Algebraic identities: alternatives to relying on memory Unit 1:

Teacher Education through School-based Support in Indiawww.TESS-India.edu.in

Algebraic identities: alternatives to relying on memory

Unit 1:

Secondary Mathematics

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The TESS-India project (Teacher Education through School-based Support) aims to improve the classroom practices of elementary and secondary teachers in India through student-centred and activity-based approaches. This has been realised through 105 teacher development units (TDUs) available online and downloaded in printed form. Teachers are encouraged to read the whole TDU and try out the activities in their classroom in order to maximise their learning and enhance their practice. The TDUs are written in a supportive manner, with a narrative that helps to establish the context and principles that underpin the activities. The activities are written for the teacher rather than the student, acting as a companion to textbooks. TESS-India TDUs were co-written by Indian authors and UK subject leads to address Indian curriculum and pedagogic targets and contexts. Originally written in English, the TDUs have then been localised to ensure that they have relevance and resonance in each participating Indian state’s context. TESS-India is led by The Open University and funded by UKAID from the Department for International Development. Version 1.0 Except for third party materials and otherwise stated, this content is made available under a Creative Commons Attribution-ShareAlike licence: http://creativecommons.org/licenses/by-sa/3.0/

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ContentsIntroduction 1

Learning outcomes 2

1 Learning through memorisation 3

2 Imagery for developing understanding of algebraicidentities 5

3 Algebraic identities seen as special cases ofmultiplication 9

4 Spotting patterns and adjusting algebraic identities 12

5 Summary 16

6 Resources 17

Resource 1 17

References 18

Acknowledgements 19

IntroductionAlgebraic identities play an import role in the mathematics curriculum and inmathematics in general. In Class IX in the Indian secondary schoolcurriculum, eight types of identities are used when solving equations andpolynomials. Knowing and recognising these identities helps the learners tolearn mathematical procedures. It will also enable them to develop fluencywhen applying these procedures in algebraic manipulations and problemsolving. In order to use the power of identities fully, it is important to beable to spot variations in the algebraic identities. The main issue whenlearning and applying identities is that, for most students, the work is purelya question of memorising and regurgitating them. This unit will exploredifferent approaches to learning algebraic identities. These approaches relyless on memorisation and instead build on understanding the concepts ofidentities.

In this unit you are invited to first undertake the activities for yourself andreflect on the experience as a learner; then try them out in your classroomand reflect on that experience as a teacher. Trying for yourself will mean youget insights into a learner’s experiences, which can influence your teachingand your experiences as a teacher. This will help you to develop a morelearner-focused teaching environment.

The activities in this unit require you to work on mathematical problemseither alone or with your class. First you will read about a mathematicalapproach or method; then you will apply it in the activity, solving a givenproblem. Afterwards you will be able to read a commentary by anotherteacher who did the same activity with a group of students so you canevaluate its effectiveness and compare with your own experience.

Pause for thought

Thinking about your classroom, how are algebraic identities perceived

by your students? Do they like it? Do they struggle with it? Why do you

think this is?

Thinking back about your experiences as a mathematics learner, how

did you perceive algebraic identities?

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Introduction

Learning outcomesThis unit addresses learning outcomes in different areas relating to theteaching and learning of mathematics:

After studying this unit, you should be able to:

. understand the difference between an identity and an equation

. use images to explore and discover how identities are formed

. use and apply identities without having to rely on memory

. develop methods to reduce reliance on memory

. adjust existing tasks to allow students to focus on the process of doingmathematics.

This unit links to the teaching requirements of the NCF (2005) and NCFTE(2009) outlined in Resource 1.

TDU 1 Algebraic identities: alternatives to relying on memory

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1 Learning through memorisationLearning through memorisation, or rote learning, is a learning techniquebased on repetition. There are several arguments in favour of this learningapproach: one is that having rapid recall of certain facts in mathematics isnecessary to become fluent in other areas or in other mathematics topics. Theargument goes that if, for example, you do not know your times tables byheart, you will spend too much time and effort working out 6 × 7 instead offocusing on the procedure of working out, for example, the volume ofcombined solids, especially when there is no access to calculators. It is alsosometimes seen as a cultural way of learning. Knowing times tables by heartcould also give you a better number sense; for example, of the numbers’magnitude, of how numbers are related or of multiples and fractions. Similararguments could be used for learning algebraic identities throughmemorisation.

However, there are also many counter-arguments to the use of memorisationas a learning technique. One is about accessibility; not all students benefitfrom memorisation due to their poor school attendance, their lack of time oropportunity for the required practice, or just their poor recall. Students withspecial educational needs such as dyslexia, for example, are enormouslydisadvantaged.

Another argument concerns the kind of learning that memorisation affords.Memorisation does not focus on comprehension or building understanding;nor does it support exploration of what concepts could mean, or how theyare connected to other areas of mathematics. It focuses on memorising andaccurate reproduction, which can become problematic when studying morecomplex aspects of a subject, for example formulae and algorithms, thatentail complex steps. Understanding of meaning is not created bymemorisation, which means elements get missed out, details get muddled up,stress increases and exams are failed.

The teaching experience when using memorisation is often not exciting, andcan even be boring, because of its repetitive nature and lack of focus onunderstanding and making connections. Students mechanically ‘go through’the exercises, engaging their brains as little as possible. This is problematicfor all students, including high achievers, who tend to be categorised as suchbecause they are good at rote learning. Boredom when learning mathematics,little demand for thinking and a lack of opportunity to work on makingconnections and giving meaning to mathematics makes it hard for learners todevelop an understanding of and a love for the subject.

Pause for thought

What are your thoughts about learning through memorisation? Do you

think it works well always, sometimes or not often?

How did you experience learning mathematics by memorisation?

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1 Learning through memorisation

Think of one of your students who seems to be memorising well, and

think of one that is struggling. What is the same and what is different

about how they learn?


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2 Imagery for developingunderstanding of algebraicidentitiesThis unit works on developing ways for teaching algebraic identities thatallow an understanding of the mathematics to take place, and to rely less onmemory. An effective way to address this is through the use of imagery,which is discussed more in depth in the TDU 3, Number systems: exploringby comparing and contrasting, visualising, and imaging.

The operation of multiplication can be modelled in different ways. The onethat will be worked with in this unit is the representation of multiplication asa product table that is similar to area.

For example, the product 7 × 3 can be represented by an area of 7 by 3.From this image it is also clear that multiplication is commutative, that is,the product of 7 × 3 is the same area as the product of 3 × 7.

Hence 7 × 3 ~ 21.

The area model helps us understand how we can decompose a multiplicationproblem, because the area of a large rectangle can be easily decomposed intoareas of smaller rectangles. It is good practice not to draw the area models inproportion to the numbers used: it stimulates more abstract representationsand makes the jump to representing negative numbers smaller.

From the above figure we see how

7 × 3 ~ 4 × 2 + 3 × 2 + 4 × 1 + 3 × 1

This decomposition model is very helpful when we are finding products oflarger numbers.

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2 Imagery for developing understanding of algebraic identities

For example, 24 × 13 can be represented by:

Hence 24 × 13 ~ 200 + 40 + 60 + 12 ~ 312.

192 can be represented as:

Hence 192 ~ 400 – 20 – 20 + 1 ~ 361.

You might have noticed that in these examples the equivalence sign hassometimes been used instead of the equation sign, which would be equallyvalid. The equivalence sign can offer a certain freedom, a playfulness tomathematics, in particular if the sign is read as ‘is another way of saying’instead of ‘is equivalent to’.

The next task will support students in working with the imagery of seeingmultiplication as a product table.

Activity 1: Representing multiplication as an area

For each multiplication problem given below:

(a) model the problem as a product table

(b) if appropriate, decompose the area into smaller areas

(c) show how can you work out the products of these calculations by

using your answers to (a) and (b)

1 (105)2

2 (14.3) 2

3 4(99)

4 982

5 7(t + r)

6 (r + q) (s – r)


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Case Study 1: Teacher Habib reflects on usingActivity 1

Before starting on this activity we had a discussion on how a number

squared or multiplied could be represented by an area model, mimicking

the way I learnt it from reading this unit. I started with small numbers

like 5 × 6 and then to bigger numbers like 56 × 64 and 65 × 115. Anup

thought of using the distributive property for the question 65 × 115 = 65

(100 + 10 + 5), and so he went ahead and represented it as a one-by-

two table.

We also discussed whether it would help to do the same with the other

number and form a sort of product table. Only when we had done that

did I ask them to go ahead and try the tasks given. They worked on this

mainly on their own, although every now and then I could see students

looking at their neighbour’s work. They could do most of them well. With

the decimal question, most of them distributed it as 14 + 0.3. I asked

them if they felt it was now simpler and some of them said that it was

not, so I told them that they could think of a further decomposition to

make a three-by-three table.

When they came to 982 the ones who decided on 90 + 8 had no

problems, but one of them decided to represent it as 100 – 2 so then he

wanted to know how an area could be negative. This led to a lively

discussion about representation and modelling in mathematics and why

it can be helpful to come up with different labels. In this case, the

difference and sameness between a product table and an area

representation. We conjectured that the product table and the area

representation would be identical when working with positive numbers

but with negative numbers you would have problems with the area

model because a negative area does not actually exist. However, we

know a product can be negative, so that is why we would call it a

product table!

I did not go deeper into the 100 – 2 representation itself, but it led nicely

into discussing the next part of this unit: how to deal with negative

numbers in a product table.

Pause for thought

When you do such an exercise with your class, reflect afterwards on

what went well and what went less well. Consider the questions that led

to the students being interested and make connections between topics.

Such reflection always helps with finding ways to help you to engage

the students and help them to find mathematics interesting and

enjoyable. If they do not understand and cannot do something, they are

less likely to become involved. Use this reflective exercise every time

you undertake the activities, noting as Teacher Habib did some quite

small things that made a difference.

Good questions to trigger this reflection are:

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2 Imagery for developing understanding of algebraic identities

. How did it go with your class?

. What questions did you use to probe your students’ understanding?

. Did you feel you had to intervene at any point?

. What points did you feel you had to reinforce?

. Did you modify the task in any way? If so, what was your reasoning

for doing so?


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3 Algebraic identities seen asspecial cases of multiplicationStudents often perceive algebraic identities as some magic or a gospel truth.They rarely develop a sense of where these identities appear from or thatthey are special cases of multiplication.

One of the reasons why students tend to memorise identities is that they failto associate any meaning to the relationship being depicted by the identity.You may have noticed students making common errors when they recallidentities:

(x − y)2 = x2 − y2

(x − y)2 = x2 + 2xy + y2

(x − y)2 = x2 + 2xy − y2

Even though it is fairly simple to check if the two statements are correct (allthat is needed is to verify them for a few values of the variables), studentscontinue to make these mistakes. One reason may be that they are not awareof how easily they can verify their statements. The second, and more seriousreason, is that they have never created a physical (or geometric) meaning ofthese statements.

The next task helps students discover for themselves the meaning ofdifferent identities. The task focuses on finding, seeing and generalising thepatterns of algebraic identities.

Before beginning this activity, it would be a good idea to first check ifstudents can illustrate the lengths x + y and x − y correctly. The former iseasier to conceive and the latter may take a bit more effort.

x + y

If the green portion is x, and the blue portion is y, then the red portion is x+ y. This could also be described as 'the length of the green portion(representing x) added to the length of the blue portion (representing y) isthe same as the red portion (representing x + y)'.

x − y

If the green portion is x, and the blue portion is y, then the red portion is x− y. This could also be described as 'the length of the green portion(representing x) take away the length of the blue portion (representing y)leaves the red portion (representing x − y)'

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3 Algebraic identities seen as special cases of multiplication

Activity 2: Algebraic identities as special cases ofalgebraic multiplications

1 Describe, not draw or work out, the following mathematical

expressions as a product table to your neighbour – what would it

look like?

(a) (x + y)2

(b) (x + a)(x + b)

(c) (x − y)2

(d) (x − y)(x + y)

2 Now draw the representation of each expression as you have

described it in step 1 as a product table.

3 Now try to write an alternative mathematical expression that

represents the areas that you have drawn in step 2.

4 Observe and compare your answers to steps 1, 2 and 3 for each

of the four expressions:

(a) How many terms do you get for the expressions (a), (b), (c) and

(d) in step 3?

(b) How are these terms formed?

(c) What is the same about these expressions? What is different? It

might help if you colour in the area boxes corresponding to the

terms.

(d) Describe a rule or method for working out the algebraic identities in

this way so other students in other classes could read about it.

Case Study 2: Teacher Rashid reflects on usingActivity 2

Having represented pure numbers in the product table, as well as area

representation rectangles, the students could do parts (a) and (b) pretty

easily. They represented their ideas as both product tables and areas of

rectangles (as Mona said that it was represented in the same way for all

positives). For the negative numbers, the product table was very easily

constructed, however when representing areas of rectangles there was

a bit of confusion. I asked Mona to come to the board and follow the

instructions of others. I reminded them that they had done something

that would be useful when using numbers in the earlier task.

They found the third question easy and we discussed the significance of

equivalence: that an expression with collected terms can be equivalent

to one where the terms have not been collected yet – it just looks a bit


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messier somehow! I asked them whether they could come up with some

ideas to make it look even more messy – and they had great ideas! It

made us all laugh, which was really nice.

The fourth question was no problem at all, except the last part. The

difficulty was not in describing it, but in doing so succinctly. The

descriptions were not perfect, but we were all happy with them and

realised that to improve we would simply need practice at writing our

own descriptions and methods more.

We extended the questions to include (a + b + c)2 and then tried with

terms being assigned different signs and they could get the solutions

easily .

We also decided that to do (a + b)3 or (a − b)3 we could do it in two

parts (a + b)(a + b)2 = (a + b)(a2 + 2ab + b2) and then put this in the

product table of 2 × 3.

The students felt really happy and confident with working in this way.

One student said that he felt so relieved that he would now be able to

think of a way to work out the algebraic identities if his memory failed.

Pause for thought

How did it go with your class? What questions did you use to probe

your students’ understanding? Did you feel you had to intervene at any

point? What points did you feel you had to reinforce? Did you modify the

task in any way? If so, what was your reasoning for doing so?

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3 Algebraic identities seen as special cases of multiplication

4 Spotting patterns and adjustingalgebraic identitiesIn the previous task, students worked on developing an image ofmultiplication and algebraic identities, and so developed an understanding ofwhat these are about. Through this, students will now be aware of methodsother than memorising formulae and algorithms to work out the products ofmultiplication and algebraic identities.

The power of understanding algebraic identities in mathematics is not onlybeing able to work out their products, but also, and perhaps moreimportantly, to spot them when they are not in an easily recognisable formand be able to tweak expressions so they can be written as variations ofalgebraic identities, which means actually creating identities of their own.

The next two tasks focus on this. Activity 3 requires students to activelydevelop ways to spot patterns and manipulate expressions in the context ofalgebraic identities.

Activity 4 uses Vedic mathematics to learn to create an algebraic identitythat reflects this method of finding squares of numbers ending in 5. Oncethat is done, they compare how this identity compares with two otheridentities they already know.

Activity 3: spotting patterns

This is an activity on spotting patterns and manipulating mathematical

expressions in the context of algebraic identities.

For each of the calculations, decide whether it is a particular case of

one of the generalisations that you know as algebraic identities. You can

find these in your textbook or as a reminder, they are:

(a) (x + y)2 = x2 + 2xy + y2

(b) (x − y)2 = x2 − 2xy + y2

(c) x2 − y2 = (x + y)(x − y)

(d) (x + a)(x + b) = x2 + (a + b)x + ab

(e) (x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zx

(f) (x + y)3 = x3 + y3 + 3xy (x + y)

(g) (x − y)3 = x3 − y3 − 3xy (x + y) = x3 − 3x2y + 3xy2 − y3

(h) x3 + y3 + z3 − 3xyz = (x + y + z) (x2 + y2 + z2 − xy - yz − zx)

1 5.62 − 0.32 = 31.27

2 (x − 3)(x + 5) = x2 + 2x – 15

3 118 × 123 = 14 514

4 25/4x2 − y2/9 = (5/2x + y/3)(5/2x − y/3)


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Case Study 3: Teacher Anju reflects on usingActivity 3

Since the list was there to see, the students happily started comparing

the given questions with the identities. For the first one, they did identify

the correct identity but Suman and a few others wrote it as 5.62 − 0.32 =

(5.62 − 0.32)(5.62 + 0.32), so I asked her to come to the board and write

it down. At once Ravi said how can it be that on the right-hand side

(RHS) we have the same expression but then it is multiplied by

another? Suman immediately saw what she had done and rubbed out

the indices on the RHS leaving the correct answer.

The second one was done easily enough but for the third some

distributed it as 100 + 18 and 100 + 23. This led to a discussion on

whether that was simplified enough or was there another way of making

it simpler?

Some of the students wanted to write the last one as (5/4x − y/9)(5/4x +

y/9). There was a lot of discussion on what was right and what was

wrong with this suggestion. Then I asked them to take out their

textbook, not to ‘do the exercise’ but to now see if they could easily

identify the identities that they needed to use.

Pause for thought





Activity 4: Creating and comparing identities –Vedic mathematics

Vedic mathematics has a rule that allows you to find the square of

numbers ending with 5 without actual multiplication of the number.

According to this rule,

9952 = 99 × (99+1) × 100 + 25 = 990 025

Whilst this is often offered as a trick to remember and use, with a little

thought you can understand why it works and therefore use it whenever

it may help with calculations.

In the last task, we had computed (105)2 by modelling it as an area. In

the process we saw that:

(105)2 = 100 × 100 + 2 × (100)(5) + 5 × 5.

Now to explore the Vedic maths method.

1 Think of a number x, such that:

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4 Spotting patterns and adjusting algebraic identities

(105)2 = (x)(x + 1)100 + 25.

2 Now rewrite the left-hand side (LHS) of the equation in step 1

using the variable x. Write out the whole equation in step 1 using

your new LHS. Look at it carefully.

3 Do you think that the equation that you have obtained in step 2 is

an identity? How are you sure about your answer?

4 Compare and contrast the identity obtained in step 3 with the two

identities:

(a) (x + y)2 = x2 + 2xy + y2

(b) (x − y)2 = x2 – 2xy + y2

What is the same and what is different?

Case Study 4: Teacher Anju reflects on usingActivity 4

The students were hesitant and did not know what to do here. Harinder,

however, was doing some calculations feverishly. I gave him some time

and then went over to see what he had done. What he had done was

found the product of the RHS, i.e.:

(x2 + x)100 + 25 = 100x2 + 100x + 25

And he used the area model on the LHS:

He then compared 100 × 100 with 100x2 and decided that x2 = 100 so x

= 10.

He also applied that value in the given equation to see if it worked and

it did.

I asked him to share what he had done with the others. I told them to

take some numbers of their own and find out if it worked for all sorts of

squares. They came to the conclusion that the product rule worked only

for numbers ending in 5.

On the second part they were again stuck, so I pointed out how they

had found that 100 + 5 = 10x + 5 where x = 10 and I asked them to find

the square by the area model. We discussed what they understand by

an identity and checked out whether this one was an identity. For the

last part, they could identify the one it was associated with.


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Pause for thought





15

4 Spotting patterns and adjusting algebraic identities

5 SummaryThis unit has focused on using visual representations to facilitate workingwith complex expressions. Once the learners grasp the connections betweenarea calculations and expanding brackets they immediately have a way ofworking out identities rather than relying on their memories. These ideasallow learners to give a meaning to what they are doing and therefore to feelthat the ideas are their own. It also suggests that the students play with theseideas, asking ‘What if I do it this way instead of that way?’, or make it moremessy rather than less. In this way some of the anxieties that can be built upwhen demanding memorisation can be dissipated and memorisation canhappen more easily. This is important: when students can reproduceidentities from memory, they can more fluently solve mathematical problems– but often they are so worried about memorising that they cannot developthe fluency.

Identify three techniques or strategies you have learned in this unit that youmight use in your classroom and two ideas that you want to explore further.


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6 ResourcesResource 1

This unit links to the folllowing teaching requirements of the NCF (2005)and NCFTE (2009) and will help you to meet those requirements:

. Let students see mathematics as more than formulae and mechanicalprocedures.

. Learn through activities, discovery and exploration in a child-friendly andchild-centred manner.

. View learners as active participants in their own learning and not as mererecipients of knowledge; encouraging their capacity to constructknowledge; shifting learning away from rote methods.

. Organise learner-centred, activity-based, participatory learningexperiences.

. Engage with the curriculum, syllabuses and textbooks to criticallyexamine them rather than taking them as ‘given’ and accepted withoutquestion.

. Let students use abstractions to perceive relationships, to see structures,to reason out problems, to argue the truth or falsity of statements.

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6 Resources

ReferencesNational Council of Educational Research and Training (2005) NationalCurriculum Framework (NCF). New Delhi: NCERT.

National Council for Teacher Education (2009) National Curriculum Frameworkfor Teacher Education [Online], New Delhi, NCTE. Available at http://www.ncte-india.org/publicnotice/NCFTE_2010.pdf (Accessed 16 January 2014).

National Council of Educational Research and Training (2012a) MathematicsTextbook for Class IX. New Delhi: NCERT.

National Council of Educational Research and Training (2012b) MathematicsTextbook for Class X. New Delhi: NCERT.


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http://www.ncte-india.org/publicnotice/NCFTE_2010.pdf

http://www.ncte-india.org/publicnotice/NCFTE_2010.pdf

AcknowledgementsThe content of this teacher development unit was developed collaborativelyand incrementally by the following educators and academics from India andThe Open University (UK) who discussed various drafts, including thefeedback from Indian and UK critical readers: Els De Geest, Anjali Gupte,Clare Lee and Atul Nischal.

Except for third party materials and otherwise stated, this content is madeavailable under a Creative Commons Attribution-ShareAlike licence:http://creativecommons.org/licenses/by-sa/3.0/

The material acknowledged below is Proprietary and used under licence (notsubject to Creative Commons Licence). Grateful acknowledgement is madeto the following sources for permission to reproduce material in this unit:

Every effort has been made to contact copyright owners. If any have beeninadvertently overlooked, the publishers will be pleased to make thenecessary arrangements at the first opportunity.

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Acknowledgements


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Algebraic identities: alternatives to relying on memory Unit 1: