Expand (x+3)(x+1).

The Ancient Greeks perceived a much stronger relationship between Algebra and Geometry than we do today and because of that they often utilised Geometry to help think about, and solve, certain algebraic questions.

Area can act as a representation of multiplication and so we shall construct a rectangle to help us visualise and calculate the expansion of (x+3)(x+1) with (x+3) as our horizontal base and (x+1) as our vertical line but it is important we do this in steps. I want you to draw a horizontal line of random length and label this line X. We then extend this line by 3 units. We should try and draw this to scale, however, the important part is to make sure we are consistent with our lengths. If we add a line of length 3 to the horizontal and 3 to the vertical then we should draw the lines the same length. We then do the same process with the vertical. Draw a vertical line, starting at the beginning of the horizontal X line, making sure this vertical line is the same length as our horizontal X line and we also label this line X as well. Add a vertical line of length 1 to the end of our vertical X line. We should now have one horizontal line and one vertical line, each broken into two segments of differing lengths. To finish the question, complete the rectangle, replicating our vertical line at the end of the X line and again at the end of the line of length 3. We also replicate our horizontal line at the end of our vertical line of length X and again at the end of the vertical line of length 1.

We should now have a grid of multiple rectangles and squares, each labelled with their respective dimensions. Now we just calculate the areas of each shape, writing them in the centre of each, and add them up, collecting "like terms" (terms of the same order of magnitude). So we get: X^2 + 3X + X +3 which we tidy up, collecting like terms, to read as X^2 + 4X +3

And that is our final answer. This method can be used for more complicated questions involving minus signs with a slight variation and once we become confident with the methods we can be more relaxed in our drawings to save time without getting confused. Overall, I think this method is very good for preventing errors while simultaneously reminding us of the connections between Algebra and Geometry that the Ancients were so well versed in.