senior - western cape...7 + 6 + 3 = 13 + 3 = 16).associative properties i.e. order of operations...

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Mathematics

GRADE 8 TEACHER’S GUIDE 1 / 95

INTRODUCTION TO THE TEACHER’S GUIDE Accompanying the Work Schedule is this Teacher’s Guide which gives further detail on the Work Schedule. For each week the following information will be found in this Teacher’s Guide:

1. Core Concept

2. Resources

3. Integration

4. Teaching Tips – a few ideas for teaching the concept

5. Examples – different examples to practise the concept

6. Consolidation

7. Assessment

At the end of the document is an example of the computer software which can be used as an additional resource for teaching and learning or consolidation of a concept.

The material developed by IMSTUS (University of Stellenbosch) which is referred to weekly in the Teacher’s Guide can be accessed via a link from the WCED website for the Teacher’s Guide to the IMSTUS website. Each module covers a different concept of the Senior Phase curriculum within an integrated approach.

The Teacher’s Guide attempts to focus teaching and learning on the change in focus in the learning outcomes within the senior phase as follows:

Senior Phase Focus In Learning Outcome 1 the focus is on:

Representing numbers in a variety of ways and moving between these ways Problem-solving involving higher order reasoning Recognising and using irrational numbers

In Learning Outcome 2 the focus is on:

Finding the relationships between variables in context and representing this relationship in different forms (words; tables; flowcharts; graphs; formulas)

Expressing these relationships in algebraic language or symbols Manipulating algebraic expressions Drawing and interpreting graphs that represent relationships between variables


Drawing and constructing a wide range of geometric figures and solids in order to investigate their properties

Investigating similarity and congruency


Deriving formulae through investigation for area and volume of different geometric figures and solids


Data handling involving contexts wider than the learners’ own environment Drawing graphs best suited to represent the data Interpreting data represented by graphs with emphasis on misleading graphs Probability involving single and compound events


Built into the work schedule and teacher’s guide is time for consolidation of the concepts. Learners must be given enough time in class to practise the concepts. Homework must be given daily so these concepts practised in class can be consolidated. Homework helps learning. Learners will not be able to consolidate mathematical concepts without doing homework.

Ideas for formal assessment have been given. Exemplar assessment tasks which could be used with this work schedule will be distributed to schools in 2010 / 2011. This should further assist in the setting a proper standard of assessment in WCED schools.

The WCED hopes that these work schedules and teacher’s guides will assist in reducing the load on teachers with regard to planning. Time can be spent on the actual planning of the lesson; to make it relevant and interesting.

The WCED wishes to thank all the teachers and curriculum advisers involved in the writing of the work schedules and teacher’ guides. A special word of thanks must be expressed to IMSTUS (Institute for Mathematics and Science Teaching University of Stellenbosch) for the (free) use of their material (on website), their support and academic input into the documents.

DAILY ROUTINE At least one hour must be spent on Mathematics every day

TIME ALLOCATION 10 min Grade 7: Oral and written Mental work. (Optional for grade 8 and 9)

10 min Review and correct homework of previous day

20 min Teacher introduces the concept of the day (or continue with the development of the previous concept) through investigation or problem-solving depending on the concept

15 - 25 min Calculations and problem solving relating to the concept of the day

5 min Homework tasks are given and explained by the teacher


TERM 1

TERM 1 - WEEK 1 REVISION

TERM 1 - WEEK 2

ASSESSMENT STANDARD 8.3.10 Locates positions on co-ordinate systems (ordered grids), Cartesian plane (first quadrant) and maps, and describes how to move between positions using: • Horizontal and vertical change; ordered pairs; Compass directions

TERMINOLOGY Co –ordinate system; Cartesian plane; quadrant; horizontal; vertical; ordered pairs; grid

RESOURCES Gr 8 Text books; WCED Illustrative examples; Calculator; Poster; Internet Web sites , IMSTUS

INTEGRATION Technology, Social Science

TEACHING TIPS For ideas on developing the concept and extra examples, look at IMSTUS module 4.

Make sure that learner knows Terminology which is needed.

Use maps that relate to learners’ background.

Grids should be given to learners. (Use plastic sleeves to put grid in. Plotting can be done with koki and be wiped with tissue paper. One grid can be used for many examples without confusing learners.)

Explore the first quadrant on a Cartesian plane, where x is the horizontal axis and y the vertical axis. (Use the position where learners sit by numbering the desks using rows and columns: Question e.g. where does Peter sit.)

Understand ordered pairs i.e. (2; 3) where 2 is the x co-ordinate and 3 is the y co-ordinate.

Learners should be able to use compass directions e.g. N30°E

It is always good to use ordered pairs which, when plotted and joined together form something e.g. a bird; car; boat etc.

Teach learners that once an ordered pair has been translated to a new position, it is indicated with the same letter/s (E) as the original co-ordinate but with an accent on the top e.g. E′

EXAMPLES 1. Use a system of axes to do the following: a) Plot and connect the following points: (5 ; 5), (6 ; 7), (5 ; 6)

b) Also plot & connect the following points: (1 ; 5), (0 ; 7), (1 ; 6)

c) Do the same with the following points: (3 ; 7), (3 ; 8), (4 ; 9)

d) Plot the point (3 ; 1½)

e) Plot and connect the following points: (1 ; 3), (1 ; 6), (2 ; 7), (4 ; 7), (5 ; 6), (5 ; 3) now to (3 ; 1,5) and now to (1 ; 3)


2. a) On grid paper plot the following points and join them with a line: A (3; 1) and B (7; 9).

b) On grid paper plot and draw triangle DEF with co-ordinates D (1; 7), E (6; 7) and F (5; 5).

c) What is the horizontal change as you move from D to E and from E to D?

d) What is the vertical change as you move from D to E and from E to D?

e) What is the horizontal change as you move from D to F and from F to E?

f) What is the vertical change as you move from D to F and from F to E?

CONSOLIDATION / HOMEWORK Complete given work on grids.

ASSESSMENT Informal: class work

TERM 1 - WEEK 3

ASSESSMENT STANDARD 8.1.3 Recognises, classifies and represents the following numbers in order to describe and compare them: • integers 8.1.9 Uses a range of techniques to perform calculations including: • using the commutative, associative and distributive properties with rational numbers; • using a calculator.

TERMINOLOGY Integers, commutative, associative, distributive, rational numbers

RESOURCES Gr 8 Text books; WCED Illustrative examples; Calculator; Poster; Internet Web sites, IMSTUS material

INTEGRATION Natural Sciences

TEACHING TIPS For ideas on developing the concept and extra examples, look at IMSTUS module 15

(integers). Start by revising numbers in ascending / descending order ( e.g. lift, thermometer, etc. )

Make sure the learner knows how to use the 4 operations with natural numbers (e. g. + ; - ; × ;÷ .).

Make sure learners are aware of the difference between negative 3 and subtract 3.

Which number is smaller / bigger by using a number line (e.g. 4 > -10 or -32 < -30).

Temperature works well to interpret integers.

Represent integers on number lines to develop number concept.

Commutative properties i.e. order of the numbers (e.g. -45 + 39 = 39 + ( - 45).


Associative properties i.e. order of operations (e.g. 7 + 6 + 3 = 13 + 3 = 16).

Distributive properties i.e. apply an operation to two or more numbers (e.g. -2 x (4 + 5) = (-2 x 4) + (-2 x 5) = -8 + -10 = -18).

Learners do not have to know the names of the properties only the operations.

EXAMPLES 1. Calculate the following:

a) 100 + −30

b) 30 + −100 c) −100 + −30

d) −30 + −100

e) −100 + 30

f) −30 + 100 g) −100 − −30

h) −100 − 30

i) −30 − 100

j) 20 × −50

k) −20 × 50

2. In the following “addition wall” numbers are written on each brick of the bottom row. In the following row of bricks the sum of the numbers on the two bottom bricks are written on each brick above. (You must add the two numbers each time you move to the row above.) Find the numbers in the bottom row of bricks:

+10

-3

-5

-3

+5

+

+

+

+


3. In the following exercise you must complete the arrow diagrams. It works as follows: each time you move one block to the right or below, you must add the value indicated to the number in the previous block. With all the horizontal arrows use the same value (in this case "4"), and with all the vertical arrows use the same value (in this case "–10"), as indicated.

a) Complete the arrow diagram: b) What values belong to the arrows in this diagram? What number must be written in

the bottom right hand block?

+ ?

?

+8

-7

+14

+20

+10

0

+24

+28

+ +4

-10


4. The following arrow diagrams have triangles. A fixed value corresponds to arrows which point in the same direction. To the right of the diagram is a triangle which indicates the value corresponding to each arrow. The plus sign in the triangle indicates that it has to do with addition.

a) Complete the following arrow diagram:

b) Now you get a diagram with two unknown arrow values. (You can calculate for

yourself what the missing arrow values are.) Complete the diagram:

+?

+3

?

+8

+3

+10

+ ?

+3

+9

+16

+4


5. Find the values in the bottom row of bricks: 6. Complete the following arrow diagram with two unknown values: 7. Read the following information.

When you start with +9 and cross the –3 border in the same direction as the arrow, then you must multiply by -3 and the value on the other side of the border will be –27:

2739 −=−×+ )( When you start with -27 and cross the –3 border in the opposite direction to the arrow direction, then you must divide by -3. The value on the other side of the border is thus +9:

9327 +=−÷− )(

+96

-24

+12

+6

-3

×

×

×

×

×-10

?

?

-5 -1

+1

multiply


a) Now calculate the missing values:

REMEMBER: The arrow direction is the same for the entire line!

b) Here is another "customs sum". You must (with the help of the values supplied) calculate what values the arrows must have. Fill in the missing values.

CONSOLIDATION / HOMEWORK Give examples for home.


multiply

multiply


TERM 1 - WEEK 4

ASSESSMENT STANDARD 8.1.3 Recognises, classifies and represents the following numbers in order to describe and compare them: • numbers written in exponential form including squares and cubes of natural numbers and their

square and cube roots; 8.1.7 Estimates and calculates by selecting and using operations appropriate to solving problems that involve

TERMINOLOGY Exponential form; Squares; Cubes; Square roots and Cube roots;

RESOURCES Gr 8 Text books; WCED Illustrative examples; Calculator; Poster; Internet Web sites, IMSTUS material on website

INTEGRATION Technology, Natural Science

TEACHING TIPS For ideas on developing the concept as well as extra examples, look at IMSTUS module 18

(exponents). Understand concept of exponents by explaining: Exponential form Write exponents in expanded notation (e.g. 2 x 2 x 2 = 23)

Find the value of numbers in ( e.g. 23 = 2 x 2 x 2 = 8)

Write numbers in exponential notation (e.g. 100 = 102) .

Use prime factors if necessary , e.g. the cube root of 1728 can be found by factorising 1728 completely: 1728 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 = 2 6 x 3 3 3 1728 = 3 36 32 × = 2 2 x 3 = 12.

Only now do operations with exponents. Squares of natural numbers (e.g. square means number multiplied by itself) Cubes of natural numbers (e.g. cube means number multiplied by itself three times)

Square roots of natural numbers (e.g. 144 = 12)

Cube roots of natural numbers (e.g. 3 125 = 5)

EXAMPLES 1. a) Simone wants to draw a square with an area of exactly 10 cm2. Specify, accurate to

the nearest millimetre, how long each side should be.

b) Is it possible to state, with complete accuracy, how long each side should be?


length

2. Complete the following table:

x 11 12 13 14 15 16 17 18 19

x2

3. 212 is 441, which is a number with a last digit of 1.

222 is 484, which is a number with a last digit of 4.

a) Find three other numbers of which the squares have a last digit of 1.

b) Find three other numbers of which the squares have a last digit of 4.

c) Try to find a number of which the squares have a last digit of 2.

4. Find a number which is both a perfect square and a perfect cube.

5. Calculate:

a) 23 =

b) 34 =

c) 24 + 24 =

d) 102 × 102 =

6. Calculate:

a) (-2)7 =

b) (-10)3 =

c) (-2)101 + (2)101 =

d) (-10)3 × (-10)3 =

Example 7 and 8 shows the integration of exponents with other LO’s

7. Length can be represented by a number. In this case it is the measured length of the sides of a cube.

a) Write down the formula for the volume of a cube. Write it in power (exponential) form.

b) A cube has side lengths of 8 cm. What is its volume?

c) A cube has a volume of 8 m3. How long are the side lengths?

8. For the total outside surface area of a cube, the formula is: area = 6 x (length)²

a) Why is the number 6 in the formula?

b) Why is there “ length to the power of 2” in this formula?

c) Calculate the outside surface area of a cube with side length 3 cm.

d) Calculate the outside surface area of a cube with side length cm.

e) If the side length is doubled, what happens to the outside surface area? Does the outside surface area also double?

f) A cube has an outside area of 600 cm2. Calculate the side length.


9. Calculate:

a) 5,3 × 104 =

b) 1,98 × 106 =

c) 3,409 × 105 =

d) Write the numbers above from smallest to biggest (ascending order).

CONSOLIDATION / HOMEWORK Give examples as homework to test if learners know what is a perfect square and a perfect cube.

ASSESSMENT Assessment Task 1 e.g. Tutorial on weeks’ 2-4’s work

TERM 1 - WEEK 5

ASSESSMENT STANDARD 8.1.3 Recognises, classifies and represents the following numbers in order to describe and compare them: • Multiples and factors;

TERMINOLOGY Multiples; factors; rational numbers; composite numbers

RESOURCES Gr 8 Text books; WCED Illustrative examples; Calculator; Poster; Internet Web sites , IMSTUS material on website

INTEGRATION EMS

TEACHING TIPS Prime numbers are the building blocks of whole numbers. Each whole number is a prime

number or a product of prime numbers.

Revise and define multiples and factors e.g. Multiples of 8 = { 8; 16; 24; . ; . } e.g. Factors of 12 = { 1; 2; 3; 4; 6; 12} when 12 is divided by any one of its factors there is no remainder.

Revise and define prime factors e.g. Prime factors of 12 = { 2; 3;}

Composite numbers: Numbers with more than one factor i.e. numbers that are not prime numbers.

Product of Prime numbers: i.e. Write a number as a product of its prime factors e.g. 24 = 2 x 2 x 2 x 3

Perfect number: A number is perfect when all of its factors add up to the number itself e.g. 6 as 1 + 2 + 3 = 6

Highest common factor( HCF): The biggest number that will divide exactly (remainder is zero) into all the numbers in question e.g. factors of 12: 1, 2, 3, 4, 6, 12, factors of 30: 1, 2, 3, 5, 6, 10, 15, 30 , Common factors are 2 ,3 and 6 : HCF is 6.


Lowest common multiple (LCM): The smallest numbers that can be divided by all the numbers in the question with remainder equal to zero. e.g. multiples of 6: 6; 12; 18 ;24 ... multiples of 8: 8; 16; 24; 32; ... For both numbers 24 is a multiple and also the lowest therefor LCM is 24.

Do multiple calculations using different techniques e.g. tree diagram, etc.

Check answers e.g. with calculator.

EXAMPLES 1. Use different patterns to illustrate how to pack out a group of e. g. 12

************ 12 in a row

******

****** 2 rows of 6

****

**** 3 rows of 4

****

2. Write down all the numbers from 1 to 100 in a table with 10 columns and 10 rows. Circle multiples of 2; 3; 5; 7; 11; etc Continue in the same way until you are left with the prime numbers. Colour prime numbers to make them stand out. (You now have all the prime numbers less than a hundred.)

3. 72 can be written as a product of numbers, for example: 72 = 8 × 9 We say “8 and 9 are factors of 72”. We can also say ”72 is a multiple of 8 and a multiple of 9”. You can also write 72 = 6 × 12. Therefore 6 and 12 are factors of 72 and 72 is a multiple of 6 and 12. Adjacent you will find all factors of 72. The number 72 has thus got 12 factors.

4. a) Write down all the factors of 24.

b) Write down all the multiples of 18 less than 100.

c) Write down all the factors of 31. 5. Let’s break up 72 into factors:

72 = 8 × 9 72 = 2 × 4 × 3 × 3 72 = 2 × 2 × 2 × 3 × 3 72 = 23 × 32

a) How is 72 written as a product of prime numbers?

b) How did we find this product?

c) Write 96 as a product of prime factors.

CONSOLIDATION / HOMEWORK Learners should be able to find the multiples and factors of any number.


72987212672184722437236272721

=×=×=×=×=×=×


TERM 1 - WEEK 6

ASSESSMENT STANDARD 8.2.1

Investigates and extends numeric and geometric patterns looking for a relationship or rules, including patterns:

• represented in physical or diagrammatic form; • not limited to sequences involving constant difference or ratio; • found in natural and cultural contexts; • of the learner’s own creation; • represented in tables; • represented algebraically

8.2.2 Describes, explains and justifies observed relationships or rules in own words or in algebra.

TERMINOLOGY Numeric patterns, geometric patterns, physical form, diagrammatic form, constant difference, ratio, algebraically

RESOURCES Gr 8 Text books; WCED Illustrative examples; Manipulatives, Poster; Internet Web sites , IMSTUS material on website

INTEGRATION Technology, Arts & Culture

TEACHING TIPS For ideas on developing the concept as well as extra examples, look at IMSTUS

module 12 pg 1-6. The extension of this AS from Grade 7 is the addition of algebraic representation in

Grade 8.

Learners must be able to do the following with regard to patterns: Investigate Extend Describe and explain using different ways to represent relationships (Emphasise the algebraic expression as a way to represent relationships.) Justify the above selection

The patterns investigated can have: a constant difference e.g.1; 3; 5; 7; ……… has a constant difference of 2 or a constant ratio e.g. 3; 6; 12; 24 .... has a constant ratio of 2 or a pattern not involving constant difference or ratio e.g. 1; 2; 4; 7; 11...... i.e. add 1, then 2, then 3 etc.

In the Senior Phase the examples used for numerical patterns should be of a higher number range e.g. 103; 206; 412 .... (constant ratio 2)


EXAMPLES 1. Given: 7; 13; 19; 25; 31; 37; . . . ; . . . ; . . . ;

a) Find the next three terms.

b) Find the 10th term.

c) Find the rule for the sequence.

d) Use the rule to check that it forms the sequence (e.g. use calculator).

2. Make number patterns as described below. In each case write down the first eight terms of your number pattern. Also write down the 15th, 50th and 100th terms.

a) Make a number pattern with a constant difference between consecutive terms.

b) Make a number pattern with a constant ratio between consecutive terms.

c) Make a number pattern that does not have a constant difference, nor a constant ratio, between the terms.

3. Write down the first 10 terms of the number pattern in each of the following cases:

a) value of a term = 3 × number of the term + 13

b) value of a term = 3 × number of the term + 8

c) value of a term = 3 × number of the term + 1

d) value of a term = 3 × number of the term − 2

e) value of a term = 5 × number of the term + 13

f) value of a term = 5 × number of the term + 8

4. a) What do you observe with respect to the differences between consecutive terms in the number patterns in question 3? Describe your observations in words.

b) Write a formula, like in question 3, for any number pattern that will have a constant difference of 4 between consecutive terms.

c) Write a formula for the number pattern that will have a constant difference of 4 between consecutive terms, and a first term of 10.

c) Write a formula for any number pattern that will have a constant difference of 8 between consecutive terms, and a first term of 5.

5. Make a formula, like the formulas in question 3, for each of the number sequences given below:

a) 15; 18; 21; 24; 27; 30 ; . . . ; . . . ; . . . ;

b) 9; 22; 35; 48; 61; . . . ; . . . ; . . . ;

6. Rewrite the formulas given in question 3, but now use letter symbols instead of words.

7. Find the next three terms of this number pattern (you may use your calculator). Round off to three decimal places.

1; 1,2; 1,44; 1,728; 20,736; . . .; . . .; . . .

8. Find the next three terms of this number pattern:

2; 4; 5; 9; 14; 20; . . .; . . .; . . .


9. Thabo links balls with rods in arrangements like this: The table shows how many balls are needed for different arrangements:

Arrangement number 1 2 3 4 5 6 7 8 20 30 60

Number of balls in arrangement 1 4 9 16 25 36 625

a) Complete the table. Show all your work.

b) Write a rule to calculate the number of balls in any arrangement.

c) How can you convince others that you are correct?

10. Sipho uses matches to build pictures like this:

Picture 1 Picture 2 Picture 3 Picture 4

The table shows how many matches are used for the different pictures.

Picture number 1 2 3 4 5 6 7 8 19 37 59 100

Number of matches in picture 3 5 7 9 11 13

Complete the table. Show all your work.

11. On another day Sipho builds squares, like this:

Picture 1 Picture 2 Picture 3 Picture 4

a) How many matches does he use to form (i) 5 squares (ii) 100 squares?

b) Sipho has 455 matches. How many squares can he form in this way?

c) What is the same and what is different in the situations in 10 and 11?

CONSOLIDATION / HOMEWORK Give learners problems to work out at home at their own pace.

ASSESSMENT Assessment Task 2 e.g. Investigation on number patterns + representation

Arrangement 1 Arrangement 2 Arrangement 3 Arrangement 4


TERM 1 - WEEK 7

ASSESSMENT STANDARD 8.2.3 Represents and uses relationships between variables in order to determine input and/or output values in a variety of ways using: • verbal descriptions; • flow diagrams; • tables; • formulae and equations

TERMINOLOGY Flow diagrams; tables; formulae; equations; mathematical models

RESOURCES Gr 8 Text books; WCED Illustrative examples; Calculator; Poster; Internet Web sites

INTEGRATION Technology, A & C

TEACHING TIPS When introducing the use of letter symbols, the utmost care should be taken to ensure that

learners understand that the letter symbol represents numbers, more specifically that it represents the values that a variable may assume. The examples below suggest ways to help learners to understand this.

EXAMPLES 1. A certain parcel transport and delivery company charges R20 per parcel and R2,80

per kg for deliveries in a local area.

a) Calculate the delivery costs for parcels of 18 kg and 34 kg.

b) Which one of the following formulas can be used to calculate the total cost for delivery of a parcel:

Delivery cost in rand = 20 × mass of the parcel + 2,80

Delivery cost in rand = 2,80 × (mass of the parcel + 20)

Delivery cost in rand = 2,80 × mass of the parcel + 20

c) Rewrite the correct formula by using the symbol y to represent the delivery cost in rand, and the symbol x to represent the mass of the parcel.

2. The passenger fares charged by a certain long distance bus company consist of two components: a fixed fare of R60 for any distance and a fare of 18c per km.

a) Make a formula, in words, that shows how the fare for any distance can be calculated.

b) Choose single letter symbols to represent the distance and the total fare, and rewrite your formula by using the letter symbols.

c) Complete the following table of fares for various distances.

Distance in km 100 200 300 400 500 600 700 800 900

Total fare in rand


3. a) Complete the table, to find out which of the given values of y will make 5 x y + 7 equal to 117.

y 5 10 20 25 24 23 22

5 x y + 7

b) Work in the same way to find out for which values of y it will be true that 23 x y − 13 = 171

4. a) Find the missing input number in the flow diagram below: b) Represent the input number by the symbol x and write an equation that sets the same

question as the above.

5. At a certain hospital, a patient has to pay an admission charge of R60 and a daily fee of R120.

a) Write a formula that can be used to calculate the cost for being in this hospital, for any number of days.

b) Make a flow diagram that presents the same information as your formula.

c) Make a table of the cost of being in this hospital, for any period between 1 and 10 days.

d) Make a point graph to show the cost of being in this hospital, for any period between 1 and 10 days.

6. The fares for travelling on a certain train service are given in the table below.

Distance in km 1 2 3 4 5 6 7 8 9 10

Fare in cent 184 190 196 202 208 214 220 226 232 238

Distance in km 10 20 30 40 50 60 70 80 90 100

Fare in cent 238 298 358 418 478 538 598 658 718 778

a) Make a flow diagram to show how the fare may be calculated for any distance.

b) Make a formula to show how the fare may be calculated for any distance.

c) What is the fare for a distance of 42 km?

d) Write an equation to represent question (c).

CONSOLIDATION / HOMEWORK Problems to prepare for test.


input number 470 + 240 × 23


TERM 1 - WEEK 8 - 10

ASSESSMENT STANDARD 8.2.8 Uses conventions of algebraic notation and the commutative, associative and distributive laws to: • classify terms as like or unlike, and to justify the classification; • collect like terms; • multiply or divide an algebraic expression with one, two or three terms by a monomial; • simplify algebraic expressions given in bracket notation, involving one or two sets of brackets and

two kinds of operations; • compare different representations of algebraic expressions involving one or more operations,

selecting those which are equivalent, and justifying own choice; • write algebraic expressions, formulae or equations in simpler or more useful equivalent forms in

context.

8.2.9 Interprets and uses the following basic algebraic vocabulary in context: term, expression, coefficient, exponent (or index), base, constant, variable, equation, formula (or rule)

TERMINOLOGY Like terms; unlike terms; monomial (one term); binomial (two terms); trinomial (three terms); polynomial (many terms); equations; algebraic expression;

RESOURCES Gr 8 Text books; WCED Illustrative examples; Calculator; Poster; Internet Web sites , IMSTUS material on website, Malati.

INTEGRATION Technology, EMS

TEACHING TIPS For ideas on developing the concept as well as filling gaps, look at IMSTUS modules 24

and 28. Make clear the meaning of coefficient, exponent, base.

e.g.

Make clear the meaning of a variable, constant and term e.g. Identify a term in an expression.

)23(5 +− xx : first term is x5 (5 is a constant and x is a variable) second term is )23(1 +− x

Teach according to the sequence in the work schedule for weeks 8 - 10

We say that expressions in the tables are equivalent if they produce the same output numbers for the same input numbers.

For example, xx 24 + and x6 are equivalent expressions, because they produce the same output values for the same input values.

We can explain the equivalence in the following way: xx 24 + means xx ×+× 24 = ( ) ( )xxxxxx +++++ = x×6 = x6

3y2 exponent

coefficientbase (in this case the variable)


The learners should be able to multiply and divide by using: Sign (positive; negative); coefficient; variable (in the base); exponent: e.g. 22x x 33x = 2 x x x x x 3 x x x x x x = 2 x 3 x x x x x x x x x x (commutative law) = 6 x x5

If learners understand this they can use the “short-cut”: if the bases are the same, multiply coefficients and add exponents.

REMEMBER the laws of exponents are formally taught in grade 9 but learners need an informal understanding of this to multiply or divide any algebraic expression (with one two or three terms) by a monomial.

EXAMPLES 1. Complete the following table:

x 1 2 5 12 19 37 45

xx 52 +

xx 43 +

512 −x

x7

xx +6

xx 29 −

a) What do you notice in the table?

b) Determine the value of xx 52 + if x = 19. Discuss your method.

c) Determine the value of x if xx 29 − = 35. Discuss your method.

2. Complete the following table:

x 1 2 5 12 19 37 45

xx 134 +

xx 413 +

413 +x

x17

x413+

x+17

a) Which of the expressions are equivalent?

b) Which of the expressions are not equivalent?

c) Try to find the shortest expression for xx 413 + . Check and discuss your answer.

d) Try to find the shortest expression for 413 +x . Check and discuss your answer.

3. Which is the unlike term:

a) 3x²y; 4x²y; -2x2y; 2xy2.

b) abc ; 2cba ; 3ab ; -5bca


4. Identify whether the following are equations or expressions.

a) 413 +x b) 14 = 4x + 3 etc.

5. Multiply an algebraic expression with one, two or three terms by a monomial

a) a x a b) 10(x + 4),

c) )53(2 2yy +− d) – 3x²(6xy – 4x²y² + 5y)

CONSOLIDATION / HOMEWORK Learn for test on work covered in weeks 1- 8

ASSESSMENT ASSESSMENT TASK 3 e.g. Test on term’s work


TERM 2

TERM 2 - WEEK 1

ASSESSMENT STANDARD 8.2.5 Solves equations by inspection, trial-and-improvement or algebraic processes (additive and multiplicative inverses), checking the solution by substitution.

TERMINOLOGY Equations; inspection; trial-and-improvement; algebraic processes; substitution.


INTEGRATION -


(solving linear equations). Follow “ Concept to be taught “ as set out in work schedule.

Equations: remind learners of the rule, if you take something away from one side you must do the same for the other side and vice versa.

EXAMPLES 1. Change the following sentences into new sentences by using the rule on equal expressions.

sentence the same operation on both sides new sentence

25010015 =+× y subtract 100

25105 =+× x divide by 5

25,015,0 =+× p multiply by 4

20210 −×=+ kk add 20

bb ×−=× 3152 add b×3 2. Solve the following equations:

a) the same operation equation

For which y is 2384 =+× y ?

subtract 8

divide by 4

Check the solution:


b) the same operation equation

For which p is 1863 =+× p ?

subtract 6

Check the solution:

c) the same operation equation

For which a is 243 +=× aa ?

subtract a

Check the solution:

d) the same operation equation

For which t is 2875 =−× t ?

add 7

Check the solution:

e) the same operation equation

For which x is 231

4=+

x?

Check the solution:

3) Solve the following equations:

a) 5x − 2 = 13

b) 2x + 7 = 17

c) −3x − 8 = 20 + 14

d) 47 = −10a + 3a + 12

e) 6x

- 5 = 12

f) −4 − 3a

= − 115


g) 3(x − 2) = 42

h) 132

5b3−=

−+

i) −2x − 5 − (x + 4) = − 9

j) 22 = 3(2x + 1) − (x − 6)

CONSOLIDATION / HOMEWORK A worksheet can be given as homework to complete.


TERM 2 - WEEK 2

ASSESSMENT STANDARD 8.2.4 Constructs mathematical models that represent, describe and provide solutions to problem situations, showing responsibility toward the environment and the health of others (including problems within human rights, social, economic, cultural and environmental contexts).


INTEGRATION Technology, Natural Sciences, Social Sciences

TEACHING TIPS In the traditional curriculum learners were usually thrown directly into meaningless

manipulations of algebraic expressions supported by a body of rules, but with very little reflection on the structure of the expressions, for example why is xxx 532 =+ ? Emphasize the development of learners' “sense-making” and avoid the “rules without reason” approach that is so prevalent.

Develop three kinds of approaches to the understanding of equivalence:

EQUIVALENCE

NUMERICAL

JUDGING BY THE CONTEXT

ALGEBRAIC


When using a context, the hidden mathematics should not be the major factor for judging equivalence, but rather that learners develop understanding by using their common-sense.

It is not about getting numerical answers but about reflecting on the structure and ultimately reflecting on generalisation. Learners should use their structural knowledge to judge equivalence.

Create own word problems based on a given mathematical model e.g. number sentence.

EXAMPLES 1. RAMAPHOSA’S APPLE ORCHIDS

Farmer Ramaphosa decides to plant Starking apple trees in one of his orchids. He decides to plant 465 Starking apple trees initially and 5 new trees every day thereafter as shown in the table below:

Number of days 0 1 2 3 4 5 6 x Number of Starking trees 465 470 475 480

Farmer Ramaphosa decides to plant Golden Delicious (G-D) apple trees in his other orchid. He initially plants 225 G-D apple trees and 15 new trees every day thereafter as shown in the table below:

Number of days 0 1 2 3 4 5 6 x

Number of G-D trees 225 240 255 270

a) What is the functional rule for finding the number of Starking apples planted in x days?

b) What is the functional rule for finding the number of G-D apples planted in x days?

c) On what day will there be the same number of Starking apples and G-D apples in the two orchids?

d) Say he decides to plant 20 new trees in the G-D orchid instead of 15. How many new trees must be planted in the Starking apple orchid so that the number of trees in the two orchids will be the same on the 24th day?

e) Say he decides to start off by planting 250 G-D apple trees instead of 225. What must the number of Starking apple trees that he planted initially be, for the number of apples trees in the two orchids to be the same on the 24th day?


2. The supervisor of the Cape Town Garden and Claremont Garden organised volunteers to help dig out weeds. The tables below show the number of people who volunteered to pull out weeds and the number of bags of weed pulled in a day in the two gardens:

Claremont Garden

Number of people 1 2 3 4 5 6 x

Number of bags of weed pulled in a day 5 9 13 17

Cape Town Garden

Number of people 1 2 3 4 5 6 x

Number of bags of weed pulled in a day 7 11 15 19

a) Write down the algebraic rules for the number of bags of weed pulled by x people in the Claremont Garden and in the Cape Town Garden.

b) How many people need to pull weeds, for the number of bags pulled in a day to be the same in both gardens?

3. Marlene wants to buy a video game that costs R489. She has R84 in the bank, and makes R15 every week from an after-school job. How long will it take her to save enough money to buy the video game?

a) Write an expression that represents how much money Marlene will have at any time.

b) Write an equation using this expression which will represent the question in the story.

c) Solve the equation.

d) How long will it take Marlene to save for the video game?

e) Verify that your solution is correct. (Source: Malati)

CONSOLIDATION / HOMEWORK Give examples for learners to try at home using his / her own pace.


TERM 2 - WEEK 3

ASSESSMENT STANDARD 8.1.3 Recognises, classifies and represents the following numbers in order to describe and compare them: • decimals , fractions and percentages; 8.1.7 Estimates and calculates by selecting and using operations appropriate to solving problems that involve: • rounding off; • multiple operations with rational numbers (including division with fractions and decimals);


8.1.9 Uses a range of techniques to perform calculations including: • using a calculator.

8.1.11 Recognises, describes and uses: • Algorithms for finding equivalent fractions

TERMINOLOGY Decimals; percentage; rational numbers; fractions; common fraction; decimal fraction; conversion.



TEACHING TIPS Represent fractions on number lines to develop number concept.

Arrange set of fractions in ascending / descending order.

Explain reciprocal of a fraction (Revision) [numerator and denominator and vice versa. e.g.

21 is the reciprocal of 2].

Estimate, calculate, solve problems with addition, subtraction and multiplication and division of common fractions.

Division of fractions is done for the first time in Grade 8.

EXAMPLES 1. Complete by using equivalent fractions / fill in missing numerator or denominator

e.g. 32 =

9 etc.

2. Compare the pairs of fractions and use the signs = ; < or > to indicate their relationship.

e.g. 43 and

34 , etc.

3. Arrange the following numbers in ascending / descending order. Indicate whether some of the numbers are actually equal.

159

43

83 0,75

2515

53

4. Give examples with addition and subtraction e.g. 87 +

41 and / or 5 -

52 , 2

87 + 3

41 and /

or 5 -523

5. Give examples with multiplication e.g. 32 x

24 , 3

54 x 2 , etc.

6. Give examples with division e.g. 32 ÷

24 , 3

54 x 2 , etc.


7. Simplify e.g. 481 -

32 x 3

43 , etc.

8. Give problem solving examples. e.g. Robin’s grandmother gave her a recipe for a pudding which she loves. The ingredients were as follows:

43 cup sugar 2 eggs 1

21 cups flour,

21 teaspoon salt

121 teaspoon baking powder

41 kg dates, 4 tablespoons oil

1 cup boiling water 1 teaspoon bicarbonate of soda 31 cup chopped nuts

a) If Robin has 6 teaspoons of baking powder, how many puddings can she make?

b) If she has 121 kg of dates how many puddings can she make?

c) If she needs to make a pudding 3 times bigger than the recipe states how many spoons of baking powder are needed.

CONSOLIDATION / HOMEWORK A worksheet with fractions can be given to learners as homework.


TERM 2 - WEEK 4

ASSESSMENT STANDARD 8.1.3 Recognises, classifies and represents the following numbers in order to describe and compare them: • fractions 8.1.7 Estimates and calculates by selecting and using operations appropriate to solving problems that involve: • multiple operations with rational numbers (including division with fractions); 8.1.9 Uses a range of techniques to perform calculations including: • using a calculator.

8.1.4. Recognises and uses equivalent forms of the rational numbers listed above.

TERMINOLOGY Fractions; common fractions; equivalent fractions; rational numbers; denominator; numerator.


INTEGRATION Technology, Social Sciences


TEACHING TIPS For ideas on REVISION for the developing of the concept and filling gaps, look at IMSTUS

module 3 and 6 (ratios and percentages). Revise Gr 7 fractions, decimals and percentages.

Use practical examples to explain the principle of fractions e.g. apple, pizza, cake, etc.

Pattern blocks can be used to help learners develop the concept of equivalent fractions, addition and subtraction of fractions with different denominators.

Convert simple common fractions to decimals by using place values e.g.

1,0101= 01,0

1001

=

(place value is very important here!!)

In Grade 7 decimals may have been limited to three decimal places. In Grade 8 learners should engage with decimals without any limitation on the number of decimal places, and they should engage with recurring decimals as an alternative representation of certain

fractions, for example 0,33333 . . . . . as an alternative representation of 31

.

Explain how a recurring decimal is shown e.g. means 0,33333... and means 0,325325325... and means 0,1353535... etc.

When solving problems try to use examples that relate to learners.

EXAMPLES 1. Look at the sketch of a string of 15 beads. Six of these beads are black.

You can write 6 of 15 as the number 156 .

You can calculate this number with your calculator. Underneath you see a computation strip. The grey squares represent the keys of the calculator.

156 = 6 ÷ 15 = 0,40

a) Nontyatyambo says: "two of every five beads are black". She may also say: "two fifths

of the beads are black" and write it as the number 52

Use your calculator to make sure that 52 is the same number as

156 :

52 = ÷ =

b) Nontyatyambo makes a string of beads, which is twice (two times) as long as the string above and with the same pattern. How many beads are there in total in this string? How many black beads are there in this string ? Complete:

......

...... = ÷ =


c) Nontyatyambo makes a string of 100 beads and with the same pattern. How many black beads are there in this string ? Complete:

100...... = ÷ =

You may say:

6 of 15 or 156 or

10040 or 0,40

2 of 5 or 52 or

10040 or 0,40

12 of 30 or 3012 or

10040 or 0,40

40 of 100 or 10040 or

10040 or 0,40

Note: Here we work with the notion of ratio and not ‘multiply the top and bottom with the same number’.

You can see there are different ways to write the same number. We often rewrite a fraction as a fraction with a denominator of a 100. This makes it easier to compare

numbers and also links to our decimal system. Therefore, 52 can be written as

10040 ,

which is seen as "40 per a 100" and can also be written as 40%.

When one reads or hears the following words or phrases: save, buy on credit, inflation, price increases, wage negotiations and many more examples, one sometimes also reads or hears the word “percent” or "percentage". Percent comes from Latin and means per hundred. Twenty percent, for example, is equivalent to

10020 = 0,20 and can be written as 20%.

2. If you’ve eaten 43 of your own birthday cake it means:

for every four equal parts, you’ve eaten three of the parts. If the cake was cut into 100 pieces you would have eaten 75 pieces. This means that you have eaten 75 of the 100 pieces, in other words, 75% of the cake!

a) Can you and your maths mate explain how the amount of 75 pieces was calculated? Write down your explanation.

b) 75 pieces sounds a lot more than 3 pieces. Is it the same amount of cake in this case? Write down your explanation.

c) Say that Jolene also ate 75% of her birthday cake but that she had a smaller cake than you. Is your 75% and her 75% the same amount of cake?


3. Use your calculator and complete:

a) 43 = 3 ÷ 4 = 0,75 = %

b)

21 = ÷ = = %

c)

8040 = ÷ = = %

d)

6012 = ÷ = = %

e)

10012 = ÷ = = %

f) ...

... = ÷ = = 80%

4. Colour 4 of the 8 small triangles: Colour 2 of the 4 small rectangles:

5. Complete the following and compare with no 4:

a) 84 = ÷ = = %

b) 4

2 = ÷ = = %

c) 2

1 = ÷ = = %

d) What percentage of the figures in no. 4 is coloured in?

6. A “centigrid” is a pattern of 10 rows with 10 blocks in each row.

a) What percentage of the left hand grid is coloured grey?

b) Colour in 30% of the right hand grid.


c) Write down, next to each figure, which fractional part of each figure is coloured in. Write it as a fraction and as a percentage. Check your answer with your calculator.

fig 1

fig 2

fig 3 fig 4

d) Colour in 25% of each of the following figures. Do it in two different ways for each figure:

e) Colour in 25% of each of the following figures: 7. You must colour in the figures according to the percentage written next to each figure.


CONSOLIDATION / HOMEWORK Give example on problem solving to develop skills.


TERM 2 - WEEK 5

ASSESSMENT STANDARD 8.4.11 Investigates (alone and/or as a member of a group or team) the relationship between the sides of a right-angled triangle to develop the Theorem of Pythagoras.

8.4.12 Uses the Theorem of Pythagoras to calculate a missing length in a right-angled triangle leaving irrational answers in surd form (√).

8.4.13 Describes and illustrates ways of measuring in different cultures throughout history (e.g. determining right angles using knotted string, leading to the Theorem of Pythagoras).

TERMINOLOGY Right-angled triangle; surd; opposite side; adjacent side; hypotenuse side; Theorem of Pythagoras


INTEGRATION Technology


(Theorem of Pythagoras). Revise properties of squares and right angled triangles, area of squares, surd notation and

area of triangles

The introduction of the Theorem of Pythagoras in Grade 8 should be done, as the Assessment Standard suggests by means of an investigation (one possible investigation is provided in examples 1 - 4). At some stage the theorem needs to be formally stated so that it can be used in the application which is prescribed by the Assessment Standard.


EXAMPLES A Century Old Discovery

Cut out (very carefully) all the squares on the separate handouts that your teacher will provide (it is at the end of this document). Work very precisely and make sure that you keep all the squares together in, for example, an envelope. The side-lengths of the squares are given in terms of “small squares”.

You already know that:

a triangle with one of the angles greater than 90° is an obtuse triangle. a triangle with one angle equal to 90° is a right-angled triangle. a triangle with all the angles less than 90° is an acute triangle.

1. It is possible to enclose a triangle with three squares. The enclosed triangle can be obtuse, acute or right-angled. Write down what types of triangles are formed in the following four examples.

2. "A person cannot enclose a triangle with all groups of three squares. "

Use the cut out squares to investigate this statement. Write down with what groups of squares you cannot enclose a triangle:

Group A: squares with side lengths (number of small squares): 12, 8 and 6

Group B: squares with side lengths (number of small squares): 12, 8 and 4

Group C: squares with side lengths (number of small squares): 12, 8 and 2

Think of a rule that states when it’s possible or not to enclose a triangle with three squares. Write it down.


3. You are now going to try to place three of your cut out squares in such a way that a triangle is formed. Certain information is given in the following table. You have to complete the table.

Lengths of the sides of the SQUARES (from small to big):

Lengths of the sides of the TRIANGLE (from small to big): Type of triangle:

7, 11, 17

6, 8, ? right-angled

8, 10, 10

6, 10, 15

10, 11, 12

2, 4, 4

3, 5, ? obtuse

9, ?, 15 right-angled

?, 13, 14 obtuse

4. When one encloses a right-angled triangle using three squares, a specific rule applies.

a) To discover this rule you need to complete the following table. (Use your cut out squares.)

Length of the sides of the squares (from small to big):

Sum of the areas of the two smaller squares:

Area of the largest square:

Type of triangle:

7, 8, 10 11387 22 =+ 100102 = acute

4, 5, 8 4154 22 =+ 6482 =

6, 8, ? 10086 22 =+ Right-angled

8, 13, 17

3, 4, 5

5, 6, 7

5, 12, 13

5, 7, 10

8, 15, 17

2, 10, 10

7266 22 =+ 3662 =

b) What do you observe (notice) about the right-angled triangles in the table above? What do you observe about the acute triangles in this table? What do you observe about the obtuse triangles in this table?


c) For the following table you don’t need your cut out squares. Complete the table by using your pocket calculator and use the answers in b.

Length of the sides of the squares (from small to big):

Sum of the areas of the two smaller squares:

Area of the largest square: Type of triangle:

20, 21, 31

7, 24, 25

12, 35, 36

60, 80, 100

20, 25, 30

9, 40, 41

d) Predict what type of triangle you will get when you use squares with side lengths of 100 cm, 225 cm and 400 cm.

You have observed that right-angled triangles have a distinctive property. This property makes a distinction between the two squares that enclose the right-angle and the third square. Therefore the sides of a right-angled triangle were given different names: the two adjacent sides and the hypotenuse. (The hypotenuse is always the longest of the three sides and lies directly opposite the right-angle.)

5. Write down which are the adjacent sides and the hypotenuses of the following triangles:

The special property of right-angled triangles that you have examined is known as Pythagoras’ Theorem.

6. Go to the library or use the internet to find out more about who Pythagoras was. Write a story about this man describing what he did, why he is still famous and about the times he lived in. Describe at least one practical application of Pythagoras’ Theorem in our daily lives.

The converse of Pythagoras’ Theorem can also be used

In Egypt the Nile floods every year; and as a result the land along the banks of the Nile is fertile agricultural soil. Long ago the plots of ground had to be measured every year after the floods. Therefore the Egyptians were very good at practical surveying. To measure a right-angle (90 degrees) they used a rope with 12 knots in it. The knots were equal distances apart (equidistant) from each other.

Adjacent side

Adjacent side

Hypotenuse


7. a) Take a long piece of string/cord and tie the ends together into a knot. Measure off 12 equal lengths on the string. Instead of knotting the cord you can make clear marks on the string with a pen / marker.

b) You can mark out triangles with this string in such a manner that each point of a triangle is precisely on a mark (knot.) How many different triangles can you make with your 12-knot string? Draw these triangles to scale: let one centimetre represent the distance between two marks (knots.)

c) Which of the triangles that you have drawn do you think the Egyptian surveyors used to measure the correct angles (90 degrees?)

8. Builders must sometimes ensure that the angle between two objects (e.g. two walls) is a

right-angle (90°.)

a) Give at least three examples of such instances.

b) Builders sometimes use the sixty-eighty-hundred rule to get two objects “square” (at 90° to each other); for this you only need a metre-stick:

Each group must now do the following:

Nail two longish rods/planks together so that they can still move a little (A);

On one rod measure off 60 cm and make a mark (B), On the other rod measure off 80 cm and make a mark (C), Nail another rod, exactly 100 cm long, fast at marks B and C.

CONSOLIDATION / HOMEWORK Learners must know the theorem by heart e.g. ‘The square of the hypotenuse is equal to the sum of the squares of the opposite and adjacent sides’ and be able to apply it in different examples.

ASSESSMENT Assessment Task 4 e.g. Investigation on Pythagoras + tutorial on part of week 1-4 work Use the examples above for the investigation.


TERM 2 - WEEK 6 AND 7

ASSESSMENT STANDARD 8.5.1 Poses questions relating to human rights, social, environmental, economic and political (own environment) in RSA

8.5.2 Selects appropriate sources for the collection of data (including peers, family, newspapers, books, managements and INTERNET

8.5.3 Designs and uses questionnaires with a variety of possible responses in order to collect data (alone + group) to answer questions

8.5.5 Organises (including grouping where appropriate) and records data using tallies, tables and stem-and-leaf displays

8.5.6 Summarises grouped and ungrouped numerical data by determining mean, median and mode as measures of central tendency, and distinguishes between them

8.5.9 Determines measures of dispersion, including range and extremes.

8.5.7 Draws a variety of graphs by hand/technology to display and interpret data including: among bar graphs and double bar graphs, histograms with given and own intervals; pie charts; line and broken-line graphs and scatter plots. 8.5.8 Critically reads and interprets data presented awareness of sources of error and manipulation to draw conclusions and make predictions sensitive to the role of: • context (e.g. rural or urban, national or provincial); • categories within the data (e.g. characteristics of different target groups age, gender, race); • data manipulation (e.g. grouping, scale, choice of summary statistics) for different purposes; • the role of outliers on data distribution • any other human rights and inclusivity issues.

TERMINOLOGY Questionnaire, responses, tallies, tables and stem-and-leaf displays, grouped and ungrouped numerical data, mean, median, mode as measures of central tendency, measures of dispersion, range and extremes, bar graphs; double bar graphs; histograms; pie charts; line and broken-line graphs and scatter plots.

RESOURCES Data Handling in the GET Band, Gr 8 Text books; WCED Illustrative examples; Calculator; Poster; Internet Web sites, IMSTUS material

INTEGRATION Life Orientation; Social Sciences: Natural Sciences


TEACHING TIPS Use data in contexts that learners can identify with or which fits in with their frame of

reference.

One very useful site for collecting municipal, provincial and national data is the Statistics South Africa website (www.statssa.gov.za).

The “Population Census 2001” (http://www.statssa.gov.za/census01/html/default.asp) link on the Statistics South Africa takes the user to a site with information on the 2001 census. The link to the “Digital Census Atlas” (http://www.statssa.gov.za/census2001/digiAtlas/index.html) takes the user to a wonderful resource with a wide range of data

Use the book ‘Data Handling in the GET Band’ by Jackie Scheiber and Meg Dickson. (The book was sent to all WCED in 2008. If further copies are required contact Edumedia at 021-6892641) The following page references are from the book: Collect data: pages 2 - 3 Organise/ sort data: pages 3 – 6 and 37 - 39 Represent data: pages 6 - 27 Group data – measures of central tendency: pages 28 - 36 Measures of dispersion range: page 40 – 41 Critically read and interpret data

There are different kinds of data:

- Quantitative or numerical data is data in the form of numbers e.g. age, the number of learners in a family and the height of the building. It is possible to determine the mean, median, mode and range of such data

- Data that is not numerical in nature but which describes attributes or categories is called categorical data e.g. hair colour, method of transport, favourite television programme. Categorical data: Cannot be summarised using a stem and leaf plot. Cannot be represented by means of a histogram or a line/broken line graph.

- Data that is referred to as discrete can only have countable (positive whole number) values e.g. number of siblings you have and the number of times you shower a day.

- By contrast to discrete data continuous data has values that can be any number within a range of numbers e.g. the time taken to run 100m and the amount of rain that falls on a day.


The table below illustrates the organising, summarising and representing techniques mentioned in the Assessment Standards and the kinds of data to which they can be applied.

Qua

ntita

tive

data

Cat

egor

ical

dat

a

Dis

cret

e da

ta

Con

tinuo

us d

ata

Summarising data

Mean

Median

Mode

Measures of dispersion

Representing data

Bar and double bar graphs

Histograms

Pie charts

Line and broken line graphs

Scatter plots

EXAMPLES 1. Use examples from the book ‘Data Handling in the GET Band’.

Both the examples given in the above book and the illustrations provided below have addressed human rights and inclusivity issues such as gender, population group, rural vs. urban etc.

2. Water usage in the Western Cape

The main source of water for dwellings in the Western Cape by the population group of the head of the household

Black African Coloured

Indian or

Asian White

Piped water inside dwelling 88760 407025 9788 286125

Piped water inside yard 110792 86438 392 9829

Piped water on community stand: distance less than 200m from dwelling 56028 15750 106 2404

Piped water on community stand: distance greater than 200m from dwelling 53471 19414 384 6537

source: Census 2001 (www.statssa.gov.za)

The table above lists data collected in the 2001 population census by Statistics South Africa.

a) Calculate the total number of people per source of water for the Western Cape Province.

b) Draw a bar graph illustrating the total number of people per source of water for the Western Cape Province.


c) Draw a bar graph for each of the population groups (by head of household) illustrating the source of water for the Western Cape Province.

d) By studying your graphs determine whether or not sources of water are equitably distributed across population groups in the Western Cape Province.

3. Truck drivers recently went on strike demanding a minimum wage of R14,50 per hour. A newspaper reporter decided to investigate for herself what the truck drivers earned and decided to interview a sample of 10 drivers. She asked each driver to show her their most recent weekly pay slip. She recorded the number of hours worked in a particular week and the wages earned. Her data is recorded in the following table.

Driver Hours worked in a week Wages earned (in Rand)

1 32 492

2 38 537

3 43 370

4 34 537

5 36 504

6 37 509

7 43 599

8 46 636

9 39 597

10 47 727

a) Determine the hourly wage rate being paid to each of the drivers.

b) Calculate the mean hourly rate for the drivers in the sample.

c) Draw a scatter plot for the data and identify any outliers.

d) Recalculate the mean hourly rate for the drivers in the sample leaving out any outliers that you identified in the scatter plot.

e) Based on this sample do you think that the drivers have cause to strike?

4. For other ideas and extra examples, look at IMSTUS module 14 and 23

CONSOLIDATION / HOMEWORK Practise the other ways of collecting, organising and representing data i.e. not covered in the project.

ASSESSMENT Assessment Task 5 e.g. Project on Data – Start in week 6 and submit in week 7

The Project can just cover one way of collecting and organising data. The project can just cover one or two types of graphs and the drawing of conclusions. During class and/or homework the learners must practise the different ways of collecting and organising data, the drawing of all the graphs and their interpretation.

TERM 2 - WEEK 8 - 10 CONSOLIDATION


TERM 3

TERM 3 - WEEK 1 REVISION

TERM 3 - WEEK 2

ASSESSMENT STANDARD 8.1.3 Recognises, classifies and represents the following numbers in order to describe and compare them: • Large numbers in scientific notation; • Irrational numbers in the context of measurement (e.g. π and square and cube roots of non-

perfect squares and cubes).

TERMINOLOGY Scientific notation, pi, square, cube roots, non-perfect squares, cubes


INTEGRATION Technology, Natural Sciences


(Scientific notation). SCIENTIFIC NOTATION:

Explore distances from earth to moon and other planets.

Revise exponents and expand to 220 , 315, etc.

Convert large numbers into scientific notation and from scientific notation to large numbers.

IRRATIONAL NUMBERS Remember pi – day 14 March (First term).

Knowledge and use of recurring decimals provide access to the important understanding (knowledge) that all fractions can be represented as (finite or recurring) decimals and vice versa. This in turn, provides access to understanding that there are other numbers that cannot be represented as fractions, or as finite or recurring decimals. One example of such a number is 2 , which may occur in calculating the length of a side of a triangle with the Theorem of Pythagoras (assessment standard 8.4.12).

The ancient Greek mathematicians were so perplexed when they could not find/express the number that describes the lengths of certain sides of triangles, or of squares with certain areas, that they called such a number an irrational number, which literally means “illogical number”.

Pi can be done in detail here or it can just be mentioned here and done in detail when area of the circle is covered in week 8.


EXAMPLES 1. (a) Express the following as ordinary numbers:

3,4 × 104, 3,4 × 106 and 3,4 × 109

(b) Express the following numbers in the scientific notation:

45 000 000 120 000 3 090 000 000

2 The sun is the star (in our galaxy) closest to the earth. The second closest star is on a distance of one light-year. A light-year is the distance that light covers in ONE year: kilometresyearlight 3652460600003001 ××××=−

a) Write down what each number represents in the conversion of one light-year kilometres.

b) Check whether 1 light-year = 9 460 800 000 000 km.

c) A person who knows a lot about stars says: "When you look at the stars, you are looking back in time." What do you think she means by that ?

3. The production figures for a certain mine are given below, for a period of six months. How much ore was produced in total over this period?

January: 7,34 × 105 kg February: 3,23 × 106 kg March: 4,08 × 106 kg

April: 9,67 × 105 kg May: 8,27 × 104 kg June: 7,66 × 106 kg

PI DAY IS ONE OF TWO UNOFFICIAL HOLIDAYS HELD TO CELEBRATE THE MATHEMATICAL CONSTANT π (PI).

IT'S PI DAY AGAIN CELEBRATED IN 200...

ON 14 MARCH

AT 1:59 PM

WHY? Pi Day is observed on March 14 (3/14 in American date

format), due to π being equal to roughly 3,14.

Pi Minute is celebrated on March 14 at 1:59, and

Pi Second will be on March 14 at 1:59:26 pm (π =3,1415926)

WHAT IS π ? THE BASICS OF: π

π is the 16th letter of the Greek alphabet. The pronunciation of this letter in Greek is like the English word "pea" (the same way they say the name of the letter "P". It is never pronounced like the English word "pie" in Greece.

π is the number of times a circle’s diameter will fit around its circumference


A CIRCLE OF π : π is the ratio of a circle's circumference to its diameter

the area of a circle is π X radius2 (A = 2r×π )

the circumference of a circle is π X diameter or (C = )dπ 2π X radius (C= rπ2 )

π TO 100 DECIMAL PLACES IS:

3,14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164

06286 20899 86280 34825 34211 70679

INTERESTING TO KNOW.....

there is no zero in the first 31 digits of Pi

if a billion decimals of pi were printed in ordinary type, they would stretch from New York City to the middle of Kansas

Taking the first 6 000 000 000 (billion) decimal places, this is the distribution:

0 occurs 599 963 005 times 1 occurs 600 033 260 times 2 occurs 599 999 169 times 3 occurs 600 000 243 times 4 occurs 599 957 439 times 5 occurs 600 017 176 times 6 occurs 600 016 588 times 7 occurs 600 009 044 times 8 occurs 599 987 038 times 9 occurs 600 017 038 times

at position 763 there are six nines in a row

the sequence 314159 re-appears in the decimal expansion of Pi at place 176 451. This sequence appears 7 times in the first 10 million places(not including right at the start)

all the digits of π can never be fully known

Simon Plouffe was listed in the 1975 Guinness Book of World Records for reciting 4096 digits of π from memory.

If one were to find the circumference of a circle the size of the known universe, requiring that the circum-ference be accurate to within the radius of one proton, only 39 decimal places of π would be necessary.

3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074462379962749567351885752724891227938183011949129833673362440656643086021394946395224737190702179860943702770539217176293176752384674818467669405132000568127145263560827785771342757789609173637178721468440901224953430146549585371050792279689258923542019956112129021960864034418159813629774771309960518707211349999998372978049951059731732816096318595024459455346908302642522308253344685035261931188171010003137838752886587533208381420617177669147303598253490428755468731159562863882353787593751957781857780532171226806613001927876611195909216420198938095257201065485863278865936153381827968230301952035301852968995773622599413891

249721775283479131515574857242454150695950829533116861727855889075098381754637464939319255060400927701671139009848824012858361603563707660104710181942955596198946767837449448255379774726847104047534646208046684259069491293313677028989152104752162056966024058038150193511253382430035587640247496473263914199272604269922796782354781636009341721641219924586315030286182974555706749838505494588586926995690927210797509302955321165344987202755960236480665499119881834797753566369807426542527862551818417574672890977772793800081647060016145249192173217214772350141441973568548161361157352552133475741849468438523323907394143334547762416862518983569485562099219222184272550254256887671790494601653466804988627232791786085784383827967976681454100953883786360950680064225125205117392984896084128488626945604241965285022210661186306744278622039194945047123713786960956364371917287467764657573962413890865832645995813390478027590099465764078951269468398352595709825822620522489407726719478268482601476990902640136394437455305068203496252451749399651431429809190659250937221696461515709858387410597885959772975498930161753928468138268683868942774155991

855925245953959431049972524680845987273644695848653836736222626099124608051243884390451244136549762780797715691435997700129616089441694868555848406353422072225828488648158456028506016842739452267467678895252138522549954666727823986456596116354886230577456498035593634568174324112515076069479451096596094025228879710893145669136867228748940560101503308617928680920874760917824938589009714909675985261365549781893129784821682998948722658804857564014270477555132379641451523746234364542858444795265867821051141354735739523113427166102135969536231442952484937187110145765403590279934403742007310578539062198387447808478489683321445713868751943506430218453191048481005370614680674919278191197939952061419663428754440643745123718192179998391015919561814675142691239748940907186494231961567945208095146550225231603881930142093762137855956638937787


Here’s a π limerick:

Three point one four one five nine two

Its been around forever –its not new

It appears everywhere in here and in there

Its irrational I know but its true!!!!

CONSOLIDATION / HOMEWORK Examples for homework



ASSESSMENT STANDARD 8.1.5 Solves problems in context including contexts that may be used to build awareness of other Learning Areas, as well as human rights, social, economic and environmental issues such as: • financial (including profit and loss, budgets, accounts, loans, simple interest, hire purchase,

exchange rates); 8.1.10 Uses a range of strategies to check solutions and judges the reasonableness of solutions.

TERMINOLOGY Profit and loss; budgets; accounts; loans; simple interest; hire purchase; exchange rates; discount; reduction; retailer; wholesaler; cost price; selling price; currency weak/strong currencies; income; expenditure


INTEGRATION EMS


Learners must know:

- How to calculate the percentage if the amount is known. - How to calculate the amount if the percentage is known. - How to calculate the original amount if the amount after a certain percentage is

subtracted, is known. - How to calculate the original amount if the amount, after a certain percentage is

added, is known. Make learners justify their answers e.g. Should the new amount be more or less than the

original amount.

π


Example:

A shop owner gives 30% discount on CD’s. After discount you have to pay R 105,00. What is the price for a CD without the discount ?

Amount 105 ?

Percentage 70 1 100

A 105 1,50 ?

P 70 1 100

Therefore the CD would have cost R150,00 without the discount.

The financial maths section lends itself to a project which could include starting up of a business involving buying and selling, profit and loss, working to a budget, having to raise a loan and pay accounts etc. Refer to the book ‘My Business: Steps to Financial Freedom grade 8 and 9”. The book was sent to all WCED schools in 2007.

When studying accounts, use accounts relevant to the households from which the learners belong e.g. lights and water, telephone, cell phone, clothing stores etc. Link parts of financial maths together e.g. if the account is not paid on time, how much more will you have to pay if the municipality adds 12% interest to any overdue account.

Let learners investigate how money can be ‘loaned’ and the different interest charged by these ‘loaning’ companies. Let them compare this to hire purchase. Which would be cheaper? Borrowing the money or buying the item on hire purchase?

When teaching exchange rates- work from a scenario of planning a trip overseas and deciding in hindsight when the best time would have been to exchange your money from rand to pounds or dollars. If learners are battling with this concept give them ‘play money’ to enact the exchange of money between a tourist and a banking clerk.

EXAMPLES 1. Mpho borrows R6 000 from a bank at 13% interest per year, for a period of 15 months.

How much should he pay back at the end of the period?

2. Rashied borrows R6 000 from a bank at 14% interest per year, for a period of 27 weeks. How much should he pay back at the end of the period?

3. A shop owner gives 35% discount on CD’s. After discount you have to pay R 95,00. What is the price for a CD before the discount ?

4. Eddie borrowed money for one year and has to pay 20% interest per year. At the end of the year he has to pay back R1 800 to clear his debt. How much did he borrow at the beginning of the year?

5. Sipho buys a fridge for R4 499 (cash price) on credit agreement. (hire purchase). He has to pay a deposit of R400. His monthly payment is R 273,00 for two years. If the monthly payments go into arrears, the fridge will be repossessed.

a) Calculate the total amount that Sipho will have paid for the fridge?

b) How much extra has Sipho paid for the fridge? Give your answer as a percentage of the cash price.

× 100÷ 70

× 100÷ 70

Amount 0 105 ?

0% 70% 100% Percentage

× 100÷ 70


6. On 21 July 2009 the following exchange rates were: 1 US dollar = R7,85; 1 British pound = R14,20 and 1 euro = R11,00

What will the exchange rates be if the value of the rand increases by 2% against all these currencies?

7. With the birth of his first grandchild Louis opened a savings account for this grandchild with a deposit of R500,00 . The average interest earned on this account is 10% per annum (per year).

a) If Louis deposits no more money into this account, what would the balance be in this account when the grandchild is 9 years old? (Remember: A person receives interest on the previous year’s interest... and you don’t have to take bank charges into consideration.) You may complete the following table to help you get your answer:

year interest total 0 R500,00 1 R50,00 R550,00 2 R55,00 R605,00 3 R60,50 R665,50 4 R66,55 5

b) After how many years did the opening balance double?

c) After three years the account balance is R665,50 . After how many years is the total amount double this amount?

8. Stanley wants to buy a bicycle for R972,00 . He entered into a lay-bye contract with the shop. Stanley must pay R162,00 per month for half a year.

a) Stanley keeps record of his payments. Complete:

date payment payment to date 1 April 162,00 162,00 1 May 162,00 324,00 1 June 162,00 1 July 162,00

1 August 162,00 1 September 162,00

b) If Stanley pays on the first of the month, every time, when can he get his bike?


c) Every month the shop owner banks the money that is paid to him. He receives 0,55% interest per month on the money in his bank account. The interest is added to his account on the last day of each month. Complete the following table to see what influence the interest has on the shop owner’s bank balance:

date Stanley’s payment interest balance 1 April 162,00 162,00

30 April 0,89 162,89 1 May 162,00 324,89

31 May 1,79 326,68 1 June 162,00 488,68

30 June 2,69 491,37 1 July 162,00 653,37

31 July 3,59 656,96 1 August 162,00

31 August 1 September

30 September

9. If it’s not going so well in a particular month, then the payment for that month may be in arrears; but then the delivery of the bike is also postponed for a month. Mousa also bought a bike for R972,00 on lay-bye on the same day as Stanley. Mousa skipped his payments for the month of July and August.

a) When can Mousa get his bike from the shop? Write down your calculations.

b) Complete the following table to see what influence the interest has on the shop owner’s bank balance:

date Mousa’s payment interest balance 1 April 162,00 162,00

30 April 0,89 162,89 1 May 162,00 324,89

31 May 1,79 326,68 1 June 162,00 488,68

30 June 2,69 491,37 1 July 0,00 491,37

31 July 2,70 494,07 1 August 0,00 494,07

31 August 2,72 496,79 1 September 162,00 658,79

30 September 3,62 662,41 1 October 162,00 824,41

31 October 1 November

30 November

c) Which kind of payment do you think the shop owner prefers? Your answers must be supported by meaningful reasons.


10. Nkululeko plans to save R90,00 every month so that he has extra cash in case something unexpected happens and he needs the money. The money earns 0,6% interest per month and he gets interest on the previous month’s interest. Calculate what the balance will be in the account after one year. (Nkululeko does not have to pay bank charges.) Use the table:

Month Deposit Balance at

beginning of month

Interest Balance at end of month

1 R90,00 R90,00 R0,54 R90,54

2 R90,00 R180,54 .... ....

CONSOLIDATION / HOMEWORK Examples which will build skills towards completion of Assessment Task 7

ASSESSMENT Assessment Task 7 e.g. Tutorial / Project on financial maths. Choose a context relevant to the learners e.g. let learners select a business to start - from this starting point include questions to cover buying and selling, profit and loss, discount, accounts, loans, hire purchase, interest etc.

TERM 3 - WEEK 5 AND PART OF WEEK 4

ASSESSMENT STANDARD 8.3.2 In contexts that include those that may be used to build awareness of social, cultural and environmental issues, describes and classifies geometric figures and solids in terms of properties, including: • sides, angles and diagonals and their interrelationships, with focus on triangles and quadrilaterals

(e.g. types of triangles and quadrilaterals).

8.4.10 Estimates, compares, measures and draws angles accurate to one degree using protractors.

8.3.3 Use vocabulary to describe parallel lines cut by a transversal, perpendicular lines, intersecting lines and triangles in terms of angle relationships (e.g. vertically opposite, corresponding).


TERMINOLOGY Angles: acute; right-angled; obtuse; straight; reflex Intersecting lines; perpendicular lines; vertically opposite angles; supplementary angles; complementary angles; parallel lines; transversal lines; corresponding angles, alternate angles, co-interior angles.

Isosceles triangle; equilateral triangle; right-angled triangle; interior angles; exterior angles


INTEGRATION Natural Sciences


(angles). Revise angles and use of the protractor from week 4.

The use of the protractor to measure and draw angles can be practised first. Learners can then classify angles. This will lead into the drawing of nets and the properties of geometric figures and solids.

Context: Social, cultural and environmental issues.

Learners require practice in measuring and drawing angles. Some may have done this quite comprehensively in grade 7 others may have never individually used a compass or protractor.

Plan ahead. If learners do not have protractors or compasses get the school to buy enough for a class, so that the instruments can be shared among the classes.

Grade 8 curriculum is only asking for learners to identify and know the names of angles and triangles and draw/construct them. The grade 9 curriculum asks learners to use the geometry of straight lines, parallel lines and triangles to solve problems and justify relationships. (That is the reason for only one week being given for this section).

Remember that parallel lines never intersect. The (perpendicular) distance between parallel lines is the same everywhere. (Remember: distance is the shortest space between two points).

A parallelogram is a quadrilateral in which the opposite sides are parallel.

When parallel lines are cut by another line (transversal), we can indicate angles that are the same.


EXAMPLES 1. a) Below are four diagrams of lines cutting each other. Use your protractor to measure

the sizes of all the angles in each sketch. Clearly mark them on each diagram. (Remember to write the ‘degree sign’)

b) What do you observe if you look at the size of each angle?

c) Look at the diagrams below. The angles that are marked the same are equal. These angles are directly opposite each other and are called vertically opposite angles or X-angles (to help you remember how they look). Investigate to show that vertically opposite angles are equal.

d) Which angles would you call adjacent angles ?

e) The sum of all the angles around a point is called a revolution. What is this sum ?

2. a) Look around in your classroom and record where you see parallel lines.

b) Give everyday examples of situations where we would find parallel lines.

3. a) Draw two parallel lines such that the distance between the lines is 3 cm.

b) Describe how you drew the two parallel lines 3 cm apart.


4. a) Draw a parallelogram in which an internal angle is 50o and where one side is 4 cm long and another 5 cm long. Measure the distance(s) between the opposite sides.

b) Draw a parallelogram in which the distance between one pair of sides is 3 cm and the distance between the other pair of opposite sides is 5 cm. (There are various solutions.)

5. Look at this drawing of part of a tiled floor. Each tile has the shape of a parallelogram. Indicate with arrows which lines are parallel on your sketch. What do you do if there are different pairs of parallel lines?

REMEMBER: LINES ARE PARALLEL WHEN THE (PERPENDICULAR) DISTANCE BETWEEN THEM IS THE SAME EVERYWHERE.

a) Angles marked the same are equal. You know that vertically opposite angles are equal. Indicate, using different markers, all angles that are equal. Use the same mark to denote equal angles. You may use your protractor to check.

b) Below we have the same tiled floor but line segments are drawn on the floor to form the letter F. Some of the F’s look a bit funny. In each F-diagram, three angles are marked. What can you note about these angles in the F?


c) Below we have the same tiled floor, but line segments are drawn on the floor to form the letter Z. Some of the Z’s look a bit funny. In each Z-diagram, two angles are marked. What do you notice about these angles in the Z? First indicate with arrows which lines are parallel on your sketch.

6. In the diagram below we also drew a F-figure and Z-figure.

a) Are there any parallel lines in the figures above?

b) Are there angles that are equal in the F-figure and Z-figure ?

c) What conclusion can you draw about the angles in the F-figure and Z-figure ?

7. An F-figure and a Z-figure consist of three line segments. The three line segments include angles. Indicate which lines are parallel to each other.

When two of the three line segments are parallel, we will find angles that are equal. In the F-figure, the corresponding angles are equal and in the Z-figure, the alternate angles are equal.


8. Construct triangles:

When an architect makes a drawing, it is done very accurately. An ordinary sketch will not be acceptable. We are now going to construct accurate drawings. When the lengths of the sides of a triangle are given, e.g. triangle ABC, with sides 3 cm; 4 cm; 5 cm, one can construct this triangle in the following manner: Draw any one of the sides (remember to be accurate). In the example, the side with length 3 cm is drawn first. Draw a circle (an arc is sufficient) using A as the centre of the circle and a radius (distance that the legs of the compasses are apart) of 5 cm . Point C must lie somewhere on this arc. Draw a second circle (an arc is sufficient) using B as the centre of the circle and a radius (distance that the legs of the compasses are apart) of 4 cm. Point C must lie somewhere on this arc. Where the two arcs intersect is a point 5 cm from A and 4 cm from B. This is the point C.

9. a) Construct a triangle with side lengths 3 cm; 4 cm and 5 cm

b) Does your triangle have a right angle ?

10. a) Construct a parallelogram with sides 9 cm; 5 cm and a diagonal of length 7 cm.

b) Measure the internal angles of your parallelogram using a protractor.

c) Did all the learners in your class obtain angles of the same size?

11. a) Construct a right angled triangle with sides adjacent to the right angle being 6 cm and 8 cm.

b) Cut out your construction and place it as accurately as possible onto another learner’s construction. Does it fit?

CONSOLIDATION / HOMEWORK Draw the different triangles from given information



TERM 3 - WEEK 6

ASSESSMENT STANDARD 8.3.1 Recognises, visualises and names geometric figures and solids in natural and cultural forms and geometric settings, including: • those previously dealt with; • the platonic solids (tetrahedron, cube, octahedron, dodecahedron, icosahedron).

8.3.2 In contexts that include those that may be used to build awareness of social, cultural and environmental issues, describes and classifies geometric figures and solids in terms of properties, including:

• sides, angles and diagonals and their interrelationships, with focus on triangles and quadrilaterals (e.g. types of triangles and quadrilaterals).

8.3.4. Uses a pair of compasses, ruler and protractor to accurately construct geometric figures for investigation of own property and design of nets.

8.3.5. Designs and uses nets to make a model of geometric solids studied up to and including this grade

TERMINOLOGY Square; rectangle; parallelogram; rhombus; trapezium; kite; polygon; polyhedra; prism; pyramid; tetrahedron; cube; octahedron; dodecahedron,; icosahedron; vertices; edges; platonic sollids




(Geometric Solids). The properties of the geometric solids can be taught by investigation of the concrete

objects themselves or the teacher can introduce the investigation of the properties whilst the learners are designing /drawing the models of the 3D objects. This means that normally assessment standards 8.3.2, 8.3.3., 8.3.4, 8.3.5, 8.3.9 and 8.4.10 are all integrated and will be taught together over a two week period.

Most learners will need to start with the construction of triangles and quadrilaterals. Investigate their properties and then move onto 3D objects and nets.

Let learners flatten different shape boxes and trace their nets. Learners could do this in a group- each leaner in the group traces the net of a different box and then they compare their nets. The construction of the models could also be done in groups with each group member constructing a different platonic solid and they could compare their solid’s properties.


EXAMPLES 1. Working as a member of a team: a) Using a pair of compasses, a protractor and a ruler: design and make the nets for

each of the five platonic solids: Tetrahedron Cube Octahedron Dodecahedron Icosahedron

b) Construct each of the objects from the nets you have designed

c) Draw and complete the following table:

Platonic solid Number of faces Number of vertices Number of edges

Tetrahedron

Cube

Octahedron

Dodecahedron

Icosahedron

d) Study the table you have completed for the platonic solids and describe, in words: Any patterns that you observe between the entries in the table

e) Reflect on how you completed the table. Did you count the number of faces, vertices and edges each time? Look again at each of your objects and this time see if you can predict the number of vertices and edges for each object based on you knowledge of the number of faces, the shape of its faces and the number of faces that meet at a vertex. Describe your method in words.

2. Let learners do exercises from IMSTUS module 13 pages 3-14

CONSOLIDATION / HOMEWORK Complete model construction for homework (not the one for assessment)

ASSESSMENT Assessment Task 8 e.g. Investigation on building models and properties of the geometric solid/s built. Choose the investigation from the IMSTUS material.


TERM 3 - WEEK 7

ASSESSMENT STANDARD 8.3.9 Draws and interprets sketches of geometric solids from different perspectives with attention to the preservation of properties

8.4.1 Solves problems involving:

length;

perimeter and area of polygons

8.4.3 Solves problems using a range of strategies including: • estimating; • calculating to at least 2 decimal places; • using and converting between appropriate S.I. units. 8.4.5 Calculates, by selecting and using appropriate formulae: • perimeter of polygons • area of triangles, rectangles, squares and polygons by decomposition into triangles and

rectangles 8.4.6 Converts between: mm² ↔ cm² ↔ m² ↔ km² 8.1.5 Solves problems in context including contexts that may be used to build awareness of other Learning Areas, as well as human rights, social, economic and environmental issues such as: • measurements in Natural Sciences and Technology contexts.

TERMINOLOGY Perspective; perimeter; area; polygons; estimating; S.I. units; convert.

RESOURCES Gr 8 Text books; WCED Illustrative examples; Calculator; Poster; Internet Web sites, IMSTUS material on website.



and 13 on the website. PERSPECTIVE

Do practically. Build a model with blocks at different levels. Let learners work in groups. Each learner in the group draws the model from a different view. The sketches get mixed and learners must identify from which side the sketch was drawn.

From this learners can progress to just the sketch.


PERIMETER AND AREA Start practically by using the relevant measuring instruments to measure classroom /

school environment (length).

The models built in week 6 can be used to calculate perimeters and areas.

Use the correct SI units.

Everything learners have learnt previously this term relating to lines, angles, triangles, quadrilaterals etc. will be revised whilst doing measurement.

WHEN SOLVING A PROBLEM; TEACH THE LEARNERS A STRATEGY E.G. Read problem thoroughly- underline key words

Draw sketches of situations described in problem. Operation signs can be included

Decide what calculations are necessary

Estimate the answer- 100 or 1000’s

Do the calculations

Check that all steps in the calculations have been done

Check that the correct SI unit has been used or if necessary convert between SI units

Re-read the question to see if you have answered the question.

EXAMPLES 1. a) By making sketch drawings of each of these sets of blocks determine which sets of

blocks have the same:

Front view Back view Left view Right view Top view Bottom view

b) Hence, or otherwise, answer the following question. Can two objects that are not the

same have all their views the same?

back view

bottom view

top view

right view

left view

front view

FEDCBA


side view

top view

front view

G

F

E

D

C

B

A

2. Consider the object drawn below: a) What kind of quadrilateral is:

ABCD CDFG ADFE

b) When the object is viewed from above (top view) will quadrilaterals ABCD and ADFE

have the same shape as in question a? Explain. c) When the object is viewed from in front (front view) will quadrilaterals ABCD and

ADFE have the same shape as in question a? Explain.


3. Use a piece of squared grid paper and draw a number of different squares as shown below. In each case cut out the two copies of each of the squares and cut one of them into the four triangles and smaller square along the lines shown.

Next re-arrange the cut out pieces as shown below. Investigation:

a) What can you say about the area of the square and total area of the hexagon?

b) Consider the dotted line drawn across the hexagon. It divides the hexagon into two squares. What can you say about the area of these two squares and area of the original square?

c) By using the names of the sides of one of the triangles write the relationship that you have observed above in words.

4. In the rectangular based prism alongside:

AB = 2cm, BC = 3cm and BF = 4cm

Calculate the length of the diagonal BH.

H

G

F

E

D

C

B

A


5. If you want a formula for the area of a parallelogram you can deduce it from the formula for the area of a rectangle:

In the diagrams above a triangle is cut off of a rectangle. This triangle is then replaced on the other side of the rectangle.

a) What can you state about the area of the rectangle and the last figure?

b) What can you state about the perimeter of the rectangle and the last figure?

c) Write down, in your own words, how you would calculate the area of a parallelogram.

CONSOLIDATION / HOMEWORK 3- 4 problems for consolidation of perimeter and area


TERM 3 - WEEK 8

ASSESSMENT STANDARD 8.4.1 Solves problems involving: • perimeter and area of circles 8.4.3 Solves problems using a range of strategies including: • estimating; • calculating to at least 2 decimal places; • using and converting between appropriate S.I. units. 8.4.4 Describes the meaning of π and uses π in calculations involving circles and discusses its historical development in measurement 8.1.2 Describes and illustrates the historical and cultural development of numbers (e.g. irrational numbers).

TERMINOLOGY Area of circles; π (pi); perimeter of circle; circumference; diameter; radius




TEACHING TIPS For ideas on developing the concept as well as extra examples, look at IMSTUS modules

11 and 13 on the website. The details about Pi can be done here or in term 3, week 2.

Measure the perimeter (circumference) and measure the diameter of the circle. Divide the perimeter by the diameter. The answer that you get is close to a number called PI (π ). (Note: Use any practical form of a cylinder e.g. can, round CD holder, etc.)

The circumference of a circle is given by c = 2π r. ( unit e.g. m, cm, km , etc)

The area of a circle is given by A = π r 2 ( unit e.g. m 2 , cm 2 , km 2 , etc.)

The surface area of a cylinder = 2 x area of the base + the area of the curved surface; therefore: surface area = 2 π r 2 + 2π r h = 2 π ( r + h )

EXAMPLES 1. Determine the perimeter and area of a circle with radius of 5 cm.

2. Determine the area of e.g. labels of cans.

3. In a historical text it is stated that Hiram, a bronze smith, had to make various articles for a temple: “He made a cast water bowl, completely round, five metres in diameter, two and a half metres high, and with a circumference of fifteen metres.”

In the text the circumference (the perimeter of a circle) and diameter of a circle is given.

a) What is the diameter of the water bowl? How many times longer than the diameter is the circumference?

b) What can you say about the relationship between the diameter and circumference of this circular water bowl?

c) Examine a number of circular objects: for example, a soup bowl, the centre ring of a soccer pitch, a bicycle wheel, etc.

Circular object Measured diameter

Measured circumference diameter

ncecircumfere

Is the relationship between circumference and diameter in the historical text also valid for the circular objects you examined?

d) What formula can you use to calculate the circumference of a circle if you know the diameter?

CONSOLIDATION / HOMEWORK 3 - 4 examples for consolidation. Relate it to a context the learners work with.



TERM 3 - WEEK 9

ASSESSMENT STANDARD 8.4.5 Solve problems with volumes and surface area of rectangular prisms and cylinders Calculates, by selecting and using appropriate formulae: • volume of triangular and rectangular based prisms and cylinders

8.4.3 Solves problems using a range of strategies including: • estimating; • calculating to at least 2 decimal places; • using and converting between appropriate S.I. units. 8.4.6 Converts between: • mm³ ↔ cm³ ↔ m³ • mℓ (cm³) ↔ ℓ↔ kℓ

8.1.5 Solves problems in context including contexts that may be used to build awareness of other Learning Areas, as well as human rights, social, economic and environmental issues such as: • measurements in Natural Sciences and Technology contexts.

TERMINOLOGY Volumes; surface area; rectangular prisms; cylinders; triangular based prisms and rectangular based prisms.


INTEGRATION Technology

TEACHING TIPS When calculating areas and volumes it is often easier to break the figure into parts e.g.

triangle and square.

Revise: The circumference of a circle is given by c = 2π r. ( unit e.g. m, cm, km , etc) The area of a circle is given by A = π r 2 .(unit e.g. m 2 ; cm 2 ; km 2 , etc.) The surface area of a cylinder = 2 x area of the base + the area of the curved surface; therefore: surface area = 2 π r 2 + 2π r h = 2 π ( r + h )

Teach concept by using practical examples e.g. coke can, etc.

Capacity is the volume of a liquid measured in ml, l etc.


The following rules are valid for capacity /volume: The volume of spatial figures like the following three (two prisms and a cylinder) are: area of the base Χ height.

EXAMPLES 1. a) Calculate the capacity (volume) of your classroom.

b) Measure and calculate the volume of a milk carton.

c) Measure a can of soup and establish by means of calculation whether the volume agrees with what is stated on the label.

d) Suppose that 5mm of rain has fallen. A flat roof of 3 by 4 metres has a drainpipe that leads into a tank. How many litres of water went into the tank? (The tank is closed at the top.)

2. a) Calculate how much paper you need to wrap a cube of 1 litre.

b) A small box is 2 cm high, 15 cm wide and 20 cm long. How much material is needed to make the box (excluding the glue tabs and closure flaps?) What is the box’s volume?

c) Compare the cube to the box. Which one has the largest capacity (biggest volume), and for which one have you used the most material?

3. The dam below is in the form of a rectangular prism. The dam wall is 2,2 m high, the outer length is 4,5 m and the outer width is 2,7 m. The wall is 36 cm thick.

a) Calculate the length of a steel band that is put right around the dam to strengthen the wall.

b) The inner surface of the dam (walls and floor) must be coated with a special paint to make it leak proof. Calculate the area that must be covered.

c) How much water will the dam hold when it is full?

height


4. The dam shown is in the shape of a cylinder. The wall is 1,8 m high and 0,41 m thick, and the inside diameter of the dam is 4.1 m.

a) Calculate the length of a steel band that is put right around the dam to strengthen the wall.

b) The inner surface of the dam (walls and floor) must be coated with a special paint to make it leak proof. Calculate the area that must be covered.

c) How much water will the dam hold when it is full?

CONSOLIDATION / HOMEWORK 3 - 4 problems on surface area and volume.

ASSESSMENT Assessment Task 9 e.g. Test on third term’s work.

TERM 3 - WEEK 10

ASSESSMENT STANDARD 8.1.6 Solves problems that involve ratio and rate.

8.4.2 Solves problems involving more complex problems involving time, including relating time, distance and speed. 8.4.3 Solves problems using a range of strategies including: • estimating; • calculating to at least 2 decimal places; • using and converting between appropriate S.I. units.

TERMINOLOGY Ratio; rate; speed; distance



TEACHING TIPS This assessment standard also appears in grade 7, as 7.1.6.

The intention is clearly that learners should in Grade 8 progress to higher levels of understanding of ratio and rate than what they have achieved in Grade 7, and become more sophisticated in solving problems involving these concepts. This is illustrated in the differences between the examples given for grade 7 and 8 below.

Questions 1, 2 and 3 below are similar to questions 1, and 2 in the illustrative examples for assessment standard 7.1.6, yet the questions below require higher levels of understanding and sophistication.


Note that in question 1 (for 7.1.6), the ratio is given and language informs the learner that the comparison is in terms of “how many times bigger?” In question 1 below the learner has to decide to compare by ratio and not by difference.

Question 2 (for 7.1.6) can actually be done without consciously thinking in terms of the ratio between the amount of concentrate and the amount of water: the learner may notice that 200 mℓ is one-fifth of 1 000 mℓ and correspondingly calculate one-fifth of 250 mℓ to find the required amount of concentrate. Question 2 for grade 8 below cannot be solved that easily.

EXAMPLES Grade 7

1. A photograph is 8 cm long and 5 cm wide. An enlarged copy of the photograph is 112

times as long as the original. How wide is the enlarged copy?

Grade 8

1. A photograph is 8 cm long and 5 cm wide. An enlarged copy of the photograph is 12 cm long. How wide is the copy?

Grade 7

2. The recipe for making a certain fruit drink specifies that 250 mℓ of concentrated juice should be added to 1 ℓ of water. How much concentrated juice should be added to 200 mℓ of water to make fruit drink of the same strength?

Grade 8

2. The recipe for making a certain fruit drink specifies that 250 mℓ of concentrated juice should be added to 1 ℓ of water. How much concentrated juice should be added to 300 mℓ of water to make fruit drink of the same strength?

TEACHING TIPS (CONT.) Distance, speed and time are taught through problem solving. Encourage learners to draw

diagrams with annotations to solve the problem. If at all possible, introduce this section practically e.g. example 10 could be done practically on the athletic track with different learners’ times.

Encourage learners to underline the key words and estimate the answer.

Encourage learners to check the unit in which the answer must be written.

EXAMPLES (CONT.) 3. Geologists have determined that when you go vertically down a certain mine shaft, the

temperature of the air increases at a rate of 0,3° C per 100 metre. At a depth of 56 m, the air temperature is 24,6° C on a certain day. What will the air temperature be at a depth of 312 metres?

4. A motorcar is in motion for 15 minutes and travels at a speed of 90km per hour.

a) What distance does the motorcar travel in 15 minutes? Use the given table (and equivalent fractions) to calculate the answer. (Remember: You need not use all the columns in the table.)

distance in km 90 time in minutes 60 30


b) A motorist wants to travel from Cape Town to Port Elizabeth. The distance between the two cities is approximately 800 kilometres. He thinks that he should travel at an average speed of 90 km/hour. Use a ratio table (and equivalent fractions) to determine how many hours the trip will take. What values must go into the first column? Complete the table:

distance in km 800 number of hours ?

5. A recipe to bake bread indicates that the following ingredients are needed:

1 cube of yeast 15 ml salt 30 ml sugar 1 litre boiling 1,5 kg cake flour water

a) What quantities of these ingredients do you need to bake 2, 4, 10, or 12 breads? Calculate these by means of a ratio table:

number of breads 1 2 4 10 12

cubes of yeast 1

litres boiling water 1

ml sugar 30

kg cake flour

ml salt

b) Every morning a certain baker bakes 100 breads. Calculate what quantities of ingredients he requires by means of a ratio table. Verify your answer.

6. Nelson wants to use black and white tiles to tile the floor of a room. Below a small section of the pattern to be used is given:

a) The floor is bigger than the section drawn here. What is the ratio of number of white tiles to number of black tiles?

b) The dimensions of the tiles are 10 cm by 10 cm. The rectangular floor is 2 metres wide and 3 metres long. How many tiles in total does Nelson have to buy to tile the floor?

c) In order to determine how many of each colour tile he needs, Nelson makes use of a ratio table. By completing the table you would be able to advise Nelson. Make use of the ratio white as to black tiles.

number of black tiles 2 number of white tiles total number of tiles


7. For the tile pattern below Nelson uses 120 black tiles.

a) How many white tiles does he need to buy in order to complete the pattern with 120 black tiles.

b) These tiles are 30 cm by 30 cm in size. Nelson has to use these black and white tiles to tile a section of the surface of the playground according to the above pattern. What is the size of the tiled surface? Write down all your calculations.

8. A vine-grower has to spray his grapes with certain poisons. The poison needs to be dissolved in / diluted with water in the ratio: 2 gram of poison per litre of water. A maximum of 3 kg of poison per spray session may be sprayed on one hectare of vine.

a) How much poison does a vine-grower need to buy if he wants to spray 1,5 hectare of land?

b) What volume of water is required to dissolve this much poison?

9. A man finds that when he walks on level ground, he normally covers about 4 km in an hour. However, when he walks uphill in the mountain behind his house, he covers only about 1,5 km in an hour. On a certain day, he plans to walk to his brother who lives on top of the mountain. To get to his brother he has to walk 3,6 km on level ground, and then another 1,4 km up the mountain. Approximately how long will it take him to make the journey?

10. An athlete runs in a 10 000 m race around an athletics track. One lap around the track is a distance of 460 m. The athlete takes 58 sec to complete the first lap, 62 sec to complete the second lap, 61 sec for the third lap and 59 sec for the fourth lap. How long will he take to complete the whole race, if he continues to run at more or less the same pace as he ran over the first four laps?

CONSOLIDATION / HOMEWORK 4 - 5 examples for consolidation.



TERM 4

TERM 4 - WEEK 1 REVISION of transformations, length, perimeter and area.


ASSESSMENT STANDARD 8.3.6 Uses transformations (rotations, reflections and translations) and symmetry to investigate (alone and/or as a member of a group or team) properties of geometric figures

8.3.7 Uses proportion to describe the effect of enlargement and reduction on properties of geometric figures

(Can be clustered with LO 4 8.4.1 and 8.4.5 Length, perimeter, area and volume)

TERMINOLOGY Transformations; Translation; Rotation; Reflection; Symmetry; Enlargements; Reductions; Properties; Geometric; Proportion; Asymmetry.




on the website. One of the important reasons for studying transformations is that the properties of shapes

are preserved under transformation—that is, rotations, reflections and translations do not change the angle sizes or the lengths of the sides of a shape. Transformations involving a triangle can be used to create a tessellation that gives a grid of intersecting parallel lines. The tessellation can be used to illustrate various angle relationships including the relationship between angle pairs when a pair of parallel lines is cut by a transversal, the angle sum of a triangle etc.

When we ask learners to tessellate on a grid, we are hoping that they will with time begin to do so without cutting out an actual triangle. When they reach the stage of being able to tessellate without the cut out triangle they are automatically addressing the expectation of Assessment Standard 8.3.10 in that they are “locating position” on grids using motions that involve “horizontal and vertical change.”

The “properties of geometric figures” referred to in the Assessment Standard are the lengths of the sides, the sizes of the angles and the area of the figure, while the word “figure” refers to 2-D shapes. In meeting the requirements of this assessment standard learners should come to realise that enlargements and reductions of figures will change the lengths of the sides and the area of the figure but not the sizes of the figure’s angles. While the lengths of the sides change in proportion to the scale factor the change in area is in proportion to the square of the scale factor (i.e. if we enlarge a figure by a factor of 3, we increase the area by a factor of 32 = 9; if we reduce a figure by a factor of 4, we reduce the area by a factor of 42=16)


In Grade 6 learners were expected to be able to: “Draw enlargements and reductions of two-dimensional shapes (at least quadrilaterals and triangles) using grid paper to compare their size and shape” (6.3.5). It follows that learners should understand what it means to enlarge and reduce figures and are able to perform such enlargements and reductions. Learners should also have an understanding that while the lengths of the sides of a figure change when we reduce and enlarge figures, the angle sizes do not change.

In Grade 7 learners were expected to be able to: “Calculate, by selecting and using appropriate formulae: perimeter of polygons; area of triangles, rectangles and squares; volume of triangular and rectangular based prisms” (7.4.5). In Grade 8 the figures whose areas can be calculated are increased to include polygons and circles. The areas of polygons are determined by decomposing into triangles and rectangles.

Example 8 brings a lot of the ideas together. The following would be a desirable answer to the second bullet of question a):

Scale factor = 3

area of the original rectangle = length × breadth

area of the enlarged rectangle = (3 × length) × (3 × breadth)

= 3 × length × 3 × breadth

= 3 × 3 × length × breadth

= 32 × (area of the original rectangle)

This is also a good time to revise co-ordinate systems (done in term 1) and co-ordinate systems can be integrated with transformations. By integrating the Assessment Standards that deal with co-ordinate systems and those that deal with Transformations we have a powerful mechanism for teaching about geometric figures and for investigating their properties including congruence and similarity.

Make learners aware that reductions are indicated by a scale factor less than 1 e.g.½ which means the diagram will be half the original size.

Very important: although there is no specific assessment standard in grade 8 stating the words “similarity” or “congruency” like in grade 7 and 9, these concepts must be consolidated from the grade 7 assessment standard i.e. learners must know the difference between similarity and congruency. They do not need to know the tests for congruency of triangles until grade 9.. The must know that:

Similar figures have the same shape, the corresponding angles of the two figures are equal and the corresponding sides are in equal proportions.

For figures to be congruent they must be similar and the corresponding sides must have exactly the same lengths.


EXAMPLES 1. Which of the following triangles are similar? Explain!

B

C D

E

F

G H

I

J

K L

A


2.

a) Use the grid above to tessellate triangle ABC so that the whole grid is covered with triangle.

b) Describe, in terms of transformations, how you moved the triangle in order to create the tessellation.

c) By highlighting a pair of parallel lines and a transversal that cuts them use the tessellation to justify the properties of the following angle pairs: Alternate angles Corresponding angles Co-interior angles

d) By highlighting the intersection of two lines use the tessellation to justify the properties of vertically opposite angles.

e) Use the tessellation to justify the rule which states that the angle sum of the interior angles of a triangle is 180°.

f) Use the tessellation to justify the rule which states that the exterior angle of a triangle is equal to the sum of the interior opposite angles of the triangle.

3. a) On the grid (next page) draw two triangles that are congruent to triangle ABC in such

a way that together they form a kite.

b) On the grid (next page) draw two triangles that are congruent to triangle ABC in such a way that together they form a parallelogram.

c) For both the kite and the parallelogram describe how you made the quadrilateral in terms of transformations. In other words how did you transform the first triangle to get the second triangle?

d) For both the kite and the parallelogram write down as many properties of each as you can. Discuss how the diagrams that you have made confirm each of the properties you list.

B

A

C


4. a) On the grid below draw four triangles that are congruent to triangle ABC in such a

way that together they form a rhombus.

b) On the grid below draw two triangles that are congruent to triangle ABC in such a way that together they form a rectangle.

c) For both the rhombus and the rectangle describe how you made the quadrilateral in terms of transformations. In other words how did you transform the first triangle to get the other triangles?

d) For both the rhombus and the parallelogram write down as many properties of each as you can. Discuss how the diagrams that you have made confirm each of the properties you list.

C

B

A

B

CA


5. a) Translate ∆PQR 2 units to the right and four units up. Label the new triangle ∆P′Q′R′.

b) Show how the transformation you did can be used to illustrate the statement that when a pair of parallel lines is cut by a transversal the corresponding angles are equal.

c) Show how using a different transformation of ∆PQR can be used to illustrate that when a pair of parallel lines is cut by a transversal the alternate angles are equal.

6. The left hand grid shows a sequence of two transformations that Tammy did on the figure

ABCDE while the right hand grid shows how Jill made a sequence of two different transformations to get to the same position.

a) Describe, in detail, the sequence of transformations made by Tammy.

b) Describe, in detail, the sequence of transformations made by Jill. c) Eddie says that he can achieve the same result using a single transformation.

Describe, in detail, the transformation that Eddie would use.

3 3

2

1

21

109

C

D

B

E

A

8765421

10

9

8

7

6

5

4

3

2

1

109

C

D

B

E

A

8765421

10

9

8

7

6

5

4

3

2

1

3

B

A109

RQ

P

8765421

10

9

8

7

6

5

4

3

2

1


7. a) What will happen to the area of a rectangular figure that is enlarged or reduced by each of the following scale factors?

Scale factor 2 Scale factor 3 Scale factor 4

Scale factor 12

Scale factor 13

Scale factor 14

b) What will happen to the area of a triangle shown alongside if it is enlarged or reduced by each of the following scale factors?


Scale factor 12

Scale factor 13

Scale factor 14

c) What will happen to the area of the triangle alongside if it is enlarged or reduced by each of the following scale factors?


Scale factor 12

Scale factor 13

Scale factor 14

d) Based on the work above write a generalisation that describes how the area of a rectangle and the area of a triangle change in terms of the scale factor of the enlargement or reduction when a figure is reduced or enlarged.

e) Will the same relationship hold true for any polygon? Does the relationship work for circles? Justify your answer.

8. In the figure alongside the small prism has been enlarged by a scale factor of 3 to create the larger prism.

a) Discuss how the following properties of the two figures are related:

The lengths of the corresponding sides The sizes of the corresponding angles The areas of the corresponding faces

b) How are the volumes of the two figures related? Justify your answer.

CONSOLIDATION / HOMEWORK Use examples as class or homework.

ASSESSMENT Assessment Task 10 e.g. Tutorial / Investigation on enlargements and reductions.

C

B

A

5 cm

4 cm

5 cm

C

BA


TERM 4 - WEEK 4

ASSESSMENT STANDARD 8.2.7 Describes a situation by interpreting a graph of the situation, or draws a graph from a description of a situation, with special focus on trends and features such as: • linear or non-linear; • increasing or decreasing; • maximum/minimum; • discrete or continuous

TERMINOLOGY Linear, non-linear, discrete, continuous, increasing, decreasing

RESOURCES Gr 8 Text books; WCED Illustrative examples; Calculator; Poster; Internet Web sites, Magazines, Newspapers

INTEGRATION Technology, EMS, Social Science, Natural Sciences

TEACHING TIPS For ideas on developing the concept and extra examples, look at IMSTUS module 19 on the website.

Learners must be given practice in both skills i.e.

The graph is given and questions are asked which show that the learners understand what is happening in the situation represented by the graph.

The situation is given in words and numbers and learners can draw the graph to represent the situation.

Use contexts associated with the content of the other learning areas and/or relevant information from magazines and newspapers.

Emphasise the relevance of scale to project the correct information.

Make sure learners understand the difference between continuous and discrete graphs. (If all real numbers are allowed as input values then the graph will be a solid line otherwise it will be separate points.)

EXAMPLES Question 1 below addresses increasing and decreasing values of variables, as well as maximum and minimum values.

1. Water is supplied to a small rural community from two boreholes, and from a dam in the river that flows past the town. Water is pumped from the boreholes into two reservoirs, one for each borehole. The table (next page) shows data on the water levels in the two reservoirs over a period of 22 days. The community normally uses water from the dam. Water is only taken from the reservoirs when the dam runs low. The data was collected by measuring the water level in the two reservoirs at 06:00 each morning.


Day 1 2 3 4 5 6 7 8 9 10 11

Water in reservoir A in m3 234 250 265 281 297 313 329 345 361 377 394

Water in reservoir B in m3 655 655 655 655 655 683 691 697 708 712 734

Day 12 13 14 15 16 17 18 19 20 21 22

Water in reservoir A in m3 390 384 381 369 354 351 343 359 375 391 406

Water in reservoir B in m3 736 734 688 654 623 601 563 543 523 511 480

Draw the graph and answer the following questions:

a) When was the water level the highest, in each of the two reservoirs, during the above period of time?

b) When was the water lowest, in each reservoir?

c) When was the water level increasing in reservoir A?

d) When was the water level decreasing in reservoir A?

e) When was the water level increasing in reservoir B?

f) When was the water level decreasing in reservoir B?

Question 2 addresses linear (approximately linear) and non-linear change.

2. This question refers to the data given in question 1. Compare the way in which water flowed into reservoir A from days 1 to 11 with the way water flowed into reservoir B during days 6 to 12. Describe the difference in words.

3. At 10:00 one morning, the pump that feeds water into a dam A is switched on. At that moment the water in the dam is 1,1 m deep. No water is taken out of the dam for the rest of that day. Information about the water level during the day, up to 17:30, is given in the point graph below.

1,5

2

1,0

2,5

17:0015:00 16:0012:0010:00 13:00 14:0011:00

Wat

er le

vel i

n m

etre

Time of the day

Water level in dam A


a) What was the water level at 12:00?

b) At approximately what time did the water level reach 2,1 m?

c) By how much did the water level rise in the period of 1 hour from 14:30 till 15:30?

d) By how much did it rise in the period of 1 hour from 15:30 till 16:30?

e) During what part of the day was the pump not working?

f) Was there any difference in the way water flowed into the dam before 12:30 and after 14:30? Write your ideas down in a short clear paragraph.

g) By how much did the water level rise each hour before 12:30?

h) By how much did the water level rise each hour after 14:30?

i) If you wish, you may revise your response to question f now.

4. Linear relationship (rate of change is constant) In order to calibrate the above beakers, it is necessary to know how the height of the

liquid depends on the volume in the beaker. The sketch graphs below represent the height-volume relationship for each beaker on the same system of axes. In your group discuss which graph best represents the height-volume relationship for each beaker. Explain your choice in each case.

Beaker A Beaker X Beaker B

Height

Volume

h g f


6. Water drips simultaneously into the above beakers from three different taps which deliver water at the same constant rate (millilitres per second).

On the same system of axes sketch graphs to show how the height of the water in each

beaker varies with time.

Beaker A Beaker X Beaker B

Height

Time (in minutes)


7. Express in words what information is given about the gradients of each of the following situations.

8. In some communities people used to pay a flat rate for electricity, but this has mostly

been changed to a metered rate. A flat rate means that a person pays the same amount every month, no matter how much electricity is used.

a) John pays a flat rate of R50 per month to the city council for electricity. David pays the metered rate of R20 per month for the service plus R1,25 per kWh. Complete the table to show what they pay for different usages per month.

Units used (kWh) 0 10 20 30 40 50 60 70

John (flat rate)

David (metered rate)

b) Draw a graph illustrating John and David's electricity bill for different amounts of electricity used.

c) Which system (model) do you prefer? Is it fair? Discuss!

Petrol (in litres)

Cost in Rand

Time (in minutes)

No. of words typed

Distance traveled (in km)

Petrol in tank (in litres)

Time (in sec)

Distance (in metres)


9. Non linear; (rate of change is not constant) When a motorist brakes, the car does not come to an immediate stop. While he brakes, the car still moves a little further. This distance covered by the car is called the braking distance. The braking distance covered depends on the speed of the car. The braking distance for a speed of 120 km/h is greater than for a speed of 60 km/h. There is a formula which helps us to estimate the braking distance. The formula is:

braking distance 1004

3 2speed×=

The speed must be in kilometres per hour, then the braking distance will be in metres.

a) Use the formula to calculate the braking distance if someone drives at 60 km/h.

b) Use the formula to calculate the braking distance if someone drives at 120 km/h.

c) Is the braking distance for 120 km/h twice as much as 60 km/h?

d) Complete the table:

speed in km/h 0 20 40 60 80 100 120 140

braking distance in m

e) Draw/plot the corresponding graph for the table above. (Draw a continuous line through the points you have plotted on the axes below.)

f) A motorist had to break because of an accident in the road ahead. The car’s breaking distance was 48 metres.

How fast was the motorist driving? Show on the graph where we can read this off. Show, using the formula, how you worked this out.

Braking distance in m

Speed in km/hour


g) For a braking distance of not more than 50 m, read off from the graph what the speed should be.

h) Describe what is meant by the statements below:

“The greater the speed, the greater the braking distance!” “The braking distance for a speed of 120 km/h is twice as long

as for a speed of 60 km/h?”

10. A crane can be used to pick up/lift very heavy objects. The mass which the crane can lift depends on the hanging position of the pulley on the arm of the crane. This is called the arm length. The following boards hang on the arm of a crane:

10 m 12000kg

15 m

8000 kg

24 m 5000 kg

30 m 4000 kg

40 m 3000 kg

48 m 2500 kg

60 m

2000 kg

The top number indicates the arm length in metres.

The bottom number indicates the mass that can be lifted at that specific arm length.

A mass of 8000 kg can thus be lifted if the pulley is 15 m from the vertical part of the crane. If it hangs further than 15 m, the crane might break or fall over.

a) Use the information on the boards to draw/plot a graph.

A formula can also be used to calculate the greatest mass that can be lifted at a particular arm length. The formula for this crane is:

lengtharmmassgreatest 120000

=

greatest mass is measured in kilograms and the arm length in metres.

b) How can you determine if the information on the boards is in agreement with the formula?

c) Read off from the graph, what board should be hung up for an arm length of 35 metres. Show on the graph where we can read this off.

Mass in kilograms

Arm length in metres


d) Calculate using the formula, what board should be hung up for an arm length of 35 metres.

e) Read off from the graph, what the arm length should be for a mass of 6000 kg. Show on the graph where we can read this off.

f) Calculate using the formula, what the arm length should be for a mass of 6000 kg.

CONSOLIDATION / HOMEWORK Exercises involving drawing and interpreting graphs.

ASSESSMENT Assessment Task 11 e.g. Tutorial on graphs

TERM 4 - WEEK 5

ASSESSMENT STANDARD 8.2.6 Determines, analyses and interprets the equivalence of different descriptions of the same relationship or rule presented: • verbally • in flow diagrams • in tables • by equations or expressions In order to select the most useful representation for a given situation.

TERMINOLOGY Learnt already



TEACHING TIPS For ideas on developing the concept and extra examples, look at IMSTUS module 19 on the website.

Although the wordings are the same for 7.2.6 and 8.2.6, letter symbols as placeholders for variables are introduced in Grade 8, hence expressions (formulas) and equations may now be represented in symbolic form, for example: Rental cost = 2,30 × x + 240, where x represents the number of km

This assessment standard belongs together with 8.2.7, since a graph is another way of describing a relationship between variables.

Let learners use the same relationships or rule, but express it verbally, in a flow diagram, in tables and equations.

Let them decide which the most useful way to represent a situation is.


EXAMPLES 1. a) Find the missing input number in the flow diagram below: b) Represent the input number by the symbol x and write an equation that sets the same

question as the above.

2. At a certain hospital, a patient has to pay an admission charge of R60 and a daily fee of R120.

a) Write a formula that can be used to calculate the cost for being in this hospital, for any number of days.

b) Make a flow diagram that presents the same information as your formula.

c) Make a table of the cost of being in this hospital, for any period between 1 and 10 days.

d) Make a point graph to show the cost of being in this hospital, for any period between 1 and 10 days.

3. The fares for travelling on a certain train service are given in the table below.

Distance in km 1 2 3 4 5 6 7 8 9 10

Fare in cent 184 190 196 202 208 214 220 226 232 238

Distance in km 10 20 30 40 50 60 70 80 90 100

Fare in cent 238 298 358 418 478 538 598 658 718 778

a) Make a flow diagram to show how the fare may be calculated for any distance.

b) Make a formula to show how the fare may be calculated for any distance.

c) What is the fare for a distance of 42 km?

d) Write an equation to represent question (c)

4. Amanda has started a biscuit bakery. She sells her biscuits in boxes. She has employed you to calculate the costs for the various orders. Amanda charges R8,00 per box plus an administration fee of R15,00 per order.

a) How much would you charge for an order of 3 boxes? 10 boxes? 100 boxes? Write a sentence describing how you found these answers.

b) How many boxes could a customer purchase with R111? R500? R2000? Write a sentence describing how you found these answers.

c) A competitor advertises that they will sell the same boxes of biscuits for R5,50 per box plus an administration fee of R49,95 per order. When Amanda sees this advertisement she tells you to figure out how this might affect her business.

d) In order to give Amanda good answer, clearly identify the problem situation, produce tables of values, graphs and algebraic rules.

(Source: Malati)

CONSOLIDATION / HOMEWORK Examples for consolidation.


+ 240 × 23 input number 470


TERM 4 - WEEK 6

ASSESSMENT STANDARD 8.5.8 Critically reads and interprets data presented awareness of sources of error and manipulation to draw conclusions and make predictions sensitive to the role of: • context (e.g. rural or urban, national or provincial); • categories within the data (e.g. characteristics of different target groups age, gender, race); • data manipulation (e.g. grouping, scale, choice of summary statistics) for different purposes; • the role of outliers on data distribution • any other human rights and inclusivity issues

TERMINOLOGY Outliers, data manipulation


INTEGRATION Technology, EMS, Social Sciences, Natural Sciences

TEACHING TIPS Refer to the book ‘Data Handling in the GET Band’ Misleading diagrams: pages 42-44

EXAMPLES

1. CONTEXT (E.G. RURAL OR URBAN, NATIONAL OR PROVINCIAL) - Consider the graphs below that are based on the data established in Census 2001

(www.statssa.gov.za). They illustrate the dominant home language in a rural municipality (Prince Albert), an urban municipality (City of Cape Town), a province (Western Cape) and for South Africa. These four graphs very clearly illustrate (by being very different from each other) choosing correct sample of the population can have a great impact on the conclusions and predictions that can be made. e.g. if the population of Prince Albert is taken as a representative sample of the population of the Western Cape, then the conclusion is that nearly everybody in the Western Cape speaks Afrikaans and there are no isiXhosa speaking people. In this way the wrong conclusion is drawn because the choice of sample was incorrect.

Prince Albert

0

2000

4000

6000

8000

10000

12000

Afrikaa

ns

Englis

h

IsiNde

bele

IsiXho

saIsi

Zulu

Seped

i

Sesoth

o

Setswan

a

SiSwati

Tshiv

enda

Xitson

gaOthe

r

Num

ber o

f peo

ple


City of Cape Town

0200000400000600000800000

100000012000001400000

Afrikaa

ns

Englis

h

IsiNde

bele

IsiXho

saIsi

Zulu

Seped

i

Sesoth

o

Setswan

a

SiSwati

Tshiv

enda

Xitson

gaOthe

r

Num

ber o

f peo

ple

Western Cape

0

500000

1000000

1500000

2000000

2500000

3000000

Afrikaa

ns

Englis

h

IsiNde

bele

IsiXho

saIsi

Zulu

Seped

i

Sesoth

o

Setswan

a

SiSwati

Tshiv

enda

Xitson

gaOthe

r

Num

ber o

f peo

ple

South Africa

0

2000000

4000000

6000000

8000000

10000000

12000000

Afrikaa

ns

Englis

h

IsiNde

bele

IsiXho

saIsi

Zulu

Seped

i

Sesoth

o

Setswan

a

SiSwati

Tshiv

enda

Xitson

gaOthe

r

Num

ber o

f peo

ple


CATEGORIES WITHIN THE DATA (E.G. AGE, GENDER, RACE) - Example 2 below demonstrates how categories within data can influence the

conclusions and predictions that can be made.

2. The table that follows lists data collected in the 2001 population census by Statistics South Africa.

Income category by gender for the North West province Male Female

No income 8427 6129

R 1 - R 400 69156 56206

R 401 - R 800 73632 52464

R 801 - R 1600 129070 52119

R 1601 - R 3200 121612 43736

R 3201 - R 6400 53064 32008

R 6401 - R 12800 25697 10009

R 12801 - R 25600 8526 1663

R 25601 - R 51200 2433 535

R 51201 - R 102400 975 414

R 102401 - R 204800 501 168

R 204801 or more 286 60 source: Census 2001 (www.statssa.gov.za)

a) Calculate the total number of people per income category for the North West province.

b) Draw a histogram illustrating the total number of people per income category for the North West province.

c) Draw a histogram illustrating the number of males people per income category for the North West province.

d) Draw a histogram illustrating the number of females people per income category for the North West province.

e) By studying your graphs determine whether or not income in the North West province is even distributed by gender.

DATA MANIPULATION (E.G. GROUPING SCALE, CHOICE OF SUMMARY STATISTICS) FOR DIFFERENT PURPOSES

- Example 3 below demonstrates quite clearly how using different grouping scales when grouping data can change the impact on the summary statistic. Learners should be aware that the scale chosen for grouping can have a profound impact on the conclusions and predictions that can be made.


3. Hayley is preparing to present some data. She has decided that she wants to use a histogram with grouped data. She is concerned about whether or not her choice of group size will affect the conclusions she can draw from the data.

40 20 28 26 42 34 12 14 3 37

37 17 20 39 27 1 45 29 35 8

18 21 20 3 1 4 30 13 10 11

37 35 12 12 31 30 29 11 7 42

25 5 32 30 41 7 5 9 35 30

5 29 13 2 14 29 43 38 37 30

1 30 2 29 33 24 14 5 26 29

13 28 2 42 1 5 15 43 6 4

45 24 37 25 20 4 10 20 38 16

33 37 10 27 41 5 28 41 45 25

16 28 12 30 36 25 10 5 1 5

a) Organise the data above by making a table and grouping the data in intervals of 5, i.e. 1 to 5, 6 to 10 etc.

Draw the corresponding histogram Use the grouped data to determine the mean, median and

modal group.

b) Organise the data above by making a table and grouping the data in intervals of 9, i.e. 1 to 9, 10 to 18 etc.

Draw the corresponding histogram Use the grouped data to determine the mean, median and

modal group.

c) Are any of the statistics influenced by the choice of interval for the group? If so which ones?

THE ROLE OF OUTLIERS ON DATA DISTRIBUTION - The term outlier is well defined in mathematics, though its formal definition is

way beyond the scope of Grade 8 mathematics. For our purposes, in Grade 8, it is enough for a learner to recognise that within data sets there may exist one or more values that are so different from the others that when we include them in the calculation of say the mean they distort the statistic. Consider the numbers 1; 2; 3; 2; 3; 1; 1; 3; 3; 2; 1; 1 and 31 clearly 31 is very different from the others in the set. If we calculate the mean of all 13 numbers it is 4,15. If, however, we take the 31 out of the set and calculate the mean for the remaining 12 numbers it is 1,9. Outliers typically arise from measurement and or other errors. Learners should be sensitive to the possible presence of an outlier which can, in turn, distort the findings. Outliers can sometimes be easily identified through the use of a scatter plot—where such a graph can be drawn. See example 2.

ANY OTHER HUMAN RIGHTS AND INCLUSIVITY ISSUES - Both the examples given in the remarks above and the examples provided

below have addressed human rights and inclusivity issues such gender, population group, rural vs. urban etc.


4. Blood carries oxygen from the lungs to the rest of the body. The blood is pumped through the body by the heart. If your heart beats faster the blood circulates around the body and back to the lungs more quickly and thus more oxygen reaches your muscles so they can work harder. We measure this heart rate in beats per minute (bpm).

a) Find your heart rate by placing your figures on your wrist and feeling the beat. Count how many times your heart beats in 1 minute. Record the result.

b) Now run energetically on the spot for 2 minutes and measure your heart rate again. What do you notice?

c) Run for another minute and measure you heart rate again. Is there any change?

5. Good sports people have to train hard so that their hearts get used to pumping lots of blood around their bodies. However they must be careful that they don't strain their hearts as this can lead to a heart attack. Below is a graph that sports people use to determine how hard they are exercising by measuring their heart rate. The line shows what 60% of their maximum heart rate is.

80

90

100

110

120

130

140

20 30 40 50 60

Age (years)

Hea

rt ra

te (b

pm)

a) As you get older what happens to your heart rate?

b) Calculate the maximum heart rate of a 20 year old?

c) If an athlete exercises at 60% of his maximum heart rate, his heart rate is 115 bpm, estimate the age of the athlete.

d) Estimate what 60% of your maximum heart rate should be. Explain how you calculated this.

e) When your grandfather is 80 years old what will 60% of his maximum heart rate be?

60%

96


f) Elana Meyer is 30 years old, what is 60% of her maximum heart rate? During a race her heart rate goes to 95% of her maximum heart rate, how many times per minute will her heart beat?

g) Paul Adams is 20 years old. When he is bowling in a cricket match his heart rate is 80% of his maximum heart rate, what will his heart rate be in beats per minute?

(Source; Malati)

CONSOLIDATION / HOMEWORK Graphs to critically read and interpret.


TERM 4 - WEEK 7

ASSESSMENT STANDARD 8.5.4 Perform simple experiments using random number generators , coins, spinners, dice and cards in order to collect data

8.5.10 Considers a simple situation (with equally likely outcomes) that can be described using probability and:

• Lists all the possible outcomes;

• Determines the probability of each possible outcome using the definition of probability (see

• Mathematics Learning Area Glossary);

• Finds the relative frequency of actual outcomes for a series of trials;

• Compares relative frequency with probability and explains possible differences;

• Predicts with reasons the relative frequency of the possible outcomes for a series of trials based on probability.

TERMINOLOGY Relative frequency; actual outcomes; series of trials; possible outcomes


INTEGRATION Social Sciences

TEACHING TIPS For ideas on developing the concept as well as extra examples, look at IMSTUS module 26 on the website.

Assessment standard 8.5.10 is a very natural extension of the chance concept that has been developed in the Intermediate Phase where learners were expected to be able to predict the likelihood of daily life-based events and to rank them from impossible to certain; list the possible outcomes for simple experiments and count the actual frequency of outcome for a series of trials and in Grade 7 where learners considered a simple experiment and collected data on the frequency of outcomes.

This was in 1997! What about now?


In Grade 8 learners are expected to be able to determine the probability of possible outcomes for a given situation. The Assessment Standard expressly states that the possible outcomes of the situation should all be equally likely. This is to ensure that the complexity does not get too great. However, it is also important for learners to realise that there are many situations for which the outcomes are not equally likely and some class discussion around this would be very helpful.

ONE OF THE IMPORTANT IDEAS THAT LEARNERS MUST GRAPPLE WITH IN GRADE 8 IS WHAT “PROBABILITY” ACTUALLY MEANS. IT IS IMPORTANT THAT THEY REALISE THAT:

Even though they know that the probability of tossing a head with a fair coin is 12 this does

not enable them to predict what the outcome of a particular experiment will be.

If they have tossed a coin 3 times, irrespective of what the outcomes were they have absolutely no way of predicting what the outcomes of the 4th or subsequent experiments will be.

If they toss a coin 50 times and 21 of those times the coin lands with the head up and 29 times with the tail up then there is neither an error in their calculation not a problem with the coin.

All that knowing that the probability of tossing a head with a fair coin is 12 enables us to do

is to predict that if we tossed a coin a very large number of times then we would expect

that the relative frequency of heads would get closer and closer to 12 .

EXAMPLES 1. Make a spinner that resembles the one alongside. Push a drawing pin through the centre

point from below and assemble the spinner as shown alongside. Spin the arrow 30 times and record the colour of the segment in which the spinner lands on the table below.

Spin number Outcome Spin

number Outcome Spin number Outcome

1 11 21

2 12 22

3 13 23

4 14 24

5 15 25

6 16 26

7 17 27

8 18 28

9 19 29

10 20 30


2. Cut out five identical shapes from a piece of sturdy cardboard. On each piece of cardboard write the name of a member of your class. Place all of the names inside a large paper packet or black plastic bag. Shake the packet or bag a few times.

a) Make a list of all of the possible outcomes when a shape is drawn from the bag.

b) Determine the probability of each name being drawn (possible outcome) from the bag.

c) You are about to conduct 60 experiments each of which consists of drawing a name from the bag, recording the result of the experiment, replacing the name in the bag and shaking it. What do you predict the relative frequency to be for each name? Justify your answer.

d) Conduct 60 trials and record the outcomes of each trial. Calculate the relative frequency of each outcome.

e) Compare your predicted relative frequency with the actual relative frequency. Are they the same? If not, why not?

3. A learner has been given a bag with some identical counters in it. She has been told that each counter has a different letter on it, but she has not been told how many counters there are. She systematically takes out a counter, records the letter, replaces the counter shakes the bag and starts again.

a) After three experiments she has recorded the following: A, B, A. How many counters are there in the bag and what are the letters on the counters? How sure are you of your answer? If not very sure, what do you know?

b) After another three experiments her record looks as follows A, B, A, A, C, B. How many counters are there in the bag and what are the letters on the counters? How sure are you of your answer? If not very sure, what do you know?

c) After thirty experiments she has the following record:

A, B, A, A, C, B, B, C, C, B, C, B, C, D, C, B, B, B, B, C, B, D, B, D, B, D, D, B, C, A

d) How many counters are there in the bag and what are the letters on the counters? How sure are you of your answer? Are you more or less confident now compared to after 6 experiments?

e) After another thirty experiments she has the following record:

A, B, A, A, C, B, B, C, C, B, C, B, C, D, C, B, B, B, B, C, B, D, B, D, B, D, D, B, C, A,

A, A, A, C, A, A, A, B, B, A, C, B, A, A, D, A, B, B, B, B, B, D, A, D, A, A, B, C, D, D

f) How many counters are there in the bag and what are the letters on the counters? How sure are you of your answer now?


4. Mpande is a contestant in the Zama Zama Game Show. She is told that a box contains three balls of the same size: a black ball, a white ball and a grey ball. She cannot see the balls in the box. In this game she must remove a ball from the box and then put it back before selecting another one.

a) What is the chance of Mpande removing the black ball? And the grey ball?

b) Show your answers on the likelihood scale below.

c) Now write your answers as fractions.

d) Craig said that the white ball is most likely to be removed, because when he last watched Zama Zama, every contestant removed the grey ball.

Craig’s mathematics teacher suggested that the Grade 8’s conduct an experiment to test Craig’s claim. Each group was given a box containing three balls of equal size and the learners had to remove a ball without looking. They recorded the results after 10 tries and then again after another 20. The results of the experiment conducted by Craig’s group are shown below:

Try black white grey

1 a

2 a

3 a

4 a

5 a

6 a 7 a

8 a

9 a

10 a

Total after 10 5 4 1

Impossible Certain 50% chance

Unlikely LikelyVery unlikely Very likely


Try black white grey

11 a 12 a

13 a

14 a

15 a

16 a

17 a 18 a

19 a 20 a

21 a 22 a 23 a

24 a

25 a

26 a

27 a

28 a 29 a

30 a

Total after 30 9 14 7

e) Which ball was removed most often after 10 tries? Write the result for each ball as a success fraction.

f) Which ball was removed most often after 30 tries? Write down the success fraction for each ball after 30 tries.

g) Write down anything interesting you notice about the success fractions after 10 tries and after 30 tries.

h) Do the results of this group’s experiment agree with your prediction in question 1? Explain. Do the results agree with Craig’s prediction?

Craig’s teacher then took each group’s results after 30 tries and combined them. The results are shown in this table:

No of Tries black white grey

300 93 119 88

i) Compare the success fraction for each ball with those obtained in question 3. What do you notice?

j) Another Grade 8 class conducted this experiment and obtained the results for 300 tries. The two classes combined their results. Write down what you think the success fraction for each ball will be might be after 600 tries. And after 1000 tries?

(Source: Malati)


CONSOLIDATION / HOMEWORK Complete examples for homework.


TERM 4 - WEEK 8 CONSOLIDATION


ASSESSMENT Assessment Task 12 : EXAMINATION

SQUARES TO CUT OUT

Work Schedule Week

other LO and AS Concepts

WK 1WK 2 INTERNET 8.3.10 Cartesian plane

WK 38.1.3, 8.1.9 laws on rational

numbers

WK 4 8.1.3, 8.1.7 exponents

WK 58.1.3

multiples and factors

WK 6MS Excel 8.2.1, 8.2.2 numeric patterns

WK 7

MS Excel 8.2.3 relationships between variables

WK 8

WK 9WK 10

Work Schedule Week


WK 1 8.2.5 solving equations

WK 2 8.2.4 mathematical models

WK 3 Ms PowerPoint; MsExcel

8.1.3, 8.1.7, 8.1.9,8.1.11 fractions

WK 4 Ms PowerPoint; MsExcel 8.1.3, 8.1.7,8.1.4, 8.1.9 decimal fractions and

percentages

WK 5 INTERNET 8.4.11, 8.4.12, 8.4.13 Pythagoras

WK 6 INTERNET 8.5.1, 8.5.2, 8.5.3, 8.5.5, 8.5.6, 8.5.9 data

WK 7 INTERNET 8.5.7 draw and interpret

WK 8

WK 9WK 10

Work Schedule Week


WK 1 revisionWK 2 INTERNET 8.1.3 scientific notationWK 3 MINDSET 8.1.3, 8.1.5, 8.1.10 budgetingWK 4 MINDSET (FIN) 8.1.5, 8.1.10, 8.4.10 finances / anglesWK 5 8.3.3, 8.3.4 lines and angles

WK 6 8.3.1, 8.3.2, 8.3.4, 8.3.5 quadrilaterals

WK 7 8.3.9, 8.4.1, 8.4.3, 8.4.5, 8.4.6, 8.1.5

length and perimeter / area

WK 8 8.4.1, 8.4.3, 8.4.4, 8.1.2

circumfences and area of circles

WK 9 8.4.5, 8.4.3, 8.4.6, 8.1.5

surface area and volume

WK 10 8.1.6 ratio and rate

8.3.7, 8.4.2, 8.4.3 distance, time and speed

8.2.8, 8.2.9

AF12; AG46;

AF12; AF40; AG48

AF13

AF29; AF30; AF31; AF32; AF33

(HOEKE) - AF02; AF03AF04; AF05; AF07

conventions of algebra

Master Maths

AF16; AF17; AF19; AF20

AF16; AF17; AF19; AF20

AF18;

AF09

AF11

AF41

AF001

Consolidation

Master Maths

AF01; AF06; AF09;

AF08

9.7.1, 9.7.2

2.6.1.5, 2.6.2.1, 2.6.2.2

9.5.1.2, 9.4.1

2.8.6.1 - 4

AF15; AG18

AF33

GRADE 8 TERM 1 COMPUTER SOFTWARE PLANNING

10.1.1, 10.1.2,

2.5.1.1 - 2.5.1.6

2.5.1-5, 1.8.6.1

Cami Exercises


4.2.1, 4.2.1, 4.2.1.1

1.8.4.1

2.5.1.1 - 2.5.1.6

4.1.1.9, 4.1.3.3 - 4.1.3.6

2.3.1.1 - 2.3.1.4, 2.3.3.2, 2.4.3.1 -2.4.3.4, 2.4.4.1 - 2.4.4.2

Cami Exercises

4.2.1.1

2.1.1.- 7, 2.2.1 - 7

7.1.1. - 7.1.3

Assessment

AF34; AF36; AF37;

4.4.1.1, 4.4.1.2, 4.4.2.2, 4.4.3.2, 4.4.3.3, 4.1.6.3, 4.1.7.1,4.1.7.2,.4.1.7.3,

The Maths software programmes which are in the Khanya schools e.g. Cami and Robo, have been linked to the grade 7-9 work schedules. The information is in a from of a table with the weeks and learning outcomes/assessment standards exactly the same as in the work schedules. The exact exercise e.g Cami is then listed for that concept. All the teacher must do is open the Cami programme and type in the 4-digits and the exercise will open.

Master Maths

AG21

AF28

Revision

Cami Exercises

9.3.4.1, 9.3.4.3,9.3.4.4, 9.3.4.5

8.2.1.1 - 8.1.1.2, 8.2.2.1, 8.2.2.2, 8.2.3.1, 8.2.3.2, 8.2.5.1, 8 1 5 2

1.8.5.1, 1.8.6.12.8.2.1, 2.8.2.7, 2.8.5.5


10.1.3.1, 10.1.4.1, 10.3.2.1, 10.3.2.2

8.4.1.1, 8.4.1.2

9.3.3.6, 9.3.3.7, 9.3.3.8, 9.3.6.2,

Work Schedule Week


WK 1

WK 2&3INTERNET/ MINDSET 8.3.6, 8.3.7 transformations

WK 4 8.2.7 graphsWK 5 8.2.6 flow diagrams

WK 6 8.5.8interpretation of data

WK 7 8.5.4, 8.5.10probability

WK 8, 9, 10 Revision, Assessment

AF10

AG57; AG58;

Master Maths

10.2.1, 10.2.2

8.3.1.3, 8.2.2, 8.3.7.1,8.3.7.2


AF106.1.2.1 - 6.1.2.3

revision

10.1.3.1, 10.1.4.1

Cami Exercises

AG41

senior - western cape...7 + 6 + 3 = 13 + 3 = 16).associative properties i.e. order of operations...

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