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Level Seven Maths
Teaching Guide
Shazia Asad
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.Curriculum
A 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 pitfalls when planning to teach maths according to a
curriculum.
2.InstructioNS
Instruction 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 shortsighted 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.Overview
The 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.Objectives
This 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 Frame
Determining 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.Methodology
This 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.
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•
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 work
A 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.Evaluation
As 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 / peercorrected 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
Mathematics
Sara
Ahmed
Ameera
Abdullah
English
Alia
Maya
Myra
Marium
Kanwal
Wajiha
Alia
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
SUGGESTED TIME LIMIT
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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KEYWORDS / TERMINOLOGIES
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.
1 1 3 1
2. Write a sum: e.g. 4 ÷ 2 – 4 + 4 .
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.
SUGGESTED TIME LIMIT
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 th
It 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
527643
If the number called out is 3 hundredths then the student will place the
decimal point after 6.
5276.43
Similarly,
807945
If the number called out is 5 thousandths then the student will place the
decimal point after 7.
807.945
This 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.
SUGGESTED TIME LIMIT
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 = 169
Thus, 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 dividend
Now 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.
SUGGESTED TIME LIMIT
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.
KEYWORDS / TERMINOLOGIES
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
2
b 1 l = reciprocal would be 2 and the power will stay as 3
2
3
2 =8
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Similarly, if the same value had a negative exponent, the same rule would
apply.
Example
–2
1
4–2 = b l
4
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.
SUGGESTED TIME LIMIT
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
RECALL
This 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.
Examples
a. 4 : 8
1:2
b. 400 g : 1000 kg
400 g : 1000,000 g
YY : 1000, 0YY
00
400
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.
Activity
Examples 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.
Examples
1. 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?.
METHODOLOGY
Direct 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
ApplesCost
5
Rs 25
x
Rs 45
When we cross multiply
5 × 45 = 25 × x
x=9
The number of apples costing Rs 45 is 9.
Example
Number of pipes
4
5
Time
30 minutes
x
The time taken would be:
Since it is inverse variations we would multiply horizontally.
4 × 30 = 5 × x
x = 24
The 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:
4
Ali= 9 × 72,000 = 32,000
3
Maha= 9 × 72,000 = 24,000
2
Neha= 9 × 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.
SUGGESTED TIME LIMIT
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 – CP
Loss = CP – SP
It 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
Profit % = CP × 100
Loss
Loss % = CP × 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 amount
Material costs = y amount
Labour= z amount
Total cost
= w amount
Sale price
= v amount
Profit= v – w amount
v–w
Profit %
= 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 600
Profit % = 20%
Then,
CP = 100%
SP = 120%
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1
600 is 100%
X is 120
600 × 120 = 100X
600 × 120
X=
100
X = 720
Similarly, if there is a loss, the percentage of 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.
SUGGESTED TIME LIMIT
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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17
KEYWORDS / TERMINOLOGIES
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
Discount % = MP × 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 Fashions
Item 1:
Marked price: Rs 400
Discount percentage: 30% off
30
Sale price (after discount) = 100 × 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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1
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.
SUGGESTED TIME LIMIT
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.
KEYWORDS / TERMINOLOGIES
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
PRT
Simple Interest = 100
where:
P denotes the principal amount
R denotes the rate per year
T denotes the time in years
Amount = Principle + Simple Interest
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19
If the interest is given, and alternate values are asked, the formula will
have to be manipulated.
SI × 100
T = PR
SI × 100
PT
Zakat 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.
R=
Activity
Students 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 9
The 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 / RECALL
It 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 LIMIT
This 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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1
KEYWORDS / TERMINOLOGIES
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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21
Example
Add 5x³ + 3x² + 2x and 6x³ + 2x² + x
5x³ + 3x² + 2x
6x³ + 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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1
SUGGESTED TIME LIMIT
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 cutout of a square. How the formulae evolved should then be explained.
Example
4
(4 + 2)² = 36
Also:
(4)² + (4)(2) + 4(2) + (2)²
= 16 + 2(4)(2) + 4
= 16 + 16 + 4
= 36
2
4
KEYWORDS / TERMINOLOGIES
4
2
2
4
2
‘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.
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23
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.
SUGGESTED TIME LIMIT
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 + 10xyz
2xy (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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1
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 - 3x
3x(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.
SUGGESTED TIME LIMIT
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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25
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
1
1
8x × 8 = 3 × 8
3
x= 8
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: 3x
Sum of the number and six: x + 6
Thus
3x = x + 6
Similar 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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1
Exercises 13a and b
Alternate 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 MISTAKES
Most 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 LIMIT
This chapter should cover at least 8 lessons.
Chapter 14 Perpendicular and Parallel Lines
RECALL
The 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.
METHODOLOGY
The 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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27
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
g
f
h
Pairs adding up to 180°: a + b, b + c, c + d, d + a, e + f, f + h, g + e,
g+h
Alternate ∠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.)
KEYWORDS / TERMINOLOGIES
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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1
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.
SUGGESTED TIME LIMIT
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
centre, while the diameter touches it at two points passing through
centre. The radius and diameter are constant values as the radius is
measure of the centre to the circumference of the circle, whereas
diameter is the measure of the circle across.
the
the
the
the
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
F
d
r
cho d
i
D
a quadrant
semi O m
e
radius
circle
t
e
r
sector
C
A
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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1
A
A
P
L
L
M
O
M
O
L
M
O
Q
B
C
A
O
D
O
A
B
B
C
D
O
A
A
●
B
O
B
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 15
This exercise requires pictorial representation. The teacher should make
sure that the students draw clear diagrams before they work out the
answer.
FREQUENT MISTAKES
Students 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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31
relevant sums step-by-step. If all the properties are done in one go,
confusion will ensue.
SUGGESTED TIME LIMIT
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 segment
Bisection of an angle
Constructing angles of 60°, 90°, and 120°
Construction of triangles
The 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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1
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.
Square 3
5 cm
4 cm
A
25 cm2
Square 1
16 cm2
4 cm
5 cm
3 cm
B
3 cm
Square 2
C
9 cm2
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33
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.
SUGGESTED TIME LIMIT
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
KEYWORDS / TERMINOLOGIES
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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1
best cut-outs and presentation could be given prizes. By listening to
different presentations, the students will be able to learn the properties
effortlessly.
A
D
A
D
Trapezium
Parallelogram
B
C
A
B
C
D
A
Rhombus
B
Isosceles
Trapezium
C
A
D
B
C
D
A
Rectangle
B
D
Square
C
B
C
A
B
Kite
D
C
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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35
FREQUENT MISTAKES
The students may find ASSOCIATION and RECALL, for this chapter,
tedious. The activity stated above will help avoid this.
SUGGESTED TIME LIMIT
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
1
b is the base and h is the height
A = 2 bh
A is the area
P = S + S + S
S is the measure of each side
P is the perimeter
Area and perimeter of a rectangle
A = l × b
l is the length and b is the breadth
P = S + S + S + S
S is the measure of each side
Area and perimeter of a square
A=l×l
= l²
P = S + S + S + S
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1
S is the measure of each side
Area and perimeter of a parallelogram
A=b×h
P = 2(S +S)
a parallelogram has two equal breadths and two
equal lengths
Area and perimeter of a rhombus
1
d1 and d2 are the two diagonals
A = 2 d1 × d2
P=S+S+S+S
S is the measure of each side
Area and perimeter of a trapezium
1
a and b are the parallel sides
A = 2 h (a + b)
P=S+S+S+S
S is the measure of each side
Area of shaded regions
The 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 scissors
Ask 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.
1
37
Similarly, for a rhombus, the diagonals are perpendicular to each other
and therefore become part of the formula.
D
C
d2
d1
BD = d1
AC = d2
O
A
B
Derivation
Area of ∆DAB + ∆CDB
1
1
(factorize)
2 × BD × OA + 2 × BD × OC
1
2 BD (OA + OC)
You will note that OA + OC forms a diagonal AC, and BD is the second
diagonal.
Therefore,
1
Area of a rhombus = 2 BD × AC
1
= 2 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.
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1
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.
SUGGESTED TIME LIMIT
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 crosssectional 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 litres
1 litre
= 1000 cubic centimetre
Volume of a cuboid = length × breadth × height
Total surface area of a cuboid = 2LB × 2BH × 2LH
Volume 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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39
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 in
area 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.
SUGGESTED TIME LIMIT
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.
SUGGESTED TIME LIMIT
This chapter should not take more than 2 or 3 lessons. The activity could
be given as a homework assignment.
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NOTES
NOTES
NOTES