New Edition
Countdown
Level Eight Maths
Teaching Guide
Shazia Asad
Contents
Introduction.................................................................................................. iv
Format of the guide.........................................................................................v
Lesson Planning............................................................................................. vi
Chapter 1: Operation on Sets.....................................................................2
Chapter 2: Square and Square Roots.........................................................4
Chapter 3: Cube and Cube Roots..............................................................6
Chapter 4: Binary System of Numbers.....................................................8
Chapter 5: Exponents and Radicals...........................................................9
Chapter 6: Compound Interest............................................................... 11
Chapter 7: Stock and Shares.................................................................... 12
Chapter 8: Averages.................................................................................. 14
Chapter 9: Operations on Polynomials.................................................. 15
Chapter 10: Some Simple Formulae......................................................... 17
Chapter 11: Factorization of Algebraic Expressions.............................. 18
Chapter 12: Basic Operations on Fractions............................................. 19
Chapter 13: More Simple Equations......................................................... 21
Chapter 14: Axioms and Propositions..................................................... 23
Chapter 15: Practical Geometry............................................................... 26
Chapter 16: Areas....................................................................................... 28
Chapter 17: Cylinders, Cones and Spheres............................................. 29
Chapter 18: Symmetry............................................................................... 31
Chapter 19: Statistics.................................................................................. 33
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 they are using the textbook. As a reference source, it preempts 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
iv
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.
v
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 it 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 pitfallsfalls when planning maths lessons according
to a curriculum.
2.InstructioNS
Instruction methods used include verbal explanation, material-aided
explanation, and the 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 not only tests the
student, but also indicates how well the lesson has been taught.
vi
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
the students’ ability 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 as well. 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.
vii
2.Overview
The initial concern, while planning a topic, should be ‘how much do the
students already know about the topic’? If this topic has already been
studied, though in a basic form, review the previously learnt concepts.
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 skills to be taught, or reinforced in 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 class’ receptivity 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.
The teacher may find that although the introduction takes longer, as the
session progresses, the students learn faster and the initial-time frame
may actually be reduced. For example, one should realise in advance that
the topic on cylinders, 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 in to the classroom.
viii
•
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 down 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 sums varying in their
level of difficulty. Homework should be handed out once the teacher is
assured that the students are capable of independent work.
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, this 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.
1
Chapter 1 Operation on Sets
RECALL
Before starting a chapter, it is essential for the students to revise the basic
concepts on the topic that they had been introduced to in previous
grades. There are various ways of revising. The teacher can initiate a class
discussion by prompting the students to describe and explain concepts
they already know. As a further recalling technique, they could write
them on the board. For example, the students should know the basics of
sets and the kind of sets. This, called the Socrates way of teaching by
asking, is a very successful way of revising.
KEYWORDS / TERMINOLOGIES
The students should revise complements, union, and intersection, as well
as the signs and symbols of subsets. Union is the joining of two sets,
where the common and the uncommon elements are written once. A
Union Set is denoted by ‘∪’, which is easy to remember as union begins
with ‘∪’.
Intersection of a set includes the common elements only, which is obvious
from the term itself. It is represented by an inverted ∩.
The complement of a set (A´) contains the elements that are not in the
set. For example, A´ would have all the elements that are not the members
of set A.
Example
Universal Set: {1, 2, 3, 4, 5, 6, 7, 6, 9, 10}
Set A: {1, 2, 3, 4}
Then Set A´: {5, 6, 7, 8, 9, 10} ....... complement of Set A
An interesting concept: the intersection of a set with its complement will
always be a null set.
Example
Set A : {1, 2, 3, 4}
Set A´ : {5, 6, 7, 8, 9, 10}
A ∩ A´ : ø or { }
2
Similarly the union of a set with its complement will give the universal
set, provided that set is the only set of the universal set.
Example
A
A ∪ A´ = ∑
In this chapter the concept of the power set is introduced. It tells us how
to ascertain the number of subsets that can be made from a given set.
Example
If A = {1, 2, 3, 4}. Then the number of subsets formed would be
P(A) = 2k = 24 = 16. Thus a set of 16 sets can be formed for Set A.
Exercise 1
This exercise contains problems related to power sets, and combines all
previously learnt concepts of sets. It should not be difficult for them to
solve the exercise once the concepts have been revised.
FREQUENT MISTAKES
Students sometimes forget that a null set is always a subset of any given
set. This concept should be emphasized.
SUGGESTED TIME LIMIT
This chapter is the first chapter of the term, and as only one new concept
of sets is being introduced, it should not take long. The teacher should
take the opportunity to revise other concepts also. It should be completed
within five lessons.
3
Chapter 2 Square and Square Roots
RECALL
This topic has been covered by students in previous grades, but revising
the concepts is important in order to move ahead. Finding the square
root of a number, by either the prime factorization method or the division
method, is an important concept and it should be revised with the help
of board examples and revision worksheets.
METHODOLOGY
The rules for long division method are given on page 10 of the textbook.
The students should be asked to revise the steps of mathematical
computations in previous grades and then introduced to the square roots
of decimals and vulgar fractions. The writing of finding square root of a
non-square number is primarily done by making bars and bringing them
down in pairs. Once this is done the calculation is the same. In the
division, the first divisor and the new divisor are put together and then
used for division.
Example
1
+ 1
28
+ 8
364
+ 4
3687
18. 4 7
3t 41.1409
–1
241
–224
1714
– 1456
25809
– 10192
15617
Activity
This topic may be done as a quiz on the board. The class could be divided
into four groups. Each group should select one of its members to go to
the board. Prompting by other members of the team may be allowed in
the first round, but in the final round, the team member will have to
attempt the sum on his/her own. This is a fun way to learn the steps of
4
mathematical computation by indirect peer participation and
instruction!
KEYWORDS / TERMINOLOGIES
Some of the terms introduced in this chapter are: vulgar fractions, square
roots of decimals, radicals, index of the radicand. The teacher should
stress that the square root of any square number will have both a positive,
and a negative, value.
Exercise 2
This should be a relatively easy exercise for the students to attempt but
one has to follow the steps carefully. The word problems will have to be
explained by the teacher, and the students should be told to avoid rote
learning of the sum.
Example
Question number 8 on page 13 of the textbook gives the dimensions of
a field as:
Square 1 = 2.4 m
Square 2 = 1.8 m
The students will have to perform multiple computations. First they will
have to find the radical whose square root is required, and the square
root.
Area of square 1 = 5.76 m2
Area of square 2 = 3.24 m2
Total area
= 9 square metres
The dimension= 9 = 3 meters
FREQUENT MISTAKES
Students, while doing long division, generally face problems in taking the
cue for the next step. The teacher should make them write the steps in
the exercise books, and encourage and offer help whenever they face any
difficulty.
SUGGESTED TIME LIMIT
This chapter, and the recap, should be completed in three lessons.
5
Chapter 3 Cube and Cube Roots
CONCEPT LIST
This chapter is the extension of square numbers. A square number is
obtained by multiplying the number by itself, a cube number is obtained
when the number is multiplied by itself three times over.
Activity
The best way to explain cube numbers is with the use of a 3-dimensional
object. A square is a 2-dimensional shape with equal dimensions.
Similarly a cube has three dimensions which are equal.
A square has two basic dimensions:
l × b = l × l = l2
l
l
l
l
A cube has 3 basic dimensions:
l × b × h = l × l × l = l3
l
Example
A 3 cm cube has a volume of:
3 × 3 × 3 = 27 cubic centimetres
However 28 is neither a 3D cube nor a cube number. (It can be a cuboid
28 = 2 × 2 × 7)
The teacher should make or get cubes and cuboids of different dimensions
and explain the difference.
6
METHODOLOGY
Recall that the students learnt LCM, by the prime factorization method,
in grade 6. Determining cubes and cube roots involves a similar
procedure. Now, all they have to do is to make groups of threes instead
of groups of twos as they did for square numbers.
Example
Cube root of 729
3729
3243
381
327
39
33
1
3×3×3×3×3×3 = 3 × 3 = 9
Thus 9 is the cube root of 729.
The teacher should also specify that although there are no negative square
numbers, cube numbers can be negative because three minus signs
multiply to give a minus.
Example
–8 is a cube number.
= –2 × –2 × –2 = –8. It is a cube of (–2)3.
Exercise 3
This exercise is relatively simple and can be done in class. Have the
students solve some of the sums in the class; give alternate sums for
homework.
7
FREQUENT MISTAKES
The students tend to forget to make groups of threes, since they are used
to finding square roots.
SUGGESTED TIME LIMIT
This chapter, with the activity, should be completed in three lessons.
Chapter 4 Binary System of Numbers
CONCEPT LIST
This system of numbers is generally used in computer technology. It is a
new system that the students are being introduced to, and needs to be
explained carefully with diagrams and concept as given on page 23 of the
textbook.
METHODOLOGY
A lot of explanation should be done on the board as this is a new concept.
Once the students understand the representation it will be easier to move
on to binary addition, subtraction, multiplication, and division.
Activity
This chapter can involve a lot of oral class quizzes. The teacher can write
the numerals on the board and ask each student as they raise their hands.
It can be a fun activity and the students can learn by play.
KEYWORD / TERMINOLOGIES
Binary system, binary number reader, octal system, penta-base system,
base 10 system.
Exercise 4a and 4b
Exercise 4a is very brief as it as an introductory topic exercise. Exercise
4b gives practice sums of the four operations in the binary system.
SUUGGESTED TIME LIMIT
This chapter should be covered in at least four lessons.
8
Chapter 5 Exponents and Radicals
RECALL
This chapter is a continuation of the work done in grade 7. Revision of
concepts is necessary for the students to move on to the next level.
This chapter is rule-based; students need to study this chapter through
the use of examples. The teacher should break the exercise up while doing
this chapter. When introducing each stage and rule, the teacher should
reinforce the concept with solved examples.
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 now to add all the products.
Whoever gets the highest score is the winner. The students may be timed
with a stopwatch; declare the one who finishes first, the winner.
This activity not only develops the students’ ability to organize date (data
collection) but is also an indirect way of explaining what exponents /
index / power actually signify.
KEYWORDS / TERMINOLOGIES
Continued product, sixth power, surd, mixed surds
9
CONCEPT LIST
In this chapter, the students are taught the radical and surd form and
their operations.
Example
1
3 27 actually means 27 to the power of 3 .
Once the radical form is solved the students are able to proceed with the
operation.
Example
3
27 +
2
1
1
4 = 33 × 3 + 2 2 × 2 =3 + 2 = 5
Similarly it is important to point out that any negative fractional power
becomes a positive fractional power only when the base variable is
switched from the numerator to the denominator or vice versa.
Example
–1
– 23
1
3
^ 5h 2 × ^ 5h
1 2
1 2
d
n ×d
n
5
5
1
1
1 1 ×
1 3
2
2
5
52 2
×
×
1
1
3
1 ×
4
5
54
5
1
1+3
4 4
=
1
1
4 =
5
4
5
Exercise 5
This exercise must be done progressively. After revision, practice
questions 1 to 3 should be done in class; extended sums may be given for
homework. Once the students are clear about the concept of exponential
forms, the teacher should then proceed to question 4. 5-minute class
quizzes are a good way of knowing whether or not the students have
understood the concepts.
10
SUGGESTED TIME LIMIT
This chapter should be completed within five lessons.
Chapter 6 Compound Interest
Compound interest is an extension of the concept of simple interest. This
is a system employed by most banks and financial institutions.
CONCEPT
The basic concept of compound interest is that when the tenure is of
more than a year, the principal changes as the ‘simple interest’ charged is
added to it.
The investor gains more ‘interest’ as the ‘principal’ amount increases
every year. In simple interest, however long the time period, the principal
and therefore the interest accrued remains fixed.
Activity
This concept can be reinforced by organizing a field trip to a bank. A
representative of the bank can give the students an orientation of how the
banking system works, and explain the difference between simple and
compound interest.
If a field trip cannot be organized, a banker could be invited for a
classroom lecture. This will help the students gain an understanding of
how a bank works, what its departments are, and the calculations bankers
perform on a daily basis.
The students should be encouraged to research and learn which banks in
Pakistan use compound interest.
KEYWORDS / TERMINOLOGIES
Principal, amount, time, and rate are the terminologies to be understood
by the students.
11
FORMULA
Compound interest formula is given below:
C.I = A – P
r t
A = P b1 + 100 l
where:
A is the amount
P is the principal
R is the rate
T is the time in years
The students must be told that sometimes the rate or time can also be
asked for a given amount and the principal.
Solve examples 1 to 6 on the board with active participation from the
students. Additional practice sums in class, as well as homework, should
give practice in calculating compound interest and/or other terms
in the formula.
Exercise 6
This is a progressive exercise comprising of word problems. The students
should be asked to write the given data first, then the formula, and lastly
substitute the values.
FREQUENT MISTAKES
Students tend to make mistakes while substituting values in the
formula.
SUGGESTED TIME LIMIT
This chapter should be completed in two lessons.
Chapter 7 Stocks and Shares
This chapter explains the way a financial institution works, and requires
students to use their knowledge of percentages.
12
CONCEPT
The concept of how the stock market works, shareholders’ dividends,
market value, bank draft, dividend, debenture, below par, at par, broker
etc. are the keywords to be explained, and understood, by the students.
How the shares of a company are floated should be explained. They
should know how dividends are distributed among the shareholders when
a company starts to grow and makes a profit.
KEYWORDS / TERMINOLOGIES
Stocks, banking system, saving, current fixed deposit, and recurring
accounts are some of the terminologies used relating to the banking
sector. The students should write these definitions down in their exercise
books.
Activity
The students should be encouraged to bring newspaper clippings of
the stock market. The teacher can monitor the movement of the stock
value of a number of companies weekly. The students can quote the
previous day’s stock market value, and whether it was a bullish or a
bearish run.
This will encourage some interactive conversation and the students will
be able to relate their mathematics to real-life.
Exercise 7
This exercise consists of word problems related to the stock market. These
test the students’ skills in calculating percentages as well as general
arithmetic.
SUGGESTED TIME LIMIT
Although the topic can be stretched to five lessons, it should be completed
in three or four lessons. The teacher can move on, to the next chapter,
but should continue to encourage conversation on the stock market for
the entire week.
13
Chapter 8 Averages
This is an interesting topic where the students collect actual data from
various sources. Averages is a topic that students will have come across
as the class pass average is often discussed when a test result is
announced.
CONCEPT
The concept of averages is given by a simple formula:
Averages = sum of the quantities / number of such quantities
The students can apply this formula easily. It will be advisable not to
mention frequency in place of number at this stage. The concept of mean
can be taught at a later stage.
However, the teacher should be conscious of the fact that averages lead
to mean, median and mode, and therefore the difference between the sum
of the subject, and the number, should be made clear to the students.
Activity
The students could be asked to list ten different kinds of data for example:
cricket scores, height of students, ages, and then be asked to find their
averages. These can be represented on chart paper, and displayed on the
soft board. These activities help the students understand the concept
better as they take much more interest in hands-on activities. Relating
and bringing in real-life data into the classroom motivates the students
to understand mathematics better.
METHODOLOGY
This chapter is very practical. The students can be asked to collect
different types of data similar to the ones given below.
Examples
•
•
•
•
14
Runs scored by Shahid Afridi in his last 7 appearances in the
20 Twenty matches
Temperature of Karachi in the last seven days
Stock market value for the last week
Currency exchange rate of dollar to rupee for a week
These interesting collections of data can then be used in the class by
listing them on the board and calculating their averages. The students
can copy the data into their exercise books.
Exercise 8
This exercise is also relatively easy. However, one important point that
should be stressed is that the values should all be in the same units.
Key points for the students to take note of are:
•
•
•
•
•
All values that are to be added should be in the same unit
The data should be written first
Formula should follow next
Substitute the values in the formula
Lastly, perform the calculations carefully.
FREQUENT MISTAKES
Students tend to make minor mistakes while adding and dividing
resulting in incorrect answers. The teacher should discourage the use of
calculators, and ask them to do the sums themselves. This helps develop
their computational skills.
SUGGESTED TIME LIMIT
This chapter should be completed in two to three lessons.
Chapter 9 Operations on Polynomials
RECALL
In this chapter, students will deal with algebra. Students generally enjoy
algebra as it is easy to understand and they tend not to make careless
computational errors. The teacher should revise the following topics
before teaching this chapter:
•
•
Ordering of polynomials
Addition and subtraction of polynomials
15
CONCEPT LIST
The rules given on pages 55 and 57 of the textbook need to be discussed
thoroughly in the class. Also, ask the students to write them down in their
exercise books. They must follow the step-by-step rules of multiplication
and division. The sign rules learnt in grade seven, will also apply.
The following sign rules are applied in multiplication and division:
+ and + = +
+ and – = –
– and – = +
(Although the operation (× and ÷) is done any way.)
METHODOLOGY
The teacher should ensure that the steps for performing multiplication
and division are displayed in the classroom for referral, by the students,
while they attempt the sums. Initially, the students will need to refer to
the steps but they will soon become proficient. Extra practice sums may
also be given.
Example
Horizontal multiplication can also be taught as an alternate method.
Example
m2 + mn + n2, m2 + n
(m2 + n) (m2 + mn + n2)
= m2 (m2 + mn + n2) + n(m2 + mn + n2)
= m4 + m3n + m2n2 + m2n + mn2 + n3
It is important to note that while multiplying, the powers of the same
variable are added e.g. m2 × m2 = m4.
Exercises 9a and 9b
These exercises need to be solved gradually; alternate sums can be done
in classroom and the rest given for homework.
16
FREQUENT MISTAKES
Students tend to skip steps resulting in careless errors.
SUGGESTED TIME LIMIT
This chapter should be completed within five lessons.
Chapter 10 Some Simple Formulae
RECALL
The formulae learnt in grade 7 should be revised in class.
Example
(a + b)2 = a2 + 2ab + b2
(a – b)2 = a2 – 2ab + b2
a2 – b2 = (a + b) (a – b)
CONCEPT LIST
The cube of the sum, and the difference of two terms, are learnt in this
chapter. The basic concept would be to expand the cube into a square
expression and a simple expression.
Example
(a+ b)3 = (a + b) (a + b)2
This is a very logical way of multiplying as the square will open in the
same way, by squaring the terms on the sides and multiplying them by
the two in the centre. Once the square is expanded, the then the
expression can split with each term.
The net computation result would be:
a3 + b3 + 3ab (a + b)
The cube of the difference of two terms also works in the same way.
(a – b)3 = (a – b) (a – b)2
= a3 – b3 – 3ab (a – b)
17
Exercises 10a and 10b
These two exercises deal with the above rules and also extend onto the
concept of expansion. The sums will require a lot of concentration as the
working for each sum is quite extensive and covers many steps. The
teacher should do a number of sums on the board, clearly spelling out
each step. Also, remind the students not to miss any steps as this would
result in a lot of errors.
FREQUENT MISTAKES
The students many mistakes with signs and while missing steps, or
missing a term. The teacher should ensure that these sums are
meticulously done.
SUGGESTED TIME LIMIT
5 to 6 classes will be needed to complete this chapter.
Chapter 11 Factorization of Algebraic Expressions
METHODOLOGY
This chapter deals with advanced calculations. Students are expected to
compile, and apply, all the rules of factorization.
They have to learn two things in this chapter:
•
•
Resolve into factors: cubic terms
Factorization of expressions: ax + bx2 +c
Example
(a + b)3 – (a – b)3
If the students understand that they must treat (a + b) as x and (a – b) as
y, expansion becomes quite simple.
In quadratic expressions, the treatment is totally different. The teacher
should explain to the students that there is no set formula for such
expressions. They will need to follow a few steps:
•
•
18
First, multiply the coefficient of x square with the constant.
Find the factors of the product that would either add or subtract, to
give the middle x term.
•
•
Students have to ensure that while breaking up the expression and
replacing the middle term, there should be pairs of signs. For
example, two sets of + and – signs or all plus or minus signs. One
cannot have one minus sign and the remaining three expressions with
plus signs as the pair grouping cannot be done then.
Students then have to factorize the expressions.
Activity
This chapter is a little difficult for students to comprehend, and practicing
these sums becomes quite tedious. A lot of interactive class activity might
be a good idea. The teacher can give a 5-minute quiz.
The class can be divided into groups, and sums written on the board. One
member from each group can be asked to attempt the questions. The
other group members can help when the student solving the sum makes
a mistake.
Such group activities help students develop a working relationship with
one another. They also give them a break from the lectures teachers tend
to give day-in and day-out.
Exercises 11a and 11b
The teacher should ensure that each sum is done on one page and that
not a single step is missed. These sums should be done very carefully.
FREQUENT MISTAKES
It may take the students a while to understand this chapter. The teacher
should give individual attention to the students to ensure that they are
following all the steps and formulae.
SUGGESTED TIME LIMIT
This chapter should be completed in six to eight lessons.
Chapter 12 Basic Operations on Fractions
This chapter gives an advanced level of knowledge of algebra, involving
fractions. So, if earlier concepts of algebra are still unclear the students
will face difficulty in handling these sums. A revision lesson might be a
good idea.
19
CONCEPT
Fractions in algebra
Example
a 2 + 7ab + 12b 2 a + 4b
÷
a–b
a2 – b2
(Factorize each term after the reciprocal)
a 2 + 4ab + 3b + 12b 2 (a – b)
×
(a + b) (a – b)
(a + 4b)
(a + 4b) (a + 3b) (a – b)
×
(a + b) (a – b)
(a + 4b)
Reduce the common brackets.
(a + 3b)
Answer
(a + b)
Example
a 2 – 5a + 6
a 2 – 6a + 6
The numerator and denominator are to be factorized separately and then
reduced.
Students will be required to do middle-term factorization, difference of
two squares, and the square of the sum.
Once that is done, the LCM is found. The teacher should stress that the
common brackets are only taken once, and that the uncommon terms are
also taken just like number LCM. Reducing, where the students have to
cancel the common brackets, is the easiest part.
METHODOLOGY
The teacher should revise the sums of the revision chapter on page 82 of
the textbook, before beginning this chapter as the students will need to
apply all concepts in this chapter. A lot of practice drills in class should
be done as this is a skill-based chapter.
20
Exercise 12
Questions 1 to 6 should be done first. If the students are able to tackle
these sums without any difficulty, then the teacher should proceed to
sums 7 to 15. The boxed terms have to be solved first. Sums 16 onwards
are quite lengthy and must be done very carefully.
The teacher could do each sum on the board and then erase it. The
students should then, do it on their own. Once the students are
comfortable with these sums, the teacher should let them do the rest of
the exercise without help. It is advisable that alternate sums are done in
class. In this way, the level of difficulty is spread out. For example, all odd
numbered sums from 17 to 25 could be done in class, and the even
numbered sums given for homework.
FREQUENT MISTAKES
Students get confused while factorizing terms, especially when it is the
square of the sum as this does not open the usual three-term way in
simplification. Instead the term is written twice so that it reduces easily
with the numerator or the denominator.
Example
(x + 2) 2 (x + 2) (x + 2) (x + 2)
=
=
(x + 2) (x – 2) (x – 2)
x2 – 4
Example
(a – b)2 = (a – b) (a –b)
SUGGESTED TIME LIMIT
This chapter should be completed in six to eight lessons.
Chapter 13 More Simple Equations
RECALL
The rules of transposing, learnt earlier, applies in algebraic equations. The
term, whose value is needed, is taken to the LHS of the equation. The
other terms are transposed, the positive term as negative and the
multiplicand as divisor, and vice versa.
21
METHODOLOGY
The basic rule the teacher needs to stress is that one term each should be
reduced on either side of the equation, so that the denominators can cross
multiply. The denominators can go across the equation only if there is
only one term.
For example:
17x – 5 14
12x = 2
The 2 can now multiply with the term (17x – 5) and 14 with 12x.
Example
For sums involving expressions in the denominator, first take out the
LCM and then simplify.
3
1
4
x – 1 + x +1 = x
3 (x + 1) + 1 (x – 1) 4
=x
(x – 1) (x + 1)
3x + 3 + x – x
4
= x
(x – 1) (x + 1)
x(3x + 3) = 4(x2 – 1)
3x2 + 3x = 4x2 – 4
x2 – 3x – 4 = 0
x2 – 4x + x – 4 = 0
x(x – 4) + 1(x – 4) = 0
(x –4) (x + 1) = 0
x–4=0 x+1=0
x = +4
x = –1 Answer
The LCM would be both x + 1 and x – 1. Next, 3 will be multiplied by
x + 1 and 2 by x – 1.
The LCM will be cross-multiplied with 4.The sum is then carried on with
the regular transposing that the students have done in grade 7.
22
Forming the equations of a problem sum will require patience from the
teacher; a number of sums could be done on the board. Ask the students
to break-up the statement into phrases, and then make the equation.
Example
A father is twice as old as his son. The son’s age is x and the father’s 2x.
20 years ago, the father was four times as old as his son.
son: x – 20
father: 2x – 20
2x – 20 = 4 (x – 20)
It is important that the students should have developed the ability to
convert a word statements into a mathematical statement.
Exercises 13a and 13b
The sums in both these exercises are progressive. Once the students have
understood how to solve them, they can easily be done in class, and also
be given as homework.
FREQUENT MISTAKES
Students generally make mistakes in transposing. For example, they
forget to change the sign when taking a positive term to the RHS of the
equation. The teacher should do their corrections individually, and point
out the step containing the error.
SUGGESTED TIME LIMIT
This chapter should be completed in five to six lessons.
Chapter 14 Axioms and Propositions
RECALL
Students in grade 7 have been introduced to parallel lines, transversals,
and the angles formed by them. It is important that the teacher revises
the concept of alternate, corresponding, and interior angles.
METHODOLOGY
This chapter deals with postulates, or theorems, termed axioms and
propositions in the chapter.
23
The students are given a set of data with the help of which they have to
prove a fact. This is called mathematical proof. Derivations of formulae
and proving theorems are done simultaneously. The students learn to
apply mathematical proofs systematically, improving their logic and
mathematical thinking. The teacher must ensure that the students do the
derivation in a very systematic and organized manner.
The theorems given on page 86 of the textbook should be done in detail
in the classroom.
They should be explained with diagrams on the board, and put up on the
class soft-board to refer to. As a further practice tool, ask the students to
note them in down their exercise books.
Exercises 14a and 14b
It is important that the teacher stresses the steps of mathematical proof,
labelling, and recognizing the angles. Also, it is important to mention the
reasons for the mathematical proof.
Example
C
A
O
B
90°
D
If ∠AOD = 90°
∴ ∠BOC = 90° (vertically opposite)
∠BOC + ∠AOC = 180° (angles on a straight line/supplementary)
Since ∠BOC = 90°
∴ ∠AOC = 90°
∠AOC = ∠BOD (vertically opposite angle)
Similarly ∠BOD = 90°
∠AOD = ∠BOC = ∠AOC = ∠BOD = 90° (Proved)
∠AOD + ∠BOC + ∠AOC + ∠BOD = 4(90°) = 360°
24
Question 2 of Exercise 14a
While proving that all angles are equal to 90°, it is important to mention
supplementary and vertically opposite angles as reasons for the
mathematical proof.
Example
Question 6 of Exercise 14a
4
4
5 of angle POR = 5 × 180
Therefore, reasoning done systematically is important to solve exercises
given in the chapter.
Activity
Students could be encouraged with the following activity that will make
them learn better.
You will need:
Three, two cm diameter bamboo sticks, markers, protractors, A4 paper,
and tape.
You can ask the students to place the 3 bamboo sticks in the form of an
intersection, and to label the angles with the marker and paper.
Keep on changing the measure of the angles, and ask them to prove the
questions given by measuring the angles.
If a board protractor is not available, the students can measure the angles
by tracing the angles made by the sticks on to a chart paper and then
measuring them.
Students will enjoy this activity as it will give them practical experience
of these theories.
FREQUENT MISTAKES
The students generally name the angles incorrectly.
Example
Angle PQR could be written wrongly as angle QPR. It should be
explained, to the students, that whenever an angle is formed, the
vertex is always written in the centre.
25
SUGGESTED TIME LIMIT
This chapter should be completed in 5 days.
Chapter 15 Practical Geometry
RECALL
As this is a construction chapter, the students need to develop the relevant
skills. In the previous grade, they had learnt to construct circles of given
radii, bisectors, and triangles. It is suggested that the students be given a
revision worksheet of these constructions before they proceed to this
chapter.
KEYWORDS / TERMINOLOGIES
Parallel lines, bisectors, line segments are a few key terms that the
students will be dealing with during the course of this chapter. The
students will also be required to revise the properties of these
quadrilaterals: rectangles, squares, and rhombus. Since they have to do
construction, they should be aware of these properties:
Square: all sides are equal and all angles are 90°.
Rectangles: lengths are equal and so are the breadths, the angles are
all 90°.
Rhombus: all sides are equal but the adjacent angles are supplementary.
METHODOLOGY
This chapter not only develops the students’ skill in construction, but also
makes them link their mathematical proofing and computations. The
students are expected to be involved in a ‘thinking process’ while bisecting
a line segment into ratios or while drawing an incircle or a circum-circle.
The teacher should explain to them during the working, that incircle
angle bisectors will give the centre of the angle. Similarly, a circum-circle
will require perpendicular bisectors to be constructed to yield the
centre.
The following topics are discussed in this chapter:
•
26
Drawing parallel lines: the arcs should be of the measure of the
distance asked.
•
Dividing a line segment into given ratios : ratios are to be understood;
equal parts are first constructed and then the ratios fit. For example,
2 : 3 means two and three equal parts. Subsequently, the line segment
will have to be cut into 5 parts.
•
Circum-circles and incircles: once the centre is constructed for the
incircle, the radius is taken as the perpendicular distance from the
centre to the sides. The radius of the incircle is simply from the centre
to the vertex of the triangle.
•
Quadrilaterals: various cases are explored. For example, all sides
could be given, or the diagonals and angles could be provided. It can
be suggested that the students make a rough sketch of the given case,
and then proceed.
•
Circles: tangents are to be constructed, with the point either outside
or on the circle. The students should understand that tangents are
lines touching the circle only at one point.
Exercises 15a, 15b, and 15c
These exercises should be done progressively. It can be suggested that the
teacher draws each sum on the board, and then explains to the students
how they should proceed and construct it.
FREQUENT MISTAKES
This chapter is mainly skill-based, and does not require as much
mathematical reasoning. However, stress should be given to neatness and
accuracy. Pencils should be well-sharpened, and good compasses should
be used which are neither too stiff nor too loose. Students sometimes are
not able to make perfect constructions while drawing circum-circles and
tangents. The teacher should sort-out students’ concerns individually.
SUGGESTED TIME LIMIT
This chapter will require a lot of time and patience. Two weeks is the
suggested time to complete it. Students could be given ten minute revision
worksheets on algebra or arithmetic to break the monotony, which would
also serve as revision.
27
Chapter 16 Areas
RECALL
The formulae for triangles, rectangles, squares, trapezium, and circles
should be revised. These formulae are given on pages 105 and 107 of the
textbook. It is important that, while revising the formulae, the teacher
explains the dimensions of each shape, as these will be used to substitute
in the formulae.
KEYWORD / TERMINOLOGY
Perpendicular heights are slightly difficult for students to identify as they
are sometime drawn outside the triangle. Similarly, shaded regions are
worked out by first finding the total area; then, the unshaded area is
found, and subtracted from the total area.
10 cm
5 cm
3 cm
6 cm
Area of the shaded region = Area of triangle – Area of Rhombus
1
= b 2 × 10 × 6l – (5 × 3)
= 30 – 15
= 15 cm2
METHODOLOGY
This chapter is an extension of the concepts already taught in grade 7.
The teacher now has to develop the students’ ability to visualise word
problems and then sketch them in order to work out the values and
formulae. Just as in word problems data representation is the key to the
solution. Similarly diagrammatic representation is very important in
chapters related to area and volume.
28
Students should be able to comprehend that a circle’s revolution covers,
linearly, a distance equal to its circumference:
one revolution = circumference of the circle.
revolution.
circumference
Exercises 16a and 16b
These exercises can be used for class quizzes and be done on the board.
With this activity based method, the whole class will participate
enthusiastically and the least-able students will be better able to
understand the concepts. Students may be asked to do these sums in their
exercise books as homework.
FREQUENT MISTAKES
As mentioned earlier, dimension recognition and diagrammatic
representation are the keys to success in this chapter. Students generally
fail to visualise the correct triangle, or may not understand the word
problem related to a circle, and therefore are unable to solve the sum.
SUGGESTED TIME LIMIT
This chapter should take at least six days to complete.
Chapter 17 Cylinders, Cones, and Spheres
RECALL
The students are well-versed with the concepts of volume and surface
area. It may be an idea for the teacher to revise the concepts and explain
the link between 2- and 3-dimensional figures. It is important to re-teach
the concept of height/depth, that differentiates a 2-dimensional figure
from a 3-dimensional figure.
Cubes and cuboids were done earlier in grade 7; the formulae could be
revised and the students could be given a quiz. This will help the teacher
to gauge whether or not to move on with the topic.
29
KEYWORDS / FORMULAE
In this chapter, cylinders, cones, and spheres have been discussed. Before
the formulae are given out, it is imperative that the teacher explains the
dimensions of each shape.
Cylinder: this shape has radius and height as its key dimensions.
Volume = rh2
Curved surface area = 2 rh2
Total surface area = 2 rh + r
Cone: this shape has a radius and height, but also slant height ‘l’.
1
Volume: 3
Curved surface area = rl2
Total surface area = rl + r
Sphere: this shape has a radius that is used in the formula.
4
Volume: 3 43
Surface area: 4r2
METHODOLOGY
This chapter is formula and substitution based. If the students understand
the formulae and can identify the dimensions, it should not pose a
problem for them. Continuous revision of formulae is therefore very
necessary.
Activity
The students will understand it better if the shapes are explained with the
help of net diagrams.
To make net diagrams you will need: A4 paper, geometry box, and
markers.
Cylinder: this shape is formed by folding a rectangle into a roll. The
breadth of the rectangle becomes the height of the cylinder, and the
length the circumference of its base.
a
Cylinder net diagram
‘a’ is the height of the cylinder
‘b’ is the circumference of the base of the cylinder
30
b
Cone net diagram
‘a’ is the circumference of the base of the cone
‘b’ is the slant height of the cone
a
b
a
Cones: this shape is formed by folding a semicircle or any fractional part
of a circle (sector), and a circle as its base.
The students should be encouraged to first cut out the 2-dimensional
figure and then fold it to form the 3-dimensional figure.
Exercise 17
This exercise should be done progressively. The cylinder should be
explained first, and then questions 1 to 4 should be done. Similarly, the
cone should be explained and questions 5 to 8 done. Spheres should be
done at the end, when the previous two shapes are perfectly clear to the
students.
FREQUENT MISTAKES
Students tend to confuse the volume formulae with the surface area
formula. The activity mentioned above will help them understand and
remember the formulae. Also, students’ sometimes substitute the wrong
dimensions in the formula, especially vis-a-vis the difference between the
height and slant height in cones.
SUGGESTED TIME LIMIT
This chapter should be completed within six to eight classes for the
students to thoroughly understand the concepts.
Chapter 18 Symmetry
METHODOLOGY
This is a relatively easy chapter as the concept is very simple. Symmetry
literally means an equal or mirror image. This chapter can be explained
practically by the teacher in class. Cut-outs on chart paper, can be folded
symmetrically as many times as the shape folds equally along the lines of
symmetry.
31
There are four theorems mentioned in the textbook on pages 115 and 116
that should be written by the students’ in their exercise books. The teacher
can prove these postulates on the board and explain them. The students
then need to learn the postulate so that they can apply the concepts.
Examples
i)
the letter ‘Z’ has no line of symmetry but
has a point of symmetry
ii)
the letter ‘A’ has one line of symmetry
iii)
A parallelogram has no lines of symmetry
but has a point of symmetry
KEYWORD / TERMINOLOGIES
Collinear points are points that lie on the same line. Linear symmetry and
point symmetry will also be new terms for the students to understand.
Point of symmetry is a point that is equidistant from all sides of a shape; it
is actually the centre of the shape.
Exercise 18
Students can do this exercise easily. The teacher should stress on the fact
that the students should draw the diagrams concisely. As symmetry is all
about mirror images, the image has to be drawn perfectly.
FREQUENT MISTAKES
This, as stated earlier is an easy chapter to comprehend and so the students
should not make mistakes.
32
SUGGESTED TIME LIMIT
This chapter will take two to three classes to complete.
Chapter 19 Statistics
RECALL
The students are well aware of statistics. It is the collection and
organization of data. There are various ways of representing the data; the
students are well-versed with bar graphs. The teacher should give them
a revision worksheet of bar graphs where they interpret / read and draw
bar graphs.
KEY WORDS / TERMINOLOGIES
In this chapter, not only are bar graphs revised, but histograms are
introduced as well. Interestingly, histograms of grouped data are also
introduced. Class intervals or class widths, frequency, and frequency
distribution are terminology that the students will come across in this
chapter.
METHODOLOGY
The teacher while introducing histograms will have to give a clear
differentiation between bar graphs and histograms.
The drawing of histograms will also require clear instructions. The bars
of the histograms have no gaps, but the scale chosen will have to be
accurately represented in the graph.
It will have to be explained that the ‘x’ value is sometimes given in the
form of groups, and that the values will be written in the graph on the
sides of each bar rather than the middle (as is done in bar graphs.)
x
f
50 – 60
60 – 70
70 – 80
5
7
3
33
A grouped data histogram
7
6
5
f 4
3
2
1
50
60
x
70
80
Tally marks are bundles of value of frequency. These are bundles of fives,
where the fifth bundle is slashed across.
Example
||||
Averages progresses to ‘mean’ at this level as frequency has to be
individually multiplied with the value of ‘x’ to get the ‘fx’ product.
Example
2, 2, 2, 3, 3, 5, 6, 6, 6
Mean = ∑
fx (2) (3) + (3) (2) + (5) (1) + 6 (3) 26
=
= 9 = 2.88 or 2.9
9
f
Activity
This chapter could be used as a project topic where the students can be
encouraged to collect data and then represent this on a histogram. The
data collected could be:
•
•
•
34
Number of people investing in four or five types of shares
Batting statistics of four or five famous batsmen
Age in years and number of teachers in school
Exercise 19
Questions 1 to 6 are related to converting raw data into tables and then
histograms. After this is covered thoroughly the ‘mean’ which is a natural
progression of averages should be explained and then questions 6 to 11
should be done.
FREQUENT MISTAKES
Students generally make mistakes while counting the raw data for
frequency. Ask them to add up all the frequencies; this should tally with
the numbers on the raw data. While computing the ‘fx’ for this mean,
they tend to make errors in addition.
SUGGESTED TIME LIMIT
This chapter is quite simple to understand. It should not take more than
5 classes to complete. If the students take longer to make histograms, the
teacher should extend the number of classes.
35
NOTES