K20 LEARN | Traditional Transformations, Part 3

Summary

In this lesson, students will explore the culture of the Plains tribes and their Star Quilts. They will then use patterns to explore rotations and discover algebraic rules for common rotations. Students will apply what they learn to create their own quilt block and demonstrate their understanding of rotations and rotational symmetry. Prerequisite vocabulary knowledge for this lesson includes transformation, preimage, image, and rigid motion, which are all included in the Traditional Transformations, Part 1 lesson. This is the third lesson of five in the "Traditional Transformations" lesson series.

Essential Question(s)

How are transformations and symbolism used through indigenous cultures?

Snapshot

Engage

Students watch a video about the tradition of Plains Star Quilts.

Explore

Students make observations and discover patterns for rotations about the origin.

Explain

Students complete guided notes with the class and formalize their understanding of common and uncommon rotations.

Extend

Students apply what they have learned to identify different types of transformations and create their own quilt blocks using rotations.

Evaluate

Students demonstrate their understanding by rotating a point 180° about another point.

Instructional Formats

The term "Multimodality" refers to the ability of a lesson to be offered in more than one modality (i.e. face-to-face, online, blended). This lesson has been designed to be offered in multiple formats, while still meeting the same standards and learning objectives. Though fundamentally the same lesson, you will notice that the different modalities may require the lesson to be approached differently. Select the modality that you are interested in to be taken to the section of the course designed for that form of instruction.

Face-to-Face

Materials

Lesson Slides (attached)
Tribal Transformations handout (attached; one per student; printed front/back)
Guided Notes handout (attached; one per student; printed front/back)
Guided Notes (Teacher Guide and Model Notes) document (attached; for teacher use)
Star Quilt Transformations handout (attached; one per student; printed front only)
Rotation Exit Ticket handout (attached; one quarter per student; printed front only)
Pencils
Paper
Graph paper
Creating Quilt Blocks handout (optional; attached; one per student; printed front only)
Compass (one per student)
Protractor (one per student)
Patty Paper (optional; 1–2 per student)

Online

Materials

Hybrid

Materials

Desmos account
Guided Notes handout (attached; one per student; printed front/back)
Guided Notes (Teacher Guide and Model Notes) document (attached; for teacher use)
Star Quilt Transformations handout (attached; one per student; printed front only)
Pencils
Paper
Graph paper
Creating Quilt Blocks handout (optional; attached; one per student; printed front only)
Compass (one per student)
Protractor (one per student)
Student devices with Internet access

Hybrid

Engage

15 Minute(s)

Teacher's Note: Desmos Classroom Activity Preparation

To access this Desmos Classroom activity, go to the ”Traditional Transformations, Part 3” Desmos page.

Use the buttons on the page header to create an account or sign in. After signing in, the green “Assign” dropdown button will be active. Select the arrow next to the word “Assign,” then select “Single Session Code.” After making some setting selections, select “Create Invitation Code'' and provide the session code to students. For more information about previewing and assigning a Desmos Classroom activity, go to our Desmos Classroom page.

For more detailed information about Desmos features and how-to tips, go to our Desmos Resources page.

To set up the activity’s pacing for students, select “View Dashboard” (next to the session code). In the upper-left corner of the screen, select the icon above the word “Pacing.” Desmos Classroom should then prompt you to select the first and last screens that you want students to see. When prompted to set a range, select screens 1 and 4. To confirm your selection, select “Restrict to Screens 1–4”. This allows students to access only screens 1–4 at this time. For more information about teacher pacing, go to our Pacing Activities page.

Provide students with your session code. Then, have students go to student.desmos.com and enter the session code.

Introduce the lesson using screens 1–2 of the Desmos Activity. Review the lesson series’ essential question and learning objectives on screens 1–2 to the extent you feel necessary.

Direct students’ attention to screen 3 and introduce the GramIt instructional strategy. Let students know that they are about to watch a video of Crystal Pewo Lightfoot, a member of the Apache Tribe of Oklahoma and a descendant of the Kiowa Tribe, sharing her knowledge of her tribal culture and how she uses rotations in her Star Quilt creations. Explain that, after watching the video, they will be asked to create two hashtags: one representing an aspect of the culture they learned about, and another representing a mathematical concept they learned.

Then, have students watch the “Plains Star Quilts and Rotations” video that is linked on the screen.

After watching the video, direct students’ attention to screen 4 and ask them to enter their two hashtags, each representing a cultural and mathematical aspect they learned from the video.

If time allows, ask for volunteers to share their answers with the class.

Online

Engage

15 Minute(s)

Face-to-Face

Engage

15 Minute(s)

Introduce the lesson using the attached Lesson Slides. Review the lesson series’ essential question on slide 3 and the lesson’s learning objectives on slide 4 to the extent you feel necessary.

Display slide 5 and introduce the GramIt instructional strategy. Let students know that they are about to watch a video of Crystal Pewo Lightfoot, a member of the Apache Tribe of Oklahoma and a descendant of the Kiowa Tribe, sharing her knowledge of her tribal culture and how she uses rotations in her Star Quilt creations. Explain that, after watching the video, they will be asked to create two hashtags: one representing an aspect of the culture they learned about, and another representing a mathematical concept they learned.

Show slide 6 and play the “Plains Star Quilts and Rotations” embedded video on the slide.

Move to slide 7 and have students take out a piece of paper. Then, ask them to write two hashtags, each representing a cultural and mathematical aspect they learned from the video.

If time allows, ask for volunteers to share their answers with the class.

Face-to-Face

Explore

15 Minute(s)

Show slide 8 and pass out a copy of the attached Tribal Transformations handout to each student. Share with students that, for this activity, they will work with an arrow symbol from a Plains-style quilt design.

Ask students to partner up or assign them their partners to complete the Tribal Transformations handout. For question 1, direct them to complete the table next to the coordinate plane. Then, ask students to use that table to try to figure out an algebraic rule for that transformation and to then describe the transformation.

Have students repeat this procedure for questions 2–3.

Online

Explore

15 Minute(s)

Hybrid

Explore

15 Minute(s)

On the Dashboard, press the orange plus sign to allow students to progress to screen 5. Give students a couple of minutes to familiarize themselves with the Desmos sketch tools if this is their first time using them.

On the dashboard, press the orange plus sign three times to allow students to progress to screens 6–8.

Share with students that, for this activity, they will work with an arrow symbol from a Plains-style quilt design. Direct students to use the sketch tools to label image 2, complete the table, then answer the prompts on their screen:

Write the algebraic rule to describe the relationship between the preimage and image 2.
How would you describe what happened visually when the preimage was transformed into image 2?

Ask students to partner up or assign them their partners. Direct them to compare their work and responses from screen 6. If time allows, ask for a few volunteers to share their answers with the class.

Have pairs work together through screens 7–8 to find patterns between the preimage and rotated images. Then, have them attempt to write the algebraic rule.

To help students enter their coordinate rules, guide them to copy the mapping symbol from the “Type Math” button. If students see “\to” when they paste the mapping symbol, have them highlight that text, then press the “Type Math” button to convert that syntax into the symbol.

If time allows, ask for a few volunteers to share their responses with the class.

Hybrid

Explain

20 Minute(s)

Give each student a copy of the attached Guided Notes handout, then press the orange plus sign on the Dashboard to allow students to progress to screen 9.

Introduce the vocabulary of rotation and angle of rotation to the class and guide them to write those vocabulary words on their handout. Remind students of the difference between clockwise and counterclockwise and let them know that all rotations are assumed to be counterclockwise unless otherwise stated. This is communicated both on screen 9 and on the handout. Then ask for a volunteer to answer, “Is a rotation an example of rigid motion?” Make sure to ask the student to provide their reasoning. Have students record the answer (“yes”) with justification on their handout.

On the Dashboard, press the orange plus sign three times to allow students to progress to screens 10–12. Now, go through the algebraic rules of special rotations for rotations of 90°, 180°, and 270° about the origin. One special rotation is provided per screen. Consider having students take a moment to discuss with their partner what they think the rule should be.

Have students enter their responses into the Desmos Classroom activity, then bring the class together for a discussion on each rotation. To develop the rules, have students use the pictures on their Guided Notes or Desmos Classroom activity as well as their work from screens 6–8.

The activity on screens 10–12 will provide feedback on students’ responses. Be sure to help students see that the rule is already started for them using (a, b). Help students see that a is the x-value and b is the y-value. Have students record the notation that makes sense to them on their handouts.

On the Dashboard, press the orange plus sign three times to allow students to progress to screens 13–15. Direct students’ attention to the bottom of their handouts. Ask them to enter their answers to the following prompts on each screen:

Rotating a figure 90° CCW is the same as rotating the figure …
Rotating a figure 180° CCW is the same as rotating the figure …
Rotating a figure 90° CW is the same as rotating the figure …

Then, ask them to record their answers on their handout:

Screens 13–15 will not provide feedback for incorrect selections but will let students know when they have selected the correct options (a “Well done!” text will appear). Use the Dashboard to assess students’ answers. Make sure everyone has made the correct selections before proceeding.

Direct students’ attention to the back of their handouts. Then press the orange plus sign on the Dashboard twice to allow students to progress to screens 16–17. Ask students to follow the directions on the screen and use the Desmos Classroom activity to complete examples 1–2 on their handouts. Here, students will see a “Rotate” button once they complete their table. Pressing the button shows the preimage rotate to where the image should be and provides students feedback.

Circulate the room or use the Dashboard to assess students’ understanding. Consider bringing the class together after each example to answer questions.

Give each student a compass and protractor and ask them to set aside their devices. With example 3, guide the class through rotating a figure that is not on a coordinate plane.

Have students add their completed Guided Notes to their math notebooks if it is a classroom norm.

Online

Explain

20 Minute(s)

Face-to-Face

Explain

20 Minute(s)

Display slide 9 and provide the attached Guided Notes handout to each student.

Introduce the vocabulary of rotation and angle of rotation to the class and guide them to write those vocabulary words on their handout. Remind students of the difference between clockwise and counterclockwise and let them know that all rotations are assumed to be counterclockwise unless otherwise stated. Then ask for a volunteer to answer, “Is a rotation an example of rigid motion?” Make sure to ask the student to provide their reasoning. Have students record the answer (“yes”) with justification on their handout.

Now, go through the special rotation algebraic rules for rotations of 90°, 180°, and 270° about the origin. Have students use the illustration on their Guided Notes and their work from the Explore part to develop the rules.

Ask the class to finish the following prompts at the bottom of their handout:

Rotating a figure 90° CCW is the same as rotating the figure …°.
Rotating a figure 180° CCW is the same as rotating the figure …°.
Rotating a figure 90° CW is the same as rotating the figure …°.

Help students see that their algebraic rules work when they rotate a figure about the origin. Moreover, they can still apply the same rules to other figures about any point.

Give each student a compass and protractor. Then, with example 3, guide the class through rotating a figure that is not on a coordinate plane.

Have students add their completed Guided Notes to their math notebooks if it is a classroom norm.

Additional Scaffolding

For the Explain and Extend portions of this lesson, consider handing out pencils and patty paper to help students who struggle visualizing transformations. Patty paper is great for this activity because of its see-through property and low graphite adhesion, which means that pencil marks made on the paper can be easily smudged onto another surface. Alternatively, you can also use parchment paper, wax paper, or tracing paper for this activity.

Here are the steps to use patty paper to capture and rotate a figure’s preimage:

Place the patty paper on top of the handout.
On the patty paper, trace the figure’s center of rotation and sketch short horizontal and vertical line segments that intersect at the center of rotation (e.g., when the center of rotation is the origin, sketch part of the x- and y-axes as the horizontal and vertical lines). These segments will help students visualize 90° rotations later.
Trace the preimage (or at least the vertices) of the figure with a pencil.
Label the vertices.
Flip the patty paper over.
On the back of the patty paper, use its see-through property to trace and draw the figure’s vertices. The vertices need to be drawn boldly so that the lead of the pencil will later transfer onto the handout paper later. During this step, make sure students move away their patty paper from their handout to prevent lead transfer.
Once the vertices are drawn boldly, flip the patty paper to its original side.
Align the traced preimage and center of rotation with those on the handout.
Press the pencil tip on the center of rotation and use the pencil to secure the patty paper in place as you rotate it.
Use the sketched vertical and horizontal line segments to help rotate the preimage in 90° increments, following the given transformation.
Once the figure reaches the desired rotation, retrace the vertices on the patty paper. The pressure from the pencil will transfer the pencil lead from the back of the patty paper onto the handout.
Label the new vertices on the handout. To keep track of the original vertices, consider just lifting part of the patty paper when writing on the handout.
Lift the patty paper and use the edge of the protractor to connect the vertices on the handout.

Face-to-Face

Extend

35 Minute(s)

Display slide 10 and remind students what rotational symmetry and the center of symmetry mean.

Transition to slide 11 and facilitate a whole-class discussion regarding the question, “Where else do you see rotational symmetry?”

Show slide 12 and give each student a copy of the attached Star Quilt Transformations handout and four coloring utensils that students can share. Have students work individually to color code their Star Quilt pattern: one color should be the preimage, and the other three colors should each indicate either a translation, reflection, or rotation. Let students know that not every rhombus needs to be fully colored by the end of this activity.

Encourage students to collaborate with a partner to help each other see the different transformations.

Once students complete the activity, get them into groups of four and compare their results. As time allows, ask for volunteers to share any observations with the whole class.

Transition to slide 14 and give each student a piece of graph paper or the attached Creating Quilt Blocks handout. If students are using graph paper, ask them to draw the x- and y-axes in the center of the page.

Share the Pass the Problem strategy with the class. Explain that each group will work together to create four unique quilt blocks. Explain that each person will create their own preimage in Quadrant III, then pass the problem around to complete the quilt block design collaboratively. Tell students that their preimage creation can be a tessellation—like the Plains Star Quilt pattern—but is not required to be. Take a moment to introduce or remind students of the vocabulary tessellations, which is the repetition of one or more shapes such that the shapes cover a surface with no gaps or overlap.

Show slide 15 and direct each student to create their own preimage in Quadrant III. Then instruct them to label at least four vertices.

After a few minutes, transition to slide 16 and have everyone pass their paper to their right. On the paper they received, have them draw the image by rotating the preimage 270° about the origin, then have them label the corresponding vertices.

After a few minutes, show slide 17 and have everyone pass their paper to their right. Ask them to check the work of the previous student, draw another image by rotating the preimage 180° about the origin, and label the corresponding vertices.

After a few minutes, display slide 18 and have everyone pass their paper to their right, check the previous student’s work, draw the image by rotating the preimage 90° about the origin, and label the corresponding vertices. Encourage students to work together to adjust any points that need to be corrected.

After a few minutes, transition to slide 19 and have everyone pass their paper to their right. The paper should now return to the student who drew the preimage. Have them check the previous work and for rotational symmetry.

Consider having students display their quilt blocks on the wall to make a class quilt.

Online

Extend

Hybrid

Extend

35 Minute(s)

Now it is time for students to apply what they have learned. On the Dashboard, click the orange “Stop” button. Students can now complete the Desmos activity at their own pace. Direct their attention to screen 18 and remind students what rotational symmetry and center of symmetry mean.

Direct students’ attention to screen 19 and facilitate a whole-class discussion regarding the question, “Where else do you see rotational symmetry?”

When students get to screen 20, have them set aside their Desmos Classroom activity. Give each student a copy of the attached Star Quilt Transformations handout and four coloring utensils that students can share. Have students work individually to color code their Star Quilt pattern: one color should be the preimage, and the other three colors should each indicate either a translation, reflection, or rotation. Let students know that not every rhombus needs to be fully colored by the end of this activity.

Encourage students to collaborate with a partner to help each other see the different transformations.

Once students complete the activity, get them into groups of four and compare their results. As time allows, ask for volunteers to share any observations with the whole class.

Now give each student a piece of graph paper or the attached Creating Quilt Blocks handout. If students are using graph paper, ask them to draw the x- and y-axes in the center of the page.

Share the Pass the Problem strategy with the class. Explain that each group will work together to create four unique quilt blocks. Explain that each person will create their own preimage in Quadrant III, then pass the problem around to complete the quilt block design collaboratively. Tell students that their preimage creation can be a tessellation—like the Plains Star Quilt pattern—but is not required to be. Take a moment to introduce or remind students of the vocabulary tessellations, which is the repetition of one or more shapes such that the shapes cover a surface with no gaps or overlap.

Direct each student to create their own preimage in Quadrant III. Then instruct them to label at least four vertices.

After a few minutes, have everyone pass their paper to their right. On the paper they received, have them draw the image by rotating the preimage 270° about the origin, then have them label the corresponding vertices.

After a few minutes, have everyone pass their paper to their right. Ask them to check the work of the previous student, draw another image by rotating the preimage 180° about the origin, and label the corresponding vertices.

After a few minutes, and have everyone pass their paper to their right, check the previous student’s work, draw the image by rotating the preimage 90° about the origin, and label the corresponding vertices. Encourage students to work together to adjust any points that need to be corrected.

After a few minutes, and have everyone pass their paper to their right. The paper should now return to the student who drew the preimage. Have them check the previous work and for rotational symmetry.

Consider having students display their quilt blocks on the wall to make a class quilt.

Hybrid

Evaluate

5 Minute(s)

Use the Exit Ticket strategy to individually assess what students have learned from the lesson. Have students go back to their Desmos Classroom activity and direct their attention to screen 21. The screen will ask students to enter their responses to a practice question—only the teacher can see the feedback on students’ answers.

Collect student responses and use them to determine if your students need additional practice or if they are ready for the next lesson. If students need additional practice, consider having students practice with more basic shapes, like rotating triangles or even just individual points about the origin 90°, 180°, or 270°.

The “Traditional Transformations, Part 4” lesson will discuss dilations and beadwork.

Online

Evaluate

5 Minute(s)

Face-to-Face

Evaluate

5 Minute(s)

Display slide 20 and use the Exit Ticket strategy to individually assess what students have learned from the lesson. Give each student a quarter sheet of the attached Rotation Exit Ticket handout. Alternatively, give students a sticky note or an index card to write their responses. Use the hidden slide 21 for a sample response.

Collect student responses and use them to determine if your students need additional practice or if they are ready for the next lesson. If students need additional practice, consider having students practice with more basic shapes, like rotating triangles or even just individual points about the origin 90°, 180°, or 270°.

The “Traditional Transformations, Part 4” lesson will discuss dilations and beadwork.

Resources

K20 Center. (n.d.). Bell Ringers and Exit Tickets. Strategies. https://learn.k20center.ou.edu/strategy/125
K20 Center. (n.d.). Desmos classroom. Tech Tools. https://learn.k20center.ou.edu/tech-tool/1081
K20 Center. (n.d.). Gramit. Strategies. https://learn.k20center.ou.edu/strategy/2554
K20 Center. (n.d.). Pass the Problem. Strategies. https://learn.k20center.ou.edu/strategy/2554
K20 Center (2023, July 5). Plains star quilts and rotations [Video]. YouTube. https://youtu.be/YJxmt0dt3mw

Authentic Lessons for 21st Century Learning

Traditional Transformations, Part 3

Rotations: Plains Star Quilts

Summary

Essential Question(s)

Snapshot

Instructional Formats

Materials

Materials

Materials

Engage

Engage

Engage

Explore

Explore

Explore

Explain

Explain

Explain

Extend

Extend

Extend

Evaluate

Evaluate

Evaluate

Resources

Publisher

Downloads

Standards

Facilitator Resources

Learner Handouts

Authentic Lessons for 21st Century Learning

Summary

Essential Question(s)

Snapshot

Instructional Formats

Materials

Materials

Materials

Engage

Engage

Engage

Explore

Explore

Explore

Explain

Explain

Explain

Extend

Extend

Extend

Evaluate

Evaluate

Evaluate

Resources

Publisher

Downloads

Standards

Facilitator Resources

Learner Handouts

Related Resources