In the spirit of giving the answer to the crucial question of “why” learning algebraic expansions I will give two key reasons: by multiplying out the brackets and combining like terms we will get an expression which is often much simpler than the original situation. Unfortunately, the big “why” will be appreciated only in the later stages of the students’ learning. So the small “why” is that this method provides the tools to deal with bigger macro–numerical and algebraic reality on a much smaller scale, which can be examined on the basis of already known, prior learning.
All algebraic expansions in this collection are given in parallel formats on each slide: the abbreviated version and a full version. The abbreviated version on the left side represents how such concepts are taught in the majority of learning environments (supported by published school textbooks), and unfortunately not understandable to the smaller or greater portion of the student population. Because of that, on the right side a full, geometrically supported explanation of those mathematical statements is given in several steps which help build algebraic comprehension. Even those students who can properly perform algebraic expansions as a set of prescribed steps appreciate the epistemological explanations on the right, since algebraic visualisations (rectangles and squares) give meaning to the something they can do. Each slide has its own twin, where like terms are added, representing a simplified (or summarised) explanation.
Teachers like students’ “aha” moments in their learning and I found them often coming after the presentation of the concepts geometrically. Now algebraic expansions are perceived and understood as repeated additions of the same groups of elements, which represent the next step in conceptualising basic numeracy.
Students can see that any chosen sequence is a part of a greater continuum and it gives clues about the unseen steps. For me the greatest achievement when students follow this way is that they are now not only able to do algebraic expansions, but they can explain what they do and why they do it that way.
Building algebraic literacy usually starts with binomial expansions. The great majority of available resources introduce the distributive law as a way to convert an algebraic product into a sum (or a difference) of terms, explaining how to do it, without any indication why it is done that way, how it was developed and what it is for. It just appears as an extra mathematical rule for its own sake.
When I want to introduce this concept to my students I present the fact that by now the majority of them know (or should know) their times timetables from 1 × 1 to 12 × 12. Then I begin by asking the questions: What about multiplying bigger numbers? How can we do this easily without using a calculator? And another investigation starts.
Usually I instruct students to multiply a single digit number by a two digit number, 3 × 14 for example. 14 represents 10 + 4 so that it is not difficult to convince students that 3 × 14 = 3 × (10 + 4). I illustrate it as a rectangle whose height is 3 units and whose length is 14 units. This product represents the numerical value of the rectangle’s area. The total area can be represented as a sum of the two smaller rectangles whose dimensions are 3 × 10 and 3 × 4.
The next step is to use the same example of 3 × 14, but now I would split 14 into a sum of 9 and 5. This time 3 × 14 = 3 × (9 + 5) and the total rectangular area can be represented as a sum of the two smaller rectangles whose dimensions are 3 × 9 and 3 × 5.
The third step would split 14 into a sum of 8 and 6. Now 3 × 14 = 3 × (8 + 6). The total rectangular area could be represented as a sum of two smaller rectangles whose dimensions are 3 × 8 and 3 × 6.
If there is a need to continue, I would use the same pattern, but this would be enough for the majority of students to make a good generalisation: a times sum of the segments b and c is sum of the areas consisting of the rectangles a b and a c. Algebraically a (b + c) = a b + a c. It is an illustrated distributive law, which can be remembered, because it makes sense in learners’ heads.
I would take advantage of the common ground we have established. I would use 3 × 14 again. 14 represents 15 – 1 so the problem 3 × 14 becomes 3 × (15 – 1). It helps if this situation is illustrated as a difference of two rectangles. The total area can be represented as a difference of the two rectangular areas whose dimensions are 3 × 15 and 3 × 1.
The next step would be to use the same example of 3 × 14, but now I would split 14 into a difference of 16 and 2. 3 × 14 becomes 3 × (16 – 2). The total area can be represented as a difference of the two rectangular areas whose dimensions are 3 × 16 and 3 × 2.
The third step would split 14 into a difference of 17 and 3. 3 × 14 becomes 3 × (17 – 3). The total area could be represented as a difference of the two rectangular areas whose dimensions are 3 × 17 and 3 × 3.
If there is a need to continue, I would use the same pattern, but this is usually enough for the majority of students to make a good generalisation. a times difference of the segments b and c is the difference of the areas consisting of the rectangles a b and a c. Algebraically a (b – c) = a b – a c. Again, it is an illustrated distributive law, which can be remembered, because it makes sense in students’ minds.
After this traditional introduction of basic algebra, I switch to the presentation and explanation of electronic resources. They provide not only a variety of teaching tools and methods, but the whole purpose of their development was to include all the types of problems that can appear through the process of algebraic generalisations. Without the visual resources, learning these principles cannot be achieved relatively quickly (i.e. within a given number of teaching periods) and effectively (engaged students who are able to reach and retain the desirable learning outcomes). The teachers who best know their students can make an appropriate selection of the grouped resources to keep their students actively involved, judging how many more examples are needed to be presented in order to keep the learning pace sufficiently stimulating to reach optimal results.
The beauty of the binomial expansion materials is that they can be used to extend all students, because even at this level, our students can be exposed to basic identities and the meaning of equivalents. The key principle is that everything remains the same if multiplied by 1. 1 is the product of (- 1) × (- 1), hence both binomial factors can be multiplied by (- 1). This transforms the original product of two factors into the product of the two opposite factors, which generates the same final result. With the help of the coloured tools students can deduce the next set of identities given in Algebraic Visualisations 1a and Algebraic Visualisations 1b:
a (x + b) = - a (- x – b) = a x + a b;a (b x + c) = - a (- b x – c) = a b x + a c; a (x – b) = - a (- x + b) = a x – a b;a (b x – c) = - a (- b x + c) = a b x – a c; - a (x – b) = a (- x + b) = - a x + a b;- a (b x – c) = a (- b x + c) = - a b x + a c; - a (x + b) = a (- x – b) = - a x – a b;- a (b x + c) = a (- b x – c) = - a b x – a c.
And not only that; we can go back to close the circle by asking our students why both ways (both sides of the equation representing the distributive law) are equivalent.
The attractiveness of algebraic generalisations is in their universality. A symbol (variable) can be replaced with either any number or any object. The mathematical truth will remain.
After presenting the resources contained in Algebraic Visualisations 1a and 1b, students can be exposed to the further generalisations present in Algebraic Visualisations 1c (from 1c1 to 1c7). They illustrate the next set of algebraic expansions:
When p and q have the same size we have further (quadratic) generalisations:
Each set can be used in a general mode, vertical grouping mode or in a summarised mode. From my own experience, the best learning outcomes are achieved when the materials are used in the sequence from general explanations to vertical grouping and finally to summarised situations.
Once expansion skills have been mastered, students can be introduced to the next learning experience. We use our newly acquired skills to simplify something that at first glance looks difficult or cluttered. And while teaching this process, we provide our students with the reason why we do it. Simplifications after expansions help establish simplicity and order to a more complex mathematical reality. The squares and rectangles are used to illustrate different (unlike) terms (qualities), the “opposite” colours are chosen to indicate opposite signs and through a process of grouping we sort out the proper quantities (sizes) for the same qualities (like terms). All these skills are needed to solve equations. Different types of learners demand a variety of representations that help equip them well for the next learning step in their mathematical cognitive development. Algebraic visualisations in set 1d illustrate the next set of algebraic expansions and simplifications:
a (b x + c) + d (e x + f),a (b x + c) + d (e x – f),a (b x – c) + d (e x – f), a (b x + c) – d (e x + f),a (b x + c) – d (e x – f),a (b x – c) – d (e x – f).
Developing algebraic literacy continues with trinomial expansions. The distributive law, already introduced in its simplest form as a way to convert an algebraic product into a sum (or a difference) of terms a (b ± c) = a b ± a c can get a meaningful extension. Another investigation in the form of a playful activity brings the desired learning outcomes, without a huge effort.
When I introduce this concept to my students I ask them to bring spare coins from home and then get them to make stacks of $1.30 using one dollar, 20 cent and 10 cent coins. Once the stacks are formed, I ask students to say what the total is when they add two, three, four or more stacks together, expressing their result as a total of coins with the same value. (This can also be achieved using “Monopoly” money or similar types of “educational currency”.)
Very soon, most students can make good generalisations. Then I show them Algebraic Visualisations, set 1e using only positive terms in brackets to illustrate that a times the sum of the segments b, c and d is a sum of the areas consisting of the rectangles a b, a c and a d. Algebraically a (b + c + d) = a b + a c + a d. It is an illustrated extended distributive law, which can be remembered and retained, because it can be linked with the students’ daily experience and makes sense in pupils’ heads.
The next learning step utilises the common ground that has been established. $1.30 can be created using one dollar, 50 cent and a paper “negative 20 cent” coins which I prepare before the lesson. Finding how much it is in total when they add two, three, four or more lots together represents a numerical simulation which serves as a preparation for the algebraic illustration that a (b + c – d) = a b + a c – a d. Set 1e of Algebraic Visualisations can be effectively used to illustrate this type of mathematical reality by choosing negative d value in expanding trinomials.
Next, I show that $1.30 can be created using a two dollar coin, a “negative 50 cent” coin and “negative 20 cent” coin which I prepare before the lesson. Finding the total when they add two, three, four or more lots together represents a numerical simulation which serves as a preparation for the algebraic illustration that a (b – c – d) = a b – a c – a d. I use set 1e of Algebraic Visualisations with negative second and third term of the trinomial to help students visualise and remember this type of mathematical reality.
Algebraic Visualisations are very effective tools which alleviate the acquisition of needed insights when learning algebra. The teachers who best know their students can make an appropriate selection of the resources to keep their students actively engaged in the process of learning. The resources contain more examples than a great majority of students would need; hence teachers can provide meaningful variety when illustrating new algebraic concepts to their student population. The inclusion of negative factors in this set of resources helps incorporate cases that appear in the process of expansion of non-quadratic trinomials.
Some teachers devote more time to binomial and less time to trinomial expansions; some do the opposite. The choice sometimes reflects personal preference, but most often the time available and the learning progress of the student population dictate how much time we spend with different groups of learners on different concepts.
All trinomial expansions in this collection are given in four different modes: general mode, horizontal grouping, vertical grouping and summarised mode. As a novelty, horizontal grouping is a meaningful mid–step between general and vertical grouping modes. Each mode illuminates the same matter from different angles, hence providing additional ways of looking at the same reality and elaborating it with more different insights.
By the end of learning this section, trinomial expansions are perceived and understood by students as repeated simultaneous additions (and/or subtractions) of the same groups of elements (like terms). This represents expressive generalisation in conceptualisation of basic numeracy. The introduction of trinomial expansions helps students make meaningful extensions to this learning experience of extending the distributive law not only to three, but to four or more unlike terms. From my experience, through the process of deductive and inductive learning, sooner or later, students reach a point when they are able to generate (induce) a much bigger picture of the distributive law in the format a (b ± c ± d ± … ± z) = a b ± ac ± ad ± … ± az. This is the main argument which justifies introducing the expansion of trinomials into students’ learning.
After presenting the resources contained in Algebraic Visualisations 1e, students can be exposed to the final generalisations (at this stage of learning) present in Algebraic Visualisations 1f (from 1f1 to 1f3). They illustrate the next set of algebraic expansions:
When p and q have the same size we have further (quadratic) generalisations:
Each set can be used in a general mode, horizontal grouping mode, vertical grouping mode or summarised mode. From my experience, the best learning outcomes are achieved when the materials are used sequentially from general explanations, followed by the horizontal grouping, then by the vertical grouping and finally by summarised situations.
The textbooks which I have used in my classrooms do not contain this aspect of algebraic expansions, but somehow I found this mid-step useful and beneficial before teaching quadratic expansions. If you have time, I would wholeheartedly recommend you to give it a go and offer your students this learning experience. It sets a very receptive atmosphere for introducing not only quadratic expansions (which are a special case of two - stage expansions), but also the multiplication (and later division) of polynomials.
I introduce this concept to my students starting with the multiplication of a two digit number by another two digit number: 17 × 24 for example. 17 represents 10 + 7 and 24 represents 20 + 4 so that it is not difficult to convince students that 17 × 24 = (10 + 7) × (20 + 4). It helps if this is illustrated as a rectangle whose height is 17 units and length 24 units. This product represents the numerical value of the area of the rectangle. Then I split this rectangle horizontally and vertically into 10 + 7 by 20 + 4 units, creating four different rectangles. The total area can be represented as a sum of the areas of the four rectangles whose dimensions are 10 × 20, 10 × 4, 7 × 20 and 7 × 4. The total numerical value of the area is 200 sq. units, plus 40 sq. units, plus 140 sq. units, plus 28 sq. units, making a grand total of 408 squared units.
To keep your students engaged and willing to get into algebraic generalisations, you can offer them one or two extra examples to be done on their own, implementing the same method of breaking the big rectangular reality into 4 smaller ones, for example
23 × 36 = (20 + 3) × (30 + 6) = …
48 × 51 = (40 + 8) × (50 + 1) = …
With the skills already developed using binomial and trinomial expansions, without great difficulty it can be shown that this situation can be algebraically represented as (a + b) times (c + d). The big product can be broken down into four chunks of a × c + a × d + b × c + b × d. As a summary:
(a + b) (c + d) = a c + a d + b c + b d
The next step is to use the same initial example of 17 × 24, but now I would create a slightly different mathematical reality. 17 represents 10 + 7 and 24 represents 30 – 6 so it is not difficult to convince students that 17 × 24 = (10 + 7) × (30 – 6). Then present this situation creating four different rectangles whose base is 30 – 6 units and whose height is 10 + 7 units. The total area can be represented as a sum of the areas of the four rectangles whose dimensions are 10 × 30, 7 × 30, 10 × (- 6) and 7 × (- 6). The total numerical value of the area is 300 sq. units, plus 210 sq. units, minus 60 sq. units, minus 42 sq. units, making a grand total of 408 squared units. The beauty of this approach is that students can see that the final result remains the same.
For pupils this is a kind of ‘algebraic magic’ that keeps them cooperative and engaged. Offer them to check the validity of two extra examples, done on their own, implementing the same method of breaking the big rectangular reality into 4 smaller ones, for example:
25 × 36 = (20 + 5) × (40 – 4) = …
42 × 53 = (40 + 2) × (60 – 7) = …
Without great difficulty it can be shown that this situation can be algebraically represented as (e + f) times (g – h). The big product can be broken down into four chunks of e × g + e × (– h) + f × g + f × (– h). As a summary:
(e + f) (g – h) = e g – e h + f g – f h
The next step is to use the same example of 17 × 24, but now I would create another mathematical reality. 17 represents 20 – 3 and 24 represents 30 – 6, so it is not difficult to convince students that 17 × 24 = (20 – 3) × (30 – 6). Then present this situation creating four different rectangles whose base is 30 – 6 units and whose height is 20 – 3 units. The total area can be represented as a sum of the areas of the four rectangles whose dimensions are 20 × 30, (- 3) × 30, 20 × (- 6) and (- 3) × (- 6). The total numerical value of the area is 600 sq. units, minus 90 sq. units, minus 120 sq. units, plus 18 sq. units, making a grand total of 408 squared units. The beauty of this approach is that students can see again that the final result remains the same.
These algebraic manipulations help to keep students interested. Offer them to check the validity of two extra examples, done on their own, implementing the same method of breaking a big rectangular reality into four smaller ones, for example:
23 × 36 = (30 – 7) × (40 – 4) = …
48 × 51 = (50 – 2) × (60 – 9) = …
Without great difficulty it can be shown that this situation can be algebraically represented as (i – j) times (k – l). The big product can be broken down into four chunks: i × k + i × (- l) + (- j) × k + (- j) × (- l). As a summary:
(i – j) (k – l) = i k – i l – j k + j l
These learning experiences create motivation for students to learn new mathematical insights about the same, “already known and well researched” reality. To deepen and widen their learning experiences, expose students to the materials present in Algebraic Visualisations set Two-stage expansions containing algebraic generalisations expressed in the following forms:
(a + b) (c + d);(a + b) (c – d);(a + b) (- c + d);(a + b) (- c – d); (a – b) (c + d);(a – b) (c – d);(a – b) (- c + d);(a – b) (- c – d).
Pedagogically it is justifiable to implement this way of teaching, because this algebraic operation represents two binomial expansions merged together and a new concept is explained as an addition of two binomial expansions. Students can notice this reality based on their prior learning and they can (with some teacher’s assistance if needed) derive the correct conclusions.
Enrich students’ learning by showing them a mirror effect created by the algebraic opposites, using the already explained principle of multiplying both brackets by - 1 to produce the same, identical outcomes:
(-a – b) (-c – d);(-a – b) (-c + d);(-a – b) (c – d);(-a – b) (c + d); (-a + b) (-c – d);(-a + b) (-c + d);(-a + b) (c – d);(-a + b) (c + d).
When I introduce this concept to my students, I start by multiplying a two digit number by another two digit number, 13 × 14 for example. 13 represents 10 + 3 and 14 represents 10 + 4 so it is not difficult to convince students that 13 × 14 = (10 + 3) × (10 + 4). It helps if this is illustrated as a rectangle whose height is 13 units and length 14 units. This product represents the numerical value of the rectangle’s area. Then I split this rectangle vertically and horizontally into 10 + 4 by 10 + 3 units, creating one square and three different rectangles. The total area can be represented as a sum of the areas of this square and the three smaller rectangles whose dimensions are 10 × 10, 10 × 3, 4 × 10 and 4 × 3. The total numerical value of the area is 100 sq. units, plus 30 sq. units, plus 40 sq. units, plus 12 sq. units, making a grand total of 182 squared units.
The next step is to use the same example 13 × 14, but now I would split this rectangle vertically and horizontally into 9 + 5 by 9 + 4 units, creating one square and three different rectangles. The total area can be represented as a sum of the areas of this square and the three smaller rectangles whose dimensions are 9 × 9, 9 × 4, 5 × 9 and 5 × 4. The total numerical value of the area is 81 sq. units, plus 36 sq. units, plus 45 sq. units, plus 20 sq. units, making a grand total of 182 squared units.
The third step would use the same example 13 × 14, but now I would split this rectangle vertically and horizontally into 8 + 5 by 8 + 6 units, creating one square and three different rectangles. The total area can be represented as a sum of the areas of this square and the three smaller rectangles whose dimensions are 8 × 8, 8 × 5, 6 × 8 and 6 × 5. The total numerical value of the area is 64 sq. units, plus 40 sq. units, plus 48 sq. units, plus 30 sq. units, making a grand total of 182 squared units.
If there is a need to continue, I would use the same pattern, but this would be enough for the majority of the students to make a good generalisation: (x + a) times (x + b) gives a total area which can be represented as the sum of the areas of the square and the three rectangles whose dimensions are x × x, a × x, b × x and a × b. The total numerical value of the area is x2 sq. units, plus a × x sq. units, plus b × x sq. units, plus a × b sq. units.
It does not take a long time to convince learners that this type of operation represents two binomial expansions placed together. If you have time to do two - stage expansions, then the transition into quadratic expansions can be done easily using that concept as well.
Quadratic expansions can be performed either horizontally or vertically; the end product is the same.
(x + a) times (x + b) is x times (x + b) plus a times (x + b).
In the same way it can be shown that the identical solution is obtained if we do expansion
(x + b) times (x + a) is x times (x + a) plus b times (x + a).
Each term is multiplied by each term.
Nearly all textbooks offer the FOIL rule when it comes to performing quadratic expansions: multiply the First term in the first bracket by the First term in the second bracket, then multiply the Outside terms (the first term in the first bracket by the last term in the second bracket), after that multiply the Inside terms (the second term in the first bracket by the first term in the second bracket) and finally multiply the Last terms (the second term in the first bracket by the second term in the second bracket). While learning this way is good for obtaining procedural knowledge, it is much better to build structural knowledge showing that FOIL actually can be any of:
FILO, FIOL, FLIO, FLOI, FOIL, FOLI,IFLO, IFOL, ILFO, ILOF, IOFL, IOLF, LFIO, LFOI, LIFO, LIOF, LOFI, LOIF,OFIL, OFLI, OIFL, OILF, OLFI, OLIF.
Not all permutations are good for making an appropriate mnemonic that can be easily memorised; however, they are good to indicate that each term is to be multiplied by each term; the order is not essential. This aspect is vital (a bit later) for the multiplication (and division) of polynomials. Visualisation helps not only to see, but to comprehend and remember this mathematical reality.
The quadratic expansions set contains systematised examples of quadratic expansions expressed in forms: (x + a) (x + b); (x + a) (x – b); (x – a) (x – b).
The difference of two squares set contains systematised examples of quadratic expansions expressed in forms: (x + a) (x – a) or (x – a) (x + a).
The squared sum or difference set contains systematised examples of quadratic expansions expressed in forms: (x + a)2 and (x – a)2.
The last two sets represent special cases which are formed when b = - a and when b = a. All the examples in sets 2b2 and 2b3 are contained in the first set 2b1, but they are extracted here and put in the same group (family) to visualise and emphasise once again the guiding principles behind this type of quadratic expansions.
The reason for creating set 2b2 (the difference of two squares) is because it is another beautiful mathematical model which we add to our already rich mathematical toolbox. I involve students to discover it, first numerically, and later they can take ownership of finding the general algebraic rule for their numerical calculations.
The usual warming up activity would be to find the answer to the simple multiplication problems using the “classical” multiplication method and, after that, algebraically:
9 × 11 = 99;9 (10 + 1) = 90 + 9; 8 × 12 = 96;8 (10 + 2) = 80 + 16; 7 × 13 = 91;7 (10 + 3) = 70 + 21; 6 × 14 = 84;6 (10 + 4) = 60 + 24; 5 × 15 = 75;5 (10 + 5) = 50 + 25.
After that I would encourage students to be thinking mathematicians who can do this using another elegant way:
9 × 11 = ? ... = (10 – 1) (10 + 1) = 100 + 10 – 10 – 1;
8 × 12 = ? ... = (10 – 2) (10 + 2) = 100 + 20 – 20 – 4;
7 × 13 = ? ... = (10 – 3) (10 + 3) = 100 + 30 – 30 – 9;
6 × 14 = ? ... = (10 – 4) (10 + 4) = 100 + 40 – 40 – 16;
5 × 15 = ? ... = (10 – 5) (10 + 5) = 100 + 50 – 50 – 25.
The smart “bright sparks” usually start making comments that these examples can be simplified to:
9 × 11 = (10 – 1) (10 + 1) = 100 – 1;
8 × 12 = (10 – 2) (10 + 2) = 100 – 4;
7 × 13 = (10 – 3) (10 + 3) = 100 – 9;
6 × 14 = (10 – 4) (10 + 4) = 100 – 16;
5 × 15 = (10 – 5) (10 + 5) = 100 – 25.
Then I would ask those students to explain this multiplication procedure to the other students (using their own language). Usually they would articulate it as: “the ’middle number’ squared minus the squared difference between the larger multiple and the ’middle number’”. In essence, this is the short-cut or the spelled out rule which is found in textbooks as an algebraic rule called “the difference of two squares”:
(a – b) (a + b) = a2 – b2
To confirm the validity of this rule I would ask students to generate more examples to be multiplied and without difficulties they create heaps of illustrations like:
15 × 25 = (20 – 5) (20 + 5) = 202 – 52 = 400 – 25 = 375;
100 × 300 = (200 – 100) (200 + 100) = 2002 – 1002 = 40000 – 10000 = 30000.
but to fully open students’ eyes and show them the universality of this well-designed rule I would ask them to multiply:
12 × 14 = ? ... = (13 – 1) (13 + 1) = 132 – 12 = 169 – 1 = 168;
11 × 15 = ? ... = (13 – 2) (13 + 2) = 132 – 22 = 169 – 4 = 165;
10 × 16 = ? ... = (13 – 3) (13 + 3) = 132 – 32 = 169 – 9 = 160.
Mathematics would not be so graceful without what if questions. What if we were to multiply:
12 × 13 = ? ... = (12.5 – 0.5) (12.5 + 0.5) = 12.52 – 0.52 = 156.25 – 0.25 = 156;
13 × 16 = ? ... = (14.5 – 1.5) (14.5 + 1.5) = 14.52 – 1.52 = 210.25 – 2.25 = 208;
14 × 19 = ? ... = (16.5 – 2.5) (16.5 + 2.5) = 16.52 – 2.52 = 272.25 – 6.25 = 266.
The validity of this rule can be illustrated on any scale, even when it is not practical to use it, but show it to your students anyway:
13.3 × 13.8 = ? ... = (13.55 – 0.25) (13.55 + 0.25) = 13.552 – 0.252 = 183.6025 – 0.0625 = 183.54;
6.98 × 7.07 = ? ... = (7.025 – 0.045) (7.025 + 0.045) = 7.0252 – 0.0452 = 49.350625 – 0.002025 = 49.3486.
For those readers who would like to know genesis of my Algebraic Visualisations I can say that the early stages began with squared sums (set 2b3). I discovered that for many students who were coming into my class (x + a)2 was equal to x2 + a2 and I needed something solid that would destroy this misconception in their algebraic learning. To eliminate this problem, I remember that I had started with numerical examples that created obvious differences.
32 = 9;42 = 16;42 = 16;
(1 + 2)2 = 9;(1 + 3)2 = 16;(2 + 2)2 = 16;
12 + 22 ≠ 9;12 + 32 ≠ 16;22 + 22 ≠ 16;
1 + 4 ≠ 9;1 + 9 ≠ 16;4 + 4 ≠ 16;
5 < 9;10 < 16;8 < 16.
In the second and third example (splitting fours) the students spotted that when squaring 4, the difference between segments (1 + 3) compared to (2 + 2) was creating different outcomes. They guessed that “the trick” was in the size of the parts that made the original side of the square. While it was a correct observation in order to guide the students in the desired direction, I would insist that for this learning discovery we would use only natural numbers. [0.00012 + 3.99992 gives a relatively small difference compared to 42 and this could lead to another investigation in relation to the concept of limits in the Calculus course later on]. So to go in the desirable direction I created some more numerical examples:
62 = 36;62 = 36;62 = 36;
(1 + 5)2 = 36;(2 + 4)2 = 36;(3 + 3)2 = 36;
12 + 52 ≠ 36;22 + 42 ≠ 36;32 + 32 ≠ 36;
1 + 25 ≠ 36;4 + 16 ≠ 36;9 + 9 ≠ 36;
26 < 36;20 < 36;18 < 36.
It became obvious that the catch was not only in the size of constitutive components, because there was always a missing part. The constitutive components did play part in this search for “the missing part” because “the missing part” was made of double value of the components’ product. Then, after this scaffolded discovery I gave the students a chance to “see” the complete missing part. I was sketching squares with numerical values representing (x + a)2. They liked it because it was convincing and they remembered it! The rest is history: what is represented to you today is a systematised set which illustrates this mathematical reality.
Teachers should decide on the optimal selection of available materials for students, since the electronic resources contain more than an average group of learners would require.
The beauty of the spiral curriculum which we have in New Zealand is that we can address the same issue several times (for example after introducing algebraic graphs). While y = (x + a)2 represents a horizontal translation of a basic parabola to the left for a units, y = x2 + a2 represents a vertical translation of a basic parabola up for a2 units. It is a completely different type of transformation of the basic parabola.
Two-step two-stage expansions set represents a further extension of the two-stage expansion principles, obtained when two brackets are multiplied by a constant c ≠ 1. The resources contain the three most frequent situations (positive signs in both brackets; positive sign in one bracket and negative sign in another, and negative signs in both brackets):
n (a + b) (c + d);n (a + b) (c – d);n (a – b) (c + d);n (a – b) (c – d);[- 3 ≤ n ≤ 3].
These algebraic concepts are presented in two equivalent formats: expanded horizontally (if the constant multiplies the first bracket) and expanded vertically (if the constant multiplies the second bracket). The different outcomes illustrate the commutative law for multiplication present in these situations. The development of this set of algebraic illustrations was initiated by a frequent question posed every year by some slower learners: why are both brackets not multiplied by a constant? The algebraic reason could not convince a great majority of those students; the reason became obvious only when these situations were numerically and geometrically represented.
Two-step two-stage quadratic expansions represent a further extension of the two-stage quadratic expansion principles, obtained when two brackets are multiplied by a constant c ≠ 1. The resources contain the three most frequent situations (positive signs in both brackets; positive sign in one bracket and negative sign in another, and negative signs in both brackets) and present them in two equivalent formats: expanded horizontally (if the constant multiplies the first bracket) and expanded vertically (if the constant multiplies the second bracket). The different outcomes illustrate the commutative law for multiplication present in these situations.
The two-step two-stage quadratic expansions set contains systematised examples of quadratic expansions expressed in forms:
n (x + a) (x + b);n (x + a) (x – b);n (x – a) (x + b);n (x – a) (x – b);[1 ≤ n ≤ 3].
The two-step two-stage difference of two squares set contains systematised examples of quadratic expansions expressed in form:
n (x + a) (x – a) [1 ≤ n ≤ 3].
The two - step two - stage squared sum or difference set contains systematised examples of quadratic expansions expressed in forms:
n (x + a)2andn (x – a)2;[1 ≤ n ≤ 3].
In relation to materials presented in Quadratic expansions (Algebraic Visualisations 2b) where both brackets contained a single x value, here both brackets have x value multiplied by a constant c ≠ 1. This situation creates a myriad of different possibilities. Examples of quadratic expansions (Algebraic Visualisations 2e) illustrate the next set of algebraic situations:
This type of algebraic expansions represents a qualitatively new moment in acquisition of algebraic skills and this way of elaborating quadratic expansions provides more illustrations of how a change of different elements within brackets affects the overall outcome.
After elaborating on typical situations met in learning environments I have created their mirror images (obtained when both brackets are multiplied by - 1) and fused them together into this set. Such an “enriched” set generates numerous opportunities for the further extension of students in our care.
This small set of algebraic visualisations contains three examples which help to illustrate what differences brackets make when they are present and when they are inadvertently or deliberately omitted. These examples comparison of differences 1, comparison of differences 2, and comparison of differences 3 show the most common errors when students who lack structural understanding expand brackets.
The teachers who attended my workshops about visualisation of algebraic processes and structures and who encouraged me to persist with the systematisation of then existing non - systematised materials, gave me the greatest compliments for algebraic expansion materials located in this set. In their words, this is a grand final which completely opens students’ eyes when doing basic school algebra. This set contains:
At this stage, after presenting the existing materials, through a process of mathematical induction, students are able to generalise even more complex structures. At this point it is worthwhile to look back retrospectively at the expansion of binomials, and evaluate students’ accomplishments in light of seeing their starting point.
For me, this set is the final point in visualisation of algebraic expansions. Principles learned so far can be confidently implemented into the multiplication of not only two but three or even more brackets to obtain correct algebraic results. However, we have to be honest that the obtained squared and rectangular representations are geometrically true only for two dimensional reality.
Since the multiplication of three brackets requires entering into the third dimension, to create such representations I had to use isometric paper. Personally I am satisfied with the three examples which I put into this set. They persuasively visualise the components from Pascal’s triangle that are present when cubing binomials. These constitutive components are shown within the structure of cubes and as the separate entities.
The three exemplars (a + b)3, (x + 1)3 and (x + 2)3 show the direction in which a new generation of authors can go. Their works may contain not only perfect cubes, but cuboids elongated in all three directions as well. They might create the possibility to rotate those cuboids to see the concealed reality (as I indicated in these three exemplars), and to show algebraic parts in the negative octants in relation to the 3D origin as well. Although I was pleased with the final outcome of those three exemplars, it took so much time to me that I had to stop here (the time - benefit ratio was too high). The synergy of modern technology and new, skilled graduates willing to further advance the reality of algebraic visualisations will give more convincing and deeper insights into the world of three dimensional algebra.