Countdown Maths Oxford

CountdownLevel Seven Maths

Teaching Guide

Shazia Asad

New Edition

Contents

Introduction ................................................................................................. iv

Format of the guide ........................................................................................v

Lesson Planning ............................................................................................ vi

Chapter 1: Operation of Sets .....................................................................2

Chapter 2: Rational Numbers ...................................................................4

Chapter 3: Decimal Numbers ...................................................................7

Chapter 4: Square Roots of Positive Integers ..........................................9

Chapter 5: Exponents .............................................................................. 10

Chapter 6: Direct and Inverse Variations ............................................. 13

Chapter 7: Profit and Loss ...................................................................... 15

Chapter 8: Discount ................................................................................ 17

Chapter 9: Simple Interest ...................................................................... 19

Chapter 10: Algebraic Expressions .......................................................... 20

Chapter 11: Simple Algebraic Formulae ................................................. 23

Chapter 12: Factorization of Algebraic Expressions ............................. 24

Chapter 13: Simple Equations .................................................................. 25

Chapter 14: Perpendicular and Parallel Lines ........................................ 27

Chapter 15: Circles .................................................................................... 30

Chapter 16: Geometrical Constructions ................................................. 32

Chapter 17: Quadrilaterals ....................................................................... 34

Chapter 18: Perimeters and Areas ........................................................... 36

Chapter 19: Surface Areas and Volumes ................................................. 39

Chapter 20: More about Graphs .............................................................. 40

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Introduction

Professional development improves a teacher’s depth, knowledge, and instructional decision-making. Judgement and leadership skills are two of the many facets of a professionally trained teacher.To be effective, the teacher must be actively engaged in content learning. While delivering lessons in the classroom, the teacher needs to be open to learning at the same time. Many clever ideas can be picked up from the students, and the lesson plan modified accordingly.Mathematics should become part of ongoing classroom routines, outdoor play, and activities involving day-to-day life.Teachers benefit from working with colleagues who can question, challenge, support, and provide a network of resources, for each other.This teaching guide has been developed keeping in mind the needs of the teachers while using the textbook. As a reference source, it pre-empts potential queries the teacher may have during the course of this series. Use it as a ‘guide by your side’, and not as a ‘sage on the stage’.The guide follows the format given on the following page. I hope teachers will find the guide useful and enjoy their teaching even more.

Shazia Asad

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Format of the guide

IMPROVISE / SUPPLEMENT / METHODOLOGY / ACTIVITY

Teaching techniques and activities mentioned in the manual are to be utilized by implementing, improvising or supplementing.

KEYWORDS / TERMINOLOGIES

Usage of mathematical language and alternative terms.

EQUATIONS / RULES / LAWS

Quick recall of numbers, facts, rules, and formulae.

ASSOCIATION

Ability to adapt the aforementioned facts to mathematical usage.

CONCEPT LIST

Flow charts, pictorial representations, and steps of mathematical procedure.

BACKTRACK QUESTIONS

An adaptable approach to calculations, investigating various techniques and methodologies. FREQUENT MISTAKES

To be able to pre-empt pitfalls the students might encounter in the course of a particular topic.

SUGGESTED TIME LIMIT

Suggested duration of classes will be mentioned, but it is entirely up to the teachers to evolve their own time limits considering the level of the students.

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Lesson Planning

Before we begin to discuss lesson planning, it is important to talk about teaching and the art of teaching.

A. Furl

First understand by relating to day-to-day routine and then learn. It is vital for teachers to realize the need to incorporate meaningful teaching, by relating to daily routine. Another R is re-teaching and revising, which is covered under the supplementary/continuity category.Effective teaching stems from engaging every student in the classroom. This is only possible if you have a comprehensive lesson plan. How you plan your work and then work your plan, are the building blocks of teaching.There are three integral facets to lesson planning: curriculum, instruction, and evaluation.

1. CURRICULUMA curriculum must meet the needs of the students and the objectives of the school. It must not be over-ambitious, or haphazardly planned. This is one of the major pitfallsfalls when planning to teach maths according to a curriculum.

2. INSTRUCTIONSInstruction methods used include verbal explanation, material-aided explanation, and teach-by-asking philosophy. The methodology adopted by a teacher is a reflection of the teacher’s skill. I will not use the term experience as even the most experienced teacher can adopt flat and short-sighted approaches; the same can be said for beginner teachers. The best teacher is the one who works out the best plan for the class, customized to the needs of the students. This only happens when the teacher is proactive, and is learning and re-learning the content throughout the year, reinventing the teaching methodology on a regular basis.

3. EVALUATION

This is the tool that shows the teacher how effective he or she has been in teaching the topic. The evaluation programme is not just a test of the student, but also indicates how well the lesson has been taught.

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B. Long-term Lesson Plan

The long-term lesson plan encompasses the entire term. Generally, the schools’ coordinators plan out the core syllabus, and the unit studies.Core syllabus comprises of the topics to be taught during the term. Two important considerations while planning this are the time frame and whether the students have prior knowledge of the topic.An experienced coordinator will know the depth of the topic, and the ability of the students to grasp it in the given time frame. Allotting the correct number of lessons for a topic is essential, as extra time spent on a relatively easy topic could affect the time needed for a difficult topic.

C. Suggested Unit Study Format

Week Dates Month Number of Days Remarks

D. Short-term Lesson Planning

This is where the course teacher comes in. The word lesson comes from the Latin word ‘lectio’ which means ‘action of reading’. The action of conducting a topic (what and how the students are taught) takes place in the classroom.The following is a suggested format for planning a topic. It should be noted that each school and each teacher may have their own ways of doing things. This should be respected, but at the same time the teaching can be developed by improvising from other sources. Study the following outline to understand this.

1. TOPIC

This could just be the title of the topic e.g. Volume of 3–D shapes.

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2. OVERVIEWThe initial concern while planning a topic should be ‘how much do the students already know about the topic’? If it is an introductory topic, recall a preceding topic that may lead to the topic being taught. For the topic given above, the teacher could write the following:The students have prior knowledge of the properties of 3-D shapes e.g. cube, cuboid, cylinder. They can identify the dimensions of each shape viz. their length, breadth, height, and radius.

3. OBJECTIVESThis highlights the aims of the topic.

Example:• to calculate the volume of a cube and cuboid• to calculate the surface area of a cube and cuboid• to calculate the volume of a cylinder• to calculate the surface area of a cylinder

4. TIME FRAMEDetermining the correct time-frame makes or breaks a lesson plan. Generally, class dynamics vary from year to year, so flexibility is important.Teachers should draw their own parameters, but they can adjust the time frame depending on the receptivity of the class to the topic at hand.The model plan could say• 5 classes to teach the volume of a cube and cuboid,• 6 to 7 classes to teach the surface area of a cube and cuboid,• 5 classes to teach the volume and surface area of a cylinder.Note that the introduction takes longer but as the sessions continue, the teacher may find that the students learn faster and the initial-time frame may actually be reduced. For example, for the topic on cylinders, one should realise in advance that it will take less time.

5. METHODOLOGYThis means how you demonstrate, discuss, and explain a topic.• The introduction to calculating the volume of a cube and cuboid

should be done using real-life objects that the students should be encouraged to bring.

• Students must be encouraged to feel the surfaces and the space within the objects, and to measure the dimensions of the shapes.

• The formula can then be introduced on the board, with the students noting it in their exercise books.

The same procedure may be followed for the topic on cylinders.

6. RESOURCES USED• Everyday objects and models of cubes, cuboids, and cylinders.• Exercises A, B, and C.• Worksheets made from source X.• Assignment / project on drawing diagrams of the shapes.• Test worksheets made from source X.

7. CONTINUITY• Alternate sums will be done from Exercises A, B, and C for class work.• The remaining alternate sums to be done for homework.It may be noted that class work should comprise of all easy to difficult sums. Once the teacher is assured that the students are capable of independent work, homework should be handed out.

8. SUPPLEMENTARY WORKA project or assignment can be organized. It can be group work or individual research to complement and build on what the students have already learnt in class.• The students can be organized into groups of threes, and assigned to

making net diagrams for the shapes discussed.• They can do this work at home and then conduct a presentation in class.

9. EVALUATIONAs stated earlier, it is an integral teaching tool. Evaluation has to be ongoing while doing the topic, and also as an ‘end of topic’ formal test.• Students can be handed a worksheet, on day 3, covering the volume

of a cube and cuboid. It should be a 15-minute quiz and self / peer-corrected in class.

• Similarly, a quiz worksheet on the volume and surface area of a cylinder can be handed out.

• A formal test of 20 marks can be given to the students at the end of the chapter.

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Chapter 1 Operation of Sets

This chapter is a direct link to the chapter on sets done in level 6.

RECALL

The teacher will need to revise the earlier concepts before introducing new ones. The easiest way would be to redo the activity done in class 6, but with a different example.

Activity

SaraAhmedAmeeraAbdullah

AliaMayaMyra

MariumKanwalWajihaAlia

Mathematics English

The teacher could make a table of the students who got As in Mathematics and English. While doing so, students will observe that some of the students got As in both subjects.Set of students achieving A in mathematics:{Ali, Myra, Maya, Sara, Ahmed, Ameera, Abdullah}Set of students achieving A in English:{Marium, Kanwal, Wajiha, Ali, Myra, Maya, Alia}The teacher will highlight the fact that three students were common to both sets, Ali, Myra, and Maya.An interesting 10-minute game can be an alternate activity. The teacher will require two hoola hoops. Place them on the floor and label them ‘English’ and ‘Mathematics’ (make sure the hoops overlap).Call out BEGIN as the students take their respective positions in the hoops. Change the labelling of the hoops, to any other two subjects, for different sets of students.This activity will enable the students to understand the concept of sets completely. Sets are basically a method of representing groups.

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KEY WORDS / TERMINOLOGIES

This chapter not only recalls earlier concepts but also builds upon new concepts. Union of sets, intersection of sets, and complement of a set are important topics, and their symbols should be given special importance.Union is basically the conjoining of two sets, where the common and uncommon elements are written once. It is denoted by ‘∪’ which is easy to remember as union begins with ‘∪’.Intersection is basically the common elements only and this is obvious from the term itself. It is denoted by ‘∩’.Complements are the elements that are not in the set.

Example

Set A would have all the elements other than the elements of set A.

Example

Universal Set: {1, 2, 3, 4, 5, 6 …… 10} Set A: {1, 2, 3, 4}Then Set A : {5, 6, 7, 8, 9, 10}An interesting concept would be that the intersection of a set with its complement will always be a null set.Similarly, the union of a set with its complement will give the universal set, provided that the set is the only set of the universal set.

CONCEPT LIST

The best way of introducing new concepts on sets would be to use the SYMBOL TABLE on page 5 of the textbook. The students should be asked to copy it into their exercise books. The letter ‘n’ signifies the number of elements in a set.

Example

Set A {a, b, c, d}Then n(A) = 4

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Exercise 1a

This is a very brief exercise; the teacher can supplement this exercise with more practice sums. It basically involves the concept of the number of elements.

Exercise 1b

This exercise involves the new concepts introduced in this chapter. The teacher could do alternate sums on the board in class to ensure that the students understand the concept, and then give the rest for homework. The teacher can even ask the students to come to the board to attempt the sums.

FREQUENT MISTAKES

Students tend to overlook the ‘n’, and instead of mentioning the number of elements, might just state the elements. Sometimes students have difficulty in remembering the symbols and so fail to get the cue. The teacher should make sure that students memorize the SYMBOL TABLE.

Example

Set A = {a, b, c, d, e}n(A) = 5


This topic is relatively easy to comprehend and so it should not take more than 3 lessons. The activity should not take more than 10 minutes of class time.

Chapter 2 Rational Numbers

RECALL

This chapter is a continuation of what the students did in level 6 under this heading. It is important that the teacher revises the concepts of whole, natural, and real numbers. The concept of rational numbers is introduced at this level, and the order of operations is discussed in this chapter.

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Integers, rational and irrational numbers, standard form, reciprocals, comparing rational numbers of like and unlike denominators

CONCEPT LIST

Rational numbers are numbers that can be expressed in the form of p/q. Irrational numbers cannot be expressed in the form of fractions; they are numbers that do not end e.g. √ 2. While doing mathematical operations, the facts and rules should be explained with the help of examples solved on the board.

METHODOLOGY

It is important that the teacher makes the students write all the rules, as mentioned on pages 14 to 17 of the textbook, in their exercise books and also the examples that are done on the board. These will serve as a reference guide for the students when they attempt the exercises on these concepts.

Activity

This activity can be done on the board as a fun game.1. Divide the students into groups of threes.2. Write a sum: e.g. 4

1 ÷ 12 – 4

3 + 41 .

3. Ask one group to attempt the sum left to right.4. Ask the next group to follow the order of operations.5. See who gets a higher value.6. Continue with various sums. At the end, tally which group’s answer

adds up to the highest score.

NOTE: It is not necessary that the order of operations will yield the highest value or vice versa.

This activity will not only make the students practise together but will also make them realize the significance of the order of operations. Since they will be working in groups, the students will pinpoint the mistakes the group members make while working on the board.

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This activity can also be used as a timed activity, with each group attempting the maximum number of sums in a given time-frame. This will act as a maths drill—the students will learn to solve sums quickly in the allotted time.

Exercise 2a, b, and c

These exercises work progressively for each concept that is being taught. The teacher should not do these exercises in one go. In exercise 2a the students need to know the definition of ‘rational numbers’, and what it signifies.While doing exercise 2b, the following rules should be kept in mind:For addition and subtraction:+ and a + = addition with a plus sign.+ and a – = subtract and put the sign of the larger term.– and a – = add and put the minus sign.

For multiplication and division:+ and a + = multiply or divide with a plus sign.+ and a – = multiply or divide with a minus sign.– and a – = multiply or divide with a plus sign.In exercise 2c, the topics of reciprocals and ordering are covered. The students should be told that the numerators in rational numbers can only be compared if their denominators are the same.

FREQUENT MISTAKES

This chapter is the basis for algebra and advanced maths. If the students do not appreciate the importance of the order of operations and the related rules now, they will face a lot of difficulty later.


This chapter should be taught slowly, with extra time given to clarifying concepts. The teacher should spend at least 4 lessons on this chapter. A lot of 5-minute quizzes should be given to ascertain the absorption of the rules and methodology used in solving problems. More time could be spent making sure that the students have fully understood the concepts given in this chapter.

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Chapter 3 Decimal Numbers

The students are familiar with the concept of decimals from their earlier classes.The place value of decimals is a good way of revising this chapter.

RECALL / ASSOCIATION

The best way to revise decimals would be to recall the definition of decimals: decimals are numbers expressed as powers of ten with a decimal point.The place value of decimals should be revised as well. It is important that the students know the values: tenths, hundredths, thousandths etc.H T U . Tth Hth Th thIt is also important that the teacher reviews the facts that decimals: (i) have a decimal point and (ii) decimal places.

Activity

Revision could be done with a 10-minute activity similar to the ‘pinning the tail on the donkey’ game. The teacher calls out a 5- or 6-digit number, and the student has to place the decimal point at the right place.

Example

527643If the number called out is 3 hundredths then the student will place the decimal point after 6.5276.43Similarly,807945If the number called out is 5 thousandths then the student will place the decimal point after 7.807.945This activity should be done, on the board, and for not more than 5 to 10 minutes. It is a quick and fun way to revise the basics.

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CONCEPT LIST

In this chapter, students are introduced to the concepts of terminating and recurring decimals, and to the rounding-off of decimals.Students should be taught the literal meaning of the terms ‘terminating’ and ‘recurring’. ‘Terminate’ means ‘end’; terminating decimals do not go on to infinity but are finite with exact decimal places. Similarly, ‘recurring’ means ‘repeating’; recurring decimals have a number pattern that repeats itself continuously.Rounding-off of decimals has a simple methodology: circle the place value to be rounded off and then check the digit after it. If it is 5, or more than 5, the circled digit becomes one value bigger.The teacher should do a few examples on the board, and ask the students to copy them into their exercise books.

Exercise 3

Questions 1 to 8 give enough practice to students to convert fractions into decimals and to differentiate between terminating and recurring decimals.Questions 9 and 10 deal with rounding-off decimals. These questions should be introduced after the conversion concept is clearly understood. The teacher should not do this exercise in one go, but should divide it into two sections. Introduce rounding-off after doing questions 1 to 8.

FREQUENT MISTAKES

This chapter is relatively easy. However, stress should be laid on the place value of decimals, which is the root cause of the mistakes made by students. If they understand that ‘tenth’ means one decimal place, ‘hundredths’ means two decimal places, and so on, they will have understood the concept of decimals.


This chapter should not take more than 4 lessons.

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Chapter 4 Square Roots of Positive Integers

RECALL / ASSOCIATION

Students already know what square and cube numbers are. The fact that square roots can be both negative and positive should be revised.

Example

13 × 13 = 169–13 × –13 = 169Thus, both positive and negative numbers can be the square root of a square number. The teacher could also take this opportunity to revise the rules of multiplication / division (done in the previous chapter). It should not take more than a few minutes for them to recall the rule that two negative numbers, when multiplied or divided, will have a positive answer.The fact that all concepts converge and build up to form new concepts, has to be recognized by the teacher. This is imperative to create a network of mathematical concepts for the students.

METHODOLOGY

Students are already aware of prime factorization, by the division method, to find the square root. This chapter shows an alternate way of finding the square root. Students should be asked to revise the method at home, and then given a quiz in class. This will work not only as a revision tool but will also give the teacher a clue as to whether or not the students are ready for the new method.Once the revision is over, a fresh lesson should be taken for the introduction of this method. The teacher should follow the steps, given on pages 31 and 32 of the textbook, on the board. At least 3 to 4 sums should be solved, with the teacher attempting some of the exercises from the book. The students should be made to copy the steps, and the worked examples, into their exercise books.

Activity

Since this is a totally computation-based chapter, you can make it into a fun game.

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You will require different coloured chalks, a stopwatch, a clean floor, and a few students in pairs. On the floor:• write the square number with a green chalk• put bars with a different colour• use red for the divisors and blue for the dividendNow blindfold a pair of students and ask them to pick a colour. Clock them, with a stopwatch, till they finish the sum. Continue with other pairs of students. Write the finishing time of each pair on the board. Judge the winning pair by whoever finishes a sum correctly in the shortest time period.This activity will involve the participation of the entire class while each sum is being solved, will give a lot of practise, besides quickening the pace with which the students do their mathematical computation.

Exercise 4

This exercise has enough practice sums for the students. Alternate sums for the exercise can be done in the class, and the remaining for homework. Questions 2, 3, 4 and 5 are word problems. The teacher should explain the exercise to the students, with the help of a diagram, and let them come to a conclusion about the required methodology.


How many lessons this topic will take to complete depends on how fast the students understand, and remember, the steps of working. If they tend to miss out on steps, the teacher should begin each class with an oral revision of the steps and then give sums to be solved in class. The students should be allow to refer to the rules written in their exercise books. It may be a good idea to write the steps on a chart paper, and put up on the soft board, for the students to refer to whenever they need. At least 5 lessons will be needed to finish the chapter.

Chapter 5 Exponents

This concept had been informally introduced to the students while doing prime factorization. At this level, exponential notation and the rules governing it are taught. The students will use the same rules and laws, under the terms indices or index notation later on.

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Activity

The concept of exponential notation, can be re-introduced with a fun activity.You need X pack of cards, with the ace, king, queen, jack, and joker removed.Divide the students into groups of fours. Ask one student, from each group, to deal the cards.Each student then organizes his cards in the exponential form.

Example

If the student has 3 fives he will write it as:53

Next, ask the students to find the product of their exponential list.

Example

53 = 5 × 5 × 5 = 125 and so on.Now ask each student to add all the products.Whoever gets the highest score is the winner. Also, clock them with a stopwatch, and you will have another winner in terms of timing.This activity not only develops their ability to organize data, but is also an indirect way of telling the students what exponents / index / power really signify.


The students are aware of reciprocals, but may face difficulty in understanding reciprocals with exponents. The teacher needs to tell the students that the same rule applies; it is just that you need to reciprocate the value first and keep the power.The same applies to reciprocals with negative exponents; reciprocate the value and keep the power.

Example

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2b l = reciprocal would be 2 and the power will stay as 3

23 = 8

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Similarly, if the same value had a negative exponent, the same rule would apply.

Example

4–2 = 14

2–

b l

Note that the exponential value remains the same.

EQUATIONS / RULES / LAWS

There are six laws of indices, which are introduced to the students on page 40. The teacher should make sure that the students write them down in their exercise books. The teacher should also put the rules on the soft board, for the students to refer to during the week that they are doing this chapter.

METHODOLOGY

The chapter is rule-based, and the students need to study it by ASSOCIATION. The teacher should break-up the chapter’s exercises while doing this chapter. When introducing each stage and rule, the teacher should reinforce the concept with solved examples.

Exercise 5

Questions 1 to 4 should be done after the initial introduction of the chapter and activity. Once the reciprocal concept is clear, through repetitive activity on the board, questions 1 to 4 should be done in the exercise books.Similarly, the remaining questions should only be done once the laws are learnt thoroughly.


This chapter should take at least 5 lessons. The teacher should take 5-minute quizzes, in class, before moving on to the next rule / law.

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Chapter 6 Direct and Inverse Variations

RECALLThis chapter is a continuation of the topic of ratios (from class 6). The teacher should conduct a recall session whereby the rules of ratio are revised, i.e. ratios are always in the simplest / reduced form, they have the same units, and are always placed in the order of new : old.

Examplesa. 4 : 8 1 : 2b. 400 g : 1000 kg 400 g : 1000,000 g 400YY : 1000, 000YY or 1 g : 2500 kg

KEYWORDS / TERMINOLOGIES‘Direct and inverse proportion’ is the new topic related to ratios which will be done this year. This topic is termed as ‘direct variation’ and ‘inverse variation’. Proportional parts in ratios have also been discussed in this chapter.

ActivityExamples on the side bar of pages 44 and 45 of the textbook can be incorporated during the class as real-life examples. The teacher can then encourage the students to list other such examples.

Examples1. Cost and number of apples.2. Speed of a car and the time.3. Stacks of hay and the days they would last.4. Number of pipes filling up a tank and the time it would take.This can be elaborated into direct or inverse variations. For example, if 6 apples cost Rs 70, what would 8 apples cost?.

METHODOLOGYDirect variation involves a simultaneous increase and decrease. If the number of apples increases so does the cost.

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Inverse variation involves increased and decreased parity. If the speed of a car increases, the time taken to complete the journey decreases.Examples on pages 44 and 45 of the textbook can be done on the board.

Example

Apples Cost 5 Rs 25 x Rs 45When we cross multiply5 × 45 = 25 × x x = 9The number of apples costing Rs 45 is 9.

Example

Number of pipes Time 4 30 minutes 5 xThe time taken would be:Since it is inverse variations we would multiply horizontally.4 × 30 = 5 × x x = 24The students will realise, while working with inverse variations, that if one value increases the other value decreases.

Example

Proportional parts are a way of dividing proportionally according to the ratios.Suppose an amount of Rs 72,000 is to be divided amongst Ali, Maha, and Neha in the ratio of 4:3:2 respectively, then:

Ali = 94 × 72,000 = 32,000

Maha = 93 × 72,000 = 24,000

Neha = 92 × 72,000 = 16,000

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RULES / EQUATIONS

In this chapter, the students need to develop the skill of identifying direct and inverse variation. The application of the methodology is then possible by either cross multiplication or horizontal multiplication. Proportional parts require a fraction to multiply with the total amount. If the above mentioned rules and methods are memorized, this chapter will be very easy for the students.

Exercise 6

This exercise should be done in steps. The questions on direct and inverse proportion can be alternated and solved.

FREQUENT MISTAKES

Students sometimes confuse the parity, whether it is increase / increase, or increase / decrease. The teacher needs to do a lot of oral practice in the class before the students attempt the exercise.


This chapter can be completed in 4 lessons.

Chapter 7 Profit and Loss

RECALL

This chapter has already been covered, at the introductory level, in grade 6. It would be advisable to revise the concept at this level.

EQUATIONS

Profit = SP – CPLoss = CP – SPIt is important to note that when profit is incurred, the sale price is more than the cost price; but when a loss is incurred the cost price is more than the sale price.

Profit % = CPProfit × 100

Loss % = CPLoss × 100

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It is important to know these equations when attempting this chapter. After relating them to day-to-day examples, the teacher should ask the students to make a note of the formulae in their exercise books.

SUPPLEMENT

Students can create their own hypothetical situations, produce a business feasibility plan, and present it in class.

Activity

The teacher could organize a trip to a manufacturing unit, e.g. a shoe factory. The manager could make a scaled-down break-up of the production costs, and give a presentation to the students.

Overhead costs = x amountMaterial costs = y amountLabour = z amountTotal cost = w amountSale price = v amountProfit = v – w amount

Profit % = –w

v w × 100

CONCEPT LIST

In this chapter, the concept that CP is always taken as 100% and SP as ± 100% is introduced.

Example

If a manufacturer makes a profit of 20%, and the CP is Rs 600, then the SP (X) would be:

CP = Rs 600Profit % = 20%

Then,CP = 100%SP = 120%

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600 is 100% X is 120

600 × 120 = 100X

X = ×100

600 120

X = 720

Similarly, if there is a loss, the percentage loss will be subtracted from 100.

Exercise 7

This exercise gradually takes the students to various levels. However, it will be advisable to revise some of the exercises from level 6 first.

FREQUENT MISTAKES

Students tend to get confused with multiple association word problems. They first might have to find the cost price, and then with another set of data find the profit or loss. Exercise 7.6 is an example of this.Activities and day-to-day examples are very important in the teaching of this topic. If these are not incorporated, students may not comprehend the formulae and so resort to rote learning. This would not be helpful when learning advanced concepts of mathematics.


This chapter might take longer to complete as the revision is lengthy. It is important that the lesson on Profit and Loss, as done in grade 6, is revised before beginning this chapter as otherwise the class will face difficulties in solving exercise 7. This chapter can be completed in 5 lessons.

Chapter 8 Discount

This chapter deals with day-to-day experiences and the teacher can take advantage of this aspect. The students will become totally involved in the arithmetic as real-life examples are always interesting and concepts are more easily understood.

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This chapter includes some technical terms which the teacher should explain carefully. Discount is never calculated on the cost or selling price. It is a reduction on the marked / list price. Marked price should be introduced to the students, and explained that it is neither the selling price nor the cost price.

EQUATIONS

Discount = marked price – net selling price

Discount % = MPdiscount × 100

The teacher should also explain that sometimes there are multiple discounts, and one has to solve the problem step-by-step. The new, net selling price is found and it is then taken as the marked price for the next discount. Successive discounts are calculated this way. Examples 4 and 5 of this chapter can be done on the board for the students.

Activity

The teacher will ask the students to go to a shop with their friends or parents, which is offering items on sale for example a clothing outlet. They can then write an assignment on their findings.

AZEE FashionsItem 1:Marked price: Rs 400Discount percentage: 30% off

Sale price (after discount) = 10030 × 400

= 120 = 400 – 120 = 280

The students will list as many as 10 items in their report. This activity will be very valuable as the students will understand the concept well and enjoy mathematics.

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Exercise 8

This exercise has a table that must be completed. The less able students could be asked to write each question separately, and then work out the answers. Questions 5, 6, and 7 are to be done carefully as they involve multiple sets of calculations.

FREQUENT MISTAKES

As stated earlier, the students get confused with new terminologies, for example marked / list price, net selling price. The teacher should ask them to make a note of these in their exercise books. Also, while attempting the questions, a lot of importance should be given to data representation.


This chapter should take at least 3 to 4 lessons.

Chapter 9 Simple Interest

This topic has been covered in level 6. Now, additional terms related to Simple Interest are dealt with. including Zakat.It is important that real-life examples are used extensively so that this topic is understood by the students.


Debtor, creditor, principal, amount, rate of interest per annum, and Zakat are some of the terms the students will have to make a conscious effort to understand and learn.

FORMULA / EQUATIONS

Simple Interest = PRT100

where:P denotes the principal amountR denotes the rate per yearT denotes the time in yearsAmount = Principle + Simple Interest

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If the interest is given, and alternate values are asked, the formula will have to be manipulated.

T = P×

RSI 100

R = PSI 100×

TZakat has been introduced in this chapter as it is a certain percentage of the accrued amount. It can also be explained by correlating it to taxation. This topic is about percentages and how a certain percentage of an amount is calculated to determine whether it is Zakat or interest.

ActivityStudents should be asked to prepare a mock report on some assets such as gold jewellery, cash in hand, land etc, and to calculate the Zakat incurred.

Exercise 9The questions in this exercise progressively test the students on the concepts they have learnt. The students are required to read through the word problems and then organize the data. The formula should be written and then the relevant values substituted.

ASSOCIATION / RECALLIt is important that students be given worksheets of ratios, percentages, profit and loss, and simple interest sums. The revision exercise on page 62, and the Test Paper on page 63, of the textbook satisfies this requirement. This will help the students’ application skills, ability to go through the data, and apply the right formula to solve the sums.

SUGGESTED TIME LIMITThis topic should not take more than 2 to 3 lessons. The activities planned may be done as homework. The material brought in should be discussed, and pinned on the soft board.

Chapter 10 Algebraic Expressions

Algebra was taught at an introductory level in Level 6; a more detailed study is done in Level 7.

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In this topic, the identification of key terminologies is very important.Base is the variable which is denoted by a letter. It is called a variable because, depending on the value of the variable, the quantity changes.

Example

For 4x, if x is equal to 3, then the value is 12.Co-efficients are the numbers placed just before the variable.Algebraic terms and polynomials are interlinked as more than one term forms a polynomial.

METHODOLOGY

This chapter can be done on the board. The teacher should encourage the participation of each and every student. The topic is generally a favourite among students. While ordering the polynomials, the teacher should ensure that the students understand that a power / exponent determines the order of the term and not the coefficient.

Example

4x³, 5x², x5, 7

According to descending order:x5, 4x³, 5x², 7

CONCEPT LIST

In this chapter, all four operations are taught. The teacher should realise that although it is an easy topic, it should be done step-by-step.

Addition

Addition requires collection of like terms, and application of the rules of integers (already done in an earlier chapter).

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Example

Add 5x³ + 3x² + 2x and 6x³ + 2x² + x

5x³ + 3x² + 2x6x³ + 2x² + x 11x³ + 5x² + 3x

Subtraction

The same rearranging of like terms is done. However, in subtraction the term that has to be subtracted is arranged in such a way that all the signs are changed.Subtract 5x³ + 3x² + 2x from 6x³ + 2x² + x 6x³ + 2x² + x–5x³ – 3x² – 2x x³ – x² – x

Multiplication

The rules of multiplication and division are the same, but the exponents of like terms get added. Students should be asked to do the multiplication as usual, but the powers must be added. Examples on pages 68 to 70 of the textbook should be solved on the board.

Division

Division requires the powers to be subtracted and the numbers divided. Examples 15 to 21, on pages 72 to 73 of the textbook, should be solved on the board.

Exercises 10a and 10b

These exercises have enough practice sums to enable the students to grasp the concept of all four operations thoroughly.

FREQUENT MISTAKES

Students tend to judge a polynomial on the basis of the co-efficient and not the power. Plenty of practice sums on the board could help them to avoid that pitfall.

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This topic should take at least 6 lessons.

Chapter 11 Simple Algebraic Formulae

This chapter is essentially a chapter on factorization.The area of a square should be used to help students understand the topic. Otherwise, they tend to learn the factorization formulae by rote.

METHODOLOGY

This topic requires a square. The teacher should use a big cardboard cut-out of a square. How the formulae evolved should then be explained.

Example

(4 + 2)² = 36

Also:(4)² + (4)(2) + 4(2) + (2)²= 16 + 2(4)(2) + 4= 16 + 16 + 4= 36


‘square of the sum of two terms’‘square of the difference of two terms’‘product of the sum and difference of two terms’The above phrases are self-explanatory. However, the students should make a note of these terms and their formulae in their exercise books, to help them to understand and learn. The teacher should make a chart of these formulae, and put it on the soft board for the week the students are studying this topic.

Exercise 11

This exercise should be broken up into stages. Questions 1 to 6 should be done together as they deal with ‘the whole square’ formula.

4 2

4 2

2

4 4

2

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Questions 7, 11, 12, 13, 14, and 15 deal with the difference of two squares formula.Question 8 might be challenging for the least able students, and the teacher should try to explain it to them. However, if they understand that a square expression will always open up as two square terms, and that one term is the product of the two terms, then the students will not face any difficulty while attempting this question.

FREQUENT MISTAKES

Students find it difficult to differentiate the factorization of the terms, and to remember the formulae. Once again, this topic can be done on the board. The more oral practice the students get, the better they will be able to understand the concepts.


This topic should take at least 6 lessons. The teacher should give a lot of 5-minute quizzes to the students to ensure that they have understood the concepts being taught.

Chapter 12 Factorization of Algebraic Expressions

Factorization is a new topic for students of algebra, but the teacher should not have any difficulty in explaining it to them as they are well aware of the term and of the definition of factors. However, incorporating factorization into algebra can be quite a challenging task to teach.

METHODOLOGY

The teacher should first clarify that factors are terms that when multiplied give the same term back again.

Example

4x²y + 6xy² + 10xyz2xy (2x + 3y + 5z)2xy is the factor of the above term.It is not difficult to factorize polynomials but it is confusing when the formulae of factorization have to be used and applied.

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CONCEPT LIST

The first and foremost rule to teach is to check for common numbers as factors and then common variables. The common variables and numbers have to be the smallest.

Example

In the term 4xy + 6xy + 10xyz, z is the common factor, and the smallest common variables are x and y. Thus 2xy is the factor.Sometimes, multiple factorization must be done. Here, the students must first factorize and then apply ‘the difference of two squares’ formula.

Example

27x³ - 3x3x(9x² – 1)3x(3x +1)(3x – 1)3x is the common factor.

Exercises 12a and 12b

These two exercises should first be solved on the board, and then additional sums given to the students for practice.The students might face some difficulty in solving questions 12b 12 to 19. The teacher should check that most of the class are able to cope with it. Otherwise, these sums can be carried over to level 8 with the permission of the Coordinator/ Head etc.


This chapter could take longer, but the teacher should not spend more than 8 lessons on it.

Chapter 13 Simple Equations

This chapter deals with the formation, and solving, of equations. Students have been introduced to equations in earlier grades but the teacher will need to revise the rules.

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RULES

A + (plus) sign term transposes to the other side as a – (minus) sign term.A coefficient in the numerator will go on the other side as a denominator.All variable terms are always collected on the LHS, and the constants on the RHS.Rules on pages 85 and 86 of the textbook must be memorized by the students, and should be written in their exercise books.

METHODOLOGY

Since this chapter involves mostly algebra and mathematical computations, the teacher should first incorporate the following steps:1. The equation should be opened, and the brackets done away with.2. If there are fractions, the LCM should be found first and then the

equation simplified.3. The denominators must then be cross multiplied.4. Transposing of signs should then be carried out.

Example

2(4x + 1) = 5 8x + 2 = 5 8x = 5 – 2 8x × 8

1 = 3 × 81

x = 83

For word problem, the students must be able to translate the English language into mathematical terms.

Example

Three times a number is equal to the sum of the number and 6.Three times a number: 3xSum of the number and six: x + 6Thus3x = x + 6Similar examples, solved one at a time on the board, will help the students solve the exercises in this chapter with ease.

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Exercises 13a and bAlternate sums can be chosen to be solved in class, and the remaining given as homework.The word problem should be done after a lot of oral construction of equations has been done in class.

FREQUENT MISTAKESMost students find it difficult to construct equations when solving word problems. This can be remedied if the construction of equations is explained extensively orally and then worked on the board. Solving equations can be a problem if the students forget the rules of transposing. Therefore, these should be reviewed as much as possible before they are used.

SUGGESTED TIME LIMITThis chapter should cover at least 8 lessons.

Chapter 14 Perpendicular and Parallel Lines

RECALLThe students have been introduced to geometry earlier (level 6). It is important to recall the correct usage of the geometrical instruments. The recognition and definitions of lines, rays, and line segments were also taught and should be revised again.

METHODOLOGYThe construction of perpendicular and parallel lines is taught in this chapter. The steps for the construction of these drawings, as given on pages 96, 97, and 101 of the textbook, should be emphasized by the teacher and followed by the students. The teacher should use the board, and the board geometrical tools, so that the correct handling of the set square is demonstrated.After enough practice in the exercise books, the teacher should move on to the main topic of the chapter which is angles formed by parallel lines.

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Activity

A fun way, to explain parallel line angles, would be with an activity.Every student will need:A4 paper, three straws, a pair of scissors, a glue stick, a protractor.Ask each student to paste two straws on the A4 paper as parallels and one as a transversal.Label the angles formed as: a, b, c, d, e, f, g, and h.Then compare and measure.

a b

d c

e f

g h

Pairs adding up to 180°: a + b, b + c, c + d, d + a, e + f, f + h, g + e, g + hAlternate ∠s: c and e, d and f (These are equal to each other.)Interior ∠s: c and f, d and e (These are not equal but add up to 180°.)Corresponding ∠s: b and f, a and e, d and g, c and h (These are equal pairs.)


Perpendicular and parallel are terminologies that students are already aware of. Transversal, alternate, interior, and corresponding angles, as well as perpendicular and parallel, are terminologies whose definitions the students should make a note of in their exercise books.

Exercise 14

This exercise can be done in stages. Questions 1 and 2 should be done first. After the activity is done in the class, and the students have been introduced to the concept thoroughly, the remaining sums should be attempted.

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FREQUENT MISTAKES

Students get confused when identifying alternate, corresponding, and interior angles.You can explain the terms by giving them the following rules:

Alternate angles form a Z.

Interior angles form a U.

Corresponding angles form an F.


4 lessons, with an additional class for the activity, should be sufficient to complete this chapter.

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Chapter 15 Circles

RECALL

Parts of the circle are simple. However, the difference between a chord, diameter, and radius should be explained and practised in diagrams.The radius touches the circumference of a circle at one point, from the centre, while the diameter touches it at two points passing through the centre. The radius and diameter are constant values as the radius is the measure of the centre to the circumference of the circle, whereas the diameter is the measure of the circle across.A chord touches the circle at two points, but does not pass through the centre.A semicircle is half a circle, subtended (meeting at two points) by a diameter. A quadrant is a quarter of a circle subtended by two radii.Arcs and two radii subtend major and minor sectors.Sectors are a fractional value of the entire circle.The circumference of a circle is its perimeter and the circular measure of its boundary.

B

A

C

D

F

Osemicircle

chord

radius

sector

quadrantdiameter

METHODOLOGY

This chapter discusses the properties of a circle. Diagrammatic representation of the four properties (given on pages 103—105 of the textbook) should be done on the board by the teacher.Students should be given worksheets of diagrams of properties so that they can measure the angles and prove the theorems.

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L MO

AP

QB

L MOL M

O

A

C

O

A

B

D

A B

O

AB

DO A B

O●

C

The teacher should give a lot of quizzes, involving different diagrams, to groups of students and give points for correct answers. This activity helps in group work and peer teaching, which will be a useful approach for this chapter. The students’ ability, to apply the concepts they have learnt, is tested when they pick the correct theorem to be applied to find the answer.

Exercise 15This exercise requires pictorial representation. The teacher should make sure that the students draw clear diagrams before they work out the answer.

FREQUENT MISTAKESStudents find this topic very different from the mathematics they are used to. The teacher will have to deal with each property slowly, and solve the

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relevant sums step-by-step. If all the properties are done in one go, confusion will ensue.


The teacher should devote a lesson for each property. Therefore, 4 lessons, with an additional lesson for revision, are recommended.

Chapter 16 Geometrical Constructions

This chapter deals with basic construction, and takes the students progressively to a higher level. It will be useful for the students to recall the instruments of the geometry box. They should be able to identify the instruments by name. The handling of these instruments is very important. The teacher should observe the handling, by each student, very minutely.Importance should also be given to neat, clear, and well-labelled drawings. It will be easier for the teacher to do so now as the students have already been introduced to geometrical constructions at level 6.

METHODOLOGY

The following constructions are dealt with in this chapter:Bisection of a line segmentBisection of an angleConstructing angles of 60°, 90°, and 120°Construction of trianglesThe steps for construction are clearly stated on pages 107—111 of the textbook. The teacher should make the students copy these down in their exercise books and ask them to learn the steps. Recalling and writing the steps every time they do a construction will allow them to learn the rules by heart. Statistics have proven that formatting and following a methodology in mathematics plays a vital role in learning the subject.

CONCEPT LIST

The students should be asked to make a rough diagram when the data is given to construct the triangle as there are three different cases of construction of triangles.

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Verification of a right angled-triangle property by the name of Pythagorean Theorem can be done by filling out the table given on page 112.

Activity

The Pythagorean Theorem can be verified with the help of an activity. This can be done as a group activity in the class.You will need A4 paper, markers and a ruler.Ask the students to draw a triangle ABC with AB = 4 cm, BC = 3 cm, and AC = 5 cm.Next, ask them to make adjoining squares; they will make 3 squares.Calculate the areas of the 3 squares.They will discover that the area of the square formed by AC will be equal to the combined areas formed by BC and AB.

16 cm2

Square 1

9 cm2

Square 2

25 cm2

Square 3

4 cm5 cm

3 cm

3 cm

4 cm

5 cm

A

B C

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RESEARCH

Groups of students could be asked to write papers on the Greek mathematician Pythagoras and share their research in class by reading them out in class.


This is a lengthy chapter and requires the students to develop skills in construction. The teacher should assign at least 8 lessons for this chapter.

Chapter 17 Quadrilaterals


This chapter deals with various types of polygons. Polygons are shapes enclosed by 3 or more line segments. The table on page 115 of the textbook elaborates the types of polygons. 4-sided figures have various shapes with their own particular properties.Rhombi, rectangles, squares, parallelograms, and trapeziums are the various types of quadrilaterals. The properties of each shape are clearly stated in the textbook.Isosceles trapeziums, and kites are slightly more advanced shapes that can be explained once the students have understood the properties of the more common 3- and 4-sided shapes.The students need to be familiar with equilateral, equiangular, and regular polygons.

METHODOLOGY

Each shape should be explained individually, and sufficient sums should be given to reinforce the properties of the shapes. It would be advisable for the teacher to draw big and clear diagrams on the board and to work out the sums on the board with the class’ involvement.

Activity

Students could be asked, for homework, to make big chart paper cut-outs for each shape. They could clearly identify the properties in the cut-outs and make a presentation on them in class. The student or group with the

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best cut-outs and presentation could be given prizes. By listening to different presentations, the students will be able to learn the properties effortlessly.

IsoscelesTrapezium

A D

B C

A D

B C

Rhombus

A D

B C

ParallelogramTrapezium

A D

B C

A D

B C

Square

A D

B C

Rectangle

A

C

B DKite

Exercise 17

As stated earlier, the teacher should break-up the exercise in stages and let the students attempt one question at a time as they might take a while in absorbing all the properties. The students will need to associate the property with the diagram, and then apply the concepts.

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FREQUENT MISTAKES

The students may find ASSOCIATION and RECALL, for this chapter, tedious. The activity stated above will help avoid this.


This chapter should take at least 4 or 5 lessons.

Chapter 18 Perimeters and Areas

ASSOCIATION

This chapter is a continuation of earlier chapters on perimeter and area. The teacher will not need to do any ground-breaking as the students will already be well-versed in spatial geometry.

CONCEPT LIST

The definitions for area and perimeter should be recalled in class; students could note them down. In this chapter they will deal with the formulae of:

Area and perimeter of a triangle

A = 12 bh b is the base and h is the height

A is the areaP = S + S + S S is the measure of each side P is the perimeter

Area and perimeter of a rectangleA = l × b l is the length and b is the breadthP = S + S + S + S S is the measure of each side

Area and perimeter of a squareA = l × l = l²P = S + S + S + S S is the measure of each side

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Area and perimeter of a parallelogramA = b × hP = 2(S +S) a parallelogram has two equal breadths and two

equal lengths

Area and perimeter of a rhombus

A = 12 d1 × d2 d1 and d2 are the two diagonals

P = S + S + S + S S is the measure of each side

Area and perimeter of a trapezium

A = 12 h (a + b) a and b are the parallel sides

P = S + S + S + S S is the measure of each side

Area of shaded regionsThe formula for calculating this is given in example 3 on page 132 of the textbook.

Activity

The formulae derivations of a parallelogram and rhombus can be done in class as a group activity. Students need not see these derivations as mathematical computation alone; when they experience practical examples related to them, they will readily understand the derivations and their application to real-life.Things you will need:2 coloured chart papers, thick markers, ruler, and a pair of scissorsAsk the students to draw a parallelogram and a rhombus on separate chart papers. The shapes should be drawn to the size of the chart paper. Trim the chart paper along the lines drawn to form the shapes. Label each shape.They will see that the shapes are divided into two triangles each.Now, show them the working on the board, and explain. Ask them to understand and then write, the derivations inside the shapes.In the case of a parallelogram, the students will realise that the height of a parallelogram plays an important role in calculating its area area.

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Similarly, for a rhombus, the diagonals are perpendicular to each other and therefore become part of the formula.

BD = d1 AC = d2

DERIVATION

Area of ∆DAB + ∆CDB12 × BD × OA + 1

2 × BD × OC (factorize)

12 BD (OA + OC)You will note that OA + OC forms a diagonal AC, and BD is the second diagonal.Therefore,

Area of a rhombus = 12 BD × AC

= 12 d1d2

FREQUENT MISTAKES

Students sometimes do not recognize the height of a parallelogram. The derivation activity will help them do so. Shaded regions, and combined shapes’ areas, sometimes pose a problem for students. The teacher should teach them how to break-up the combined shapes and label them. When they find the area step-by-step, they will not face any difficulties. Recognizing the external and internal dimensions, in shaded area problems, is important; the teacher should explain the method.

D C

A B

O

d1

d2

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Exercise 18

Questions 1 to 6 can be given to the students, to solve, as a whole. For question 7, take a few minutes to recall the formulae, and then attempt the question. Ask the students to first identify the different shapes in 7b and then apply the formulae.Diagrams should be drawn first for the word problems.


This chapter should take 8 lessons to complete.

Chapter 19 Surface Areas and Volumes

This chapter deals with the concept of volume, involving cubes and cuboids.

METHODOLOGY

The students have already done 3-D shapes at Level 6. The teacher should allot a time slot to recall cubes and cuboids and their dimensions. The rectangular faces of a cuboid, and square faces of a cube, are important to stress upon. The height is a dimension of a 3-D shape that joins cross-sectional areas of cubes, cuboids, and prisms.

CONCEPT LIST

The conversion table on page 173 of the textbook is important and students need to learn it carefully. Also, the conversion of cubic metres to litres should be learnt thoroughly.1 cubic metre = 1000 litres1 litre = 1000 cubic centimetreVolume of a cuboid = length × breadth × heightTotal surface area of a cuboid = 2LB × 2BH × 2LHVolume of a cube = L3

Total surface area of a cube = 6L2

Activity

The identification of dimensions is very important as students are required to be able to change the values in a formula. The best way to

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ensure that the students understand this is to make net diagrams of a cube and cuboid. The teacher could make the net diagrams on a worksheet to ensure a uniform size, and ask the students to cut them out, and fold them to form the shapes.

Cube (Net diagram)Since all the sides are equal,the faces are all equal inarea and dimensions.

Cuboid (Net diagram)Since the dimensions are different,2 faces each are the same in area and dimensions.

Exercise 19

This exercise challenges the students. They have to first visualise the spatial dimensions and then apply the formula.


This chapter can easily be covered in 4 lessons.

Chapter 20 More about Graphs

This chapter is a continuation of the chapter from level 6. The students are able to read bar graphs.

METHODOLOGY

Statistics is the branch of mathematics that deals with various ways of collecting data. The textbook restricts itself to bar graphs at this level. This chapter is a good winding-up chapter as it provides more fun and play than mathematical computations.

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CONCEPT LIST

Bar graphs can be both horizontal and vertical. The concept of a scale is important. A better way of explaining ‘scale’ would be to link it with RATE. If 2 cars cost Rs 5000, then one car will cost Rs 2500. Similarly, a centimetre in a bar graph can represent a grading of Rs 5000.A key point to be stressed is that the bars are always of equal width, and that the gaps are always uniform. Ask the students to follow the steps given in the textbook on page 145.

Activity

Plan a collection of data with the students.

Examples

Data on the matches played by the school football team (won, lost, and drawn).Number of ‘As’ in mathematics in the last four years.Ask the students to represent this data as bar graphs on chart paper.This activity will involve all the students and will be fun!

Exercise 20

The teacher should make sure that the representation is clear and concise. The scale should be used carefully.


This chapter should not take more than 2 or 3 lessons. The activity could be given as a homework assignment.

Countdown Maths Oxford

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