Approximating Square Roots Lesson Plan

Warm Up

Students should review what they already know about perfect squares and square roots. Have students write in the number they are squaring, and the perfect square that gives us. This can be a good time to determine where students may have gaps in their understanding with square roots and perfect squares. This is also a good time to remind students that squares and square roots are opposites. You can do that with the example when they find `\sqrt{225}`. Tell students that we know `\sqrt{225} = 15` because `15^2 = 225`.

Approximating square roots

Once students have shared and explained their answers, ask them how they might approach estimating the square root of a number, like `\sqrt{20}`, without using a calculator. While some students will be tempted to incorrectly just divide `20` by `2`, students should also recognize that `20` is not in their list of perfect squares. 

If students get stuck, ask students if they could approximate the square root by using their list of perfect squares from earlier. Let students share their reasoning to determine what two numbers `\sqrt{20}` would fall between. Consider writing down the list of perfect squares so that students can visually see the square roots and perfect squares, like this:

Giving a visual cue can help students recognize that `20` falls between `16` and `25`, so the square root would fall somewhere between `4` and `5`. If needed, have the students find the actual square root using a calculator to show that the `\sqrt{20}` is approximately `4.472…`, which falls between `4` and `5`.

Approximating with larger numbers

The next three examples use larger numbers `(\sqrt{77}, \sqrt{130}, \sqrt{200})`. Using their warm up, students should be able to determine the two values the square root would fall between. 

Depending on the values given, students will sometimes think the answer falls between two different numbers. For example, `\sqrt{77}` may cause some students to say it is between `7` and `8` because they will inadvertently divide by `7` or `11`. Refer students to their list of perfect squares from the warm up to help them approximate the square roots they are given.

Number line activity

For this number line activity, you can have students work independently or with a partner. The idea is that they approximate each square root and plot it between the correct two whole numbers on the number line. Go around while students are working and see what strategies they’re using. Some students may choose to write the numbers on the number line as their perfect square, and plot the non-perfect squares based on that. Have students explain their strategies to the class once they’re finished.