Estimating square roots using visual models video

How do you estimate square roots without a calculator? Do you use a number line, visuals, manipulatives or something else? In this post is a video for estimating square roots using visual models. There is also a free set of printables (the ones used in the video) linked in the post.

A teacher messaged asking if I had a video showing how to estimate square roots using manipulatives. I didn't, but it sounded like a lot of fun to make a video on this topic. 

To prep for the video, I typed up and printed a sheet of visual models. If you want to show this method in class, you don't need anything more than paper, but here is the sheet of visuals that I use in the video if you'd like them.

There are 3 examples in the video. The first example shows how we can approximate the square root of 19 and, most importantly, why it works. As it turns out, what we're really doing is finding a square with a fractional side length that is close to the benchmark square root of 16.

How to estimate square roots using visual models - approximating the square root of 19 using cut paper squares
Estimating the square root of 19

e'd need to add 9 to 16 to get to the next perfect square 25. The non-square number 19 is 3 more than 16, so it fills 3 of those 9 extra boxes needed to get to 25. This leads to an approximation of 4 and 1/3 for the square root of 19. The 1/3 comes from 3 of the 9 extra boxes filled between the square 16 and the square 25. We can then cut up those 3 little paper squares into thirds to show this better.

How to estimate square roots using visual models - approximating the square root of 19 using manipulatives
Estimating the square root of 19

When we line up those 9 thirds around the 16 square, we can see 4 and 1/3 is the approximate side length of a square with an area of 19. In other words, the square root of 19 is approximately 4 and 1/3. There is a small piece left over, which gives us even more information. Since that little piece doesn't quite fit, the square root of 19 must be a bit more than 4 and 1/3. That extra little piece can also lead us to see that the square root of 19 must be irrational. There doesn't seem to be a good way to cut up that extra piece to make it fit evenly. 

I really like using manipulatives to show how numbers come together. To me, manipulatives make the numbers feel more real, more concrete, and almost like actual building blocks. As part of an online conference in summer 2020, I made this algebra tiles tutorial video. Also linked in the post is free set of digital algebra tiles (they're set up for factoring quadratics now but you can paste the tiles on a new blank page to use them to show more topics). For more math using cut paper, an even/odd investigation and a prime number investigation are both linked in this number sense post.

Simplifying Radicals with Variables Digital Math Escape Room
Simplifying radicals digital math escape room

Students simplify square root expressions with variables in this simplifying radicals digital math escape room. After simplifying each expression, students find their answers in the answer grid and type in the 4-letter code. A correct code unlocks the puzzle and allows students to move to the next puzzle. The entire self-checking activity is housed in one answer-validated Google Form.


  1. I love this! Thank you for the video!! So helpful. I’ll be sharing it soon😊

    1. Thank you, Mary! I hope you have a great week.

  2. This method is a great way to not only get an initial estimate but can be done iteratively to get a better estimate. That excess that doesn't fit in the corner can be cut off and then divided equally along the sides (requires a bit more math to account for the extension).

    I have made an example for the square root of 5 in the link below. I plan to eventually try to make a video out of it and color code the excess block and its subdivisions.
    The process you describe in the video is iterative and is a perfect way of demonstrating irrational values.

    I have made a geogebra display that I eventually plan to color code and turn into a video.