Lesson 6

In this warm-up, students measure four line segments. They discuss the different aspects of making and recording accurate measurements. It is important to highlight the fractional markings and fraction and decimal equivalents used as students explain how they determined the length of the segment.

Give students 2 minutes of quiet work time followed by whole-class discussion. 

Students may struggle with the ruler that is not pre-partitioned into fractional units. Encourage these students to use what they know about eighths and tenths to partition the ruler and estimate their answer.

Invite students to share their responses and record them for all to see. Ask the class if they agree or disagree with each response. When there is a disagreement, have students discuss possible reasons for the different measurements.

Students are likely to have different answers for their measure of the third segment. The ruler shown is not as accurate as the question requires as it has not been pre-partitioned into fractional units. Ask 2–3 students with different answers to share their strategies for measuring the third segment. There will be opportunities for students to use measuring strategies later in this lesson.

The purpose of this activity is for students to see that translations, rotations, and reflections preserve lengths and angle measures. Students can use tracing paper to help them draw the figures and make observations about the preservation of side lengths and angle measures under transformations. While the grid helps measure lengths of horizontal and vertical segments, the students may need more guidance when asked to measure diagonal lengths. It is important in the launch to demonstrate for students how to either use the tracing paper or an index card to mark off unit lengths using the grid (MP5). 

Since students are creating their own measuring tool, they can only give an estimate, and some flexibility should be allowed in the response. During the discussion, highlight different reasonable answers that students find for the lengths which are not whole numbers.

As students work individually, monitor and ask them to explain how they are performing their transformations and finding the side lengths and angle measures. During the discussion, select students who mention corresponding sides and angles, which they learned in grade 7 when making scaled copies, to share. Also select students who estimated the side lengths for Figure C correctly using either the tracing paper or index card.

Students may try to count the grid squares on the diagonal side lengths. Remind students to measure these lengths with their tracing paper or index card. Students may also struggle estimating the diagonal side lengths on their self-marked index card or tracing paper. Remind students of how they estimated the lengths for the questions in the warm-up where the ruler was not marked.

Ask selected students to share how they performed the given transformation for each question. After each explanation, ask the class if they agree or disagree. Introduce students to the idea of corresponding sides and corresponding angles. Ask students to identify the corresponding angles in the first question and the corresponding side lengths in the second since they were not asked about these attributes the first time. The point here is not to find the actual values but to note that the corresponding measurements are equal. Since it is sometimes not possible to measure angles or side lengths exactly, student estimates for these values (both corresponding sides and corresponding angles) may be slightly different.

Point out that for each of the transformations in this activity, the lengths of the sides of the original figure equal the lengths of the corresponding sides in the image, and the measures of the angles in the original figure equal the measures of the corresponding angles in the image. For this reason, we call these transformations rigid transformations: they behave as if we are moving the shapes around without stretching, bending, or breaking them. An example of a non-rigid transformation is one that compresses a figure vertically, like this:

Tell them that a rigid transformation is a transformation where all pairs of corresponding distances and angle measures in the figure and its image are equal. It turns out that translations, reflections, and rotations are the building blocks for all rigid transformations, and we will explore that next.

The purpose of this activity is to decide if there is a sequence of translations, rotations, and reflections that take one figure to another and, if so, to produce one such sequence. Deciding whether or not such a sequence is possible uses the knowledge that translations, rotations, and reflections do not change side lengths or angle measures. The triangles \(ABC\) and \(CFG\) form part of a large pattern of images of triangle \(ABC\) that will be examined more closely in future lessons.

Monitor for students who use different transformations to take triangle \(ABC\) to triangle \(CFG\) and select them to share during the discussion. (There are two possible sequences in the Possible Responses section, but these are not the only two.)

Ask a student to explain why triangle \(ABC\) cannot be taken to triangle \(DBE\). (We are only using rigid transformations and therefore the corresponding lengths have to be equal and they are not.) If a student brings up that they think triangle \(DBE\) is a scale drawing of \(ABC\), bring the discussion back to translations, rotations, and reflections, rather than talking about how or why triangle \(DBE\) isn’t actually a scale drawing of \(ABC\).

Offer as many methods for transforming triangle \(ABC\) as possible as time permits, selecting previously identified students to share their methods. Include at least two different sequences of transformations. Make sure students attend carefully to specifying each transformation with the necessary level of precision. For example, for a rotation, that they specify the center of rotation, the direction, and the angle of rotation.

If time allows, consider asking the following questions:

• "Can triangle \(ABC\) be taken to triangle \(CFG\) with only a translation?" (No, since \(CFG\) is rotated.)
• "What about with only a reflection?" (No, because they have the same orientation.) 
• "What about with a single rotation?" (The answer is yes, but this question does not need to be answered now as students will have an opportunity to investigate this further in a future lesson.)