Visualizing Algebraic Expressions 1

At the early stages of learning Algebra, students may get confused looking at algebraic expressions. Students may struggle to internalize the difference between x + y & xy and x2 & 2x!

A geometric interpretation of algebraic expressions makes it easier for students to get a visual understanding of the difference between various algebraic expressions.

First degree expressions - Lines

An algebraic expression of first degree can be thought of as a line whose length represents the value of the variable.

1. “x” & “y” can be thought of as lines of length x & y respectively.

2. “x + y” can be thought of as a line with length x+y.

3. “2x” can be thought of as a line of length 2x or x+x.

4. “2x + 3y” can be thought of as a line of length 2x + 3y

Second Degree Expressions – Rectangles & Squares

An algebraic expression of second degree can be thought of as the area represented by a rectangle (or a square as the case may be)

1. x2 - can be thought of as a square of side x & area equal to x2

2. xy can be thought of as a rectangle with sides x & y and area xy

3. 2yz can be thought of as a rectangle with sides 2y and z (or 2z & y) and area 2yz

Using language syntax to understand operations

In language we say 4 apples and 3 apples equals 7 apples

We can extend the analogy to 4 hundreds and 3 hundreds is 7 hundreds

We can also say 4 one-fifths and 3 one-fifths equals 7 one-fifths

Similarly we say 4 xy and 3 xy equals 7 xy.

This can be extended to cover and collection of "like" terms. Even if the variable is itself a collection of other variables, it can be treated as one variable as long as it is being considered with another similar "like" term.

Understanding Bracket Operations

Representing algebraic expressions visually also helps in understanding bracketing & unbracketing operations.

It can be seen that two areas x2 and xy can be seen together as the area of a rectangle with sides x and (x+y). This is the same as x2 + xy = x(x + y)!

Plotting on Square Dotted Sheets

These ideas will become clearer if students draw lines and rectangles using a Square Dotted sheet. All the dots are plotted so as to form equal squares in both the horizontal & vertical direction. Lines should always be drawn connecting the dots, either in the horizontal or vertical direction. (Chapter 28.7 gives some examples of these representations)