Lesson 3: Finding unknowns Starting points and learning goals

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

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Universitas Sriwijaya Table 4.5 Overview of hypothetical learning process in the bartering marbles (part 3)

Conjectures of students’ thoughts Suggestions for teachers The students might first have in mind

that the weight of the other two medium and the big marbles must be more than 3 gram.

 Teacher encourages students to write that information, so that, they can reflect their answers with that information.

 The teacher might encourage students to use the list of balanced conditions (equations) they have made in the previous meeting

Students, who neglect to use algebraic expressions in the previous meeting, might have difficulty.

Other students will use the lists of equations of the balances (from the previous meeting), they may select certain equations and change (substitute) S (symbols for the small marbles) with

„3‟.

Some selections may be helpful, and some others may not.

 Encourage the students to try to use work with algebraic expressions (or symbols).

 Teacher encourages the students to make a new equation, after they substituted the S. This will ease them think to find the solution, since the unknown things in the expression became less.

(This step is important to promote the use of variables to represent quantities).

 Discussion about the students‟ selection of equations should also be conducted.

So, the students can recognize which representations are more helpful to find the solutions.

Students who fail to make a good selection probably will do guess-and-check. Here, they will substitute „S‟ with

„3‟, and then guess arbitrary values for

„M‟ and „B‟. Afterward, they check the result.

 Ask the students to rethink of their selections, like: “why do you choose this relation?”, or “why do not you choose another relation?”, or “what do you think make you hard to find the solution?”.

Big probability that the students will start to find the „M‟ instead of the „B‟ due to direct relations that „M‟ and „S‟ have.

Here, they may do the following:

Afterward, they may do the same thing to find the B from M, or from M and S.

 In the discussion, the teacher should encourage students to show (write) these steps completely which show how changes happen.

 The words “we do the same to both sides” or “to maintain the balance, once

Conjectures of students’ thoughts Suggestions for teachers

we add or remove something to certain arm, then we have to do the same with the other arm” is important to

constantly say during the process.

In the discussion, the teacher might also raise issue of treating „2 ‟ as „ ‟.

This activity would help students to find the unknown by identifying the relationship among quantities. Systematic substitutions would become the modest strategy promoted in this activity.

Weighing beans

This activity would start with a story about efforts to find the weight of two bags of beans with a two-armed balance scale and combination of 50g masses. Here, the students would not do the real balancing; they would just see how people in the story struggle to find the balance. The final balance condition is given in figure 4.5. The task for students was to find out the weight of one bag of beans.

Figure 4.5 Balance conditions in the weighing beans problems

Some conjectures about the students‟ thinking when solving this problem are presented in table 4.6. Here, we focus on how the students might represent a balanced condition. There was no explicit clue about how the students should make their representations, which was intended to see the students‟ development in representing situations on a balance scale.

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Universitas Sriwijaya Table 4.6 Overview of hypothetical learning process in the weighing beans activity

Conjectures of students’ thoughts Suggestions for teachers The students will recognize the efforts

that Iman (subject in the story) used to make a balance.

The teacher might show pictures of each step and asks some students to interpret the pictures.

The students might look at the balance conditions, and start to move from certain things to find a new balance.

Unfortunately, they will find unfriendly situation that does not allow them perform so:

Some might still do removing, but find a wrong answer, like following:

Removing similar things is also an idea of equivalence that the teacher should emphasize in the discussion.

Regarding some students‟ decisions to keep removing things, the teacher might bring it into classroom discussion. The teacher might ask, “do you think it is true? can you explain?” or “why do you remove one bag of beans and one 50g mass?” or “do you think the result will still be balance?”

Another option is that the teacher can ask to students to reflect on their conclusion. The teacher can ask, “so how weigh do you think is one bag of beans? Is it 50g, as you removed, or 100g as it stays on the scale?”.

Further, the teacher might suggest to change the object with numbers or work with equations, by asking “how many gram are the masses in total?” or can explicitly offer “I think we have to find another way”.

The students replace the masses with numbers. They may also do the same for the beans; they might change it into variables or just leave them as drawings.

The teacher encourages both ways;

drawings and equations. In the discussion, he should relate those two representations.

Some students might be able to find the answer with this representation, but some others might have difficulty, especially if they do not really think of

To help the students who find difficulty to solve the expression, the teacher might ask them to interpret the picture or the representation. The teacher can

Conjectures of students’ thoughts Suggestions for teachers

the interpretation of the representation ask, “can you say it in words what did you understand from the picture?”

The students might say “the weight of 2 bags of beans is 150 gram”.

This will help them think to find the weight of 1 bag of beans.

The teacher can ask, “so, what is the weight of only 1 bag of bean?”

Activities in the third meeting might become the students‟ first experiences in solving for unknowns in a linear equation problem with one variable. The rules to maintain the balance will be really employed here, either to find the answer or to reflect the strategies that the students take.

4.5. Lesson 4: Formative Evaluation