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INTRODUCTION
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TO ALGEBRA
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PADMAPRIYA SHIRALI
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INTRODUCTION
Introduction to school algebra can happen through varied approaches. Some prefer to start
with an unknown in an equation, while some prefer to start with a formula and some others
may prefer to use a pattern based approach. Does it make a difference which approach one
uses? Is one approach better than the others? These questions can be debated. However,
each of these approaches relates to different conceptions of algebra.
The unknown in an equation conceives algebra as a study of procedures for solving certain
kinds of problems requiring simplification; the formula approach conceives algebra as the
study of relationship among quantities which vary. The pattern based approach conceives
algebra as generalized arithmetic leading to generalization of known relationships among
numbers.
Algebra thus is all of these: generalized arithmetic, a procedure for solving certain problems,
and a means of understanding relationships and mathematical structures.
In school algebra, the term ‘variable’ typically appears first in the form of a letter that
represents an unknown in an open sentence or an equation (e.g., 4 + x = 9), followed by
formulas (e.g., A = L × B), as a generalized property (e.g., a + b = b + a), later as an identity
(e.g., (a + b)2 = a2 + 2ab + b2) and as a function (e.g., y = 3x). Students learn to use variables
to solve various types of problems.
However, does algebraic thinking take place in a child’s mind well before he/she encounters
a variable? For instance, when a child says ‘I have 6 toffees; if there were 4 more I would
have 10” or when a child is able to abstract a pattern from numerical relationships, or when
a child is able to guess the tenth figure in a pattern of figures, can one say that the child has
begun to think algebraically?
The late Shri P. K. Srinivasan had developed an approach to the teaching of algebra titled
‘Algebra – a language of patterns and designs’. I have used it for several years at the Class 6
level and found it to be very useful in making a smooth introduction to algebra, to the idea
and usage of concepts such as variable and constant, to performing operations involving
terms and expressions. This approach steadily progresses from studying numerical patterns
to line and 2-D designs, finally leading to indices and identities. Over the years, I have
adapted this material to meet the needs and interests of the students. However, the basic
structure has remained largely the same. I share here the adapted approach.
Patterns, numerical or visual, have an inherent appeal to children and adults alike. It may
have to do with the aesthetic feeling present in the human psyche. We are able to recognise
and sense patterns in nature, patterns in the movements of the heavenly bodies, patterns in
time (seasons) – patterns on a macro-scale as well as on a micro-scale.
Patterns make a very good starting point for the introduction of algebra. They arise easily
from the mathematical knowledge that students have already acquired by Class 6 (even and
odd numbers, multiplication tables, behaviour of certain numbers, number relationships).
In this pull-out, I focus on the first step of working with patterns as an introductory step to
the usage of concepts such as Variable, Constant, Term and Expression and also operations
involving these concepts. In the second pull-out I will take up design language and depict
the usage of the same concepts and operations. In the third pull-out I will take up indices
and identities. Subsequently, approaches to equations will be taken up.
Keywords: Algebra, unknowns, equations, expressions, patterns, activities, manipulatives.
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Azim Premji University At Right Angles, March 2018
ACTIVITY 1
Objective: To expose students to different kinds of patterns
Pattern recognition is innate to the brain and
happens quickly and naturally. However, if students
have been taught earlier through a rote and
mechanical approach, one may need to reawaken
their observation and thinking powers.
Pattern problems in numbers and designs are
available in plenty as resources. The teacher will
need to make an appropriate graded selection
suited to the needs of the upper primary kids.
1. What is the pattern here? What goes into the
blank space?
a. 7, ____, 24, 34, 45, 57, 70
b. 71, 70, 73, 72, 75, _____, _____, _____
2. Find the odd one out. Justify your answer.
a. 252, 72, 1, 275, 24, 488
I have given here a few model problems.
Figure 1
1. Here are the first five triangular numbers: 1, 3,
6, 10, 15.
2. Can you see a pattern?
3. Can you predict the next triangular number?
4. What would the tenth triangular number be?
Figure 2
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1. I used 3 matches to make 1 small triangle.
2. How many matches do I need to build a second
row of triangles under that?
3. How many matches do I need to build a third
row of triangles under that?
4. How many matches will I need to make the
sixth row?
5. Can you make out how many matches I will
need to make the twentieth row?
Figure 4
1 block is needed to make an up-and-down
staircase, with 1 step up and 1 step down.
4 blocks are needed to make an up-and-down
staircase with 2 steps up and 2 steps down.
How many blocks would be needed to build an
up-and-down staircase with 5 steps up and 5 steps
down?
Explain how you would work out the number of
blocks needed to build a staircase with any number
of steps.
Figure 3
1. Make a box around a set of nine numbers (a 3 × 3
square) in the tables square.
a. Add the numbers in the shaded squares.
b. Add the corner numbers.
c. Multiply the centre number by 4. What
happens?
2. Make a box around another set of nine numbers
and try this again.
Figure 5
A hundred square has been printed on both sides
of a piece of paper. One square is directly behind
the other just like in the pages of a book.
What number lies on the other side of 100? The
other side of 58? Of 23? Of 19?
Do you see a pattern?
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Azim Premji University At Right Angles, March 2018
ACTIVITY 2
Objective: Introduction to pattern language and the usage of a letter for changing numbers.
Introduction to the notion of a changing number
(variable) and an unchanging number (constant)
can begin in familiar settings. We introduce the
words constant and variable, term and expression
a little later.
Students already know about even numbers and
about multiples and square numbers. This activity
helps them to learn to write pattern language in
the context of their prior knowledge of number
relationships.
Give students a set of even numbers. For example,
12, 22, 8, 44.
Pose the question, “What are these?” They will
notice that they are all even numbers.
What else can be said about them? They are all
multiples of 2.
Now the teacher can rewrite all these numbers as
multiples of 2.
22 = 2 × 11
8=2×4
44 = 2 × 22
Now pose the question ‘What do you notice about
the right hand side?’ What is the first number?
It is always 2. What is the second number? It is
changing each time.
So how can we describe an even number? It is 2
times some number.
Since the second number changes or varies, we
represent it using a letter.
An even number can be now written as 2 times ‘n’
or 2 × n. (Mention to the students that we drop the
multiplication sign as it looks like the letter ‘x’. So
2n means ‘2 times n’).
We can take up another example using multiples.
44,11, 220, 121.
What are these? They are all multiples of 11.
They can be written in this way.
44 = 11 × 4
11 = 11 × 1
220 = 11 × 20
121 = 11 × 11
What do we see on the right side? The first number
is always 11. The second number is changing.
This pattern can be written as 11x or 11y. (Tell the
children that any letter can be used to stand for the
changing number).
Let us take up a slightly different example where
there is no constant factor.
16, 49, 4, 81.
What are these numbers? Square numbers.
They can be written in this way.
16 = 4 × 4
49 = 7 × 7
4=2×2
81 = 9 × 9
What can we say about the numbers on the right
hand side? Help the students articulate this. ‘The
first number is changing. The second number too
is changing. But the first number and the second
numbers are always the same.’ So, how does one
describe such a pattern?
It can be described as ‘y’ times ‘y’ or ‘y × y’or ‘yy’.
(Note: At this point, we do not write yy as y2 as we
have not yet introduced indices to them.)
Let us take up another type of situation where
both the factors are different variables.
Here are some numbers. Can you write them
as product of two numbers without using 1 as a
factor?
65, 14, 6, 77.
We write them as products:
65 = 5 × 13
14 = 2 × 7
6=2×3
77 = 7 × 11
On the right hand side, what can we say about the
first number? The second number? They are both
changing. The first changing number can be called
‘x’ and the second changing number can be called
‘y’. The pattern here can be described as ‘x’ times
‘y’ or ‘x × y’ or ‘xy’.
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The teacher can ask the students to come up with
more such examples of their own.
Students can also work in pairs. Each student can
create a pattern using multiples of slightly larger
numbers, say, between 10 and 20 and ask another
to describe the pattern using pattern language. Or
they can do the same with cube numbers.
ACTIVITY 3
Objective: Patterns with two terms and two operations.
Let us look at these numbers.
21, 43, 7, 101.
69 = 10 × 7 – 1
What are these numbers? They are odd. How do
we describe them? Students may take time to
respond to that question.
19 = 10 × 2 – 1
Another question which can help is: ‘What is their
relationship to even numbers?’ They are either 1
more or 1 less than even numbers.
So we write them initially as follows:
89 = 10 × 9 – 1
The pattern here is 10n – 1.
Let us look at another pattern which uses place
value.
36, 75, 49, 81, 19.
7=6+1
What pattern can one see here? They are not all
composite. They are not multiples of any single
number. They are all double digit numbers. They
can be written as follows:
101 = 100 + 1
36 = 10 × 3 + 6
At this point, we can describe them as n + 1. Is
there something further we can do? How did we
describe the even numbers earlier? So now we
write these numbers as follows:
75 = 10 × 7 + 5
21 = 20 + 1 = 2 × 10 + 1
19 = 10 × 1 + 9
21 = 20 + 1
43 = 42 + 1
43 = 42 + 1 = 2 × 21 + 1
7=6+1=2×3+1
101 = 100 + 1 = 2 × 50 + 1
Now we describe the pattern as 2n+1.
49 = 10 × 4 + 9
81 = 10 × 8 + 1
This pattern can be described as 10m+n.
How about this set?
94, 99, 91, 95.
They could be expanded as follows:
Students can be shown that the same numbers,
expressed differently, can be described as 2n –1.
94 = 100 – 6 = 10 × 10 – 6
Note: At this point the teacher can introduce the
words variable, constant, term and expression to
the students.
91 = 100 – 9 = 10 × 10 – 9
Here is another pattern.
49, 69, 19, 89.
All the numbers end with a 9 in the units place.
They can be written as follows.
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49 = 10 × 5 – 1
Azim Premji University At Right Angles, March 2018
99 = 100 – 1 = 10 × 10 – 1
95 = 100 – 5 = 10 × 10 – 5
Hence the pattern becomes 10 × 10 – n.
Students may also see it as 90 + n.
Game: Pattern detective
Objective: To detect the pattern created by another.
Here is another example of this game between the
two students.
Materials: Black board or blank paper
This game can be played by the whole class or by
small groups of 5 students or even in pairs.
Student I calls out any number between 1 and 10,
say 5. Student II performs any two operations on
the given number to generate a new number, say
12. This exchange between student I and student II
is repeated at least four times. Each time student II
performs the same operations in the same order to
generate corresponding numbers.
This is how it goes.
Student I
Student II
5
12
3
8
8
18
10
22
5
24
3
8
8
63
10
99
Here student II is squaring the number and
subtracting 1 from the product.
The pattern can be described as ‘nn – 1’.
Here is one more example of this game between
two students.
The pattern needs to be detected by either student
I or the group or the class that is watching.
The pattern can be described as 2n+2.
Student II
What is student II doing with the numbers given
by student I?
What is student II doing with the numbers given
by student I?
Here student II is doubling the number and adding
2 to the product.
Student I
Student I
Student II
1
3
2
7
3
11
4
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What is student II doing with the numbers given
by student I?
I will leave it to you to figure out!
Note: Initially it is better for the students to use
two specified operations, i.e., either ‘× and +’ or
‘× and –’.
ACTIVITY 4: PATTERNS IN EXPRESSIONS
Objective: To describe given patterns and create patterns for a given expression
To observe addition of like terms with a single
variable
2×3+3×3
2×5+3×5
2×2+3×2
2×1+3×1
How do we describe the pattern here?
Let the students state that it is 2a + 3a.
Now ask the students to work out the sum for each
expression and write it as shown.
Ask them to find a pattern in the answers. They will
see that they are multiples of 5.
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Let them write the answer as multiples of 5.
Provide a number pattern like this:
2×3+3×3
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5×3
3×3+2×4
17
2×5+3×5
25
5×5
3×5+2×2
19
2×2+3×2
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5×2
3×2 +2×7
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2×1+3×1
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5×1
3× 1+2×3
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How will this pattern be described? It will be 5a.
Teacher can point out to the students the fact that
2a and 3a have summed up to 5a.
Now pose the question: “What would 3x and
4x add up to?” Let the students guess and build
patterns to verify their answer.
It is important at this point to show that when
unlike terms are added, the answer cannot be
‘simplified’.
How will the pattern on the left hand side be
described? It is 3a + 2b.
Can the students find any pattern in the sums of
these numbers?
They can now try to guess the answer for a
subtraction situation, e.g., 5x – 2x, and build a
pattern to verify the answer.
ACTIVITY 5: PATTERNS IN EXPRESSIONS
Objective: To describe given patterns and create patterns for a given expression
To observe addition of like terms with more than
one variable
What is the pattern of the answers in the final
column?
How will this pattern be described?
It is 2ab.
2×4+4×2
3×6+ 6×3
Again draw the students’ attention to the addition
of ab + ba which equals 2ab.
5×2+ 2×5
Are ab and ba like terms? Why?
8×3+ 3×8
Discuss more examples of ‘like’ and ‘unlike’ terms
in two variables.
It is of the form ab + ba.
Here again the students can sum them and observe
the results.
What is the pattern in the answers? They are all
multiples of 2.
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As a practice exercise, students can be asked to set
up a number pattern for a given pattern language,
using only like terms initially.
Ex. Create number patterns for xy + xy + xy.
Let the students write them initially as multiples of 2
(16 = 2 × 8, etc).
What does it become?
As a second step, they can write the factors of the
second number as well (16 = 2 × 2 × 4, etc).
Create number patterns for 5cd – 2cd.
2×4+4×2
2×8
16
2×2×4
3×5+5×3
2 × 15
30
2×3×5
6×7+7×6
2 × 42
84
2×6×7
3×3+3×3
2×9
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2×3×3
Azim Premji University At Right Angles, March 2018
Would it be different for xy + yx + xy?
What does it become?
Let the students also create patterns for addition
and subtraction of unlike terms.
Example: Create number patterns for each:
(i) abc – cde (ii) ab + bc + ca.
ACTIVITY 6: LAWS OF COMMUTATIVITY AND ASSOCIATIVITY
Objective: To establish commutativity and associativity
What do we notice here?
Property: a + (b + c) = (a + b) + c.
3+2=2+3
In a similar manner, the teacher can build patterns
to demonstrate properties of multiplication and
division, properties of 0 and 1 by studying the
patterns.
5+1=1+5
6+4=4+6
Property: a + b = b + a
Pose the question to the students: “Can I replace
the + sign with – sign?” “Can I replace the + sign
with ×?” “Can I replace the + sign with ÷?”
What do we notice here?
Property: a × b = b × a, a × (b × c) = (a × b) × c,
a × (b + c) = (a × b) + (a × c).
Properties of 1: 1 × a = a, a ÷ a = 1, a ÷ 1 = a.
Properties of 0: a + 0 = a, a – 0 = a, a – a = 0,
a × 0 = 0, 0 ÷ a = 0.
2 + (3 + 5) = (2 + 3) + 5
1 + (4 + 2) = (1 + 4) + 2
5 + (2 + 1) = ( 5 + 2) + 1
ACTIVITY 7
Objective: To discover some number properties and describe them as expressions
Create a pattern with consecutive numbers.
Tell the students to sum the numbers in the pattern
to discover and state the property using pattern
language.
11 + 12
It can be described as n + n + 1 which becomes
2n + 1.
The students can set up patterns and discover the
answers for the following questions. The answers
can be stated as expressions.
What is the difference between any pair of
consecutive numbers?
2+3
7+8
What is the sum of three consecutive numbers?
10 + 11
The sum of two consecutive numbers is always an
odd number.
This pattern can be rewritten as follows:
11 + 12
11 + 11 + 1
2 × 11 + 1
2+3
2+2+1
2×2+1
7+8
7+7+1
2×7+1
10 + 11
10 + 10 + 1
2 × 10 + 1
Can they state a property of the product of two
consecutive numbers?
Can they state a property of the product of three
consecutive numbers?
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ACTIVITY 8
Objective: Exploring challenging problems through algebraic thinking
Let the students take any two digit number, say 53.
Ask them to reverse the digits, i.e., 35. Let them
find the difference between these numbers.
They can do this with some more numbers to spot
a pattern.
3. “If you double each of those numbers what
happens to the total? Why?”
53 – 35
Are they able to use expressions to answer these
questions?
74 – 47
One final challenge!
21 – 12
Here is an interesting result.
63 – 36
552 – 452 = 1000
Can they describe the pattern that emerges?
1052 – 952 = 2000
Here is one more question.
852 – 652 = 3000
Ask the students to take a set of five numbers, say
5, 12, 4, 20, 6. Let them total it.
Now pose the following questions:
1. “If you take 2 away from each of those numbers
what happens to the total? Why?”
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2. “If you add 3 to each of those numbers what
happens to the total? Why?”
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How do we describe this pattern?
Are there any other pairs which give multiples of
1000?
Padmapriya Shirali is part of the Community Math Centre based in Sahyadri
School (Pune) and Rishi Valley (AP), where she has worked since 1983, teaching a
variety of subjects – mathematics, computer applications, geography, economics,
environmental studies and Telugu. For the past few years she has been involved
in teacher outreach work. At present she is working with the SCERT (AP) on
curricular reform and primary level math textbooks. In the 1990s, she worked
closely with the late Shri P K Srinivasan, famed mathematics educator from
Chennai. She was part of the team that created the multigrade elementary
learning programme of the Rishi Valley Rural Centre, known as ‘School in a Box’
Padmapriya may be contacted at padmapriya.shirali@gmail.com
Padmapriya Shirali
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Azim Premji University At Right Angles, March 2018