Correlation Spiral Lesson

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Objective

Students will be able to work with a variety of correlation questions

Big Idea

The tighter packed the data, the better the correlation

Overview and Lesson Structure

This is not a test prep unit (although it could be construed as such). My goal is not to simply expose students to test questions and format (although they will get an amazingly wide variety of exposure and practice). I want to help my students look back on the year and fill in the gaps and build on their foundation through repeated practice. All students need a chance to recall the things they have forgotten (or never mastered in the first place). Our curricula often push forward, build up and look ahead. But we can’t forget to spiral back and give students an opportunity to revisit topics from past units.  These lessons represent my attempt designing a rewarding and fulfilling review unit for algebra. It is an attempt to prepare students for the summative exams that we use to asses our curriculum design and mark student progress.

The lessons in this unit are built around a simple format:

  • A 15-minute start up problem, where we introduce the basic ideas of the concept they are about the review.
  • A 45-minute chunk of time devoted to giving students in depth, meaningful practice, where students use very short videos to cover appropriate problems

The work reserved for the final 45 minutes of class is connected to a series of homework assignments. The materials are designed so that whatever students don’t finish in class, they can do at home.  If they finish the current assignment, they can move ahead and complete future assignments. 

Each day, the lesson starts with some follow up on their practice work. Based on the previous lesson, I will be prepared to ask specific local and global questions about their work. Local questions can be anything specific to a certain aspect of a problem, like "how did you get that number there?" Global questions can be things like, "how does this example connect to the problem before it?"

Start Up

15 minutes

I like to display a bit of humor around correlation and causation at the start of this lesson: 

I think it is really important to distinguish between correlation and causation. Helping students understand the difference between each concept is critical. I review questions that show correlated and causal and correlated but not causal relationships, like "a rooster crows and the sun rises." Here We need to ask, "does the rooster cause the sun to rise?" More importantly, we might ask, "how could you prove the rooster does not cause the sun to rise?" This leads to the idea that if you are able to see the sun rise when the rooster does not crow, then it is clear that the rooster does not cause the sun to rise. This technique, of showing that the absence of once variable doesn't effect the occurrence of the other variable, is an effective way of distinguishing between correlation and causation. 

Similar types of problems can be found here

I also like to use the problems from this JMAP pdf.

Although it doesn't apply on this problem, I like to offer the graphing calculator as a resource: Correlation on TI84

Practice

45 minutes

We give students a set of correlation problems and a helpful template to follow along. The templates are set up to help me recognize when a student needs help. They rate how they feel about each problem as they finish and I look at these numbers to figure out if they feel comfortable with the material. I try and keep this rating system simple. Something like, give yourself a 4 if you really understood the question and give yourself a 1 if you felt really overwhelmed.

With correlation, I ask students to either explain the different types of correlation in each graph or I ask them to sketch a graph that matches the types of correlation offered in the choices of a question. This deepens the mathematical experience around the questions and pushes them to think critically about correlation. The first problem serves as a useful example, in which they ask which graph represents the correlation between x, the score on a test and y, the number of incorrect answers on the test. 

YouTube video of the question

This question is interesting as they switch the independent and dependent variables from their natural position, where x is number of incorrect responses and y is the score of the test and then offer different graphs to match the situation. This switch is challenging and it is even tough to describe the correlations provided in each graph. I encourage them to write little stories to match each graph, especially the undefined and zero slopes. 

Since all of the work is accessible via video on  you tube and/or internet archive (which is not blocked at schools and can run on very slow networks) students start a video, pause it, try the problem and check the worked example. These videos are more worked example than instruction, but they seem to really help.

I use this time to conference with groups of students on issues they might have had during the start up or questions they might have as they attempt the problems through video. If I see a common issue, I might give them an exit ticket to bring home and return the next day (it would be very similar to the start up problem but with different numbers).