HYPOTHETICAL LEARNING TRAJECTORY (HLT)
4.6. Lesson 5: Manipulating Balance Starting points and learning goals
Table 4.9 Overview of hypothetical learning process in „finding mass‟
Conjectures of students’ thoughts Suggestions for teachers The students might translate the picture into
the following expressions:
OR
OR
The teacher might let the students continue with those representations.
But, in the discussion, the shows his preference to use letters by constantly using that kind of representations.
Realizing the need to keep the balance, the students might think to just remove same amounts of the masses. Thus, the remaining work would be:
Big possibility that the students will get confused to decide their next step.
The teacher might emphasize this removal by writing something like
“minus 200 from both sides”.
The teacher should stimulate the students‟ thinking by asking, like
“what do you think you can do?” or “if it is not a ½ do you think you can solve the problem? Why do not you do the same?” or “what do you think you can relate from the equation?”
Some students might think to remove the half part of C from both sides. Thus, they will arrive into the result:
“ ”. So, “ ”
The teacher can ask the students to clarify their steps.
He should emphasize the steps that maintain the equality of the expression, for example, the change from to .
Other students might see the relation between a half and a full unit. So, they can come up with:
which means that and
4.6. Lesson 5: Manipulating Balance
52
Universitas Sriwijaya we still have no idea whether the students can recognize a linear equation problem given in a non-weighing context. Therefore, the purposes of the present lesson are to:
1. Build students‟ flexibility in manipulating quantitative relationships
2. Introduce the students to linear equation problems in a non-weighing context
Activities, conjectures of students’ thoughts, and suggestions for teachers
During the fifth meeting, the students will be given two problems, namely, combining balances and buying fruits. Classroom discussions in this meeting would be focused more on the mathematical aspects rather than the context. This would guarantee the applicability of the students‟ knowledge in a more general situation.
Combining masses
The first problem that the students would solve is „combining balance‟. In this problem, the students would be asked to arrange different masses on a balance scale to weigh mung-beans (700 gram). The provided masses weigh: 1 gram, 2 gram (2 pieces), 10 gram, 20 gram, 50 gram, 100 gram, 200 gram (2 pieces), and 1 kg (normally, a 500-gram mass also exists, but in the story we said that it was gone).
Figure 4.10 Masses and a balance scale given in „combining masses‟
This activity perhaps builds students‟ flexibility in relating quantities. This also might give the students insights of how to find unknowns without solving it; or also called acceptance of lack of closure. Students‟ possible thoughts when working with this activity are given in table 4.10.
Table 4.10 Overview of hypothetical learning process in combining masses Conjectures of students’ thoughts Suggestions for teachers Some students might put the mung beans (700
gram) on one side alone, and then try to combine the existing masses to get the 700.
(They may think that they should not involve the 1kg mass, because it weighs more than 700 gram).
These students will end up with combining all the other masses, but the 1kg, on one left of the balance scale, and found the following:
This might lead them into thinking that, „even if we combine all masses, but 1 kg, the weight does not reach 700 gram‟. So, it might be impossible to find to make 700 gram.
The teacher opens a discussion by asking whether it is really impossible to weigh 700 gram at a time, using the provided masses.
Student‟s doubt answer should be utilized to bridge them think.
Teacher might ask question like
“why do we only manipulate one arm?” or “can‟t we put another mass together with the 700?”
If none of them appears with the idea of manipulating both arms, the teacher can ask simpler problem to think of, like, “how do you think we can make 30gram from 20 and 50 gram masses?”
The students will find the idea of manipulating both sides and come up with the result:
OR
Teacher asks the students to explain their strategy.
It is also possible that the students still draw the masses and the mung beans in their answer. However, it is suggested to focus discussion more on the mathematical aspect rather than the representation in this phase.
54
Universitas Sriwijaya Buying Fruits
The context raised in this problem would relate weights of certain fruits with their prices. This task aims to facilitate the students‟ movement from weighing contexts. Although, they can still find the word „weight‟ in the question, the meaning of variables appeared in the representations would tell „the price‟ instead of „the weight‟ of each kg fruits. The meaning of variables should be emphasized in classroom discussion. Students‟ possible performances when dealing with this activity are summarized in the table 4.11.
Table 4.11 Overview of hypothetical learning process in „buying fruits‟
Conjectures of students’ thoughts Suggestions for teachers The students might have no problem in
translating “the price of 1 kg of grapes, and 5 kg of duku is Rp.64.000,- into mathematical words”.
They probably translate into:
The teacher might give a simpler illustration by asking, “if the price of 1 kg duku is Rp. 5.000,-, how much do you think the price of 1 kg grape is?”, or “how may you interpret the question”, or “what information have you got from the story in the question?”.
However, some students might have difficulty in understanding, “the price of 1 kg grape is three times as much as the price of 1 kg duku”.
The students might get the idea of the relation between the price of the duku and the grape, that is:
For the next step, some students might do guess-and-check.
Here, they first guess the price of the duku, and then find the price of the grape using relation (2).
Afterward, they may check the result by substituting the prices into relation (1).
The teacher might let these students go with their ideas. However, later in the discussion, they might discuss and compare this strategy with other strategies.
Other students might substitute the equation Discussion about the most efficient
Conjectures of students’ thoughts Suggestions for teachers (2) into equation (1), like what they have done
in lesson 2 and 3.
Here, the result would be:
And the price of 1 kg G would be Rp.24.000,-.
and effective strategy should be conducted.
“efficiency implies the fastest way to the solution, while the effectiveness implies the correctness of the answer”.
4.7. Lesson 6: Balancing across contexts