One example is using squares and rectangles to factor an equation. Here is a link to problems with the option of different levels of difficulty to factor an equation using the knowledge of area of squares and rectangles being length times width to find what a polynomial factors into. We use the units of length multiplied by the units of height to find the two factors. Similarly, We can use our knowledge of area and/or patterns when looking at certain patterns to find an algebraic equation. Here is an example:
Next, we can look at the perimeter among the different dimensions of shapes. We can use the pattern of the values to find an equation that will allow us to find any perimeter based on dimension or find the dimension based on the perimeter. I saw an activity in a 8th grade classroom where students used toothpicks to find the perimeter of one hexagon. Then, connected another hexagon to one side of the previous hexagon, then counted the perimeter of the figure created from both hexagons and how many total, and found a pattern creating an algebraic expression that allowed them to find the amount of toothpicks that would be needed for 10 hexagons and how many toothpicks would be needed for the perimeter. Another activity was finding the perimeter of different sized squares based on looking at one square. This example is posted below.
These are just three of many more relationships among algebra and geometry. In a way, geometry is just a visual representation of the algebra. It has been proven that using visuals and discovery based activities students are able to learn more conceptually instead of a procedural approach which is more beneficial for the students. Realizing the connection among geometry and algebra is beneficial for everyone!