Activity 4: Playing Card Games

80 In the end of the third lesson, there was no student who was able to really say the relationship. They knew that those two equations involved the same number and each number had their own meaning. However, they were still struggling to see the relationship.

81 Figure 28. Students were playing with card games

Students were reading the cards one by one. They highlighted at the numbers used in a card and tried to make a group of cards having different colors which had the same number. In less than 5 minutes, there were some groups of students who had matched the card correctly. The focus group was the runner-up of the game.

After playing with the game, the teacher guided students to look at the two equations using multiplication and division which were on the red and blue cards.

She made another table for multiplication and division operation on the whiteboard and asked students to mention the equations so she could write them down in the table.

Figure 29. Discussing the relation between multiplication and division

82 Like what she did in some last minutes of the third meeting, she prompted students to look carefully at the two equations in multiplication and division. She asked the pattern of the numbers used in those two equations and guided students to give meaning for the three numbers. In those two equations, the three numbers were for the total length of ribbon, the number of parts, and the length of each part made.

After having discussion with the whole students, there were some students, not from the focus group, trying to give comments on the equations. They looked at the first pair of multiplication and division equations written in the table,

.

With the guide of the teacher, they gave meanings of each number in the two equations. They said that the 5 in the equation is the number of small ribbons,

½ is the length of each ribbon, and the 2 ½ is the total length of ribbon. From the division equation, they mentioned the dividend, the divisor, and the result. Finally, they concluded that if the result of the division is multiplied with the divisor, the result is the dividend.

During the discussion, most students were only listening at the discussion.

Some of them gave response to some small questions asked by the teacher, for example, “is there any similarity of the multiplication and the division equations?”, “are there any new numbers which are not used in the multiplication equation which are used in the division equation?” Most students responded that there were something “similar” from the two equations and they knew that the two

83 equations had three same numbers. They knew that the order of the numbers used in the two equations was different but they couldn’t say out loud their thinking.

When leading the discussion, the teacher gave an illustration how the multiplication and the division work in the whole numbers. She gave a division equation and she put meanings in each numbers used in the equation.

Figure 30. Looking how the division and multiplication in whole numbers

In the whiteboard, the teacher wrote a division equation, . She gave meaning that if 10 is the total length of a ribbon (in meter) and 2 is the length of each part of ribbon which is made from making some partitions of the ribbon, then there will be 5 partitions of the ribbon.

There was a piece of paper given to students where they could write down what they knew relating to the relation of the multiplication and division equations. They solved it individually. We call the paper as a reflection sheet, where they could reflect or recall anything they know relating to the topic which had been learned. They could give an example of problems and the strategy to

84 solve it, and also the place to write down the relation between multiplication and division equations written in the table in the whiteboard.

The four students in the focus group had similar answer. In their reflection sheet, they only gave some examples of multiplication and division statements which involve the same number. For instance:

i. For multiplication: I multiply 5 number of ribbons with ½ and the result is 2 ½

For division: I divide 2 ½ by ½ and the result is 5 number of ribbons ii. For multiplication: I multiply 13 number of ribbons with ¼ and the

result is 3 ¼

For division: I divide 3 ¼ by ¼ and the result is 13 number of ribbons

Figure 31. The conclusion made by the students in the focus group

85 Students in the focus group only gave examples of statements relating to multiplication and division involving fractions and whole numbers. They knew that both the multiplication and the division used the same numbers and their own functions. However, they couldn’t make the real mathematical equation for the statement.

There are some interesting answers from other students from other groups.

Rafly could make a general conclusion in his reflection sheet.

Figure 32. Rafly’s conclusion about the inverse relation

The translation and some interpretation of the answer sheet which Rafly wrote:

“The conclusion is that if the length of a ribbon is divided by the length of parts having the same length, the result is the result of partitioning. Then, the result of partitioning is multiplied with the length of parts having the same length, the result is the length of a ribbon.

86 Dividing a dividend with a divisor gets the quotient. Multiplying the quotient with the divisor gets the dividend.”

Looking at the conclusion which Rafly had made, he didn’t use some small examples to show the relationship. He was able to see the inverse relation between the multiplication and the division and he was also able to make a good generalization of the inverse relation.