Algebra as Pattern

Algebra is viewed as an abstract and symbolic component of mathematics however, algebraic thinking begins as soon as students notice consistent change and seek to explain it. For instance, in the early years, algebraic thinking was often represented through everyday situations like balancing concrete materials using balance baskets. This progresses to the utilization of more symbolic representations within the upper levels when letters were used to generalise thinking or to think about situations using variables. Algebra patterns support thinking, reasoning and dealing mathematically. This strand draws together the elemental properties and relationships that guide arithmetic thinking to algebraic thinking. It involves the event of the knowledge, procedures and methods related to two topics:

• Patterns and Functions: which develops understandings of consistent change and relationships.

• Equivalence and Equations: which develops understandings of balance and therefore the methods related to solving equations. Algebraic notation enables us to represent problems within the sort of equations (number sentences) that involve unknown quantities and to unravel them efficiently

What is The Relation between Algebra and Patterns (Algebra Patterns)?

To understand the connection between patterns and algebra(algebra patterns), let’s try something different. We can make use of pencils to construct a simple pattern and understand how we can create a general expression to describe the entire pattern that we will construct. You will definitely need a lot of pencils for this task. 

You need to find a solid surface and arrange any two pencils parallel to each other with some space in between the pencils. Now add a second layer on top of it and another on top of that just as shown in the image given below.

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There are a total of six pencils in this arrangement shown above. Here, there are three layers and each layer shown in the picture has a fixed count of two pencils. The number of pencils in each layer never changes, but you can build as many layers you wish.

Current Number of Layers of pencils = 4

Number of Pencils per Layer in the picture above = 2

The total Number of Pencils used = 2 x 4 = 8

What happens when you increase the number of layers from 4 to 10? What will happen if you keep building up to a layer of 100? Is it easy to sit and stack those many layers? Obviously you cannot. Let’s try to calculate.

Number of layers is equal to 100 and the number of pencils per layer is equal to 2.

Total Number of pencils is equal to 2 x 100 = 200

There are obvious algebra patterns hidden here. The number of pencils in a single level is equal to two and this remains constant, regardless of the number of levels built. So to get the total number of pencils,we need to multiply 2 (the number of pencils per level) with the number of levels we have built. Let’s take for example, if we want to construct 30 levels, then we will need 2 multiplied 30 times which is equal to 60 pencils.

Conclusion: To make a building of ‘x’ number of levels, according to the previous calculation we have done, we will require 2 multiplied ‘x’ times which is 2x the number of pencils. We just created algebraic expressions based on the pattern of pencils.

Let’s discuss a few algebra patterns .See the table given below and recognize the algebra patterns. 

Examples of Algebra Patterns

The rules determining the position of each element might be easier to see by going through the points below: 

• The alphabet ‘C’ has positions 3, 6, 9, 12, … ( which are multiples of 3) . So the algebra pattern obtained is 3n (n = 1, 2, 3, 4 etc).

• The elements ‘A’ and ‘B’ have different starting positions, but then ‘+ 3’ to those starting positions each time.

• The element ‘A’ has positions 1, 4, 7, 10, 13, … and ‘B’ has 2, 5, 8, 11, … These position numbers basically become growth algebra patterns that students can analyse and interpret.

Questions to be solved

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Let’s look at other patterns. Let’s say we have a triangle , now flip the triangle downwards and complete this image to form a complete triangle as given below-

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The pattern is still a triangle but the number of smaller triangles increases to the number 4. This bigger triangle now has two number of rows. 

What will happen when we increase the number of rows, and then fill in the gaps, with smaller triangles, to complete that big triangle shown in the picture above?

Now as we proceed to the 3rd row, what is the total number of rows? Now,the number of smaller triangles in the triangle is 9.

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Now you might think what is the pattern here?

When, row number = 1, Total no of triangles, let it be ‘n’ = 1

When, row number = 2? Then the number of triangles  n = 4

And when, row number = 3, n = 9

We observe that the size of the triangle increases , the number of smaller triangles increases too. This means that in this case n = r² which is 1², 2², 3² = 1, 4, 9…

So algebraically the pattern  or algebra patterns here can be represented as n=r².