Central Tendency Spiral Lesson
Lesson 2 of 7
Objective: Students will be able to work with a variety of central tendency questions
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?"
In this lesson we review what my students know about measures of central tendency. My goal is to make sure that they connect their understanding of measures of center to their knowledge of algebra. In order to achieve this goal, We will solve problems relating to the mean of a set of numbers.
Specifically we look at problems that ask things like, "give the following data set, what number would we need to add to raise the average by __________ points."
We start the lesson by giving students the following problems:
They have calculators available and often use various guess and check strategies to converge on the answer. It is important to encourage these algorithms and use them as a springboard for the efficient algebraic approach.
Once I notice that students have generally reached an answer for each of the questions, I ask them, "could you set up an equation with variables to solve this problem?" The question I usually get at this point is, "why bother?" I remind them that not only is this a great opportunity to sharpen their algebraic skills, but having an algebraic technique will allow them to deal with any problem and any number (not just the friendly ones).
The second problem offers a great opportunity to demonstrate the effectiveness of the algebraic approach:
The mean of three numbers is 25. The second number is four less than twice the first. The third number is two more than four times the first. Find the smallest number.
A conversation might go like this:
"If the mean of three numbers is 25, how could we write these statement as an equation?" In response, my students usually give some equation with three variables (this is a great way to start this problem). They might say something like, "(a + b + c)/3 = 25."
Then we review the importance of the second piece of information: "The second number is four less than twice the first. The third number is two more than four times the first." I like to ask questions about this statement, like "why did they tell us this information? Could we have solved the problem without this information?" We usually talk about the difficulty of working with three variables in one equation and the importance of finding a way of writing the three variables as one.
I want students to understand that this statement gives us a way of writing our equation with only one variable. If they understand the necessity of this step, they will also recognize that every problem must give a way of relating the variables. By simply looking for this in every problem, they will have a wonderful step to solving problems of this type. They simply need to ask themselves, "how can I define my problem in terms of one variable?" If they can find a way to do this, then they can solve the problem.
In this case, we can write everything in terms of a."The second number is four less than twice the first. The third number is two more than four times the first." So we can write the following:
a = a
b = 2a - 4
c = 4a + 2
To stimulate conversation around these equations, I ask them questions that reflect common misconceptions, like "why can't we write c = 2a + 2? How do we know that we need to write 4a?"
I finish the start up by asking them to analyze every question with algebraic statements. They can work with their own algorithms, but I at least ask them to supplement their algorithm with the algebra.
We give students a set central tendency questions 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.
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.
Here are the links to the videos: