# 6-8

### Tangram Polygons: Composing and Decomposing

In my last post, Tangrams: A World of Geometry, Part Two, I talked about the thirteen convex polygon shapes that can be formed with the seven tangram pieces. In the video, I showed how to make five of them, and then I left a challenge for you to look for the remaining eight convex shapes. By way of encouragement, I provided downloads of two of the eight shapes, but left it to you to put the puzzle pieces together to form these two shapes.

In the following video, I review putting together the five shapes. You’ll see that I’ve made the tangram pieces in two different colors. I think it makes it easier to notice patterns and relationships between the shapes and the way the pieces go together to form the shapes.

Now we’ve reviewed putting the five shapes together, and you’ve seen how the colors help us think about the different ways the pieces can be put together. The next video will start by showing those two shapes for which I provided you with downloads in my previous post. Then we follow that up with finding the remaining six shapes. For some of these shapes, there may be multiple ways they can be put together. I don’t claim to have exhausted all of those ways.

Below are several attachments that you can download. The first shows all of the convex polygon shapes that are possible; the second shows one way to put the pieces together to form each shape. Then, there are three pages that have templates for all 13 of the shapes, and finally there are two pages of multiple copies of the tangram pieces in case you want to run them off on two different colors of cardstock.

It is my hope that many of you will find ways to use the tangrams as way to challenge students to look at composing and decomposing shapes. Each of these quadrilaterals, pentagons, and hexagons are composed of the same pieces and so have the same area.

For students in seventh and eighth grade it might be interesting to look at the perimeters of these thirteen shapes. If we took a side of the square tangram piece as the unit of length measure, what would be the lengths of the sides of each of the pieces? Then we could ask about the perimeters of each of the shapes.

Well, maybe that will be a future post.

### Have You Done a Good Math Problem Lately?

In work or social settings it is common to hear the question, “Have you read a good book lately?” The question often starts a lively sharing session about books that elicit pleasure, profundity, or insight. A population that regularly engages in these discussions is an indicator of a literate society. As those appointed by society… Continue Reading

### The Forty-Yard-Line is Opposite the Forty-Yard-Line?

One of the Common Core Standards for Mathematical Practice includes a focus on students knowing and using correct mathematical language and using clear definitions in discussions with others. There are times when everyday words are used in special ways in school mathematics, and it is important that students come to understand the precise mathematical meaning… Continue Reading

### Do We Really Understand What Math Is?

What would you or your students say math is? Some common answers could be numbers, addition, subtraction . . . Below are the posters a group of AIMS trainers created answering that question. Most people don’t understand what math really is. If you have read some of my previous posts, you know my elementary and… Continue Reading

### Friday Institute: A Common Core Resource

I want to share with you two very helpful, quite extensive Common Core Math resources that are available from the Friday Institute for Educational Innovation at North Carolina State University. The first resource is an interactive map of all of the Common Core Content Standards organized into 18 learning trajectories or progressions http://www.turnonccmath.net/index.php?p=map. For example,… Continue Reading

### Do Comics Have a Place in Your Classroom?

One feature of the AIMS Essential Math Units, a series that is targeted for middle school, is the inclusion of comics as a way to show students engaged with some of the activities in a unit. Our hope for the comics was that they would help to make explicit the content knowledge that is the… Continue Reading

### One Object Three Shapes: The Solution

In an earlier post, One Object Three Shapes, we posed the problem of finding an object that would appear to be a triangle when viewed in one way, a square when viewed in another way, and a circle when viewed in yet a third way. The challenge was to create an object that would fit… Continue Reading

### One Object Three Shapes: Circle Square Triangle

Front and back, top and bottom, and left and right are ideas that we use to describe objects in our three-dimensional world. Young children learn these positional words in Kindergarten. In fact, learning the meaning of these words is one of the Common Core Kindergarten geometry standards. In reality these are big ideas that are… Continue Reading

### Month, Day, Year: A Number Sequence

Daily journaling is one of the constants in my life. Over many years, I don’t believe there has been a single day when I did not make at least half a dozen entries in my journal. Each entry is dated, so on any given day I will write the date at least half a dozen… Continue Reading

### Frank Lloyd Wright, Froebel Geometric Gifts, and Hands-On Learning

This post is a continuation of the story of Froebel’s geometric gifts that was introduced in my previous post. I ended with a promise to tell a story about the famous architect, Frank Lloyd Wright. In 1876, when Wright was eight or nine years old, his mother attended the Philadelphia Centennial Exposition. Wright describes in… Continue Reading