CHAPTER IV HYPOTHETICAL LEARNING TRAJECTORY
A. Identifying the Attributes
a. Conservation of area Goals:
Students are able to identify the attribute of area
Students are able to compare and order the area
Mathematical activity: Telling the size of the cake
In this activity, students will tell about cakes that are brought by teacher. There are two different sizes of cakes with the same thickness. Students are asked what they can tell about those two cakes and what the differences between those cakes. The students will discuss about the differences especially about the size of the cakes. After that, students are given figure of three pieces of cakes with different size. They are asked to order the cake in the figure based on the size. They will be provided with scissor if they feel need to cut the figure (See figure 1)
Figure 1: The cakes with different size
The tasks are:
1. The figure above is three pieces of cakes. Sort the cake based on the size.
2. What is your strategy to sort the cakes? Explain your answer.
Conjectures:
C
B A
In discussion about the size of two cakes, students can compare the cakes by sight.
They will react about the size in which one is big and another one is small. Then they have to explain why they think the cake is big or small. The students might think that the side of the big cakes is longer than another one. They might also think that the big cake can be cut into small pieces like the small one. Then, to order the cake in the figure, students might order from the smaller to the biggest or another around. Then, they have to explain why they think so. The problem is how they decide which one is the biggest/the smallest since the figures do not differ much in size so that they cannot compare by sight. Some children might say that the biggest is the longest one, but they have to consider the width of the cake. To decide which one is the biggest/the smallest they might directly compare the cake by cutting the figure and putting one cake on the top of other cakes. Then, they will see part of the cakes that sticks out. Students can reshape the cake in such way one cake is covered by another so that they can see which one is the biggest. They might compare the cake one by one, for example compare cake A and cake B first, after they know which cake is bigger, they will compare it with cake C. after that they can order the cakes from the biggest to the smallest or another around.
Discussion:
In discussion students usually begin by describing the sizes of objects as big and small.
They gradually learn to discriminate in what way an object is big or small. They will use specific term such as long, short, large, wide, etc. By describing the size of objects as big and small, students can develop awareness of what area is, and of the range of words that can be used to discuss it. In here, students will use words that represent
quantity or magnitude of attribute by comparing the differences of the cakes based on the size. For the next task, student must order the cakes based on the size that they have discussed before. Conflicts will emerge when students compare the figure by cutting one cake and putting on the top of another but the biggest cake cannot be said certainty. What they have to do is reshape the cake so that one cake can cover another cake, so that the biggest can be said certainty. Through this problem, students become more aware that the larger piece of cakes, deals with the largest area. During this activity, students can acquire experiences with comparing strategies related to physical quantity area. The use of words such as greater, larger and smaller will focus on the attribute of area. It is also expected that they will realize that the area of a plan object does not change if it is reshaped.
b. Identical unit Goal:
Students are able to compare area by using same kind of unit Mathematical activity: Choosing the chocolate
In this activity, students will work in small group (3 or 4 students). They will pretend to buy chocolate. There are two chocolate that have same price but different in size (See figure 2). The figure of the chocolates will be given to them. Students are asked to choose which chocolate that they want to buy. After that they will present their work in front of class and compare their method in choosing the chocolate with other groups.
The figure of the chocolates as following:
Figure 2: The chocolates with different sizes
The tasks are:
1. The price is those chocolate above is same. If you want to buy one of those chocolates, which one would you buy? Explain your answer.
2. Describe the method that you use in choosing the chocolate that you want to buy?
3. How many unit chocolate in each chocolate? Explain your answer.
Conjecture:
Some students might think that the biggest is the best choice, but there is possibility that some students choose the small chocolate. It is become a good discussion among them because they have to explain their decision. To decide which one is big/small, they might think that the chocolate that has more units chocolate is the big one. Then, they will count the number of unit chocolate in each chocolate. They will find that the first chocolate has 24 chocolates unit and the second has 16 chocolates unit. However, if they cut the figure of the chocolate and put the first on the top of the second chocolate like what they do in the first activity, they will find that the second chocolate is bigger that the first chocolate. To convince which one is big/small, they could use the small unit of the chocolate and measure another one with them or the other way around by iterating the unit so that the object can be compared.
A
B
Discussion:
It is expected that there will be a discussion in group to decide which chocolate is big/small. It will be a conflict for students when they have different answer in discussion. Some group might decide the first chocolate is the biggest since they count the number of unit in each chocolate. Other groups maybe have different answer because they compare directly by cutting and putting on the top of another and decide the second chocolate is the biggest. It is expected that students will discuss about the unit used in each chocolate. They cannot compare the chocolates because the units are different. So, students can understand that the area of objects can be easy to compare if the similar unit is used.
B. Comparing Area