Category Archives: Launch

Data kids might find relevant

I know this is a non-starter. But I hope it sparks some others to think about something important with me.

In the Connected Math unit Data about Us at sixth grade, students collect information about themselves as a class, they represent the data in a variety of ways and they draw some rudimentary inferences.

Two subsequent units in the curriculum draw on these ideas. In Bits and Pieces I, students use percents to summarize survey data. In How Likely Is It? students use data analysis and proportional/fraction reasoning to study probability.

In HLII, there is an Investigation involving inherited traits, such as the ability to curl one’s tongue, attached earlobes, curly v. straight hair, etc. Many of these are standard chestnuts of Mendelian genetics; nearly all have been debunked.

So future versions of the curriculum will not use the Punnett-square for theoretical analysis of trait inheritance. But they remain (I think) reasonable areas for data collection. They are age-appropriate and interesting to middle school kids.

So we get rid of the genetics lesson and focus on descriptions of populations instead. Fair enough.

But frankly, how interesting is the following task?

table of data from a survey of genetic traits in the US

The original task from Connected Mathematics 2: How Likely Is It?

Answer: Not very.

It’s fun to think about our own attached/detached earlobes. But not so fun to look at survey data on the matter.

So I started thinking about what kinds of rudimentary data inference kids could do instead. And what if we took Dan Meyer’s challenge seriously and applied it to this problem?

I will reiterate that my first idea is a non-starter. No way is this the right problem. But consider what an interesting question can result.

The task

Here is a fifth-grade class, circa 1984.

photograph of a fifth-grade classWhere in the United States is the school located?

I would love help thinking about this problem on two levels:

  1. This problem. What do you notice in the picture that might help answer the question?
    What data do you attend to?
    How do you find yourself wanting to answer? Regionally? By state? Rural/suburban/urban?
    How sure are you of your answer? It’s probably…? It might be…? It’s got to be…?
  2. This kernel of an idea. My hunch is that this task as stated now is too sensitive for sixth grade math classrooms. But do you agree that it’s a more compelling question than the original? If so, what might the mashup of this task with the original look like? Is there a version of this idea that poses as intriguing a question, without setting off political-correctness alarm bells?

Resources

Some potentially useful census data.

The answer.

Acknowledgments

I Googled “Class pictures” and this site was the first one that had pictures of classrooms full of kids. I didn’t set out to find a classroom with any particular characteristics (other than being in 5th-8th grade).

A visual launch-probably interesting only to CMP teachers

Connected Mathematics-a 6th through 8th grade mathematics curriculum-is designed with a teaching model in mind: Launch-Explore-Summary. In my professional development work, I help teachers to understand the teaching model in general, and I demonstrate effective techniques for particular problems.

For a few summers now, I have been demonstrating a Launch for a problem in the 8th grade unit Frogs, Fleas and Painted Cubes. I came up with the idea after leaving the middle school classroom, so I have not used it with students. But every summer, teachers ask where they can find written instructions for it and I have had to say that they do not exist.

My summer teaching partner has reported to me that it has been effective in her classroom, so I am now motivated to write it up and share it widely. In what follows, I will assume that the reader has access to the Student and Teacher Editions of the unit. The problem in question is Problem 2.1 of Frogs, Fleas and Painted Cubes.

The original launch

In the original problem, students consider the effects of a land swap in which a square piece of land is swapped for a rectangular one with the same perimeter. So a 5 unit by 5 unit square, might be swapped for a 3 unit by 7 unit one. Students record their information in a table and look for a pattern in the relationship between the areas of the two pieces of land.

In designing my alternate launch, I wanted to achieve two ends: (1) increasing access for visual learners, and (2) increasing the generality of the results.

(1) was important to me because I have been working in my own teaching on visual representation in mathematics, and because the problem asks us to represent something geometric in a table. It seemed like a natural place to increase the visual components of the problem.

(2) is important as a teaching principle. The original problem asks students to consider a variety of squares, but to always change each dimension of the square by 2 units. Thus a 5 by 5 square becomes 3 by 7, and a 6 by 6 square becomes 4 by 8. But there is nothing special about 2 units. A more general pattern emerges if we consider different numbers of units.

The new launch

For this problem, students will work in groups of 3 or 4. Each group receives a bunch of colored (say pink) inch grid paper and a bunch of white inch grid paper. After setting up the context as in the Student and Teacher Editions, have each student draw a square (with whole number side lengths) on a sheet of the white grid paper. Students should coordinate to make sure that each student has a square of a different size from their groupmates.

Each group is assigned a number from 1 to 6, repeating numbers if there are more than 6 groups.

Using the pink grid paper and the group’s assigned number, each student draws a new rectangle. Say my group’s assigned number is 3. Then I will transform my original square (say it was a 5 by 5 square) into a new rectangle by increasing one dimension by and decreasing the other by 3. My new rectangle would be 2 by 8. This preserves the perimeter. What does it do to the area?

To investigate this question, each student tries to cover his/her original white square with his/her pink rectangle. Students will need to cut these pink rectangles apart in order to cover as much as possible of the white square.

Then each group puts their covered squares onto a large poster paper and these are displayed around the room, in order of the assigned numbers (see below).

Now the class is ready to do the mathematical work of investigating the relationship between the assigned number, the size of the original square, and the area of the pink rectangles. Some possible observations that students might make and questions they might pose include:

  • It does not matter what size square we start with; the difference between the area of the original square and the area of the pink rectangle is the same within each group.
  • The group’s assigned number matters-as the assigned number increases, so does the difference between the areas of the original square and the pink rectangle.
  • The area of the pink rectangle is always less than the area of the original square. We can see this because we never quite cover the white with the pink.
  • Several of these images show a white square peeking out from behind the pink. Is it always possible to rearrange the pink so that a white square peeks out from behind?
  • When will it be possible to arrange the pink so that it is in the shape of a square (with whole-number side lengths)?
  • etc.

After some work observing patterns and asking questions, now students should be ready to make the table in the text. However, students should alter the table to match their group’s assigned number, and they should conjecture how the tables of groups with other assigned numbers should look.

Launch

Deep in my professional heart, I am a math teacher. Right now I am in higher education but I got my start in the middle school. Part of my job has been and will be teaching how to teach math, through methods courses and professional development. I enjoy this work, but really I am happiest when I am teaching mathematics, not when I am teaching about teaching mathematics.

My favorite part of this other aspect of my job (the teaching about teaching mathematics aspect) is having professional conversations about teaching and learning. I don’t enjoy being the expert. I do enjoy trying out new ideas and hearing how others think about mathematics, and about teaching and learning mathematics.

Ultimately, I have more to say than any one person ever wants to hear so I am starting a blog. I can talk and others who are interested can listen.

I write for publication, but I have more to say than any one journal probably wants to publish so I am starting a blog. I can try out my short-form ideas and decide which are worth sending on to other, more formal venues.

I come into contact with hundreds of current and future teachers every year through my courses, my conference presentations and professional development workshops. I often think about conversations I have had with these teachers long afterwards, but I have no way of following up with them so I am starting a blog. I hope to invite others into these conversations.

In the coming weeks, I’ll write about some of my current activities, including Connected Mathematics (CMP) workshops, my new job at Normandale Community College and my recent work at Minnesota State University, Mankato. I hope to write weekly and to attract readers who will share their ideas and constructive criticism. So set a bookmark for christopherdanielson.wordpress.com and let me know what you think.