### Summary

In this lesson, students will discover how matrices are used in computer graphics. Students will learn how to perform the matrix operations of addition, subtraction, multiplication, and scalar multiplication. They will then learn how multiplying matrices relates to transformations and, in turn, animation of computer graphics.

### Essential Question(s)

How do we use matrices?

### Snapshot

**Engage 1**

Students watch a video about the relationship between computer graphics and matrices.

**Explore**

Students find the connection between data and matrices.

**Explain 1**

Students start guided notes with the class and formalize their understanding of adding and subtracting matrices and scalar multiplication.

**Extend 1**

Students apply what they have learned to solve equations involving matrix operations.

**Engage 2**

Students watch a more in-depth video about the relationship between computer graphics and matrices.

**Explain 2**

Students complete guided notes with the class and formalize their understanding of multiplying matrices.

**Extend 2**

Students apply what they have learned to transformation matrices.

**Evaluate**

Students demonstrate their understanding of the criteria needed to perform a matrix operation.

### Materials

Lesson Slides (attached)

Missing Information handout (attached; one per pair; printed front-only)

Guided Notes handout (attached; one per student; printed front/back)

Problem Solved handout (attached; one half-sheet per student; printed front-only)

Transformation Matrices handout (attached; one per group of 4; printed front-only)

Transformation Cards (attached; one set per group of 4; printed front-only)

Exit Ticket handout (attached; one half-sheet per student; printed front-only)

Pencils

Paper

Graph paper

Student devices with internet access

Scientific calculators (optional)

### Engage 1

5 Minute(s)

Introduce the lesson using the attached **Lesson Slides**. **Slide 3** displays the lesson’s 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 show the video titled "Matrices in Computer Graphics."

Inform students they are going to learn some of the basics of moving figures using matrices during this lesson.

### Explore

10 Minute(s)

Instruct students to find a partner or assign students partners. Show **slide 6** and pass out the attached **Missing Information** handout to each pair of students. Instruct students to match the matrices (options A–D) with the given data (questions 1–4). Guide students to use their matches to answer questions 5–8 on the handout and find the unknown values.

When they are finished, show **slide 7** and allow students time to check their work. Ask for volunteers to share their reasoning.

### Explain 1

15 Minute(s)

Pass out copies of the attached **Guided Notes** handout and display **slide 8**. Complete the front side of the handout as a class.

After completing only the front side of the handout, direct students to set it aside. Students complete the back side later in the lesson.

### Extend 1

5 Minute(s)

Display **slide 9** and give each student a copy of the attached **Problem Solved** handout. Have students work in pairs to solve each equation using the matrix operations learned during the Explain 1 portion of the lesson. Encourage students to use their Guided Notes handout for reference.

When students finish, transition through **slides 10–11** and allow students time to check their work. If time allows, ask for volunteers to explain their process for solving each equation. Use student responses to identify any misconceptions.

### Engage 2

10 Minute(s)

Display **slide 12** and introduce the video "How Rendering Graphics Works in Games." As they watch, students will see more clearly that matrices are used in computer graphics. Prepare the class for the reflective question that they are asked to respond to after watching the video.

What did you learn from the video that you did not know?

Discuss students’ responses to the reflective question: *What did you learn from the video that you did not know?* Ask for volunteers to share.

Transition into the next activity by telling the class that they are now going to learn how to perform matrix multiplication, which is different from scalar multiplication.

### Explain 2

15 Minute(s)

Show **slide 13**. Start completing the back side of the Guided Notes handout as a class by teaching students how to multiply matrices using the information at the top of the page and in example 3.

Before starting example 4, read the following to the class to introduce the example:

Sports are an excellent source of data. Think about all of the numbers used to describe a basketball player’s performance, like their offense, defense, and rebounding. These data can be used to rank players. For example, EA Sports uses data for rankings in NBA 2K video games. But how do they do it? When processing that amount of data, computers are performing the calculations using matrices. They multiply the players’ ratings by the weights of the categories to yield the overall player ranking.

Complete example 4 as a class. Have students add their completed Guided Notes handouts to their math notebooks if that is a classroom norm.

### Extend 2

20 Minute(s)

Direct students to get into groups of four or assign students to groups. Show **slide 14** and pass out a copy of the attached **Transformation Matrices** handout and a set of **Transformation Cards** to each group. Direct students to read and follow the steps on the handout. Here students are to select a figure, select four transformation cards, and use matrix multiplication to transform their selected figure.

As students work, circulate around the room. Once you notice most students are on the back side of the handout, transition to **slide 15**. This slide helps clarify step 5 by showing students how to write their vertices as matrices to use for matrix multiplication.

After students finish multiplying their transformation matrix, instruct them to use the graph paper to draw their original figure and their new figure. As students finish their sketches, share **slide 16** and direct them to use their devices to access the GeoGebra activity: geogebra.org/m/re5ajfet. Students should click the transformation buttons in the order in which they want them applied. Remind students that this order is the opposite order of how they multiplied their matrices.

### Evaluate

10 Minute(s)

Use the Exit Ticket strategy to individually assess what students have learned from the lesson. Go to **slide 17** and pass out copies of the attached **Exit Ticket** handout. Students are asked to perform the possible indicated matrix operations. If the operation is not possible, they are to write *impossible*. Use student responses to identify what misconceptions still exist and guide further instruction.

### Resources

Bitbinge. (2017, December 19). Kingfisher [Illustration]. Pixabay. https://pixabay.com/illustrations/bird-halcyon-kingfisher-eisvogel-3027224/

Huffman, C. J. (2019). CGI matrix multiplication activity. Open Educational Resources - Math. Pittsburg State University Library Services. https://digitalcommons.pittstate.edu/oer-math/16

K20 Center. (n.d.). Bell ringers and exit tickets. Strategies. https://learn.k20center.ou.edu/strategy/125

K20 Center. (n.d.). GeoGebra. Tech Tools. https://learn.k20center.ou.edu/tech-tool/2352

K20 Center. (2022, July 21). Matrices in computer graphics [Video]. YouTube. https://youtu.be/g8sMoZ2fgZw

McLaughlin, A. [TheHappieCat]. (2015, July 22). How rendering graphics works in games!