Lesson 3: Letters on the tiled floor models

4.3.1 Starting points

This lesson intended to give students a further justification about angles similarity in a parallel-transversal situation. We chose mathematical explorations on the tiled floor models as a way for the students to be able to prove their conjectures about angles similarity that they have acquired from the previous lesson. We assume the students can perform the following activities before they work with the tasks and the questions in the worksheet (see worksheet 3 in the appendix).

a. The students can reason with the line patterns from the given geometrical figures.

b. The students understand the terms of lines such as, parallel, perpendicular, and intersect each other.

4.3.2 The learning goals Main goal

The students are able to explain angles similarity by utilizing the uniformity of tiles on the tiled floor models.

Sub-goals

a. Enable students to identify the lines patterns on the tiled floor models by analyzing the gaps between adjacent tiles.

b. Enable students to examine the angles on the tiled floor models.

c. Enable students to determine the magnitude of angles on the tiled floor models to get further justification of angles similarity on the letters that have parallel sticks on them (students‟ conjecture from the second lesson).

d. Enable students to relate the magnitudes of angles on two situations; letters from matchsticks and letters on a tiled floor model.

e. Enable students to describe the parallel lines using the similarity of angles and vice versa.

4.3.3 Description of activity

The teacher start the lesson by telling a story about a girl named Ana that found the patterns of her name on a tiled floor when she observed the gaps between adjacent tiles in her kitchen. After telling the story, the teacher display two pictures of tiled floors and ask the students to determine which floor that Ana refer to (see figure 4.3). Our intention in presenting the story is to raise students‟ expectation that they will do some explorations on the presented situation. At this moment, it is not obligatory for students to have the sophisticated explanations for their opinions. When working with the worksheet (see worksheet 3 in the appendix) the students will have more room to explain their idea related to the presented situation.

We divide this lesson into three stages. In the first stage, students should perform a mathematical exploration related to the patterns like letters on the two floor models. The second stage, students compare the letters on a tiled floor model (kitchen floor) with the letters on the matchsticks activity (second lesson) to justify angles similarity in a parallel-transversal situation by using the uniformity

of tiles. In the last stage, students should explain about angles similarity that they have justified. Students can utilize the uniformity of tiles and connect it to their knowledge about angles magnitudes on some letters (F, X, and Z) to justify their claims from the second meeting (letters from matchsticks).

First stage: exploring the angles on the tiled floors

There are several instructions in the worksheet that ask students to perform the tasks such as, showing their opinions to the story that presented earlier, finding as many letters as possible from the kitchen floor, and comparing the angles magnitudes on the letters on the kitchen floor with the angles on the letters from matchsticks activity. Teacher can orchestrate a classroom discourse that simultaneously covers these tasks in one compact discussion. The main goal of the discussion is to make students aware that they can calculate the magnitude of an angle without using a protractor in some special situations.

Second stage: justify angles similarity using the uniformity of tiles

Students should work in their previous group on the lesson two to perform this task. The task requires students to compare, analyze, and explain the angles on the letters in two situations; matchsticks and the kitchen floor. In this stage, teacher should stress the discussion on comparing the shape of some letters (E, F, N, X, and Z) from the poster in lesson two with the letters on the kitchen floor. Teacher also should help students to justify their previous conjectures about the

similarity between angles on these letters. Conducting a classroom discourse that focus on the fact that the shape and the orientation of the lines do not affect the similarity between corresponding angles may help students to justify their conjectures. In addition to that, it is also important to ask students to recall why the vertical angles (X-angle) have the same magnitude, even this not really related to the task on this stage. However, the students need to understand this fact in order to be able to explain the similarity between angles on a straight line that falling across two parallel lines.

Third stage: explaining the similarity between angles magnitudes using the uniformity of the tiles

In the worksheet 3, there is another picture of a tiled floor model (Figure 4.4) and some questions related to this floor model. Students will carry out simple mathematical explorations that beg them to applying their current knowledge. It is rather more complex situation compare with the previous activities, where the patterns of the gaps between tiles not clearly depict the shape of letters. However, if the concepts from the previous explorations are well understood, then it is more likely that they will arrive at a consensus where they are agree that parallelity and angles similarity are strongly connected.

4.3.4 Conjecture on students’ reaction

a. In the first task, students will highlight the gaps between tiles that form a word „ANA‟ but they may use different amount of gaps to construct the word. b. In the second task, some students may find all the letters on the kitchen floor

and some may not.

c. In the third task, students may find out the relation between the parallel orientation of the gaps and the parallel orientation of the matchsticks resulting the same consequence; similarity between angles in both situations. They may also figure out that they can easily see the similarity of angles on the tiled floors situation compare with the letters from the matchsticks activity.

d. In answering the first question, students may indicate all the angles with the same mark (symbol) and produce the ambiguity when we ask them which angle that equal to which angle.

e. In answering the second question, some students may use equal length symbol to indicate the parallelity.

f. In answering the third question, students would have different opinion related to the existence of the right-angle on the figure.

g. In answering the fourth question, students may realize that there is a connection between the parallelity and the similarity of angles on a situation when a straight line falling across a pair of parallel lines.

4.3.5 Discussion

This lesson is designed to create an adequate learning environment to allow students to test their own conjectures related to the angles similarity in a parallel- transversal situation. The magnitudes of angles in the lesson two are uncertain and limit the possibility for students to have satisfied proofs about angles similarity. However, in this lesson, the context is more suitable for the students to justify what they already infer from the lesson two. The magnitudes of angles on the tiled floor models are easy to determine. For instance, if there are six tiles that have a common point, students can carry out some simple calculations to find out that each corner of the tile will be 60 . The certainty of angles magnitudes can help

students in the process of justification. In addition to that, the appearance of the letters on both situations also can help students to justify their conjectures.

It is important to understand that the focus of this lesson is on the aspect of reasoning about angle magnitude. We focus our attention mainly on how students‟ reasoning about angles magnitudes helps them to prove their previous conjectures. As we can see, students should perform some calculations related to the angles magnitudes. We are fully aware that, students need to have some strategies on how to calculate the magnitude of angles in the presented situations. Therefore, in the next lesson we provide the students with a learning context that will help the students to sharpen their mathematical ability in reasoning about angles magnitude.