New Way To Teach Square Roots

One of the workshops at the math conference was on teaching students to find and simplify square roots.  It begins using concrete before moving to pictures to abstract.

It was awesome and I learned so much.  The activity uses small square of certain areas.  There are squares ranging from 2 cm squared to 10 cm squared.  These are not 2 x 2 but sqrt 2 by sqrt 2 so the area is 2 cm squared.  The same applies to each square.

The introduction starts with a square that has an area of 6 cm squared.  The diagram shows how each side is the sqrt of 6 cm.  The professor running the workshop, emphasized using the proper language every time including the unit.  He stated if there is no unit listed use units squared and the square root of 6 units because using the language of math properly helps.

After the introduction, students are given a square that is 12 cm squared.  They are expected to use smaller squares to cover the area exactly.  After checking things out, they'll discover they can use four   squares with an area of 3 cm^2.  So one side is 2 sqrt 3 and that is the square root of 12 simplified.  This part of the exercise has every step written out along with questions.

The students repeat the exploration using a square with an area of 45 cm^2.  After a lot of trying different possibilities, they'll discover they need 9 squares of 5 cm^2 to cover the square.  If they count things, they'll find the simplified version is 3 sqrt 5.

As they work through the packet, they move from having to fill the whole square to find the answer, to only needing to do the edges, to understanding it. For each part a step is removed so people are doing one more step during the practice.  About half way through, they are expected to draw in lines so they are creating a picture of the squares rather than using them physically.  By the end, they can do square roots.

In addition, students can use these same squares to learn more about the Pythagorean theorem.  Students take the squares and work on putting them together so they form the edges a right angle triangle.  For instance, if the two smaller squares have sides of area 4 and 4, the larger square is 8, it shows they have a right triangle.  As they play with the area squares, they might notice that the total of the two smaller squares is less than the larger square and it forms an obtuse triangle.  If the total of the two smaller squares is bigger than the larger square, they have an acute triangle.

So cool.  Students see how the Pythagorean theorem is saying when you add the areas of the sides together, you end up with the area of the hypothenuse but that is not shown as often. This led me to wonder if I could use the length of the sides to show the triangle inequality theorem which says that the length of two sides added together must always be longer than the third side to have a triangle.

I really loved this activity and can hardly wait to use it in my classroom.  I'd love to hear what you think about it.  I got the information from a Professor who teaches in North Carolina.  If you want his information, leave a note and I'll be happy to share.  Have a great day.