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I have really enjoyed the 4 corners activities that we have tried. They help the students to see problems written in the 4 different formats. I believe the students get a deeper understanding of the skill as well as exposure to different styles of questions which is important for the AP exam. -Jessica Beck

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I like the idea of having students get the same problem, but written in different ways that they may see it in the AP. They can work alone, then together, to help them fully understand the notation and the material. -Effie O.

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Unit 6: Look for a pattern; Unit 7 Collaborative poster; Unit 8: Round Table

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Unit 6: Look for a pattern; Unit 7 Collaborative poster; Unit 8: Round Table

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Look for a pattern:
I like the fact that while the students doing this activity identify the patterns of error they will eventually establish the foundation for integrating using substitution.

Collaborative poster:
Posters can be a way to get students show their creativity, therefore I fully felt attracted to this activity where students will be assigned different roles and the solve a free response question. Great idea

Unit 8: Round Table
This collaborative activity is interesting because by passing the problem to their classmates, they will be able to give feedback to each other and check what they have done previously. I like that the activity comes with links with the FR problems. Very handy!

Luis Alvarez

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Unit 6: Look for a Pattern; Unit 7: Collaborative Poster; Unit 8: Scavenger Hunt

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Unit 6: Look for a Pattern; Unit 7: Collaborative Poster; Unit 8: Scavenger Hunt

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Look for a Pattern:
I like Look for a Pattern because it has the students doing the antiderivatives and checking what is wrong with the original. I have that when students' can identify mistakes in problems, then can usually find their own when doing them by themselves.

Collaborative Poster:
I like this one because it allows the students to work together and individually looking at different parts of the problems.

Scavenger Hunt:
I like this one because it allows the students to walk around the room and they can start at any point. Then you can go over the questions after to see who got them correct.

Jennifer J.

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Additional Strategies: Debriefing, Distractor Analysis, and Identify a Subtask

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Additional Strategies: Debriefing, Distractor Analysis, and Identify a Subtask

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The end of the CED has a plethora of extra instructional strategies that I like more than the sample strategies given in Units 6, 7, and 8 (these start on page 206).

Debriefing asks students to summarize the steps used to solve a problem, and I think this can be applied especially well in Unit 6 when students are introduced to Riemann sums and antidifferentiation strategies, especially in regards to u-substitution.

Distractor Analysis asks students to look at the incorrect responses on multiple choice questions to identify the mistakes or errors in thinking that might lead to them. Although this strategy is useful throughout the year, I think it would work especially well in Unit 7 as students look at differential equations, slope fields, and translating between the two.

Finally, Identify a Subtask asks students to break down more complicated problems into discrete units. While this strategy is also useful throughout the course, I recall often feeling overwhelmed by solids of rotation when I was in my calculus courses. Having been instructed to break down these problems into subtasks would have been exceptionally helpful when I was learning these techniques, so I'd like to show my students how to do this.

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This is a great activity because students work in groups of 4, but each person in the group has a different role, basically responsible for different steps within the integration process. Groups are then assigned one of four FRQs to work through. I think that collaborative activities are great for students as they get to hear how their classmates process things. I also really like that this activity is an opportunity for students to get exposure to more FRQs. (Tim Carlson)

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Mathematical Practice 4: Use of correct notation, language, and conventions to communicate results. pg. 219

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Mathematical Practice 4: Use of correct notation, language, and conventions to communicate results. pg. 219

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I feel like I have seen a great deal of activities, which is wonderful. So, I decided to look through the use of proper notation and justification which I believe will help my students get as many points as possible on the AP exam. There is a list of great questions that students should ask themselves when practicing problems. I am thinking about listing a few important ones on the board as a reminder. I might change up the questions as needed. -Alanda

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Unit 6 Odd One Out, Unit 7 Collaborative Poster, Unit 8 Round Table

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Unit 6 Odd One Out, Unit 7 Collaborative Poster, Unit 8 Round Table

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Odd one out is a great way to have students identify the appropriate strategy for integrand. Having them work through long division and completing the square. Practicing between strategies will help them identify the best strategy.

Collaborative poster is a great way to have students work together and be creative. I like that the roles in the group are already laid out.

Round table an awesome way to have students complete all the steps of solving a problem but work together at the same time. I think it would be great to have students ask each other questions, check each others work and learn from each other.   - Kaley

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Unit 6: Quickwrite, Unit 7: Match Mine, Unit 8: Scavenger Hunt

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Unit 6: Quickwrite, Unit 7: Match Mine, Unit 8: Scavenger Hunt

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Quickwrite is a great warmup activity. Students only get 5 minutes to write the common elements between expressions then they need to share out with the class. This gets students the feeling of the time crunch they will have during the AP exam. It also gets students listening to observations from classmates on similarities they missed.

Match Mine is a great way for students to use mathematical vocabulary and test their knowledge. Using an activity where students have to describe graphs to get their partner to get the graphs in the same grid tests the knowledge of graph descriptors.

Scavenger Hunt is a good activity to get students out of their seats and moving. I like how they suggest it so that the problems are in a circle. A student can start with any problem and when they are done, they will end up where they started.
-Stephanie Yung

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This activity involves students passing around their papers checking the different parts of a free response question and continuing on until the process is complete. I want to do extra activities and practice on the topic mentioned here - Volumes with Cross Sections. I remember learning this myself towards the end of curriculum in Calculus and not feeling as strong about it. This activity would be a great way to get students further practice on that topic and further practice on FRQs. - Sarah Cossack

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I had to go to the end of the CED for a better explanation of this activity. It seems like this activity works best in a group of about 4 students. Each student would receive the same problem and they would have some time to work independently on the problem. Then you would have the students get together in their groups to discuss what they each did and fix any mistakes that were made. Once everyone agrees on an answer, you would have one person from each group share out to the class. I like this idea especially because it allows students to think independently first. Too often, group work is taken over by the "smart" kid or the kid who knows it the quickest. Giving them individual time first allows for more students to be able to equally participate in the group discussion.

Theresa Graben

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I actually saw this later in the CED but I love the idea. It seems like conferences always have you do this. But making one for differentiation techniques that the students add to throughout the unit could be a fun visual reminder/review. -Jessica Beck

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I really like this activity because I think students are always working with equations and have less experience with graphs that are presented with no equation.  Being able to solve questions without the equation requires different skills and is valuable when preparing for the AP test.
- Jennifer L

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Unit 3&4: Round Table, Unit 5: Critique Reasoning

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Unit 3&4: Round Table, Unit 5: Critique Reasoning

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Round table is a great idea to get students to check their classmate's work. This could also be collaborative learning because students have the responsibility to check classmates' work and if they are all responsible in the end they know that their paper will have all the correct work. This puts the responsibility of their assignment on the whole group and not just one person. In unit 4 the suggested round table gives students multiple representations of the same problem which can come in handy for studying or just to see that one concept can have multiple representations.

Critique Reasoning is a great way for students to have mathematical conversations and learn how to give and get constructive criticism. Students don't always know how to give feedback to their peers in a mathematical setting and this strengthens that process for them.
Stephanie Yung

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This activity was posted in Unit 3 in the CED. I like this activity because it is a quick efficient way for students to get a lot of review and practice. It gives students an opportunity to not only be tested on what they know with regards to the content, but it also gives students a chance to hear how their classmates process their own understanding. (Tim Carlson)

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Unit 2. Error Analysis; Unit 3. Scavenger Hunt; Unit 4. Create representations; Unit 5. Predict and Confirm

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Unit 2. Error Analysis; Unit 3. Scavenger Hunt; Unit 4. Create representations; Unit 5. Predict and Confirm

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Error Analysis: I find this activity to be very interactive amongst students since they get to check their peers work, spot the error and explain why it its wrong. This helps them obtain a deep understanding of the process they follow to solve a problem.

Scavenger hunt: At this age students may find classes less interactive as they were in the past, so bringing a nice activity that involves competition is always a good point for them to energize, in this case mixing a scavenger hunt with derivatives, students will solve the problems and look for the card with the solution and so on, they can be divided in teams and achieve a common goal.

Create Representations: I liked this activity because students will also they listening skills through an engaging activity where they will create a roller coaster, graph, identify how it behaves (velocity acceleration, position). Interesting activity.

Predict and confirm: students will be able to confirm their knowledge about the behavior of f'given f by sketching and debating among themselves why the graph is like they think it is, after they do so use techonogy, an activity I will be using in my class.

Luis Alvarez

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Odd One Out for Unit 2: Differentiation (Definition and Fundamental Properties)

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Odd One Out for Unit 2: Differentiation (Definition and Fundamental Properties)

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I'd like to use Odd One Out during this unit because it allows students to work through various problems and have a discussion as a group as to why each person thinks one problem is different than the rest. Because this is the beginning of the course, these nuances are helpful to recognize in such an informal setting. -Alanda

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Graphic Organizer, Round Table, Find the Error

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Graphic Organizer, Round Table, Find the Error

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These are all identification strategies that can be employed throughout the units. I like the graphic organizer to summarize the progression of differentiation. It will also be useful when integration is introduced so they can see all parts in both orders.

From there, it's about creating work, receiving feedback, and understanding how to present your knowledge. Find the error activities are some of my favorite and I definitely plan to use these. Sometimes errors can be so small and this really forces the students to reflect on the topic and consider all aspects. This is a little different from the Round Table work because you are already assuming there is an error with the work. The Round Table is a great way to support each other and learn from each other. You can see peers' work, ask questions about different approaches, and begin to widen your own approaches. The student will still have the opportunity to review for errors but it's a little different than the Error Analysis.

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Unit 2 Graphic Organizer, Unit 3 Quiz-Quiz- Trade, Unit 4 Quickwrite, Unit 5 Predict and Confirm 

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Unit 2 Graphic Organizer, Unit 3 Quiz-Quiz- Trade, Unit 4 Quickwrite, Unit 5 Predict and Confirm 

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I love any chance to be creative in math. I think all students will benefit from a graphic organizer. Having the students make a place for all the derivative rules in a clean and clear way will allow them to reference back quickly if needed. Having them include the mathematical definition, examples, pictures and hints for each rule will hopefully have them understand and comprehend each rule. Having them add hints and examples will hopefully really be the point that helps them comprehend each rule.

For unit 3 Quiz-Quiz-Trade is a great idea that can be applied to all units! Students working with each other is something I always look for. I think they learn best from each other. I like that they first have to solve it then quiz their partner. They can become an "expert" on their questions and build confidence.

For unit 4 Quickwrite is the perfect way to get students interested and applying the topic in a meaningful way. Students are more connected and engaged when they care about what they are learning about or can apply it to their current life.

Unit 5 Predict and Confirm is a great way to challenge students. I like that they have to try graphing before getting the answer of the rule of the function. I also like the technology piece to the activity.
- Kaley

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Unit 2 Graphic organizer/ Unit 4 Quick write

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Unit 2 Graphic organizer/ Unit 4 Quick write

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I like the graphic organizer creative activity because the process of creating the work will help them solidify the concepts.  And the quick write also gives them the creative license to make the problems their own by finding things that change and creating problems from them.
-Effie O.

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This activity gets students to think through the graphical relationships between functions and their derivatives. I think working with multiple representations really deepens student understanding and helps break the monotony of all the algebraic methods used in calculus.

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The Round Table activities provide students the opportunity to work with peers and critique the work of others. I think this can build in stronger communication skills while reinforcing procedural fluency in problem solving. The strategy can be applied to multiple units, turning it into an instructional routine that students can come back to repeatedly.

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The focus of this activity is to have students describe graphs using mathematical vocabulary for their partner in order to get their partners graphs in the same grid as theirs. This activity is good for collaborative learning and using appropriative mathematical terms in order to describe a graph. I can see this being important because full understanding of math vocabulary is show through describing graphs to their partner. You can get the students who are struggling to learn from their classmates. You could also tap into someone’s competitive side if you put a time limit that is less than the class period to see which pair finished first.
-Stephanie Yung

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The focus of this activity is for students to explain to others their reasoning for their answers, and then the listeners provide feedback. I like this activity because it will help the students practice explaining their reasoning (with correct vocabulary) in preparation for those Free Response Questions on the AP exam. - Sarah Cossack

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I plan to incorporate this strategy into my classroom. I like that the students are grouped in a round table manner of four students with 4 questions. The students all complete the first question on the sheet and then pass it to the right/left. The next student checks the answer of the previous question before completing the second question and so on. This gives students a chance to not only compute but analyze methods of other students and deepen their understanding of content. ~T. Durica

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I really like the idea of using the graphic organizer to create a means for students to keep all of the derivative rules in one place.  They would then be able to refer back to it any time they get stuck on a problem moving forward.  Once integration is introduced, I notice that students sometimes confuse the two, especially when it comes to the trig derivatives! ~T. Durica

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I think that a scavenger hunt around the classroom is a good way to change up the instruction and get the students to be more involved. Then after everyone is finish we can go over to make sure that the students did the derivatives correctly.
Jennifer J

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Error analysis is an activity that I try to integrate in other classes that I teach. In the past, I give students a worked-out problem that has an error and they have to find the error. I like how the description in the CED takes it beyond that. Students first must work out the problem correctly and then go back and redo it with an error. I like that the students are the ones deciding what error to make and whereas I think this leads to a deeper level of understanding. I also like that it suggests that only half of the class redo their problem with an error. I like that when the problems are shuffled, students will not know if they got a problem that is correct or one that has an error.

Theresa Graben

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The graphic organizer activity in the CED is very similar to the derivative rules project that Horvath uses with her class. I especially like this for two reasons: First, it gets students to think creatively and transcribe the rules, and the more we write something, the more deeply we internalize it. Second, it provides students with a tool that they can refer back to and use while studying; this is a skill they can take to future classes as well.

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I like the Four Corners strategy because I believe it will help students deepen their understanding of any topic. Students would be given a sheet of paper divided into 4 sections with the topic name in the middle. Each section of the paper would have a problem from the topic in a numerical, analytical, graphical and verbal setup. I believe this will help students make connections and deepen their understanding, as well as help them create a better visual of the concept. Going back to the matching limits activity, I thought the different representations really helped be gain a better understanding so I am hoping this strategy will help my students.

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I like that the students have to come up with a list of all the strategies used to evaluate limits and then create problems based on that list.  It gets them communicating and verifying their thought process as well as thinking outside the box to create their own problems.  Anytime they can bounce thoughts and ideas off of each other is a great way to gauge their understanding. I would also add in that each group would have to create a specific number and type of problem based on that list so that they all do not choose the same ones to create their sample problems. I may also use this as a World Cafe strategy where students have to explain their problem to students as they move around the room.

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I like that odd one out requires each problem in a group to be solved, but then requires further discussion to see why one problem is the odd one out. This would be a great way to intro a new technique or to review a tricky special case.
-Jessica Beck

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This is one of my favorite practices to focus on in Algebra 2 and Pre-Calculus as well!
I think having students connect multiple representations gives them a toolbox for problem solving and deepens conceptual understanding.
--Kristin Terrigno

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Notation Read aloud and Discussion Groups

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Notation Read aloud and Discussion Groups

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I really like both these ideas and think you could combine them as well. Sometimes students are more ready to make a mistake in a small group than in a large group so practicing the vocabulary in small groups while practicing a learned skill is something I think I will try this year. -Effie

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This activity stuck out to me since I have a heavy geometry background. In geometry, I focus on drawing pictures and visually representing the problem that is given. I love that this activity ties everything together for limits. I'm sure some students will feel stronger about analytical representations while others will find graphical representations easier to read. I think this would be a great group activity so that students can lean each other and come together to make sense of the different representations. Theresa Graben

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To often students can not read a mathematical expression. That is the first step in solving! Students need to know what the question is asking. This activity of having students read aloud gets them in the headspace of how to read limit in analytical form. Literacy is a huge push in many schools and this is a perfect way to incorporate literacy in the math classroom. I like that this activity is a partner one too. Having students work together is the best way for them to learn. I feel like this can be made into a fun activity if you have challenges where you try to read aloud completed/"hard" expressions!
-Kaley

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I think it is important for students to understand the different ways that graphs and limits can be represented. By them coming up with their own equations, graphing them, and looking at the limits, the students can get a better understanding of the material.
Jennifer Joyce

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Notation Read Aloud is something that needs to be done in every class. Students need to learn and talk in mathematical vocabulary. This is a great instructional activity because it helps students communicate math. When the students are struggling, instead of using generic words, they can use correct vocabulary so the teacher is guessing what the student is trying to ask. Also, when the teacher responds, the students know what they are talking about. This is also good for students to have mathematical conversations with their classmates. This way they can help each other when stuck while using correct terms.

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I think it is important to be able to think about a problem in different ways. This would especially be helpful for someone who is stuck on a particular problem. We saw this happen last night with our discussion about limits. It was confusing to understand why Theresa was solving the limit from a particular direction, and someone pointed out how a table could help us see why. - Sarah Cossack

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I am a big fan of notation read alouds and I've used them for years in my Precalculus class for expressions as varied as logarithms and sigma notation. I think it's important that students are able to translate symbolic/analytic expressions into verbal expressions for a number of reasons:

1. If students cannot do this, they are likely to skim over symbolic expressions and not actually take time to look at them because they don't have the vocabulary to process them

2. Knowing how to verbalize an expression allows students to effectively communicate with each other and with teachers

3. Reading notation precisely reinforces conceptual understanding; for example, when talking about Riamann sums, reading an expression as "the sum of f(x) from x=0 to x=2" reminds us that what we're calculating is a sum (addition) and that our x varies between different values (0 to 2)

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This is one of my favorite instructional activities. I like to give students an answer to a problem and then have them work in reverse. Automatically, there will be some students with a response of, "well, that's just the answer." The justification is the key to this activity as "it just is" doesn't work for legitimate reasoning. They have to fill in the gaps and connect both sides of the problem, teaching them how to think about where they're going as they progress through work.