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?  

In these crazy times, I wanted to share a positive lesson that actually had the impact I was looking for! Imagine!
I wrote last year about how I was on a proportional reasoning kick. The lesson didn't get the "wow," I was looking for, so I switched it up this year.
Both started with an old project I used one year teaching 7th grade Humanities in an International School in Ghana:
The 100people site translates the world population into a village of 100 people. I asked the students to to do 3 things: Translate our school population to a country of their choice, present their findings and create a poster with at least 4 info graphics. The students had to determine how many people each student at WHS represented in their country and had to translate 10 aspects of their lives to those of WHS students. We used the above website, that was updated for 2017(!), the CIA World Factbook and the Worldbank Data page to translate a country of their choice to the population of Windsor High School--our suburban high school of 1712 students. It was the start of the new semester and a way to front load proportional reasoning in Geometry.

The presentations were so-so. A few got how to move from "proficient" to "exemplar," (Rubric)with some examples that I provided from my chosen country, Cuba. (Imagine if you were a Freshman an d you weren't allowed to use the bathroom at school. That is how many people in Cuba have no access to indoor plumbing.) I haven't graded the posters yet, but they look amazing, pictures to come. What has got my attention at the moment is the reflection papers. I have been making phone calls home to express to parents my appreciation for what their students have taken away from this project.

Angela on Brazil," We are extremely fortunate for all we have here in the United States and this project opened my eyes to that."
Brandon on his project about Switzerland," These types of projects make proportions fun to learn...This project should continue to be done every year because it will help people understand proportions a bit better."
Colton on his project on Italy, "The main thing I have taken from this project is how lucky I am to have grown up in Windsor with the luxuries I've had. So many times people take electricity, sanitation and drinking water for granted."
Tatiana on Greece, "I found this project to be a cool, eye-opening project and I definitely enjoyed doing it."
(Please note, the first names here sound very Anglo, however, the majority of them have Hispanic surnames.)

The students are now going around doing their presentations in their science classes that are studying food, food access, and contemplating how we will feed our 7.5 billion inhabitants.

Here is the project hand-out and Rubric:If Windsor High School Were...

Please let me know how it goes for you if you try it, or if you have any suggestions. Check back for pictures!