4 Triangles Exploration

This exploration comes from Marilyn Burns, and it goes very nicely with her book, The Greedy Triangle.  

In this exploration, small groups work together to see how many polygons they can make with four triangles. The trick is:  the construction of the polygons must follow a specific rule.

To get started, put your students in pairs or groups of three.  The students will need a large paper on which to place their findings. Students must use all four triangles to create their polygons.  The easiest way is to use post it notes and cut them diagonally to form congruent triangles.  You might want to give them two different colors of post it notes so that each polygon they form has two triangles of two different colors.  (The colors just help to make the lines between the triangles more distinguishable.)

As far as the rules, you might want to show them some that don't follow the rules and some that do and see if they can figure out the rules.   

 These two do not follow the rule.

This one follows the rule.

Here is the rule that their 4-triangles must follow:

Sides that touch must be the same length and match up exactly

As students make polygons that match this rule, have them tape onto their paper.  The goal is for them to find as many polygons as they can.  (There are 14).  

Be sure to build in time to have a discussion about what shapes were formed.  Consider taking the time to sort them and have the students decide what rule you used to sort them.  This end of exploration discussion is key to the students making sense of the exploration by talking about it, and it gives you more time to infuse more vocabulary into the lesson!

What do they think would happen if you gave them five triangles?  How would that change the number of polygons?

Possible ways to sort:  number of sides, types of angles, lines of symmetry, convex/concave, perimeter... Do your students know why you can't sort them by area?

Give this exploration a try and let me know what you think--better yet: What your students think!!

DIfferentiating with Fractions

Differentiation can be a challenge. And a lot of work.  Over time, however, I have learned that it doesn't mean that I have to have 3 or 4 different lessons over different content.  In fact, we are encouraged to extend the lesson rather than teach a different lesson.  Below is an example of what I did with Mrs. Bainbridge's fraction kite idea.

I love to color and have lots of those cool design books. I took those design books and used them to design my kites.  The simpler kite ( top design) is made up of 100 equal pieces, and the students had to use 6 different colors.  The more complicated kite did not have equal pieces.  Using the smaller size as a unit, the kite had 650 pieces.  Students then had to use at least 4 different colors for their design, and then had to count and convert the big pieces into small pieces.

Here are examples of what the grids looked like:

The students received a kite shape already made out of these patterns, and they were told how many equal parts there were.  The kids had fun, and it was an appropriate challenge for those kids who needed it.  When the students were done writing their fractions under the kite, they had to check their work by adding the numerators of each fraction to be sure it equaled the appropriate denominator (100 or 650).  Maybe you make kites, or maybe you decide to make holiday ornaments or something else that matters to your class--the idea stays the same.

I have also used Mary from Pitner's Potpourri idea.  We looked at Ed Emberley's Picture Pie and created our own picture pie.  Differentiation was natural on this one.  Certain kids were told they must use 4ths and 8ths in a certain number of colors while others just used 4ths.  The artwork created was fun and beautiful!



Toying with Tangrams

Do you enjoy the challenge of a good puzzle?  Sometimes, people really like them, and sometimes--not so much.  However, most people believe that puzzles are a necessary component to child development.  They help children to build spatial relationships as well as problem-solving skills.  Besides that, the hands-on component of puzzles is an important form of learning for some of our students.

Tangrams are the puzzles that especially seem to connect with math because of their geometric shapes.  If you want to introduce  your students to tangrams, this book is a great start!

You might decide to read and discuss this book as part of reading workshop, or it might be part of your listen to reading activities when you send students to this Youtube video of the book.  No matter how you decide to use this book, it is one that students are sure to enjoy!

Another great book to use with younger kiddos is this one:

If your students are older, you might be able to review the legend of the tangram, and then have them move into building with tangrams.  You might have them as a recess activity, an activity that they can choose when they have completed their work, or a math station.

Here are some resources that might help you put tangrams in your students' hands:

Tangram pattern cards

More Tangram pattern sheets

Interactive Tangram puzzles

A web unit for students in grades 4-5

Online Tangram puzzles

More online Tangram puzzles

A friend of mine used to make each of her students a tangram set for the holidays each year.  She would then print out a couple of patterns that they could do at home, too. What a great idea!

Once your students become used to tangrams, you might try a literacy activity I used to use as an option for my kiddos:

After reading a book, they would make a tangram picture that represented the beginning of the book and then write a sentence or two about the beginning.  They would do the same for the middle and the end.  It was a fun option for some of the kids!

Do you have other ideas for Tangrams?  I'd love to hear them. Please comment below if you do.