Lesson 2: Finding balance Starting points and learning goals

developed. Afterward, they demonstrate their trick, and let other students participate.

During this activity, the students may come up with ideas, as given in table 4.2.

Table 4.2 Overview of hypothetical learning process in Secret Number (part 2) Prediction of students’ thinking Suggestions for teachers Students might make an arrow

representation to formulate their trick. Here, they may use symbols to control over the sequences.

Teacher asks the students to try out their secret numbers. Afterward, he/ she might challenge the other groups to find out the trick.

Teacher might ask the students to write their trick formula on the blackboard and let other groups comment on their trick. Teacher might rise question like, “are you sure that your trick would be applicable for any numbers people might choose?” “Can you explain, why?”

The students might come up with answer that represents the general applicability of the symbol, like,

“because this actually represents whatever number you want”.

Teacher highlights the keyword, and asks students to make conclusion of what they have learned.

With the two main activity presented in lesson 1, we expect that the students can build up their understanding of the uses of symbols, and basic algebraic expression. It might be that the students have successfully produced an equation, here. But, the teacher should not force the students for it. In this phase, some students might have used letters to represent the generalized number given in their trick formula, however, it is possible that other students still neglect to use it, and are more convenient to make a drawing. For this phase, the teacher may let the students use their own representations.

4.3. Lesson 2: Finding balance

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Universitas Sriwijaya equation. In this phase, the students would do balancing activities with a real balance scale. Thus, the students‟ experiences with the balance scale contribute to students‟

prior knowledge for this lesson. The activities given in the second lesson aim to:

1. Introduce the rules of balancing to students

2. Introduce the use of letters (or symbols) to represent objects 3. Build the students‟ understanding of equivalent equations

Activities, conjectures of students’ thoughts, and suggestions for teachers

Activities in lesson 2 were all related to balancing activities. There were two main activities that the students should do in this meeting, namely, bartering marbles (part 1), and bartering marbles (part 2). In those activities, the students were first introduced to a problem in context, where two children with their collections of three-different-sized marbles. The problem would lead the students to the idea of bartering by considering the weight of the marbles.

Bartering marbles (part 1)

In the first part, the students would compare the weight of 3-different-sized marbles in a real balance scale. The task would be to make as many balanced conditions as possible on a balance scale, and then record it in their worksheets. The task was set to perform in group. Some conjectures have made, as can be observed in table 4.3. These cover the students‟ strategies to make the balanced conditions and the way they represented the balanced conditions.

Table 4.3 Overview of hypothetical learning process in the bartering marbles (part 1) Conjectures of students’ thoughts Suggestions for teachers The strategy the students might use to find

the balance conditions:

 Some students might do arbitrary trials.

They might combine two kinds of marbles first, and then estimate how those might relate. They would do combination of others by considering the result of the previous trial.

 Teacher suggests the students to record every trial they have made, to prevent them repeat measuring.

After a certain time, the teacher might ask the progress of the group, and encourage them to be more flexible in combining.

Conjectures of students’ thoughts Suggestions for teachers

 Other students might combine the same objects on one arm, and combine the other two objects on the other arm.

 Teacher might ask the students to explain their strategy to make the balance, like, “what did you do to get into the balance” OR “what will you do if the scale tends to the right?

The way the students represent the balanced conditions might vary, such as:

 Ordinary listing

 Making tables

 Draw the balance and the marbles

 Write an equation of the balanced conditions

(Some students may not use the plus sign

„+‟. Instead, they use the word „and‟ to state the combination of objects)

(It is also possible that the students do not use the equal sign yet „=‟, but use word „is balance‟, or „has equal weight‟, or

„equals‟)

 Teacher should let the students with these different representations show and explain their representations.

 The teacher can offer the students to comments on the benefits of each other‟s representations, and then ask the students‟ preferences. The teacher might address question like: “if you are asked to do the same tasks, which representation do you think will you use? Why?”.

 This moment should also be used by the teacher to discuss whether the plus and the equal signs are appropriate to use in this case.

By the end of this activity, the teacher would provide the students with a poster hang on the blackboard, where students can write any balanced conditions they have made. This poster will be used in the next activity.

Bartering marbles (part 2)

In this part, the students would be asked to predict more balances, given the same conditions as in part 1. Here, the students would not be allowed to use balance scales anymore, and would not be restricted by the number of marbles they have. Thus, they could only rely on the list of balanced conditions they have made in the previous

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Universitas Sriwijaya activity. Table 4.4 gives predictions of what the students might come up with when performing this task.

Table 4.4 Overview of hypothetical learning process in the bartering marbles (part 2) Conjectures of students’ thoughts Suggestions for teachers

It might not be really easy for students to think of any new relations. They may keep trying to combine the existing marbles, and can probably reveal a new relation.

 Teacher encourages students to keep imagining the balance. (If necessary) he might give clue to combine the existing balanced conditions.

He can ask, “what do you think make them balanced?” or “what does it mean if (or taking another example from the known balance)

 The keywords are „imagine‟, „relate‟, and „think of the balance‟.

The students might look at relationship between the balance conditions they have found, like and , where they can see that the latter is double the former one, and try to generalize this into “finding new balanced combinations by multiplying or dividing the quantities in each balance with the same amount”

 This reveals the equivalence under the multiplication and division. Teacher must note this notion and highlight it in the discussion and conclusion part.

The students might think to add the same amounts to both sides from a balanced condition, like:

into

 This will lead the students to the equivalence under the addition and subtraction.

The students might think to exchange certain marbles with other combinations as they realized that they have the same weight.

For instance, from into

 Teacher must note and emphasize the point of understanding the consequence of the balance. The idea will lead the students to substituting with the same quantity

It is important that the teacher highlighted each idea, and invited students to formulate/ summarize any actions that maintained the balanced situation on a balance

scale. The teacher may also ask students to compare the use of letters in the current meeting (representation of objects) with that in the previous meeting (representation of numbers).

4.4. Lesson 3: Finding unknowns