## Authentic Lessons for 21st Century Learning

Michell Eike, Laura Halstied, Teresa Lansford, Patricia McDaniels-Gomez | Published: December 18th, 2023 by K20 Center

• Subject Mathematics
• Course Geometry
• Time Frame 85-100 minutes
• Duration 2-3 class periods

### Summary

In this lesson, students will explore the culture of Native Americans and their beadwork. They will then use patterns to explore dilations and discover the relationship between the center of dilation, the preimage, and the image. Students will apply what they have learned to create their own beadwork design and demonstrate their understanding of dilations. Prerequisite knowledge for this lesson includes the following vocabulary: transformation, preimage, image, and rigid motion, which are all included in the Traditional Transformations, Part 1 lesson. This is the fourth lesson of five in the "Traditional Transformations" lesson series.

### Essential Question(s)

How are transformations and symbolism used through indigenous cultures?

### Snapshot

Engage

Explore

Students make observations to find the relationship between the center of dilation, the preimage, and the image.

Explain

Students complete guided notes as a class and formalize their understanding of dilations, scale factors, and centers of dilation.

Extend

Students apply what they have learned to create a beadwork design and trade with a friend to dilate the friend’s design.

Evaluate

Students demonstrate their understanding by finding the scale factor and using it to dilate a 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)

• Exploring 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)

• Designing Beadwork – Student A handout (attached; one per student; printed front only)

• Designing Beadwork – Student B handout (attached; one per student; printed front only)

• Dilation Exit Ticket handout (attached; one quarter per student; printed front only)

• Pencils

• Paper

• Compass (one per student)

• Straightedge (one per student)

• Calculator (one per student)

• Coloring utensils (optional)

• Graph paper (optional)

Face-to-Face

### 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.

Show slide 5 and introduce the “Beadwork” video on the slide, which is a video of Laverna Capes, a member of the Kiowa Tribe of Oklahoma and of Wichita descent, sharing her knowledge of her tribe and her beadwork creations.

Display slide 6 and introduce the Elbow Partner instructional strategy. Give students a few minutes to discuss what they learned from the video, using the questions below to guide their discussion:

• How could someone represent their culture with beadwork?

• Why do you think she uses a larger bead to make a larger design?

Face-to-Face

### Explore

25 Minute(s)

Show slide 7 and provide students with the link to the GeoGebra activity: geogebra.org/m/ecyvtdfg. Give students a few minutes to explore the applets and see how things work, especially if this is their first time using GeoGebra. As students are experimenting, pass out a copy of the attached Exploring Transformations handout to each student. Share with students that during this activity, they will be working with a nine-point star design that is an example of a design found on beaded medallions.

Direct students to follow the directions on their handout to complete the tables and use the directions in the GeoGebra activity for how to use the applets. Have students begin working independently.

Ask the class to be thinking about the following questions as they work through the front side of their handout:

• What does k seem to do?

• What does point Z seem to do?

As students are working, circulate the room. As you notice students finishing the first table: Part A: k > 1, have students find a partner and compare their results and what they think point Z does.

After discussing for a couple of minutes, have students independently complete the second table: Part A: 0 < k < 1.

Again, as students complete the second table, have them discuss again with their partner and answer the questions on their handout about the proximity of point Z to the preimage and what they think k seems to do.

Transition to slide 8 and direct students’ attention to Part B. Tell students that are going to still adjust k, but now move the preimage instead of point Z in the GeoGebra applet.

Have students continue to work with their partner. They are to compare what they see with what they have written so far on their handout. In other words, when k > 1 and point Z was to the left of the preimage, is the image still to the right of the preimage in Part B like they found using the Part A applet?

Encourage students to make adjustments to what they have written on their handout as needed. Let students know, “We want to see if our observations are always true or just sometimes true.” Tell students that they should record the things that are always true.

If time allows, ask for a few volunteers to share their observations and conclusions.

Show slide 9 and direct their attention to the back of their handout. Direct students to use the Part C applet to draw lines through the corresponding vertices. Ask for a few volunteers to share what they noticed about the lines.

After students conclude that the three lines intersect at the same point, have them complete the table to find the ratios of corresponding side lengths. Ask for a volunteer to share what they noticed about the ratios.

Once students notice that the ratios were all two, move to slide 10 and direct their attention to Part D. Have them complete the table to find the ratios of distances such as from point Z to point A and from point Z to point A'. Ask for a volunteer to share what they noticed about the ratios.

Depending on your class, transition to slide 11 before or after having students close the GeoGebra activity and ask the following question: What do you think happens when k < 0?

You can have students use the GeoGebra applet (Parts A or B) to answer this question, or you can have kids wonder about it and continue to stoke their curiosity. Regardless of your choice, have students jot down their hypothesis on scratch paper or in the margins of their handout. This question will be resolved during the Explain portion of the lesson. If time allows, consider facilitating a short discussion on why they think what they think.

Face-to-Face

### Explain

25 Minute(s)

Display slide 12 and provide the attached Guided Notes handout to each student.

Introduce the vocabulary of dilation, scale factor, and center of dilation to the class and guide them to write those vocabulary words on their handout. Then ask the class what they think the center of dilation and the scale factor were in the GeoGebra activity.

After students conclude that the center of dilation was point Z or the point where all the lines through the corresponding vertices intersected and that the k-value was the scale factor, ask for a volunteer to answer the following question: Is a dilation an example of rigid motion? Be sure to have the student provide reasoning. Have students record the answer (no) with justification on their handout.

Now go through how different k-values affect the preimage and the words one could use to describe these transformations:

• When k > 1, then the image is an enlargement of the preimage.

• When 0 < k < 1, then the image is a reduction of the preimage.

Ask the class the following questions: What do you think would happen if k = 0? What do you think would happen if k = 1? Students do not need to write down the answers to this information as they are not values of k that they would ever see. Use these questions as a way to help students find a pattern and understand the relationship between k, the preimage, image, and the center of dilation.

Once students feel confident about the center of dilation, ask the class: What do you think happens when k < 0? Have students use the graph on their handout and try to describe what they see. Have students compare what they see with what they thought would happen from the Explore portion of the lesson.

Help students see that the lines through the corresponding vertices still all intersect at the center of dilation. Use this to transition to the algebraic rule. Consider asking the class where they have seen (–1·x, –1·y) before. Use this time to remind the class of the algebraic rules of rotations for 180° about the origin to help describe what they are seeing. The graph on the handout has k = –1. Share with students how that graph would differ if k = –2, for example.

Then help students understand why the ratio for k is the image over the preimage by having them algebraically solve for k. If we multiply the coordinates of the preimage to get the image: k·(preimage) = image, then k = (image) / (preimage).

Direct students’ attention to the back of their handout and complete the examples together as a class. After example 1, ask the class if it was an example of a reduction or an enlargement. Have students justify their answer. Then consider asking the students to work with their partner to try example 2 before bringing the class back together to ensure that everyone understands.

After example 2, ask the class if that was an example of a reduction or an enlargement. Have students justify their answer.

Give each student a compass and protractor, then guide the class through how to complete a dilation 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.

Face-to-Face

### Extend

15 Minute(s)

Display slide 13 and facilitate a whole-class discussion regarding the following question: Where else do you see dilations?

Show slide 14 and give each student a copy of the attached Designing Beadwork – Student A handout. Share the idea that many native dancers wear matching pieces and the design on a pair of earrings is likely a reduction of the design on a medallion with students.

Instruct students to create their own design for a cuff, which would be worn around the wrist. Let students know they have approximately five minutes to create their design. Tell students that their design needs to be a polygon and that they must include six labeled vertices. If time allows, give students coloring utensils and ask them to imagine that each square represents a bead.

Transition to slide 15 and have students trade designs. Distribute a copy of the attached Designing Beadwork – Student B handout. Tell the class that they are to create a dilation of their classmate’s design by a scale factor of two; this dilated image design is intended to be worn on a vest. Remind them to label the corresponding vertices and that the center of dilation is at the origin.

Face-to-Face

### Evaluate

5 Minute(s)

Display slide 16 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 Dilation Exit Ticket handout or give students a sticky note, an index card, etc. for them to write their response. Use the hidden slide 17 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 dilating triangles or even just individual points with a center of dilation of the origin.

The “Traditional Transformations, Part 5” lesson is about compositions of transformations and fashion design.