Lesson 4

In this lesson, students begin to transition from understanding square roots simply as side lengths to recognizing that all square roots are specific points on the number line. This understanding takes time to develop because students have previously only worked with rational numbers, which can be found by dividing the segment between two numbers into equal intervals (MP1). The purpose of this lesson is to encourage students to reason about square roots and reinforce the idea that they are numbers on a number line. This lesson continues students’ move from geometric to algebraic characterizations of square roots.

In the first activity, they still find \(\sqrt{10}\) by relating it to the side length of a square of area 10 square units, but then are asked to approximate the value of \(\sqrt{10}\) to the nearest tenth. In the second activity, students find a decimal approximation for \(\sqrt{3}\) by looking at areas and also computing squares of numbers. This lesson shows students that irrational numbers are numbers—specific points on the number line—and we can find them by rotating a tilted square until it is sitting "flat."