### Summary

In this lesson, students will explore the culture of the Osage tribe and their ribbonwork. Students will apply what they have learned about math and indigenous cultures to create their own ribbonwork design and demonstrate their understanding of reflections and reflective symmetry. Prerequisite knowledge for this lesson includes the following vocabulary: transformation, pre-image, image, and rigid motion, which are all included in the Traditional Transformations, Part 1 lesson. This is the second 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 Osage ribbonwork.

**Explore**

Students make observations about and discover patterns of reflections over the axes.

**Explain**

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

**Extend**

Students apply what they have learned to identify lines of symmetry and create their own ribbonwork design.

**Evaluate**

Students demonstrate their understanding by reflecting a point over a vertical line.

### 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.

### Materials

Lesson Slides (attached)

Exploring Ribbonwork (Part A) handout (attached; one per student; printed front only)

Exploring Ribbonwork (Part B) 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)

Seeing Symmetry handout (attached; one per student; printed front only)

Perfecting Patterns handout (attached; one per student; printed front only)

Over the Line handout (attached; one quarter per student; printed front only)

Pencils

Coloring utensils (colored pencils, markers, etc.; 4 per student)

Paper

Graph paper

Scissors (one pair per student)

Compass

Straightedge

Patty Paper (optional; 1-2 per student)

Plastic reflective device, such as GeoMirror, Mira, etc. (optional; one per student)

Construction paper (optional; 4 different colored strips per student)

### Engage

15 Minute(s)

Introduce the lesson using the attached **Lesson Slides**. **Slide 3** displays the lesson series’ essential question. **Slide 4** identifies the lesson’s learning objectives. Review each of these with your class to the extent you feel necessary.

Display **slide 5** and let students know that they are about to watch a video of Dana Daylight sharing her knowledge of her tribe and how she uses reflections in her Osage ribbonwork creations. Explain that after watching the video, they will be asked to share something new they learned from the video and something that they already knew.

Show **slide 6** and play the “Osage Ribbonwork and Reflections” video on the slide.

Transition to **slide 7** and introduce the Elbow Partners strategy to the class. Have students discuss the following questions with their partner, focusing on mathematics of ribbonwork and the Osage culture:

What is one new thing you learned?

What is one thing that you already knew?

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

### Explore

15 Minute(s)

Show **slide 8** and pass out a copy of the attached **Exploring Ribbonwork (Part A)** handout to each student. Share with students that during this activity, they will be working with an arrow pattern from Osage ribbonwork.

Have students work individually to complete the pattern in Quadrant III, then ask them to answer the following prompts on the handout:

How did you complete the pattern?

Describe how the pre-image transformed into

*image 2*.Describe how the pre-image transformed into

*image 4*.

Have students find partners or assign students partners. Direct them to compare their work and responses. If time allows, ask for a few volunteers to share with the whole class.

Transition to **slide 9** and distribute a copy of the attached **Exploring Ribbonwork (Part B)** handout to students. Direct their attention to the top of the page: *Reflect Over the y-Axis*. Instruct pairs to complete the table and answer the following questions:

Does your rule apply when

*image 4*is reflected over the*y*-axis to get*image 3*?Does your rule apply when

*image 2*is reflected over the*x*-axis to get*image 3*?

Here, students are looking for patterns to write an algebraic rule to describe a reflection over the *y*-axis. If time allows, ask for a few volunteers to share their responses with the whole class.

Direct their attention to the back of the page: *Reflect Over the x-Axis*. Instruct pairs to complete the table and answer the following questions:

Does your rule apply to the other pair of reflections over the

*x*-axis?What else do you think we could reflect a figure over?

Here, students are looking for patterns to write an algebraic rule to describe a reflection over the *x*-axis and consider reflections over something other than an axis. If time allows, ask for a few volunteers to share their responses with the whole class.

### Explain

20 Minute(s)

Display **slide 10** and provide the attached **Guided Notes** handout to each student.

Introduce the vocabulary of *reflection* to the class and guide them to write this vocabulary word on their handout. Then ask for a volunteer to answer the following question: *Is a reflection an example of rigid motion?* Be sure to have the student provide reasoning. Have students record the answer (yes) with justification on their handout.

Go through the algebraic rules for reflections over the *x*-axis, the *y*-axis, the line *y* = *x*, and the line *y* = –*x*. Have students use the pictures on their Guided Notes as well as their work from the Explore portion of the lesson to develop the rules.

Direct students’ attention to the back of their handout and complete the examples as a class. After example 1, consider asking the students to work with their partner to try example 2, before bringing the class back together to ensure everyone is understanding.

Help students see that example 2 is a vertical reflection and just like the vertical reflection over the *x*-axis, the *x*-values of the corresponding vertices did not change.

Give each student a compass and straightedge, then guide the class through how to complete a reflection not on the coordinate plane with example 3.

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

### Extend

25 Minute(s)

Now it is time for students to apply what they have learned and to recall some prior knowledge. Give each student a copy of the attached **Seeing Symmetry** handout. Display **slide 11** and remind students what a *line of symmetry* is.

Transition to **slide 12** and have students independently work to write the equation(s) for the line(s) of symmetry of the given figure for question 1.

As students complete question 1, transition to **slide 13** and give students time to check their work and ask questions.

Show **slide 14** and have students again work independently to write the equation(s) for the line(s) of symmetry for question 2.

Once students complete question 2, have them find a partner and compare their equations. Then, display **slide 15** so pairs can check their work. Give pairs time to discuss their work and make changes as needed.

Display **slide 16** and have pairs write the equation(s) for the line(s) of symmetry for question 3.

As students finish the last question, transition to **slide 17** and allow time for pairs to check their work.

Once students feel comfortable with finding lines of symmetry and writing their equations, move to **slide 18** and facilitate a whole-class discussion regarding the following questions:

Where else do you see reflective symmetry?

What is something in this room or that is familiar to you that only has one line of symmetry?

What is something in this room or that is familiar to you that has zero lines of symmetry?

What is something in this room or that is familiar to you that has two lines of symmetry? More than two lines of symmetry?

Show **slide 19** and give each student a copy of the attached **Perfecting Patterns** handout. Instruct students to get into groups of 3-4 or assign groups. Then share with the class the Pass the Problem strategy.

Explain to the class that they are now to follow the directions for “Student A” and to write their name at the top of the upper-left box. Instruct everyone to think about the ribbonwork designs that they have learned about and create their own polygon design (pre-image) in Quadrant II.

Move to **slide 20** and have everyone pass their paper to the person on their right within their group. Explain that they are now to follow the directions for “Student B” and to write their name at the top of the upper-right box. Direct everyone to label the vertices of Student A’s design. Remind students that the letters should go in alphabetical order but can go clockwise or counterclockwise.

Then direct the class to use the space in their handout to create a table of the corresponding points if the pre-image was reflected over the *x*-axis. In other words, students are expected to use what they have learned during this lesson to write the new ordered pairs for the corresponding vertices of the image without drawing the reflection.

Move to **slide 21** and have everyone pass their paper to the person on their right within their group. Explain that they are now to follow the directions for “Student C” and to write their name at the top of the lower-left box. Direct everyone to check Student B’s table. Give time for students to check and talk through and correct any mistakes.

Now direct the class to use the space in their handout to create a table of the corresponding points if the pre-image (from Student A) was reflected over the *y*-axis. In other words, students are expected to use what they have learned during this lesson to write the new ordered pairs for the corresponding vertices of the image without drawing the reflection.

Show **slide 22** and have everyone pass their paper back to Student A. Give each student a piece of graph paper.

Instruct everyone to copy their pre-image to their graph paper and then plot the points from Student B’s table and Student C’s table, then work together to adjust any points that need to be corrected.

Display **slide 23** and direct students to complete the ribbonwork design by completing the pattern in Quadrant IV. Then let them know that instead of appliquéing their designs onto an article of clothing, they will be using their pattern to make a bookmark.

Have students use 4 coloring tools to color their design. Remind the class to use contrasting light and dark colors like the Osage tribe does. Consider giving students the challenge of using more than 4 colors while keeping the symmetry. Give students scissors and time to cut out their design, which they can use as a bookmark.

### Evaluate

5 Minute(s)

Display **slide 24** 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 **Over the Line** handout or give students a sticky note, an index card, etc. for them to write their response. Use the hidden **slide 25** for a sample response.

Collect student responses and use them to determine if your students need additional practice or are ready for the next lesson. If students need additional practice, consider having students practice with more basic shapes, like reflecting triangles or even just individual points over a line. Also, consider giving students problems where the line of reflection goes through the polygon instead of being next to the polygon.

The “Traditional Transformations, Part 3” lesson is about rotations, rotational symmetry, and Lakota star quilts.

### 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.). Elbow Partners. Strategies. https://learn.k20center.ou.edu/strategy/116

K20 Center. (n.d.). Pass the Problem. Strategies. https://learn.k20center.ou.edu/strategy/151

K20 Center. (2023, July 5).

*Osage ribbonwork and reflections*[Video]. YouTube. https://youtu.be/CCU7hBirn9c?si=hDNjlL8FbxcZELAVOsage Nation. [OsagenationnsnGovmedia]. (2020, July 21).

*Grow Gather Hunt Virtual Camp 03 - Ribbon work Bookmarks*[Video]. YouTube. https://youtu.be/8IofzcZUXzg