senior - western cape · let learners e.g. calculate a multiple calculation on their calculators...

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Mathematics

GRADE 7 TEACHER’S GUIDE 1 / 72

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

ASSESSMENT STANDARD 7.1.2 Describes and illustrates the historical and cultural development of numbers (e.g. integers, common fractions).

TERMINOLOGY None

RESOURCES Gr 7 Text books; WCED Illustrative examples; Calculator; Poster; Internet Web sites

INTEGRATION Languages, Social Sciences

TEACHING TIPS The historical and cultural development of numbers is a vast and complex issue, and only

fragments of it can possibly be addressed at school. The ideas of integers and fractions occurred independently in several different parts of the world (including China, Egypt and Mesopotamia) more than 2300 years ago.

Teach only integers and common fractions of two different number systems e.g. Mayan, Chinese etc. (Do a different number system than what they did in the Intermediate Phase).

Don’t spend too much time on this.

EXAMPLES 1. Use the table of Egyptian numerals to write

down the following number in modern numerals.


2. Very long ago the Egyptians depicted fractions in the following way:

Use the information above as well as the table in example 1 to write the following

Egyptian numbers as improper fractions:

a)

b)

CONSOLIDATION / HOMEWORK Examples for homework

ASSESSMENT Informal: class work

TERM 1 - WEEK 2

ASSESSMENT STANDARD 7.1.9 Uses a range of techniques to perform calculations including: • using the commutative, associative and distributive properties with positive rational numbers and

zero; • using a calculator.

7.1.11 Recognises, describes and uses: • algorithms for finding equivalent fractions; • the commutative, associative and distributive properties with positive rational numbers and zero

(the expectation is that learners should be able to use these properties and not necessarily know the names of the properties).

TERMINOLOGY Order of calculation; Commutative property; Associative properties; Distributive properties


INTEGRATION -

TEACHING TIPS OPERATION TECHNIQUES

By the time they get to Grade 7, learners have been using the commutative, associative and distributive properties for many years, although they may not be aware of it. Assessment standards 7.1.9 and 7.1.11 require that learners become aware of these properties of operations and how they use them (albeit “automatically”) when they perform


calculations, so that they can learn to apply the properties to formulas (algebraic expressions) from Grade 8 onwards. Make sure that learners know and can apply the order of operations correctly.

Let learners e.g. calculate a multiple calculation on their calculators (e.g. 7 + 8 x 4 – 11). Ask for all the answers. There will be mainly two answers, i.e. 28 and 49. Discuss why they have different answers. This can lead up to the rules for order of calculation.

First teach commutative properties, then associative and then distributive. The one leads up to the other.

Don’t force these properties, only use specific properties when it helps to make the calculation easier e.g. associative property: 59 + 37 + 21 (by grouping the 59 and 21, you can simplify the calculation).

There are certain conventions in Mathematics (ways to do things that we have agreed upon – you could almost compare it to spelling rules) which are VERY important. We call it the ‘order of operations’.

The following are the rules for the order of operations:

- When there are brackets in the mathematics problem, you always have to first simplify that which is placed inside brackets.

- Powers have the highest priority. - When the problem only involves division and/or multiplication, you have to

proceed from left to right. - When you have to do a problem that only involves addition and/or

subtraction - with no brackets - you have to proceed from left to right. - In case of a calculation problem that involves “something of everything”, and

there are no brackets, always first tackle the parts that require powers, division and multiplication.

Teachers often use the mnemonic ‘BODMAS’ to teach learners the order of operations, but note division does not necessarily come before multiplication. If both operations are in the calculation the ‘left to right’ rule applies.

USING THE CALCULATOR This element of assessment standard 7.1.9 means that learners should sometimes use

calculators to do calculations, and they that should know how to use a calculator correctly and how to guard against and check for mistakes. It also means that teachers should exercise sound judgment in their decisions to use a calculator or not, and specifically that they should not allow the availability of calculators to sabotage the development of learners’ number skills and knowledge of basic number facts.

Make sure learners know the basic functions on the calculator.

Show learners specific calculations when doing the different concepts, e.g. cubes and squares; fractions (scientific); decimals.

Even though learners have a calculator, they still have to decide how they will obtain their answer. With any sum, it is important that they decide which number(s) and which operation(s) they will key in, AND IN WHICH SEQUENCE. They must think clearly about the problem and how they will calculate the answer before pressing the keys. They must also concentrate so that no errors occur when keying in the numbers. They must also look critically at the answer; by either estimating or working backwards.

It is important to note that different calculators have different keys, thus when doing calculations they must be able to correctly use the keys on their calculator!! Always remember: ‘A calculator is merely there to assist you in making your mathematics life easier.’ It would also help if they familiarise themselves with the instruction manual (normally in book form or on the back of the packaging) of the calculator.


EXAMPLES USING THE COMMUTATIVE, ASSOCIATIVE AND DISTRIBUTIVE PROPERTIES Questions 1(b) and 2(b) below provide learners with opportunities to demonstrate awareness of using the properties of operations in doing calculations. Questions 3 and 4 are different ways of assessing learners’ awareness of properties of operations.

Learners who do not use the distributive property to replace the expression in question 4 by 2,7 × (3,4 + 5,2 − 1,9 + 6,7), then calculate 3,4 + 5,2 − 1,9 + 6,7 and finally 2,7 × 13,4, clearly have little or no awareness of the distributive property and its use.

Some learners may use the distributive property for question 4 yet do not give good responses to questions 3(a), (e) and (f), and 1(b). By doing so, such learners demonstrate some, but not strong awareness of the properties of operations.

(Use smaller numbers if the level of examples is too high.) 1. a) Calculate 7 × 465 without a calculator.

b) Describe your method to calculate 7 × 465 in detail.

2. a) Calculate 3235 + 5341 without a calculator.

b) Describe your method to calculate 3235 + 5341 in detail.

3. Will any of the following sets of calculations produce the same answers? Answer this question without doing calculations in any way.

a) 23 × 348 + 23 × 2674 + 23 × 6129 + 23 × 2357 + 23 × 2738 + 23 × 107

b) 2 389 + 5623 + 1388 + 9 09 + 2 355 + 1 898

c) 2 389 + 5 623 − 1 388 + 909 + 2 355 − 1 898

d) 2389 + 5623 + 1388 − 909 − 2355 + 1898

e) 23 × 348 + 2674 + 6129 + 2357 + 2738 + 107

f) 23 × (348 + 2674 + 6129 + 2357 + 2738 + 107)

4. Calculate 2,7 × 3,4 + 2,7 × 5,2 – 2,7 × 1,9 + 2,7 × 6,7 without using a calculator. Show all your workings.

5. Do each of the following calculations without using a calculator:

a) 23 × 3,47

b) 34 087 + 15 879 + 10 080

c) 17,26 − 9,67

d) 15 × 60

e) 9 × 25

f) 9 × 8

g) R3,40 + R5,90

6. For which of the calculations in question 5 would you normally use a calculator?

CONSOLIDATION/HOMEWORK Examples for homework



TERM 1 - WEEK 3

ASSESSMENT STANDARD 7.1.8 Performs mental calculations involving squares of natural numbers to at least 10² and cubes of natural numbers to at least 5³.

7.1.3 Recognises, classifies and represents the following numbers in order to describe and compare them: • factors including prime factors of 3-digit whole numbers; 7.1.7 Estimates and calculates by selecting and using operations appropriate to solving problems that involve: • exponents

TERMINOLOGY Exponential form; exponents, squares, cubes, square root, cube root, multiple operations, factors, prime factors; base and power of exponents

RESOURCES Gr 7 Text books; WCED Illustrative examples; Calculator; Poster; Internet Web sites; Number board; 100 block

INTEGRATION Technology

TEACHING TIPS CUBES AND SQUARES

When working with cubes and squares in mental maths, make sure that you only work up to 102 and 53.

When doing calculations, you can use much bigger numbers.

The following two number sequences are at stake here:

Squares: 1 4 9 16 25 36 49 64 81 100

Cubes: 1 8 27 64 125

As a learning activity, learners may be asked to investigate the differences between consecutive terms in these sequences, and to look for patterns in these “sequences of differences”. In fact, the sequences of differences may be investigated in exactly the same way, i.e. by forming their sequences of differences.

Learners should know the squares of at least the first ten natural numbers by heart (1, 4, 9, 16, 25 etc up to 100), and preferably the squares of the natural numbers up to 25. This is not difficult to learn. The squares of the natural numbers up to 10 form part of the multiplication tables that learners should know already, as early as in grade 4.

Similarly, learners should know the cubes of at least the first five natural numbers (but preferably more) by heart.


FACTORS Introduce factors in the context of the times tables.

Factors are always number pairs that make a number when multiplied.

e.g. 72 = 1 x 72 72 = 2 x 36 72 = 3 x 24 72 = 4 x 18 72 = 6 x 12 72 = 8 x 9 i.e. the factors of 72 are 1; 2; 3; 4; 6; 8; 9; 12; 18; 24; 36; 72

PRIME FACTORS Remember that 1 is not a prime number.

The prime factors of any number can be determined by completing a factor tree:

e.g. 72 72

2 x 36 9 x 8

2 x 18 3 x 3 2 x 4

2 x 9 2 x 2 3 x 3 Therefore 72 as a product of its prime factors is 2 x 2 x 2 x 3 x 3

The prime factors can also be determined in the form of a ladder.

Start dividing by the smallest prime factor.

e.g.

Therefore 72 as a product of its prime factors is 2 x 2 x 2 x 3 x 3

EXPONENTS Write a number as a product of its prime numbers. (e.g. 8 = 2 x 2 x 2)

- This can be written in a shorter way. (e.g. 23) - The shorter form is called an exponent. - We read it as two to the power of three.

The number at the bottom is called the base and the small number at the top is called the power of the number.

Let learners explore the difference between 23 and 32?

2 72

2 36

2 18

3 9

3 3

1


EXAMPLES Questions about factors and prime factors may be phrased at different language levels.

CUBES AND SQUARES 1. Express each of the following numbers in the exponential form: 25 125 256 2. Calculate each of the following, without using a calculator: 92 142 53

3. Write down the next three numbers in each case:

64 81 100 121 144 . . . . . . . . . 27 64 125 216 . . . . . . . . .

FACTORS AND PRIME FACTORS 4. Find three different numbers so that when you multiply them the answer will be 90. 5. Find three other different numbers (than those you found in question 4), so that when you

multiply them the answer will be 90. 6. Find all the different combinations of three numbers, none of them 1, that will give 90

when you multiply them. 7. Write down all the different factors of 90. 8. Write down all the different prime factors of 90. 9. Express 90 as a product of prime factors.


ASSESSMENT ASSESSMENT TASK 1 – Activity 1 e.g. tutorial / assignment (properties, factors, prime factors and exponents) or just do one activity at the end of week 5.

TERM 1 - WEEK 4 & 5

ASSESSMENT STANDARD 7.1.1 Counts forwards and backwards in the following ways: • in decimal intervals; • in integers for any intervals.

7.1.3 Recognises, classifies and represents the following numbers in order to describe and compare them: • integers; 7.1.7 Estimates and calculates by selecting and using operations appropriate to solving problems that involve: • multiple operations with integers; 7.1.9 Uses a range of techniques to perform calculations including: • using the commutative, associative and distributive properties with positive rational numbers and

zero; • using a calculator 7.1.10 Uses a range of strategies to check solutions and judges the reasonableness of solutions.


TERMINOLOGY Negative and positive integers, intervals

RESOURCES Gr 7 Text books; WCED Illustrative examples; Calculator; Poster; Internet Web sites; Flashcard; Number lines; Thermometer; Bank Statements; Newspaper article; Weather report

INTEGRATION EMS, Natural Sciences

TEACHING TIPS Introduce integers in the context of measurement (temperatures below zero – weather

map); sport (golf – below par); financial (bank statement); lift (below ground level).

The term “integers” refers to positive and negative whole numbers, i.e. the set

. . . . −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, . . .

What is at stake is that learners recognise negative numbers like −4 as numbers like ordinary (positive) whole numbers or fractions (i.e. that they do not simply regard −4 as an instruction, e.g. “subtract 4”.

Highlight the fact that a minus sign has two meanings at this stage namely “negative number” and “subtract a number”.

Make sure learners understand the concept by referring to the number line.

A vertical number line can be used because it links directly to the real life examples of the lift and thermometer.

Practical activity:

- Give 11 learners flashcards numbered from -5 to 5 (including 0). - Ask learners to arrange themselves according to the number line. - Learners will be confused initially. - Explain concept in this way.

Counting forwards and backwards in integers will enforce learners’ understanding of negative integer concept.

In grade 7 it is expected that all 4 operations be taught.

The phrase “multiple operations with integers” (rules) in assessment standard 7.1.7 is clearly not intended to mean more difficult calculations like −5 + (−4) × (−8) ÷ (−2). However, grade 7 learners should learn to do certain simple operations with positive and negative numbers, as illustrated in the examples below.

Although there are algorithms (rules) for the different operations, it is extremely important to start on the number line to understand the different rules.


The work schedule refers to finding methods (rules) for the different operations. The following are possible methods (rules):

Addition : 16 + 13 = 29 Subtraction: 29 – 16 = 13 -16 + 13 = -3 29 - -16 = 29 + 16 = 45

(subtracting a negative number is the same as adding a positive number)

16 + -13 = 3 -29 - -16 = -29 + 16 = -13

-16 + -13 = -29 -29 – 16 = -45

Multiplication: 8 x 7 = 56 Division: 72 ÷ 8 = 9

-8 x 7 = -56 -72 ÷ 8 = -9

8 x -7 = -56 72 ÷ -8 = -9

-8 x -7 = 56 -72 ÷ -8 = 9

Assessment standard 7.1.7 is quite clear about the level of independence that learners are required to demonstrate in solving problems: they have to be able to decide for themselves which operations are required to solve a specific problem

EXAMPLES 1. The temperature at 19:00 was 5°C. Between 19:00 and 24:00 the temperature drops by 8

degrees. What is the temperature at midnight?

2. Write the next five numbers: 10; 7; 4; 1; . . . . . . . . . . . . . . .

3. Write down the number that will satisfy the number sentence 10 + ? = 3

4. How much is 20 + −8?

5. What is the difference between 4 and −4?

6. During a certain 24-hour period, the highest and lowest temperatures in a certain town are 8°C and -9°C. What is the difference between the highest and lowest temperatures?

7. Calculate 8 × -10 and -8 × 10.

CONSOLIDATION / HOMEWORK Examples for homework.

ASSESSMENT Assessment Task 1 – Activity 2 e.g. : Tutorial (negative integers) or Activity 1+2


TERM 1 - WEEK 6 & 7

ASSESSMENT STANDARD 7.1.1 Counts forwards and backwards in the following ways: • in decimal intervals; • in integers for any intervals.

7.2.1 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. 7.2.2 Describes, explains and justifies observed relationships or rules in own words

TERMINOLOGY Numeric and geometric patterns, input and output

RESOURCES Gr 7 Text books; WCED Illustrative examples; Calculator; Poster; Internet Web sites; Concrete Material (e.g. counters, sticks); 2-D shapes; 3-D objects; Number lines; Beads; Hundred block

INTEGRATION Technology, Arts & Culture, Natural Sciences

TEACHING TIPS Start teaching patterns in context of environment (petals of flowers); cultural (jewels, huts);

architecture (paving, tiling, and carpets); design (material, graffiti).

Then patterns with a constant difference e.g. 2, 4, 6, 8, 10, ..... (constant difference is 2).

Then patterns with a constant ratio e.g. 2, 4, 8, 16, 32, ..... (constant ratio is 2)

There are also other number patterns For example:

Starting no. Second no. Third no. Fourth no.

6 → 15 → 33 → 69

Starting no. Second no. Third no. Fourth no. Fifth no. Sixth no.

1 → 2 → 4 → 7 → 11 → 16

A number pattern can be represented in a table. This is useful to develop the idea of numbering the terms. (See no. 3 and 4 below) (7.2.2)

Multiply the

number by 2 and

add 3

Multiply the

number by 2 and

add 3

Multiply the

number by 2 and

add 3

add 1 add 2 add 3 add 4 add 5


EXAMPLES 1. Write down the next numbers in each of the following cases: In each case describe in

words how you calculated the numbers.

a) 100 77 54 31 . . . . . . . . .

b) 2 6 18 54 162 . . . . . . . . .

c) 1 4 9 16 25 36 . . . . . . . . .

d) 2 5 10 17 26 37 . . . . . . . . .

2. Make number patterns as described below. In each case write down the first eight numbers of your pattern. You may start with any number of your own choice in each case.

a) Make a number pattern by adding the same number each time.

b) Make a number pattern by subtracting the same number each time.

c) Make a number pattern by multiplying with the same number each time.

d) Make a number pattern that cannot be continued by adding or subtracting the same number each time, or by multiplying or dividing by the same number.

3. a) How many matches are there in the first figure below?

b) How many matches are there in the second figure?

c) How many matches are there in the third figure?

d) How many matches are there in the fourth figure?

4. Complete the following table, for the situation described in question 3.

Figure number 1 2 3 4 5 6 7

Number of

matches




TERM 1 - WEEK 8 & 9 &10

ASSESSMENT STANDARD 7.1.8 Performs mental calculations involving squares of natural numbers to at least 10² and cubes of natural numbers to at least 5³.

7.3.1 Recognises, visualises and names geometric figures and solids in natural and cultural forms and geometric settings, including those previously dealt with as well as focusing on: • similarities and differences between different polyhedra; • similarities and differences between all quadrilaterals including kites and trapeziums.

7.3.2 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: • faces, vertices and edges; • sides and angles of polygons (with focus on, but not limited to, triangles and quadrilaterals); • parallel and perpendicular sides.

7.3.4.

Uses a pair of compasses, ruler and protractor to accurately construct geometric figures for investigation of own property and design of nets (7.3.1, 7.3.2., 7.3.4., 7.3.5. to be clustered)

7.3.5 Designs and uses nets to make models of geometric solids studied up to and including this grade

7.3.8 Recognises and describes the properties of similar and congruent figures and the difference between them (Cluster with 7.3.2.)

TERMINOLOGY Polygon, polyhedra, quadrilaterals, kite, trapezium, parallel lines, perpendicular lines, faces, vertices, edges, similarity, congruency

RESOURCES Gr 7 Text books; WCED Illustrative examples; Calculator; Poster; Internet Web sites; Geo-board; 2-D shapes; 2-D objects

INTEGRATION Technology, Arts & Culture

TEACHING TIPS Start by teaching how to measure an angle with a protractor. (Types of angles plus further

practise with the protractor is in term 4, week 1)

In the sketch (on the next page) an angle (in bold), measured by means of a protractor is indicated. Ensure that the centre of the protractor fits exactly onto the vertex. Furthermore, ensure that the zero line of the protractor fits exactly on top of the one leg of the angle. Now read off through which value on the protractor the other leg goes.


Start teaching 2-D shapes and 3-D objects in context of different containers (cereal boxes); cultural (beads – different shapes); architecture (paving, tiling, carpets); design (material, graffiti).

In the Senior Phase 2-D shapes are referred to as geometric figures and 3-D objects are referred to as geometric solids.

In Grade 7 we want learners to combine these skills. They must (a) use accurate construction techniques (7.3.4) to draw shapes in order to (b) make nets (7.3.5) and from those nets to make objects (the range of these objects is listed in 7.3.1), and (c) to investigate the properties and relationships (the particular properties are listed in 7.3.2).

INVESTIGATING PROPERTIES Teach properties of shapes (2-D) and objects (3-D) through investigation.

Work with concrete objects during these investigations.

Use a geo-board and elastic bands to construct different polygons (e.g. triangles, squares, rectangles, kites, hexagons, pentagons, etc.)

Dotted paper can also be used instead of the geo-board.

Construct frames of 3-D objects by using jellytots and toothpicks or straws and prestik to investigate properties.

Make sure that learners know properties by heart to be able to apply it in different situations.

Make sure that learners know the symbols used to indicate properties on sketches. (>> on a sketch means parallel lines; in a sketch means sides of same length; in the corner of an angle to indicate a right angle).

CONSTRUCT 2-D SHAPES (GEOMETRIC FIGURES) To construct means to make accurate drawings of 2-D shapes or 3-D objects.

NETS Make sure learners can measure and draw right angles before constructing nets. The 90°

drawing triangle can also be used to construct nets.

You will recognise that learners should already have a sense of what it means to make a net (Grade 5: cutting open boxes), to use a net (Grade 6: nets provided by the teacher), and even to draw the shapes that make up a net (Grade 6: drawing shapes on grid paper).

Remember to add extra flaps to the sides of nets for pasting sides together.

MAKE Bring everyday objects to the classroom that learners will use during an investigation. By

making a model of the object, they will acquire geometric skills (using pairs of compasses and protractors) and understand the properties and interrelatedness of these properties.


SIMILARITY AND CONGRUENCY Ensure that learners can distinguish between concepts of similarity and congruency.

Similarity means 2-D shapes and 3-D objects of the same shape, but different sizes. (a 10c-coin and a 50c-coin)

It is important that for example, not all rectangles are similar – the sides must be in proportion:

If one rectangle has dimensions of 3 by 5 then both side lengths must be multiplied by the same number (fraction or integer) to create a similar rectangle.

Congruency means 2-D shapes and 3-D objects of the same shape and sizes. (two 10c-coins) e.g. Look at the following 3 triangles:

Δ ABC is congruent to Δ DEF (same shape and size)

Δ ABC;Δ DEF and ΔXYZ are similar (same shape; different size, in proportion)

Give learners grid paper with one 2-D shape drawn in. Ask learners to draw similar and congruent shapes.

EXAMPLES Working as a member of a team:

1. Using a pair of compasses, a protractor and a ruler: design and make the nets for the following objects:

Prisms with different bases (e.g. equilateral triangle; square; regular pentagon; regular hexagon; regular octagon)

Pyramids with different bases (e.g. equilateral triangle; square; regular pentagon; regular hexagon; regular octagon)

2. Make each of the objects from the nets you have designed.

B C

A

E

F

D X

Y

Z


3. Draw and complete the following tables:

PRISMS

Shape of the base

Number of faces

Number of edges

Number of vertices

Shape of surface

Equilateral triangle

Square

Regular pentagon

Regular hexagon

Regular octagon

PYRAMIDS

Shape of the base

Number of faces

Number of edges

Number of vertices

Shape of surface

Equilateral triangle

Square

Regular pentagon

Regular hexagon

Regular octagon

4. Study the table for the prisms and describe, in words:

a) A general relationship (or rule) that relates the number of sides in the base with the number of faces of the prism.

b) A general relationship (or rule) that relates the number of sides in the base with the number of edges of the prism.

c) A general relationship (or rule) that relates the number of sides in the base with the number of vertices of the prism.

d) A general relationship (or rule) that relates the number of sides in the base with the number of faces of the pyramid.

e) A general relationship (or rule) that relates the number of sides in the base with the number of edges of the pyramid.

f) A general relationship (or rule) that relates the number of sides in the base with the number of vertices of the pyramid.

5. How are the relationships you determined and described for the prisms and pyramids the same?

6. How are the relationships you determined and described for the prisms and pyramids different?


ASSESSMENT ASSESSMENT TASK 2 : e.g. Project (construct 3-D objects from nets – classify according to properties) Start project at the beginning of week 8 and give rubric beforehand – must be done in class

ASSESSMENT TASK 3 : TEST (WHOLE TERM’S WORK)


TERM 2

TERM 2 - WEEK 1, 2 & 3

ASSESSMENT STANDARD 7.2.3 Represent 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. 7.2.4 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). 7.2.5 Solves or completes number sentences by inspection or by trial-and-improvement, checking the solutions by substitution (e.g. 2 x - 8 = 4). 7.2.6 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. 7.2.7 Describes a situation by interpreting a graph of the situation, or draws a graph from a description of a situation (e.g. height of a roller-coaster car over time; the speed of a racing car going around a track). TERMINOLOGY Variables, flow diagram


INTEGRATION Technology, Natural Sciences

TEACHING TIPS Assessment standards 7.2.3 refers to “relationships between variables”. Learners can

only engage with relationships between variables in a meaningful and conceptual way if they know about variables.

To achieve assessment standards 7.2.2, 7.2.3 and 7.2.4 learners first of all have to distinguish between constant quantities (constants) and variable quantities (variables), as they occur in both purely mathematical situations (e.g. number patterns) and in real life situations. It is important to stress that a variable is a number and not a letter.


Introduce relationship between variables in the following contexts:

- The number of hours in a day is a constant (24). - The number of hours of daylight in a day is a variable (it varies from day to day). - The number of sides in pentagons is a constant (different pentagons do not have different

numbers of sides: each pentagon has 5 sides). - The perimeter of pentagons is a variable (different pentagons may have different

perimeters).

MATHEMATICAL MODELS Assessment standard 7.2.4 requires that learners are able to make verbal descriptions, in

their own words, of relationships between variables.

A table of values that illustrates the relationship between two variables (like the table for areas of squares, or the tables given in questions 4 and 5) is a mathematical description (also called a mathematical model) of a situation. Similarly, a rule expressed in words or otherwise, that describes how values of the one variable may be calculated for given values of the other variable, like the descriptions learners have to produce for question 3, is called a mathematical model of the situation. Hence assessment standards 7.2.4 actually require the same knowledge and skills as assessment standard 7.2.2. The only difference is that assessment standard 7.2.4 specifically requires that description of relationships between variables in real-life situations, like those in questions 3 and 6 below.

Question 6 in the examples also illustrates how one variable (the total fee in this case), may depend on more than one independent variable. At the same time this question illustrates one possible way of relating mathematical work to responsibility towards the environment.

Flow diagrams provide a non-verbal way to represent some relationships between variables. For example, the relationship in question 4 may be represented by the flow diagram below.

RELATIONSHIPS BETWEEN VARIABLES Equally important, learners need to understand that the values of one variable may depend

on the values of another variable. This is what is meant by a “relationship between variables”.

For example, the area of a square (different squares may have different areas) depends on the length of the sides, as illustrated in the table below.

+

Length of side in cm 2 3 4 5 6 7

Area of square in cm2 4 9 16 25 36 49

Flow diagrams provide a non-verbal way to represent some relationships between variables. For example, the relationship in question 4 may be represented by the flow diagram below.

1

2

3

4

5

4

7

10

….

….

x 3 + 1


A flow diagram like this shows the information that can be represented in a table, as well as the rule by which the values of the dependent variable (the “output variable” in the flow diagram) may be calculated.

One can also have a flow diagram that only represents the rule for calculating values of the dependent variable:

EXAMPLES 1. In each case state whether the quantity is a variable or a constant.

a) The number of days in a week.

b) The height of a tree.

c) The mass of a dog.

d) The number of legs that a spider has.

e) The cost of petrol.

f) The number of players in a soccer team.

g) The wages of domestic workers.

2. What information do you need in order to be able to answer the following questions?

a) How much does it cost to make a telephone call?

b) What is the area of a rectangle?

c) How tall is a tree?

3. In a certain game reserve, the entrance fee for a bus is R200 plus R30 per passenger. You have to explain to a new gate clerk how he should calculate the total entrance fee for any bus and its passengers. Write a clear short explanation for the new clerk.

4. Consider the five match-figures below.

a) Make a drawing of the next figure.

b) How many matches are needed to make a figure with 2 squares?

c) How many matches are needed to make a figure with 5 squares?

d) How many matches are needed to make a figure with 8 squares?

e) Complete the table below, to describe this situation.

Number of squares 1 2 3 4 5 6 7 8 9

Number of matches 4 7 10

f) Calculate how many matches are needed to make such a figure with 15 squares.

g) Describe in words how you did the calculation.

h) Describe a different way in which one may calculate how many matches are needed to make such a figure with 15 squares.

i) Make a flow diagram to show the same information than the table you completed in (e).

x 3 + 1Number of squares Number of matches


5. One may also make match-figures that consist of hexagons instead of squares. Here is an incomplete table that shows how many matches are needed for figures with different numbers of hexagons.

Number of hexagons 1 2 3 4 5 6 7 8 9

Number of matches 6 16 26 41

You have to find out how the number of matches needed for any number of hexagons may be calculated. To find this out, you may try the following rules to see if any of them produces the correct results for the numbers given above, and you may try some other similar rules.

Rule 1: Multiply the number of hexagons by 6 and add 3 to the answer.

Rule 2: Multiply the number of hexagons by itself and add 12 to the answer.



6. Because of the damage caused to roads and to the environment by heavy vehicles, a certain game reserve charges higher entrance fees for vehicles with more than 4 wheels. The following information is displayed on a signboard at the entrance gate:

a) Complete the following table, so that it can also be displayed at the entrance gate so that people do not need to do calculations to find out how much they have to pay.

Number of wheels

Number of passengers 4 6 8 10 12 14 16 18 20 1 2 3 4 5 6 7 8 9

10 20 30 40 50 60 70 80

b) Describe in words how one may calculate the total fee for any number of wheels and any number of passengers.

ENTRANCE FEES FOR BUSES R200 per bus plus

R25 for every extra wheel, for buses with more than 4 wheels. AND

R30 per passenger.


TEACHING TIPS SOLVES OR COMPLETE NUMBER SENTENCES (AS 7.2.5)

The idea is not that learners should learn any special methods of solving equations. They should simply try out different values of the variable until they find one that makes the equation true.

DIFFERENT DESCRIPTIONS of the SAME RELATIONSHIP between two variables: (AS 7.2.6)

A VERBAL (IN WORDS): The cost of hiring a car from a certain agency is R240 per day, plus R2,30 for every km travelled.

B FLOW DIAGRAMS:

OR

C TABLE:

Number of km 100 200 300 400 500

Total cost of rental 470 700 930 1160 1390

D FORMULA: With words: Rental cost = 2,30 × number of kilometres + 240

With variables: C = 2,30 × n + 240, where n represents the number of kilometres and C the cost

The part on the right hand side of the equal sign in the above is called an expression.

A specific description is sometimes better than another way of portraying the same information.

EQUATIONS (7.2.6)

An equation is used when a value of the dependent (output) variable is given, but the corresponding value of the independent (input) variable is unknown, for example:

2,30 × number of km + 240 = 815 or 2,30 × n + 240 = 815

Solving this equation means to look for the input value that will give an output value of R815.

An equation like this may also be represented by means of a flow diagram:

100

200

300

400

500

470

700

930

1160

1390

x 2,30 + 240

Number of kilometres

Total cost of rental

x 2,30 + 240

x 2,30 + 240? R815


GRAPHS (7.2.7)

DESCRIBE A SITUATION BY INTERPRETING OR DRAWING A GRAPH OF A SITUATION - This assessment standard requires that learners represent a relationship

between two variables by points on a coordinate graph. - A good way to introduce coordinate graphs is to start with bar graphs, which

learners already know, as illustrated in the examples below.

HINTS FOR INTERPRETING A GRAPH OF A SITUATION: - Read the heading. - Read what is represented along the horizontal axis. - Read what is represented along the vertical axis. - Look carefully at the units at the scale of the horizontal and vertical axis.

HINTS FOR DRAWING A GRAPH OF A SITUATION: - Read the description carefully until you understand what the situation is. - Choose one variable, like time, for the horizontal axis. - Choose another variable, like height, distance or volume for the vertical axis. - Draw the graph using a suitable scale on the axis. - Give a heading and label the axes. - Check that the graph describes the situation correctly.

EXAMPLES 1. 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

2. The length of a young bean stalk after various periods of time is given in the table below. The length of the bean stalk was measured at 06:00 each morning.

Day 1 2 3 4 5 6 7

Length in cm 34 37 42 48 57 63 70

a) Draw a bar graph to show the length of the bean stalk on the various days. Use a pencil, because you will rub out part of your graph later.

b) Make a dot at the top of each bar with a pen.

c) Rub out the bars that you have drawn in pencil.

3. A graph like the one that you have drawn in question 2 is called a point graph, or co-ordinate graph. The growth data for another bean stalk is given in the table below.

Day 1 2 3 4 5 6 7

Length in cm 54 57 61 64 67 69 72

a) Draw a point graph of the above data.

b) Which of the two bean stalks is growing the fastest?


4. Data on the growth of two bean stalks is given below.

Day 1 2 3 4 5 6 7 8

Length in stalk A in cm 55 57 60 62 64 67 70 72

Length in stalk B in cm 13 17 22 26 30 33 37 41

a) Draw point graphs of the above data, on the same sheet (“on top of each other”).

b) When do you think bean stalk B will catch up with bean stalk A?


ASSESSMENT ASSESSMENT TASK 4 e.g. Investigation (patterns in numbers – find equivalence of same description)

TERM 2 - WEEK 4, 5 & 6

ASSESSMENT STANDARD 7.1.3 Recognises, classifies and represents the following numbers in order to describe and compare them: • decimals (to at least three decimal places), fractions and percentages; 7.1.7 Estimates and calculates by selecting and using operations appropriate to solving problems that involve: • addition, subtraction and multiplication of common fractions; 7.1.11 Recognises, describes and uses: • algorithms for finding equivalent fractions

TERMINOLOGY Algorithms

RESOURCES Gr 7 Text books; WCED Illustrative examples; Calculator; Poster; Internet Web sites; Number rods; Fraction blocks; Dienes blocks; Fraction wall; concrete material; Number line; Counters

INTEGRATION Technology, EMS, Natural Science

TEACHING TIPS Introduce fractions in the context of real life situations focussing on equal sharing of

concrete material (e.g. share a bar of chocolate between five friends).

You need to understand Mathematics and fractions well to make the following decisions.

Would you prefer to have 65 of a chocolate or

87 of the same chocolate?

Which one is the best buy: 221 kg grape fruit @ R9,95 or 1

21 kg grape fruit @ R6,30?


Use concrete material in classroom to strengthen every concept taught (dienes blocks; fraction wall; number line; number rods; fraction blocks).

Do revision of addition and subtraction.

Methodology - Multiplication:

343

x 272

= 4

15 x

716

(convert to improper fractions and simplify and/or cross-simplify if possible)

= 760

(multiply the two denominators (15 x 4) and multiply the two numerators (1 x 7))

= 8 74

(convert from improper fraction to mixed fraction)

EXAMPLES ALGORITHMS FOR FINDING EQUIVALENT FRACTIONS Traditionally, the highly useful algorithm “multiply the numerator and the denominator by the same number”, to produce equivalent fractions, was taught in Grade 5 or Grade 6.

1. Calculate each of the following.

a) 2013 +

2512

b) 54 -

83

c) 2013 x

83 +

2013 x

41 +

52 x

83 +

2013 x

83 +

2013 x

103

2. Arrange the following numbers from smallest to biggest. Indicate whether some of the numbers are actually equal.

159

43

83 0,75

2515

53

3. A man wishes to purchase a refrigerator costing R1480,00. He must pay 41 of the price in

cash.

a) How much must he pay in cash?

b) The balance must be paid over a period of 24 months, plus R25,00 interest per month. How much will his monthly payment be?

4. A pupil sleeps 83 of a day and spends

52 of a day at school. How much time is left for

other activities ?

5. The pupils in Valley Primary have to cover their exercise books with brown paper. The brown paper is sold at the school in rolls. To help the parents decide how many rolls they must buy, the school gives the following information:

A Grade 4 pupil needs about 32 of a roll to cover all his/her books.

A Grade 5 pupil needs about 43 of a roll to cover all his/her books.

4

1


A Grade 6 pupil needs about 1 roll to cover all his/her books.

A Grade 7 pupil needs 121 rolls to cover all his/her books.

Mrs Daniels has twins in Grade 4, one child in Grade 5 and one child in Grade 7. How many rolls must she buy?


ASSESSMENT ASSESSMENT TASK 5 : Test (fractions)

TERM 2 - WEEK 7 & 8

ASSESSMENT STANDARD 7.5.1. Poses questions relating to human rights, social, environmental, economic and political (own environment) in RSA

7.5.2. Selects appropriate sources for the collection of data (incl. peers, family, newspapers , books, magazines

7.5.3. Uses simple questionnaires (variety of responses) designs uses questionnaires in order to collect data (alone + group) to answer questions

7.5.4 Distinguishes between samples and populations, and suggests appropriate samples for investigation (including random samples)

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

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

7.5.9 Identifies the largest and smallest scores in a data set and determines the difference between them in order to determine the spread of the data (range).

TERMINOLOGY Grouped and ungrouped data, numerical data, mean, mode, median, central tendency, tally, stem-and-leaf display, sample, population

RESOURCES Data Handling in the GET Band (Jackie Scheiber & Meg Dixion – RADMASTE Centre) Gr 7 Text books; WCED Illustrative examples; Calculator; Poster; Internet Web sites; Newspapers, Magazines; Books; People (e.g. family, peers)

INTEGRATION Technology, Social Sciences, Natural Sciences


TEACHING TIPS Introduce data in the context of local issues regarding human rights and / or social issues.

Use the book ‘Data Handling in the GET Band’ for teaching tips plus examples.

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.

Pose a question AS 7.5.1 & 7.5.2

The problem that focuses the Data Cycle would more than likely be introduced during a class discussion and should as far as possible be current.

ILLUSTRATION After a teacher led discussion on the water crisis in the Western Cape (2005) the following task could be posed:

Working as a member of a group, propose a question that you and your group would like to investigate with regard to the current water crisis in the Western Cape. You should also determine where and how you will collect data to answer your question.

Some questions that learners may suggest include:

Question Data source Data collection method Are people in our

neighbourhood aware of the water restrictions?

A sample of people in the neighbourhood is interviewed. Questionnaire

How much water does a typical house in our neighbourhood

use?

A sample of the head of the household in the neighbourhood is asked to look at the

water account to determine how much water is used.

Questionnaire

How much water does a typical house in our neighbourhood

use?

Each learner in the class that lives in a home with a water meter records the

daily consumption for one week or more. Recording sheet

How do different household activities contribute to water

consumption?

Measure the volume of water used by different activities (e.g. bathing, flushing the toilet, washing hands, brushing teeth

etc.). A sample of persons in the

neighbourhood are interviewed on how many times they do each of the activities

per day.

Recording sheet (for measurements)

Questionnaire for the survey.

What is the trend in dam levels in the Western Cape over the

last twenty years?

Newspapers, books and other records in the library. Records provided by the

department of agriculture.

Recording sheet with tables to record dam levels for each

of the dams over the last twenty years.

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


Collect the data AS 7.5.3. & 7.5.4

Having established a question to be answered and identified the data source for the investigation, learners must now collect their data. In Grade 6 learners were supplied with the questionnaire by the educator. In Grade 7, by comparison, they are expected to be able to make their own questionnaires for questions with yes/no type answers. In Grade 7 the educator introduces questionnaires that have a variety of responses.

Illustration 1 gives an example of a task that could be posed to ask learners to develop a questionnaire. Illustration 2 shows a questionnaire with a variety of responses that the educator has developed and which the learners must administer.

ILLUSTRATION 1 Working as a member of a group, develop a questionnaire to determine whether or not the people in your neighbourhood are aware of the current water restrictions in Cape Town. Your questionnaire should determine appropriate information to enable you to decide which group of people: school children or adults know the most about the water restrictions.

Describe the sample of people that you think should be targeted by the questionnaire.

A possible response to this task could look as follows:

Please tick the correct box

Do you go to school? YES NO

Are you aware that water restrictions have been imposed in Cape Town? YES NO

Are you observing all of the water restrictions? YES NO

Do you think that the water restrictions are effective? YES NO

In identifying the sample of people to be interviewed for the task, learners should recognise that asking very young children would give a “NO” response to “Do you go to school?” and yet these children are not adults.

This implies that the sample should not include children who are still too young to go to school. Similarly it would also make sense that there would be no need to interview more than one person per household.


ILLUSTRATION 2 Use the following questionnaire to collect data for at least 10 different people in your community.

How many times a day do you take a bath?

How full do you fill the bath? (please tick)

Half full Full

How many times a day do you shower?

How many minutes do you shower for? (please tick)

0 – 2 2 – 4 4 - 6 > 6

How many times a day do you flush the toilet?

When you brush your teeth, do you:

(please tick one option only per row) N

ever

Som

etim

es

Alw

ays

Use a cup or glass of water to rinse your mouth?

Rinse your mouth by drinking directly from the tap?

Let the tap run while you are brushing your teeth?

Organise the data (AS 7.5.5; 7.5.6; 7.5.9)

- Having collected data the next step in the cycle is to organise the data. Organising data involves counting how many of each type there are etc. This is done by means of tallies, tables and stem-and-leaf plots in Grade 7.

- What you want to do with your data determines how you summarise it - this is done through so-called measures of central tendency and measures of dispersion for numerical data. In Grade 7 we consider the following measures: mean, median and mode. While mean, median and mode are useful measures of central tendency they do not tell us whether the data was all very similar or different. To give us a sense of the spread (dispersion) of the data we determine the range of data.

THERE ARE THREE MAIN TYPES OF MEASURES OF CENTRAL TENDENCY: A The MODE is the value that occurs most often. It is the MOST COMMON or MOST

POPULAR (MOST FASHIONABLE) value. (To help you remember, notice that MOst and MOde begin with the same two letters.) e.g. The mode of 1; 1; 1; 2; 2; 2; 3; is 2 Shops and manufacturers often use the mode. It helps them to decide what to order or make.


B The MEDIAN is the MIDDLE VALUE when all the values are placed in order of size (in numerical order. There are as many values below it as there are above it.

- If there is an ODD NUMBER OF VALUES in the set of data, there will be a middle value. e.g. The median of 1; 1; 1; 2; 2; 2; 3 is 2

- If there is an EVEN NUMBER OF VALUES in the set of data, there will be two middle values and the median is taken as half-way between them i.e. add the two middle values and then divide by 2. e.g. The median of 1 ; 1 ; 1; 2 ; 2 ; 3 lies half-way between 1 and 2.

The median = ==+

23

221 1 2

1 .

- By determining the middle of a set of marks of a class you can determine which learners fall in the top 50% of the class and which learners fall in the lower 50 % of the class.

C The MEAN is the average found by SHARING OUT EQUALLY the total of all the values. (To help you remember: When SHARING OUT some sweets, it would be MEAN not to give everyone the same amount!)

The mean = valuesofnumber

valuesthe all of total

e.g. Tebogo, Rory and Jennifer work in a restaurant. Tebogo gets R60,50 in tips, Rory gets R30,10 and Jennifer gets R50,85. At the end of each day they share out their tips. (They put them all together and then share them out equally.)

R 60,50

R 30,10 R 141,45 ÷ 3 = R47,15

R 50,85 This is the mean or average.

R 141,45

MEASURES OF DISPERSION A mode, median or mean gives you one data item that represents a set of data. It gives

you an impression of the centre of the data . How spread out the values are, gives you another impression of the data.

The RANGE is the simplest measure of spread. It is the difference between the largest and the smallest values in the data.

Range = largest value – smallest value e.g.

Look at these midday temperatures recorded in July of one year:

JULY

S M T W Th F S

8º 9º 7º 10º 6º

13º 13º 13º 13º 13º 11º 12º

14º 14º 15º 17º 16º 12º 11º

10º 8º 11 12º 16º 14º 17º

14º 18º 19º 25º

The average (mean) temperature is 13ºC. But the temperatures are spread from 6ºC to

25ºC. The RANGE is 25ºC – 6°C = 19ºC. This means that the coolest temperature was 6°C and the

warmest temperature was 25°C.


Represent data (AS 7.5.7)

The next stage in organising data is to represent it—a picture tells a thousand words. We use graphs to represent data and in Grade 7 learners are expected to be able to use the following graphs to do so:

- bar graphs and double bar graphs (these have already been dealt with in Grade 6) - histograms with given intervals; - pie charts; - line and broken-line graphs.

EXAMPLES 1. The illustrations are all based on the following table of data collected by a Lynette. She

visited 31 homes and at each home asked to speak to the person who deals with the family accounts. In the table below she has recorded the answers to the following questions (note she did not ask for the gender—she could see this):

- How old are you? - Please would you look at your municipal services account and tell me how

many kilolitres of water you used last month? - Are you aware of the water restrictions in Cape Town? - Lynette used three codes to record the responses of people:

Y = “yes I am aware” N = “no I am not aware” NI = “I am not interested in things like that” (or words to this effect)

House Gender Age Kilolitres Aware House Gender Age Kilolitres Aware

1 M 32 58.6 NI 17 m 35 55.3 Y

2 F 18 25.0 N 18 m 68 33.6 N

3 M 45 43.6 N 19 f 34 21.8 Y

4 M 24 17.6 N 20 m 43 7.2 N

5 F 45 51.8 NI 21 m 19 10.9 N

6 M 28 5.8 Y 22 m 28 43.4 Y

7 M 62 30.8 N 23 m 45 30.0 N

8 m 29 32.3 Y 24 m 31 6.2 NI

9 m 45 18.2 NI 25 m 28 42.8 Y

10 m 45 21.8 Y 26 m 45 12.3 NI

11 f 34 17.3 NI 27 m 32 50.2 NI

12 f 53 35.3 NI 28 f 66 23.3 Y

13 f 32 8.5 Y 29 f 68 38.3 NI

14 f 22 7.6 NI 30 m 72 13.1 NI

15 f 45 34.6 Y 31 m 44 58.5 NI

16 m 33 54.9 Y


a) Use the following intervals to make an appropriate table to summarise the consumption of water in kilolitres:

0 to 10 kl 10 to 20 kl 20 to 30 kl 30 to 40 kl 10 to 50 kl 50 to 60 kl

b) Complete the table that you have created using tallies.

c) Draw a histogram to illustrate the water usage.

d) Organise the data on ages using a stem-and-leaf display.

e) Calculate each of the following for the ages of the people who were interviewed.

- Mean - Median - Mode - The age of the youngest person, the age of the oldest person and the difference

between them (the range)

(This section will be revised again in term 4 as a project including the drawing and interpretation of graphs.)


ASSESSMENT Informal: class work + homework

TERM 2 - WEEK 9 & 10 Assessment Task 6 : Examination (2 Papers of 50 Marks each on Term 1 & Term 2)


TERM 3

TERM 3 - WEEK 1 & 2

ASSESSMENT STANDARD 7.1.1 Counts forwards and backwards in the following ways: • in decimal intervals; 7.1.3 Recognise, classifies and represents the following numbers in order to describe and compare them: • decimals (to at least three decimal places), fractions and percentages; 7.1.7 Estimates and calculates by selecting and using operations appropriate to solving problems that involve: • rounding off numbers to at least one decimal place; • addition, subtraction and multiplication of positive decimals to at least 2 decimal places; • division of positive decimals with at least 3 decimal places by whole numbers;

TERMINOLOGY None

RESOURCES Gr 7 Text books; WCED Illustrative examples; Calculator; Poster; Internet Web sites; Money; Money games; Number line; Pamphlet; Abacus; Menus; Classified ads; Newspapers; Magazines

INTEGRATION Technology, EMS, Natural Sciences

TEACHING TIPS Introduce decimals in the context of money and measurement.

RECOGNISE AND CLASSIFY DECIMALS Place value is very important when teaching decimals.

Use place value charts constantly to re-enforce concept.

The position of the decimal comma determines the value of the number.

The decimal comma is a fixed point. When you multiply or divide by multiples of 10, the digits move one, two, three, etc places left or right and not the decimal comma.

Also remind them that 0,1 = 0,10 = 0,100 and that it relates to equivalent fractions namely,

ROUNDING OFF OF DECIMALS

Write the number to be rounded off e.g. round off 3,245 to the nearest tenth.

Encircle the digit in the tenths column. 3,245

Look at the number directly to the right of the digit to be rounded off.

This number determines whether you round up or down. i.e. 3,2


The following instructions can be given:

Round off to the first decimal digit or to the nearest tenth

Round off to the second decimal digit or to the nearest hundredth

Round off to the third decimal digit or to the nearest thousandth

CALCULATIONS WITH DECIMALS Start addition and subtraction in notation columns if learners struggle with the concept.

Remember to use 0 as a place holder when adding and subtracting.

If calculations are done vertically, make sure the decimal commas are aligned.

After exploring the rules, ensure learners can apply the rules.

Ensure that the methods are consolidated properly before mixed operations are attempted.

EXAMPLES 1. Counting in 0,05’s. Complete the number line:

a) How many 0,05’s in one whole?

b) What common fraction is 0,05 therefore?

2. Start with the given number and do the operation in brackets at least 10 times. Then do the same pattern with corresponding common fractions. Check your answers.

8,4 → → → → → → → → → →

(– 0,3)

8104 → → → → → → → → → →

(–103 )

Sometimes we do not have to write the zero because it does not change the size of the number, e.g. 0,1 = 0,10 = 0,100 Other times, we have to write the zero, otherwise the number will change. E.g. 1; 10; 100

3. In each case say whether it is crucial to write the zero. Why? (Remember we do not want to change the number.)

a) 1,04

b) 3,480

c) 0,42

d) 2,055

e) 8,80

4. How thick do you think one sheet of paper is? Can you measure it with your ruler? Dumisani has a bright idea. He measures 100 sheets of paper. The stack is 14 mm thick.

a) Calculate how thick each sheet of paper is.

b) How thick will a document of 7 pages be?

c) If 245 copies of this document are printed and stacked on top of one another, how high will the stack be?


5. The mass of the air in a container (1 m long, 1 m wide and 1 m high) is 1,322 kg. What is the mass of the air in 5 such containers?

6. A car uses 105 l of fuel over a distance of 1 000 km. How many litres per 100 km on average?

7. A CD costs R129,99. How much will 12 of them cost?

8. The thickness of the paper of a book is 0,07 mm. How thick is the book if it has 368 pages?

9. The mass of 1 l is about 1,03 times as much as that of water. What is the mass of 1 l of milk if the mass of 1 l of water is about 995 g?

10. The diameter of a thick wire is 1,4 mm and the thickness of a cable is 14 mm. How many times thicker is the cable than the wire?

11. The mass of 1 l of fuel is about 0,679 kg.

a) What is the mass of a 200l drum of petrol if the mass of the drum itself is 18 kg?

b) The transport cost on a 200l drum of petrol (over a short distance) is R3,25/km. How much is the transport cost on such a drum of petrol over a distance of 72 km?

12. A train which transports coal, consists of 72 coaches which each carries a load of 17,25 tons. What is the mass of the coal which the train carries?

13. A farmer has a harvest of 437,6 tons. He can load 11,25 tons on his truck at a time. How many loads will he have to take to the depot?

14. I have a roll of string which is 72 m long. I cut off 6 pieces of 4,88 m each. What length of string is left over?

15. An athlete’s steps are 4,78 m long.

a) How many steps does he give in a 200 m race?

b) What distance will the athlete have covered if he gives 350 steps?

16. Grandma wants to do divide R100 between her 7 grandchildren. Lynette uses her calculator and finds that 100 ÷ 7 = 14,2857142858. What could grandma do with this answer?

17. Decide what would be the best buy:

PRODUCT 1: 6 Cans of cool-drink

Option 1: 6 tins at R20,00

Option 2: 1 tin at R3,35 each

PRODUCT 2: 12 Sweetcorn

Option 1: R6,95 for 4

Option 2: R10,50 for 6

PRODUCT 3: 6 kg apples

Option 1: 1,5kg at R4,95

Option 2: 2kg at R6,95




TERM 3 - WEEK 3

ASSESSMENT STANDARD 7.1.4 Recognises and uses equivalent forms of the rational numbers listed above, including: • common fractions;

• decimals; • percentages.

7.1.7 Estimates and calculates by selecting and using operations appropriate to solving problems that involve: • finding percentages

TERMINOLOGY Rational numbers, percentages

RESOURCES Gr 7 Text books; WCED Illustrative examples; Calculator; Poster; Internet Web sites; Newspaper article; Concrete material; Containers (2 % milk bottle)

INTEGRATION Technology, EMS, Natural Sciences

TEACHING TIPS Remember that % is not something like units of measurement. If you write % next to a

number it literally means that number divided by 100. That is 7% means 7 per hundred or 7 divided by hundred.

A key element of assessment standard 7.1.3 is that learners should recognise fractions as numbers. This means that learners should not interpret fractions simply as instructions

to divide, e.g. to simply regard the fraction 2013

as the instruction to divide 13 by 20, and not

attach any other meaning to it. The narrow interpretation is also reinforced if learners are taught to convert fractions to decimals by long division, without having a clear understanding of decimals and common fractions as two alternative notations for the same numbers.

When converting a common fraction to percentage it means writing it as a fraction with a denominator of a 100.

Let the learners complete the following to make clear the link between the different ways in which we can write a number:

a) 43 = 3 ÷ 4 = 0,75 =

10075 = 75%

b) 21 = ÷ = = = %

c) 8040 = ÷ = = = %

d) 6012 = ÷ = = = %


e) 10012 = ÷ = = = %

f) ...... = ÷ = = = 80%

One may use a calculator to divide the numerator by the denominator or do long division.

e.g. 72

= 0,2857 ≈ 0,286 (rounded off to the third decimal digit)

EXAMPLES 1. Rewrite the following sentence, but now use percentages instead of common fractions:

“About two fifths of the plants in a certain area are indigenous wild plants, about one tenth of the plants are exotic wild plants (they come from other countries), and the rest are cultivated plants (e.g. crops, orchards or vineyards planted by farmers for the production of food).”

2. Rewrite the sentence in question 1 again, but now use decimals instead of common fractions.

3. Express each of the following numbers as a fraction:

0,7 0,07 0,35

4. Express the following as a single number in the decimal notation. Show all your work.

1000

51007

103

++

5. Rewrite the following numbers, from the smallest to the biggest:

85

43

2011

6. Rewrite the following numbers, from the smallest to the biggest:

0,439 0,6 0,55 0,099

7. Which of the following numbers are equal?

3 fifths 6 tenths 15 3,5 159

53

8. Which of the following numbers are equal?

0,6 60% 351 3 fifths


ASSESSMENT Informal: classwork


TERM 3 - WEEK 4

ASSESSMENT STANDARD 7.1.5 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); 7.1.7 Estimates and calculates by selecting and using operations appropriate to solving problems that involve: • finding percentages

TERMINOLOGY Interest, loan, hire purchase, exchange rates

RESOURCES Gr 7 Text books; WCED Illustrative examples; Calculator; Poster; Internet Web sites Money; Money games; Number line; Pamphlet; Abacus; Menus; Classified ads; Newspapers; Magazines

INTEGRATION Technology, Natural Sciences, EMS

TEACHING TIPS PERCENTAGES AND FINANCE

Introduce concept of percentages in financial context or start with a real life situation (e.g. marks learners received for two tests – compare by converting to percentage).

Remember that a percentage is also a fraction with a denominator of a 100.

Let learners collect all material where percentage is used or indicated (e.g. newspaper articles, magazine articles, pamphlets, advertisements, food containers).

Study different material brought by learners (e.g. food containers) and describe the meaning of the % indicators on the container (e.g. 2% milk; 100% fruit juice).

Allow learners to use their calculators initially to do different conversions (fractions %) until they can do it with confidence.

The learner must be able to use the calculator with confidence when doing different problem solving techniques with percentages.

Do problem solving with decimals and percentages in financial context (e.g. profit and loss, budgets, accounts, loans, simple interest, hire purchase, exchange rates).

The material collected from a bank, newspaper, advertisements and pamphlets can be used as resources.

HIRE PURCHASE Most people buy luxury items on hire purchase. This means that a deposit is put down and

the rest of the article is then paid off through monthly instalments. The provider charges interest over the longer term.

One must look at the total purchase price of the article at the end of the hire purchase term in comparison with the cash purchase price.


Example: Let us look at an advertisement selling a music centre. Cash price :R 1 999 Hire purchase amount : deposit R200 Monthly R2 232 (93 ×24) R2 432

Additional amount to be paid: R2 432 – R1 999 R433

EXAMPLES PERCENTAGE: 1. Change William's marks to percentages:

Mathematics 16 out of 25

Science 19 out of 20

History 28 out of 40

Afrikaans 18 out of 30

English 45 out of 60

Geography 60 out of 70

Which subject is his best subject??

2. Discount of 15% was given on a television set that cost R2 499,50. What was the selling price?

3. The population of a country was 7 695 875. After five years the number has decreased with 20%. What was the population then?

4. The value of a certain car decreased by 15% the first year. If the price of a new car is R37 963, what will the value be at the end of the first year?

5. A dealer buys eggs @ 84c for 10 eggs and sells it for R1,05 for 10. Calculate his percentage of profit on the cost price.

6. A vegetable dealer buys oranges @ R4,50 a pocket. Some of the oranges went bad and he had to sell them for R4,05 a pocket. Calculate his percentage of loss?

7. A flash light costs R37,05, VAT included. What was the amount of VAT included in this price?

8. A stove, which costs R3 136,00, is sold for R2 800. Which percentage discount was given?

9. HEWITT PACKARD COMPUTER – FOR SALE

Compare the cash price with the hire purchase price payable over 2 years.


ASSESSMENT Assessment task 7 e.g. : Project/ Investigation on finance involving percentages and decimals

R 3 999

R184 x 24 months

Deposit R400


TERM 3 - WEEK 5 & 6

ASSESSMENT STANDARD 7.1.6 Solves problems that involve ratio and rate.

7.4.2 Solves problems involving time, including relating time, distance and speed.

TERMINOLOGY Rate, ratio

RESOURCES Gr 7 Text books; WCED Illustrative examples; Calculator; Poster; Internet Web sites; Beads; Fruit juice container; Speedometer

INTEGRATION Technology, EMS

TEACHING TIPS

THE CONCEPT OF RATIO Introduce the concept of ratio in the context of recipes (e.g. double or halve a recipe).

Different colour beads can also be used to make necklaces in a specific ratio.

Two quantities of the same kind may be compared by subtracting the one from the other (to obtain the difference), or by dividing the one by the other (to obtain the quotient).

As an example, consider the salaries of a manager and a technician in a certain company. The manager earns R60 000 per month and the technician earns R4 000 per month.

One may compare these two salaries by subtraction (R60 000 − R4 000) and can then state that the manager earns R54 00 per month more than the technician, or one can compare the salaries by division (R60 000 ÷ R4 000) and can then state that the manager earns 15 times as much as the technician.

When a quotient (in this case 60 000 ÷ 4 000 = 15) is used to compare two quantities, it is called the ratio of the two quantities.

The concept of ratio is acquired when learners learn that two quantities (of the same kind) can be compared in a different way than by taking the difference (“how much is the one more than the other?”), i.e. that they can also be compared by taking the quotient (“ how many times is the one more than the other?”), and when they learn that the latter way of comparing is sometimes more appropriate.

The ratio between two variable quantities may be constant or it may vary.

THE CONCEPT OF RATE Introduce the concept of rate in the following contexts:

- The cost of petrol is currently R8,34 per litre - The cost of a long-distance telephone call is R1,85 per minute - The specific mass of a certain material is 3,4 g per cm3 - At 16:00 on a certain day, cars passed through a certain town at a rate of 840 cars per

hour - Water is pumped into a reservoir at a rate of 7 485 litres per hour.

With a rate, we describe how much of one quantity (e.g. price, mass, number of cars, amount of water) corresponds to one unit of another quantity (e.g. volume, time).


Note that the two quantities that are compared by a ratio are of the same kind (and can be expressed in the same units), while the two quantities that are referred to in a rate are of different kinds and are measured in different units (e.g. volume and time in the last example given above).

Learning to interpret and use the word “per” is a crucial element of acquiring the concept of rate.

TIME, SPEED AND DISTANCE Objects seldom move at a constant speed. For example, when you walk or travel by car,

the speed varies all the time. For this reason, a statement like the following is simply false, there is no such situation in reality. “A motorist travels at a speed of 90 km/h. How long will he take to travel a distance of 450 km?” Make sure the learners understand that this is not reality but a mathematical model of reality.

EXAMPLES

RATE AND RATIO 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? (ratio)

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 take fruit drink of the same strength? (ratio)

3. Mary makes a telephone call from a public telephone. She speaks for 7 minutes and pays R14,56. How much does she pay per minute? (rate)

4. Between 07:00 and 16:00 on a certain day, cars pass through a certain town at a rate of 845 cars per hour. (rate)

a) How many cars passed through the town between 07:00 and 16:00?

b) Can you say exactly how many cars passed through the town between 09:40 and 10:40?

5. Mouth-watering curry hamburgers!! The recipe makes 4 monster hamburgers! a) Adele wants to make 12 hamburgers. She thus requires three times as much meat as

the quantity stated on the recipe. Explain to your maths mate how Adele knows this.

b) Lynette wants to make 20 hamburgers. What do you think she needs to do with respect to the ingredients of the recipe? How much milk (in millilitre) will she use?

c) How many onions does Lynette need to make 26 hamburgers?

500g minced meat 2 t salt 1 egg ½ t ground pepper 1 slice whole wheat ½ k milk

¼ t fine nutmeg ½ t fine coriander1 diced onion


TIME, DISTANCE AND SPEED 6. A man travels by bicycle from one town to another. The table below shows how far from

his starting point he is at different times during the journey.

Time 10:00 10:15 10:30 10:45 11:00 11:15 11:30

Distance from start 0 km 11 km 21 km 33 km 48 km 58 km 69 km

When did she travel the fastest, and when did she travel the slowest?

7. A motorist finds that although he sometimes travels faster and sometimes slower, he usually covers about 90 km in each hour.

a) Approximately how far will this motorist travel in 5 hours?

b) Approximately how long will this motorist take to cover a distance of 360 km?

c) Approximately how long will this motorist take to cover a distance of 400 km?

8. A man walks at a fairly regular pace, and covers 8 km in 3 hours. Approximately how far will this man walk in 1 hour, if he walks at more or less the same pace as when he walked the 8 km?



TERM 3 - WEEK 7, 8 & 9

ASSESSMENT STANDARD (AS 7.4.1; 7.4.3; 7.4.4 and 7.4.5 will be integrated) 7.4.1 Solves problems involving: • length; • perimeter and area of polygons; • volume and surface area of rectangular prisms. 7.4.3 Solves problems using a range of strategies including: • estimating; • calculating to at least 2 decimal places; • using and converting between appropriate S.I. units. 7.4.4 Describes and illustrates ways of measuring in different cultures throughout history, including metric and other formal measuring systems.

7.4.5 Calculates, by selecting and using appropriate formulae: • perimeter of polygons; • area of triangles, rectangles and squares; • volume of triangular and rectangular based prisms. 7.1.5 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.


7.4.7 Describes interrelationships between perimeter and area of geometric figures. 7.4.8 Describes interrelationships between perimeter and area of geometric solids.

TERMINOLOGY Prisms, polygon, SI units

RESOURCES Gr 7 Text books; WCED Illustrative examples; Calculator; Poster; Internet Web sites; Measuring instrument (e.g. ruler, measuring tape); Cube blocks; Grid paper; String


TEACHING TIPS

MEASUREMENT Introduce length in the context of 2-D shapes and 3-D objects.

Make sure learners realise that perimeter is also a length.

Make sure learners know all the SI units for measuring length and can do conversions with confidence.

Make sure learners know the properties of the different 2-D shapes, especially the triangle, square, rectangle, kite, trapezium and regular polygons.

Do proper revision of perimeter, area and volume of grade 6 work.

If learners don’t understand the concept, use the 3-D objects made in the first term to re-enforce concepts.

Don’t only supply the rules, allow learners to discover rules through practical investigations.

This is the first grade where learners must know the formulas. Ensure that learners memorise formulas and know when to use the appropriate formulae.

The area of a triangle and the volume of triangular prism is done for the first time in grade 7. Investigate properly and do not just apply the rule unless learners understand the concept.

The perimeter and area of a circle is not compulsory for grade 7. It should only be taught in grade 8.

PROBLEM SOLVING: Before the learners start doing any calculations, teach them to order their problem solving

strategies.

They can organise their thoughts by drawing a diagram. e.g. Calculate the area/surface of the following shape.

12 cm

10 cm

4 cm 4 cm

3 cm


Alternative methods can also be used to solve this problem.

Learners should start doing calculations only after they have organised their thoughts constructively.

INTERRELATIONSHIPS Assessment Standards 7.4.7 and 7.4.8 are intended to provide learners with an

opportunity to investigate. In particular they come to realise that there is no relationship between the area and the perimeter of a figure.

Area and perimeter 7.4.7 There are many different figures that can have the same perimeter yet have different areas, or the same area but different perimeters. Similarly it is possible for different solids to have the same surface areas and yet have different volumes, or the same volume but different surface areas.

Volume and surface area 7.4.8 While these Assessment Standards in Grade 7 provide no more than a context for an investigation, the investigation should alert learners to the notion that a particular volume can be “packaged” in different ways each of which uses a different amount of packaging. In the FET band these problems reappear as optimisation problems in areas such as Calculus.

EXAMPLES 1. Make drawings of approximately the right size and shape of the triangular prisms with the

following dimensions:

a) height = 3 cm, width = 5 cm, depth = 8 cm

b) height = 4 cm, width = 3 cm, depth = 5 cm

2. a) Draw nets for the two prisms described in question 1, and cut the nets out.

b) Calculate the area of each net.

c) Calculate the total length of wire needed to make a frame for each of the two prisms.

3. The picture shows a closed water tank in the form of a prism. The tank is made of flat steel plates that are welded together at the edges.

Calculate lengths of

all unknown sides

Divide shape into

three rectangles

Determine area of each

rectangle

Add the three answers to

get total area Check units


The scale drawing below shows what the steel plates at the front and the back look like. The actual plates are exactly 10 times as big as the drawing below. The distance between the front and back plates is 1,8 m.

The steel plate costs R289 per m2, and the cost of welding plates together at the edges is R845 per m.

a) Calculate the total cost of steel plate and welding for the tank.

b) Calculate the volume of the tank.

4. Our living room is 6m long and 5m wide. There are 2 doors of 0,9m by 2m and two windows of 1,5m by 2m.

a) Calculate the perimeter of each window and the door to determine how much wood is

needed for the window and door frames.

b) The glass of one of the windows broke; determine the surface of glass that has to be replaced.

c) What is the total surface of the walls? (Remember to subtract the doors and windows.)

d) What will it cost to paint the walls if paint costs R30 per litre and one litre can cover 2 m2?

5. You want to make a (cubical) box out of cardboard and wire. a) How much wire (in cm) do you need to make the framework?

b) What is the minimum size of cardboard (in cm2) that you will need to cover the framework?

6 m 5 m

0,9 m

2 m 2,8 m

2 m 1,5 m

10 cm


6. (PERIMETER AND AREA) a) For each of the figures above determine the area.

b) For each of the figures above determine the perimeter.

c) What do you notice?

7. (VOLUME AND SURFACE AREA)

The solids alongside both have a volume of 12 cm3.

a) Determine the total surface area of each.

b) What do you notice? 8. Investigate: how many different figures can you find that all have an area of 24cm2? Draw

rough sketches and determine the perimeter of each.

9. How many different figures can you find that all have a volume of 48cm3? Draw rough sketches and determine the total surface area of each.

10. Can you find two figures that are not similar which have the same area and the same perimeter? If you can, draw a rough sketch of each.

11. Can you find two solids that are not similar which have the same volume and the same total surface area? If you can, draw a rough sketch of each.


ASSESSMENT ASSESSMENT TASK 8: e.g. Tutorial / Investigation on the relationship between perimeter / area and area / volume.

TERM 3 - WEEK 10

ASSESSMENT TASK 9: TEST ON WHOLE TERM’S WORK


TERM 4

TERM 4 - WEEK 1

ASSESSMENT STANDARD 7.4.9 Classifies angles into acute, right, obtuse, straight, reflex or revolution.

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

TERMINOLOGY Straight angles, reflex angle, revolution, degrees, protractor

RESOURCES Gr 7 Text books; WCED Illustrative examples; Calculator; Poster; Internet Web sites; Geo-board and elastic bands


TEACHING TIPS Introduce angles in the context of flags, architecture (paving, tiling, carpets); design

(material, graffiti, cultural murals).

Take our national flag (use the school’s one) and then give learners paper copies of the flag (large enough for learners to be able to measure all the angles).

In Grade 6 learners classified angles as being right angles, smaller than right angles and greater than right angles, in Grade 7 we simply add the vocabulary. However, it is also important to recognise that classifying angles is an important precursor to “estimating” and “measuring” them.

Let learners explore the different angles by showing them with their body (e.g. knee, elbow, etc.).

The following can be used as a summary for visual learners.

- If you do a full turn (360º) on the same spot it is called a revolution. - A quarter of a revolution is a right angle,

and a right angle is ninety degrees. We write it as: 90° Sometimes we use a symbol that looks like a little square to indicate 90°.

- An angle that is smaller than a right angle is called an acute angle. The acute angle drawn here is 60°.


- Half a revolution we call a straight angle. A straight angle is always 180°.

- An angle that is bigger than a right angle but smaller than a straight angle is called an obtuse angle. The obtuse angle drawn here is 127°.

- An angle bigger than a straight angle, but smaller than a full turn is called a reflex angle. The reflex angle drawn here, is 200°.

Another reflex angle of 310°:

REMINDER:

- In the sketch below an angle (in bold), measured by means of a protractor is indicated.

- Ensure that the centre of the protractor fits exactly onto the vertex. - Furthermore, ensure that the zero line of the protractor fits exactly on top of the

one leg of the angle. Now read off through which value on the protractor the other leg goes.


Protractors are not easy tools to use and learners experience great difficulty with them. For this reason it is important that learners should first estimate the size of an angle before measuring it. The thought process should be something like:

“that angle appears to be greater than 90°… it also appears to be less than 180°… in other words it is obtuse, let measure it… it is 143°”

Ensure that learners can use the protractor with confidence for measuring and drawing angles.

Ensure that learners understand when to use which set of the numbers when using the protractor.

Investigate angles by using the sum of the inside angles of e.g. the triangle (revision term 1).

The pie graph can also be taught here, since the calculation of angles is used when drawing a pie graph.

Assessment Standard 7.4.10 is intended to be used in conjunction with 7.3.4 (Uses a pair of compasses, ruler and protractor to accurately construct geometric figures for investigation of own property and design of nets). That is, learners should be drawing angles with a purpose.

EXAMPLES 1. Study the following sketch and say what type of angle each of the following is: a) ∠A

b) ∠ABD

c) ∠ACE

d) ∠CDE

e) ∠E 2. Use the information in the table alongside as well as a ruler, and a protractor to

accurately construct the following geometric figures:

a) An equilateral triangle with sides of 6 cm.

b) A square with sides of 6cm.

c) A regular pentagon with sides of 6 cm.

d) A regular hexagon with sides of 6 cm.

e) A regular octagon with sides of 6 cm.

A

B

C DE

Regular polygon Angle size

Equilateral triangle 60º

Square 90 º

Pentagon 180 º

Hexagon 120 º

Octagon 135 º


3. If ∠ABC = 90º then ∠ABD : ________ ∠YXZ : ________



TERM 4 - WEEK 2 & 3

ASSESSMENT STANDARD 7.3.6 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.

TERMINOLOGY Transformation, rotation, reflection, translation

RESOURCES Gr 7 Text books; WCED Illustrative examples; Calculator; Poster; Internet Web sites; Scissors; Triangles on cardboard; Grid paper

INTEGRATION Technology, Arts and Culture

TEACHING TIPS Introduce the concepts in context of 2-D shapes.

Reflectional symmetry:

Use the following example to introduce reflectional symmetry

- You can see yourself in a mirror. When you blink with your left eye, your mirror image blinks its right eye. In some mirrors, your mirror image has the same size as yourself and in some mirrors it has not. A mirror transforms (changes) a figure into a figure that looks different. In this process some things stay the same and others change.

A

B C

D

40º

V W X

Y Z

70º 50º


- In Mathematics this works almost the same when we speak of mirror images. In this module you will only work with two-dimensional shapes. Therefore the mirror is represented by a straight line.

A figure has rotational symmetry (point symmetry) if it can be rotated (turned) about a point in such a way that its rotated image coincides (fits exactly) with the original figure after turning through less that 360º. The number of times that this can be done in one revolution is called the order of symmetry. The smallest angle of rotation to let the figure fit on itself is called the angle of rotation symmetry.

For example:

Dotted paper can be used as a resource when drawing shapes for symmetry.

Mirrors can be used to investigate symmetry if learners struggle to understand the concept.

Distinguish between mirror/line and rotation symmetry.

A transformation that does not change the size or shape of a two dimensional or three dimensional object is called a rigid transformation, or an isometry. The image is always congruent to the original. The following are three types of rigid transformations and they are often referred to as a “flip”, a “slide” or a “turn”.

Translation or move: Translation means you slide the shape to a new position without turning it.

Reflection or mirror image: Reflection means that you create a mirror image of the shape by flipping it.

Rotation, turning or turning around: Rotation means turning a shape through a given angle. The point which is fixed during rotation is called the centre of rotation.

Reflection (Flip)

Rotation (Turn)

Translation (Slide)


EXAMPLES

SYMMETRY: REFLECTIONAL SYMMETRY 1. Below you see a drawing of a (match) box. Cut the box in half, approximately where the

box in the figure shows a light grey plane.

a) Can you let it appear as if the box in figure a is intact with the help of a mirror?

b) Would this also work if you had cut the box according to figure b?

2. In the sketches below, you look straight down on half a box and the mirror.

a) For each sketch, draw the reflection that you see in the mirror.

b) Does the semi-box of figure a, joined with its reflection, form a whole box?

c) Does the semi-box of figure b, joined with its reflection, form a whole box?

d) Which cutting plane in question 1 would you call a plane of symmetry?

3. By folding and cutting paper, you can produce very nice reflection symmetrical figures:

- Take a sheet of paper and fold it in half. - Cut out any figure, starting and ending at the folded side. You don't need to cut

accurately. - Unfold the cut out figure. - The result is a reflection symmetrical figure.

Create the following cut-outs using the same method:

a) Why are the halves left and right from the folding line equal in these figures?


b) The folding line is called the axis of symmetry. What do you think is meant by "axis of symmetry"?

c) Place a mirror vertically on the figure with one side on the axis of symmetry. Explain why the mirror is called the plane of symmetry of this figure.

SYMMETRY: ROTATIONAL SYMMETRY 4. Investigate each of the figures below for rotational symmetry by showing the following:

a) the centre point of rotation

b) the angle of rotation symmetry

c) the order of symmetry

5. Sometimes there are good and practical reasons to make an object symmetrical.

a) What type of symmetries do you find in each of the following 4 playing cards: 8 of clubs, 8 of diamonds, king of spades and queen of hearts?

b) Why is it convenient when a playing card is symmetrical?


c) What type of symmetries occur in this nut and spanner?

d) Why is it convenient that a nut is symmetrical?

TRANSFORMATIONS: TRANSLATION (SLIDE) 6. Below a pattern from Zambia is drawn.

Draw, by translating (sliding), another piece of this pattern. You may use tracing paper.

7. Draw a nice pattern yourself and let it repeat itself by using translation.

TRANSFORMATIONS: REFLECTION (FLIP).... NEXT PAGE


TRANSFORMATIONS: REFLECTION (FLIP) 8. Let’s reflect the “fish” below in the given order form (a) to (c):

a) first, reflect in the black horizontal line

b) then reflect in the dotted diagonal line

c) now reflect in the black horizontal line

Compare the end position of your fish with that of your maths pal.

TRANSFORMATIONS: ROTATION (TURN) 9. In the following tasks, each time after turning the figure, you must draw the "rotated"

figure. The "bold" dot at each figure is the centre of rotation. Use a piece of tracing paper if you can't visualise (see) the result immediately.

a) Rotate the figures below through 90º, 180º and 270º in a clockwise direction. (Remember to draw each time!)


b) Rotate the figures below through 120º and 240º in an anti-clockwise direction. (Remember to draw each time!

c) Rotate the figures below through 180º in a clockwise direction. (Remember to draw

each time!)

d) Use the following shapes and the rigid transformations (translation; rotation; reflection) and create your own border pattern.

This activity can also lead up to co-ordinates by doing it on grid paper instead of dotted paper.

CONSOLIDATION / HOMEWORK Examples for homework.



TERM 4 - WEEK 4

ASSESSMENT STANDARD 7.3.9 Draws and interprets sketches of solids from different perspectives

7.3.10 Locates positions on co-ordinate systems (ordered grids) and maps using: • horizontal and vertical change;

• compass directions

TERMINOLOGY Perspective, co-ordinate system, grid, map, horizontal edge, vertical edge, compass direction

RESOURCES Gr 7 Text books; WCED Illustrative examples; Calculator; Poster; Internet Web sites; Grid paper; Concrete blocks

INTEGRATION Technology, Natural Sciences, Arts & Culture

TEACHING TIPS

PERSPECTIVES Introduce in context of objects in learners’ environment (e.g. let learners draw a motor-car

from different perspectives, arrange cubes and let learners draw from angles/views.)

The matching Grade 6 Assessment Standard (6.3.7) “Draws and interprets sketches of simple three-dimensional objects from different positions (perspectives)” is very similar to this one. The progression from Grade 6 lies in the increased complexity of the objects that learners are working with.

In example 1 that follows, learners use concrete objects as the basis for their sketches.

In example 1 that follows for the prism, the front view of the regular-pentagon based prism will still look like a regular pentagon. The top view will look like two identical rectangles although their edges are not in the same proportion as they are on the model. The side view will look like two rectangles that are not identical to each other even though on the model they are.

CO-ORDINATES Investigate location in the context of maps, connect-the-dot activities.

Co-ordinate systems: What is apparent is that locating positions on co-ordinate systems (7.3.10) and describing movement between them is something that learners have already done in Grade 6 (and earlier).

By integrating co-ordinate systems and transformations we have a powerful mechanism for teaching about geometric figures and for investigating their properties including congruence and similarity. (Refer to term 1)

Start by using numbered grids. Mark the horizontal axis by using letters (can be replaced by numbers) and the vertical

axis by using numbers.

When you give the location of a point on the grid you first give the letter (horizontal axis) and then the number (vertical axis).


The co-ordinates of a point must always be in brackets, separated by a semi-colon.

e.g. (B ; 4)

Start by finding the co-ordinate of a dot on a numbered grid.

Give learners co-ordinates of points of 2-D shapes and connect the dots to form shape, progress to grids with pictures.

Let learners write a short message on grid paper. Let them determine the co-ordinates and give it to another learner to decipher.

3-D objects can also be drawn on a grid by giving learners the co-ordinates.

Use maps to apply this concept in a real life situation.

EXAMPLES Draws and interprets sketches of solids from different perspectives. 7.3.9

1. Place a regular-pentagon based prism down on one of its rectangular faces (as illustrated above). By looking at the prism from the front, the side and from above draw its front, side and top views.

2. Place a regular-pentagon based pyramid down on one of its triangular faces (as illustrated above). By looking at the pyramid from the front, the side and from above draw its front, side and top views.

3. Repeat for a regular-hexagon based prism and pyramid.

4. Plot the following points: A (3; 1), B (1; 4), C

(3; 6), and D (4; 4). What kind of figure is ABCD? List, with reasons, as many properties of ABCD that you can justify using the sketch.

5. Plot the following points: P (1; 7), Q (1; 9), R (3; 10), S (7; 10). What kind of figure is PQRS? List, with reasons, as many properties of PQRS that you can justify using the sketch.

6. Plot the following points: H (4; 8), I (7; 9), J (10; 6), K (7; 5). What kind of figure is HIJK? List, with reasons, as many properties of HIJK that you can justify using the sketch.


7. Plot the following points: L (10; 3), M (8; 1), N (6; 3), O (8; 5). What kind of figure is LMNO? List, with reasons, as many properties of LMNO that you can justify using t he sketch.

8. Draw the triangle that results from a reflection of ΔABC about the line AC. ΔABC and its reflection form a quadrilateral. What kind of quadrilateral is it? List, with reasons, as many properties of the quadrilaterals as you can justify using sketch.

9. Draw the triangle that results from half turn of ΔDEF about midpoint of line segment DE. ΔDEF and its rotated image form a quadrilateral. What kind of quadrilateral is it? List, with reasons, as many properties of the quadrilaterals as you can justify using the sketch.

(SIMILARITY AND CONGRUENCY) 10. Draw each of the following quadrilaterals on

the grid paper provided:

- A with vertices (1; 1), (1; 4), (2; 4) and (2; 1) - B with vertices (3; 1), (3; 4), (7; 4) and (7; 1) - C with vertices (8; 2), (8; 8), (10; 8) and (10; 2 - D with vertices (6; 9), (6; 10), (10; 9) and (9; 9) - E with vertices (1; 8), (1; 10), (5; 10) and (5; 8) - F with vertices (1; 7), (3; 9); (7; 6) and (5; 3)

11. State with reasons which of the quadrilaterals that you have drawn are: Congruent to each other or similar to each other


ASSESSMENT ASSESSMENT TASK 10: Tutorial/Investigation (Angles, 2-D & 3-D, Co-ordinates, Perspective)


TERM 4 – WEEK 5 & 6

ASSESSMENT STANDARD 7.5.7 Draws a variety of graphs by hand/technology to display and interpret data (grouped and ungrouped) including:

• bar graphs and double bar graphs; • histograms with given intervals; • pie charts; • line and broken-line graphs

7.5.8 Critically reads and interprets data presented in a variety of ways 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. age, gender, race); • scales used in graphs as a source of error and bias; • choice of summary statistics (mean, median or mode); • any other human rights and inclusivity issues.

TERMINOLOGY Histogram, pie chart, line and broken line graph, context, categories, scales, summary, statistics, inclusivity

RESOURCES Gr 7 Text books; WCED Illustrative examples; Calculator; Poster; Internet Web sites; Grid paper; Graph paper: Data Handling in the GET Band

INTEGRATION Social Sciences and Natural Sciences

TEACHING TIPS Use the book ‘Data Handling in the GET Band’ Revise collecting and organising data –term 2 (to be included in the project) DIFFERENT TYPES OF GRAPHS

A. BAR GRAPH: - Do not only expose learners to vertical bar graphs, but also horizontal bar

graphs. - Initially let learners use grid paper to draw bar graphs. Let them use graph

paper when they can do this with confidence.


A HORIZONTAL BAR GRAPH

Favourite TV channel

0 20 40 60 80 100

TV1

TV2

TV3

Mnet

etvN

umbe

r of p

eopl

e

TV channels

A VERTICAL BAR GRAPH

Favourite TV channel

0

20

40

60

80

100

TV1 TV2 TV3 Mnet etv

TV channels

Num

ber o

f peo

ple

- Draw bar graphs on squared paper. This makes it easier to draw and to read afterwards.

- AXES: Draw 2 axes at right angles to each other. Decide whether the bars will go up or across the page. Label each axis clearly, and describe the data.

- SCALE Try to make the diagram fill a sensible amount of the paper. Work out an easy scale, and make each ‘small space’ an easy

value to use. Number the axis in equal steps. Check that the largest value fits on the paper.

- DRAWING THE BARS Draw each bar the correct length for the data it shows. Make all the bars the same width. Leave equal spaces between them. Label each bar clearly.

B. DOUBLE BAR GRAPH: - Sometimes you need to compare different sets of data. - If you look at separate bar graphs it is not very easy to compare them. The bar

graphs can be combined into one diagram. The bars in each group are put next to each other. This makes it much easier to compare the data sets.

NUMBER OF RESIDENTS PER HOUSE

0

5

10

15

20

25

30

1 2 3 4 5 6

Number of residents

Num

ber o

f hou

ses

NorthvaleEastvale


C. HISTOGRAM: - We could draw a histogram to illustrate some data. A histogram is similar to a

bar graph with no gaps between the bars. - Notice that the intervals are NOT written 0 to 5, 5 to 10, 10 to 15, etc. If we did

that, we would have problems deciding where numbers like 5 and 10 would fit. - REMEMBER: With continuous data, information is collected by

MEASUREMENT. The variables are measurements such as lengths, masses, and times – which go up continuously, not in jumps, so once again one class begins where the other one ends.

- The grade 7 learners must be given the intervals. - This can only be used when working with whole numbers. It cannot be used

with decimals.

0123456789

1 to 5 6 to 10 11 to 15 16 to 20 21 to 25 26 to 30

Number of Hours per week

Hours spend watching TV

D. PIE GRAPH: - A pie graph is a circular diagram used to display data. It is particularly suitable

if you want to illustrate how a population is divided up into different parts, and what proportion of the whole that each part represents. This proportion can be written as a fraction, as a decimal fraction or as a percentage of the whole.

- The whole circle stands for the whole amount of data being dealt with. Each slice stands for a named part of the data. Its size represents the size of that part of the data.

- Sometimes it is easy to estimate the sizes of these fractions, as in the next activity: This pie graph shows what a bran breakfast cereal is made of.

Number of learners

Fat

Carbohydrate Fibre

Others

Protein


- Drawing a pie graph Do not give grade 7 learners data in the form of percentages if they must draw a

pie graph. (Leave for grade 8.) There are two main steps to follow when drawing a pie graph:

Calculating the angle for each sector of the pie graph: If learners are asked to draw a pie graph showing how much time a learner spends on each activity during a 24 hour day e.g. sleeping, at school, eating etc. The following procedure should be followed:

a Find the total amount to be shown on the pie graph (e.g. 24 hours).

b Divide 360º by this total (e.g. 360° ÷ 24 = 15°) this means that 15º represents one hour.

c Multiply by the number of items (e.g. for 3 hours, 3 x 15° = 45°).

d Check that the angles add up to 360º.

(Remember: when angles are rounded off to the nearest degree, the total may not be exactly 360º.)

Drawing the pie graph

e Draw a circle with a pair of compasses.

(Do not make your circle too small)

f Draw a radius as a starting line for measuring angles.

g Use a protractor to draw the angle of each sector at the centre of the circle.

(i) Make sure that the centre of the protractor is at the centre of the circle. (ii) Measure the first angle from the starting line, then measure the next angle

from the new line, and so on. (iii) Measure the size of each angle very carefully. Any mistakes will mean that

the sectors will not fit into the circle. Draw the smallest angle first, then the next smallest, and so on ... so that the largest angle is drawn last. This can sometimes help reduce the effect of any errors.

h Label each slice carefully. If it is difficult to fit the full name of each group on

each slice, label each with a letter and use a key to say what each stands for.

i Give the pie graph a suitable title.

j Draw the circle of the pie graph with a radius of at least 4 – 5 cm, otherwise it will be difficult to measure and draw the lines.

k) A key must also be added if sectors are not labeled in the pie graph.

E. LINE AND BROKEN LINE GRAPHS - A line graph can be drawn by plotting data items and then connecting the points

with a line or a broken line. Line graphs are often useful to see patterns or trends over time.

- A broken line graph is when you use broken lines to connect the dots. e.g. Thandi is in hospital. Every 3 hours her temperature is taken and the points are plotted on a graph.

Measure from here first

Measure from here

next

Starting line


- When data is collected at time intervals, it is quite usual to show them as line graphs like the one which follows

- The graph shows that Thandi’s temperature dropped slightly between 3:00 and 6:00, but it rose sharply between 6:00 and 9:00.

- Be careful – the line doesn’t say that Thandi’s temperature only dropped between 3:00 and 6:00. It is possible that her temperature rose between 3:00 and 4:30 and then dropped again between 4:30 and 6:00.

GRAPH SHOWING THANDI'S TEMPERATURE

37

37.5

38

38.5

39

39.5

40

0.00 3.00 6.00 9.00 12.00 15.00 18.00

Time

Tem

pera

ture

in d

egre

es

Cel

sius

The reason for collecting and organising the data is ultimately because we have a question that we want to answer or a prediction that we want to make. We do not collect data because there is some intrinsic value in doing so.

INTERPRETATION OF DATA

Assessment Standard 7.5.8 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. It is, however, also important that learners develop the ability (indeed habit) to reflect critically on the methods of data collection, organisation and representation and on how these can influence the conclusions that can be drawn and the predictions that can be made. The Assessment Standard lists a number of different factors that can influence these conclusions and predictions.

CONTEXT (E.G. RURAL OR URBAN, NATIONAL OR PROVINCIAL) - Consider the graphs below (on the next page) 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

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 1 that follows later demonstrates how categories within data can

influence the conclusions and predictions that can be made.

SCALES USE IN GRAPHS AS A SOURCE OF ERROR OR BIAS - The graphs used to illustrate the comment on context above all use a different

scale on the vertical axis. Learners should be aware of and sensitive to how this impacts on the conclusions and predictions that can be made. Example 2 gives another example of how scale can have an impact on the impression that graphs create.

- Teach learners to choose an appropriate scale for different number ranges, i.e. intervals of 5, 10 ,20, 100, 1000, etc.

- On some bar graphs the scale is very easy. Each ‘small space’ stands for one unit. Here are some examples of scaling of horizontal bar graphs:

0 1

2 3

0 1

2 3

4 5

6

0 5

10 15

0 10

- Some scales are not so easy. Each small space may stand for more than 1 or for a fraction. Here are some examples:

0 100

0 20

0 1

2

10 spaces 100 1 space 100 ÷ 10 = 10

10 spaces 20 1 space 20 ÷ 10 = 2

5 spaces 1 1 space 1 ÷ 5 = 0,2

- Sometimes charts and graphs are used to mislead people. Changing the scale of a diagram can have a big effect on its appearance. When you read statistical diagrams, you should always look carefully at the scale.

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


Look a these diagrams. They show how changing the scale on a graph can change the way it ‘looks’.

Company X's share of home PC market

0

2

4

6

8

10

1992 1993 1994 1995 1996

%

software hardware

Company X's share of the business PC market

05

101520253035

1992 1993 1994 1995 1996

%

software hardware

CHOICE OF SUMMARY STATISTIC (MEAN, MEDIAN OR MODE) - In term 2 learners calculated the mean, median and mode for a small data set.

(Assessment Standard 7.5.6). These three statistics were quite different from each other. Learners should recognize that to use one of these statistics rather than another one could create a very different impression about the data set.

ANY OTHER HUMAN RIGHTS AND INCLUSIVITY ISSUES - Both the examples given in the remarks above and the illustrations provided

below have addressed human rights and inclusivity issues such as gender, population group, rural vs. urban etc.

EXAMPLES 1. 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.

2. The two graphs that follow have been created using exactly the same data.

a) What has the person who has drawn the graphs done to make the graphs look so

different?

b) What are the different impressions created by the two graphs?

c) Study the above two diagrams. How does changing the scale of the second graph affect your reading of the graph?

Cars manufactured

0

20

40

60

80

100

120

140

June Ju

ly

Augus

tSep

t

Otober

Novem

ber

Decem

ber

Num

ber o

f car

s m

anuf

actu

red

Cars manufactured

90

100

110

120

130

June Ju

ly

Augus

tSep

t

Otober

Novem

ber

Decem

ber

Num

ber o

f car

s m

anuf

actu

red


3. Thabo sells bicycles. He records the number of bicycles sold each month as follows:

Month Jan Feb Mar Apr May June Number of bikes sold 26 27 29 34 44 55

He draws two different bar charts to illustrate this information. Chart A uses the whole scale. In chart B the scale starts at 25. The two graphs show exactly the same information, but they look very different because the scales are different.

Bar chart A

05

1015202530354045505560

Jan Feb Mar Apr May JunMonth

Number of bikes

sold

Bar chart B

2530354045505560

Jan Feb Mar Apr May JunMonth

Number of bikes

sold

He wants to expand his business, and needs to borrow money from the bank. He wants to

show the bank manager that his sales are growing fast. These are his sales for the last 6 months: Which chart should Thabo show the bank manager? Explain your answer.

4. This table shows the sales of a company which provides take-away meals. It lists the number of meals sold in the last eight weeks.

Week 1 440 Week 5 465 Week 2 455 Week 6 485 Week 3 450 Week 7 490 Week 4 460 Week 8 495

The manager of the company wants to employ more staff. She wants to show that sales are increasing.

The owner of the company does not want to pay for any more staff. She wants to show that the sales are about the same level.

Draw one graph for each person. Choose your scale carefully and say who would use each graph.



ASSESSMENT Assessment TASK 11 : e.g. Investigation / Project - data (investigate a problem in given context.) This project should include the work covered in term 2 weeks 7 and 8.

TERM 4 - WEEK 7

ASSESSMENT STANDARD 7.5.10

Performs simple experiments where the possible outcomes are equally likely and:

• lists the possible outcomes based on the conditions of the activity;

• determines the frequency of actual outcomes for a series of trials;

• determines the relative frequency using the definition of relative frequency

TERMINOLOGY Frequency; series of trials; relative frequency; actual outcomes

RESOURCES Gr 7 Text books; WCED Illustrative examples; Calculator; Poster; Internet Web sites;

INTEGRATION -

TEACHING TIPS Chance and probability (7.5.10)

Introduce the concept in the context of gambling to show how unlikely the chances are of winning e.g. the National Lotto.

Do simple experiments by: e.g. Tossing a coin to determine the probability of the coin falling on heads or tails and Casting a die to determine the probability of the die falling on 1 to 6.

Play the gambling game:

You are now going to simulate a gambling machine in the classroom. Instead of a machine you are going to use three dice. The playing rules of this machine are:

- The person playing the machine must throw all three dice simultaneously. - You pay one rand for each throw. - You get ten rand if you throw three sixes. - You get five rand if you throw two sixes. - You get one rand if you throw one sixes. - In all other instances you get nothing. - As a group throw three dice 40 times and record the results in a table.


Event Tally marks Total (group) Fraction Percentages 3 sixes

2 sixes

1 six

0 sixes

Totals 40 40 100%

Assessment standard 7.5.10 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 an impossible to certain; list the possible outcomes for simple experiments and count the actual frequency of the outcomes for a series of trials.

In Grade 7 learners are expected to consider:

- a particular situation, - to be able to list all of the possible outcomes, - to conduct a series of trials and - record the outcomes and - then to calculate the relative frequency of each of the outcomes for the series of

trials. e.g.:Tossing a coin: There are only two possible outcomes—the coin will land with the head facing

up or with the tail facing up If a coin is tossed 60 times and lands on heads 35 times (we say the frequency

of heads is 35) and tails 25 times (the frequency of tails is 25) then the relative

frequency of heads is 6035

or 58% and the relative frequency of tails 6025

or 42%.

Rolling die: There are only six possible outcomes—the die will either land with a 1 or a 2 or

a 3 or a 4 or a 5 or a 6 facing up. If a die is rolled 90 times and lands as shown in line 2 of the table, then the

relative frequency is shown in line 3 of the table

Number on the die face 1 2 3 4 5 6

Frequency 12 16 14 17 13 18

Relative frequency 1290 16

90 1490 17

90 1390 18

90

The only constraint that the Assessment Standard places on the teacher wanting to design the experiment with which the learner will work is that the outcomes must be equally likely—coins, die and spinners with equal regions are obvious candidates for the experiments. Another option is suggested in the illustration below.


EXAMPLES 1. 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) Using your list of possible outcomes make a table on which to record the outcomes of 60 experiments.

c) Draw a card from the bag and place a tally on your table to record the outcome. Replace the card.

d) Repeat this 60 times.

e) Add up the tallies on your table to determine the actual frequency of each outcome.

f) Calculate the relative frequency for each of the outcomes.



TERM 4 - WEEK 8 REVISE: Mathematical models (7.2.4.) from Term 2 week 2

TERM 4 - WEEK 9-10

ASSESSMENT TASK 12 I.E. EXAMINATION 2 PAPERS OF 50 MARKS EACH ON YEAR’S WORK

Work Schedule week

other LO and AS Concepts

WK 1 Encarta en Internet 7.1.2revision and historical and

cultural development of numbers

WK 2 Robo 6 akt 8 7.1.9, 7.1.11 calculator, operation techniques

WK 3 Robo 6 akt 3, 7 7.1.3, 7.1.8, 7.1.7 factors, prime factors, exponents

WK 4 Robo 6 akt 1 7.1.1, 7.1.8, 7.1.3, 7.1.7 integers

WK 5 Internet 7.1.7, 7.1.9, 7.1.10 operations with integers

WK 6 My world 7.1.1, 7.2,1, 7.2.2 geometric patterns

WK 7 Speelpark Wiskunde 7.1.1, 7.2.1, 7.2.2 numeric patterns

WK 8 Ms Excel en Ms Word 7.1.8,7.3.1,7.3.2,7.3.8

2D and 3DWK 9 Ms Excel en Ms

Word 7.3.4, 7.3.5, 7.4.10 2D and 3DWK 10

Work Schedule week


WK 1 7.2.3 patterns

WK 2 7.2.4, 7.2.5 number sentences

WK 3 Ms Excel en Ms Word 7.2.6, 7.2.7 graphs and equivalance

WK 4 Robo 6 akt 13, 18, 19 7.1.3, 7.1.11 equivalent fractions

WK 5 Jamit Fractions 7.1.3, 7.1.11 add and subtration of fractions

WK 6 Jamit Fractions 7.1.7 multiplication with fractions

WK 7 My world/ MS excel 7.5.1 - 7.5.5 data

WK 8 Internet / MS excel 7.5.6, 7.5.9 mode and median

WK 9WK 10

Work Schedule week


WK 1 7.1.1, 7.1.3 revision

WK 2 Jamit Fractions 7.1.7 round off, operations with decimals

WK 3 Robo 6 akt 7,17 7.1.4, 7.1.7 fractions, desimal fractions and percentages

WK 4 7.1.5, 7.1.7 financial mathsWK 5 7.1.6 rate and ratio

WK 6 7.1.6, 7.4.2 speed, time and distance

WK 7 7.1.4, 7.4.3, 7.4.4, 7.1.5 length and perimeter

WK 8 7.4.5,7.1.5,7.4.3,7.4.7 perimeter and area

WK 9 7.4.5, 7.4.8, 7.3.5 volume

WK 10

AE25, AE30, AE31

assessering

AE34

AE41; AE16

AE20; AE21

AE24; AE27

AE28; AE11; AE12;AE29

2.3.1 - 2.3.5

9.5.1.1 - 9.5.1.5

AE11

AE12; AE28

AE28; AE17

2.4.4.1, 2.4.4.2

GRADE 7 TERM 3 COMPUTER SOFTWARE PLANNING

9.1.1.1, 9.3.1, 9.3.2

Cami Exercises

assessment

AE39

Master Maths

AE08

AE35

8.1.6.1 - 8.1.6.4 AE09;AE13;AE42;AE24

4.1.2.1, 4.1.2.2, 4.1.2.5,

2.1.4.1-2.1.4.6


AE09;AE13;AE42;AE24

Assessering

Master Maths

AE08; AE37


2.2.5.1. - 2.2.5.3

1.7.5.3 - 1.7.5.8

Cami 1.1.7.4, 1.1.7.5, 1.1.76, 1.8.1.1, 1.8.2.2, 1.8.2.3, 2.3.6.1 - 2.3.6.3, 2.3.7.1 - 2.3.7.3, 1.8.4.1

10.1.1.1, 10.1.1.2

3.2.9.1 - 3.2.9.5, 4.1.1.10

AE37

AE08

AE44

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.

8.3.1.1, 8.3.2.1 , (Perseptueel)

Cami 2.5.1.1 - 2.5.1.6

2.2.1, 2.2.2, 2.2.3, 2,.3.1, 2.3.2, 2.3.3, 2.4.1-6

Cami 4.1.1.7-4.1.1.8, 4.1.2.10, 4.1.3.1 - 4.1.3.6 , 4.1.3.7, Perseptueel (geometriese vorme)

9.7.1, 9.7.2

2.2.2.2 - 2.2.2.7

9.3.6.1, 9.3.3.1,9.3.5, 9.4

2.3.3.1, 2.3.6.1 - 2.3.6.6, 2.3.7.2, 2.3.7.6

2.6.1.1 - 2.6.1.3

Cami Exercises

Cami Exercises

2.8.1 - 2.8.6

assessment

Master Maths

Work Schedule week


WK 1 7.4.9 angles

WK 2&3power point / ms

excel 7.3.6properties of geometric

figures

WK 4 7.3.9, 7.3.10 solidsWK 5 & 6 INTERNET 7.5.7, 7.5.8 graphs

WK 7 INTERNET 7.5.10 probabilityWK 8WK 9

WK10

Master Maths

AE03; AE04AE05

revisionrevision

Cami Exercises

10.1.2.3

8.1.3

10.2.1-3

assessment

AE24

AE24

8.1.6.1


senior - western cape · let learners e.g. calculate a multiple calculation on their calculators...

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