That's Not Fair! Fractions of a Region Part 1

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Students will be able to partition a whole into unit fractions to solve a word problem.

Big Idea

How can 2 cookies be shared by 3 hungry sisters? How can 2 peanuts be shared by 3 squirrels gathering nuts? These are the real world questions that students need to be able to answer by using their knowledge of fractions.

Warm Up/Mini Lesson

5 minutes

In order to warm the students up, put a dark green Cuisenaire rod on the board and tell the students it represents a candy bar partially eaten. Tell the students that the rod represents three-fourths (3/4) of the candy bar.  

Have the students turn and talk with their partner to figure out what the whole candy bar looked like. (4 reds, which also equals one brown).

Then place a trapezoid block on the board and and explain that it represents three-eighths (3/8) and have students build the whole shape.  In this problem, they will need to "break" the trapezoid into unit fractions of thirds by using the triangles, and then they will add 5 more triangles to the shape.  

This is a complex understanding, and it may take time for the children to understand and apply this concept.  Continue to walk those that need it, through this thinking and modeling.

It is particularly helpful to encourage students who are still stuck, to use skills previously taught to create new shapes, using known shapes (Grade 2, Critical Area - describe and analyze shapes). Exploration should lead to creating the trapezoid using triangles, and then as you remove one unit, ask the student to tell you what is being shown. As you continue to remove units, the student can then build their thinking about division of a trapezoid. But once they count the number of equal units (triangles) in the trapezoid, they will be confronted by the problem that the trapezoid has six equal pieces. So this shape - this whole - is more than 1 trapezoid. The next step then, is to facilitate the student in reasoning through the quantity - 1 whole, but broken many parts? And then to build it by counting units until they have 1 whole described by the denominator.

Some students see the denominator from the start, and this drives their reasoning. In this clip, my student works to explain how he knew to use eight triangles, even though we were discussing trapezoids. 


Group Active Engagement

20 minutes

My class is currently studying the first settlers of our state in social studies.  I am therefore reading Laura Ingalls Wilder's book Little House in the Big Woods as a read aloud.  In the book, on page 178, there is an excerpt that depicts Laura and Mary nibbling on cookies given to them by Mrs. Peterson. Every time the girls visited Mrs. Peterson, they would get a cookie to eat on their way home.  They always ate half and each gave the other half to Baby Carrie. They always felt that "somehow this wasn't quite fair," but the book never says that they solve the problem. 

Here is our real world problem for the class to work on. Even though it seems simple, having third graders to divide the full region (cookie) into thirds two times and then share the pieces among three is a big task.

I have chosen to present this problem to them in a new way.  I will display the problem on the board, after reading the chapter to them of course. Next, I will ask them to discuss with a partner how they might solve this problem.

After discussing ideas and defending thinking with drawings on the board, or with fraction circles, cuisenaire rods, or other methods, I will alert the students to a new way to present our knowledge: Drama.

I will ask the student groups of 3 or 4 to take fraction circles or other tools they deem necessary, and find a way to "act out" the solution using their words put into conversations, as well as actions. 

I will only let them have 15 minutes to plan this.

This group found a way to move from their comfortable "halving" to partitioning the whole into obvious fractions.  In this case it was sixths, even though there were only three people to share. They went on to realize each person would get 4/6 of the total cookies. 

This student went about the solution with his partners by using a white board so they could draw and erase several times before they go to their solution.

Small Group Activity

25 minutes

When the students complete working with their groups, ask them to reassemble and be prepared to share their skits. 

In this portion, before tomorrow's more structured lesson, do not give the students paragraph frames or boundaries.  This is a pre-assement to help you figure out how much structure and what type, they will need.

Allow the children to act out their solutions and take praise and critiques from the class. 

As student groups finish, begin creating a list on the board, or a chart paper, of what each group did that helped the others understand the math and the solution to the problem.

These boys worked hard to incorporate math vocabulary terms. I asked them if there was a way, if they were to redo this, if they could write a line or two about "why" thirds would work.  In this way, I push them to think about this in tomorrow's lesson. 

This group did a good job solving and modeling.  However, I realized that they are referring to the fractions as wholes.  Listen as I discuss this at the end of their skit.  Also, I will interject this as a mini lesson in the coming days. 


Wrap Up and Home Practice

10 minutes

Following the sharing and anchor chart development, explain to the students that in the next day they will be creating a skit and recording it for viewers.  The task will be to teach, through drama, how to partition a whole into fractions.

For preparation and home practice, assign the following home practice activity.

Andy was having 3 friends over after school for an afternoon of video games.  His mom put aside 4 brownies for Andy and his friends to share.  However, Andy's younger sister got home first and ate one of the brownies!

How can Andy and his friends share the remaining brownies and still be fair?  Show your work.