Proportional Reasoning and the Olympics

In light of the Olympics, I found this activity on proportional reasoning: http://mathalicious.com/lessons/on-your-mark

I'm rather bummed that we just finished our unit on proportions because this would have been a really fun thing to incorporate!  The activity poses the question: Do taller Olympic sprinters have an unfair advantage?

First students calculate how far each sprinter would run if he ran a distance proportional to his height (using the tallest runner to set the ratio).  Using this information, they are asked to calculate how long it would take for each sprinter to run his new distance.  Based on these calculations the students are asked to find the new winning order.  Will the medals still be awarded in the same order?  The answer is no!  The original gold medalist will now earn bronze.

Next students use the second tallest runner to set the ratio.  Does this effect the original winning order?  The revised winning order?  Students will find that both revised winning orders are the same.  This is a great point to talk about what it means to be proportional.

The final task is to look compare a scatterplot showing actual time vs. height results to one showing the proportional results.  Students use these to discuss whether races should be reorganized by height.  In looking at the data, students may conclude that reorganizing would not be fair to taller runners.  In other words, there seems to be no good solution.  Someone will always be at a disadvantage.

I like this activity because it could be used to introduce proportionality or can be used as an extension to the lesson.  Student interest in the topic will help with engagement.  I also like that you can extend this discussion to other sports (ie. swimming, wrestling).

Gradual Release of Responsibility

Direct Instruction vs. Discovery...two opposite approaches to teaching.  Both can be effective and appropriate in different situations.  In my experience (though it is limited), students tend to zone out during the first and get frustrated during the second.  I really like the idea of Gradual Release of Responsibility because it seems to bridge the gap and allow students to build confidence in their abilities while still asking them to be active participants in learning.  I was first introduced to this concept in a literacy course but feel it works really well in a math classroom.  Here is the general idea:

1st:  I do, you watch.
2nd:  I do, you help.
3rd:  You do, I help.
4th:  You do, I watch.

As I progress through a lesson I try to incorporate this idea in my teaching.  Since I teach 7th grade and we are building the basis of algebra, many times my notes have a number of similar problems.  That means "drill-and-kill" is often my method for presenting notes.  It's easy to just talk at the students and then ask, "Any questions?"...to which I hear mostly crickets :)  Using this method during my lectures, I find my students are more likely to ask questions and take ownership over their learning.  I like that my students are beginning to pick up on patterns of thinking and solving.  I've found that there are fewer students who say, "I got it in class but when I got home I couldn't do any of them."  During the third phase I walk around the class to catch student errors and answer questions.  Students aren't able to tune out.  During the fourth phase I have students present their solutions.  Depending on the lesson, I may have them compare with a partner first.  My goal in this is to keep them engaged, build their confidence, and have the content stick with them.