Lesson 11

In previous lessons, students have studied lines with positive and negative slope and have learned to write equations for them, usually in the form \(y = mx + b\). In this lesson, students extend their previous work to include equations for horizontal and vertical lines. Horizontal lines can still be written in the form \(y = mx + b\) but because \(m = 0\) in this case, the equation simplifies to \(y = b\). Students interpret this to mean that, for a horizontal line, the \(y\) value does not change, but \(x\) can take any value. This structure is identical for vertical lines except that now the equation has the form \(x = a\) and it is \(x\) that is determined while \(y\) can take any value. 

Note that the equation of a vertical line cannot be written in the form \(y = mx + b\). It can, however, be written in the form \(Ax + By = C\) (with \(B\) = 0). This type of linear equation will be studied in greater detail in upcoming lessons. In this lesson, students encounter a context where this form arises naturally: if a rectangle has length \(\ell\) and width \(w\) and its perimeter is 50, this means that \(2\ell + 2w = 50\). 

Take a piece of string 50 centimeters long and tie the ends together to be used as demonstration in the third activity.