There is thought amongst students that teaching algebraic notation involves the students haveing to memorise a set of abstract rules unique to the topic and separate from the rules of arithmetic. If, on the rare occasion they see the two overlap this is simply by coincidence.
My ambition when first introducing algebraic notation to students is to challenge such preconceptions by teaching the rules of algebra as an extension to the rules of arithmetic through diagrams rather than numbers.
I do this by drawing on the student’s powers of intuition. We all have this innate ability to make links between the things we know. In my opinion this ability is stronger in children as they are more open to learning.
We begin with two shapes. A square whose area is defined as x cm2 and a quarter circle whose area is given as y cm2.
We discuss the area of shape a). The shape consists of three squares and a quarter circle, x + x + x which we can write as 3x. I ask the class to consider why we never write the multiplication symbol with algebra. We also discuss how we write divisions. The quarter circle we write as y. Not 1y but y. Simply because by writing y we only see one y. Therefore the area of the shape is 3x + y. I challenge the class to use their mini-whiteboards to write the same area differently. The majority come up with y + 3x.
Interestingly none of the students wrote 3yx as they intuitively understand the square is different from the quarter circle so the two letters cannot be collected together. The x terms can be collected as there are three congruent squares.
The class attempt to find the area of shapes b) and c) without my help. Given this is a mixed ability class it was very pleasing to note the only minor misconception was to write the area of shape c) as y4 rather than 4y. We discuss why the number goes before the letters.
Shape c) does cause some confusion. Most students can see the x and y components but find it difficult to combine them to create the blue shaded region. What is pleasing however, is their difficulty does not lie in the algebraic notation but finding the right combination of the shapes. Once the class can visualise the square has a quarter circle removed from it they actually find it easier to describe the combination using algebraic notation. x – y is the resounding description of the shape.
As an extension to this I ask the class to draw two shapes with the area 4(x – y) with four lines of reflective symmetry. Most able students were able to draw both shapes whereas the least able were able to draw one shape after a little prompting. What is interesting though is that all the class intuitively recognised the equivalence between 4(x – y) and 4x – 4y.
In the plenary I wanted to assess how well the students understood the formal notation associated with algebra. How different letters represent different values (or in this case the area of a shape) and for the most able how brackets can be used to simplify terms with a common factor.Students were asked to show two diagrams for each expression on their mini-whiteboards. The base of each shape was equal but the heights different. Below are some of the most common answers for the most, least and core ability students.
This was the first in a series of lessons introducing algebra. Future lessons go on to consider writing expressions from words, simplifying expressions, basic substitution and use of function machines to solve basic equations.
You can download the lesson for free by clicking here.
Click here to see some research on teaching algebraic notation from the Nuffield Foundation.
How do you go about teaching algebraic notation? I am very interestested in other approaches. Please leave your comments.
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