Hypothetical Learning Trajectories In Mathematics Education

“Seeing from Students’ Point of Views”

By

Wahid Yunianto

SEAMEO QITEP In Mathematics

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Contents1. Introduction...................................................................................................................................3

2. HLT Construct................................................................................................................................6

3. Examples of HLT in RME research..................................................................................................8

Meeting 1: Javanese Batik Gallery (Line Symmetry)..........................................................................8

Meeting 2: Rotating the pattern......................................................................................................19

Meeting 3: Making it symmetrical..................................................................................................27

Table 4.1..........................................................................................................................................34

Table 4.3..........................................................................................................................................36

Table 4.3..........................................................................................................................................37

Table 4.5..........................................................................................................................................40

Table 4.5..........................................................................................................................................41

Table 4.6..........................................................................................................................................44

References...........................................................................................................................................46

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1. IntroductionMost teachers including mathematics teachers think that they know what is going on in their

classrooms. Yes, they know what is happening in the classroom since they have experienced

the same situation many times. The teaching and learning processes are often repeated with

no or little changes. For instance in the teaching and learning of area measurement, the

teacher tells students area formulas of geometrical figures and gives some examples.

Afterward, students should memorize the formulas and solve some problems. To check

students’ understanding, the teacher asks students certain questions such as, what the area

formula of a rhombus and a rhombus with given diagonals are. Students answer in choir and

the teacher agrees then continue to the next lesson. Whether students really understand or

just remember that formula at that time., it is still questionable.

Mathematics teachers mostly give routine tasks and one single way and solution. Routine

tasks seems not potentially let students think to reason and justify strategies hired to solve

the tasks. In Indonesia, possibly, in many Southeast Asia countries, teachers often choose

routine problems. Teachers tend to deliver the routine problems by directly giving the

algorithms or formulas, showing examples, and then practicing the problems. Through this

way, students do not have opportunities to construct mathematical concepts. Research shows

that once forgetting formulas, students would face difficulties solving problems. View of

constructivism believed that mathematics should capitalize on the inventions by the students

(Gravemeijer, 2004). It raises a question what kind of task or activities which lets students

(re)invent and develop mathematical concepts. When teachers develop and design a new

task, will they know what is going to happen in the classroom?

Let’s elaborate the two question into how teachers select or develop tasks which help

students see the world though mathematics. Professional teachers will likely be innovating

their teaching and learning. The innovations are aimed at helping students to better

understand mathematics. Teachers may develop tasks or activities, tools, media as their

innovations. Thus, teachers need to deal with the tasks that will be used in their teaching and

learning. Teachers should consider the appropriate tasks to develop certain mathematical

concepts. Teachers may use/adjust the existing tasks developed by experts or researchers or

develop the tasks by themselves. Even though teachers hire the existing tasks, they need to

conjecture how the tasks would help students learn mathematics (meaningfully). It differs

from transmitting knowledge where students receive knowledge from the teachers. Now,

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students should develop their knowledge and mathematize their world (reinventing

mathematics).

Teachers can influence their students’ inventing activity only in a more indirect manner. To

do so, teachers will have to put themselves in the shoes of the students (Gravemeijer, 2004).

The challenge for the teacher and also for us is to try to see the world through the eyes of the

student. How much these worlds may differ may be illustrated by pictures of Watson’s strip

about Calvin and Hobbes. Figure shows the world of Calvin and his tiger friend Hobbes seen

through his eyes, and figure shows how we see Calvin and his tiger doll.

Figure 1. What Calvin sees when playing with Hobbes

Figure 2.What we see when Calvin plays with Hobbes

As a teacher, we should position ourselves in students’ situation. What will we do (we

pretend as a certain age student) if we face the designed task (instructional activity). As

teacher, you might easily deal with the task and successfully find the answer. However, you

might only have one way to solve it, maybe in a formal way. For instance, in dealing with

the following task:

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There are four groups of students, A, B, C, and D taking a field trip. Group A, B, C, and D

go to Museum, Ellies Island, Statue of Liberty, and Planetarium respectively. The

Committee only provides 17 sandwiches for lunch.

(taken from http://www.clipartpanda.com/categories/sandwich-20clipart)

This is how the sandwiches are distributed to each group.

To which group are you in? Why do you choose that group?

It is not difficult for an adult or you as a teacher to find the solution. What will you do is directly divide the number of sandwiches by the number of students.

Figure3 . Direct calculation performed by adults

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Adapted from: Young Mathematicians At Work Constructing Fractions, Decimals, And Percentsby Catherine TwomeyFosnot Maarten Dolk, 2002

8students go to statue of Liberty

5students go to Planetarium

5students go to Ellis island

4students go to Museum

Most teachers solved the problem by directly find the result of division of whole numbers. Then they compared the decimal numbers. Then they will find difficulties explaining why it is more than others. They will argue that the number is larger than other numbers and stop. To some extends, it is not easy to conjecture what students might response. Even, the teacher has many year experiences of teaching, if he/she only delivered routine tasks then he predicted only students could solve it in one uniform strategy.

Figure 4.One of students’ possible answers

Gravemeijer (2004) states that to be able to plan instructional activities that may foster

certain student inventions, the teacher has to take an actor’s point of view, and to try to

anticipate what students might do. In this manner, the teacher can plan instructional activities

that may foster the mental activities of the students, and which fit his or her pedagogical

agenda. This is the idea of Simon (1995) about hypothetical learning trajectory (HLT) where

teachers have to envision how the thinking and learning would be when students participate

in the tasks relating to its learning goals.

2. HLT ConstructSimon (1995) conceptualized hypothetical learning trajectory with three components as follow:

Learning goalIt defines the direction. Where do you want to bring your students to more sophisticated knowledge from their currents?

The learning activities.Students will engage on the design activities which have potential support or means to help students construct new knowledge and (re)invent mathematics.

The hypothetical learning processIt consists of conjectures of possible students reactions toward the given activities and also teachers’ responses to students’ reactions.

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He speaks of a mathematical teaching cycle; the HLT is a big part of the cycle. The creation and ongoing modification of the hypothetical learning trajectory is depicted on figure xxx. The initial HLT is not always perfect, by using experiences, experiments, and studying the teaching and learning of mathematics educations, and deepening the mathematics concepts can refine and improve the HLT. Teachers at first set the learning goals and use or develop the activities to achieve the goals. Therefore, the decision of choosing the activities is crucial. The teacher has the dual role of fostering the development of conceptual knowledge among her or his students and of facilitating the constitution of shared knowledge in the classroom community (Simon, 1995). He explained, the notion of a hypothetical learning trajectory is not meant to suggest that the teacher always pursues one goal at a time or that only one trajectory is considered. Rather, it is meant to underscore the importance of having a goal and rationale for teaching decisions and the hypothetical nature of such thinking. The development of a hypothetical learning process and the development of the learning activities have a symbiotic relationship; the generation of ideas for learning activities is dependent on the teacher's hypotheses about the development of students' thinking and learning; further generation of hypotheses of student conceptual development depends on nature of anticipated activities

Figure 6. Teaching cycle (Simon, 1995)

Teacher may set the learning trajectory to achieve the learning goals. The learning trajectory here is still hypothetical, since we are not sure how students deal with the given problems or how they construct new knowledge (Fostnot & Dolk, 2002). Thus, teachers’ expectation may not occur as expected or unpredictable situations happened without anticipatory actions. Since, students differ in their ways of thinking, especially when the tasks are too open. Learning trajectory seems to be linear, but real learning is messy (Duckworth, 1987 in Fosnot & Dolk, 2002). Then teachers may move in some paths in order arrive the learning goals. Therefore, Fosnot & Dolk called this as landscape of learning.

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Figure7.Linear framework (left) and hypothetical learning trajectory (right) (Fosnot&Dolk, 2000)

The choice of the word" trajectory"is meant to refer to a path, the nature of which can perhaps be clarified by the following analogy. Consider that you have decided to sail around the world in order to visit places that you have never seen. One does not do this randomly (e.g., go to France, then Hawaii, then England), but neither is there one set itinerary to follow. Rather, you acquire as much knowledge relevant to planning your journey as possible. You then make a plan. You may initially plan the whole trip or only part of it. You set out sailing according to your plan. However, you must constantly adjust because of the conditions that you encounter. You continue to acquire knowledge about sailing, about the currentconditions, and about the areas that you wish to visit. You change your plans with respect to the order of your destinations. You modify the length and nature of your visits as a result of interactions with people along the way. You add destinations that prior to your trip were unknown to you. The path that you travel is your "trajectory. "The path that you anticipate at any point in time is your "hypothetical trajectory (Simon, 136-37)

3. Examples of HLT in RME researchThe following is the learning sequence in teaching and learning addition of fraction using RME design. The hypothetical learning trajectory of each lesson / meeting is developed and elaborated based on the research conducted in an elementary school in Indonesia. The topic for the following HLT is about symmetry. It makes use of batik pattern, since it embeds symmetrical and non-symmetrical patterns. There are three meetings in this lesson with two or three activities in each meeting.

Meeting 1: Javanese Batik Gallery (Line Symmetry)A. The starting point

The starting point of this first activity will be based on the students’ prior knowledge of

symmetry which already taught in grade 4, but it is particularly from their written works in

the pre-test. In the grade 4, the students already learned the following knowledge and skills of

symmetry,

The students are able to identify the symmetrical objects in daily life

The students are able to determine the symmetrical two-dimensional shapes

The students define line symmetry as a line that determines whether the objects are

symmetric.

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B. The mathematical learning goal

The first activity aims at making the students to be able to understand the notion of line

symmetry by exploring the characteristic of batik patterns. The goal can be elaborated into

these following sub-learning goals,

The students are able to differentiate the patterns which have regularity (line symmetry)

and the patterns that have no regularity.

The students are able to deduce the characteristics of line symmetry from the regular

batik patterns by using a mirror.

The students are able to differentiate between diagonal and line symmetry on the batik

patterns.

In order to achieve the learning goal, the researcher presents table 1 for giving an overview of

the main activity and the hypotheses of learning process and the details information in the

following section.

Main Activity

Table 1. An overview of the activities in meeting 1 and the hypotheses of learning process

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C. The instructional activities

a) Introducing the context of Javanese batik gallery

This first activity uses the context of Javanese batik gallery in which the gallery will held an

exhibition. As the galleryhas only two rooms, the staffs need to sort the Batik patterns into

two types based on their regularity. The teacher should make sure that all students understand

the problem and exactly know what they should do. It can be done by asking several students

to paraphrase the problem and asking the other students whether they agree with the

statement. The example of questions,

Could you explain the problem in your own words?

Do you agree with your friend’s statement? Why do you think so?

b) Doing the worksheet

After discussing what the context is about, the students will get oriented to do the worksheet

in the group which consists of three to four students.

c) Classroom discussion

Several pairs of students who have different answers in solving the problem on the worksheet

will have an opportunity to present their answers. Then, the other students will have a chance

to give comments or state their opinions whether they agree or disagree with the presentation.

The teacher will lead the discussion so that all the groups have a chance to state their answers

and keep the discussion focusing on the problem. In the end of the discussion, the teacher

reviews the answers of the regular patterns, the definition of line symmetry and the difference

between diagonal and the axes of symmetry. There are two important points of this lesson.

First point is about the notion of line symmetry which is represented by the regularity of the

motif that consists of repeating unit patterns. Second point is about the difference between the

diagonal and the axes of symmetry.

d) Closing activity

The teacher asks the students to reflect the lesson such as by asking these following questions

What does line symmetry mean?

What are the differences between the diagonal and the axes of symmetry?

D. The conjectures of students’ thinking and learning

The conjectures of students’ thinking and learning will be described based on the three tasks

on the worksheet.

1) The first task

This task asks the students to fill the table (figure 8) with their sorting result.

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Figure 8.The figure of table to fill the sorting result

In line with table 4.1, the following are the possibilities of students’ sorting result.

The students sort the patterns based on the colour

Room 1 : A, B, C, F, H, I, L

Room 2 : D, E, G, J, K

The students might answer that they sort the patterns by looking up the colour and they see

that most of the patterns are brown, so that they sort the patterns by differentiating brown

patterns and non-brown patterns.

The students sort the patterns based on the similarity in motif (living creature motif)

Room 1 : B, C, F, K

Room 2 : A, D, E, G, H, I, J, L

The students might answer that they sort the patterns by looking up the motif of the patterns.

They see that there are several patterns which have motif of flowers or animals.

The students sort the patterns based on the way of designing the motif

Room 1 : A, D, G, H, I, J, L

Room 2 : B, C, E, F, K

The students might answer that they sort the patterns by looking up the way of designing the

motif of the patterns. They see that there are several patterns which need to be equal in size

among each other. Hence, it needs lines to draw as the following figure,

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Then, the following figure is the pattern that cannot directly be drawn without making a line.

Therefore, they will answer that they sort the patterns based on the way of designing the

motif.

The students sort the patterns based on the regularity of the motif whether they consist of

repeating patterns

Room 1 : Batik A, D, G, H, I, J, L

Room 2 : Batik B, C, E, F, K

The students might answer that they sort the patterns by observing the motif of the patterns.

They see that there are several patterns which consist of repeating patterns and other patterns

that are unique. So, the whole regular pattern is created from the same unit Batik pattern.

It is also possible that students just divide the figures into two piles without any reason.

Therefore, the teacher should give some supports such as: take a look closely what makes it

special, what do you see from those figures?

2) The second task

This task asks the students to explore the regular batik patterns by using a mirror in order to

discover the notion of line symmetry. These are the possibilities of what students will do in

using a mirror and defining line symmetry.

The students put the mirror in the edge of the batik patterns

Then, the students acknowledge that the reflection on the mirror is the same with the initial pattern.

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mirror

mirror

mirror

mirror

mirror

mirror

mirror

mirror

mirror

The students put the mirror in the middle of the patterns

Then, the students acknowledge that the mirror reflects the same pattern as the pattern which

is covered

3) The third task

This task asks the students to determine the diagonals and the axes of symmetry of the batik

patterns. These are the possibilities of students’ answers.

The students draw the axes of symmetry and diagonals of the pattern correctly

Pattern A Pattern B

Pattern C Pattern D

The students perceive that the diagonal always become the axes of

symmetry of the pattern or vice versa.

Example:

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mirror

The axes of symmetry

Diagonal The axes of symmetry

Diagonal

No axes of symmetry

Diagonal

The axes of symmetry

Diagonal

DiagonalThe axes of symmetry

DiagonalThe axes of symmetry

mirror

mirror

mirror

mirror

mirror

mirror

mirror

mirror

The teacher’s reaction

By considering the possibilities of students’ answer, the teacher can do these follow-up

actions.

1) The first task

The students sort the patterns based on the colour

The teacher shows the Batik patterns which are printed in black and white and asks them to

sort the patterns. It aims at making the students realize that their way of sorting by looking up

the colours is not general enough. The students should observe the motif of the patterns

instead of their colour.

The students sort the patterns based on the similarity in the motif (living creatures motif)

The teacher takes one pattern with motif of flower like the following figure. Then, the teacher

asks the students to observe the motif more thoroughly. Then, the teacher can ask a follow-up

question

“Imagine how you will draw the patterns, do you find any

same motif inside the pattern?” Teacher guides the

students to notice that the patterns consist of repeating unit

patterns (regular).

The students sort the patterns based on the way of designing the

motif

The teacher asks the students to do further exploration to the patterns such as by asking “what

do you mean by making line, where will you draw the lines?”. The teacher also can ask the

students to draw the line of each pattern that they assume as the patterns that need lines to

draw. Then, the teacher can guide the students to notice the basic notion of line symmetry for

some particular lines. For example,

The students sort the patterns based on the motif whether they consist of repeating patterns

The teacher asks the students about the regularity that the students mean,

“Why do you determine batik A as the pattern which have regularity?”“What kind of regularity that you mean?”

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Guide the students to acknowledge that the regularity refers to the repeating pattern.

2) The second task

The students put the mirror in the edge of the regular batik pattern then they acknowledge

that the reflection on the mirror is the same with the initial pattern.

The teacher can ask the students to put the mirror in the edge of the batik patterns which nave

no regularity. Then, the teacher asks about what the students see in the mirror. The students

might answer that the reflection on the mirror is the same with the initial pattern. Then, the

teacher poses following question,

“If you think so, then there is no difference between Batik patterns which have regularity and

no regularity?”

“What do you think, do you need to re-position the mirror in order to differentiate the regular

Batik patterns and the patterns which have no regularity.”

The follow-up questions are intended to lead the students to position their mirror in the

centre of the pattern (vertically, horizontally and diagonally)

The students put the mirror in the middle of the patterns. The teacher can ask the students

to do further exploration with a mirror to the regular patterns and guide them to determine the

mirror positions of each pattern. It also can be followed up with questions such as

“Look at the mirror positions that you have already determined, what do you see from the

mirror position?“

“What do you usually name the mirror position?”

The follow-up questions are used to lead the students to relate the mirror position with the

axes of symmetry.

3) The third task

The students draw the axes of symmetry and diagonals of the pattern correctly

The teacher can give some following questions to make sure that the students understand the

difference between the axes of symmetry and diagonals

“So, what is the difference between the axes of symmetry and diagonals?”

“Do the axes of symmetry always become the diagonal of the shape?”

“Do the diagonals always become the axes of symmetry of the shape?”

The students perceive that the diagonal always become the axes of symmetry of the

pattern

The teacher can show batik pattern B or D and ask the students to draw the diagonal of each

pattern. Then, ask the students to observe whether the diagonal divide the pattern into two

same parts and reflecting each other. If the students still feel difficult in perceiving that the

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diagonal is not the axes of symmetry, then the teacher can use a mirror to make them realize

that the patterns are not reflecting each other. It is intended to make the students see and

realize that the diagonal of the pattern is not always its axes of symmetry.

Meeting 2: Rotating the patternA. The starting point

To start the second activity, the students should understand the notion of regularity which

represents the repeating unit patterns.

B. The mathematical learning goal

The second activity aims at making the students to be able to get the notion of rotational

symmetry by exploring the characteristic of batik patterns. The goal can be elaborated into

these following sub-learning goals,

The students are able to differentiate the patterns which have regularity (rotational

symmetry) and the patterns that have no regularity.

The students are able to deduce the characteristics of rotational symmetry from the regular

batik patterns by using a pin and the transparent batik cards.

The students are able to determine the characteristics of rotational symmetry (the order of

rotation, the angle of rotation and the point of rotation)

In order to achieve the learning goal, the researcher presents table 4.2 for giving an overview

of the main activity and the hypotheses of learning process and the details information in the

following section.

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Activity Mathematical Learning Goal

Conjectured of Students’ Strategies Teacher’s reaction

Sorting the Batik patterns into two types based on the regularity.

The students are able to differentiate the patterns which have regularity (rotational symmetry) and the patterns that have no regularity.

The expected strategy:

The students sort the patterns based on the regularity of the patterns in which whether the patterns consist of repeating motif and the patterns that are unique.

Example:Room 1 : Batik A, E, F, G, H, K, LRoom 2 : Batik B, C, D, I, J

Ask the students about the regularity that they mean,

“Why do you determine batik F as the pattern which have regularity?”

“What kind of regularity that you mean?”

Guide the students to acknowledge that the regularity refers to the repeating pattern.

Discovering the characteristics of rotational symmetry from the regular batik patterns by using a pin and transparent batik cards

The students are able to deduce the characteristics of rotational symmetry from the regular batik patterns by using a pin and the transparent batik cards.

The expected strategy:

The students put the transparent batik cards above the corresponding patterns and position the pin in the centre of the card. Then, they turn around the transparent batik card and count how many times the pattern fit into itself in one round angle (360o).

Example:

The teacher asks the students to observe the pattern and determine whether the pattern fit into itself as follows,

“What did happen to the pattern after you turn around?”

“How about the position of the initial pattern and after you turn it around, do they have the same position?”

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Determining the characteristics of rotational symmetry from the provided patterns

The students are able to determine the characteristics of rotational symmetry (the order of rotation, the angle of rotation and the point of rotation)

The expected answer:

The students can determine the characteristics of rotational symmetry properly such as

- the order of rotation depends on how many the pattern fit into itself in one round angle

- the angle of rotation can be determined by dividing 360o with the order of rotation

- the point of rotation is the center point of the pattern which can be determined from the intersection of the diagonals or the axes of symmetry

The teacher can guide the students to have a further investigation about rotational symmetry such as by asking

“Does the pattern which has rotational symmetry always have line symmetry?”

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Table 2.An overview of activities in meeting 2 and the hypothesis of the learning process

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C. The instructional activities

a) The initial activity

The teacher starts the lesson by showing the regular Batik patterns which already discussed in

the first meeting and asking the students about regularity in the pattern. It aims at reviewing

the students’ understanding of regularity so that it can help them to do the intended task. In

addition, the size of one full angle, right angle and other special angle should be reviewed. It

aims at supporting the students to do the worksheet and acknowledge the relation between

one full angle and the order of rotation in which it determines the angle of rotation.

b) Doing the worksheet

The students get oriented to do the worksheet in a group which consists of three to four

students. It is intended to make the students discuss and share their ideas so that they will

obtain more ideas and do the task easier than doing individually.

c) Classroom discussion

Several groups of students who have different answers and strategies will have an

opportunity to present their answers. Then, the other students will have a chance to give

comments or state their opinions whether they agree or disagree with the presentation. The

main point of the discussion is the characteristics of the regular Batik patterns which lead the

students to acknowledge the notion of rotational symmetry. Then, the students will have an

opportunity to define the meaning of rotational symmetry individually. After five minutes,

the students and the teacher discuss the meaning of rotational symmetry and its

characteristics.

d) Closing activity

The teacher asks the students to reflect the lesson such as by asking the following questions,

What do we have learned today?

Can you define rotational symmetry in your own words?

Besides, the teacher can give a Batik pattern and ask the students whether it has a rotational

symmetry.

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D. The conjectures of students’ thinking and learning

The conjectures of students’ thinking and learning will be described based on the three tasks

on the worksheet.

1) The first task

This task asks the students to fill the table with their sorting result.

In line with table 2, the following are the possibilities of students’ sorting result.

The students sort the batik patterns based on the colour, the motif or other constraints

instead of their regularity

Room 1 (blue batik patterns): B, C, G, H, J

Room 2 (brown and black batik patterns): A, D, E, F, I, K, L

The students might answer that they sort the patterns by looking up the colour and they see

that most of the patterns are blue and brown, so that they sort the patterns by differentiating

blue patterns and brown & black patterns.

The students sort the patterns based on the regularity of the patterns in which whether the

patterns consist of repeating motif or the pattern is unique.

Room 1 : Batik A, E, F, G, H, K, L

Room 2 : Batik B, C, D, I, J

The students might answer that they sort the patterns by looking up whether the pattern

consist of repeating patterns or an unique pattern. It might be happened because they already

experienced the similar activity in the first meeting.

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2) The second task

The students put the transparent batik cards above the corresponding patterns and position

the pin in the vertices of the cards and rotate it as follows

The students put the transparent batik cards above the corresponding patterns, position the

pin in the centre of the card, and rotate it. However, they rotate the transparent batik cards

for 360o in every rotation. As the result, they will perceive that the pattern will fit into itself

for many times.

The students put the transparent batik cards above the corresponding patterns

and position the pin in the centre of the card. Then, they turn around the transparent batik

card and count how many times the pattern fit into itself in one round angle

3) The third task

The students cannot determine the characteristics of rotational symmetry properly

- the order of rotation is the pattern fit into itself

- the angle of rotation is the degree of the rotation angle

The students have no idea in estimating the angle of rotation.

The students estimate the angle of rotation by seeing the movement of the initial pattern from

the first position until the pattern fits into itself.

- the point of rotation is the center point of the pattern

The students can determine the characteristics of rotational symmetry properly such as

- the order of rotation depends on how many the pattern fit into itself in one round angle

- the angle of rotation can be determined by dividing 360o with the order of rotation

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360o

- the point of rotation is the center point of the pattern which can be determined from the

intersection of the diagonals or the axes of symmetry

The teacher’s reaction

These are the description of teacher’s reaction toward the conjectures of what students do in

doing the given tasks

1) The first task

The students do not sort the batik patterns based on their regularity

If the students sort the patterns by their colour, then the teacher can ask the following

questions:

“How if the patterns are not printed in colour, how will you sort them?”

If the students already realized that their sorting strategy is not general enough, Then, the

teacher can suggest the students to observe the details of the motif. It leads the students to

notice about several patterns which consist of repeating patterns.

If the students sort the patterns based on the details of the motif or other constraints, then the

teacher can suggest them to review the regularity that they already discussed in the first

meeting. It is intended to make the students acknowledge that the regularity always refer to

the repeating patterns.

The students sort the patterns based on the regularity of the patterns in which whether the

patterns consist of repeating motif or the pattern is unique.

The teacher asks the students about the regularity that the students mean,

“In this task, what do you define regularity?”

This question is intended to know how the students sort the patterns based on their regularity.

The teacher also can give follow-up questions as follows,

“Instead of the repeating patterns, what do you notice from the regular pattern?”

The students might answer that the regular patterns have line symmetry. Then, the teacher

can refer to the pattern G and ask the following question,

“Look at pattern G, it consists of repeating pattern but does it have line symmetry?”

This question aims at guiding the students to do the next task which is discovering the notion

of rotational symmetry.

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2) The second task

The students put the pin in the vertices of the cards as follows

The students put the pin in the centre of the card, but they turn around the card for 360 o in

every rotation. As the result, they will perceive that the patterns will fit into themselves for

many times.

The teacher can tell the students that they should figure out whether the patterns fit into

themselves for degree of rotation less than 360o. The teacher can suggest the students to start

rotating the pattern from the red mark in top left of the pattern until the mark gets back to the

top left of the pattern.

“In this case, you are just allowed to turn the pattern

around for one rotation, then count how many times the

pattern fit into itself?”

The students put the pin in the centre of the cards as follows,

3) The third task

The students cannot determine the characteristics of rotational symmetry properly

The teacher can use pattern A as the example. Related to the order of rotation, the teacher can

ask the students how many times the pattern A fit into itself. Then, the teacher can refer the

students’ answer to the order of rotation

Related to the angle of rotation, the teacher can ask the students to count how many pattern A

fit into itself and review the size of one round angle (360o). It is intended to make the students

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The teacher asks a follow-up question as follows,

“What happened to the pattern after you turn it around?”

“Do you see any differences between initial pattern and the

patterns after you turn it around?”

“If I want to turn the pattern around and make the pattern

stays still in that position, where should I put the pin?”

The teacher asks follow up questions such as.

“Do you see any differences between the motif on the initial

pattern and the patterns after you turn it around?”

“How about the position of the initial pattern and after you

turn it around, do they have the same position?”

acknowledge that the angle of rotation can be determined by dividing 360o with the order of

rotation

“Doing the rotation is turning the pattern around under one full angle. Hence, if the pattern

can fit into itself for 4 times in one round, then what can you conclude?”

Related to the point of rotation, the teacher can ask the students to show how they determine

the center of the pattern F and how they can be so sure that it is the center of the pattern.

“Do you have any strategy to make sure that the point of rotation is located in the center of

the pattern?”

The questions aims at making the students draw the diagonals or the axes of symmetry of the

pattern.

The students can determine the characteristics of rotational symmetry properly

The teacher can ask them whether the patterns which have rotational symmetry always have

line symmetry. Then ask them to give some examples.

“Then, for which cases the centre of rotation is in the intersection of line symmetry or

diagonals?”

It aims at making the students aware that having rotational symmetry does not always mean

having line symmetry, for example: parallelogram

Meeting 3: Making it symmetricalA. The starting point

The students understand the concept of line symmetry in which the patterns are divided into

two parts and reflecting each other and rotational symmetry in which the pattern should be

rotated under the angle of rotation and fit into itself.

B. The mathematical learning goal

The third activity aims at making the students apply their understanding of line and rotational

symmetry. The goal can be elaborated into these following sub-learning goals,

The students are able to make the asymmetric batik pattern into the symmetric ones

The students are able to complete the pattern by considering the given axes of symmetry

The students are able to create their own batik design by using provided batik units

In order to achieve the learning goal, the researcher presents table 4.3 for giving an overview

of the main activity and the hypotheses of learning process and the details information in the

following section.

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Activity Mathematical Learning Goal

Conjectured of Students’ Strategies Teacher’s reaction

Restructuring asymmetric patterns into symmetrical patterns

The students are able to make asymmetric patterns into symmetric ones.

The students cut the pattern into several parts and paste them such that the patterns become symmetrical. For example,

the pattern are being rotated and repositioned

The teacher asks the students about other strategies in making the patterns become symmetrical.

Drawing the remaining pattern such that it becomes symmetric

The students are able to complete the pattern by considering the given axes of symmetry

The students complete the patterns by considering the unit pattern and the axes of symmetry as follows,

The teacher asks the students to explain their reasoning and heir strategy in completing the pattern.

Designing the symmetric pattern by using the provided batik units

The students are able to create their own batik design by using provided batik units

The students draw their own pattern by using the provided batik units and considering the given angle of rotation

The teacher asks the students to explain their strategies of creating the patterns and determine its symmetricalness

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Tabel3. An overview of activities in meeting 3 and the hypothesis of the learning process

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C. The instructional activities

a) The initial activity

In the beginning of the lesson, the definition and the characteristics of line and rotational

symmetry should be reviewed. It aims at supporting the students to apply their understanding

of both line and rotational symmetry in making symmetrical patterns.

b) Doing the worksheet

The students do the worksheet individually. It is intended to see how each student apply

his/her understanding of symmetry. While the students do the worksheet, the teacher will

supervise how the students do the given tasks.

c) Classroom discussion

The students will present their answers and strategies in creating symmetrical patterns.

During the discussion, the teacher should make sure that all students understand how to apply

the characteristics of line and rotational symmetry in creating symmetrical patterns. The most

important point is that the line symmetry always makes the pattern becomes two congruent

parts which reflect each other and rotational symmetry always makes the pattern fits into

itself.

d) Closing activity

In the end of the lesson, the students will reflect what they have learned from the activity.

The teacher can ask some following questions,

“What do you have to consider in making symmetrical patterns?”

“What is the role of line symmetry in creating symmetrical patterns?”

“How do you draw the angle of rotation?”

D. The conjecture of students’ thinking and learning

These are the possibilities of students’ answers and strategies in doing the worksheet.

1) The first task

The students cut the pattern into several parts and paste them such that the pattern

becomes symmetrical.

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these patterns are being rotated & repositionedthese patterns are being rotated & repositionedthese patterns are being rotated & repositionedthese patterns are being rotated & repositionedthese patterns are being rotated & repositionedthese patterns are being rotated & repositionedthese patterns are being rotated & repositionedthese patterns are being rotated & repositionedthese patterns are being rotated & repositioned

The students cut the pattern into several parts, but they paste them in the wrong way so that

the pattern does not become symmetrical. For example,

2) The second task

The students complete the pattern with the same motif without considering the axes of

symmetry as follows,

The students complete the pattern with the same motif and consider the axes of symmetry

such that the pattern becomes symmetrical as follows,

3) The third task

The students draw their own pattern by using the provided batik units and considering the

given angle of rotation

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these patterns are being rotated & repositionedthese patterns are being rotated & repositionedthese patterns are being rotated & repositionedthese patterns are being rotated & repositionedthese patterns are being rotated & repositionedthese patterns are being rotated & repositionedthese patterns are being rotated & repositionedthese patterns are being rotated & repositionedthese patterns are being rotated & repositioned

The students firstly draw the given angle in one full angle and its axes. Then, they imitate the

provided unit by considering the given angle.

The students draw their own pattern by using the provided batik units but without considering

the given angle of rotation

The teacher’s reaction

These are the description of follow-up action that the teacher can do toward the conjectures of

students’ answers and strategies,

1) The first task

The students cut the pattern into several parts and paste them such that the patterns become

symmetrical.

The teacher can ask the students about other strategies in making the patterns become

symmetrical and their reasoning in arranging the patterns. For example : “Do you think there is

any other way to arrange the patterns become symmetrical?”

“Why do you arrange the pattern in that way, what do you consider?”

The students cut the pattern into several parts, but they paste them in the wrong way so that

the pattern does not become symmetrical

The teacher suggests the students to observe the pattern more thoroughly and review the

definition of line symmetry in which the half pattern should reflect each other.

“What does line symmetry mean?”

“What do you have to consider the make the pattern has a line symmetry?”

It aims at encouraging the students to use the right concept of line symmetry in arranging the

patterns.

2) The second task

The students complete the pattern with the same motif without considering the given axes of

symmetry

The teacher asks the students to review the definition and the characteristics of line symmetry. It

aims at making the students realize that the half pattern should be similar and reflect each other.

“What is the axis of symmetry? In order to make symmetrical pattern, you should consider the

given axes of symmetry.”

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The students complete the pattern with the same motif and consider the given axes of

symmetry such that the pattern becomes symmetrical

The teacher asks the students to explain their reasoning and their procedure in completing the

pattern in that way.

3) The third task

The students draw their own pattern by using the provided batik units and considering the

given angle of rotation

The teacher will ask the students to explain their procedure in designing the Batik patterns and

determine the number of line symmetry, the order of rotation, the angle of rotation and the centre

rotation.

The students draw their own pattern by using the provided batik units but without considering

the given angle of rotation

The teacher asks the students to review the definition rotational symmetry and its characteristics.

The example of questions,

“What is the definition of rotational symmetry?”

“Now, look at your pattern more thoroughly and see whether the definitions hold

for your pattern”

“What is the angle of rotation?”

There is no single format to create HLT for your design. You may create it in a table or you may

describe it in paragraphs. The following is another HLT for teaching and learning of area

measurement. This topic is intended for grade 7 students in Indonesia. There are six meetings use

for this lesson. The lesson is to let students develop their understanding of the concepts related to

area measurement and about perimeter.

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Table 4.1.The overview of the activity and the hypothesis of the learning process in lesson 1

Activity Learning Goal Conjectured of Students’ thinking Guidance for teacherComparing leaves Students are able to

compare two leaves with their own strategies.Students grasp the idea that an irregular shape has an area.Students understand that area is region inside the boundary.

The students will try to compare the leaves by: Use their superficial judgmentjust by looking

at two leaves and say one leaf is larger than

the other because it looks bigger or wider or

the leaves are almost the same

Trace one leaf and place it over the other one

to see the non- overlapping areas

Overlap one leaf to the other one and cut

parts of non-overlapping area and paste them

to non- overlapping area of another leaf.

After the teacher distributes worksheet 1 to each group then ask students whether they understand the problem on the worksheet or not.“Could you tell me what the problem is?”If students answer based on the superficial judgment or use their perception, the teacher may ask students how they can be confident about their argument by asking:“How do you know it?” “How do you prove it?”” How do you convince others?”If students have difficulties, then the teacher asks students if they have the leaves on their hands, what they will do.“What will you do if you have those leaves in your hands?”Ask students what they can do from the leaves on the worksheet.If students reply that they need something to cut, then provide the materials. Otherwise, just provide the materials.

Comparing rice fields Students are able to compare two irregular figures by overlapping, cutting and pasting.

Students grasp the concept of

The students will use the same strategies like in comparing leaves. Use their superficial judgment by just

looking at two rice fields and say one rice

field is larger than the other one because it

The teacher may ask students about plants that produce food for human to the rice field. Tell me plants that produce food for human! The teacher asks students what they know about the shape of rice fields. What do you know about

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Activity Learning Goal Conjectured of Students’ thinking Guidance for teacherconservation of area looks longer or almost the same.

Trace one rice field and place it over the

other one to see the non- overlapping areas

Overlap one rice field to the other one and

cut parts of non-overlapping area and paste

them to non- overlapping area of the other

rice field.

the shape of rice fields?The teacher asks students whether the area of the rice fields changes or not when they cut and paste the rice fields. What happens to the leaf after you cut and paste?

Comparing dotted rice fields

Students are able to compare two with their own strategies. Students understand that area is region inside the boundary. They will grasp the idea of unit measurement but not too much focused on this meeting.

Students may count the dots one by one in each rice field and compared them.Student will make rectangles to count the dots efficiently as shown below:

or they will make a bigger rectangle covering the rice field and use the multiplication strategy to count the dots and then subtracting the result with the outsider dots.

Students combine tracing and counting strategies and count only the dots in the non-overlapping parts.

If students count the dots one by one, ask them to count the dots in a faster way. Do you know any other faster way to count the dots?If students have difficulties, ask them to cut and overlap. Can you cut and overlap the rice fields?

Table 4.3.The overview of the activity and the hypothesis of the learning process in lesson 3

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Activity Lesson 3 Learning Goal Conjectured of Students’ thinking Guidance for teacherDetermining the rice field to make a fair deal.

Grasp the idea of perimeter.Comparing the perimeter before and after the reshaping

Students are able to choose which rice field to fairly exchange the farmer’s rice field by comparing the area.Students are able to differentiate between area and perimeter.Students understand that reshaping will preserve the area not the perimeter.Students are able to differentiate between area and perimeter.

Some students may choose one of the options and maybe more than one.Students only judge superficially based on their visual. They will say that it is fair because the farmer will get a longer rice field or it is not fair because the shape is different.Students will choose rice field which has the same perimeter as the farmer’s rice field by using the string.Students will choose the rectangular rice fields because it is regular and common.Students will choose the rice field which has the same area by reshaping the farmer’s rice field.Students will compare the lengths of the string to measure the perimeter of each rice field.Students will use the string to measure the perimeter before and after reshaping and compare them

Ask students to convince their arguments“Tell me the reason why you think so?”“Can you convince others?”“Maybe the materials can help you”If students choose the rice field with the same perimeter, just let them to do so and this group will present in the classroom discussion.If students cut and paste the farmer’s rice field to fit the optional rice field, ask students“Why do you do that”, “What is your goal by doing that?”If students choose only the rectangular rice fields, ask:“Why do you choose them? Do they have the same area?”After students finish worksheet 6, tell students that they will discuss their work later.The teacher may ask students what is the difference between area and perimeter.

Reshaping quadrilaterals into a rectangular.

Students are able to do cut and paste to reshape figures into a rectangle.

Students may do trial and error. As students are able to do reshaping in previous activity, this geometrical figures are easier to reshape.

Ask students how they reshape it. What did you do first? Did you do trial and error? Or do you have your way to reshape them?Ask also: If you reshape, what will remain the same and what will not?What do you know about area and

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Activity Lesson 3 Learning Goal Conjectured of Students’ thinking Guidance for teacherperimeter?If you reshape, do you think it will help you to find the area after reshaping? Why?

Table 4.3.The overview of the activity and the hypothesis of the learning process in lesson 4

Activity Lesson 4 Learning Goal Conjectured of Students’ thinking Guidance for teacherComparing tiled floors Students are able to compare

two floors with different tiles as their unit measurements.Students understand the need of the same square unit to compare area.

Students will count the number of tiles on each floor either one by one or using multiplication strategy.Students will compare the number of tiles from the floors and says that floor A is bigger since it has more tiles or some students will say that floor B is bigger since it has a bigger tile.Students will take one tile of each floor and compare them.

Students may realize that the tile on floor B is for time as

big as the tile on floor A.

Do not provide materials like scissors or tools to cut.Let students use their strategies to compare the floors.If they found that the floors has the same size. Ask: What did you do when you see them having the same area?If students count the tilesHow do you find the number of tiles in each floor?Make a discussion if there is an answer that they judge the floor based on the number of tiles.

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Activity Lesson 4 Learning Goal Conjectured of Students’ thinking Guidance for teacherTiling / structuring array Students are able to compare

floors by the number of square units cover them.Students are able to use their cut and paste strategy to tile the parallelogram floor.

Students will easily tile rectangle floors by putting some tiles on the edges of the floors and use multiplication strategy to count the tiles as follows:

Or they continue to tile the floor fully by using a ruler and count the tiles. Students will tile with possible position of full tile unit and cut the remaining parts then paste them to the untilled parts. Students may reshape the floor into a rectangle

and then tile it. Students may do trial and error.

The teacher prepares the material such as straightedges if students need them. The teacher will not provide the tools to cut. The teacher needs to ask students: What do you do to cover the floors?When students have difficulties in covering the parallelogram floor, suggest: You may modify the tiles.

Determining area of rectangular floors

Students are able to use multiplication strategy.Students understand how the formula length x width or base x height works.

Students will tile all the surface of the floor and count one by one the tiles. After students get the number of the tiles (60 tiles), they will multiply it by 625 cm2 since each tile has an area of 625 cm2. Therefore, the area of the floor is 37500 cm2.

Ask:What do you do to cover the floors?Or suggest:You may use pencil to draw your tiles.If students only draw or tile only some parts on the edges of the floor, ask:

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Activity Lesson 4 Learning Goal Conjectured of Students’ thinking Guidance for teacherStudents will only tile the edges of the floors and multiply the number of tiles on the vertical edge with the number of tiles in horizontal edge. After students get the number of the tiles, they will multiply it by 625 cm2 since each tile has an area of 625 cm2.Students will only tile the edges of the first floor and measure the length of the vertical and the horizontal edges. They will get the length of the horizontal and vertical edges are 250 cm and 150 respectively. Therefore, they will find the area of the floor is 37500 cm2. On the second floor, the lengths of the horizontal and vertical edges are 250 cm and 150 cm. Its area is equal to the first floor.

Can you determine the number of the tiles by tiling some parts? Why?When students cover all the floors by tiles, ask them how they count the number of the tiles?How do you count the number of the tiles?If they count one by one, askdo you have a faster way to count them?What is the area of the floors? How many centimeter squares?Remind students of the standard measurement units. Is the unit in centimeter or centimeter squares?If the edges of the floor are called base and height, what can you conclude?Tell students that they will discuss their work after.

Determining area of rectangles by applying area formula

Students are able to apply area formula for rectangles.

Students will multiply the length of the base and height of each rectangle to get its area.

The teacher may ask how this way works. How does it work?Can you conclude what the area formula of rectangles is?

Table 4.5.The overview of the activity and the hypothesis of the learning process in lesson 5

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Activity Lesson 5 Learning Goal Conjectured of Students’ thinking Guidance for teacherEstimating an area of a covered floor.

The students are able to determine the area of a floor using the multiplication strategy and apply area of formula to find the area of a floor.

Students draw lines to trace the tile from the visible lines of the tiles. Then they will count the tiles one by one or by using a multiplication strategy.

The teacher just let students do with the strategies they use. If they have difficulties ask or suggest to use any tool to helpDo you need something for help? A ruler or straightedge maybe useful.If students try to count one by one, pose a question: Do you have a faster way to count? Do you still remember what you learned in previous meeting?If students use multiplication strategy, ask the students how long the length and the width of the floor are? Can you use the formula that you have in previous meeting?Always remind students the standard unit of measurement, is it in cm or cm2?

Comparing dotted rice fields and comparing leaves

Students are able to estimate the area of the dotted rice field by using square units. Students grasp the use of grid paper to estimate the area of irregular shape and combine it with the cut-paste strategy.

Students will connect the dots with a straightedge or a ruler. And estimate the square units fit the rice fields.

If the students still count the dots, suggest students to connect the dots with a ruler. Let students solve the problem, some assistance are needed if students have difficulties counting the square units. “Do you have another way to count the square units”. The teacher may remind students of the reallotment activity, you may reshape it to make it easier for you

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Activity Lesson 5 Learning Goal Conjectured of Students’ thinking Guidance for teacher

Students will make their own grid paper and put the leaves on it and count the number of squares and compare them

to estimate.

Table 4.5.The overview of the activity and the hypothesis of the learning process in lesson 6

Activity Lesson 5 Learning Goal Conjectured of Students’ thinking Guidance for teacherComparing area of sides of two buildings with rectangular and parallelogram shape

Students are able to determine the area of a parallelogram by reshaping and derive the area formula from a rectangle.

Students will count the full square units and combine the not fully square units with other not fully square units in order to get full square units. Then students will count the square unit one by one or using multiplication strategy.Students may reshape the figures into a rectangle and count the square units one by one or using multiplication strategy.

The teacher will show a building with a parallelogram shape and ask :What is the height of this building? Can you determine how many glasses to cover that side of building?The teacher distributes the worksheet 12 to each group.If students count one by one, ask:“Do you have a faster way to count the square units?Do you still remember the formula to find an area of a rectangle?”If students have difficulties in counting the number of square units in the parallelogram, suggest student

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Activity Lesson 5 Learning Goal Conjectured of Students’ thinking Guidance for teacherto reshape into another figure.Maybe, you can reshape it into another figure.After students find that the rectangle and the parallelogram have the same area. Ask:If they have the same base, height, do you think they have the same area? Can you use the area formula for rectangle?

Determining areas of parallelograms

Students are able to use the formula of rectangle to find the areas of parallelograms.

Students will reshape the parallelograms into a rectangle and apply the area formula of rectangle.Students will use directly the area formula of rectangle to find the areas of parallelograms.

After some minutes and students have finished the 1st problem, the teacher then asks to work on the next problem. The teacher may suggest,Maybe, you can try to use the formula and prove it by reshaping.If students reshape the parallelogram into a rectangle and use multiplication strategy or area formula for rectangle, the teacher asks:Can you say that the area formula of a parallelogram is same with the formula of rectangle? So, what is the general area formula of parallelogram?If students only use the formula, the teacher may ask:How can you use the area formula of a rectangle? Why does it work? Can you tell me the relation between the rectangle and the parallelogram?

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Activity Lesson 5 Learning Goal Conjectured of Students’ thinking Guidance for teacherCan you generalize the area formula of parallelogram?

Determining areas of triangles Students are able to derive the area formula of triangle

Students may count one by one the square unitsStudents may reshape the triangles into a rectangle and apply area formula of rectangle

The teacher asks students where they can find a triangular shape. The teacher reminds students about a unique building, in previous problem, a parallelogram building. The teacher shows a picture of building with triangular shape. The students are asked whether they can find the area of glasses to cover the side of the building. Could you find the area of glasses to cover the buildingIf students count one by one and have difficulties, ask:“Maybeyou reshape it into another figure?After student reshape the triangle into a rectangle, and still count one by one the square units, ask:Do you have a faster way to count them? Do you still remember the area formula of rectangle?What is the formula now?

Determining the areas of shaded areas.

Students understand that the area formula of triangle is half of the are formula of rectangle (1/2 x base x height)

The students will count one by one the square units.The students will use the formula of parallelogram and rectangle and divide by two.

If students count one by one the square unit, suggest them:What do you think the area of the shaded part? What is the relationship with the parallelogram and the rectangle?If students use the formula of

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Activity Lesson 5 Learning Goal Conjectured of Students’ thinking Guidance for teacherparallelogram and rectangle, ask So, what do you think the area of a triangle?

Table 4.6.The overview of the activity and the hypothesis of the learning process in lesson 6

Activity Lesson 6 Learning Goal Conjectured of Students’ thinking Guidance for teacherDetermining areas of isosceles trapezoid

Students are able to use the formula of rectangle to find the areas of trapezoid.

Students may count one by one the square unitsStudents will reshape the trapezoid and use the area formula of rectangle

The teacher shows a picture of building with trapezoid shape. The students are asked whether they can find the area of glasses to cover the side of the building.If students if students count one by one and have difficulties, ask:“What did you do to the parallelogram in previous meeting? Can you do it to the trapezoid?After student reshape the triangle into a rectangle, and still count one by one the square units, ask:Do you have a faster way to count them? Do you still remember the area formula of rectangle?What is the formula now?See the base of your trapezoid and the base of the rectangle What can you conclude?If students reshape the trapezoid and use the area formula of a rectangle,

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Activity Lesson 6 Learning Goal Conjectured of Students’ thinking Guidance for teacherask: How do you find the area? If you use any formula, what is the formula?Ask also the height of the trapezoid, what is the height of the trapezoid?

Determining areas of trapezoids Students are able to use the formula of triangle to find the areas of trapezoid and derive area formula of trapezoid.

Students will add the area of the triangles inside the trapezoid.Students will simplify the sum area formulas of triangles to derive the area of trapezoid

Ask students to see what figures inside the trapezoid. How to determine the area of the trapezoid with the figures.If students sum the area of the triangles, ask: Is it allowed to do it? Is the sum area equal to the area of the trapezoid, why?If students sum the area formula of triangles, remind students with the distributive law of multiplication. Ask: can you simplify the formula?Ask students to relate the formula with the parallel sides and the height of the trapezoid.When students get this formula :

A=12 (a1+a2) t

Ask: what is the a1 and a2 in the trapezoid?

Finding the area of a kite and a rhombus

Students are able to derive the formula of kite and rhombus by reshaping into a rectangle

Students will reshape the kite and the rhombus into a rectangle and use the area formula of rectangle.

After students have reshaped the kite and rhombus into a rectangle, ask them to use the area formula of rectangle. The teacher should ask:What is the relation between the lengths of the base and height of the rectangle with the lengths of the

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Activity Lesson 6 Learning Goal Conjectured of Students’ thinking Guidance for teacherdiagonals?Can you conclude what the area formula of kite and rhombus is?

References

Fosnot, C. T., &Dolk, M. (2001).Young mathematicians at work: constructing number sense, addition, and subtraction.Portsmouth, NH: Heinemann.

Gravemeijer, K. (2004). Creating opportunities for students to reinvent mathematics.In 10Th International Congress in Mathematics Education (pp. 4-11).

Simon, M.A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2), 114-145.

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