CBSE-i
UNIT - 5
ALGEBRA
CLASS VI
CENTRAL BOARD OF SECONDARY EDUCATION
Shiksha Kendra, 2, Community Centre, Preet Vihar, Delhi-110 092 India
CBSE-i
Algebra
UNIT - 5
CLASS VI
CENTRAL BOARD OF SECONDARY EDUCATION
Shiksha Kendra, 2, Community Centre, Preet Vihar, Delhi-110 092 India
The CBSE-International is grateful for permission to reproduce and/or
translate copyright material used in this publication.
The
acknowledgements have been included wherever appropriate and
sources from where the material has been taken duly mentioned. In
case anything has been missed out, the Board will be pleased to rectify
the error at the earliest possible opportunity.
All Rights of these documents are reserved. No part of this publication
may be reproduced, printed or transmitted in any form without the
prior permission of the CBSE-i. This material is meant for the use of
schools who are a part of the CBSE-International only.
Preface
The Curriculum initiated by Central Board of Secondary Education -International (CBSE-i) is a progressive step in making
the educational content and methodology more sensitive and responsive to the global needs. It signifies the emergence of a
fresh thought process in imparting a curriculum which would restore the independence of the learner to pursue the
learning process in harmony with the existing personal, social and cultural ethos.
The Central Board of Secondary Education has been providing support to the academic needs of the learners worldwide. It
has about 11500 schools affiliated to it and over 158 schools situated in more than 23 countries. The Board has always been
conscious of the varying needs of the learners in countries abroad and has been working towards contextualizing certain
elements of the learning process to the physical, geographical, social and cultural environment in which they are engaged.
The International Curriculum being designed by CBSE-i, has been visualized and developed with these requirements in
view.
The nucleus of the entire process of constructing the curricular structure is the learner. The objective of the curriculum is to
nurture the independence of the learner, given the fact that every learner is unique. The learner has to understand,
appreciate, protect and build on values, beliefs and traditional wisdom, make the necessary modifications, improvisations
and additions wherever and whenever necessary.
The recent scientific and technological advances have thrown open the gateways of knowledge at an astonishing pace. The
speed and methods of assimilating knowledge have put forth many challenges to the educators, forcing them to rethink
their approaches for knowledge processing by their learners. In this context, it has become imperative for them to
incorporate those skills which will enable the young learners to become 'life long learners'. The ability to stay current, to
upgrade skills with emerging technologies, to understand the nuances involved in change management and the relevant
life skills have to be a part of the learning domains of the global learners. The CBSE-i curriculum has taken cognizance of
these requirements.
The CBSE-i aims to carry forward the basic strength of the Indian system of education while promoting critical and
creative thinking skills, effective communication skills, interpersonal and collaborative skills along with information and
media skills. There is an inbuilt flexibility in the curriculum, as it provides a foundation and an extension curriculum, in all
subject areas to cater to the different pace of learners.
The CBSE has introduced the CBSE-i curriculum in schools affiliated to CBSE at the international level in 2010 and is now
introducing it to other affiliated schools who meet the requirements for introducing this curriculum. The focus of CBSE-i is
to ensure that the learner is stress-free and committed to active learning. The learner would be evaluated on a continuous
and comprehensive basis consequent to the mutual interactions between the teacher and the learner. There are some nonevaluative components in the curriculum which would be commented upon by the teachers and the school. The objective
of this part or the core of the curriculum is to scaffold the learning experiences and to relate tacit knowledge with formal
knowledge. This would involve trans-disciplinary linkages that would form the core of the learning process. Perspectives,
SEWA (Social Empowerment through Work and Action), Life Skills and Research would be the constituents of this 'Core'.
The Core skills are the most significant aspects of a learner's holistic growth and learning curve.
The International Curriculum has been designed keeping in view the foundations of the National Curricular Framework
(NCF 2005) NCERT and the experience gathered by the Board over the last seven decades in imparting effective learning to
millions of learners, many of whom are now global citizens.
The Board does not interpret this development as an alternative to other curricula existing at the international level, but as
an exercise in providing the much needed Indian leadership for global education at the school level. The International
Curriculum would evolve on its own, building on learning experiences inside the classroom over a period of time. The
Board while addressing the issues of empowerment with the help of the schools' administering this system strongly
recommends that practicing teachers become skillful learners on their own and also transfer their learning experiences to
their peers through the interactive platforms provided by the Board.
I profusely thank Shri G. Balasubramanian, former Director (Academics), CBSE, Ms. Abha Adams and her team and Dr.
Sadhana Parashar, Head (Innovations and Research) CBSE along with other Education Officers involved in the
development and implementation of this material.
The CBSE-i website has already started enabling all stakeholders to participate in this initiative through the discussion
forums provided on the portal. Any further suggestions are welcome.
Vineet Joshi
Chairman
Acknowledgements
Advisory
Shri Vineet Joshi, Chairman, CBSE
Shri Shashi Bhushan, Director(Academic), CBSE
Ideators
Ms. Aditi Misra
Ms. Amita Mishra
Ms. Anita Sharma
Ms. Anita Makkar
Dr. Anju Srivastava
Dr. Indu Khetarpal
Ms. Vandana Kumar
Ms. Anju Chauhan
Ms. Deepti Verma
Ms. Ritu Batra
Conceptual Framework
Shri G. Balasubramanian, Former Director (Acad), CBSE
Ms. Abha Adams, Consultant, Step-by-Step School, Noida
Dr. Sadhana Parashar, Head (I & R),CBSE
Ms. Anuradha Sen
Ms. Jaishree Srivastava
Ms. Archana Sagar
Dr. Kamla Menon
Ms. Geeta Varshney
Dr. Meena Dhami
Ms. Guneet Ohri
Ms. Neelima Sharma
Dr. Indu Khetrapal
Dr. N. K. Sehgal
Material Production Group: Classes I-V
Ms. Rupa Chakravarty
Ms. Anita Makkar
Ms. Anuradha Mathur
Ms. Kalpana Mattoo
Ms. Savinder Kaur Rooprai
Ms. Monika Thakur
Ms. Seema Choudhary
Mr. Bijo Thomas
Ms. Kalyani Voleti
Dr. Rajesh Hassija
Ms. Rupa Chakravarty
Ms. Sarita Manuja
Ms. Himani Asija
Dr. Uma Chaudhry
Ms. Nandita Mathur
Ms. Seema Chowdhary
Ms. Ruba Chakarvarty
Ms. Mahua Bhattacharya
Material Production Groups: Classes VI-VIII
English :
Ms. Rachna Pandit
Ms. Neha Sharma
Ms. Sonia Jain
Ms. Dipinder Kaur
Ms. Sarita Ahuja
Science :
Dr. Meena Dhami
Mr. Saroj Kumar
Ms. Rashmi Ramsinghaney
Ms. Seema kapoor
Ms. Priyanka Sen
Dr. Kavita Khanna
Ms. Keya Gupta
Mathematics :
Ms. Seema Rawat
Ms. N. Vidya
Ms. Mamta Goyal
Ms. Chhavi Raheja
Political Science:
Ms. Kanu Chopra
Ms. Shilpi Anand
Geography:
Ms. Suparna Sharma
Ms. Leela Grewal
History :
Ms. Leeza Dutta
Ms. Kalpana Pant
Material Production Groups: Classes IX-X
English :
Ms. Sarita Manuja
Ms. Renu Anand
Ms. Gayatri Khanna
Ms. P. Rajeshwary
Ms. Neha Sharma
Ms. Sarabjit Kaur
Ms. Ruchika Sachdev
Geography:
Ms. Deepa Kapoor
Ms. Bharti Dave
Ms. Bhagirathi
Ms. Archana Sagar
Ms. Manjari Rattan
Mathematics :
Dr. K.P. Chinda
Mr. J.C. Nijhawan
Ms. Rashmi Kathuria
Ms. Reemu Verma
Science :
Ms. Charu Maini
Ms. S. Anjum
Ms. Meenambika Menon
Ms. Novita Chopra
Ms. Neeta Rastogi
Ms. Pooja Sareen
Political Science:
Ms. Sharmila Bakshi
Ms. Srelekha Mukherjee
Economics:
Ms. Mridula Pant
Mr. Pankaj Bhanwani
Ms. Ambica Gulati
History :
Ms. Jayshree Srivastava
Ms. M. Bose
Ms. A. Venkatachalam
Ms. Smita Bhattacharya
Coordinators:
Dr. Sadhana Parashar,
Ms. Sugandh Sharma,
Dr. Srijata Das,
Dr. Rashmi Sethi,
Head (I and R)
E O (Com)
E O (Maths)
E O (Science)
Shri R. P. Sharma, Consultant Ms. Ritu Narang, RO (Innovation) Ms. Sindhu Saxena, R O (Tech) Shri Al Hilal Ahmed, AEO
Ms. Seema Lakra, S O
Ms. Preeti Hans, Proof Reader
CONTENT
Preface
Acknowledgment
1.
Syllabus
1
2.
Scope Document
1
3.
Teacher's support Material
8
? Teacher's note
9
? Activity skill matrix
14
? Warm up Activity W1 :
16
•
Observing patterns - Playing with a square
? Warm up Activity W2 :
•
Carrying cards
? Warm up Activity W3 :
•
20
Describing an algebraic expression: Variables and constants
? Content Worksheet C4 :
•
20
Defining terms and algebraic expressions
? Content Worksheet C3 :
•
19
Variables and Constants: A Tour to the Land of Unknown
? Content Worksheet C2 :
•
19
Repairing the Doodle copter of Dr. Kind
? Content Worksheet C1 :
•
19
Appreciate your knowledge
? Pre Content Worksheet P2 :
•
18
Fun with mathematics
? Pre Content Worksheet P1 :
•
18
Math lab Activity with Tiles
? Warm up Activity W5 :
•
17
Cooking mathematics
? Warm up Activity W4 :
•
16
Algebraic expressions through Pattern
22
CONTENT
? Content Worksheet C5 :
•
Words into expressions
? Content Worksheet C6 :
•
34
Math Award
? Content Worksheet C16 :
•
34
Fun Corner
? Content Worksheet C15 :
•
31
Equations using Tiles
? Content Worksheet C14 :
•
27
Balancing Equations
? Content Worksheet C13 :
•
26
Independent practice
? Content Worksheet C12 :
•
24
Let's race!
? Content Worksheet C11 :
•
24
Fun Corner
? Content Worksheet C10 :
•
23
Evaluating Expressions
? Content Worksheet C9 :
•
23
FUN CORNER : SUDOKU Square Puzzle
? Content Worksheet C8 :
•
23
Like and Unlike Terms
? Content Worksheet C7 :
•
22
34
Independent Practice
? Post Content Worksheet PC1
35
? Extended Practice 1
? Post Content Worksheet PC2
35
? Extended Practice 2
? Post Content Worksheet PC3
? Assessment of the chapter
36
CONTENT
4.
Assessment guidance plan
40
5.
Study material
42
6.
Student's support material (Student's Worksheets)
? SW 1 : Warm up W1:
•
Observing patterns - Playing with a square
? SW 2 : Warm up W2:
•
91
Words into expressions
? SW 13 :Content Worksheet C6 :
•
88
Algebraic expressions through Pattern
? SW 12:Content Worksheet C5 :
•
84
Describing an algebraic expression: Variables and constants
? SW 11 :Content Worksheet C4 :
•
81
Defining terms and algebraic expressions
? SW 10 :Content Worksheet C3 :
•
79
Variables and Constants: A Tour to the Land of Unknown
? SW 9 :Content Worksheet C2 :
•
77
Repairing the Doodle copter of Dr. Kind
? SW 8 :Content Worksheet C1 :
•
75
Appreciate your knowledge
? SW 7 :Pre Content Worksheet P2 :
•
74
Fun with mathematics
? SW 6 : Pre Content Worksheet P1 :
•
72
Math lab Activity with Tiles
? SW 5 : Warm up W5:
•
71
Cooking mathematics
? SW 4 : Warm up W4:
•
70
Carrying cards
? SW 3 : Warm up W3:
•
68
Like and Unlike Terms
96
CONTENT
? SW 14:Content Worksheet C7 :
•
FUN CORNER: SUDOKU Square Puzzle
? SW 15 :Content Worksheet C8 :
•
129
Extended Practice
? SW 22 :Post Content Worksheet PC2 : SW 23 :
•
127
Independent Practice
? SW 21 :Post Content Worksheet PC1 :
•
125
Math Award
? SW 20 :Content Worksheet C16 :
•
123
Fun Corner
? SW 18 :Content Worksheet C15 :
•
120
Equations using Tiles
? SW 17 :Content Worksheet C14 :
•
114
Balancing Equations
? SW 16 :Content Worksheet C13 :
•
110
Independent practice
? SW 19 :Content Worksheet C12 :
•
108
Let's race!
? SW 18 :Content Worksheet C11 :
•
105
Fun Corner
? SW 17 :Content Worksheet C10 :
•
103
Evaluating Expressions
? SW 16 :Content Worksheet C9 :
•
101
131
Post Content Worksheet PC3
? Assessment of the chapter
? Acknowledgments
136
? Suggested videos/ links/ PPT's
138
Syllabus
Algebra
Concept of constants and variables Introduction to
algebra, idea of an algebraic expression, terms and
coefficients, like and unlike terms, value of an
ALGEBRA
algebraic expression, operations (addition and
subtraction) of algebraic expression, linear equation
in one variable.
SCOPE DOCUMENT
Algebra
Prerequisite: Recall and review Whole numbers, four operations of Whole numbers, properties of
Whole numbers and Integers.
Concepts:
•
Concept of a constant and a variable
•
Introduction to Algebra
•
Idea of Algebraic expressions
•
Terms of an Algebraic expression
•
Like and unlike terms
•
Coefficient of a term
•
Value of an Algebraic expression
•
Addition and subtraction of like terms
•
Linear equations in one variable
1
Learning objectives
At the end of this unit the student will be able to
1. Understand and differentiate between variables and constants,
2. Define and find the terms of algebraic expressions.
3. Find the coefficients of a term
4. Investigate and differentiate between the like and unlike terms.
5. Express words into expressions,
6. Understand and evaluate algebraic expressions.
7. Simplify algebraic expressions.
8. Solve linear equations with one variable
9. Convert real life problems into an equation to find the unknown quantity.
EXTENSION: INVESTIGATE
(1)
List all of the 2-digit multiples of 9. What do you notice about the sum of their digits?
(2)
Using the fact that any 2-digit number can be represented algebraically as 10a+b, show/
explain the following:
If a 2 -digit number is a multiple of 9, so is the sum of its digits AND if the sum of the digits
of a 2-digit number is divisible by 9, then the number is a multiple of 9.
(3)
Try to state and prove a similar result for 3- and 4-digit numbers!
A number and its reversal:
(1)
To be a mathematical researcher, one needs to do what the scientific researcher does. Collect
lots of data first, then make conjectures and PROVE them!
Choose at least 5 different 2-digit numbers, in addition to the examples below, and complete
the table.
Number..........Reversal............Sum..........Difference
41.....................14.......................55...............27
33....................33......................66................0
72....................27......................99................45
Your turn - do this FIVE more times.
2
(Larger-Smaller)
(2)
Make conjectures about the how the sum and difference are related to the digits of the
original number. Using the algebraic representation 10a+b for any 2-digit number, PROVE
your conjectures (or disprove them!).
(3)
72 and 27 are not only reversals. They are also both multiples of 9.
Does this have to be true for any 2-digit multiple of 9?
Explain! Further, is there a special property for the sum of the number and its reversal in this
case. Make sure you verify conjectures for several cases before attempting to prove it.
(4)
Make a similar table for 3-digit numbers. Is there an obvious relationship for the sum of the
number and its reversal this time?
The difference? Make conjectures and PROVE them!
Activities/resources/projects:
1.
Construct algebraic tiles to show mathematical operations in algebra.
2.
List of games can be played with the students: Flash Card Game, Board Game and online
games such as Algebra Jeopardy and Basket ball algebra games at weblink 1 and weblink 6
3.
MSExcel Quiz Game
You may pose this game to your students and find out how much they have learnt! Open an
Excel sheet and fill up row 1 as shown
Write some questions in the first column as shown and enter the operations against each column
as add/subtract.
3
Write the correct answers as per the respective operations, in column F. Leave column D and E
blank as shown below.
4
Fill up cell G2 in the formula bar as shown below. Then drag to apply the formula in the entire row
You may hide column F and column G to hide your answers. The sheet now looks as below:
5
Now ask your students to write the answer in “Your Answer” column and the response of the excel
sheet shall be “correct” if the answer matches with the correct answers entered by you and
“incorrect” if it does not.
6
Cross curricular links:
Science:
•
Linking up temperature and light to investigate links between the two.
•
Comparing the efficiency of washing powders by measuring the amount of light that passed
through dirty and clean fabrics
•
Investigating the factors affecting the height to which a ball bounced
Social studies:
• In geography, to investigate the rate of flow of water in a stream.
Physical education:
•
Investigations in Physical Education on how exercise affects heart rate.
Life skills:
• Problem Solving
7
8
TEACHER’S NOTE
The teaching of Mathematics should enhance the child’s resources to think and reason, to visualise
and handle abstractions, to formulate and solve problems. As per NCF 2005, the vision for school
Mathematics include :
1. Children learn to enjoy mathematics rather than fear it.
2. Children see mathematics as something to talk about, to communicate through, to discuss
among themselves, to work together on.
3. Children pose and solve meaningful problems.
4. Children use abstractions to perceive relation-ships, to see structures, to reason out things, to
argue the truth or falsity of statements.
5. Children understand the basic structure of Mathematics: Arithmetic, algebra, geometry and
trigonometry, the basic content areas of school Mathematics, all offer a methodology for
abstraction, structuration and generalisation.
6. Teachers engage every child in class with the conviction that everyone can learn mathematics.
Students should be encouraged to solve problems through different methods like abstraction,
quantification, analogy, case analysis, reduction to simpler situations, even guess-and-verify
exercises during different stages of school. This will enrich the students and help them to
understand that a problem can be approached by a variety of methods for solving it. School
mathematics should also play an important role in developing the useful skill of estimation of
quantities and approximating solutions. Development of visualisation and representations skills
should be integral to Mathematics teaching. There is also a need to make connections between
Mathematics and other subjects of study. When children learn to draw a graph, they should be
encouraged to perceive the importance of graph in the teaching of Science, Social Science and
other areas of study. Mathematics should help in developing the reasoning skills of students. Proof
is a process which encourages systematic way of argumentation. The aim should be to develop
arguments, to evaluate arguments, to make conjunctures and understand that there are various
methods of reasoning. Students should be made to understand that mathematical communication
is precise, employs unambiguous use of language and rigour in formulation. Children should be
encouraged to appreciate its significance.
9
At the upper primary stage, students get the first taste of power of Mathematics through the
application of powerful abstract concepts like Algebra, Number System, Geometry etc. Revisiting of
the previous knowledge and consolidating basic concepts and skills learnt at the Primary Stage
helps the child to appreciate the abstract nature of Mathematics. Whether it is Number system or
algebra or Geometry, these topics should be introduced by relating it to the child’s every day
experience and taking it forward to abstraction so that the child can appreciate the importance of
study of these topics.
The mathematics curriculum during the preschool, elementary school and the middle school years
has many components. Proficiency in algebra, as in number system is an important foundation in
the mathematics education and its further use.
The students should be encouraged to use algebraic expressions to comprehend difficult problems
by finding the quantitative relationships for formalizing patterns, functions and generalizations. At
this stage it is important for the students to be able to convert the symbolic and verbal expressions
into numerical and quantitative relationships and vice-versa. The students should develop an
understanding of several different meanings and uses of a variable by representing quantities in a
variety of problem situations. The teacher may integrate the learning of algebra with the other
topics of curriculum and give the students real life situations where without the use of algebra,
problem-solving would be impossible or very difficult.
The students should be able to represent, analyze and generalize a variety of patterns with
symbolic expressions.
A simple problem like below can be used to introduce algebraic expressions. We begin with a
square of side 3x3 and colour the middle as shown. The number of squares coloured is 1. In stage
2, we have a square of size 5x5 and colour the middle, leaving the boundary. The number of
squares coloured are 32. At stage 3, the size is 7x7 and the number of squares coloured are 52. The
students may be asked to generalize this for the nth stage. The size of the square shall be
(2n+1)x(2n+1) and the number of the square coloured shall be (2n-1)2.
10
12
Stage 1
Stage 2
52
32
Stage 3
72
11
Stage 4
The students may be motivatied to think about the general stage and find out the size of the
square grid taken in the general stage. They may also find out the number of squares coloured at
the general stage. The teacher may, otherwise also ask the students to find out the respective
answers at any particular stage, say the 5th, 6th or the 7th.
Students’ comfort with the symbolic manipulation can be enhanced if it is based on extensive
experience with quantities in contexts through which students develop an initial understanding of
the meanings and use of variables and an ability to associate symbolic expressions.
The main aim of teaching algebra at this stage is to develop students’ facility with using patterns
and functions to represent, model, analyse a number of relationships in mathematics problems or
in the real world. Opportunities should be found in many other areas of the curriculum to model
relationships in everyday contexts.
To be taken care of…
Teachers need to be attentive to the conceptual obstacles that the students may face as they make
a transition from the real world of number system to the abstract world of variables. With what
they have learnt in number system, they might be tempted to add or subtract the algebraic
expressions without considering the like and unlike terms. Teacher should check to see if his/her
students have this misconception and should take steps to build their understanding.
While teaching addition and subtraction of algebraic expressions or while teaching the like and
unlike terms, the teacher may give examples of like and unlike terms as ‘just as you cannot add 4
bananas and 3 spoons to give you 7 bananas or 7 spoons, you cannot add 4x+5y to give you 9y’.
The teacher may also encourage the students to frame a real life problem from a given equation or
expression and therefore justify the applicability of algebra in real life.
12
COMMON ERRORS
Type of error
Error made
Correction
2 more than 3 minus a number X(2+3)x
x
3x
2 less than 4 times y
4y
3x+2
(2-4y) or (2-4)y
4y-2
2 added to x divided by y
X=(2+x)/y
(x/y)+2
Overview of the students’ worksheets
The first Warm up activity (W1) is a brain exerciser where the students are using their previous
knowledge of rotations and shall generalize the results for the given figures.
In Warm Up activity (W2), the students shall find out the missing numbers and a general pattern
without actually using algebraic symbols.
Warm up activities (W4) and (W5) recapitulates the properties of numbers done by the students in
earlier lessons.
In the pre content activity, the students are informally introduced to algebraic expressions without
actually defining any abstract variables.
The focus on the warm up an pre content activities shall be to refresh the previous knowledge of
the students so that they can comfortably build up the new topic. The pre content activities act as
a bridge between the previously learnt concepts and the new concepts to be studied.
The content worksheets from C1 through C16 aim at achieving the above stated learning
objectives. Not only shall the students learn the basic concepts of algebraic expressions, they shall
be encouraged to apply them to their daily lives. The teacher may encourage them to make
projects where they can appreciate the applicability of algebra in real life.
Further the post content activities are designed to assess the students’ understanding of the
concepts learnt in the unit.
13
Activity – skill Matrix
Activity
Name of the activity
Skills learnt
Warm up (W1)
Observing patterns - Playing
with a square
Geometrical understanding of
rotation
Warm up (W2)
Carrying cards
Understanding and
comprehension
Warm up (W3)
Cooking mathematics
Comparative and calculative
skills
Warm up (W4)
Math lab Activity with Tiles
Understanding of the negative of
a number
Warm up (W5)
Fun with mathematics
Appreciation of mathematics
Pre content (P1)
Appreciate your knowledge
Quantitative and calculative
Pre content (P2)
Repairing the Doodle copter
of Dr. Kind
Comprehension
Content worksheet (C1)
Variables and Constants: A
Tour to the Land of Unknown
Knowledge and understanding
Content worksheet (C2)
Defining terms and algebraic
expressions
Describing an algebraic
expression: Variables and
constants
Comprehension
Content Worksheet (C4)
Algebraic expressions through
Pattern
Reasoning and comparison skills
Content worksheet (C5)
Words into expressions
Language, Comprehension and
Problem Solving
Content Worksheet (C6)
Like and Unlike Terms
Understanding and thinking
Content Worksheet (C3)
14
Pictorials understanding
Content Worksheet (C7)
fun corner : sudoku square Appreciation of mathematics
puzzle
Content Worksheet (C8)
Evaluating Expressions
Diagramatical and mathematical
understanding
Content Worksheet (C9)
Fun Corner
Computational and language
comprehension
Content Worksheet (C10)
Let’s race!
Understanding and thinking
Content Worksheet (C11)
Independent practice
Application, language
comprehension and Problem
solving
Content Worksheet (C12)
Balancing Equations
Reasoning and critical thinking
Content Worksheet (C13)
Equations using Tiles
Reasoning and critical thinking
Content Worksheet (C14)
Fun Corner
Appreciation of mathematics
Content Worksheet (C15)
Math award
Understanding, language skills
and comprehension
Content Worksheet (C16)
Independent Practice
Application and problem solving
skills
Post Content Worksheet (PC1) Extended Practice
Knowledge and self learning
Post Content Worksheet (PC2)
Knowledge and self learning
Post Content Worksheet (PC3) Assessment of the unit
Knowledge and self learning
15
SW – 1
Observing patterns - Playing with a square
WARM UP ACTIVITY W1
Description – This is taken as a class activity. A square is painted blue on one side and red on flip
side. Teacher will distribute cut out of such a square to the students. Teacher then gives
instructions to rotate/flip the square in number of ways.
Execution - Students understand and analyze the pattern and then answer the questions asked by
the teacher.
SW – 2
Carrying cards
WARM UP ACTIVITY W2
Description – Students are invited in groups of four as shown below.
16
Four children stand behind the first one and so on four rows are added so that the last question
can be answered. These sixteen children are standing in four rows of four, one behind the other.
They are each holding a card with a number on it. Teacher explains the rule which is followed by
students to find the missing numbers on the cards.
Pre Preparation - Teacher will make flash cards as shown in the figure.
Execution – Teacher invites the students to stand in the grid formation and build the numbers on
the white board as shown below. Encourage learners to explain how they know what the number
on each board is, and draw attention to the fact that each number might be worked out in several
different ways.
SW – 3
Cooking mathematics
WARM UP ACTIVITY W3
Description – This is taken as a class activity. Teacher to get the class make Cherry buns using the
given recipe. Teacher encourages the students solve this problem by making pictures.
The students work in pairs and discuss different ways to solve the problem.
(This can be taken as a good example to introduce the term variable for the unknown quantity.)
17
SW – 4
Math lab Activity with Tiles
WARM UP ACTIVITY W4
Description – This is taken as a class activity. Teacher gives the instructions on how to use integer
tiles and gets the class play with integer tiles. This activity is used introduce addition and
subtraction of integers. With the help of integer tiles the comfort level of students with the signs of
integers increase immensely. A worksheet is followed by the activity to reinforce the addition and
subtraction of integers.
Pre Preparation – Teacher prepares sufficient integer tiles for the students to practice.
SW – 5
Fun with mathematics
Warm up activity W5
Objective – Recapitulation of the work done on whole numbers and integers.
Description – The teacher will hand out a riddle to the students. Few questions are given as clues
to solve the riddle. The questions are the recapitulations of the concepts done in the previous unit.
The teacher will be able to identify the students who are weak in the concepts of the previous unit
and will be able to work with them.
18
SW – 6
Appreciate your knowledge
PRE CONTENT ACTIVITY P1
Objective – Recapitulation of the work done on whole numbers and integers.
Description – After the discussion of worksheet given for Activity 1, the teacher will give few more
questions to enhance the basic skills required to work on the unit integers.
Execution- Teacher will give a handout to the students to complete. Each student of the class
solves the worksheet. Teacher monitors the level of knowledge and then plans the lessons
accordingly
SW – 7
Repairing the Doodle copter of Dr. Kind
PRE CONTENT WORKSHEET P 2
Objective - Explore and know about the need and importance of unknown numbers.
Description –Teacher will give the handout of a passage on repairing of the Doodle copter Of Dr.
Kind. The passage explains and guides the student to solve the puzzle and help Dr. Kind to repair
his machine.
SW – 8
Variables and Constants: A Tour to the Land of Unknown
CONTENT WORKSHEET C 1
Objective – To introduce the idea of variable, constant and algebra
Material Required - Dictionary/Picture math dictionary available online.
Description - Student would refer to the dictionary and find the meaning of the word variable,
constant and algebra. The meaning can be put up on the board to be referred to by the students.
19
Based on the meaning learnt the teacher will prepare a worksheet for the students to comprehend
what they have learnt.
SW – 9
Words into expressions
CONTENT WORKSHEET C 2
Objective – To understand the meaning of algebraic and numeric expressions and coefficients of
variables
Material Required - Dictionary/Picture math dictionary available online and display the picture on
the board as given below
Description - Student would refer to the dictionary and find the meaning of the word algebraic
expression, numerical expression, term and coefficients. The meaning is put up on the board to be
referred to by the students. This may also be written on a flash card and displayed on the board.
Teacher then displays the picture of certain objects on the board.
For example:
Teacher encourages the student to express expressions with the help mathematical operators.
Teacher may use the examples as mentioned below to explain the concepts.
Sample 1
Sample 2
20
Sample 3
This activity would reinforce the concept constants and variables. Further it will help student to
understand the difference between algebraic expression and numerical expression. Through this
activity students will also know how to find out:
•
Number of terms in an algebraic expression.
•
Coefficient in algebraic terms.
SW – 10
Describing an algebraic expression: Variables and constants
CONTENT WORKSHEET C3
Objective - Make students understand and correlate pictures with algebra
Material Required – 6 cups and 50 marbles
Description- This activity describes the representation of algebraic expressions. Cups are
containers with a number of counters in them. Teacher displays the model with some cups and
counters and encourages students to express the picture model into the algebraic expression.
21
SW – 11
Algebraic expressions through Pattern
CONTENT WORKSHEET C 4
Objective: To understand algebraic expression through patterns
Material Required – : Two bundles of 25 matchsticks/ tooth picks
Description – Teacher will ask the students to make patterns using matchsticks and then asks
questions to do as given in worksheet 11.
Sample pattern
NOTE: Teacher will make students do more patterns of letters such as c, f and n and so on. Teacher
can ask students to make their own pattern with at least three repeats and a similar questionnaire
can be given to them to answer.
SW – 12
Words into expressions
CONTENT WORKSHEET C 5
Objective - Recapitulation of the work done in content C 1.1.
Description - This is a recapitulation worksheet to make students recall and practice all the
concepts learnt in content C 1.1.
22
SW – 13
Like and Unlike Terms
CONTENT WORKSHEET C 6
Objective: To distinguish between like, unlike terms
Description – Through this worksheet teacher will discuss like and unlike terms and their
relevance. The idea of like and unlike terms is reinforced using weblink 3. As a follow up worksheet
is prepared and given to the students so that they practice the concept.
SW – 14
FUN CORNER : SUDOKU Square Puzzle
CONTENT WORKSHEET C 7
Objective – Practicing simplification of algebraic expressions
Description – Teacher encourages students to practice all the concepts learnt so far with the help
of few puzzles. The instructions are given to along with the puzzles. The students follow the
instructions given by the teacher and solve the puzzles. This will be good and interesting way of
learning and practicing simplification of algebraic expressions.
Pre preparation: Teacher will prepare the grid and magic squares and hand out to the students to
start the exercise.
Execution- Teacher will distribute hand out to the students. Teacher will then encourage the
students to solve each expression.
SW – 15
Evaluating Expressions
CONTENT WORKSHEET C 8
Objective: To learn to evaluate algebraic expressions through tiles.
23
Description: Teacher will prepare a hand out of problems involving variable tiles and number tiles.
Teacher will then encourage the students to evaluate the value of the expression using tiles when
the value of the variable is given. This will help students to understand the logic behind solving
linear equations.
SW – 16
Fun Corner
CONTENT WORKSHEET C 9
Objective – Practice evaluating algebraic expressions
Description- This worksheet has number of board games. Teacher can make students play on
weblink 4 and then give these board games to play. Through these board games teacher will help
students revise and practice the concepts of algebra. This will help them applying algebra. In one
activity teacher asks the questions in the form of a flow chart. This would increase analytical skills
of the students. In another activity the teacher will prepare a grid and two students can play the
game of knots and crosses according to the rules specified.
Pre preparation: Teacher will prepare the flow chart and the grid and hand out to the students to
start the exercise.
SW – 17
Let’s race!
CONTENT WORKSHEET C 10
Objective: Enhance the skills of evaluating algebraic expressions
Materials Required: 1 die, 1 Board per group
2 - 3 counters to move depending upon the number of players
24
Description: Teacher will play the algebra board game with the students depending upon the time
available. Teacher will make board like a sample shown below. Students will use the cards made by
the teacher to play the game and practice algebraic concepts.
Execution: The teacher will explain the games and the rules to the students. A hand out of the rules
will be given to the student also to read the rules as and when required.
Sample BOARD
Sample Algebra Cards
25
To Play:
1. Shuffle the algebra race cards, and then place them on the table, face down.
2. Player 1 rolls the die and simultaneously picks up a card.
3. Player 1 then evaluates the expression by substituting the value of the variable in the card by
the number obtained on the die.
4. If player 1 solves the problem correctly and the answer is positive then he or she moves their
chip same number of steps (equal to the answer) forward on the board. If the answer is
negative the player moves the same number of steps backwards. However, they cannot move
further backwards from the start position.
5. If Player 1 is unable to solve the problem correctly, he or she loses a turn.
6. If the players use all of the cards in the deck, they reshuffle the deck and start again.
7. The player who reaches Finish first wins.
SW – 18
Independent practice
CONTENT WORKSHEET C 11
Objective: Enhance the skills of simplification and evaluation of algebraic expressions.
Description: Teacher will prepare a comprehensive worksheet on simplification and evaluation of
algebraic expressions. This worksheet has all the questions and concepts done in the unit so far.
26
The teacher will encourage the students to apply the concepts of algebra without using any
manipulative. If the student in not able to visualize, the teacher must allow the use of Manipulative
to solve the problems.
Execution: Each student of the class solves the worksheet independently. The teacher may take
this activity as a formative assessment.
SW – 19
Balancing Equations
CONTENT WORKSHEET C 12
Objective: To learn to evaluate the value of the variable in a simple equation
Description: Teacher will show students video clip 1 and then will demonstrate the basic equation
principles with the Balance model to find the value of the unknown quantity. Teacher illustrates
few problems and then prepares a comprehensive worksheet for the students to practice.
a.
Balancing by weight:
If 4 ketchup bottles are equal to a milk pack and 2 ketchup bottles equal to 1 Jam bottle, can you
find how would milk pack and jam bottle will be related?
Let’s find out!
27
Let’s replace the other 2 ketchup
bottles also with the jam bottle.
This gives us the relation
between the milk and the jam
bottle!!!!
Let’s pick up 2 ketchup
bottles and place 1 jam
bottle instead. Balance
remains still equal!!!
After seeing and understanding this balance, Can you write which among the following is balanced
and which one is unbalanced?
b. Balancing by value:
Example 1:
If the value of 2 milk packets is $6, then can you balance the value of 4 ketchup bottles?
Let’s find out!
28
Representing
milk bottle by
m.
2 Milk = $6 or 2m = 6
1 m = $3
Can we find the value of a bottle of jam and the ketchup??
Try out balancing the value and the weights!!
We have seen that 2 bottles of jam equals a packet of milk. Also a bottle of jam equals 2 bottles of
ketchup.
Write the expression for the
a. Jam bottles and their value
b. Ketchup bottles and their value
Hence, find the value of a bottle of jam and a bottle of ketchup.
Using the information given, can you find the value of the following?
Example 2: Given that 2 cones balance 3 cylinders.
29
a. How many cones will balance 6 cylinders?
b. How many cylinders will balance 8 cones?
c. How many cylinders will balance 2 cones and 3 cylinders?
30
d. How many cones will balance 9 cylinders and 2 cones?
Execution: Each student of the class solves the worksheet independently.
SW – 20
Equations using Tiles
CONTENT WORKSHEET C 13
Activity - Math lab Activity Using Algebra Tiles
Objective: To understand solution of simple linear equations through manipulative of algebra tiles.
Description: Teacher will provide algebra tiles to the students and simultaneously demonstrate few
examples to explain how to find the value of variable using tiles.
31
Sample Example 1
32
Mathematically,
x–2+2=3–3-3+2
This gives us,
Hence,
x =–2+1
x =–1
Note: Teacher can take the students to the weblink 5 to make them practice balancing the
equations using tiles.
Execution- Teacher will distribute hand out of worksheet and encourage the students to solve each
expression.
33
SW – 21
Fun Corner
CONTENT WORKSHEET C 14
Objective – Practicing solving simple equations
Description- Teacher will hand out the crack code and grid shown below to the students. Students
follow the instructions given by the teacher and solve the grid. This will be good and interesting
way of learning and practicing subtractions of integers.
Pre preparation: Teacher will prepare the crack code and grid and hand out to the students to
start the exercise.
SW – 22
Math Award
CONTENT WORKSHEET C 15
Objective- Analysing word problems.
Description: Through this worksheet teacher will encourage students to understand that many real
life problems can be solved by translating them into algebraic equations, and then solving it.
Teacher encourages the use of algebra tiles to understand the problems. Teacher may
demonstrate few examples and then prepare a worksheet for the students
SW – 23
Independent Practice
CONTENT WORKSHEET C 16
Description: The objective of the worksheet is to give more practice on algebra to the students.
Pre preparation: Teacher will prepare a comprehensive worksheet on the concepts of algebra.
Execution: Students complete the worksheet and enhance their skills on concepts of algebra.
34
SW – 24
Extended practice
POST CONTENT WORKSHEET PC 1
Description: The objective of the worksheet is to give extended practice on algebra to the students.
Pre preparation: Teacher will prepare a comprehensive worksheet on the concepts of algebra.
Execution: Students complete the worksheet and enhance their skills on concepts of algebra.
Follow up: Teacher checks the worksheet for concepts and accuracy. A comprehensive post
content worksheet will be given to the students to test the knowledge of the concepts of algebra
acquired.
SW - 25
POST CONTENT WORKSHEET PC 2
Objective: To Practice the concepts in totality learnt in the unit
Pre Preparation: Teacher will prepare the comprehensive worksheets of the unit.
Description: Teacher will hands out the worksheets to the students
Follow up: Teacher will assess level of her students on the basis of the post content worksheets
PS1 and give remedial wherever required.
Note for the teacher:
1.
Students weak at the concepts must be given the enough practice through the basic
worksheets and then post content worksheets may be given to them.
2.
Students who have grasped the concepts very well and are able to solve regular problems
quite easily may be advised to move to extension activities.
35
POST CONTENT WORKSHEET PC3
Assessment of the unit
Objective: To test the concepts in totality learnt in the unit
Pre Preparation: Teacher will prepare the comprehensive worksheets of the unit.
Description: Teacher will hands out the following worksheets to the students
This will be timed test given to the students.
Execution: The teacher may give hand outs of the questions below as a test paper and assess
accordingly
Follow up: Teacher will assess her students on the basis of the post content worksheet PC3 as per
the rubrics given.
ASSESSMENT OF THE UNIT
Name of the student ______________________
1.
Which of these is not an algebraic expression?
(i)
2.
5y – 2
(iii) 2 of 3 – 2
(iv) 2m + 3
2x + 4
(ii)
x
+4
2
(iii) x + 2
(iv)
x
+2
4
sxs
(ii)
4s
(iii) 2s
(iv) 2s + 2
If a regular polygon has ‘n’ sides, its perimeter is expressed as
(i)
5.
(ii)
If ‘s’ be the side of a square , its perimeter is given by
(i)
4.
3x
Half of x is increased by 4 can be written as
(i)
3.
Date ______
nxn
(ii)
4n
(iii) n x side
(iv) n + side
Twice a number decreased by 4 is written as
(i)
2x + 4
(ii)
4x – 2
(iii) 2x – 4
36
(iv) 4x + 2
6.
If x = -1, y = 2, then 2y + x will be
(i)
7.
5
(iii) 4
(iv) 2
5x + 11
(ii)
5 + x = 11
(iii) 5x = 11
(iv) x + 11 = 5
If Ansh has Rs x and it is to be divided equally among 6 children, each child would get
(i)
9.
(ii)
5 more than a number equals 11 may be written as an equation
(i)
8.
3
Rs (x – 6)
(ii)
Rs 6x
(iii) Rs
x
6
(iv) Rs (x + 6)
Anoushka is x years old. Her mother’s age is 10 more than twice her age. How old is the
mother?
(i)
10x years
(ii)
2x + 10 years
(iii) 2x – 10 years
(iv) x + 10 years
10. Write an equation for the following statement:Eight times a number is 11 more than four times the number.
11. The price of 1 pen is Rs. 12 and the price of 1 pencil is Rs 5. Write an expression for the total
amount payable for buying x pens and y pencils.
12. 60 students can be seated in a bus. Write an expression for the number of students that can
be seated in t number of buses.
13. In a hostel 3 kg rice is consumed by one boy and 2 kg rice by one girl in a month. Write an
expression for the quantity of rice consumed by ‘x’ boys and ‘y’ girls.
14. Write the following into algebraic language.
(i)
The sum of x and y
(ii)
The difference of x and y when x > y
(iii) 2 more than x
(iv) 3 more than four times a number y
(v)
(vi) One–Fifth of x added to the sum of a and b
The product of a and b added to 5
(vii) Two–Thirds of y added to 6 y
(viii) Five times b added to 3 times c
(ix) The quotient of x and y, if x is divided by y, added to the product of x and y
(x) Three-Fourths of x multiplied by the difference of p and q (where p < q)
15. If the length of a rectangle is x meters, write an expression for
(i)
Its breadth which is 2m less than the length
(ii)
From (i), write an expression for the area of rectangle
16. The distance between points A and B is denoted by D
37
The front of a bus moving from B to A is 500 meters away from B. Write an expression for the
distance between points A and C in terms of ‘D’.
17. Answer the following questions. Take Ananya’s present salary as Rs x.
(i)
What will be her salary after a raise of Rs 500?
(ii)
Ansh’s salary is two times Ananya’s present salary. How much is his salary?
18. Nidhi’s present age is ‘n’ years
(i)
What was her age 5 years ago?
(ii)
What will be her age after 5 years?
(iii) Her sister is three times her age. How old is she?
(iv) Her mother is 20 years older than her sister. Express mother’s age ib terms of ‘n’.
19. A bag contains 18 blue marbles and 15 red marbles. Write an expression for the total number
of marbles left in the bag if x blue and 3x red marbles are taken out from it.
20. Write a statement from the following algebraic expressions
(i)
y–4
(ii)
2x + 2
(iii)
x
3
(iv) 5n – 3
(v)
10
–5
x
21. Write each of the following statements in the form of an equation using ‘x’ as the variable.
(i)
A number increased by 5 equals to 16
(ii) Difference of 7 and x (7 > x) equals 5.
(iii) Thrice a number subtracted from 29 is 11
(iv) Seven subtracted from a number is 9
(v) Three-Fourths of a number is 8
(vi) Twice a number is divided by 6 equals 20
(vii) Twice a number decreased by 8 is equal to 20 (viii) Four times a number is 24
(ix) A number multiplied by itself equals 64
(x) Four times a number is 20 more than the number.
24. Solve the equations:
38
(i)
(iv)
x + 5 = 14
(ii) x – 9 = 18
(iii) 5x = 45
y
=7
4
(v)
(vi)
(vii) 3x =
(x)
20
–x
7
5x + 3 = 12
(viii) 4x = x + 6
3x
=6
4
(ix) 3x – 8 = x + 4
13x – 14 = 9x + 10
25. A number when multiplied by 4 and increased by 5 becomes 17. Find the number?
39
Assessment guidance plan
Parameter
0 (LOWEST) 1
1
Can
differentiate
between
variables and
constants, is
able to find
the
terms
and
coefficient of
the terms of
algebraic
expressions
Does not
possess any
knowledge
and cannot
differentiat
e between
variables
and
constants.
2
Is
able
to Does
differentiate
not Is
and of
between
the and
3
4 (HIGHEST)
Can
differentiate
between
variables and
constants, is
able to find
the terms of
the algebraic
expression but
is unable to
find
the
coefficients of
the terms.
Can
differentiate
between
variables and
constants, is
able to find
the terms and
coefficient of
the terms of
algebraic
expressions
with
90%
accuracy.
Can
differentiate
between
variables and
constants, is
able to find
the terms and
coefficient of
the terms of
algebraic
expressions
with
100%
accuracy.
to Is
able
possess any differentiate
between the knowledge
like
Can
differentiate
between
variables and
constants but
is unable to
find the terms
and coefficient
of the terms
of
algebraic
expressions
2
able
differentiate
like between
unlike and
able
to Is
differentiate
like between
unlike and
able
differentiate
like between
unlike and
like
unlike
terms but
is terms and is term
and is able to
unable
to able
express
express words express words express words express words
into
expressions.
but
to
unlike terms, concepts.
words
to able
is terms, and is
to able
to
into
into
into
into
expressions
expressions
expressions
expressions
with
accuracy.
3
to Is
50% with
accuracy.
90% with
100%
accuracy.
Can evaluate Does not Can evaluate Can evaluate Can evaluate Can evaluate
and simplify
possess any and simplify and simplify and simplify and simplify
algebraic
expressions.
knowledge algebraic
algebraic
algebraic
algebraic
40
of
the expressions
concepts.
with
expressions
30% with
efficiency.
expressions
50% with
efficiency.
expressions
90% with
efficiency.
100%
accuracy
or
very
little
mistake.
4. Is able to
solve linear
equations
with
one
variable and
can convert
real
life
problems into
an equation
to find the
unknown
quantity.
Does
not Can
solve Can
possess any linear
knowledge
of
linear
solve Can
linear
solve
linear
equation with equation with equation with equation with
the on
concepts
solve Can
variable on
variable on
variable on
variable
but is unable but is able to but is able to but is able to
to convert real convert
life
problems life
real convert
problems life
real convert
problems life
real
problems
into
an into
an into
an into
an
equation
to equation
to equation
to equation
to
find
the find
unknown.
the find
the find
the
unknown with unknown with unknown with
50% efficiency. 90% efficiency. 100%
efficiency.
41
ALGEBRA - Math
Unit : 5
Introduction
Till now, you have been studying numbers, shapes, patterns etc. The branch of mathematics in which
we study about numbers is known as arithmetic. Another branch of mathematics is geometry in which
we study different shapes and their properties. There is yet another branch of mathematics called
algebra which uses letters like a, b, c…, x, y, z to represent numbers. The use of letters for numbers
helps us to create and write rules and formulas in a general sense.
In this unit, we shall make a beginning of the study of algebra by discussing various related terms and
concepts such as a constant, variable, algebraic expressions, operation on algebraic expressions,
equations etc.
The word algebra is derived from the title of the book algebra w’al
almuqabalah, written in about 825 AD by Mohammed ibn Al Khowarizmi
an Arab mathematician.
Ancient median mathematicians also made use of symbols to denote the
known quantities. Great Indian mathematicians Aryabhata ( born 476 AD)
Brahmagupta ( born 598 AD ) , Mahavira (around 850 AD ) Sridhara (around
1025 AD ), Bhaskara (born in 1114 AD ) are among those who contributed a
lot to the study of algebra.
“(1)“ and “2
Use of letters to denote numbers
Recall that
Perimeter of a square = 4 x length of the side
s
If we use letters – P for perimeter, s for the length of side of the square, we can express
(i)
as
4
Similarly in case of a rectangle, we have
42
Area of a rectangle = length x breadth
b
l
It can also be expressed as
A=lxb
Where A stands for area, l for length, and b for breadth of the rectangle.
Again, perimeter of a rectangle is given by the formula:
Perimeter of a rectangle = 2(length + Breadth)
It can the expressed, using letters, as
P = 2 (l +b)
Where P stands the perimeter, l the length and b the breadth.
The formulas (i), (ii) and (iii) are true the all systems of units and for all possible values of the letters
involved.
The above examples show that the use of letters to represent numbers helps us to think in more
general terms. Using these formulas, we can solve a numbers of different problems of the same type
by just substituting the value(s) of letter(s) in the formula.
The letters which are used to represent numbers are called literal numbers or simply literals.
Since the literal numbers are used to represent numbers, it is expected that they obey all the rules of
operations of addition (+), subtraction (-), multiplication (×) and division (÷) of numbers. However, on
additional symbol, i.e. dot (.) is also used in algebra to denote multiplications in place of the symbol
(x) which looks like letter x.
We have the following notations and rules:
(i)
the sum of two literal numbers x and y is writer as x + y
(ii)
If the literal number y is subtracted from literal number x, we denote the difference by x – y
43
(iii) The product of x and y is x ×y and is writen as x . y or xy
(iv) The repeated product of a literal number with itself is writen as
exponential form).
(v)
(1)
If the literal numbers
(in
is divided by y (≠0), we write it as
Concept of variable:
We have seen that perimeter of a square is given by.
4
Here 4 is a fixed number whereas P and s are literal numbers and their values are not fixed as
can be seen from the following:
If s = 1 cm,
P = 4 x1 cm
if s = 2 cm
P = 4 x 1 cm = 4 cm
if s = ½ cm,
P = 4 x 2 cm = 8 cm
if s = 10 cm P = 4 x ½ cm = 2 cm
P = 4 x 10 cm = 40 cm
So, as s takes different values, we get different values of P (i.e. perimeter).
Thus, values of p and s vary.
We call P and s as variables. On the other hand, value 4 is fixed. We call 4 as constant. Thus,
A quantity which takes on various numerical values is called a variable, and a
quantity which takes on a fixed numerical value is called a constant
In A = l x b
A, l, b, all are variables.
In P = 2 (l + b)
P, l, b, are variables while 2 is a constant.
44
Similarly in the expression
x+5
x is a variable and 5 is a constant.
(2)
Algebraic Expressions
You are familiar with expressions in arithmetic, for example,
1 + (2 + 3 ), 2 x 6 – ( 4 x 5 – 2), 10 – (15
3) etc.
Expressions can be formed from variables and constants also.
For example, x +5 is an expressions which uses x (variable) and 5 (constant).
Some more examples of expressions using variables and constants are:
5 x, 2 y, 2 x + 3, 5 x – 9 etc.
These expressions which contain variables and constants are called as algebraic expressions
Thus,
An algebraic expression contains both variables and constants connected to each other by one
or more fundamental operations (+, -, x, ÷)
Some more examples of algebraic expressions are:
2 x + 3,
a + b/2,
6y,
2
3
,
,
2t+ 3 ,
4
3
9
,
Forming an algebraic expression
Let as see how algebraic expression are obtained from phrases using variables and costants
Example 1: Give an expression for
(i)
5 subtracted from
(ii)
a multiplied by 10
(iii)
multiplied by 16 and then 7 added to the product
45
(iv)
multiplied by 8 and 1 subtracted from the result
(v)
Twice a number added to 4
(vi)
a number divided by 3
(vii)
times the sum of a and b
(viii)
4 times a number minus 3 times the number
(ix)
5 times p subtracted from 2 times q
(x)
the difference between the sum of p and q and the difference of p and q ( where p is greater
them q)
Solutions:
(i)
5
(ii)
10
10
(iii)
16
7
16
7
(iv)
8
1
(v)
Let the number be a. Twice the number = 2a. So , required expression is 4 + 2a
(vi)
Let the number be a. so the expression is
(vii)
Sum of a and b = a + b
q times the sum =
(viii)
Let the number be y
4 times the number = 4y, 3 times the number = 3y
so, the expression is 4y – 3y
(ix)
(x)
2
5
-(p-q)
46
Example 2: Write the following statements using variable (s) and constant (s):
(i)
the area of a triangle is product of half its base and height
(ii)
the diameter of a circle is twice its radius
(iii)
the area of a rectangle is the product of its length and breadth
(iv)
Speed of a vehicle is the distance covered by the vehicle divided by the time taken by it to
cover the distance.
(v)
the selling price of an item is equal to the sum of cost price and profit earned
Solutions:
(i)
Let a denote area, b denote base and h denote the height of the triangle
(ii)
Let d denote the diameter and r, the radius of the circle. Then
D=2×r=2r
(iii)
Let a denotes the area, l denote length and b breadth
a = l x b = lb
(iv)
Let s denote the speed, d denote the distance covered and t, the time taken by the vehicle.
Then
Speed =
s=
(v)
Let s denote the selling price, c the cost and p the profit
s=c+p
•
Monomial: When an algebraic expression consists of only one term, it is called a monomial.
Mono, in Greek means ‘Single’
Some examples of monomials are: 2 , 5 , , ,
•
Binomial: When an algebraic expression consists of two terms, it is called a binomial. Some
examples of binomials are:
,
3 ,
29
,
10
2
,
47
5
2
In Latin ‘ bi’ means ‘double’
•
Trinomial: When an algebraic expression consists of three terms, it is called a trinomial. Some
examples of trinomial are:
,
3
2
,
3
4
1,
In Latin, tri means ‘ three’
Can you tell what a quadrinomial is? An expression containing four terms such as
•
2
Terms of an algebraic expression
Parts of an algebraic expression along with with their signs ‘+’ or ‘-‘ are called terms of the
,
are terms; in a + b –c,
expression. For example, in 2x + 3 , 2x and 3 are terms; in
a, b and – c are terms and so on.
Example 3: Write the terms of the algebraic expression:
(i)
2
5
8
(ii)
Solution : (i) 2
,5
,
,8
(iii)
Example 4: State which of the following are monomials, binomials or trinomials:
(i)
3
4
(ii)
(iii)
3
(iv)
2
(v)
(vi)
yz+1
(vii)
2
1
(viii)
48
(ix)
-ab + bc
(x)
5
(xi)
4
4
Solutions :
Monomials : (iii), (iv) and x
Binomials:
(i), (v) and (vi), (xi) here, after simplification it is binomial
Trinomials
(ii), (vii) (viii) and ix)
•
Factors of a term
In a term like 3xy, 3,x and y are called factors of the term. x and y are literal factors and 3 is a
numerical factor.
Note that 3 , 3 ,
3
.
Similarly in the term 8xyz, literal factors are , , 8
. and 8 is a numerical factor.
Again in – 3xy; the factors are -3, x y, xy etc. Here -3 is numerical factor whereas x, y, xy etc.
are literal factors xy etc
In the term 3 x y, the numerical factor 3 is called the coefficient of xy. It is also called
numerical coefficient of the term, similarly in the tern 5
numerical coefficient is -5 and
in the term xy, numerical coefficient is +1 or simply 1.
In the term
, numerical coefficient is (-1)
When coefficient of a term is + 1 or -1 then ‘1’ is usually omitted. For example we
write 1p as p and 1
.
Usually, numerical coefficient of a term is taken as coefficient of the term. For example,
, is taken as -7.
coefficient of 7
49
Example 5 : Find the coefficient of each terns in the following algebraic expression
1
2
9
Solutions:
Coefficient of 9
3
4
9
Coefficient of –
Coefficient of x=1
Coefficient of –y =-1
Coefficient of
(coefficient of a constant term is constant itself)
•
Like and unlike terms
When the terms have the same literal factors they are called like terms (or similar terms )
otherwise they are called unlike terms (dissimilar terms )
For example, in the expression
4
3
3
7
9
Two constant terms are considered as like terms. for example 9 and -7 are
like terms
4 xy and 7 xy are like terms because literal factors of both the terms are x and y and 3x, -9 are
unlike terms as they have different literal factors.
5
In the expression: 4
4
, 11
11
are like terms
as the literal factors of 4
11
5
3
3
, , .
, , which are same.
?
50
4
3
5
•
?
4
?
Value of an algebraic expression
Consider an algebraic expression 2
11
This involves the variables x and y.
To find value of this expression say when
numerical values and simplify.
1,
Thus, when
2,
2
Example 6 :
Find the value of the expression 3
Solution:
Replacing x by-1 in
3
5
2,
3
5
2
1,
2, we just replace the variables by their
11
= 2(1)(2)-11(12)(2) = 4-22=-18
5
3
1
2
1.
5
1
3
5
2
Let us take same more examples
Example 7:
If x =-4 and y = 5, fluid the value of 2 x + 7 y
Solutions :
Replacing the variables with their values, we get
2
7
8
2
4
7 5
35
27
1,
Example 8 : If
2 and
2
(i)
1
(ii) 2
3
Solutions: Substituting
3
(i)
3, find the value of
2
1,
2
3 in the given expressions, weget
(1) (2)3-2(1)2 (2)+1
1
=8-4+1
=5
(ii) 2
3
2(1)3 –(2) + (-3)2 -3 (1) (2) (-3)
51
10
= 2 – 8 + 9 + 18
= 29 - 8
= 21
Example 9 : If a =0, b = ½ , c = ¾ , find the value of 2a2 + b2 – c2
Solutions : Substituting x = 0, b = ½ , c = ¾ in the given expression, we get
2
= 2 (0)2 + (
=0+¼=¼=
Addition and of subtraction of algebraic expressions
In order to understand addition and subtraction of algebraic expressions we need to know how to add
or subtract like terms.
Adding like terms
Let us add 2x and 9 x
We know that 2x and 9x are numbers, as x represents a number.
2
9
= 2
2
9
9
(using distributive property)
= 11
= 11
Thus, 2
9
11
Let as add
, 4
7
4
7
1
= 1
1
4
4
2 ,9
(i)
4
11 ,
7
7
7
52
4
(ii)
4
,7
4
Thus
7
Similarly let us add 3
3
4
= 3
4
4
4
,4
5
,
, 5
3
4
5
5
=2
(iii)
3
so, 3
4
4
, 5
2
5
2
From (i), (ii) and (iii), we find that
The sum of two or more like terms is a like term whose coefficient is the sum of the coefficients of all
the terms.
Subtracting like terms
Let us subtract 7
10
7
10
10
10
7
7
=3x
So, 10
5
3
5 , 10
3 ,
Similarly subtractt 10
5
= 5
10
5
.
5 .
10
10
= 5
(ii)
10 , 5
5
53
From (i) and (ii), we are that
The difference of two like terms is a like term whose numerical coefficient is the difference of the
coefficients of the two like terms.
Adding algebraic expressions
We explain addition of algebraic expressions through some examples.
Examples 10: Add 3
Solution:
4
5
2
5
we combine the like terms together as shown below
3
4
5
2
5
3
2
4
we can also perform addition as shown below
3
4
Example 11: Add the following:
3
4
6
7
23
15
24
Example 12. Add the following expressions:
3
3
Solution
8
4, 8
3
4
3
4
3
4
54
4
5
5
5
9
5
(a)
Subtracting Algebraic expressions
Let us understand subtraction of algebraic expressions through examples.
Example 13.
Subtract 3
Solution
4
3
4
4
3
4
3
4
3
3 .
4
3
3
4
4
7
[ Since there is a negative sign before the bracket so, sings of the terms have been
changed]
Columnwise
4
3
3
4
[sign of each terms has been changed]
7
Example 14. Subtract 12
6
Solution
7
5
5
9
10
6
7
2
9
,3
from
10
12
−
3
[4
−
+
11
2
55
4
7 ]
Example15. From the sum of 10
Subtract 5
6
6
8
6
3
6
6
5
,
2
Solution
Step 1:
Let us find the sum of fast two expressions
10
6
8
6
6
5
+
4
−
+
0
3
4
Step 2:
3
3
+
2
2
(i)
Let subtract 5
6
4
3
2
5
6
2
+
2
6
−
3
6
−
0
3
6
6
Equations
You have already come across statement of the following types:
3
4
7
1
5
3
15
2
6
9
4
12
7
54
5
4
12
3
7
12
5,
.
4
All these statements are called equality statements because each of theem involves the symbol ‘=’. It
can also be seen that they involve only numbers. You have already seen that letters like x,y,z etc. are
also used to represent numbers.
Therefore, equality statements can be obtained involving letters like x,y,z etc also.
56
They are obtained when you equate two algebraic expressions. For example,
Sum of two numbers x and y is 9 gives
9
(5)
y more than 7 is 12 gives 7
12
(6)
A number x divided by 6 gives thrice the number is
6
3
7
The sum of two number x and y is two same as the sum of two number y and x gives
8
The product of a literal number
with itself is 49 gives
49, . .
49
9
Note that all the statements (1), (2), (3), (4) ,(5) ,(6) ,(7), (8), and (9), involve symbol ‘=’. They are
called equalities
Equalities (5), (6), (7), (8), and (9) involve letters or literal numbers or literals or variables. Equalities
involving literals or variables are called equations.
It can be verified final equation (8) is true for all values of the variables x and y, while other equations
are true for only some value of variable(s)
In view of two above, equation (8) is referred to as an identical equation as simply an identity and
other equations are referred to as conditional equations or simply equations. Thus, by an equation,
we always wean a conditional equation
You can see that equation (5), ie,
9 is equation in two variables x and y while equation (6) i.e
7
12 is an equation in one variable y.
6 3
an equation in one variable and also =49 is an equation in one variable . You can
also observe that in equation 7
12 and
6 3 , exponent of the variable is 1 only.
Such equations are called linear equations
Thus,
49 is not a linear equation
9 is also a linear equation (why)
(It is linear equations in two variables
Further 7
12 and
6
and y)
3 are linear equations is one variable
and y respectively.
Every equation has two sides, namely left hand side (LHS) and right hand side (RHS).
In 7
12, 7
LHS is 7+y and RH.S is 12.
57
•
Solutions of a linear Equation
Consider two following equation:
7
12
It is a linear equation in one variable y. Let us substitute different values of y is the above equations
and record our observations in the form of a table as shown below.
Value of
LHS
RHS
Is L H S = R H
S?
0
7+0=7
12
NO
1
7+1=8
12
NO
2
9
12
NO
3
10
12
NO
4
11
12
NO
5
12
12
YES
6
13
12
NO
Note that for y = 5, L H S = R H S.
In such a case we say that y = 5 satisfies the equation 7
12.
Alternatively, we also say that y = 5 is a solution of the equation 7 + y =12
5
Example 16: Check whether
3,
2
2
the solutions of th following equations:
2
9
(ii) 3
5
(i)
3
1
1
58
Solution (i)
For
5
LHS = 2
5
9
and RHS = 3
1
5
1
14
5 is not a solution
Thus, LHS ≠ RHS. Hence,
for
3,
LHS
2 3
9
RHS= 3 3
1
9
15 and
10
3 is not a solution
Thus, LHS ≠ RHS. Hence,
2
For
LHS =2(-2)+9=5 and RHS = 3
Thus, LHS= RHS. Hence
(iii)
for
5
LHS= 3
5
5
1
5
2 is a solution
20
5
1
4
5 is not a solution
Thus, LHS ≠ RHS. Hence
For
2
3
LHS=3 3
5
4 and RHS
3
1
4
Thus, LHS = RHS, Hence, x =3 a solution
For
2
LHS =3
2
5
11
Thus LHS ≠ RH S. Hence,
For
2
= -2 is not solution
2
LHS = 3(2)-5 and RHS = 2+1 =3
Thus LHS ≠ RhS. Hence
1
= 2 not a solution
59
1
•
Solving linear Equation
An Equation may be compared with a balance used for weighing objects. The two sides LHS and RHS
can be considered as the two pans of the balance and the equality symbol indicates that the balance
is in equilibrium.
If we add equal weight in the two pans, then it can be seen that the equilibrium of the balance
remains undisturbed. Similarly if we remove equal weights from the two pans then again the
equilibrium of the balance remains undisturbed
Inspired from the above two situations, we adopt the following rules for solving an equation:
(i)
we can add the same number on both sides of the equation,
(ii)
We can subtract the same number from both sides of the equation,
(iii)
We can multiply (ie. repeatedly add) both sides of the equation by the same non –zero
number.
(iv)
We can divide (i.e repeatedly subtract ) both sides of the equation by the same non zero
number.
We explain the above process thought some examples
Example 17: Solve the following equations in one variable:
(i)
3
5
7
(ii)
2
9
12
3
(iii)
(iv)
6
2
4
20
6
3
7
(v)
(vi)
3.35
12.5
Solution: (i) 3
5
1.25
7
or 3
5
5
or 3
0
12
or 3
12
or
6
7
5
4
60
4
or
4 is the required solution of the equation
Thus,
(ii)
2
9
12
or 2
9
9
12— 9
or 2
9
9
12
or 2
0
3
or 2
3
2
9
4
or
or
the reuired solution of the eqution
Thus,
(3)
3
2
or 3
(4)
4
4
6
2
or
2=6
or
2
2
or
0
4
4
6
4
2
6
1
2
or
4
Thus,
= 4 is the required solution of the equation
6
20
3
or 6
20
or 6
0
or 6
3
7
20
3
3
7
20
1
27
27
or 6
3
3
or 3
0
27
27
3 (Rule(2))
61
or 3 +27
or
or
(rule(4))
9
9 is the required solution of the equation
Thus
(v)
or
(Rule (1))
0
or
or
or
or
(Rule (1))
0
or
or
(Rule (3))
or 3
or
(Rule (4))
or
Thus,
(vii)
is the required solution of the equation.
3.35z – 12.5 = 1.25z + 6
or 3.35z – 1.25z – 12.5 = 1.25z – 1.25z + 6 (Rule (2))
or 2.10z – 12.5 = 0 + 6
or 2.1z – 12.5 = 6
or 2.1z – 12.5 + 12.5 = 6 + 12.5 (Rule (1))
or 2.1z + 0 = 18.5
62
or 2.1z = 18.5
or
.
.
.
.
(Rule (4))
or
Thus,
Check: (i)
is the required solution of the equation.
Substituting x = 4 in the original equation 3x – 5 = 7, we have
LHS = 3 x 4 – 5 = 7
and RHS = 7
So, LHS = RHS. Hence checked.
(ii)
Substituting
in the original equation
2x + 9 = 12, we have
LHS = 2
9
12
and RHS = 12
Thus, LHS=RHS. Hence checked.
(iii) Substituting x = 4 in the original equation
-3x + 2 = -4x + 6, we have;
LHS = -3 x 4 + 2 = -12 + 2 = -10
and RHS -4 x 4 + 6 = -16 + 6 = -10
Thus, LHS = RHS. Hence, checked.
You are advised to check the solution of the remaining equations in a similar manner.
Applications
You have already seen how some practical situations can be expressed in the form of algebraic
expressions and equations. In fact, a number of daily life problems can be easily solved by first
converting them into equations and then solving them. We shall explain the process through some
examples.
63
In general, the following steps will prove to be helpful to you in solving such problems:
Step 1:
Read the problem carefully word by word and note down what is given and what is
unknown.
Step 2:
Denote the unknown by some letters x, y, z etc.
Step 3:
Translate the problem word by word and step by step in to the form of a mathematical
expressions.
Step 4:
Look for the expressions which are equal and form an equation by equating these
expressions.
Step 5:
Solve the obtained equation, which gives the solution of the problem.
Step 6:
Check the solution with the original conditions of the problem.
Example 18: Sum of three consecutive natural numbers is 24. Find the numbers.
Solution:
Let the numbers be x, x+1 and x+2
As per given condition:
x + x +1 + x + 2 = 24
or 3x + 3 = 24
or 3x + 3 – 3 = 24 – 3
or 3x = 21
or
or x = 7
So, the required numbers are 7, 8 and 9.
Example 19: Two trains start from the stations A and B at the same time towards each other with
speeds whose difference is 10 km/h. If the distance between A and B is 195 km and the trains meet
each other after 1½ hours, find the speeds of the trains.
64
Solution:
Let the slower train start from A and the two trains meet at P. Let the speed of the
slower train be x km/h.
So, speed of the faster train = (x+10) km/h
now,
1
and
10
10
1
Also, AP + BP = 195 km
So,
10
or
195
10
195
or x + x + 10 = 130
or 2x + 10 = 130
or 2x + 10-10 = 130-10
or 2x = 120
or x = 60
So, speed of the slower train is 60 km/h.
and that of faster train = (60+10)=70 km.
Check: 70 km/h – 60 km/h = 10 km/h
Also, 60 x 1 ½ + 70 x 1 ½ = 90 km + 105 km = 195 km
Hence checked.
65
Example 20: Age of a father is three times the age of his son. After five years, sum of their ages would
be 70 years. Find the ages of the father and son.
Solution:
Let the age of son be x years,
So, age of father = 3x
After 5 years, age of son = (x+5) years
and age of father = (3x+5) years
As per given condition,
(x+5) + (3x+5) = 70
or x+3x+5+5=70
or 4x +10 = 70
or 4x + 10 – 10 = 70 – 10
or 4x = 60
or
or x=15
So, age of son is 15 years and that of father is 3x15 = 45 years
Check:
45 = 3 x 15 years
Also, after 5 years their ages are
15 + 5 = 20 years and 45 + 5 = 50 years
Now, (20+50) = 70 years, which is the same as given.
Hence, checked.
66
STUDENT’S
WORKSHEET
67
STUDENT’S WORKSHEET – 1
Observing patterns - Playing with a square
WARM UP ACTIVITY W1
Name of the student ______________________
Instructions:
flip side
•
Take a square of any size.
•
Paint it blue on one side and red on the flip side.
•
A
B A
B
D
C
C
D
Denote the rotation by 90 degrees clockwise by R and 90 degree anti clockwise rotation by
L.
•
Date ______
Example: L stands for :
D
A D
A
C
B
B
Write RR for the motion you get when you do R twice.
Now answer the following question
C
What is the effect of RRRR on the square?
Is RRR same as L? Draw and explain.
How is RR different from LL?
__________________________________________________________________________
__________________________________________________________________________
68
Can we get the flip of the square using just R and L? Explain.
__________________________________________________________________________
__________________________________________________________________________
Use F for flip and try out different combinations to see how the square is unaffected.
_________________________________________________________________________
_________________________________________________________________________
Try the same activity using a regular pentagon or a rectangle. How are they different?
Discuss with your friends.
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
_________________________________________________________________________
69
STUDENT’S WORKSHEET – 2
Carrying cards
WARM UP ACTIVITY W2
Name of the student ______________________
Date ______
Instructions –
Each child in blue is holding a number which is five more than the child in the same row wearing
red.
The children in yellow shirts each have a number which is double the number of the child in the
same row wearing red.
Some of the numbers that the children in red, blue and yellow shirts are holding got erased.
Now answer the following questions:
1.
What should the numbers be in the blank cards?
Working space
70
2.
Explain how the numbers that the children in green are holding have been worked out? What
are the two missing numbers?
3.
If there were another row of four children standing behind the fourth row, what numbers
would they be holding?
STUDENT’S WORKSHEET – 3
Cooking mathematics
WARM UP ACTIVITY W3
Name of the student ______________________
Adam's grandmother has an old recipe for cherry buns.
•
To make them, she weighs three eggs.
•
Then she takes the same weight in flour, in sugar and in butter.
71
Date ______
•
She mixes all this together and then adds the weight of 2 eggs of chopped glace
cherries.
•
She has enough mixture to put 49 grams in each of 12 paper cake cases.
What was the weight of one egg?
Oral Questions:
To make 36 cherry buns
1.
How many “egg weights” of mixture is needed altogether?
_____________________________________________________________________________
2.
How many "egg weights" is needed for each of the ingredients?
_____________________________________________________________________________
3.
How much mixture is available to pour in 36 paper cake cases?
_____________________________________________________________________________
4.
How much of each ingredient is used?
_____________________________________________________________________________
_____________________________________________________________________________
STUDENT’S WORKSHEET – 4
Math lab Activity with Tiles
WARM UP ACTIVITY W4
Name of the student ______________________
Date ______
Instructions for integer tiles.
FLIP
FLIP
72
1.
All positive tiles are of the same colour.
2.
All negative tiles are of the same colour.
3.
The red tiles are additive inverses of the yellow tiles. A “zero pair is created” when they are
used together or in other words added together.
+
=
+1
0
-1
The cancelled pairs are then removed.
4.
Numbers are represented as follows:
To represent 2
To represent - 2
Q1. Using the tiles, solve the following questions:
a.
(-5 ) + 3
__________________________
b.
-9 – (- 4)
__________________________
c.
6+(-3)
__________________________
d.
– 8 + ( - 2) __________________________
e.
7+2
__________________________
Q2. What does the following tiles arrangement represent?
a.
b.
= ___________
= ____________
c.
= ___________
73
d.
= ________________
STUDENT’S WORKSHEET – 5
Fun with mathematics
Warm up activity W5
Name of the student ______________________
Date ______
He is a German mathematician, physicist and astronomer. He, Archimedes and Isaac Newton are
considered to be the greatest mathematician who ever lived. His well-known quotation is
“Mathematics is the queen of the sciences.”
Solve the following questions and write the corresponding alphabets in the boxes below by
matching the answers to the questions with those in the boxes to find out who he is.
Evaluate the following
F
595 x 8 + 595 x 2
D
6954 x 195 – 6954 x 95
I
1879 x 5 x 20
H
12356 x 25 x 40
E
2 + (-315) + (-285) + 800 + (-100)
A
(-52) + 36 – 48 + 14
R
The sum of the successor of -111 and additive inverse of -111
S
Absolute value of the predecessor of -100
U
4 – (- 8) + (-7) – (-8)
L
least integer among these -240, - 420 and -315
G
(-555) x (-233) x 0 x (-1)
74
C
-263 – 0
STUDENT’S WORKSHEET – 6
Appreciate your knowledge
PRE CONTENT ACTIVITY P1
Name of the student ______________________
1.
Date ______
A medicine bottle contains 50ml of a certain medicine. What will be the quantity of
medicine in 200 such bottles?
2.
A box of grapes contains 750g grapes. Find the quantity of grapes in 2500 such boxes. If
grapes are sold at Rs 40 per kg, find the cost of all these grapes.
3.
4.
Add the following using most convenient combinations:(a)
2841+1001+1259+9999
(b)
87+119+757+381+413+243
Andrew deposited $540555 in the bank on 10th January. He withdrew $230810 on 21st
January and again deposited $87000 on 30th January. Find his bank balance on 31st January.
5.
Mr. Paul received Rs 8156420 after selling a house. He gave Rs 2050000 each to his two
sons. He also purchased a car for Rs 652800. The rest he gave to his wife. How much money
did the wife receive?
6.
Find the product of following by suitable rearrangement
(i)
5 x 263 x 20
(ii)
8571 x 4 x 2 x 125
75
7.
8.
A table costs $ 270 while a chair costs $ 55. How much will it cost:
(i)
To buy 6 sets of a table and a chair.
(ii)
To buy one table and six chairs.
If a # b represents an operation such that a # b = 2a + 3b + 1, find the value of
(i)
9.
– 3 # 4 (ii) (- 5) # (- 3)
While selling fruits, the vendor made a profit of Rs 450 on Tuesday, a loss of Rs 200 on
Wednesday and a loss of Rs 100 on Thursday. Find the net profit or loss on three days.
10.
Write a numerical expression for each of the following using brackets. Also solve and get
the answer.
11.
(i)
Six multiplied by the sum of 5 and – 3.
(ii)
The difference between eleven and eight multiplied by 7.
(iii)
Eighty one divided by three times the difference of two and five.
Calculate :
1 + (-3) + 5 + (-7) + 9 + (-11) + 13 + (-15)
12.
If
Find the values of each
.
13.
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STUDENT’S WORKSHEET – 7
Repairing the Doodle copter of Dr. Kind
PRE CONTENT WORKSHEET P 2
Name of Student____________________
Date_____________
“I am Dr. Kind and this is my magnificent Doodle copter, a highly versatile utility vehicle. It runs on
Uranium 235 and produces incredible amount of useful radiations that can take you on a tour to
our solar system. Unfortunately, it needs a complete overhaul. I need the assistance of an Earthling
with 23 paired chromosomes, which means, a human. So you will have to perform the repairs!
Below are the formulas you will need to tune up all five of the Doodle copter’s part. Remember,
each part that is to be repaired has its own specifications.
For your ease, use the following:
a = the position of planet Mars from the Sun.
b = the number of planets in the Solar system.
77
Good Luck with your repairs!!”
To tune up each part 1, 2, 3, 4 and 5 of the Doodle copter, convert the formulas into numeric
values. Put the answer in the square next to the formula.
Once the machine is repaired, its fuel Uranium 235 is to be enriched by a chemical process. It is a
complex process of reactions and produces large quantities of the fuel which must remain under
constant supply. A simple formula is used to monitor the process. Get the fuel value to run the
machine.
78
STUDENT’S WORKSHEET – 8
Variables and Constants: A Tour to the Land of Unknown
CONTENT WORKSHEET C 1
Name of the student ______________________
Date ______
A lady Mary constant and a gentleman Connie Variable would like to invite each of you on a
journey into the Land of Unknown. Mary constant is your tour director. Connie variable is her
assistant and will help her as well all of you at different places during the journey into the
Unknown. Before we begin our journey into the Land of Unknown in the year of changing times,
there are a few questions you must answer.
Q1.
In your own words, write a definition for the word constant.
____________________________________________________________________________
____________________________________________________________________________
Q2.
Use the word constant in a sentence
____________________________________________________________________________
____________________________________________________________________________
Q3.
In your own words, write a definition for the word ‘vary’.
____________________________________________________________________________
Q4.
Use the word ‘vary’ in a sentence.
____________________________________________________________________________
79
Q5.
Think about the word variable. It is a noun. Variable is not a place or a person, so it must be
a thing. With the word vary (verb) in mind, write a definition for variable in your own
words. (Think about your definition of ‘vary’ before you begin writing.)
____________________________________________________________________________
____________________________________________________________________________
Q6.
Use your Math dictionary to define the terms:
Variable
____________________________________________________________________________
Constant
____________________________________________________________________________
Algebra
____________________________________________________________________________
Q7.
Find out whether the quantity mentioned below is a variable or a constant.
If the quantity is a variable, use any letter of the word to represent it.
If it’s a constant write its value.
a.
How many eggs you had in your breakfast today?
b.
How many pens you have in your geometry box?
c.
How many calories are there in various food items?
d.
How much time did you spend doing your home work yesterday?
e.
Length of sides of a rectangle.
f.
How many eggs are there in a dozen?
g.
No. of planets in our solar system.
h.
No. of squares on a chess board.
i.
No. of sides of a rectangle
j.
No. of days in a week.
80
STUDENT’S WORKSHEET – 9
Defining terms and algebraic expressions
CONTENT WORKSHEET C 2
Name of the student ______________________
1.
Date ______
On the way Tour operators stopped at a restaurant for lunch. Can you help the waiter with
this food order? Serve variable expression to Connie Variable and numerical expression to
Mary constant.
Both trays are
examples of
algebraic
expressions
2.
Mary Constant got unhappy with her order. So she asked the waiter to tell her in sentences
what she is being served in her platter. Help waiter express each of the three items served to
the lady in word form. For example 2 + 3 is expressed as three more than two.
3.
Mary unhappy with her order asks the waiter to take a new order and serve her as fast as
possible. Can you complete the order of numerical expressions waiter wrote in his notepad
while taking the order?
81
Order by the lady
Expression written by the
Value of the item obtained by
waiter
solving the numerical
expression
(i) Product of 5 and 3
15
(ii) Sum of 5 and 7
7+5
or we can say,
5 more than 7
(iii) Difference of 19 and 9
Or
Take away 9 from 19
Or
9 less than 19
(iv) Twice of 7
Or
Double of 7
Or
2 times 7
(v) Half of 8
Or
Quotient of 8 divided by
2
(vi)
7 less than the product
9×5–7
of 9 and 5
(vii)
Sum of the three
consecutive
11 + 12 + 13
terms
after 10
82
12
4.
Connie variable, delighted by his order, called the waiter and gave him more order to be
packed for his home. Man gave the order in sentences with variables where as waiter in a
hurry to write the order used variable expression. Can you help waiter to form algebraic
expressions correctly.
Man’s order in words
Variable expression written by the
Waiter in his notepad
1.
Double a number then add 7. Take the number to
2 × x + 7 or 2x + 7
be x.
2.
Triple a number m and subtract 5 from it.
3.
Take away 5 from p.
4.
7 subtracted from – m.
5.
y multiplied by – 8 and then 10 is added to the
result.
6.
– b divided by 6
7.
Three consecutive numbers if x is the first
x, x + 1, x + 2 ( as long as x is a whole
number.
number)
8.
5.
Sum of three consecutive numbers.
Connie requested the waiter to separate the terms in each of the algebraic expression and
also tell the coefficient of the literals given. Mary got interested in the constants in the
algebraic expressions and asked the waiter to separate constants for her. Can you help the
waiter to separate the terms, coefficient of the literals and constants in the following
algebraic expressions?
83
S.No.
Algebraic
Expression
1.
4x + 3y - 5
2.
- 3y + 2
3.
9x – 3y +1
4.
2a - 3
5.
2b – 3a + 7
literals
Terms
Coefficients
Constant
of variables
Terms
x, y
4x, 3y , -5
4 of x
3 of y
-5
STUDENT’S WORKSHEET – 10
Describing an algebraic expression: Variables and constants
CONTENT WORKSHEET C3
Name of the student ______________________
Date ______
Cups and Counters Model to describe algebraic expression
Observe the following model and answer the question that follows:
Picture of Model
No. of counters
1.
84
2.
3.
Q1. Write algebraic expression that describes how many paperclips are there in total.
Each envelope contains x paperclips.
Picture
Expression
1.
2.
3.
4.
85
Q2. Illustrating Algebraic expressions:
Draw cups and counters to illustrate the following algebraic expressions. Let each cup contain
‘n’ counters.
Algebraic Expression
Illustration
1. n + n + n + 7
2. n + n + 1
3. n + 5
4. n + n + n + n + n
86
Q3. Discuss
Can you draw a picture for 4 – n or n + n – 1?
If you have a cup containing n counters, what would you do to leave yourself with n – 2
counters? Think and discuss with your friend.
____________________________________________________________________________
____________________________________________________________________________
Q4. A bag contains certain number of candies. Riama bought two identical bags containing the
same number of candies. She also bought three loose candies.
a.
Draw a diagram to illustrate this.
b.
Write an expression to represent the total number of candies.
Let ____ represent the number of candies in a bag.
Total number of candies = ____________
c.
If each bag contains 12 candies, what is the total number of candies? ___________
d.
If each bag contains 25 candies, what is the total number of candies?____________
e.
John bought 3p + 2 candies. If p represents the number of candies in a bag,
¾
What does 3 represent? ______________________________________
¾
What does 2 represent? ______________________________________
87
¾
What does 3p represent? _____________________________________
STUDENT’S WORKSHEET – 11
Algebraic expressions through Pattern
CONTENT WORKSHEET C 4
Name of the student ______________________
Date ______
a. Form a letter L using the matchsticks
(i)
(ii)
(iii)
1. How many matchsticks did you use in fig (i)? _______
2. How many matchsticks did you use in fig (ii)? _______
3. Complete the following table showing the number of matchsticks required to form the
sequence of letter L
No. of letter ‘ L’,
n
1
No. of matchsticks, m
2
2
3
5
8
10
4. In words, write the relationship between the number of matchsticks (m) and the number of
letter ‘L’.
___________________________________________________________________________
5. How many matchsticks do you need to make a sequence of 30 L’s?
____________________________________________________________________________
88
6. How many L’s can you form with 36 matchsticks?
____________________________________________________________________________
Form the following sequence of pentagons using matchsticks.
Fig 1
Fig 2
Fig 3
Fig 4
1. Complete the following table showing the number of matchsticks required to build the
pentagons.
No. of pentagons ,
p
1
No. of matchsticks, m
2
3
5
5
2. In Fig 2, how many more matchsticks are required than Fig 1? __________________
3. Similarly in Fig 3, how many more matchsticks are required than Fig 2? ____________
4. With every new addition of a pentagon in the sequence, by what multiple is the number of
matchsticks required?
5. Can we say, For 1 pentagon we need 4 × 1 + 1 matchsticks.
For 2 pentagons we need 4 × 2 + 1 matchsticks.
Explain.
89
6. In words, write the relationship between number of pentagons built (p) and the number of
matchsticks used (m).
_____________________________________________________________________
7. Complete the algebraic expression using the variables to represent the above pattern:
No. of matchsticks = _______ times the no. of pentagons + ________
8. How many matchsticks are required to build 100 pentagons? _____________________
9. How many matchsticks are required to build ‘p’ pentagons? _____________________
b. Investigate a repeating pattern:
Select any one of the above repeating units to build out of toothpicks and answer the
following questions:
1. How many toothpicks did you use?
2. Complete the following table showing the number of toothpicks to build the pattern.
No. of units in pattern , n
1
No. of toothpicks, t
5
2
3
5
10
40
3. Describe the relationship between the number of toothpicks (t) and the number of
repeating units (n).
4. Use algebra to write this relationship rule.
5. How many repeated units could you build with 600 toothpicks?
90
STUDENT’S WORKSHEET – 12
Words into expressions
CONTENT WORKSHEET C 5
Name of the student ______________________
1.
Date ______
Form an expression using y, 2 and 7. Every expression must have y in it. Use only two number
operations. These should be different. Also, translate the expression in words.
____________________________________________________________________________
____________________________________________________________________________
____
2.
Write algebraic expressions for the following using numbers and pronumerals (
pronummerals are letters that represent a number in a question):
(i)
a added to b ______________________ (ii)
a less 7 _________________
(iii) b more than 4 _____________________ (iv) 5 times of b added to
a__________________
(v)
b subtracted from 3 times of a____________________________________
(vi) Twice x added to two times of y___________________________________
(vii) x increased by 8 times of itself____________________________________
3.
Write these without a multiplication or division sign.
(i)
3 × n ____________________________ (ii)
4________________________ (iii)
(a + x) ÷
a + 2 × 4 × b ______________________ (iv)
3 ÷ 6n __________________________ (v)
a__________________________
91
2 - 1×
4.
For each of these, copy and finish the equivalent statement for algebra.
Arithmetic
5.
Algebra
a. 3 + 5 = 5 + 3
a+b=
b. 7 × 6 = 6 × 7
ab =
c. 7 × 9 × 3= 9 × 7 × 3
pqr =
d. 2 + ( 5 + 3) = (2 + 5) + 3
x + (y + z) =
Angela and Limei make the following table to compare their ages.
a.
Angela’s age
Limei’s age
6
8
7
9
8
10
9
11
10
12
Who is older and by
how many years??
When Angela is 12 years old, how old is Limei?
__________________________________________________________________________
b.
When Angela is 15 years old, how old is Limei?
__________________________________________________________________________
c.
How is Limei’s age related to Angela’s age?
__________________________________________________________________________
d.
When Angela is ‘ n’ years old, how old is Limei?
__________________________________________________________________________
6.
Jim has $5 more than Hasan.
a.
If Jim has $20, how much money does Hasan have?
92
__________________________________________________________________________
b.
If Jim has $m, how much money does Hassan have in terms of m?
__________________________________________________________________________
7
Mrs Li bought w kg of flour. She used 5 kg of it.
a.
Express the amount of flour left in terms of w.
__________________________________________________________________________
b.
If Mrs. Li bought 8 kg of flour, how much flour had she left?
__________________________________________________________________________
8.
There are four apples in each packet.
a.
Fill the following table with the total number of apples in the packets:
Number of packets
Total number of apples
1
4×1=4
4×n means 4n
2
3
4
5
n
9.
b.
If n = 8, how many apples are there altogether?_______________________
c.
If n = 11, how many apples are there altogether?_____________________
There are 3 boxes of chicken wings. Each box contains p chicken wings.
a.
Express the total number of chicken wings in terms of p.______________________
93
b.
If each box contains 7 chicken wings, how many chicken wings are there altogether?
________________________________________________________________________
10. Mahua bought 3 books.
a.
If the total cost of the books is $12, find the cost of a book?
_______________________________________________________________________
b.
If the cost of the books is $m, what will be the cost of one book?
_______________________________________________________________________
11. Mr. Li had $50. He gave $y to his son. The remainder was then shared equally between
his two daughters.
a.
Express each daughter’s share in terms of y.
______________________________________________________________________
b.
If y = 12, how much money did each daughter receive?
______________________________________________________________________
12. In the following questions, there is a rule that tells you how to get the bottom number from
the top number. Identify that rule.
a.
1
2
3
4
5
6
6
7
8
9
10
11
__________________________________
b.
5
8
16
21
34
56
2
5
13
18
31
53
___________________________________
94
13. The number of cans of cola on a supermarket shelf is found by counting the number of packs
and multiplying by 4.
a.
Write the algebraic expression that connects the number of cans with the number of
packs.__________________________________________
b.
At the end of a day an assistant counts 23 packs on the shelves. How many cans are
there?____________________________________________
c.
She wants to put another 60 cans on the shelves. How many packs is this?
_________________________________________________________
14. First class stamps cost 25p each and are sold in books of 10. Write the algebraic
expression that gives
a.
the number of stamps in terms of the number of books.
_____________________________________________________________
b.
the cost, in pence, of the stamps in terms of the number of books
_____________________________________________________________
c.
the cost, in pounds, of the stamps in terms of the number of books
_____________________________________________________________
d.
the number of books that can be bought in terms of the number of £5 notes paid
_____________________________________________________________
e.
the number of stamps in terms of the number of £5 notes paid.
_____________________________________________________________
15. Write the coefficient of y in the following:
a.
3xy + 7b – 8
__________
b.
-2aby – 4xy + 7 __________
c.
x+5
__________
95
STUDENT’S WORKSHEET – 13
Like and Unlike Terms
CONTENT WORKSHEET C 6
Name of the student ______________________
Date ______
Activity 1 Like and Unlike Terms
1.
2.
Help this cook to make his job simpler.
Shown are three boxes containing fruits, vegetables and sports equipments.
List the items in each of the box using appropriate variables representing each of the items.
96
Activity 2: Algebraic expressions through tiles
With the help of following explanation solve Q3 and Q4
Flip to get -1
Materials required: 18 unit tiles (yellow) for positive
18 unit tiles (red) for negative
Flip to get -
8 ‘x’ tiles (pink) for positive x
x
8 ‘x’ tiles (red) for negative x
The x tile is a rectangle whose sides measure 1 by x
•
Any size tile when paired with its corresponding size red tile we get a zero, as all the
different colour tiles have their corresponding red colour tile as its additive inverse.
i.e. -x + x
•
=
-1 + 1 = 0
A different tile of same or different colour but different size can be made to represent
another variable say y. But every tile has its counterpart which is its additive inverse
represented in red colour.
y
x
z
All the Tiles are of different lengths but same width equal to 1 unit.
97
3. Group the like terms together. Remember colour of the tile does not matter but size does.
Answer the questions below to find the value of the terms in the two boxes.
a)
b)
For both the parts above, answer the following questions
(i)
How many unit square tiles are there?
(ii) What is their total value?
(iii) How many rectangular x tiles are there?
(iv) What is their total value?
98
(v) What is the total value of all the tiles shown above?
i
ii
iii
iv
V
5. Represent the following algebraic expression using the tiles. Also write the simplified
expression too. First one is done for you.
(i) x + 2 - x – x
These three tiles are left.
Now seeing their colour,
we write the expression
These two add up to zero as
they are the additive inverse
of each other.
=2–x
Mathematically, we write as
x+2-x–x+=x-x–x+2
= x (1 -1 -1 ) + 2
------ combining all like terms of x
= - x + 2 or = 2 – x
99
(ii) -1 + 2 – x – x + x
(iii) 4 – x + x + x + x – x
(iv) 2 – 3 + x – x – x – x
(v) 4 x – 3 + 1
(vi) y - 2x + 2y + 6
(vii) 2 – x + 3y + 2x – 1
100
STUDENT’S WORKSHEET – 14
FUN CORNER : SUDOKU Square Puzzle
CONTENT WORKSHEET C 7
Name of the student ______________________
Date ______
Cut out each of the nine squares and rearrange them so that the touching sides of the squares are
equivalent expressions.
101
Magic All around
a. Magic Square:
Write an expression for each square to make this a magic square with a magic sum of
3x – 3. In other words, the sum of each of the rows and the columns (not including the
diagonals) should come out to be equal to 3x – 3.
b. Magic Cross
Cut out the 9 small yellow counters given below and arrange them in the cross so that the
horizontal and vertical totals are both 6y – 5.
102
STUDENT’S WORKSHEET – 15
Evaluating Expressions
CONTENT WORKSHEET C 8
Name of the student ______________________
Date ______
Activity: Evaluating Expressions through math lab activity using tiles
Observe the following example
Find the value of 2x + 3 if x = 4
Replacing each of
the two x by four.
By counting the total
number of unit squares.
=
= 11
If we substitute x = - 2, find the value of 2x + 3
Replacing each of
the two x by -2.
= -1
103
Now solve the following
a)
using tiles
b)
without using tiles
Substitute the value of x and y to find the value of
(i)
3y – 2x + 3 when x = -2 and y = 1.
a)
b)
(ii)
- 5x – 2y – 3 when x = -1 and y = 2
a)
b)
(iii)
6y + 3 when y = - 4.
a)
b)
104
STUDENT’S WORKSHEET – 16
Fun Corner
CONTENT WORKSHEET C 9
Name of the student ______________________
FLOW CHART
Follow the steps of each flowchart and evaluate the end result.
1.
2.
105
Date ______
3.
4.
106
Tic- Tack –Toe
Toby and Ellie played Tick-Tack-Toe. They followed the order of operations to solve the problems
that are shown on the grid. Toby went first. He played X, and Ellie played O. All of Toby's answers
came out to be the same number.
Who was the winner? At the end of the game, which boxes had X's and which had O's?
107
STUDENT’S WORKSHEET – 17
Let’s race!
CONTENT WORKSHEET C 10
Name of the student ______________________
BOARD
108
Date ______
Materials Required: 1 die,
1 Board per group
2-3 counters to move depending upon the number of players
To Play:
1. Shuffle the algebra race cards, and then place them on the table, face down.
2. Player 1 rolls the die and simultaneously picks up a card.
3. Player 1 then evaluates the expression by substituting the value of the variable in the
card by the number obtained on the die.
4. If player 1 solves the problem correctly and the answer is positive then he or she moves
their chip same number of steps (equal to the answer) forward on the board. If the
answer is negative the player moves the same number of steps backwards. However,
they cannot move further backwards from the start position.
5. If Player 1 is unable to solve the problem correctly, he or she loses a turn.
6. If the players use all of the cards in the deck, they reshuffle the deck and start again.
7. The player who reaches Finish first wins.
109
STUDENT’S WORKSHEET – 18
Independent practice
CONTENT WORKSHEET C 11
Name of the student ______________________
1.
Date ______
Mr. Shaun bought y pencils. He distributed the pencils among 40 pupils. How many pencils
did each pupil receive?
2.
Jessica bought 10 similar dresses. Each dress cost $6. How much did she pay in all?
3.
Suzi bought ‘z’ oranges. She gave 6 oranges to her friends. How many oranges was she left
with?
4.
Gordon has m stamps. Michael has twice as many stamps as Gordon. Len has thrice as many
stamps as Michael. How many stamps does Len have? How many stamps do they have in all?
5.
Peter’s car consumed y litres of petrol on Monday. It consumed twice the amount of petrol
on Tuesday. How much petrol was left if Peter had 5y litres of petrol on the Monday
morning?
6.
Simplify the following algebraic expression:
(i)
a + a + 2a
(ii) 5b – 3b + 6b
(iii) 3c + 2c – 4c
(iv)
6d – d – 2d
(v) 4 + 2e + 5
(vi) 9 + 5 + f + 4f
(viii) 7h + 4 – 5h – h
(ix) 10 + 10k + 3 + 14k
(vii) 4g – 2 – 3g
7.
Evaluate the expressions when y = 6
(i)
1+y+9
(iv) 8y ÷ 12
8.
(ii) y ÷2
(iii) y – 3 + 2
(v) 3y + 4y – 3
(vi) 10 – 4y + 21 – 13
The parking charges at a shopping centre was $2 for the first hour and $y for every
subsequent hour or part thereof. Sandy parked his car at the shopping centre from 1430
hours to 1740 hours. How much did he have to pay for the parking charges? Express your
answer in terms of y.
110
9.
Jenny has 100 sweets and Betty has p sweets less than Jenny.
a.
How many sweets must Jenny give to Betty so that both have an equal number of
sweets? Express your answer in terms of p.
b.
If p = 30, how many sweets do they have all together?
10. A rectangular tank 18 cm long and 12 cm wide was filled with water to a height of x cm.
The water increased to 12 cm when some marbles were placed into the tank.
a.
What was the increase in water level in terms of x?
b.
If x = 8 cm, find the increase in the volume of water after marbles were placed into
tank?
11.
After spending $a, Mrs Roy found that she still had $135 left.
a.
How much money did Mrs Roy have at first? Express your answer in terms of a.
b.
If a = 150, how much money did Mrs Roy have at first?
12. Write down the expression for the perimeter of the following figures:
13.
At a fast food restaurant, a cup of sundae costs $s. A chicken burger costs 4 times as much as
the sundae. A packet of French fries costs $f more than the sundae.
a. Find the total cost of a sundae and a chicken burger. Express your answer in terms of s.
b. If s = $ 3 and f = $ 2, how much do 2 chicken burgers and a packet of fries cost?
14.
There were x children and half as many adults as children at a library in the morning. An hour
later, thrice the number of children and 40 adults visited the library.
a. Express in terms of x, the number of people at the library altogether?
b. If x = 56, find the total number of people at the library.
15.
Write a simplified expression for the perimeter of the rectangle or triangle.
111
16.
Circle the coefficient, put a box around the variable and a triangle around the constant in the
following expressions:
a.
8c – 9
b.
6x + 5y – 3
c.
2z + 8
17.
Evaluate 15/y + 7 – 2y , if y = 3
18.
Simplify this expression by collecting like terms : 4z – 9z + z – 3 + 4
19.
Complete the table where
represents y ,
represents 1and
represents x.
(these are the same tiles you have used before)
Tile Display
Simplified
Form Expression
Using Algebra Tiles
Substitute
x=4
into the
expression
and evaluate
- 2x + 3
-2 x 4 + 3
=-8+3
=-5
112
20. a. Write a sentence to represent this picture.
b. Write an expression to represent the picture above.
21. The original cost of a cell phone is y dollars.
The phone is on sale for 25% off.
Write the simplified expression for the sale cost.
22. a.
Model the following expression using algebra tiles:
6x – 5 – 2x + 3 – 8x
b.
Simplify the illustration and write the simplified expression below.
23. Jack and Bill collect model cars and together they own 42.
If you multiply the number of cars Bill has by 10 and add 4, this is the number of cars they
have all together.
Write an expression to represent this problem.
113
STUDENT’S WORKSHEET – 19
Balancing Equations
CONTENT WORKSHEET C 12
Name of the student ______________________
Date ______
a. By weight:
If 4 ketchup bottles are equal to a milk pack and 2 ketchup bottles equal to 1 Jam bottle, can you
find how would milk pack and jam bottle will be related?
Let’s find out!
Let’s replace the other 2
ketchup bottles also with the
jam bottle. This gives us the
relation between the milk
d h j
b l !!!!
Let’s pick up 2 ketchup
bottles and place 1 jam
bottle instead. Balance
remains still equal!!!
Relation between the milk pack and the jam bottle are related?
114
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
After seeing and understanding this balance, Can you write which among the following is balanced
and which one is unbalanced?
b. Balancing by value:
Example 1:
If the value of 2 milk packets is $6, then can you balance the value of 4 ketchup bottles?
Let’s find out!
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Representing
milk bottle by
M.
2 Milk = $6 or 2M = 6
1 M = $3
Can we find the value of a bottle of jam and the ketchup?
_________________________________________________________________________________
_________________________________________________________________________________
Try out balancing the value and the weights!!
We have seen that 2 bottles of jam equals a packet of milk. Also a bottle of jam equals 2 bottles of
ketchup.
Write the expression for the
a.
Jam bottles and their value
___________________________________________________
b.
Ketchup bottles and their value
_________________________________________________
Hence, find the value of a bottle of jam and a bottle of ketchup.
____________________________________________________________________________
116
Using the information given, can you find the value of the following?
Example 2: Given that 2 cones balance 3 cylinders.
a. How many cones will balance 6 cylinders?
___________________________________________________________________________
___________________________________________________________________________
b. How many cylinders will balance 8 cones?
___________________________________________________________________________
___________________________________________________________________________
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c. How many cylinders will balance 2 cones and 3 cylinders?
___________________________________________________________________________
___________________________________________________________________________
118
d. How many cones will balance 9 cylinders and 2 cones?
___________________________________________________________________________
___________________________________________________________________________
119
STUDENT’S WORKSHEET – 20
Equations using Tiles
CONTENT WORKSHEET C 13
Name of the student ______________________
Date ______
Math lab Activity : Using Algebra Tiles
Try this out yourself now.
1. Write down the algebraic equation that is represented by the pictures below and solve for x.
2. Fill in the chart below, first one is done for you.
Original equation
Isolate the variable x
x+7=3
x +7 – 7 = 3 – 7
Inverse operation
used to isolate x
–7
x–2=-5
2x = 8
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Solution
x = -4
x / 4 = -1
3x = - 3
2x + 3 = 5
3. Find the missing weights by writing and solving an equation for each problem given below.
First one is done for you.
a.
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4. Illustrate and explain how algebra tiles can be used to show an algebraic process for
simplifying these equations:
a. 4x – 3 = 5 b. - 6x + 8x = 4
c. 5x – x + 2x = 18 – 6
122
STUDENT’S WORKSHEET – 21
Fun Corner
CONTENT WORKSHEET C 14
Name of the student ______________________
1.
Date ______
Crack the code
Solve the following equations. Write variables in the space provided above their values ( see
below) to find the name of a famous volcano!
M
+ 22 = 26
9
1. 4e – 8 = 32
2.
4. 6t + 13 = 7
5. i – 2 = 0
6.
7. 25 + u = 17
8. 6V – 62 = 52
9. 22n – 64 = 24
2. Cross word Puzzle
Across:
123
3. a + 19 = - 31
s
= -1
10
1. replace a variable with a possible solution
4. an amount ( sometimes contained within parentheses)
5. an amount that can change or is unknown.(often displayed as a letter)
6. a value for a variable that makes a statement true
7. the result of a multiplication problem
8. result of a division problem
10. an amount that does not change
11. a mathematical phrase that contains at least one variable
12. part of a sentence or a short mathematical statement.
13. the result of an addition problem
Down:
2. a mathematical sentence that says that two quantities are equal
3. the answer to a subtraction problem
9. mathematical operations that cancel each other (opposites)
Algebraic
constant
difference
equation
inverse
phrase
Product
quantity
quotient
solution
substitute
sum
variable
124
STUDENT’S WORKSHEET – 22
Math Award
CONTENT WORKSHEET C 15
Name of the student ______________________
Date ______
Ver E Bright won the First-Place entry in the Mathematics Fair. Young Ver was to be awarded the
key to the city by the mayor. Ver asked his math teacher what he should say upon receiving the
award. Instead of answering directly, his teacher handed him a sheet of problems and said,
‘Translate each of problems into algebraic expression. One of the expressions will give you the
answer you seek.’
Help Ver out. Write the algebraic expression for each problem. Then make a check next to the
answer that he needs.
1. Rachel’s dog had puppies. She sold 3 of them. Let n be the number of puppies she has now.
Write an expression that tells the original number of puppies.__________
2. Jake ran 5 more miles this week than he ran last week. Let t be the number of miles he ran
last week. Write an expression that tells how many miles he ran this week._______
3. Mr. Stellar hired 3 students to deliver advertising fliers. He paid each student the same
amount. Let x be the total amount the students earned. Write an expression that tells how
much one student earned.___________
4. There are 42 members of the Drama Club. Let t be the amount of money each earned for
the spring trip. Write an expression that tells the total amount the club raised.______
5. The Rocket Club is 7 years old. It now has 10 times its original number of club members.
Write an expression that tells the number of members now. _________
6. Wes is 4 years older than his sister. Let x be his sister’s age. Write an expression that tells
how old Wes is._______
7. A radio is on sale for half price. Let y be its original price. Write an expression that tells its
sale price._________
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Sonya and Maya created a secret code to keep Ver’s award party a surprise. In order to
know the day and time of the party, you must crack the code. Solve the equations and fill in
each box with the letter that goes with the number under it.
a + 2 = 14
t + 5 = 15
n–4=5
3e = 15
2j = 38
i + 5 = 20
h + 10 = 14
f + 18 = 18
u÷2 = 9
r-8=3
2a = 24
d÷5 = 5
y – 6 = 30
o–4=4
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STUDENT’S WORKSHEET – 23
Independent Practice
CONTENT WORKSHEET C 16
Name of the student ______________________
Date ______
1. 3 is a factor of x and 24 is a multiple of x. What could x be?
a. 8
b. 12
c. 56
d. 72
2. The total mass of 3 boxes is 9.6 kg. The mass of two of the boxes is x kg each. What is the
mass of the third box?
a. (9.6 – x) kg
b. (9.6 + x)kg
c. (9.6 – 2x)kg d. (9.6 + 2x)kg
3. Mrs Brook had p eggs. She used 6 to bake a cake and gave 5 eggs to her neighbor. How
many eggs is she left with?
a. (p+11) eggs
b. (11-p)eggs
c. (p - 11) eggs
d. 11p eggs
c. y
d. 4 – 4 y
4. y + y + y + y means the same as
a. 4y
b. 4 + y
5. Linda is y years old. How old will she be in 3 years’ time?
a. 3y years
b. (y – 3 )years
c. (3 + y) years
d. (3 – y) years
6. A square has a perimeter of 16m. What will be its side?
a. 4m
b. 8m
c. 16m – 4
d. 2m
7. John had a m of rope. He cut off from it 5 equal pieces of rope, each 20 cm long. What was
the length of the rope left?
a. (a – 10) cm
b. ( a – 20)cm
c. (100a – 100) cm
d. (100a – 20)cm
8. The total mass of 2 boxes is 22 kg. If Box I is y kg heavier than Box II, find the mass of Box II
in terms of y. If y is 4kg, find the mass of the two boxes.
9. The average mass of 8 girls is x kg. The mass of 3 of them is 24kg. Find the average mass of
the rest of the girls in terms of x. If x is 10kg, find the mass of the rest of the girls.
10. Sam’s mass is w kg. His mother is 5 times as heavy. His sister’s mass is one third his mass.
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a. How much is the mass of his mother in terms of w?
b. What is the total mass of Sam and his mother in terms of w?
c. If mass of the mother is 85 kg, how much is the mass of his sister in grams?
11. A shopkeeper bought 100 eggs for $12. He broke 10 of them and sold the rest at 10 for $q.
a. How much did he collect from the sale of the remaining eggs in terms of q?
b. Find the amount of money earned from the sale of the eggs in terms of q.
c. If q = $1.3, find his profit or loss?
12. David saved $125 from January to June. If he saved $ y each month from January to May,
a. How much did he save in June in terms of y?
b. If he saved $ 110 from January to May, find the value of y.
13. I think of a number, add 8 and get 21 as the answer. Find the number.
14. Kim is n years old and Toni is 2 years older than Kim. If Toni is 14, how old is Kim?
15. A cup of coffee costs 15 pence more than a cup of tea. If a cup of tea costs x pence and a
cup of coffee cost 75 pence, how much is a cup of tea?
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STUDENT’S WORKSHEET – 24
Extended practice
POST CONTENT WORKSHEET PC 1
Name of the student ______________________
Date ______
1. I think of a number, multiply it by 4 and subtract 8. The result is 20. What was the number?
2. The lengths of the three sides of a triangle are x cm, x cm and 6 cm. The perimeter is 20 cm.
Find x.
3. Three boys had x sweets each. They gave 9 sweets to a fourth boy and then found that they
were left with 18 sweets among three of them. Find x.
4. I have two pieces of ribbon each y cm long and a third piece 9 cm long. Altogether there are
31 cm of ribbon. What is the length of each of the first two pieces?
5. Jen is x years old and her mother is 27 years older. Together their ages total 45 years.
a. In terms of Jen’s age, how old is her mother now?
b. Form an equation in x and solve it to find Jen’s age and her mother’s age now?
6. A fishing rod is 27ft long and consists of three parts. The first part is y ft long, the second
part is 1 ft longer than the first and the third part is 1 ft longer than the second part. Form
an equation in y and solve it to find the length of each part.
7. Sonya is t years old now and Claudia, her sister, is 2 years older. In 5 years time the sum of
their ages will be 20.
a. How old is Sonya now?
b. How old will Claudia be in 5 years time?
8. The ratio of Mandy’s mass to Jim’s mass is 4: 7. If their total mass is 99 kg, find Mandy’s
mass.
9. George and Ryan donated a sum of money in the ratio 10:7 to charity. If George donated $6
more than Ryan, how much did they donate altogether?
129
10. Billy, Alan and Leon share 35 stamps. If Billy gets twice as many stamps as Alan, and Leon
gets twice as many stamps as Billy, how many stamps does Leon get?
11. The ratio of the length of a rectangle to its breadth is 5:2. If its breadth is 8cm, find its
length. And hence, find its area.
12. A tailor bought 180 buttons. He used some of the buttons and packed the remainder into 3
boxes in the ratio 3: 1: 2. There were 26 buttons in the box with the least buttons.
a. How many buttons were there in the box with the most buttons?
b. How many buttons did he use?
13. Find the two numbers such that one of them exceeds the other by 18 and their sum is 92.
14. The sum of three consecutive odd numbers is 21. Find the numbers.
15. A man is four times as old as his son. After 16 years he will be only twice as old as his son.
Find their present ages.
16. The number of boys in a school is 334 more than the number of girls. If the total strength of
the school is 734, find the number of girls in the school.
17. A bag contains 25 paise and 50 paise coins whose total value is Rs 30. If the number of 25
paise coins is four times that of 50 paise coins, find the number of each type of coins.
18. A horse and a cart together cost $ 1600. If the ratio of the cost of the cart to that of the
horse is 1:3, what is the cost of the horse?
Activity 2:
Given below are a set of equations. Convert them into some real life statements:
1. 2x+4=x-2
________________________________________________________________________
____________________________________________________________________
2. 5x-7=2x+4
________________________________________________________________________
________________________________________________________________
3. x+4=3x+2
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________________________________________________________________________
___________________________________________________________________
4. 9x=45
________________________________________________________________________
____________________________________________________________________
STUDENT’S WORKSHEET - 25
POST CONTENT WORKSHEET PC 2
Name of the student ______________________
1.
Rita wrote down the perimeter of her bedroom as x + x + y + y .
a. Write down a different expression for the perimeter of her room.
b. Show that the expressions are equivalent.
2.
a. Write an expression for the perimeter of this rectangle.
b. A square has the same perimeter as this rectangle. Write
an expression for the length of the side of the square.
c. Write an expression for the area of the square.
d. Which has the larger area, the rectangle or the square and by how much?
3.
A teacher has 5 full packets of mints and 6 single mints.
The number of mints inside each packet is the same.
131
Date ______
The teacher tells the class:
‘Write an expression to show how many mints are there altogether. Call the number of mints
inside each packet y’
Here are some of the expressions that pupils wrote:
5+6+y
5y6
5y + 6
6 + 5y
5 + 6y
(5 + 6)× y
a. Write down two expressions that are correct.
b. A pupil says:’ I think the teacher has a total of 56 mints’.
Could the pupil be correct? Explain how you know.
4.
Simplify these
a. 3a + 4a – 2a
b. 7x + 3x + 2 – 4
c. 8y – y
d. 3b + 2b + 4b
e. 5n + 3n – 3n
f. – 2z + 3x – 4y + 6z + x – 3y
g. 7 – x -6 – 3x
h. 30x + 2 – 15x – 6 + 4
i. 7x + 3 – 9 – 9x + 2x – 6 + 11
j. 6x – 5y + 2x
5.
Use a copy of this.
Join pairs of algebraic expressions that have
The same value when a = 3, b = 2 and c = 6.
One pair is joined for you.
Find the box which does not have a pair.
6. (i). A yard is 15 m long. A fence 50 m long is needed to go
right around the perimeter of the yard. If w is the width of
the yard, which of these is correct?
a. 30 + 2w = 50
b. 15 + w = 50
c. 50 + 2w = 30
d. w = 50 – 30
(ii) Solve the correct equation to find the width of the yard.
132
7.
A teacher has a large pile of cards.
An expression for the total number of cards is 6n+8
a.
The teacher puts the cards in two piles.
The number of cards in the first pile is 2n + 3.
Write the expression to show the number of cards in the second pile.
b.
The teacher puts all the cards together.
Then he uses them to make two equal piles.
Write an expression to show the number of cards in one of the piles.
c.
The teacher puts all the cards together again, then he uses them to make two piles.
There are 23 cards in the first pile.
How many cards are in the second pile? Show your working.
8.
Jackie made these patterns with strips of felt.
a.
Copy these and fill this table.
b.
Describe the sequence and how it continues.
c.
Explain how you could find the number of strips
needed for the nth tree.
133
9.
If x = 3 and y = 5 find the value of these expressions.
a. x y
b. 2 x – 3
c. 4 ( x + y )
d.
x+2
y
e.
3y − 3
y
10.
For these coloured rods we could write the equations, r = b + g.
a.
Which of these are true?
b= p + y
b.
r-g=b
2g = r 2p + y = r
r - y = 2p
Write three more equations that are true for these rods.
11. You can work out the cost of an advert in a newspaper by using this formula:
C = 15n + 0.75
C is the cost in pounds
n is the number of words in the advert
a. An advert has 18 words. Work out the cost of the advert. Show your working.
b. The cost of an advert is £615. How many words are in the advert?
12. Marilyn made 80 pieces of fudge. She put n pieces in one box. In the second box she put 12
more pieces than in the first box. In the third box she put 4 fewer pieces than in the second
box. Write and solve an equation to find the number.
13. I think of a number, add 7, multiply by 3, subtract 3, divide by 6, and then multiply by 12. The
answer is 72. What was the number?
14. In this number pyramid, each number is the sum of the two numbers immediately above it.
Find the missing number, n, which makes the bottom number correct.
134
15. The number in each square is the sum of the numbers in the two circles either side. Find the
missing numbers. Write and solve an equation to help.
16. Two years ago, Adam was five times as old as his daughter. The sum of their ages now is 52.
How old is the daughter?
17. Tom, Jack, Charlotte and Grace got a total of 101 points. Tom got 5 more than Jack,
Charlotte got 8 more than Jack, Grace got 25. How many points did Tom, Jack and
Charlotte get?
18. For every 100 pamphlets Wasim delivers, he gets £5. Altogether last month he earned £75.
How many pamphlets did he deliver?
19. Meggie buys 4 ice creams and a drink and this cost the same as 2 ice creams and 4 drinks. If a
drink costs 80p, how much does an ice cream cost?
20. Two sisters, Janet and Nora, each have a box holding 20 chocolates. Janet eats 5 and gives
some away to her friends. Nora gives 1 away and eats three times as many as
Janet has given away. When they compare boxes, each sister still has the same number of
chocolates left as the other. If Janet gave x chocolates away, form an equation in x and solve
it. How many chocolates did Nora eat?
135
Acknowledgement:
Websites Referred to:
1.
www.hotmaths.com.au
2.
http://www.wbrschools.net/techcds/webpage%20math%2078/algebra/the%20land%20of
%20the%20unknown%20g7gle14.pdf
3.
http://mathbits.com/mathbits/AlgebraTiles/AlgebraTilesMathBitsNew07ImpFree.html
4.
http://newtonanddescartes.com/protected/content/ipe_na/grade%207/02/g7_02_01.pdf
5.
http://www.youtheducationservices.ca/algebra.html
6.
http://www.phschool.com/webcodes10/index.cfm?fuseaction=home.gotoWebCode&wcpr
efix=ask&wcsuffix=9901
7.
http://nrich.maths.org/public/index.php
8.
http://www.learningwave.com/abmath/drp99/plenobius/worksheet.html
9.
http://mathstar.lacoe.edu/lessonlinks/menu_math/var_food.html
10.
http://www.eduplace.com/kids/hmm/bt/6/6_04-1q.html
Reference Books:
1.
NCERT Mathematics for class 6
2.
National Framework 8+ Mathematics By Tipler and Vickers
3.
Composite Math for class 6
4.
Mathematics Young achievers Book 6 By SAP
5.
STP Mathematics 7A By Nelson and Thornes
Acknowledgement:
Websites Referred to:
11.
www.hotmaths.com.au
12.
http://www.wbrschools.net/techcds/webpage%20math%2078/algebra/the%20land%20of
%20the%20unknown%20g7gle14.pdf
13.
http://mathbits.com/mathbits/AlgebraTiles/AlgebraTilesMathBitsNew07ImpFree.html
14.
http://newtonanddescartes.com/protected/content/ipe_na/grade%207/02/g7_02_01.pdf
136
15.
http://www.youtheducationservices.ca/algebra.html
16.
http://www.phschool.com/webcodes10/index.cfm?fuseaction=home.gotoWebCode&wcpr
efix=ask&wcsuffix=9901
17.
http://nrich.maths.org/public/index.php
18.
http://www.learningwave.com/abmath/drp99/plenobius/worksheet.html
19.
http://mathstar.lacoe.edu/lessonlinks/menu_math/var_food.html
20.
http://www.eduplace.com/kids/hmm/bt/6/6_04-1q.html
Reference Books:
6.
NCERT Mathematics for class 6
7.
National Framework 8+ Mathematics By Tipler and Vickers
8.
Composite Math for class 6
9.
Mathematics Young achievers Book 6 By SAP
10.
STP Mathematics 7A By Nelson and Thornes
137
Video Links
Name
Tiltle/Link
Video clip 1
X finds out its value
http://www.youtube.com/watch?v=J2TYyUftI8k
Weblink 1
Formation of Algebraic expressions
http://www.math-play.com/Algebraic-Expressions-Millionaire/algebraicexpressions-millionaire.html
Weblink 2
Flash cards on Algebraic expressions
http://www.quia.com/jfc/319817.html
Weblink 3
Working with Like and Unlike terms
http://www.quia.com/mc/332031.html
Weblink 4
Game to evaluate the algebraic expressions
http://www.bbc.co.uk/education/mathsfile/shockwave/games/postie.html
Weblink 5
Solving Linear equations with beam balance
http://nlvm.usu.edu/en/nav/frames_asid_201_g_4_t_2.html?open=instructions
Weblink 6
One step equation game
http://www.math-play.com/One-Step-Equation-Game.html
138