During the warm up section of the lesson, I call the group to the carpet to demonstrate and review how items the same shape can be folded and lines drawn on the folds to show different unit fractions of halves fourths and eighths.
I begin by cutting a sheet of copy paper in half to create two equal pieces. Next, I fold one of the half sheets to create fourths, and compare it to the unfolded half sheet. I ask the students to explain if the fractions of 1/2 and 2/4 are the same. Remember to allow students think time, and if needed give them the models you've created. Because we have already worked with ordering fractions on a number line, the students were able to explain that 2/4 is the same as 1/2, and that 4/8 is also equivalent.
Next, I continue with drawing equal-sized rectangles on a whiteboard to show students how a shape can be divided into halves, fourths, and eighths. During this part of the lesson I also color in different parts of the shapes to show that the pieces do not have to be shaded in order to show three fourths to still be equivalent to 3/4.
This step is necessary for the students to see that fractions sections do not have to be touching to still be considered a fraction. I connect these drawings to real life by discussing the drawings as parts of a candy bar. It can be broken apart to share differently eating small pieces, and the piece they are given is still a fraction of the whole candy bar.
I explain that the image I'm sharing represents a garden that has been divided into smaller sections for different types of plants. Each rectangle represents a different section, and there are several unplanted sections. The green sections are the planted sections, and the tan sections are unplanted areas. The objective is for the students to determine how much of the garden is already planted, and how much needs to be added so that the entire garden is planted to 3/4 full.
One of the students may ask, why only to 3/4? I explain that different plants need to be planted at different times during the year, and some plants take longer to grow than others. By planting it to 3/4, we are allowing for other plants to be added later in the growing season. This real life connection to the Common Core provides a context and a purpose for understanding fractions and their use in daily life.
The students work with a partner to try and solve how many more rectangular sections need to be planted so 3/4 of the entire garden is planted. First, I ask the students, "How many sections are already planted?"
Students begin counting the green sections, and I ask them to "Think about how many sections could be planted in the entire garden."
In previous fraction lessons, the students have created and used fraction strips to compare and find equivalent fractions. I remind students to use these fraction strips to give them a clue about how to solve this problem.
Because I know my students are ready for this challenge, this is the only information I provide at this time. I want them to begin discussing and coming up with their own plans to solve the problem.
Partners work together using the image, their fraction strips, individual whiteboards, and math journals. Teams are encouraged to use any of these tools or reference materials to create a plan and try different strategies to solve this problem.
Students ask if they can cut apart the image, and I ask the students to first draw it into their journal for reference in case a piece is misplaced.
Some groups are counting rectangles, and another group folds the garden image into fourths. The group folding the image locates a section where it is easy to count how many rectangles in each fourth (six). They use this section to reason that the entire garden needs three sections with six rectangles or 18 rectangles to be 3/4 planted. The students count the eleven green sections of the garden, and they determine that seven more sections need to be planted for the garden to be at 3/4.
Other students are counting in groups of four. One student determines that if there are three green sections there needs to be one tan section. He decides to cut apart all the rectangles and sort them into groups of three green and one tan. It is this looking at the patterns to make sense of the problem that addresses the math practice of looking for the structure to solve the problem. During this time students are using the model of the garden, folding and cutting precisely, and exploring different combinations to determine how much is needed for 3/4 of the garden.
Some of the students are in need of a clue, so I provide a drawing on the whiteboard for the students to use as a clue. I draw two rectangles the same size, one above the other. The first rectangle is divided into fourths, and the second rectangle is divided into 24 separate rectangles equal to the number in the image. The rectangles I draw on the whiteboard are aligned. I color in eleven of the rectangles, and I also clearly mark where the fourth marks from the image above would align to clue the students to the 3/4 mark.
As students begin to solve the problem, they begin to share with their partners and other groups how many more sections need to be green to reach 3/4.
I ask the students to explain to the class about how they worked on the problem and reached a solution. Because the students already shared with each other, many of the solutions are the same. This provides the students an opportunity to use the math language and clarify their own thinking. Their sense of accomplishment in overcoming the challenge of this problem was evident in their excitement during this point of the lesson.
Working with a new partner, students are asked to write a second explanation in their journals that is different from their own. Students then have two solutions to the same problem. This teaching strategy is intended to connect the level of complexity of this lesson to the rigor of the Common Core.