The purpose of this Warm-up is for students to consider 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.
Monitor for students who use what they know about eighths and tenths to partition the last ruler and estimate their answer.
Student Lesson in Spanish
Launch
Give students 2 minutes of quiet work time followed by whole-class discussion.
Activity
None
Student Task Statement
For each question, the unit is represented by the large tick marks with whole numbers.
Find the length of this segment to the nearest of a unit.
A line segment measured by a ruler above it. The ruler has units represented by large tick marks with whole numbers from 1 to 6. Each unit is evenly divided into 8 parts. The segment measures 4 and the fraction 5 over 8 units.
Find the length of this segment to the nearest 0.1 of a unit.
A line segment measured by a ruler above it. The ruler has units represented by large tick marks with whole numbers from 1 to 5. Each unit is evenly divided into 10 parts. The segment measures 4 and the fraction 7 over 10 units.
Estimate the length of this segment to the nearest of a unit.
A line segment measured by a ruler above it. The ruler has units represented by large tick marks with whole numbers from 1 to 5. The segment measures near 3 and the fraction 3 over 4 units.
Estimate the length of the segment in the prior question to the nearest 0.1 of a unit.
Student Response
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Building on Student Thinking
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.
Activity Synthesis
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.
6.2
Activity
15 mins
Sides and Angles
Standards Alignment
Building On
Addressing
8.G.1.a
Lines are taken to lines, and line segments to line segments of the same length.
The purpose of this activity is for students to see that translations, rotations, and reflections preserve side lengths and angle measures. Students can use tracing paper to help them draw the figures and make observations. While the grid helps measure lengths of horizontal and vertical segments, students may need more guidance when asked to measure diagonal lengths. It is important in the launch to elicit strategies from students to either use tracing paper or an index card to mark off unit lengths using the grid as they use tools strategically (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 are not whole numbers that students find for the lengths.
Monitor for students who use corresponding side lengths and angle measures as well as students who estimate the side lengths using tracing paper or an index card to share.
Launch
Provide students with access to tracing paper or index cards. Display the image of Pentagon C for all to see.
In order for students to measure the side lengths, students will need to create a ruler. Ask students, “How can you create a ruler to measure the side lengths on this pentagon?” (Mark the lengths of the grid lines, then fold the index card to get markings like halves, fourths, and eighths.) If students ask why they need to create their own ruler, point out that they are measuring in grid length units, which may not match up with inches or centimeters.
Give students 5–8 minutes of quiet work time. Select students with different strategies, such as those described in the Activity Narrative, to share later.
Activity
None
Student Task Statement
Translate Polygon A so point goes to point . In the image, write the length of each side, in grid units, next to the side.
Rotate Triangle B clockwise using as the center of rotation. In the image, write the measure of each angle in its interior.
Reflect Pentagon C across line .
In the image, write the length of each side, in grid units, next to the side. You may need to make your own ruler with tracing paper or a blank index card.
In the image, write the measure of each angle in the interior.
Activity Synthesis
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 angles. Ask students to identify the corresponding angles for Polygon A and its image ( 90 degrees and 270 degrees) and the corresponding side lengths for Triangle B and its image (2 units, 4, units, and about 3.5 units). The point here is not to find the actual values but to note that the corresponding measurements are equal. 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.
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 the shapes move around without stretching, bending, or breaking. An example of a non-rigid transformation is one that compresses a figure vertically, like this:
Tell students that a rigid transformation is a transformation where all pairs of corresponding distances and angle measures in the figure and its image are equal. Translations, reflections, and rotations are the building blocks for all rigid transformations, and we will explore that next.
MLR8 Discussion Supports. Use multimodal examples to show the meaning of corresponding sides and corresponding angles. Use verbal descriptions along with gestures, drawings, or concrete objects to show how the parts of the figures are related through rigid transformations. Advances: Listening, Representing
6.3
Activity
10 mins
Which One?
Standards Alignment
Building On
Addressing
8.G.1.a
Lines are taken to lines, and line segments to line segments of the same length.
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.
Monitor for students who use different transformations to take triangle to triangle and select them to share during the discussion.
Activity Synthesis
Ask a student to explain why triangle cannot be taken to triangle . (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 is a scale drawing of triangle , bring the discussion back to translations, rotations, and reflections, rather than talking about how or why triangle isn’t actually a scale drawing of triangle .
Offer as many methods for transforming triangle as possible as time permits, inviting previously selected students to share their methods. Include at least two different sequences of transformations. Display student work or these diagrams for all to see as students share their sequences.
Make sure students attend carefully to describing each transformation with the necessary level of precision. For example, for a rotation, specifying the center of rotation, the direction, and the angle of rotation.
If time allows, consider asking the following questions:
"Can triangle be taken to triangle with only a translation?" (no, since is rotated)
"What about with only a reflection?" (no, because they have the same orientation)
Lesson Synthesis
Remind students that a rigid transformation is a transformation for which all pairs of corresponding lengths and angle measures in the original figure and its image are equal. Translations, rotations, and reflections have this property, so they are rigid transformations. Sequences of these are as well—for example, if a figure is translated and then the image is reflected, the side lengths and angle measures stay the same.
Ask students to think of ways they could look at two shapes and tell that one is not the image of the other under a rigid transformation. Give a moment of quiet think time, and then invite students to share their ideas (if two shapes have different side lengths or angle measures then there is no rigid transformation taking one shape to the other).
When there is a rigid transformation taking one figure to another, there are many ways to do this. Ask students:
"What are some good ways to tell whether one shape can be taken to another with a sequence of rigid transformations?" (Measure all of the side lengths and angle measures and ensure that corresponding measurements are equal. Use tracing paper to see if one shape matches up exactly with the other.)
"What are the three basic types of rigid transformations?" (rotations, translations, and reflections)
Student Lesson Summary
The transformations we’ve learned about so far, translations, rotations, reflections, and sequences of these motions, are all examples of rigid transformations. A rigid transformation is a move that doesn’t change measurements on any figure.
Earlier, we learned that a figure and its image have corresponding points. With a rigid transformation, figures like polygons also have corresponding sides and corresponding angles. These corresponding parts have the same measurements.
For example, triangle was made by reflecting triangle across a horizontal line, then translating. Corresponding sides have the same lengths, and corresponding angles have the same measures.
Measurements in triangle
Corresponding measurements in image
angle
angle
angle
angle
angle
angle
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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.
Launch
Provide access to geometry toolkits. Give students 4 minutes quiet work time, followed by 2 minutes to discuss with a partner. Select students with different sequences of transformations to share during the whole-class discussion.
In the digital version of the activity, students use an applet to perform transformations. The applet allows students to test multiple transformations. The digital version may be helpful for precisely describing transformations.
MLR2 Collect and Display. Circulate, listen for, and collect the language students use as they work with a partner to transform figures. On a visible display, record words and phrases such as “rotate triangle 90 degrees counterclockwise around point ” and “translate triangle 7 units right.” Invite students to borrow language from the display as needed, and update it throughout the lesson. For example, a student may rephrase “the triangles do not go together because the sides do not match” as “the triangles are not connected by rigid transformations because their corresponding sides are not congruent.” Advances: Conversing, Reading
Action and Expression: Develop Expression and Communication. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their ideas. For example, “Triangle is a rigid transformation of triangle because . . . ,” “I agree/disagree because . . . ,” or “Another transformation is because . . . .” Supports accessibility for: Language, Organization
Activity
None
Student Task Statement
Here is a grid showing triangle and two other triangles.
Three triangles A B C, D B E and C F G on a grid. Let (0 comma 0) be the bottom left corner of the grid. Then the coordinates of triangle A B C are: A(4 comma 2), B(4 comma 5) and C(8 comma 5). The coordinates of triangle D B E are D(4 comma 7), B(4 comma 5) and E(1 comma 5). The coordinates of triangle C F G are C(8 comma 5), F(10 comma 5) and G(10 comma 5).
You can use a rigid transformation to take triangle to one of the other triangles.
Which other triangle? Explain how you know.
Describe a rigid transformation that takes to the triangle you selected.
