's website
http://www.k12training.com/JamesRiser/Science/seasons/seasons2.htm
and David Stern's From Stargazers to Starships,
http://www-istp.gsfc.nasa.gov/stargaze/Sintro.htm. Scroll down the menu
for Astronomy of the Earth's 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.
EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING
(page 355)
CHAPTER REVIEW (page 356) a collection of problems to revise the
chapter.
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.
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Mathscape 7 Teaching Program
Page 21
Chapter 10. Properties of geometrical figures
Text reference
CD reference
Substrand
Mathscape 7
Chapter 10. Properties of geometrical
figures (pages 362–406)
Plane shapes
Properties of geometrical figures
Duration
2 weeks / 8 hours
Key ideas
Outcomes
Classify, construct and determine properties of triangles and quadrilaterals.
Complete simple numerical exercises based on geometrical properties
SGS4.3 (page 154) Classifies, constructs, and determines the properties of
triangles and quadrilaterals.
Working mathematically
Students learn to
• sketch and label triangles and quadrilaterals from a given verbal description (Communicating)
• describe a sketch in sufficient detail for it to be drawn (Communicating)
• 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)
• determine if particular triangles and quadrilaterals have line and/or rotational symmetry (Applying Strategies)
• 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
Teaching, learning and assessment
Students learn about
Notation
•
•
•
labelling and naming triangles (eg ABC) and quadrilaterals (eg ABCD) in text and
on diagrams
•
using the common conventions to mark equal intervals on diagrams
Triangles
• 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)
justifying informally by paper folding or cutting, and testing by measuring, that
TRY THIS
Triangle trouble (page 378): can an equilateral triangle be rearranged to form
a square?
Angle sum of a polygon (page 395)
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
http://www.kidwizard.com/GamesNumberMagic/Pentagon.asp, 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
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Mathscape 7 Teaching Program
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the interior angle sum of a triangle is 180º, and that any exterior angle equals the
sum of the two interior opposite angles
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
•
•
•
•
distinguishing between convex and non-convex quadrilaterals (the diagonals of a
convex quadrilateral lie inside the figure)
establishing that the angle sum of a quadrilateral is 360º
constructing various types of quadrilaterals
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 parallel
– opposite sides equal
– adjacent sides perpendicular
– opposite angles equal
– diagonals equal in length
– diagonals bisect each other
– diagonals bisect each other at right angles
– 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 180°, whereas for six and eight sides it is 360°. 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.
EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING
(page 405)
CHAPTER REVIEW (page 406) a collection of problems to revise the
chapter.
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.
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Mathscape 7 Teaching Program
Page 23
Chapter 11. Measurement, length and perimeter
Text reference
CD reference
Substrand
Mathscape 7
Chapter 11. Measurement, length and
perimeter (pages 410–444)
Measuring plane shapes
Perimeter and area
Duration
2 weeks / 8 hours
Key ideas
Outcomes
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.
MS4.1 (page 124) Uses formulae and Pythagoras’ theorem in calculating
perimeter and area of circles and figures composed of rectangles and triangles.
Working mathematically
Students learn to
•
•
•
•
•
•
consider the degree of accuracy needed when making measurements in practical situations (Applying Strategies)
choose appropriate units of measurement based on the required degree of accuracy (Applying Strategies)
make reasonable estimates for length and area and check by measuring (Applying Strategies)
select and use appropriate devices to measure lengths and distances (Applying Strategies)
discuss why measurements are never exact (Communicating, Reasoning)
solve problems relating to perimeter, area and circumference (Applying Strategies)
Knowledge and skills
Teaching, learning and assessment
Students learn about
Length and Perimeter
•
•
•
•
•
•
•
estimating lengths and distances using visualisation strategies
recognising that all measurements are approximate
describing the limits of accuracy of measuring instruments ( ± 0.5 unit of
measurement)
interpreting the meaning of the prefixes ‘milli’, ‘centi’ and ‘kilo’
converting between metric units of length
finding the perimeter of simple composite figures
•
TRY THIS
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
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
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Mathscape 7 Teaching Program
Page 24
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•
reproduce a scale drawing of the field. Teachers can simplify the task at their
discretion. The pitcher's plate is 10 inches (about 25 cm) above the other
bases. Students may like to discuss why. The website http://www.baseballalmanac.com/rule1.shtml is a good source for teachers who wish to access
the rules of baseball. However, the major league website
http://www.mlb.com will be more than sufficient. You will need to scroll
down to the bottom of the page to locate 'baseball basics'.
EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING
(page 442)
CHAPTER REVIEW (page 444) a collection of problems to revise the
chapter.
Technology
Measuring Plane Shapes: this file contains hyperlinks to a number of interactive geometric diagrams.
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Mathscape 7 Teaching Program
Page 25
Chapter 12. Solids
Text Reference
CD Reference
Substrand
Mathscape 7
Chapter 12. Solids (pages 447–479)
Solids
Properties of solids
Duration
2 weeks / 8 hours
Key ideas
Outcomes
Determine properties of three-dimensional objects.
Investigate Platonic solids.
Investigate Euler’s relationship for convex polyhedra.
Make isometric drawings.
SGS4.1 (page 147) Describes and sketches three-dimensional solids including
polyhedra, and classifies them in terms of their properties.
Working mathematically
Students learn to
•
•
•
•
•
•
interpret and make models from isometric drawings (Communicating)
recognise solids with uniform and non-uniform cross-sections (Communicating)
analyse three-dimensional structures in the environment to explain why they may be particular shapes, for example buildings, packaging (Reasoning)
visualise and name a common solid given its net (Communicating)
recognise whether a diagram is a net of a solid (Communicating)
identify parallel, perpendicular and skew lines in the environment (Communicating, Reflecting)
Knowledge and skills
Teaching, learning and assessment
Students learn about
•
•
•
•
•
describing solids in terms of their geometric properties
– number of faces
– shape of faces
– number and type of congruent faces
– number of vertices
– number of edges
– convex or non-convex
identifying any pairs of parallel flat faces of a solid
determining if two straight edges of a solid are intersecting, parallel or skew
determining if a solid has a uniform cross-section
•
•
Investigations for class discussions and presentations
1. Students create a scale model of a solid from an isometric drawing or net
TRY THIS
Painted cube (page 460): problem solving
The Soma Puzzle (page 475)
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
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Mathscape 7 Teaching Program
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Page 26
classifying solids on the basis of their properties
– a polyhedron is a solid whose faces are all flat
– a prism has a uniform polygonal cross-section
– a cylinder has a uniform circular cross-section
– a pyramid has a polygonal base and one further vertex (the apex)
– a cone has a circular base and an apex
All points on the surface of a sphere are a fixed distance from its centre.
identifying right prisms and cylinders and oblique prisms and cylinders
identifying right pyramids and cones and oblique pyramids and cones
sketching on isometric grid paper shapes built with cubes
representing three-dimensional objects in two dimensions from different views
confirming, for various convex polyhedra, Euler’s formula F + V = E + 2
relating the number of faces (F), the number of vertices (V) and the number of
edges (E)
exploring the history of Platonic solids and how to make them
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
http://home.teleport.com/~tpgettys/platonic.shtml
The University of Utah has an interactive site where students can rotate and
examine Platonic solids:
http://www.math.utah.edu/~alfeld/math/polyhedra/polyhedra.html
Mathworld has a nice diagram of the conic sections at
http://mathworld.wolfram.com/ConicSection.html
EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING
(page 478)
CHAPTER REVIEW (page 479) a collection of problems to revise the
chapter.
Technology
Solids: two files are included – the Solids file, which includes animated diagrams of various solids and the Paper Folding file, which includes nets of six interesting
polyhedral solids.
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Mathscape 7 Teaching Program
Page 27
Chapter 13. Area
Text reference
CD reference
Substrand
Mathscape 7
Chapter 13. Area (pages 401–509)
Measuring plane shapes
Perimeter and Area
Duration
2 weeks / 8 hours
Key ideas
Outcomes
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.
MS4.1 (page 124) Uses formulae and Pythagoras’ theorem in calculating
perimeter and area of circles and figures composed of rectangles and triangles.
Working mathematically
Students learn to
•
•
•
•
•
•
•
•
•
•
consider the degree of accuracy needed when making measurements in practical situations (Applying Strategies)
choose appropriate units of measurement based on the required degree of accuracy (Applying Strategies)
make reasonable estimates for length and area and check by measuring (Applying Strategies)
select and use appropriate devices to measure lengths and distances (Applying Strategies)
discuss why measurements are never exact (Communicating, Reasoning)
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 side lengths (Reflecting)
Knowledge and skills
Teaching, learning and assessment
Students learn about
Areas of squares, rectangles, triangles and parallelograms
•
•
•
•
developing and using formulae for the area of a square and rectangle
developing (by forming a rectangle) and using the formula for the area of a
triangle
finding the areas of simple composite figures that may be dissected into rectangles
and triangles
•
Open ended questions
1. Draw a rectangle with an area of 12cm²
2. Compare rectangles with the same area. Do they have the same perimeter?
(Questioning, Applying Strategies, Reasoning)
TRY THIS
How many people are there in your classroom? (page 492): estimation
exercise
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Mathscape 7 Teaching Program
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developing the formula by practical means for finding the area of a parallelogram
eg by forming a rectangle using cutting and folding techniques
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
FOCUS ON WORKING MATHEMATICALLY
Goal! The world cup 2002 (page 506): the website
http://fifaworldcup.yahoo.com 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 http://www.fifa.com/en/game/.
EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING
(page 508)
CHAPTER REVIEW (page 509) a collection of problems to revise the
chapter.
Technology
Measuring Plane Shapes: this file contains hyperlinks to a number of interactive geometric diagrams.
Published by Macmillan Education Australia. © Macmillan Education Australia 2004