The Sound of Polynomials

Samaya Williams, Michell Eike, James Doyle, Sherry Franklin, Kelsey Willems, Dustin Ragland | Published: November 6th, 2023 by K20 Center

• Grade Level 9th, 10th, 11th
• Subject Mathematics
• Course Algebra 1, Algebra 2
• Duration 3 days
• ICAP Resource Individual Career Academic Plan (ICAP)

Summary

In this lesson, students practice adding and subtracting polynomial expressions. They watch a video about a complicated food order and discuss the idea of combining like items to simplify the order. Students use math manipulatives to explore how to use addition and subtraction to simplify polynomial expressions. By participating in a Scavenger Hunt Notes activity, students create their own notes about adding and subtracting polynomials. Students watch an ICAP video of an education specialist from Ableton, who was a former tour musician. He shares his knowledge of the music industry, using technology to engineer and produce music, and he explains how Chebyshev polynomials of the first kind can be used to manipulate sound and music. After watching the video, students apply adding and subtracting polynomials to hear the sound of the simplified polynomial. Finally, students demonstrate what they have learned about adding and subtracting polynomials in an Exit Ticket.

Essential Question(s)

How can you use addition and subtraction to simplify polynomial expressions?

Snapshot

Engage

Students make the connection of grouping like items and figure out how that concept applies to simplifying polynomial expressions.

Explore

Students use a digital math manipulative to explore using addition and subtraction to simplify polynomial expressions.

Explain

Students participate in a Scavenger Hunt Notes activity to complete notes on using addition and subtraction to simplify polynomial expressions.

Extend

Students watch an ICAP video in which a former tour musician, who is now an Education Specialist for Ableton, discusses how polynomial expressions are used to affect or change sound. They apply their knowledge of adding and subtracting polynomial expressions to see what changes to sound the simplified expressions make.

Evaluate

Students demonstrate what they have learned about adding and subtracting polynomials in an Exit Ticket.

Materials

• Lesson slides (attached)

• Polynomials and Algebra Tiles handout (attached, 1 per student, 2-sided)

• Polynomials and Sounds handout (attached, 1 per student)

• Frayer Model Handout (attached, 1 per student)

• Scavenger Hunt QR Codes handout (attached, print 1-sided, print on colored paper)

• Exit Ticket handout (cut in half, one half page per student)

• Alternative Frayer Model Handout (optional, attached, 1 per student)

• Alternative Scavenger Hunt QR Codes (optional, attached, print 1-sided, print on colored paper)

• Frayer Model Teacher’s Guide (optional, attached)

• Learning Synths Playground handout (optional, attached)

• Pencils

• Clipboards (one per student)

• Devices with Internet access

• Algebra tiles (optional)

• Green cups (1 per group)

• Yellow cups (1 per group)

• Red cups (1 per group)

• Blank sheet of paper (1 per student)

• Earbuds (1 per student, optional)

• Colored pencils

• Markers

Engage

10 Minute(s)

Present the lesson using the attached Lesson Slides.

Display slide 3 and ask students to watch a video about ordering at a fast-food restaurant. Show students the MADtv Fast Food Ordering video.

Once the students have watched the video, display slide 4 and ask students to figure out an easier way the customer could have communicated his order.

If students don’t mention grouping things that have been ordered, guide them to the possible answer of grouping like items.

Display slides 5 and 6 to introduce the essential question and the learning objective to connect the idea of grouping like items to simplifying polynomials.

Explore

30 Minute(s)

Display slide 7 and inform students they will be using an online math manipulative to answer questions about the given polynomials. Have students navigate to k20.ou.edu/algebratiles to access the Mathsbot dienes blocks, also known as base-ten blocks. Follow the steps itemized below:

1. Instruct students to use these resources to represent terms of the polynomial expressions.

2. Inform students that to create a negative, they will need to click the ” x-1” button at the top of the tool bar.

3. Advise them to combine like terms by dropping a negative (or red block) on top of the positive block.

4. Let them know that this action will cause the blocks to disappear or cancel each other out.

5. Instruct them to get rid of a block by clicking on the block and selecting “delete.”

6. Instruct them to hit “refresh” in the web browser to clear their screen.

7. Advise them that the blocks on the screen are very large and can be made smaller by using the “negative magnifying glass” in the top right corner of their screen.

Consider modeling these steps for the class and have the students mimic at the same time. Note that this modeling will give students the opportunity to practice using the tool they will use for the activity.

Display slide 8 and introduce the Try It, Talk It, Color It, Check It instructional strategy. Let students know that as they work on the problem, they will first try it on their own, discuss it with a partner, and then choose a colored cup to represent their confidence in efficiently summarizing how many blocks of each kind they have all together. They will have three colored cups. Tell the students that the colors represent the following confidence levels:

• Green: They can summarize their thought processes.

• Yellow: They are a bit uncertain about how to summarize their thought processes.

• Red: They need help summarizing their thought processes.

Pass out the Polynomials and Algebra Tiles handout, the green, yellow, and red cups, and a device to each student. Inform students to complete only the addition side of the handout. Give students time to work through the addition side of their handout before moving on. Clear up any misconceptions as they work.

After 10 minutes, you can unhide slide 9 for students to check their answer and share their process.

Display slide 10 and inform students they will continue working on simplifying polynomials using algebra tiles and the Try It, Talk It, Color It, Check It instructional strategy. But this time, they will focus on subtraction on the back of the handout. Remind students that, as they work on the problem, they should first try it on their own, discuss it with a partner, and then choose a color to represent their confidence in efficiently summarizing their thought process.

Have students use the dienes blocks on the Mathsbot website they used for addition. Give students time to work through the subtraction side of their handouts before moving on. Clear up any misconceptions as they work.

After 10 minutes, you can unhide slide 11 for students to check their answer and share their process.

Explain

40 Minute(s)

Start day 2 with a review of the essential question and learning objects (slides 5 and 6). Then, go through the following steps:

1. Inform students that they will be taking notes on what they explored the day before.

2. Display slide 12 and introduce the students to the

instructional strategy.

3. Explain to the students that there are QR codes taped around the room to help them take notes on their Frayer model.

4. Have the students start anywhere they would like. Encourage them to spread out.

5. Pass out a Frayer Model handout and clipboard to each student.

6. Make sure each student or group of students has a device to scan the QR codes.

7. Using the QR codes on the wall, have students walk around the room filling in the boxes of their Frayer Model with the information from the corresponding videos.

8. During the scavenger hunt, in the top-left corner of the Frayer Model handout, ask students to write a definition of polynomial expression. This

concept is addressed in the video at station 1.

9. In the top-right corner of the Frayer Model handout, have students write the steps for simplifying polynomials by adding or subtracting.

10. Ask them to identify what is the most important thing to remember when subtracting polynomials. This concept is addressed at stations 2 and 3.

11. In the bottom left and right corner of the Frayer Model handout, ask students to simplify the polynomial expressions on the Scavenger Hunt QR code using either the vertical or horizontal methods.

12. Inform students that both station 4, adding, and station 5, subtracting, have two QR codes.

13. Instruct them to watch both videos before deciding whether they want to solve the problem horizontally or vertically on their Frayer Model.

14. Provide students with 20-30 minutes to complete the activity.

Once all students have completed the Scavenger Hunt, display slide 13 and facilitate a class discussion to review the Frayer Model.

Ask students to share what they put on their Frayer Model to help them understand what a polynomial expression is. Use slide 14 to reveal the answers; then have them check their work.

Then, display slides 15 and 16 to go over the steps of simplifying polynomials. Remind students of the most important thing to remember when subtracting polynomials. Click slide 16 a second time to share the answer with the students.

Use slides 17 - 20 to show the answers of the problems in the addition and subtraction sections of the Frayer Model.

Extend

30 Minute(s)

Display slide 21 and introduce the students to the Chebyshev Polynomials and the effects these polynomials have on music when they are added and subtracted.

Display slide 22 and inform students they will watch The Sounds of Polynomials with Dustin Ragland, an ICAP video of Dustin’s sharing his knowledge of the music industry, using technology to engineer and produce music. He explains how Chebyshev polynomials of the first kind can be used to manipulate sound and music.

Go to slide 23. Divide students into groups of no more than four using the method you prefer.

Instruct students to navigate to the Polynomials and Sound website on their devices as you pass out the Polynomials and Sounds handout.

For each problem of the handout, students will add or subtract polynomials from the Chebyshev sequence. Once they have the result, they will place their answer or simplified expression into the website by typing the coefficients and exponents of their expression into the corresponding parameters to create a new polynomial filter. The image below shows where each part of the expression goes on the website:

Have students turn to the back of their Polynomials and Sounds handout to walk through the first sequence together. If you choose you can unhide slide 24 to show the directions on the screen.

1.  Ask students to click the number “49” key at the top of the page. (49 is the MIDI number for the note C#3).

2. Have them turn the effect volume all the way down and slide the noeffect volume all the way up. This is the unfiltered sound.

3. Have them slide the effect volume all the way up. This is the sound being filtered by the T3 Chebyshev polynomial.

4. Have them change the preset to T5, then repeat the process clicking on “49,” turning the effect volume all the way down and sliding the noeffect volume all the way up. Then set the effect volume to “102.” This is the sound being filtered by the T5 Chebyshev polynomial.

5. Students should notice that the noeffect is the same, but the effect sound is different as the T5 sound is more nasally and lower in pitch than the T3 sound.

6. Have them go to the worksheet and simplify the first set of polynomials.

7. After getting the result, have students leave the sound on as they input the answers in the boxes. Upon changing the 3rd coefficient, they should hear the sound change.

Ask students to repeat this process with problems 2-4 using a different starting pitch (52, 56, or 63). If they do it correctly, their final sounds should always sound the same.

Unhide slide 25 to have students check their work.

Navigate to slide 26. Direct the students’ attention to Part 2 of the Polynomials and Sounds handout. Hold a quick class discussion and have a couple of students share their answers about how they describe the change of their sound. Use the following questions to help with sharing out for one subtraction and one addition problem:

• How are the sounds different?

• What descriptive words would you use for each sound? Examples include words like nasally, cold, scary, happy, etc.

Evaluate

10 Minute(s)

Display slide 27 and use the Exit Ticket strategy to individually assess what students have learned from the lesson. Give each student a half-sheet of the attached Exit Ticket handout or give students a sticky note, an index card, etc. for them to write their response.

Collect student responses and use them to determine whether they need additional practice or are ready for the next lesson.