This lesson continues the transition of student understanding from a geometric characterization of square roots as side lengths to an algebraic characterization of square roots as specific points on the number line. 

Students begin with an image of a diagonal line segment that is also a radius of a circle centered around the origin. They can calculate the exact length of the line segment by drawing a square and finding its area, and they can approximate its length by looking at where the circle intersects the -axis (MP7).

The next image of a diagonal line segment is similar to the first but without the circle. Students should notice that they can use the same strategies as earlier to find the exact and an approximate length. An optional activity gives students an image that starts with a square that has its sides already sitting along the - and -axes. Students are asked to explain whether or not the square’s side length of 2.5 is a good approximation for (MP3). This type of reasoning allows students to think flexibly about the relationship between squares and square roots. 

Finally, students think about as the solution to equations of the form and plot these points on a number line.