senior - edulis

Seni

or P

hase

Te

ache

r’s G

uide

Grade

9

Mathematics

GRADE 9 TEACHER’S GUIDE 1 / 55

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: Use module 8 of IMSTUS material on website.

TERM 1 - WEEK 2

ASSESSMENT STANDARD 9.3.10. Uses various representational systems to describe position and movement between positions, including: • ordered grids; • Cartesian plane (4 quadrants); • compass directions in degrees; • angles of elevation and depression

TERMINOLOGY Quadrants; origin; angle of elevation; angle of depression

RESOURCES Gr 9 Text books; WCED Illustrative examples; grid; protractor, compass, Calculator; maps; Internet Web sites; IMSTUS

INTEGRATION Social Sciences; Technology

TEACHING TIPS Revise the number line.

Teach the four quadrants of the Cartesian plane.

Teach terminology: quadrants, numbering quadrants; origin, horizontal axis (x-axis), vertical axis (y-axis), ordered pair (x;y).

Plot ordered pairs on the Cartesian plane.

Compass directions must be expressed in degrees from the position of the observer, e.g. W 30°S.

Teach angles of elevation and depression.


EXAMPLES 1. Plot the following points on a Cartesian plane. Join them in order

(3;1) (1;7) (-3; 7) (-3;3) (3;1) (4; -1) (6; -2) (7; -5) Stop

(4; -1) (3; -1) (4; -2) (4; -1) (4; 0) (5; -1) (4; -1) Stop

(6; -2) (5; -2) (5; -3) (6; -2) (6; -1) (7; -2) (6; -1) Stop

(7; -5) (6; -6) (7; -6) (7; -5) (8; -4) (8; -5) (7; -5) Stop

2. Compass directions: Use map of South Africa. Measure the direction from Cape Town to Bloemfontein with a protractor. Express the answer in terms of compass directions.

3. Angles of elevation and depression:

a) A girl is standing 2 km from the foot of Table Mountain which is 1000m high. Use a scale of 5cm :1 km to draw a realistic diagram. Measure the angle of elevation to the top of the mountain.

b) A bird is sitting on the mountain. What is the angle of depression to the girl?

4. IMSTUS module 8 on website

CONSOLIDATION / HOMEWORK Examples for homework

ASSESSMENT Informal: class work

TERM 1 - WEEK 3

ASSESSMENT STANDARD 9.1.3 Recognises, uses and represents rational numbers, moving flexibly between equivalent forms in appropriate contexts.

9.1.7 Estimates and calculates by selecting and using operations appropriate to solving problems and judging the reasonableness of results (including measurement problems that involve rational approximations of irrational numbers).

9.1.11 Recognises, describes and uses the properties of rational numbers

TERMINOLOGY Recurring decimals; rational numbers; irrational numbers

RESOURCES Gr 9 Text books; WCED Illustrative examples; Calculator; Internet Web sites, IMSTUS

INTEGRATION Technology


TEACHING TIPS Classify numbers:

Natural numbers i.e. {1, 2, 3, 4, ...}; Counting (whole) numbers i.e. {0, 1, 2, 3, 4, ...}; Integers i.e. {...., -3, -2, -1, 0, 1, 2, 3, ...};

Rational numbers (all numbers that can be written as );

Irrational numbers (all real numbers that cannot be written as ; e.g. π ; 8 )

Know that recurring decimals are rational numbers because they can be written as an integer divided by an integer.

EXAMPLES 1. Write down a number that is:

a) positive and rational but not an integer;

b) negative and irrational

2. Decide whether each number is rational or irrational. Tick the correct column.

number Rational Irrational

30

16

8

-5

π2

4,0 &

3. Find a rational number between 2½ and 2,6.

4. Place the following numbers on a number line; 2; 3,5; π ; - 5 ; 32 ; 5

41 ; 32,0 &&

5. Write 2 correct to 2 decimal places.

6. Write the following numbers down in the applicable columns. A number may appear in more than one column. Here are the numbers:

722 π

rational numbers

irrational numbers integers counting

numbers natural

numbers

545454553214910015027

221003330 5

873 −

−−

−− ,,25−

144 088 2 20


CONSOLIDATION/HOMEWORK Examples for homework

ASSESSMENT Assessment Task 1 e.g. Tutorial / Investigation on weeks’ 2 and 3 content

TERM 1 - WEEK 4 AND 5

ASSESSMENT STANDARD 9.1.9. Uses a range of techniques and tools (including technology) to perform calculations efficiently and to the required degree of accuracy, including the following laws and meanings of exponents (the expectation being that learners should be able to use these laws and meanings in calculations only):

x0 = 1

9.2.10 Uses the laws of exponents to simplify expressions and solve equations

9.1.3. Recognises, uses and represents rational numbers e.g. scientific notation.

TERMINOLOGY Exponents

RESOURCES Gr 9 Text books; WCED Illustrative examples; Calculator; Poster; Internet Web sites, IMSTUS

INTEGRATION Technology, Natural Science

TEACHING TIPS For revision, look at IMSTUS module 18 (exponents).

Use examples that show the meanings to formulate the laws. First write the example in expanded notation so that the meanings become clear. Do each law separately. Practise examples of the law.

e.g. x2 . x3 = (x.x.x)(x.x) = x5 e.g. xxxxxx

xx

==...

2

3

∴ x2 . x3 = x2 + 3 = x5 ∴ xxxx

== − 232

31

Now do mixed examples

Although the other two exponential laws are not mentioned, it is recommended that they are taught

i.e. (xm)n = xm x n = xmn

e.g. (x5)3 = x5. x5 . x5 = x5+5+5= x3x5

(x. y)m = xm . ym


Revise scientific notations with big numbers. Use an example to show the meaning and then practise converting from decimals to scientific notation and vice versa. Use the context of Natural Sciences

e.g. 0,002 = 33 102

102

10002 −×==

It is not recommended to teach exponential equations at this stage. Use problems in context which involve exponents.

EXAMPLES 1. Simplify and write with positive exponents:

a) 512 . 515 e) -6m0

b) 50

100

22

f) 3-2

c) x3 . x5 . x g) 23

6

4.8yxyx

d) 2

3

39aa

h) ba

ba22

34

)2(48

2. Write in scientific notation:

a) 0,0014

b) 0,000000762

3. Write the following numbers as decimals:

a) 2,8 x 10-3

b) 9,15 x 10-8

4. Solve the following problems:

a) A son is a certain age. If he multiplies his age by itself then he is as old as his father. His father is 36 years old. How old is the son?

b) Sheets of paper that are 0,167 mm thick are laminated on both sides with a plastic film that is 0,083 mm thick. How thick are the composite sheets? Express your answer in scientific notation.

c) Three extremely thin layers of optical coating are applied to certain microscope lenses. The layers are respectively 2,3 × 10−5 m, 3,1× 10−4 m and 9,2 × 10−6 mm thick.

- How thick are the three layers together? - Express the thickness of each layer in mm, in scientific notation.




TERM 1 - WEEK 6

ASSESSMENT STANDARD 9.2.1. Investigates, in different ways, a variety of numeric and geometric patterns and relationships by representing and generalising them, and by explaining and justifying the rules that generate them (including patterns found in natural and cultural forms and patterns of the learner’s own creation). 9.2.3. 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 Input; output; formulae; equations; expressions; constant; variable

RESOURCES Gr 9 Text books; WCED Illustrative examples; Calculator; Poster; Internet Web sites; IMSTUS website

INTEGRATION Technology, Natural Sciences, Arts and Culture

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

module 32. Find some patterns in the environment, nature and cultural designs that can be extended.

e.g. beadwork, paving.

Give the learners some examples of visual patterns that they can extend. This will enable them to see the pattern and enable them to generalise and write the rule.

Generate number patterns that the learners can extend and find the rule.

It is suggested that you include patterns that do not have a common difference. For these patterns deduce the formula by inspection.

Use tables as well as flow diagrams to find missing input and output values

EXAMPLES 1. Drawn below is a regular tile pattern:

2 3 4 5 6 7

a) Colour in pattern number 7.


b) Complete the following table:

Pattern number 2 3 4 5 6 7 8 9

No. grey tiles

c) How many grey tiles does pattern number 20 have?

d) Write down the rule, which can be observed, for each of the above patterns.

2. Given:

A 1; 4; 7; 10…

B 1; 4; 9; 16…

For each of the above sequences:

a) Add 2 more terms to the sequence

b) Find the 10th term of the sequence

c) Write a rule for the sequence.

d) Use the rule to check that it generates the sequence

3. Find the first three terms, and the missing terms, of each of the sequences below:

a) . . . . . . . . . 12 21 . . . 39 . . . 57

b) . . . . . . . . . 1024 2048 4096 . . . . . . . . .

4. a) Find the missing values in the flow diagram. b) Use the values of the flow diagram to complete the table.

Input values 1 2 3 4 10

Output values 5 2 -1 -28

5. Four number-machines are given:

OUT

machine A

OUT

machine C

OUT

machine B

OUT

machine D

Four formulas are given:

156

92 × 8 − 4

1

15

63

)IN(OUTformulaINOUTformula

)IN(OUTformulaINOUTformula

10241023522521

+×=+×=+×=+×=


a) Which machine and which formula belong to each other?

formula 1 2 3 4

machine

In the previous exercise there are formulas (and therefore also number-machines) that give exactly the same results. Although these formulas and machines are different, they all give the same output value for every input value.

b) Complete a table for each formula (above):

FORMULA 1

IN 0 1 2 5 10 100

OUT

FORMULA 2

IN 0 1 2 5 10 100

OUT

FORMULA 3

IN 0 1 2 5 10 100

OUT

FORMULA 4

IN 0 1 2 5 10 100

OUT

c) Which formulas (machines) give the same results?

d) Complete the following: 2 × (IN + 5) =


ASSESSMENT Assessment Task 2 e.g. Investigation: patterns- fractals

TERM 1 - WEEK 7

ASSESSMENT STANDARD 9.2.5 Solves equations by inspection, trial-and-improvement or algebraic processes (additive and multiplicative inverses, and factorisation) checking the solution by substitution.

TERMINOLOGY Inspection; trial-and-improvement; factorisation; additive inverses; multiplicative inverses


INTEGRATION -


TEACHING TIPS REVISE METHOD OF SOLVING EQUATIONS:

By inspection – without showing any written work.

By trial and improvement – guess values; substitute to see whether it makes the equation true.

Algebraic processes:

a) additive inverse 5p + 3 = 4p + 6

5p + 3 – 3 = 4p + 6 – 3 5p = 4p + 3

5p – 4p = 4p -4p + 3

p = 3

b) multiplicative inverses 632

=a

×23 6

32

=a23

×

9=a

c) factorisation The assessment standards use the word ‘factorisation’. It is not appropriate for the learners to be taught quadratic equations at this time as it is taught in Grade 10. It is recommended that this can be used for extension for the advanced learner.

Revise equations with: variables on both sides of the equal sign, e.g. 3a + 6 = a + 9 brackets, e.g. 3(x + 2) = -2(x + 3) + 2

Introduce equations with fractions. The denominator must be a number, e.g. 14

23

=−xx

EXAMPLES 1. IMSTUS module 33 pg 1-7

Solve for x:

2. 732

2=+

xx

3. 4

132

13 −=−

+ xx

4. 2

)32(3433

3)1(2 +

+=+− xx




TERM 1 - WEEK 8

ASSESSMENT STANDARD 9.2.4. 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)

TERMINOLOGY Mathematical model


INTEGRATION Technology, EMS, Natural Sciences

TEACHING TIPS The learners often get the answer to problems without showing any work. They must be

encouraged to explain their method.

Sometimes it is easier to start by drawing up a table or plotting a graph to understand the situation.

Write an equation or formula. Solve the equation to get the solution to the problem.

The learners must check the reasonableness of their answers.

EXAMPLES 1. Jane has allowed herself 300 calories per day on snacks. 4 of the snacks have a total of

15 calories. How many snacks can she have to stay in her daily limit?

2. Food goes rotten when there is too much bacteria present. The number of bacteria increases very quickly at room temperature. If the food is kept in a fridge, the bacteria do not increase that much. The health services visit an old age home at 10 am to inspect a prepared meal in the fridge. They confirm that at that stage, there were approximately 1000 bacteria per gram in the meal. For calculating the number of bacteria, the following formula is quite reliable:

time21000bacteriaofnumber ×= number of bacteria is the number of bacteria per gram food. time is the number of hours after 10 o’clock in the morning.

a) At 11:00 AM there is already approximately 2000 bacteria per gram in the food. On what can one base this statement?

b) Complete the table:

time of day 10 o’clock 11 o’clock 12 o’clock 1 o’clock 2 o’clock

number of bacteria

c) Sketch the graph of time21000bacteriaofnumber ×= on graph paper.

It is not advisable to eat food containing 100 000 bacteria per gram food. d) Determine if the prepared meal in the fridge at the old age home will still be edible by

lunch or supper. Explain your answer.



ASSESSMENT Assessment Task 3 e.g. Test: Exponents, equations etc.- term’s work


ASSESSMENT STANDARD 9.5.1. Poses questions relating to human rights, social, economic, environmental and political issues in South Africa. Context: Political, and economic in RSA

9.5.2. Selects, justifies and uses appropriate methods for collecting data (alone and/or as a member of a group or team) which include questionnaires and interviews, experiments, and sources such as books, magazines and the Internet in order to answer questions and thereby draw conclusions and make predictions about the environment. 9.5.7. Draws a variety of graphs by hand/technology to display and interpret data including: • bar graphs and double bar graphs; • histograms with given and own intervals; • pie charts; • line and broken-line graphs; • scatter plots. 9.5.6. Organises numerical data in different ways in order to summarise by determining: • measures of central tendency; • measures of dispersion

TERMINOLOGY Central tendency; mean; mode; median; stem and leaf; ascending order; descending order; dispersion; range; histogram; intervals; scatter plots; broken-line graphs; double bar graph; ungrouped data; grouped data

RESOURCES Gr 9 Text books; WCED Illustrative examples; Calculator; Poster; Internet Web sites; Data Handling in the GET Band.

INTEGRATION Choose context from Technology, EMS, Natural Sciences

Languages for questionnaire

TEACHING TIPS Co-ordinate with language teacher to practise posing questions.

Co-ordinate with other learning areas so that you do not repeat the same activity. Combine the assessment with other learning areas.


Data Cycle It is critical to recognise that the Assessment Standards of Learning Outcome 5 (Data Handling) represent a sequence of stages in a data handling cycle. The writers of the Learning Area Statement believed that learners should address the Assessment Standards as they worked through the cycle rather than on an individual basis. The cycle is represented alongside.

The Assessment Standards 9.5.5 and 9.5.7 do not show any progression in terms of knowledge and skills from Grade 8. The reason for this apparent lack of progression is that learners are expected to have acquired the skills in Grades 7 and 8 in order to be able to use them in an extended project involving all of the Assessment Standards related to the data cycle at once. In other words it is expected that Grade 9 learners should be able to integrate the knowledge and skills described in the Assessment Standards in dealing with an extended task that deals with the data cycle.

One important idea that is not mentioned explicitly in the Assessment Standards is the notion of different kinds of data. There are different kinds of data and the kind of data we are dealing with will influence the ways in which we can organise the data, whether or not we can determine measures of central tendency and the kinds of graphs we can use to represent the data:

Quantitative or numerical data is data in the form of numbers e.g. age, the number of children 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.

PROBLEM

POSE QUESTIONA question is posed regarding the problem and appropriate data sources are identified

COLLECT DATAA data collection method is

chosen and data is collected

ORGANISE DATAThe collected data is

organised, summarised and represented

INTERPRET DATAOrganised data is interpreted, conclusions are drawn and/or predictions made in order to

solve the problem


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

Given that the Assessment Standards do not show progression from Grade 8 and given that they are intended to be assessed in the context of an extended data project it is not possible to illustrate the assessment of these on their own. However, the illustrations that do follow show the kinds of thought processes we would like Grade 9 learners to use when making choices about the types of graphs they will use and the ways in which they will organise and summarise the data they have collected in their project.

EXAMPLES 1. See Data Handling in the GET Band p2 – 41. 2. For extra examples, look at IMSTUS module 23. 3. Collecting Data

After a teacher led discussion on some of the social justice, human rights and/or healthy environment issues under discussion in the media at the time of the project the following question could be posed: Working as a member of a group:

a) Propose a question that you and your group would like to investigate relating to an issue that is current in the community or nation.

b) Identify the source from which you plan to collect data in order to answer the question.

c) Develop an appropriate data collection method and collect as much data as you need in order to be able to develop a response to the question you have posed.


d) Some questions that learners may pose and collect data for include:

Question Data source Data collection method

What are the ten things that the youth of our country would most like the President to address?

The youth, organised by age, gender and socio-economic background.

Interviews and questionnaires.

How do the salaries of workers in our community compare by gender and compare to the declared minimum wage in South Africa?

Members of the community that work in a variety of different industries.

Confidential questionnaires to be completed by the participants.

Note how the question about minimum wages would almost certainly imply collecting (and later analysing) grouped data.

4. Organising Data

a) Why is it important to know about the range of the data we are working with? To understand this question better; show, by determining two sets of actual data, that it is possible to have two different sets of data with exactly the same mean, median and mode but with very different measures of dispersion.

b) Show, by creating two sets of data, that it is possible to take one number from one of the lists and include it in the other and in so doing to increase the mean of both data sets. Be careful of averages!

5 The following situation occurred at the University of Berkley (USA) in 1973. 12 763 people applied for admission. Of the 8 442 male applicants 3 738 were admitted and of the 4321 female applicants 1494 were admitted.

a) Calculate the percentage of male and the percentage of female applicants who were admitted.

b) Do you think that there could have been gender bias in the admissions process?

Now consider the following detailed discussion of the admissions by department

c) In department A: 512 of the 825 male applicants were admitted and 89 of the 108 female applicants were admitted. Calculate the percentage of male and the percentage of female applicants who were admitted to department A.

d) In department B: 313 of the 520 male applicants were admitted and 17 of the 25 female applicants were admitted. Calculate the percentage of male and the percentage of female applicants who were admitted to department B.

e) In department C: 120 of the 325 male applicants were admitted and 202 of the 593 female applicants were admitted. In department D: 138 of the 417 male applicants were admitted and 131 of the 375 female applicants were admitted. In department E: 53 of the 191 male applicants were admitted and 94 of the 393 female applicants were admitted. In department F: 22 of the 373 male applicants were admitted and 24 of the 341 female applicants were admitted. Calculate the percentage of male and the percentage of female applicants who were admitted to departments C, D, E and F.


f) Which of these departments is most responsible for the gender bias apparent in the overall admissions statistics? Can you explain this apparent contradiction?

This is a paradox! Such paradoxes involving averages are called Simpson’s paradoxes. There is another example of such a paradox in recent medical research in South Africa—this one initially appears to show racial bias—you may be interested in reading about it. Search for Simpson’s Paradox on the internet.

6. The choice we make when selecting the type of graph to use to represent data makes an impact on the interpretations that we can make. Consider the following data on government spending on three essential services:

Government spending 2000 2005

Education R 34,86 million R 22,33 million

Health R 26,15 million R 11,16 million

Pensions R 26,15 million R 3,72 million

a) Draw a pie chart to illustrate the spending pattern in 2000.

b) Draw a pie chart to illustrate the spending pattern in 2005

c) Draw a double bar graph to illustrate and compare the spending in 2000 and 2005

d) What do the two pie charts show very clearly when you compare them?

e) What do the two pie charts not show when you compare them?

f) Discuss the difference between using either two pie charts or the double bar graph to compare the spending on services in the two years.

g) Discuss who might like to use the pie charts to illustrate their argument and who would prefer to use the double bar graph to illustrate their argument.

7. Consider the following scenario, in an opinion poll for the upcoming election 2352 of the respondents indicated that they would definitely vote for candidate A, 1698 respondents indicated that they would definitely not vote for candidate A, and 1830 respondents said that they were undecided about whether to vote for candidate A or candidate B.

a) Draw a pie chart to illustrate this data.

b) Draw a bar graph to illustrate this data.

c) Which of the two ways of illustrating the data might give the candidate greater confidence about winning the election. Do you think that this method best represents the situation?

CONSOLIDATION / HOMEWORK Collect data from questionnaire.

Practise drawing all the different graphs at home.

ASSESSMENT Assessment Task 4 e.g. Project on Data (Start project in week 9)


TERM 2

TERM 2 - WEEK 1

ASSESSMENT STANDARD Data –same as for weeks 9 and 10 in Term 1

TERMINOLOGY Same as for weeks 9 and 10 in Term 1

RESOURCES Gr 9 Text books; WCED Illustrative examples; Calculator; Poster; Internet Web sites

INTEGRATION Choose context from Technology, EMS, Natural Sciences

TEACHING TIPS Revise data and complete project on data.

CONSOLIDATION / HOMEWORK For extra examples, look at IMSTUS module 23. Draw different graphs.

ASSESSMENT Assessment Task 4 e.g. Project on Data (Complete project in class and hand it in)

TERM 2 - WEEK 2

ASSESSMENT STANDARD 9.4.12 Uses the Theorem of Pythagoras to solve problems involving missing lengths in known geometric figures and solids.

TERMINOLOGY Hypotenuse (longest side of a right-angled triangle); sum of the squares; adjacent sides.

RESOURCES Gr 9 Text books; WCED Illustrative examples; Calculator; Poster; Internet Web sites, IMSTUS module 22

INTEGRATION Technology; Life Orientation

TEACHING TIPS Look at IMSTUS module 22 pg 12-20. Revise Pythagoras theorem.

Grade 9 is when learners must use the theorem to calculate the length of the sides and prove that triangles are right angled.


"home"

Second base

First base Third base

EXAMPLES Learners are expected to be able to use the Theorem of Pythagoras in problem contexts.

1. Here is the ground plan of a basketball court. It is not drawn to scale.

a) What is the direct distance from A to B?

b) And from C to B? (C is precisely in the middle of the sideline.)

2. One of the teachers drew lines with whitewash to mark out a field for mini-soccer. He brought the following measurements, on a piece of paper, to your mathematics teacher to find out if the field is rectangular. What would your teacher’s answer be? Show all your calculations.

Adjacent is a diagram of a baseball field.

(Remember: a baseball field is square.)

a) You throw the ball from first base to "home". Approximately how far do you throw the ball?

b) You throw the ball from second base to "home". Approximately how far do you throw the ball?

c) You throw the ball from first base to third base. Approximately how far do you throw the ball? (Show all your calculations.)

d) Why do we use the word “approximately?”

3. An aeroplane flies 8 km in a northerly direction to point P, then it flies 26 kilometres in an easterly direction and then flies directly back to point P. What is the total distance flown (in kilometres)?

4. The formula for the volume of a pyramid is given by:

Volume = 13 × the area of the base × height

If the lengths of sides of the isosceles triangles in the pyramid alongside are 4cm, 6cm and 6cm, calculate the volume of the rectangular based pyramid.



ASSESSMENT Informal: (Written in homework book)

TERM 2 - WEEK 3

ASSESSMENT STANDARD 9.2.8 Uses the distributive law and manipulative skills developed in Grade 8 to: • find the product of two binomials

TERMINOLOGY Binomial


INTEGRATION Life Orientation


module 32 pg 1-11. (on the website) Use the distributive law to explain the multiplication of two binomials

e.g. (x + y)(a + b) = x(a + b) + y(a + b) = ax + bx + ay + by

Use different types of examples including different variables, same variables, fraction coefficients, squaring the binomial

Explain, using the areas of rectangles, that it is true that 22)2()1( ++×+×=+×+ bababa

EXAMPLES 1. Find the product and simplify the answer where possible:

a) (m - n)(k + p) b) (a + 3)(a – 4)

c) (2p – 3q)(p – 4q) d) (a2 – 2b)(4a + 5b2)


e) (0,5m + n)(0,75m + n) f) (p + q)2

g) (2a – 0,5b)2 h) (d + e)(d – e)

i) 2(3p – 7q)(p + 2q)



TERM 2 - WEEK 4

ASSESSMENT STANDARD 9.2.8 Uses the distributive law and manipulative skills developed in Grade 8 to: • factorise algebraic expressions (limited to common factors and difference between squares)

TERMINOLOGY Factorise, difference between squares


INTEGRATION -

TEACHING TIPS Revise highest numeric common factor by giving a few numbers that have common factors

and ask learners to identify the common factor that is the biggest number.

Start with simple examples using variables without exponents; then use examples with exponents and then use more than one variable with exponents. (See examples.)

Taking out common factors in brackets can be taught as an extension exercise e.g. example (e).

Also changing of the sign of a term in order to find a common factor can be done as an extension e.g. example (f).

EXAMPLES 1. Factorise by taking out the highest common factor.

a) 4ab – 8a b) 7d2 + 21d3

c) 12m2n – 18mn2 d) 2a3b2c – 4a2bc3 + 6ab3c2

e) a(p – q) + b(p – q) f) a(p – q) + (q – p)




TERM 2 - WEEK 5

ASSESSMENT STANDARD 9.2.8 Uses the distributive law and manipulative skills developed in Grade 8 to: • find the product of two binomials; • factorise algebraic expressions (limited to common factors and difference of squares).

TERMINOLOGY Difference between two squares


INTEGRATION -

TEACHING TIPS For ideas on developing the concept and extra examples, look at IMSTUS module 32

pg 12-16. Revise square roots of numbers and algebraic terms e.g. 4a2b6

Revise binomial multiplication of (a + b)(a – b)

Factorising is the opposite operation to multiplying e.g. recognise the format of the expression to be factorised by difference of 2 squares: Two terms only: both terms must be squares; there must be a minus sign between the terms, e.g. a2 – b2

Consolidate factorising by giving a mixed exercise e.g. example 2.

EXAMPLES 1. Factorise by using the difference between two squares: a) x2 – y2 b) a2 – 16

c) 25x2 – 9y2 d) 36a4 – 1

e) x2y4 – 81p6

2. Factorise fully:

a) 4xy3 + 8xy b) 4a2 – b4

c) 5m3 – 20m d) -8a3b + 2a2b3c

e) 16x2 – 4 f) 2a(k – 5) + 50(k – 5)




TERM 2 - WEEK 6

ASSESSMENT STANDARD 9.2.11. Uses factorisation to simplify algebraic expressions and solve equations.

TERMINOLOGY Expressions; factorise


INTEGRATION -

TEACHING TIPS

Use factorisation to simplify algebraic expressions, e.g. a

aa3

63 2 −

The Grade 10 Assessment Standards limit the simplification of algebraic fractions to a monomial denominator. Therefore the examples given in Grade 9 should not be too complicated. It may be a good idea to practise simple simplifications as in examples 1b) – 1d).

Use factorisation to solve equations e.g. 522

86−=

+x

x

It is not recommended that quadratic equations are introduced at this stage. They are taught in Grade 10.

EXAMPLES 1. Use factorisation to simplify the following expressions:

a) xy

xyxyyx3

396 22 ++ b) 224

2yxyx

−+

c) 22993abba

−+

d) qpqp

105205

22

+−

2. Use factorisation to solve for x:

a) 123

63=

−x b)

2461

3129

21 −=+

− xx


ASSESSMENT Assessment Task 5 e.g. Tutorial on products and factors.


TERM 2 - WEEK 7

ASSESSMENT STANDARD 9.5.8. Critically reads and interprets data with awareness of sources of error and manipulation to draw conclusions and make predictions about: • social, environmental and political issues (e.g. crime, national expenditure, conservation,

HIV/AIDS); • characteristics of target groups (e.g. age, gender, race, socio-economic groups); • attitudes or opinions of people on issues (e.g. smoking, tourism, sport); • any other human rights and issues

TERMINOLOGY Manipulation; sources of error


INTEGRATION Choose context from Technology, EMS, Natural Sciences, Life Orientation

TEACHING TIPS Assessment Standard 9.5.8 deals with learners “reading and interpreting” the data that

they have collected and organised in order to “draw conclusions and predictions” about the question that they posed. In Grades 7 and 8 learners developed a sensitivity to the role that different data gathering, summarising and representing techniques can have on the conclusions and predictions that can reasonable be made from data gathered. In Grade 9 learners are expected to both remain sensitive to these issues and also to become sensitive to the profile of the sample from which the data was gathered.

Social, environmental and political issues (e.g. crime, national expenditure, conservation, HIV/AIDS) Social, environmental and political issues such as crime, national expenditure, conservation, HIV/AIDS to name but a few can be highly emotive. People have opinions on crime and the way that a survey question is posed may well influence the response of the person being interviewed.

Characteristics of target groups (e.g. age, gender, race, socio-economic groups) Different socio-economic groups are likely to think about different issues differently. While people in a lower income bracket may be more concerned about basic wages, people in higher income brackets may be more concerned about the personal tax rate. If you interview people in the lower income bracket about tax rates and people in higher income brackets about basic wages you are will get very different responses to interviewing each of the two groups the other questions.

Attitudes or opinions of people on issues (e.g. smoking, tourism, sport) South Africa has introduced strict laws with respect to smoking in public places. If you wanted to test the opinions of people on these laws and you interviewed only smokers you would almost certainly get a different reaction to interviewing non-smokers only.

Any other human rights and inclusivity issues Examples in the grade 7 and 8 Teacher’s Guide.


EXAMPLES 1. A wide range of data on people living in the Western Cape is provided in the table that follows.

Census 2001: number of households and individuals Black African Coloured Indian or Asian White

Households 319187 536954 10714 306452

Male 600388 1170949 22341 398641

Female 607041 1268027 22687 434261

Census 2001: household goods by population group in the Western Cape Black African Coloured Indian

or Asian White

Radio Yes do have a radio 206449 417943 9764 293766

No do not have a radio 112738 119011 948 12685

Refrigerator Yes do have a refrigerator 131084 419778 10279 301197

No do not have a refrigerator 188103 117175 432 5254

Television Yes do have a television 153619 418241 9876 287199

No do not have a television 165569 118712 835 19253

Telephone in dwelling Yes do have a telephone 60388 261920 8807 261441

No do not have a telephone 258799 275034 1904 45011

Computer Yes do have a computer 7705 54934 4573 145929

No do not have a computer 311482 482020 6139 160523

Cell Phones Yes do have a cell phone 86295 169124 7479 222397

No do not have a cell phone 232893 367830 3233 84054

Census 2001: household water source by population group in the Western Cape 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

Census 2001: income category by population group and gender in the Western Cape Black African Coloured Indian

or Asian White

Male No income 3802 7143 163 3803

Male R 1 - R 400 16076 23544 185 2163

Male R 401 - R 800 45485 81370 329 3362

Male R 801 - R 1600 78296 120095 1114 10524

Male R 1601 - R 3200 31594 112910 2375 26283

Male R 3201 - R 6400 9924 59513 2639 47368

Male R 6401 - R 12800 3578 22600 1782 51012

Male R 12801 - R 25600 1273 5108 904 31966

Male R 25601 - R 51200 446 1222 263 11040

Male R 51201 - R 102400 114 527 61 3245

Male R 102401 - R 204800 83 236 41 1727

Male R 204801 or more 59 137 21 1115

Female No income 3541 6792 192 4030

Female R 1 - R 400 16575 36035 168 2729

Female R 401 - R 800 41420 87852 418 4117

Female R 801 - R 1600 41567 107749 977 13478

Female R 1601 - R 3200 13639 76181 1735 35002

Female R 3201 - R 6400 7794 42237 1800 54583

Female R 6401 - R 12800 2541 11408 853 29191

Female R 12801 - R 25600 521 1497 245 8879

Female R 25601 - R 51200 194 502 63 2136

Female R 51201 - R 102400 57 289 22 1165

Female R 102401 - R 204800 39 125 13 474

source: www.statssa.gov.za


Determine the following:

a) What fraction of the Western Cape population is Black African, Coloured, Indian or Asian and White?

b) The mean number of persons per household for each of the different population groups.

c) The mean, median and modal income by gender and population group.

d) The percentage of homes that have radios, televisions and computers by population group.

e) The percentage of homes that have water inside their home, inside their yard, less than 200m from their home and more than 200m from the home by population group.

f) Use the data in the table and the analysis of the table that you have done above to describe the population of the Western Cape.

2. Further examples in the book ‘Data Handling in the GET Band’ p42 – 44

CONSOLIDATION / HOMEWORK Give questions relating to graphs for interpretation.


TERM 2 - WEEK 8 CONSOLIDATION

TERM 2 - WEEK 9 Assessment Task 6: EXAMINATION e.g. 2 one hour papers

TERM 2 - WEEK 10 Assessment Task 6: EXAMINATION e.g. 2 one hour papers


TERM 3

TERM 3 - WEEK 1 REVISION Go over examination and misunderstandings arising from the questions in the examination.

TERM 3 - WEEK 2

ASSESSMENT STANDARD 9.3.1. Recognises, visualises and names geometric figures and solids in natural and cultural forms and geometric settings, including: • regular and irregular polygons and polyhedra; • spheres; • cylinders

9.3.9. Recognizes and describes geometric solids in terms of perspective, including simple perspective drawing.

9.3.4. Draws and/or constructs geometric figures and makes models of solids in order to investigate and compare their properties and model situations in the environment

TERMINOLOGY Regular polygons; irregular polygons


INTEGRATION Technology, Arts and Culture

TEACHING TIPS For revision, look at IMSTUS module 13. Use books, magazines or the internet to find pictures of polygons. Discuss their usage in

everyday life. Comment on the usefulness of the particular shape in its context, e.g. Why are roof trusses triangular?

Use books, magazines or the internet to find pictures of polyhedra. Discuss their usage in everyday life. Comment on the usefulness of the particular shape in its context, e.g. why are some yoghurt containers oval shaped and not round or square?

EXAMPLES 1. The pictures below show gourds that grow in trees and also some that have been

decorated by artists. Gourds are almost completely spherical in shape.


Groups of rural women decorate the gourds and sell them as a source of income. Recently a large number of tourists have visited a particular group of women and it would appear as if their gourds are becoming very popular curios. The women have approached you to help them to design and manufacture a sturdy container that will hold four of these gourds.

INFORMATION: Although gourds vary a great deal in size the women have found that they seldom work with gourds that are bigger than 10cm. in diameter. For this task we will assume that all gourds are exactly 10cm in diameter.

Working alone or as a member of a group

a) Design and construct the following three containers to hold four gourds (you should build half size models in order to save on cardboard):

A square based prism in which the four gourds lie side by side.

A regular-hexagon based prism in which the gourds will be stacked on top of each other.

A cylinder in which the gourds will be stacked on top of each other.

You should be ready to hand in the following:

A sketch of each of the nets you used to construct the container showing all of the dimensions.

Evidence of the calculations you did to determine the various dimensions for the net—you can show these on the nets.

The actual models.

b) Determine the following for each of your containers:

Total surface area (i.e. the amount of cardboard used to manufacture the container)

The volume of the container.

c) Make a proposal to the women on which of the containers you think they should use explaining why you think it is the best choice.

2. The drawing below is an artists’ impression of a room with a square floor.

a) How has the artist created the perception of depth in this sketch?

b) What types of quadrilateral are A, B and C in the diagram and what shapes do they represent in reality?

c) Why have some shapes been distorted while others have not?


a)

b) c) d)

d) In reality the squares on the floor are created by two sets of parallel lines. Are both sets also parallel in this diagram? Discuss.

3. Which geometric solid is represented by each of the nets below?

4. A match box is 3cm long, 2cm wide and 1 cm high. Draw a net of the part of the match box in which the matches lie and the part in which the matches will slide into.

5. Draw a net of the inner cardboard roll of a toilet paper roll.

6. Draw a net of a pyramid of which the surfaces/faces are triangles of the same size and shape. Cut it out and fold it to form a pyramid. If your net was not properly drawn and cut, you must attempt another one. Write down the name of this special pyramid.

CB

A


7. Here we have two partly drawn nets of rectangular prisms. Complete each of these nets.


ASSESSMENT Assessment Task 7: e.g. Investigation on geometric solids

TERM 3 - WEEK 3

ASSESSMENT STANDARD 9.3.3 Uses geometry of straight lines and triangles to solve problems and to justify relationships in geometric figures.

TERMINOLOGY Complementary; supplementary; equilateral, isosceles; scalene


INTEGRATION Technology

TEACHING TIPS Revise angle types taught in Grade 7.

Grade 8 learners have only been introduced to the vocabulary of the relationships.

For ideas on developing the concept as well as extra examples, look at IMSTUS module 16 from pg 6.

Teach the angle relationships: Angles on a straight line add up to 180°; vertically opposite angles are equal; angles about a point add up to 360°.

Parallel lines cut by a transversal: Corresponding and alternate angles are equal; co-interior angles add up to 180°. Look for the F U and Z (or N) shapes. Let the learners highlight the angle pairs. (See Gr 8 TG p 53.)

Teach the angle relationships in triangles. Sum of the angles of a triangle is180°; exterior angle of a triangle is equal to the sum of the interior opposite angles.

Classify triangles according to their sides and angles.


Let learners first do the following:

On a clean piece of paper draw a large triangle and cut it out. Use the letters A, B and C and mark off the angles on the triangle..

Neatly tear the angles from the triangle, as shown in the first

sketch. (NB: Do not cut the angles, because you might forget which angle is the angle of the triangle.

Paste (as shown in third sketch) the three torn angles of the triangle next to each other. Write down what you observe about angles A, B and C.

EXAMPLES 1. Find the values of the angles marked a – f, in that order. Give a reason for each answer.

2. Solve for x by writing an equation. Give a reason for your statement

3. Find the value of the angles marked a – e, giving reasons. Show your working.

6x

11x 7x2x + 10°

3x - 30°

42°b

a d c

110°

Q

R

O

P N

M

e 125°

F

E

DC

B A

f

E D

CB

A

108°

b

a

T

S

c

145°

R Q

Pe

4d5d d

E

FG

O

D





ASSESSMENT STANDARD 9.3.6 Uses transformations, congruence and similarity to investigate, describe and justify (alone and/or as a member of a group or team) properties of geometric figures and solids, including tests for similarity and congruence of triangles

9.3.2 In contexts that include those that may be used to build awareness of social, cultural and environmental issues, describes the interrelationships of the properties of geometric figures and solids with justification, including: • congruence and straight line geometry; • transformations

TERMINOLOGY Similarity; congruency; rotational symmetry

RESOURCES Gr 9 Text books; WCED Illustrative examples; Calculator; Poster; Internet Web sites; IMSTUS

INTEGRATION Technology, Arts & Culture

TEACHING TIPS Use examples from the environment as an introduction to show the difference between

similarity and congruency.

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.

For ideas on developing the concept and extra examples, look at IMSTUS module 30


EXAMPLES 1. Write down the names of the following three figures below.

a) Indicate which angles are equal and which line segments are equal in each figure.

b) Write down the transformation which will ensure that the equal sides and angles will fit exactly onto each other.

2. Quadrilateral ABCD alongside is a parallelogram - that is, both pairs of opposite sides are parallel to each other.

a) By showing that ∆ADC ≡ ∆CBA justify the property that the opposite sides of a parallelogram are equal (AB = DC and BC = AD).

b) Hence, or otherwise, justify the property of a parallelogram that the diagonals of a parallelogram bisect each other (AM = MC and DM = MB)

M

D

C

B

A


3. Use different colours to draw the following transformations:

a) Translate the figure 6 units to the right and 0 units down.

b) Translate the figure 0 units to the right and 6 units up.

c) Translate the figure 6 units to the right and 6 units up.

4. Use different colours to draw the following transformations:

a) Rotate the figure to the left (anti-clockwise) through 90o about the point (0 ; 0).

b) Rotate the figure to the right (clockwise) through 90o about the point (0 ; 0).

c) Rotate the figure through 180o about the point (0 ; 0). Does it make any difference whether you do it clockwise or anti-clockwise?

axis

axis

axis

axis


5.

a) Determine the co-ordinates of triangle P′Q′R′ the image of triangle PQR after it has been translated down 4 units and left 3 units

b) Determine the co-ordinates of quadrilateral A′B′C′D′ the image of quadrilateral ABCD after it has been translated down 5 units and right 7 units

c) Determine the co-ordinates of quadrilateral A′′B′′C′′D′′ the image of quadrilateral ABCD after it has been reflected across the x-axis.

d) Determine the co-ordinates of hexagon J′K′L′M′N′O′ the image of hexagon JKLMNO after it has been reflected across the y-axis.

e) Based on the examples above, describe what happens to the co-ordinates of a figure when

it is translated.

f) Based on the examples above, describe what happens to the co-ordinates of a figure when it is reflected about the x-axis.

g) Based on the examples above, describe what happens to the co-ordinates of a figure when it is reflected about the y-axis.

6. P and Q are points on the circle with centre M. S is the midpoint of PQ. Marina states: “I can see, using a construction, that the angles at S are equal to 90º. Will it always be true or is it coincidently like that? Prove to Marina that this is always true, by proving two triangles congruent.


ASSESSMENT Assessment Task 8: e.g. Tutorial on geometry of straight lines, congruency and similarity.

-2

y

x

N

O

RA

L

K

M

J

D

-7

-6

-5

-4

-3

-1

P

C

B

-7 -6 -5 -4 -3 -2 -1

Q

7654321

7

6

5

4

3

2

1


TERM 3 - WEEK 6 - 8

ASSESSMENT STANDARD 9.2.6 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; • by graphs on the Cartesian plane in order to select the most useful representation for a given situation

9.2.7 Draws graphs on the Cartesian plane for given equations (in two variables) or Determines equations or formulae from given graphs using tables where necessary

TERMINOLOGY Gradient/slope; general formula of a linear equation; y = mx +c / y=ax + b

RESOURCES Gr 9Text books; WCED Illustrative examples; Calculator; Poster; Internet Web sites

INTEGRATION Technology, Natural Sciences

TEACHING TIPS Revise plotting ordered pairs on the Cartesian plane.

For ideas on developing the concept as well as extra examples, look at IMSTUS module 31 on the website.

Discuss the different methods of representing the relationships (words, formulas, tables and graphs) and decide which representation is the most useful.

There is more than one way of drawing the graphs of linear functions for example: A: Draw the graph of: y = 0,5x – 1 The top row of the table shows some x values that could be used. Calculate the corresponding y values:

x values -2 -1 0 1 2 3

y values

Now write these as ordered pairs: (-2 ; ), (-1 ; ) (0 ; ), (1 ; ), (2 ; ), (3 ; ) Plot these points on the Cartesian plane. Join them and extend the line. B: Draw the graph of: y = 0,5x – 1 Calculate the intercepts: x intercept: let y = 0 i.e. 0 = 0,5x – 1 ∴ x = 2 (2;0) y intercept: let x = 0 i.e y = 0,5(0) – 1 ∴y = -1 (0;-1) Plot these points on the axes. Join them and extend the line.


Special linear graphs: Horizontal line; e.g. y = 2 is a horizontal line, parallel to the x axis, passing through 2 on the y axis Vertical line: e.g. x = -1 is a vertical line, parallel to the y axis, passing through -1 on the x axis

Use this activity as an introduction:

Two motorists A and B each travelled at a constant speed. The distances (S km) they covered after t hours are represented in the graphs below.

1. Use the graphs to make a table of time-distance values for each motorist.

2. What is the speed of motorist B between points:

a) C and D

b) C and E

c) D and E

Explain your answers. What do you notice?

3. What is the speed of motorist A?

4. Write an algebraic formula that represents the relationship between time and distance for each motorist.

5. Which part of the algebraic formulae represents the speed of each motorist?

6. Explain how you can tell from the tables at which speed each motorist is driving.

7. Explain how you can tell from the graphs at which speed each motorist is driving.

A B E

D

C


horizontal step

vertical step angle of inclination

horizontal step

vertical step

angle of inclination

GRADIENT A way to introduce gradient is to use the

analogy of climbing stairs. Climbing stairs can sometimes be exhausting and sometimes not. It depends on how steep the stairs are. Uncle Bertus has a picture of a stair case in his house. It indicates what is meant by the horizontal step of the stairs and the vertical step of the stairs. A way in which the steepness of the stairs can be measured is to calculate or measure the angle of inclination. The bigger the angle of inclination, the steeper the stairs. This means that the bigger the angle of inclination the greater the gradient. The smaller the angle of inclination the less the gradient is.

EXAMPLE: A carpenter has to build three sets of wooden stairs, according to the following

specifications:

horizontal step vertical step angle of inclination

Set of stairs A 40 cm 20 cm

Set of stairs B 12 cm 45o

Set of stairs C 20 cm 23 cm

a) Which of these three sets is the steepest? Draw a scale drawing of each set of stairs.

b) For stairs that are easy to climb, the following condition is valid: 2 × vertical step + horizontal step = 63 (vertical step and horizontal step is measured in cm). Which set of stairs A, B or C will be the easiest to climb, according to this condition?

c) A carpenter builds a set of stairs that is easy to climb and he does that by making the vertical step and the horizontal step of the stairs the same. How big is the angle of inclination of this set of stairs?


Introduce the general formula of a linear equation; y = mx +c; where m is the slope and c is the intercept on the y axis.

Teach slope where xinchangeyinchangem = by linking to the previous example.

From a given graph, determine the gradient (m) and read off the y intercept (c). Now write the equation in the form y = mx +c / y= ax + b. Try the following:

A B

C D

5

–3

4

4

–2

3


EXAMPLES 1. A relationship can be represented by a table, a graph or a formula. You will have to

complete the table, graph and formula for each of the following linear relationships.

a) Fill in everything that is missing: TABLE:

x 0 -2 -3 1 3

y 1 0

FORMULA: y = 1 +0,5 × x

GRADIENT:

INTERCEPT ON THE VERTICAL AXIS:

b) Fill in everything that is missing: TABLE:

time 0 10 20 35

height 0

FORMULA:

GRADIENT:


GRAPH:

GRAPH: Height in cm

Time in hours


c) Fill in everything that is missing: TABLE:

A 0 -1 -2 2 3

B -1 0

FORMULA: B =

GRADIENT:


d) Fill in everything that is missing:

TABLE:

amount 0 10 50 70 90

price

FORMULA: price = 50 + 5 × amount

GRADIENT:


GRAPH:

Amount

Price in rand

GRAPH:


e) Fill in everything that is missing: TABLE:

P -2 -1 0 1 2

Q

FORMULA:

GRADIENT: 0,5

INTERCEPT ON THE VERTICAL AXIS: -1

f) Fill in everything that is missing: TABLE:

x -2 -1 0 1 2

y 5 -1

FORMULA:

GRADIENT:


2. Plot y = 3x + 2 using a table.

3. Plot y = -2x -1 using the intercept method.

GRAPH:

GRAPH:


4. Draw sketch graphs, using any method:

a) y = x + 1

b) y = 2

c) x = -5

d) y = 0,5x

5. Determine the equations of the linear functions represented by the following graphs:

a) b) c)

6. Answer the following questions for each of the 4 graphs below:

a) Where does the graph cut the vertical-axis?

b) What is the gradient of each of the above graphs?

c) Write an algebraic formula for each of the above graphs.

d) Where can you see the y-intercept and the gradient in the formula?

x

y

3

2 x

y

3

-3 x

y

-4

-2

D

5

C

–3

B

4

4

A

–2

3


7. Draw a graph to illustrate the following relationship between two variables, for input numbers ranging from –10 to 10.

8. Make a flow diagram to show how the y-value on the graph can be calculated for any given x- value, in each of the following cases. Also write a formula for the same in each case.


ASSESSMENT Assessment Task 9: e.g. Investigation + test on graphs


ASSESSMENT STANDARD 9.1.5 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 and compound interest, hire

purchase, exchange rates, commission, rentals and banking); 9.1.7 Estimates and calculates by selecting and using operations appropriate to solving problems and judging the reasonableness of results (including measurement problems that involve rational approximations of irrational numbers).

TERMINOLOGY Compound interest, commission, rentals, banking


Input variable 5 10 15

Out

put v

aria

ble

210

20

0

40

60

80

120

100

160

180

140

200

value of independent variable value of dependent variable + 2,4 × 0,5


INTEGRATION Technology, EMS

TEACHING TIPS Revise profit and loss, budgets, accounts, loans, simple interest, hire purchase and

exchange rates.

The new topics in this section are: Compound interest, commission, rentals, banking

The emphasis is on understanding the concept of compound interest and not just applying the formula.

EXAMPLES 1. The price of a certain article is R291,84, VAT included. What was the price before VAT

was added?

2. An amount of R6 800 is invested at 8% interest, compounded annually, for a period of three years. Calculate the value of the investment at the end of this period.

3. Mr Naidoo buys a new car. The cash price of the car is R186 000. Mr Naidoo pays a deposit which is 30% of the cash price, and then pays 48 monthly instalments of R4 400. What is the total finance charge?

4. Calculate the final amount that I have in the bank if I invest R600 for 4 years at a rate of 5% p.a. simple interest.

5. Calculate the final amount that I have in the bank if I invest R600 for 2 years at a rate of 6% p.a. compound interest.

6. Jen comes back from Japan with 5000 yen. How many Rands will she exchange it for, if the bank charges her 4% commission? [1¥ = R0.064]

7. Tebogo starts his own bookshop in a shopping mall. He has to pay R210 per square metre each month. Calculate the rental costs if his shop has an area of 12m2.




TERM 4

TERM 4 - WEEK 1 Revision on graphs and financial mathematics using Gr 9Text books; WCED Illustrative examples; Calculator; Poster; Internet Web sites; IMSTUS material

TERM 4 - WEEK 2

ASSESSMENT STANDARD 9.4.3 Solves problems – including problems in contexts that may be used to develop awareness of human rights, social, economic, cultural and environmental issues – involving known geometric figures and solids in a range of measurement contexts by: • measuring precisely and selecting measuring instruments appropriate to the problem; • estimating and calculating with precision; • selecting and using appropriate formulae and measurements.

9.1.7 Estimates and calculates by selecting and using operations appropriate to solving problems and judging the reasonableness of results (including measurement problems that involve rational approximations of irrational numbers).

9.4.4 Describes and illustrates the development of measuring instruments and conventions in different cultures throughout history.

TERMINOLOGY


INTEGRATION Technology, Social Sciences, Life Orientation

TEACHING TIPS Make sure that measurements are done accurately.

Emphasise when the term ‘volume’ is used- more for solids (e.g. m³) and ‘capacity’ term used for the volume of liquids (e.g. ml)

Convert measurements to the same units when you start a problem.

Always write the correct units with your answer, e.g. length – m; area – m2 ; volume – m3

For ideas on developing the concept as well as extra examples, look at IMSTUS module 11


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

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

c) Measure a can of soup and establish by means of calculation whether the capacity 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 2cm high, 15cm wide and 20cm 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 volume (biggest volume), and for which one have you used the most material?

3. Complete the following:

the area of a newspaper is 0.5 ........ the area of a wall tile is 225 ........ the area of your classroom floor is .............. m2 the length of a ruler is ........ the width of a road is ........ the height of the Eiffel Tower is 310 ....... the distance from Cape Town to Johannesburg is ......... the volume of a bottle of Coca Cola is ........ the volume of a mattress is ........ the volume of your classroom is .............. m3 the volume of a jerry can is ......... litres

4. On a farm a paddock is made for young horses. The fencing is made of planks. Two planks are placed, on top of each other, between two poles. The plan of the paddock, shown here, is to the scale 1:4000.

a) Calculate how wide and how long the paddock actually is.

b) How many metres of planking are needed for the fencing?

c) A pole is planted every five metres. How many poles are needed for the fencing?

d) Grass seed is planted in the paddock. The seed is supplied in 5kg packages and each package is enough to plant 100m2. How many packages of grass seed are required?




TERM 4 - WEEK 3

ASSESSMENT STANDARD 9.1.6 Solves problems that involve ratio, rate and proportion (direct and indirect).

TERMINOLOGY Direct proportion; indirect proportion


INTEGRATION Technology, EMS

TEACHING TIPS Teach ratio by revising fractions and showing that a fraction is a ratio.

Use a context familiar to learners.

EXAMPLES 1. A certain fruit drink is made by adding 250 mℓ of concentrated juice to 1 ℓ of water. How

much concentrated juice is there in 200 mℓ of the fruit drink?

2. A school hires a bus for an educational trip to the Kruger National Park. The total cost for hiring the bus for the 10 days is R14 800. The arrangement is that learners who go on the trip contribute equally to the cost of hiring the bus.

Complete the following table to show how much each learner has to pay, for different numbers of participants in the trip.

Number of passengers 10 20 30 40 50 60

Cost per passenger

3. Cell phone company A charges a monthly rental of R80, and R1,60 per minute of airtime.

a) A client uses 124 minutes of airtime during a month. What does he actually pay, for each minute of airtime that he uses?

b) The same client uses 238 minutes of airtime during the next month. What is now his actual cost per minute of airtime?

c) Make a table that will illustrate the relationship between the number of minutes of airtime used during a month, and the actual cost per minute.


4. Complete the following (ratio) table on VAT:

price before VAT 0 50 100 150 200 250 300 400

price includingVAT 57 114 171

5. A finance manager draws a graph of cost per copy. For this graph he uses the formula:

Cost per copy 5,04000+=

copiesofamount

a) How much does one copy cost if they make 2000 copies per month?

b) Show on the graph where we can read this off.

c) Show, using the formula, how you worked this out.

d) The director mentions that the cost per copy is lower when more copies are made. Is this correct? Describe how the graph indicates this.

Prys met BTW

Prys sonder BTW Price before VAT

Price including VAT

Amount of copies

Cost per copy


6. Complete the corresponding tables and write down the corresponding formula for each graph.

Table A Table B

P 0 1 2 3 10 X 0 1 2 3 10

Q 0 2 4 Y 5 7 9

formula: formula:

Table C Table D

A 0 10 20 30 50 T 0 10 20 30 50

K 0 9 18 S 10 19 28

formula: formula:

Which of these graphs represent a direct proportion ?

THE GRAPH OF A DIRECT PROPORTION IS ALWAYS A STRAIGHT LINE GRAPH THAT GOES THROUGH THE POINT (0;0) AND THE FORMULA IS ALWAYS OF THE FORM K = NUMBER Х A




TERM 4 - WEEK 4

ASSESSMENT STANDARD 9.4.2 Solves ratio and rate problems involving time, distance and speed.

9.1.2 Describes and illustrates the historical development of number systems in a variety of historical and cultural contexts (including local).

TERMINOLOGY -


INTEGRATION Technology, EMS, Natural Sciences

TEACHING TIPS Relate time, distance and speed to every day or current activities e.g. In August 2009, Bolt

ran the 100m in 9.58 sec therefore at what speed was he running?

Remember to keep the units the same when doing calculations e.g. in the above example the speed would be m/sec as the distance is in metres and the time in seconds. For a car it would be km/hr i.e. distance in kilometres and time in hours.

EXAMPLES 1. Two women, Natasha and Simpiwe, start out on a road race at the same time. From

previous experiences they know that Natasha runs much slower than Simpiwe. In fact, when Simpiwe has run 3 km, Natasha has normally covered only about 2 km.

a) Two hours after the start of the race, Natasha has covered a distance of 9,3 km. Approximately what distance can one expect Simpiwe to have covered in the first two hours?

b) Three hours after the start of the race, Simpiwe has covered a distance of 16,8 km. Approximately what distance can one expect Natasha to have covered in the first three hours?

c) If the race is over 42 km, approximately how long will Simpiwe take to finish the race?

d) Approximately how far behind her will Natasha be when Simpiwe reaches the end point?

2. A man travels 88 km during the first hour of a journey, 109 km during the second hour, 112 km during the third hour, and 108 km during the fourth hour.

a) What total distance has the man covered over the four hours?

b) Suppose he could travel at the same speed all the time (this is actually impossible), at what speed would he have to travel to cover the same distance as above in 4 hours?





ASSESSMENT STANDARD 9.5.10 Considers situations with equally probable outcomes, and:

• Determines probabilities for compound events using two-way tables and tree diagrams; • Determines the probabilities for outcomes of events and predicts their relative frequency in simple

experiments; • Discusses the differences between the probability of outcomes and their relative frequency.

TERMINOLOGY Compound events; relative frequency; probability of outcomes, equally probable outcomes


INTEGRATION Natural Sciences


module 26 from pg 16 When you throw with two dice the following are the possible outcomes:

We want to find out what is the probability of throwing one six with two die.


Let us draw a probability diagram:

Determine the following chances from the photo of the dice:

event chance fraction percentage

2 sixes 1 in 36 361

1 six

0 sixes

Remember: * represents the other 5 possibilities

*

6

*

6

6

*

1 out of 6

5 out of 6

1 out of 6

5 out of 6

1 out of 6

5 out of 6

first die

second die

6 6

6 *

* 6

* *


EXAMPLES 1. What are the probabilities of each of the possible outcomes when you toss a coin and

draw a card from a bag with three cards marked A, B and C?

Drawing one card from a bag that contains the cards A, B and C

P(A) = 13 P(B) = 13 P(C) = 13

P(Head) = 12 P(A and Head) = 13 × 12 = 16 P(B and Head) = 13 × 12 =

16 P(C and Head) = 13 × 12 =

16

Toss

ing

a co

in

P(Tail) = 12 P(A and Tail) = 13 × 12 = 16 P(B and Tail) = 13 × 12 =

16 P(C and Tail) = 13 × 12 =

16

Both techniques have revealed that there are exactly six different outcomes for the situation and each outcome has a probability of 16 .

Result of tossing a coin

Result of drawing the card

Head

Tail

A

B

C

A

B

C

12

12

13

13

13

13

13

13

Outcome Probability of the outcome

Head and A P(Head and A) = 12 × 13 = 16

Head and B P(Head and B) = 12 × 13 = 16

Head and C P(Head and C) = 12 × 13 = 16

Tail and A P(Tail and A) = 12 × 13 = 16

Tail and B P(Tail and B) = 12 × 13 = 16

Tail and C P(Tail and C) = 12 × 13 = 16


2. Cut out three 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. You will also need a coin. We are going to conduct a series of experiments each experiment consists of drawing a name from the bag and tossing a coin.

a) Use a two-way table to make a list of all of the outcomes that are possible when we conduct the experiments.

b) Use your two-way table to determine the probability of each of the possible outcomes.

c) You are about to conduct 60 experiments. In each case you will record the result of the experiment, replace the name in the bag and shake it. What do you predict the frequency to be for each of the possible outcomes after completing the 60 experiments? Justify your answer.

d) Conduct 60 trials and record the outcomes of each trial. Determine the frequency of each outcome.

e) Compare your predicted frequency with the actual frequency. Are they the same? If not, why not?

3. In a game show contestants must: draw a ball from a bag that contains three balls: a green one, a blue one and a red one and spin a spinner like the one drawn alongside.

a) Use a tree diagram table to make a list of all of the outcomes that are possible when we the participants draw a ball and spin the spinner.

b) Use your tree diagram to determine the probability of each of the possible outcomes.

c) The organisers of the game show want to give out two kinds of prizes. They want to give expensive prizes to contestants who get the same colour on the spinner as the colour of the ball they draw from the bag and less expensive prizes to contestants who get different colours on the spinner and ball. Do you think that this is a good way to allocate the expensive and less expensive prizes? Justify your answer.

d) If the show plans to host 200 contestants in the next month and the expensive prizes cost R 100 each while the less expensive prizes cost R 20 each, determine (showing your calculations) how much the show organisers should budget for prizes for the show.

e) Will the organisers spend exactly what you have budgeted? Explain the thinking behind your answer.



TERM 4 - WEEK 7 - 10 ASSESSMENT

Work Schedule Week

other LO AND AS CONCEPTS

WK1

WK 2 9.3.10 position and movement

WK 3 9.1.3, 9.1.7, 9.1.11 angles

WK 4 9.1.9, 9.2.10 exponents

WK 5 9.2.10, 9.1.3 exponential laws, scientific notations

WK 6 MINDSET 9.2.1, 9.3.2 geometric patterns, numeric patterns

WK 7 9.2.5 solving equations

WK 8 MINDSET 9.2.4 mathematical models

WK 9 MINDSET ; INTERNET; 9.5.1, 9.5.2, 9.5.7 graphs

WK 10 MINDSET 9.5.6 median / modus

Work Schedule Week


WK 1

WK 2 9.4.12 Pythagoras

WK 3 9.2.8 product of two binomials

WK 4

WK 5

WK 6 9.2.11 algebraic expressions

WK 7 9.5.8 dataWK 8

WK 9 & 10

Work Schedule Week


WK 1

WK 2 mindset 9.3.1, 9.3.9, 9.3.4 geometric 2D shapes and 3D objects

WK 3 9.3.3 straight line geometry

WK 4

WK 5

WK 6 ms excel 9.2.7 graphs

WK 7 ms excel 9.2.7 graphs

WK 8 9.2.6 equivalence

WK 9 MINDSET 9.1.5, 9.1.7 financial mathsWK 10 MINDSET 9.1.5, 9.1.7 financial maths

GRADE 9 TERM 3 COMPUTER SOFTWARE PLANNINGMaster Maths

revision

AF11

AG49

10.1.4.1, 10.3.2.1,10.3.2.2

1.8.5.1, 1.8.5.2

4.3.1.1

4.1.3.3, 4.1.2.7, 4.1.2.4 Perseptual (2.2.5, 2.2.6)

GRADE 9 TERM 1 COMPUTER SOFTWARE PLANNING

6.1.3 - 6.1.7

8.3.1.3, 8.3.2.2

CAMI exercises

4.4.4.1 - 4.4.4.5;

10.1.3.1, 10.1.4.1

revision

CAMI exercises

The Maths software programmes which are in the Khanya schools e.g. Cami and Master Maths, 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.If the Cami activity is too difficult;, click on 'previous exercise'. If too easy, click on the 'next exercise.'

AF01; AF06; AF09;AG03; AG05; AG06; AG07

AG05; AG06;AG07; 4.3.1.1, 4.3.1.2, 1.8.5.1

4.2.1.1

Master Maths

revision

AE03; AE04; AE05; AF02; AF03

CAMI exercises

4.4.6.1 - 4.4.6.4, 4.4.5.1, 4.5.1.1, 4.5.3.1

AG24; AG27; AG44; AG001

AG01; AG04; AG20;AG60

Master Maths

GRADE 9 TERM 2 COMPUTER SOFTWARE PLANNING

7.1.1. - 7.1.3

AG10; AG12; AG14; AG16; 9.2.8 factorisation of algebraic expressions

AG10; AG12; AG14; AG16; AG24; AG27;AG44

AG60;AG002consolidation

assessment

4.2.1.1 - 4.2.1.3

AG18; AG19AG17

AF04; AF05; AF07

8.3.5.1; 8.3.5.2 power point / mindset AG41

Cami Perceptual 4.1.2.7, 4.1.2.4, 4.1.2.9, 4.1.1.4

9.3.6, 9.3.2

2.8.1.7 - 2.8.2.72.8.1.1 - 2.8.1.6

6.1.3 - 6.1.7

transformations

AG22; AG23; AG25; AG26; AG29; AG30; AG3`

AG22; AG23; AG25; AG26; AG29; AG30; AG3`

AF10

Work Schedule Week


WK 1

WK 2power point /

internet9.4.3, 9.4.4, 9.1.5,

9.1.7 2D and 3D

WK 3 ms exce. 9.1.6ratio, rate and

proportion

WK 4 9.4.2, 9.1.2time, distance, speed

WK 5 9.4.2ratio and rate

WK 6 mindset 9.5.10 probabilityWK 7

WK 8 en 9

GRADE 9 TERM 4 COMPUTER SOFTWARE PLANNINGMaster MathsCAMI exercises

hersieningAG11; AG13; AG15; AG17;

AG18; AG19Cami Perceptual ; 8.1.2.1; 8.1.4.1

AF29; AF30; AF31; AF32; AF33

AG01; AG20

AF33

2.6.4 - 2.6.6

9.7.1, 9.7.2, 9.7.3

AG57; AG58;CTACTA

9.7.1, 9.7.2, 9.7.3

10.2.1.4

senior - edulis

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