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1-2: Learning Goals

Let’s be more precise about describing moves of figures in the plane.

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1-2-1: A Pair of Quadrilaterals

Quadrilateral A can be rotated into the position of Quadrilateral B.

Estimate the angle of rotation.

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Launch - slide is animated - 10 minutes

Arrange students in groups of 2–4. Provide access to geometry toolkits. Display the two quadrilateral figures for all to see. (They should also look at the task statement in their workbooks.) Ask students to give a discreet hand signal when they have an estimate for the angle of rotation. Give students 2 minutes of quiet think time and then time to share their thinking with their group before a whole-class discussion.

Anticipated Misconceptions

Students may not be sure which angle to measure. They may measure the acute angle between Shape A and Shape B. Ask these students to trace Shape A on tracing paper and rotate it by that angle to see that this does not give Shape B.

Activity Synthesis

Invite students to share their estimates for the angle of rotation. Ask students how they knew, for example, that the angle is more than 90 degrees (because the angle is obtuse) but less than 180 degrees (because the angle is less than a straight line).

Introduce or reiterate the language of clockwise (for rotating in the direction the hands on a clock move) and counterclockwise (for rotating in the opposite direction). In this case, the direction of rotation is not specified but it is natural to view Figure A being rotated counterclockwise onto Figure B. Make sure to introduce the language of the center of rotation (the vertex shared by A and B is the center of rotation) and tell students that we call Figure B the image of Figure A for the rotation.

It may be helpful to display the picture from the task statement to support this discussion, and if possible, show the 120∘ counterclockwise turn dynamically.

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1-2-2: How Did You Make That Move?

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Launch - 10 minutes

Keep students in the same groups, and maintain access to geometry toolkits. Give students 5 minutes of quiet work time, and then invite them to share their responses with their group. Follow with a whole-class discussion. Tell students that they will be describing moves as they did in the previous lesson, but this time there is a new move to look out for. Recall the words the class used to describe slides and turns.

Anticipated Misconceptions

Students may see a reflection as a translation especially since the figures are not on the same frame. Ask these students to trace Frame 2 on tracing paper. Is there any way to turn it into Frame 3 by sliding it? What do they have to do to turn it into Frame 3? (They have to flip the tracing paper over, so, this is a new kind of move.)

In describing reflections, students may confuse the terms horizontal and vertical. Consider posting the terms horizontal and vertical with examples in your room.

Activity Synthesis

The purpose of this discussion is an initial understanding that there is a third type of move that is fundamentally different from the moves encountered in the previous lesson, because it reverses directions. Some possible discussion questions to help them identify these are:

“How is the motion from panel 2 to panel 3 different than the ones we discussed yesterday?”
“Is there anywhere else that happens in this cartoon?”
“What features of the image help us to see that this move is happening?”

To help answer these questions, tell students to pay attention to the direction that the “beak” of the polygon is pointing, left or right. Draw a dotted vertical line in the middle of Frame 2, and say, “Here is a mirror. The polygon in Frame 3 is what the polygon in Frame 2 sees when it looks in the mirror.”

Demonstrate using tracing paper or transparencies to show they are mirror images. Then ask students if there are any other mirror lines in other frames. For the second reflection, from Frame 5 to Frame 6, point out that again left and right is reversed: In Frame 5 the head is on the right of the body, and in Frame 6 it is on the left, with the beak pointing down in both cases. Contrast this with a rotation through 180º, which would leave the beak pointing up. Demonstrate with tracing paper or transparencies.

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1-2-2: How Did You Make That Move?

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Activity Synthesis

The purpose of this discussion is an initial understanding that there is a third type of move that is fundamentally different from the moves encountered in the previous lesson, because it reverses directions. Some possible discussion questions to help them identify these are:

“How is the motion from panel 2 to panel 3 different than the ones we discussed yesterday?”
“Is there anywhere else that happens in this cartoon?”
“What features of the image help us to see that this move is happening?”

To help answer these questions, tell students to pay attention to the direction that the “beak” of the polygon is pointing, left or right. Draw a dotted vertical line in the middle of Frame 2, and say, “Here is a mirror. The polygon in Frame 3 is what the polygon in Frame 2 sees when it looks in the mirror.”

Demonstrate using tracing paper or transparencies to show they are mirror images. Then ask students if there are any other mirror lines in other frames. For the second reflection, from Frame 5 to Frame 6, point out that again left and right is reversed: In Frame 5 the head is on the right of the body, and in Frame 6 it is on the left, with the beak pointing down in both cases. Contrast this with a rotation through 180º, which would leave the beak pointing up. Demonstrate with tracing paper or transparencies.

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1-2-3: Move Card Sort

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Launch - Digital Access for students - 15 minutes

Arrange students into groups of 3, and provide access to geometry toolkits. Give each group the first 9 cards. Reserve the last 3 cards for use if time permits.

Tell students that their job is to sort the cards into categories by the type of move that they show. After they come to consensus about which categories to use, they take turns placing a card into a category and explaining why they think their card goes in that category. When it is not their turn, their job is to listen to their partner's reasoning and make sure they understand. Consider conducting a short demonstration with a student of productive ways to communicate during this activity. For example, show what it looks like to take turns, explain your thinking, and listen to your partner's thinking.

Give students about 10 minutes to sort the cards. Do not explicitly instruct students at the beginning to use the words translations, rotations, and reflections. Monitor for a group who uses these categories, even if they use different names for them. If time permits, distribute the remaining 3 cards. Follow with whole-class discussion.

If using the digital activity, ask the students to close their devices, at first. After they have come to agreement about how their cards should be sorted, they can open their devices and use the applets to help them refine the way they describe the moves.

Anticipated Misconceptions

Students may struggle to differentiate between the three moves, confusing reflections with either translations or rotations. After they make their best decision, encourage these students to use tracing paper to justify their response. In Card 10, students may be confused when the translated figure overlaps the original. For Card 4, students may first think that this is a rotation (much like Cards 6 and 9). Encourage these students here to use tracing paper to check their answers.

Activity Synthesis

Select one or more groups to share the names of their categories. Select one or more groups to share how they sorted the cards into the categories. Ask the class if they disagree with any of the choices, and give students opportunities to justify their reasoning (MP3).

Introduce the terms translation, rotation, and reflection. It may be helpful to display an example of each to facilitate discussion:

Alternatively, you may wish to display the geogebra applets used in the digital version of the student materials to facilitate discussion:

Translation: ggbm.at/wYYvZH7A
Rotation: ggbm.at/RUtdpQmN
Reflection: ggbm.at/nKQmSnDW

Point out ways to identify which type of move it is. Translations are a slide with no turning. Rotations are a turn. Reflections face the opposite direction. Introduce the term corresponding points. If we see the figures as rabbits, then the ear tips in the original figure and the ear tips in its image are corresponding points, for example. Also introduce the term image for a figure after a transformation is applied: for each of the cards, one figure is the image of the other figure after a translation, rotation, or reflection has been applied.

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1-2-3: Move Card Sort

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Activity Synthesis - each card (1, 2, 3) is linked to the corresponding Geogebra applet

Select one or more groups to share the names of their categories. Select one or more groups to share how they sorted the cards into the categories. Ask the class if they disagree with any of the choices, and give students opportunities to justify their reasoning (MP3).

Introduce the terms translation, rotation, and reflection. It may be helpful to display an example of each to facilitate discussion:

Alternatively, you may wish to display the geogebra applets used in the digital version of the student materials to facilitate discussion:

Translation: ggbm.at/wYYvZH7A
Rotation: ggbm.at/RUtdpQmN
Reflection: ggbm.at/nKQmSnDW

Point out ways to identify which type of move it is. Translations are a slide with no turning. Rotations are a turn. Reflections face the opposite direction. Introduce the term corresponding points. If we see the figures as rabbits, then the ear tips in the original figure and the ear tips in its image are corresponding points, for example. Also introduce the term image for a figure after a transformation is applied: for each of the cards, one figure is the image of the other figure after a translation, rotation, or reflection has been applied.

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1-2: Lesson Synthesis

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Questions for discussion:

“We encountered a new type of move that was different from yesterday. What can you tell me about it?” (It's like a mirror image, you can't make the move by sliding or turning, the figure faces the opposite direction.)
“We gave mathematical names to the three types of moves we have seen. What are they called?” (The “slide” is called a translation, the “turn" is called a rotation, and the mirror image is called a reflection.)
“What do we mean by corresponding points?” (A point that is in the same part of the figure in both the original figure and the image.)
“What do we mean by a figure's image?” (The resulting figure after a move has been performed.)

Consider creating a semi-permanent display that shows these three terms and their definitions for reference throughout the unit.

A translation slides a figure without turning it. Every point in the figure goes the same distance in the same direction. For example, Figure A was translated down and to the left, as shown by the arrows. Figure B is a translation of Figure A.

A rotation turns a figure about a point, called the center of the rotation. Every point on the figure goes in a circle around the center and makes the same angle. The rotation can be clockwise, going in the same direction as the hands of a clock, or counterclockwise, going in the other direction. For example, Figure A was rotated 45∘ clockwise around its bottom vertex. Figure C is a rotation of Figure A.

A reflection places points on the opposite side of a reflection line. The mirror image is a backwards copy of the original figure. The reflection line shows where the mirror should stand. For example, Figure A was reflected across the dotted line. Figure D is a reflection of Figure A.

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1-2: image

For example, here is triangle ABC and a translation to the right and up which is labeled DEF.

Point F in the image corresponds to point C, segment EF in the image corresponds to segment BD, and angle DEF corresponds to angle ABC.

Translations, rotations, and reflections move objects in the plane. Points, segments, and other parts of the original all have corresponding parts on the “moved object.” The moved object is called the image.

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1-2: congruent

One figure is congruent to another if there is a rigid transformation (a sequence of translations, rotations, and reflections) that moves the first figure so that it fits exactly over the second. The second figure is called the image of the rigid transformation. If a part of the first figure matches up with a part of the image, we call them corresponding parts. The part could be an angle, point, or side, and you can have corresponding angles, corresponding points, or corresponding sides.

If you have a distance between two points in the first figure, then the distance between the corresponding points in the image is called the corresponding distance.

Triangle A is congruent to triangle D. A translation takes triangle A to triangle B, a rotation takes triangle B to triangle C, and a reflection takes triangle C to triangle D.

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1-2: clockwise

An object is rotating clockwise if it is turning in the same way that the hour or minute hand goes around a clock.

The tilted square is rotated 15∘clockwise from the square sitting horizontally on its base.

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1-2: counterclockwise

An object is rotating counterclockwise if it is turning in the opposite way to the way that the hour or minute hand goes around a clock.

The tilted square is rotated 15∘counterclockwise from the square with a horizontal base.

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1-2: reflection

The reflection of a figure in a line takes every point of the figure to a point directly opposite to it on the other side of the line and the same distance from the line. In the figure, the triangle B is the reflection of the triangle A in the line ℓ.

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1-2: rotation

A rotation has a center, an angle, and a direction. It moves every point of a figure in a circle around the center, in the direction specified (clockwise or counterclockwise), and for a distance specified by the angle. For example, in the figure, triangle A is rotated 55∘ clockwise about center O to get triangle B.

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1-2: translation

A translation has a distance and a direction. It moves every point in a figure the given distance in the given direction.

The figure on the left is translated to the figure on the right in the direction from A to B, using the distance from A to B.

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1-2: Learning Targets

I know the difference between translations, rotations, and reflections.
I can identify corresponding points before and after a transformation.

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1-2-4: Is It a Reflection?

What type of move takes Figure A to Figure B?

Explain your reasoning.

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