Assignment 3a- N

1 Nermin Fialkowski Dr. Monica Kelly AIL 622 18 September, 2018 Assignment 3A- Design a PBL Learning Event 1. What is the content area? Mathematics: Pre-Calculus (both Honors and Regular) 2. What is the content area standard(s) to be met? Mathematics Content Standards for California Public Schools Calculus 1.0 Students demonstrate knowledge of both the formal definition and the graphical interpretation of limit of values of functions. This knowledge includes one-sided limits, infinite limits, and limits at infinity. Students know the definition of convergence and divergence of a function as the domain variable approaches either a number or infinity: 1.1 Students prove and use theorems evaluating the limits of sums, products, quotients, and composition of functions. 1.2 Students use graphical calculators to verify and estimate limits. 2.0 Students demonstrate knowledge of both the formal definition and the graphical interpretation of continuity of a function. 3. Inquiry Question(s) and Essential Question(s)/Problem to be addressed? Inquiry Questions 1. Before I can reach my desk from the classroom door, I must first walk halfway between the door and my desk. Then I’ll walk another halfway distance, then another halfway, and another halfway. When will I be able to reach my desk? ! " % & !)) 2. How can I find the value of " + $ + & + ' + ⋯ + "** + ⋯ without adding up 200+ numbers? ! 3. What is the value of ? !,***,***,***,***,***,***

Essential Questions 1. How can I predict the behavior of a graph? 2. What conclusions can be made about a graph that never stops growing? 3. What additional conclusions can still be made about a graph that has no value?

2 4. Set up your personalized learning environment to include resources, physical environment and other factors that contextualize the learning. Word Wall Limit

Continuous

Hole

Horizontal Asymptote

Infinity

Does Not Exist

Vertical Asymptote One-Sided Limit

Slant Asymptote Piecewise Functions

Visual Graphs- posted around room

TI-84 Graphing Calculators Desmos Online Graphing Calculator- via student laptops and smartphones Limit Activity (https://teacher.desmos.com/activitybuilder/custom/574de5cdab71b5085a2aad42) College Preparatory Mathematics (CPM) Textbook Pre-Calculus with Trigonometry (https://cpm.org/pct/)

3 Differentiated Learning Students will create their own piecewise graphs (like below) and pose various questions about the graph. Students will then quiz their classmates on the answers.

𝑓(−1)

𝑓(1)

lim 𝑓(𝑥)

4→!

lim 𝑓(𝑥)

4→7

lim 𝑓(𝑥)

4→87

lim 𝑓(𝑥)

4→8%

𝑓(−4)

5. Describe your community of learners. Southwest High School in San Diego is a highly diverse school, just five miles north of the U.S./Mexico International Border. Because of its close proximity to the border, 39% of the school’s population has limited English proficiency, while 85% come from homes where English is not spoken. Southwest High Schools is a Title I school, where 79% of the students are eligible for free or reduced lunch (Southwest High SARC). This unit lesson is intended for my Pre-Calculus classes. I teach both Honors and Regular PreCalculus. Both classes are made up of junior and seniors. The juniors in my classes are advanced learners, having taken Integrated Math II, during their freshmen year, while most take Integrated Math I during their freshmen year. The seniors in my class have a focus of studying something science/math related after high school. This way, students will be prepared to take calculus after high school. The good number of students in my Regular class lack the algebraic skills needed to have proficient procedural fluency skills. They also require more prompting and scaffolds on how to set up specific problems. None-the-less, these students still work very hard and put in their best effort into the class. Students in my Honors class, are considered advanced learners compared to students in my Regular class. These students are individual workers but work well in groups. Their cognitive processing and critical thinking skills are high. Students in my Honors class are also in other advanced classes. 6. What accommodations and adaptations will you incorporate to meet the needs of Special children? Accommodations for my Special Education students are made per their individual needs. Additional accommodations are also inclusive to provide them with success in my classroom. Such accommodations include: promoting, redirection, additional time on assessments and assignments, frequent check-ins, seating next to a supportive peer, flexibility in assignment choice.

4 7. What accommodations and adaptations will you incorporate to include children who are learning English? A large majority of my English Language Learners speak Spanish. Since I also speak Spanish, I can translate directions and further explain concepts to students in Spanish individually on a oneto-one basis. A Word Wall will also be used to help build vocabulary acquisition. Various visual graphs of different functions (continuous and non) will also be displayed around the room to reinforce the concepts of continuity and end behavior for limits at infinity. English Language Learners will also be seated next to a peer within One English Proficiency Level of them. Students will also be provided with sentence frames/starters when participating in class discussions and for explaining/proving why a function is continuous/discontinuous. 8. How do you plan to differentiate instruction to meet the needs of struggling, average and advanced learners? Grouping Students By Learning Style: Reflector, Theorist, Activist, and Pargmatist (Honey & Mumforad, 2006). This grouping allows for all students to successfully access the content material. Differentiated Learning Stations Differentiated learning stations prior to an assessment, in addition to post-assessment. This means having students review the content through various modalities at different stations. Some examples of various stations include, individual work, peer tutoring, re-teaching, and group work. These differentiated learning stations is also a form of intervention. Low performing students can go to the re-teaching station, students who have met the content objectives will go to peer tutoring and group work. Those few student in peer tutoring will help the students that were not quite there in mastering the content objectives. While high achieving students work on extension problems. Differentiated Learning Assignment Students will create their own piecewise graphs (like below) and pose various questions about the graph. Students will then quiz their classmates on the answers.

𝑓(−1)

𝑓(1)

lim 𝑓(𝑥)

4→!

lim 𝑓(𝑥)

4→7

lim 𝑓(𝑥)

4→87

lim 𝑓(𝑥)

4→8%

𝑓(−4)

9. How will you assess prior knowledge? Prior knowledge needed to be successful in this unit include: domain and range, input and output values, and determining y-values from equations and graphs. Assessing this prior knowledge will be done in a variety of different ways. One way is through the form of Daily Warm-Ups, where students will be asked to solve for y-values of functions. Students will also be asked to

5 write their own definition of domain and range and share with a peer. Additionally, students will create a table of values for a given function. Students’ understanding of these prior concepts will guide me in my instruction and pace of this unit. 10. How will you introduce your unit to gain learner interest? Inquiry Questions 1. Before I can reach my desk from the classroom door, I must first walk halfway between the door and my desk. Then I’ll walk another halfway distance, then another halfway, and another halfway. When will I be able to reach my desk? ! " % & !)) 2. How can I find the value of " + $ + & + ' + ⋯ + "** + ⋯ without adding up 200+ numbers? ! 3. What is the value of !,***,***,***,***,***,***?

Videos

1. Achilles & The Tortoise: https://youtu.be/skM37PcZmWE 2. Area of a Circle: https://youtu.be/YokKp3pwVFc 11. How will you use UDL strategies to challenge all learners and to reinforce content and skill development? Universal Design for Learning (UDL) has three main guidelines: Representations, Action and Expression, and Engagement. (CAST) Representations 1. Given a graph, students will intuitively find the limit by using Benny & Bertha the Bug as their approach. What matters is where Benny & Bertha go and if they go to the same place 𝑓(𝑎) does not have to equal lim 𝑓(𝑥) 4→;

2. Limits Student Lab http://www.mesacc.edu/~davvu41111/LimitStudent.html#MapleAutoBookmark5 3. Limits & Continuity: Desmos Activity https://teacher.desmos.com/activitybuilder/custom/574de5cdab71b5085a2aad42 Action and Expression 1. Limits & Continuity: Desmos Activity https://teacher.desmos.com/activitybuilder/custom/574de5cdab71b5085a2aad42 2. Differentiated Learning Assignment Students will create their own piecewise graphs (like below) and pose various questions about the graph. Students will then quiz their classmates on the answers.

6

𝑓(−1)

𝑓(1)

lim 𝑓(𝑥)

4→!

lim 𝑓(𝑥)

4→7

lim 𝑓(𝑥)

4→87

lim 𝑓(𝑥)

4→8%

𝑓(−4)

3. Differentiated Learning Stations 4. Differentiated learning stations prior to an assessment, in addition to post-assessment. This means having students review the content through various modalities at different stations. Some examples of various stations include, individual work, peer tutoring, reteaching, and group work. These differentiated learning stations is also a form of intervention. Low performing students can go to the re-teaching station, students who have met the content objectives will go to peer tutoring and group work. Those few student in peer tutoring will help the students that were not quite there in mastering the content objectives. While high achieving students work on extension problems. Engagement 1. Limits & Continuity: Desmos Activity https://teacher.desmos.com/activitybuilder/custom/574de5cdab71b5085a2aad42 2. Achilles & The Tortoise https://youtu.be/skM37PcZmWE 3. Area of a Circle https://youtu.be/YokKp3pwVFc The purpose of UDL has a focus on “learning” not just on teaching. It is understood that a student has learned and understood the content when they are able to transfer their knowledge. The six facets of understanding include: explain, interpret, apply, shift perspective, empathize, and self-assess (Wiggins & McTighe, 2011). In this unit students will be able to explain if a function is continuous, interpret graphs to identify their limits, interpret functions to determine end behavior, apply their knowledge of limits and functions to create their own piecewise graphs and pose various questions about the graph. In addition to self-assessments, students will then quiz their classmates on the answers of their own piecewise function (Differentiated Learning Assignment). 12. How will you check the progress of your learners? Students will be constantly monitored to check their progress of learning. This will be done formally, and informally. Informally, I will be looking for common misconceptions about limits, so that I can address those concerns early on, before students internalize this misinformation. Formally, students will also have weekly assessments to check their learning, where instruction will be modified to meet the needs of students. Students will also be asked to complete the same tasks from assessing their prior knowledge to check for growth in their understanding of the topics. Students should be able to use the formal definition of continuity to solidify their informal definition of what it means to be a continuous function. Exit Slips will also be used to

7 check for students understanding of how to find a limit, the existence of a point is irrelevant for a limit to be possible. What matters is what the function approaches. 𝑓(𝑎) does not have to equal lim 𝑓(𝑥). The Differentiated Learning Activity where students create their own graphs and pose 4→;

questions about continutity, will demonstrate students’ internalization about how find limits at various values. 13. How will your learners express their understanding of the Essential Question or pose the solution to the Essential Problem? Inquiry Questions The main focus of the Inquiry Questions is to solidify the idea of a limit. The existence of a point (or value) is irrelevant for a limit to be possible. What matters is what value is being approached. Essential Question #1/2 Given a function, students will be able to identify it’s end behavior. Without the aid of a graph, students will use leading coefficients and evaluating limits at infinity to determine and describe the end behavior of a function. Additionally, through one-sided limits and vertical asymptotes students will be able to make conclusions about a functions behavior at undefined values. The purpose of these Essential Questions is for students to definitively make conclusions about a functions’ graph and its behavior even though a visual is not provided. Essential Question #3 The main idea of Essential Question #3 is that the existence of a point is irrelevant for the existence of its limit. 𝑓(𝑎) does not have to equal lim 𝑓(𝑥). But when talking about continuity, 4→;

𝑓(𝑎) does have to equal lim 𝑓(𝑥). Students will be able to independently find the solutions to 4→;

𝑓(𝑎) and lim 𝑓(𝑥), algebracially and graphifically. Students will then make the final conclusion 4→;

if a function is continuous based on 𝑓(𝑎) = lim 𝑓(𝑥). In the process of finding 𝑓(𝑎) and 4→;

lim 𝑓(𝑥) students will be able to describe important features and charateristics of functions even

4→;

though they may be discontinuous, or hold no value/existence. References CAST. Universal Desgin for Learning Guidelines. Retrieved from: http://www.cast.org/ourwork/about-udl.html#.W6gVlhNKiRt Honey, P., & Mumforad, A. (2006). Brainbase, Keeping the Brain in Mind. Kolb’s Learning Styles. Retrieved from: http://www.ycarhe.eu/uploads/Document/learning-styles-kolbquestionnaire.pdf “Southwest Senior High School Accountability Report Card Reported 2016-17”. (2017). [PDF file]. Retrieved from: http://sarconline.org/SarcPdfs/9/37684113730124.pdf Wiggins, G., & McTighe, J. (2011). The Understanding by Design Guide to Creating High Quality Units. Alexandria, VA: Aassociation for Supervision & Curriculum Development.