When Student Struggle Leads To Better Explanations

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Call me slow. Go ahead, you wouldn't be the first.

Someone out there was calling for the worst topics we have to introduce. I would put rationalizing radicals right up there.  Then I had an epiphany. (A little late to the game in the scope of teaching, not proud, but happy that it came nonetheless)

Who cares about rationalizing radicals (and in particular, square roots)? I know from teaching both Precalculus and Geometry, why I want my students to be efficient at it, and I always struggled with; why not just use decimals?

Deep in a quandary about why my students were struggling with the Pythagorean Theorem and in particular, why is was hard for them to wrap their heads around why they needed to take the square root to find the length of the missing side, I realized I wanted my students to use simple radical form. I spent the first few days allowing the students to leave the radical unsimplified. Then I upped the anti and asked that they put all radicals in simplified radical form. (I tried starting with warm-ups of simplifying radicals, and found they were getting too hung up there when also trying to figure out the PT--yeah, some of the "getters" shake their heads and look with great sympathy (actually) at the kids who aren't there yet).

It is difficult to see the value of sqrt 192 When you rewrite it as 8 sqrt 3, and we know sqrt 3 is between 1 and 2, it gets easier to see its value when write an inequality and then build the value.

It is a start for them to place the value, and recall that the hypotenuse is the longest side.

Now we are preparing in Geometry for Special Right Triangles and Right Triangle Trig and it is time for the students to either learn or recall how to simplify radicals in the denominator. Again, why? Why can't the sqrt 2 stay in the freaking denominator.

Easy comparisons, that is why! What is
the value of

Is it more than one? Less than one?   When you start here
And move to here                      
The students give you a little, "oh." The strugglers get a number sense boost and the "Can we please just move ons" get to appreciate a little deeper why we torture need to rationalize a denominator. 

Not a big move, but forward I say!

Are your students struggling or acting needy these days? It sort has been the "water cooler" talk among the teachers. Especially in Science and Math. We are a fairly progressive school and most of the teachers are dedicated to not wasting student's time and giving them tasks that are relevant, and yet, we notice quite a few students falling apart and just asking us to help them every.step.of.the.way. Is this right? I am so confused? I don't know what to do. Students are really struggling with separating similar right triangles from one right triangle. Using Geogebra, cutting them out, using transformations all help, and yet students are reluctant to commit their findings to paper.

I can wrap my head around rationalizing radicals as developmentally a mathematically mature concept to wrap one's head around, but the Pythagorean Theorem?  What do you think for high school Sophomores and Juniors?    









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