Motivating simplest radical form

From deep in my archives – this is a lesson I used long ago to motive simplest radical form. It was one of my early ventures with Math Practice 7 – Look for and make use of structure.

So you are about to teach special right triangles, and to do that you want students to understand simplest radical form. The problem with simplest radical form from your students’ perspective is that it is a needless complication that does not add any meaning to the situation.  7.071 is much easier to understand as a quantity than 5√2. But we know that the patterns are easier to see if they use simplest radical form.

So . . .

What if you started with trying to get them to notice the pattern in the sides of a 45-45-90 triangle using decimals?  Here’s how I envision a possible intro lesson going.

  1. Give them two angles in each triangle and have them notice they are similar. Then give them one side on each triangle and have them find other two sides (isosceles, pythag).

Here are some possible triangles you could use, and the answers they would get in decimal form:

image 1

It will be handy if the Pythagorean theorem work for these is on board to come back to, so you could have students put up this work.

 

  1. Teacher says: “Using the patterns you see, can you predict the length of the hypotenuse for a 45-45-90 triangle with leg =7? No pencils allowed – predict in head.” Take guesses . . . .too high, too low, just right

Now use math to check who was closest.  Answer: 9.899

Students may use similar triangles to set up proportions and solve between triangles if they have worked with those recently. That is fine.

 

  1. Tell students: “In this next unit we will be learning how to use the consistent ratio between two sides of the one triangle (for instance hypotenuse to leg in the ones we have been looking at) to find all the angles and all the sides of any triangles similar to it. So what is the ratio in this case?” Have students find.

image 2  

 

  1. Teacher talk: So decimal is consistent but not catchy. Could we have used it?

7(1.414) =9.898

Close. Off due to rounding.

Perhaps play with including more decimals to increase accuracy.

Would be nice if there was a way to be more exact without having to write out hundreds of decimals.

 

  1. Tell students: “There is another way to express radicals to make pattern easier to see and use.” Go back to pythag work on board and break down the square roots. I would suggest for this moment that you use the greatest square method so they can understand what you are doing.

 √50= √25∙2=√25∙√2=5√2

The point now is not to teach them how, but just to let them see helps see the pattern, so that they might see some value in learning how.

triangle 1:  5, 5, 5√2         triangle 2:  3, 3, 3√2         triangle 3:  4, 4, 4√2

 

  1. Teacher: “Anyone want to guess the hypotenuse for a 45-45-90 triangle with leg 7 now? No pencil!    Answer:  7√2 !  

How accurate is that?”

(Show them how to use calculator to check.) Pretty darn accurate!

“Can anyone guess the decimal  for   √2  ?”

 

  1. Transition . . . “Before we look for more patterns, let me teach you how to rewrite square roots to help you see the patterns.” Or if you’re on a roll and feel experimental, cut a equilateral triangle in half to make a 30-60-90 and see if they can find the 30-60-90 ratio themselves in radical form. They may end up “inventing” how to break down simple square roots all by themselves.

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