**Headlines (Modification): Derivatives & Their Applications**

__Overview/Objective__: Students will use their knowledge of derivatives to write a headline that conveys their understanding of the concept of derivatives, then write a new headline (separate from the previous headlines) as each new application of derivatives is learned.

A culminating headline will be written at the end of the unit on derivatives that encapsulates all of the headlines that have already been written.

I chose this content because I found that students often lost sight of the purpose/role/origin of derivatives, especially as there were more and more applications of them that connected to other mathematical concepts. Using this thinking routine will hopefully provide students with an avenue to become adjusted to looking for the big idea throughout a unit.

The process of writing headlines will be repeated multiple times. It is estimated that with individual writing time and group sharing time, each headlines session will take approximately 10-15 minutes. The final sharing time (Invite Further Sharing) is estimated to take 15-20 minutes.

__Prior Knowledge Needed/When to Teach__: This activity will occur in Calculus throughout the unit on derivatives, beginning after the introduction to derivatives.

__Standards__: Note: there are no state standards for calculus. The specific content objectives are listed.

– determine the local or global extreme values of a function

– apply the Mean Value Theorem

– find the intervals on which a function in increasing or decreasing

– use the first and second derivative test to determine the local extreme values of a function

– determine the concavity of a function and locate the points of inflection by analyzing the second derivative

– graph f(x) using information about f'(x)

– solve application problems involving finding minimum or maximum values of functions

– solve related rate problems

__Materials__:

– Students will need a piece of paper or computer at various times throughout the process.

– Butcher paper (or other paper than can be written on and hung in the room) will be needed

__Lesson Outline__:

**Set Up**:

– After being introduced to the concept of derivatives, students will be asked to think about the big ideas related to derivatives.

**Write a Headline**:

– Students are asked to individually “write a headline for ‘derivatives’ that explains an important idea that you think is valuable to remember” in 15 words or fewer.

– As suggested by Karrie Tufts (case study from Ritchhart p. 115-118), students will also be asked to give “a little more of the story” (Ritchhart p. 116).

– These will be kept by the student, either in a notebook, binder, or digitally so that they can be accessed later (the medium is up to the student, they need to be able to be accessed later).

**Share the Thinking**:

– After students write their headline individually, students will share their headline with a partner.

– As a team, they will discuss their headlines and their reason behind choosing their words as they did.

– They will then combine their individual headlines into one and it on butcher paper to be posted in the classroom for reference throughout the unit. This will be done in order to have reminders of what derivatives truly mean as the content gets “crowded” with more and more applications.

**Rinse and Repeat:
**– These three steps of Set Up, Write a Headline, Share the Thinking will be repeated after each new application of derivatives is introduced (extreme values, Mean Value Theorem, increasing/decreasing intervals of a function, concavity of a function, optimization, related rates), using the prompt: “Write a headline for _________ (each application of derivatives listed above) that explains an important idea that you think is valuable to remember.”

– The individual headlines will all be kept in one location (chosen by the student, either digital or hard copy).

– The partner headlines will all be posted in the classroom.

**Invite Further Sharing:
**– At the end of the unit on derivatives, each student will have a collection of 7 headlines, as well as the headlines posted in class.

– Students will then work individually to write

__one__headline that encapsulates all of their previous headlines about derivatives.

– These headlines will be shared to the class, and a discussion about common themes will occur.

– Optional: A gallery walk can be done where students walk around and view their classmates’ headlines. This can be turned into a journaling assignment where they comment on their classmates’ headlines, as well as the common themes and different ideas that are focused on throughout the class.

__Assessment__:

This activity works great as informal formative assessment to see whether students are understanding the concepts as they are covered in class. Checking in with individual students to see their headlines can aid with this. By having students write their headlines individually (and give “a little more of the story”), I will be able to see if there are conceptual misunderstandings or gaps in instruction, as well as getting an idea of how deeply each student is understanding the material.

– Optional: This final piece of the activity can be used as a formal assessment at the end of this chapter.

__Notes to Teacher__:

– Remind students of some strategies of writing headlines! Fewer, more powerful words are often more effective in capturing attention than more words.

__Sources__:

Ritchhart, R., Church, M., Morrison, K., & Perkins, D. (2011). *Making thinking visible: how to promote engagement, understanding, and independence for all learners*. San Francisco, CA: Jossey-Bass.

**3-2-1 Bridge: Solving Systems of Equations**

__Overview/Objective__:

This activity is meant to help students gather their preexisting knowledge regarding solving systems of equations prior to beginning the unit. From my experience, most students have some knowledge about solving systems of equations from Algebra 1. By completing the second step of 3-2-1 Bridge at the end of the chapter, students will have the opportunity to see how their knowledge and understanding has shifted/grown as a result of the learning.

This is a 2-part activity. The first 3-2-1 and discussion is estimated to take 15-20 minutes, depending on the depth of discussion. The second 3-2-1, Bridge, discussion, and journaling is estimated to take 20-30 minutes, with the possibility of journaling at home.

__Prior Knowledge Needed/When to Teach__: Students will do this 3-2-1 Bridge activity at the beginning and end of the unit on systems of equations in Algebra 2.

__Standards Addressed in this Unit__:

A-CED.A.2 Create equations in two or more variables to represent relationships between quantities, graph equations on coordinate axes with labels and scales.

A-CED.A.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.

A-REI.C.5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

A-REI.C.6 Solve systems of linear equations, exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

A-REI.C.8 Represent a system of linear equations as a single matrix equation in a vector variable.

A-REI.D.11 Explain why the x-coordinates of the points where the graphs of the equations and intersect are the solutions of the equation ; find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations.

__Materials__:

– student computers

– projector for class

__Outline:__

**Set Up:
**– Students will record their thoughts digitally in two places:

1) they will submit a Microsoft Form (similar to a GoogleForm, but Microsoft, as my school is now using a Microsoft platform) with their answers so that I can see individual responses

2) they will submit part of their responses using a digital resource called Answer Garden (detailed later)

– Note: this can also be done using a piece of paper, but the discussion period will look different since answers cannot be displayed.

**First 3-2-1:**

Students will go to bit.ly/frydrykMTVsystemsform (sample shown). They will have 2-3 minutes to fill out the form that asks for:

– **Three Words **that come to mind when they hear the phrase “solving systems of equations algebraically”

– **Two Questions **that come to mind when they hear the phrase “solving systems of equations algebraically”

– **One Metaphor/Simile** for “solving systems of equations algebraically”

– While on that webpage, students are linked to an Answer Garden where they will submit their 3 words. The system will populate the projector screen with their classmates’ answers in the class for all to see (sample answers have been populated in this screen, shown below).– This same process will be repeated with the phrase “solving systems of equations graphically”.

**Debriefing/Discussion: **Two different pieces will be used to discuss and debrief this initial 3-2-1:

– The Answer Garden image (sample shown above)

– The 2 Questions and 1 Metaphor students provided (no names will be displayed) from an Excel spreadsheet

**Provide an Instructional Period:
**Over the next 10 – 12 instructional days, students will learn about how to solve systems of equations in both two- and three-variables in multiple ways (ex. guess-and-check, graphically, tables, substitution, elimination, matrices).

**Perform the Second 3-2-1:
**Students will do the 3-2-1’s again, this time they will write their answers on paper.

**Share the Thinking: Bridging:
**– I will pass out a print-out to each student with their answers from the first 3-2-1’s (easily done by printing out the Excel spreadsheet and cutting the rows per student into strips).

– In pairs, students will trade their papers and read their partner’s 3-2-1 responses from both before and after the unit. Each partner will comment on how they noticed their partner’s knowledge/understanding change as a result of the instruction.

– This discussion will continue individually as a journaling assignment as students explain how their own thinking evolved, as well as what insight having their partner provide feedback gave them.

__Assessment__:

The first 3-2-1’s will be used as both an individual and class pre-assessment to see if there are any topics that can be skipped, if additional depth can be added to a topic, or if differentiation is needed for specific students (high and/or low). I will have access to the results, with student names attached so that differentiation can be done as needed, via an Excel spreadsheet.

The last piece of Bridging where students individually journal (possible outline included in Notes to Teacher) about their Bridge can be used a part of a formal cumulative assessment to see students’ awareness of the content. It can also be used to launch students into an individual or group exploration of topics that they wish to further learn about based on their questions from the second bridge. A third option is to use the Bridge piece as a review before a cumulative assessment is given.

__Notes to Teacher__:

If the journal is to be an assessment, pieces to look for could include:

– Acknowledgement of how their choice of words shifted over time

– Provide an explanation of why they think their choice of words shifted over time

– If their first questions were answered, when? If their first questions were not answered, what resources could help them find the answer?

– Find the answer to one of their two final 3-2-1 questions. Include resources used (online or other).

– Explanation of their similes/metaphors in detail (so that someone who did not understand what a metaphor or simile was could understand the connections that are being made)

– Explanation of what insights their partner was able to provide

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Courtney – You’ve given me so much to think about with your two MTV lessons. As a student of calculus, I went through the motion of derivatives and really retained little. I consider this a tragedy…. By engaging in your use of MTV as well as your perfectly designed learning experience, derivatives would bring a whole new meaning to me. Thank you!

I also really like how you gave advice to teachers reading this post in terms of assessment and differentiation- powerful!

Wasn’t the “I Will Survive” video perfect for you. I agree- your students will be enthralled with it from beginning to end.

Like you, I am always saddened by students who aren’t curious and wonder “why”…. MTV helps that wonder-full demeanor come alive as if the students were toddlers again (the age where they ask “why?” all the time).

Courtney, adapting the Headlines routine to be a periodic review throughout the unit is a great idea. While I do not teach calculus, I can appreciate how it will maintain the connections and keep everything in the forefront. I also really like posting the group headlines in the room for all to see. Adapting that to topics in Algebra 2 is something I would love to try next year! I also think your adapting 3-2-1 into a before and after process is genius also. That is one every Algebra 2 teacher can use. I plan to! I am really interested in, and jealous of, your use of technology in both lessons. Thanks for two great ideas that we call can use.

I love that you used the answer garden, I had never heard of this until recently! From what I understand, the more a word is selected, the larger it gets? If would fascinating to see if elimination or substitution get more love. It always seems to be the case that individuals have a preference for one or the other (forget about graphing! 😐 ). I am curious to how the first 3-2-1 reassessment will help determine if something can skipped. Does this show that they know the words to associate as well as the mathematical process to go with those words?

Loved these lesson outlines, very informative and easy to visualize how the lesson should be carried out.

On a side note, speaking of derivatives… Enjoy this rendition of “I will Survive” if you have seen it yet! It is pretty amazing, so prepare yourself… https://www.youtube.com/watch?v=P9dpTTpjymE

Thanks for the feedback, Grady! The AnswerGarden is all thanks to Kristyn!

That’s a great point about the 3-2-1 preassessment. I think that the preassessment would be most valuable when seeing how developed students’ questions and metaphors/similes are (if they show connections between concepts). It is going to be hard to tell simply by the words that they pick, but maybe preassessment is simply not a good use of this activity?

And that video is AMAZING. My students will definitely be watching that (multiple times??) next year.

I really enjoy your use of technology in your lesson. I like that you are asking for them to write to you 2 questions and 1 metaphor. I also like your use of the visual tool Answer Garden. I think students will find it interesting and beneficial to watch it build in front of them on the projector to see what others in the class are thinking.

I completely agree with Cory’s comment. If you had asked me two weeks ago what a derivative means in mathematics, I probably couldn’t have told you. However I probably could have done some basic solving. It is scary that our students could think similarly and put focus more on computation than meaning. I love that you are helping students understand the meaning.

I also really like your “notes to teacher” section that helps with differentiation and assessment strategies.

Thanks, Audrey! I’ve found that the students in higher math classes are more likely (not always, but mostly) to simply accept what you tell them, and they don’t ask “why” nearly as often as I hope they would. I had one student this past year in Calculus who did ask “why”, and she really pushed me to make sure that I understood the concepts myself (so hard with calc!), so this lesson is dedicated to her — ha! Amy got me jump-started on considering what I would actually assess with their journal if that is the route I go, so shoutout to her!

In your objective statement, you wrote something that I believe is absolutely true. I remember at times working with derivatives became so computational, and I lost sight of what I actually was doing. I was simply preforming operations. you said “I found that students often lost sight of the purpose/role/origin of derivatives” I couldn’t agree more, and I love that you are choosing an activity that will make the dig deeper into their thinking about its meaning. I would certainly encourage the gallery walk you have as optional in the lesson plan.

Hope it goes well for you this year.

Thanks, Cory! I’m really hoping that it helps connect the concepts beyond just knowing that they need to derive in order to solve those problems.