Lesson Plan
Stage 1 – Desired Results
Content Standard(s): (Standards used for this course are the AP Calculus Standards)
EK 2.2A1: First and second derivatives of a function can provide information about the function and its graph including intervals of increase or decrease, local (relative) and global (absolute) extrema, intervals of upward or downward concavity, and points of inflection.
EK 2.2A2: Key features of functions and their derivatives can be identified and related to their graphical, numerical, and analytical representations.
EK 2.2A3: Key features of the graphs of f, f ′, and f ″ are related to one another.
EK 2.2B1: A continuous function may fail to be differentiable at a point in its domain.
Unpacked Standard(s):
Know (Nouns)
Derivative Notation (f’, f’’, dy/dx/ d^2y/dx^2)
Continuity conditions
Differentiability conditions
Concavity
Points of Inflection
Local/Global extrema
Do (Verbs)
Identify key features of a graph
Identify concavity, increasing/decreasing, extrema from the second derivative
Identify features of f from f’ and f’’ and vice versa
Draw derivative curves from a function
Standards Note
The concept of the second derivative and higher order derivatives involves many standards with many concepts to know and skills to master. However, this lesson focuses on the graphical significance of the derivative and sketching curves based upon the features of the original function.
Posted Objective
I can analyze a variety of graphs to understand the graphical significance of derivatives.
The posted objective is simplified, so that students have a single manageable goal for the individual class session. The objective reflects that in this lesson students will solely analyze graphs and understand the graphical significance of the derivative.
Students will set their own personal goals by…
Students will review their self-selected goals based on the rubric that was established at the beginning of the unit.
Progress on students' personalized goals will be monitored by…
Students will continue to use the rubric below to monitor their progress. The closure for today’s lesson will provide the first opportunity for students to demonstrate fully meeting the rubric. Students will re-read the rubric and be prompted to “evaluate” not grade their response on the closure and update their projected performance on the summative assessment as it relates to this objective.
Rules and Procedures:
Democracy Norms: Sovereignty of Public, Freedom, Equity, Individualism and Social responsibility
Student Created Rules (from the beginning of the year):
I will be productive
I will listen, to my peers and my teacher
I will be respectful to my peers, my teacher, and my classroom.
Procedures: Do Now, Closure, Think-Ink-Pair-Share, Group Expectations
Student Grouping Strategies: Students are sorted into groups along two axes communication abilities and mathematical skills. The information for the groupings was determined by the previous summative assessments and the heterogeneous groupings ensure that students are challenged on their weaknesses and can benefit from helping each other. Such groupings are based on the principles in Visible Learning Mathematics (Hattie et al. 2017).
Collaborative Expectations (posted in the classroom, used as best practices for group work)
Stage 2 – Assessment Evidence Directly Aligned to Content Standard
Pre-Assessment (including analysis of the pre-assessment results):
one formal and one informal data point will be used as a pre-assessment.
Students completed problem 3-80 as part of their homework, which leads into today’s lesson. I will evaluate student work on the homework quiz after class, but students self-evaluate using 2-checks if they were able to complete the homework problem on the first try, one check if they were able to find the correct answer after some revision or an X if they were unable to find the correct answer (Jackson 2014).
The second pre-assessment involve the real-world hook. Students will be presented with a reflected parabola representing the position of a rocket. “How could we draw the derivative of this graph and what helpful information would such a sketch provide?” The students may not understand the process of sketching a derivative graph, as it is today’s objective, but students should have rick background knowledge of the significance of the slope of the position graph. I will informally use student responses to guide the beginning of the lesson and future lessons on position and velocity.
Performance Task(s) or Assignment Description(s):
Closure -
“The graph at right will be presented to students and the following questions will be answered in the student’s preferred format:
Over what intervals is the function increasing? Is f’(x) positive or negative?
Over what intervals is the function decreasing? Is f’(x) positive or negative?
Where is f’(x)=0?
Sketch y=f’(x) from this information. “
Homework - The homework for the day consists of the following problem set: 3-87, 3-88, 3-89, 3-91
Self- or Peer Assessments:
Students self grade homework
Students are provided with the answers to all homework questions and then self-evaluate their success at the beginning of class. Students use two checks if they were able to complete the homework problem on the first try, one check if they were able to find the correct answer after some revision or an X if they were unable to find the correct answer (Jackson 2014).
Self assessment on the Closure
Students will self-evaluate their progress towards the larger learning goal by using the above rubric, after completing their Closure activity.
Formative Assessments, Summative Assessments, etC:
Problem 3-83 and Problem 3-85
These problems will be informally observed by the teacher while circulating. Students will be marked with a “+” if they demonstrate strong understanding on the problem and are able to support other students, a check if they have minor misconceptions that can be addressed by another student, and a “-” to be targeted with small group intervention on the following day.
Closure
The closure activity outlined below will comprise the main formative assessment for the day. After self-evaluating according to the process at left, students will report their level of understanding on a sticky note to be compared with the informal data described above.
Stage 3 – Learning Plan – Directly Aligned to Content Standard AND Assessments
Learning Activities:
1 - Entrance and Relevance (10 minutes):
Do Now/Entrance Activity:
Students will be presented with problem 3-80 as their Do Now Homework quiz, to be completed on their homework trackers and then will complete their homework completion on their chapter 1 overview according to the Algebra sandwich model (Jackson 2014).
Curiosity/Context - Begin with projectile motion graph and discussion:
Class will be presented with a gif of the Space-X Falcon rocket booster landing in order to be recycled, as well as a projectile motion graph.
Process Opportunity Students will be asked to draft an answer to the question “How could we draw the derivative of this graph and what helpful information would such a sketch provide?” This will allow students to connect their rudimentary understanding to the real world and process the ways in which the abstraction in the lesson can later be used in mathematical modeling.
2 - Lesson 3.3.1 Part 1 - Think-Ink-Pair-Share (10 minutes):
Connection/Coherence Students will begin working on problem 3-82: “Knowing if a function increases or decreases tells us something, but not everything, about its possible shape. (a) Draw an example of a function that is increasing everywhere. What type of function behaves like this? Is there more than one possible shape? (b) Draw an example of a function that is decreasing, then increasing, then decreasing again. What type of function behaves like this? (c) What type of function infinitely alternates between increasing and decreasing?”
Students will be prompted to Think about the problem, Ink their initial thoughts, Pair with their shoulder partner to come to a consensus and then students will be asked to share their thoughts with the class. Students will have the options to transcribe their thoughts through Pear Deck, write their responses in their notebooks or write their responses on a whiteboard.
3 - Lesson 3.3.1 Part 2: 3-83 (10 minutes)
Coherence “One of the functions below is f and the other is its slope function. Can you determine which function, A or B, is the slope function of the other? How do you know?”
The teacher will circulate monitoring student progress and tracking class understanding on a roster as outlined above.
Process Opportunity: Teacher will prompt students to process the strategies that they utilized to determine which function was the original and which was the derivative.
4 - Lesson 3.3.1 Part 3: 3-84 (10 minutes)
Concentration “Use the graph of y = f(x) at right to complete the parts below. (a) At what values of x does f change from increasing to decreasing or decreasing to increasing? What is f ′(x) at these points? (b) State the intervals where f is increasing. What is true about f ′ over these intervals? (c) Using your answers from parts (a) and (b), sketch the graph of y = f ′(x).
Students will complete problem 3-84 with their study teams and record their answers in their notes. Teacher will circulate to ensure student notebook expectations are being followed.
5 - Lesson 3.3.1 Part 4: 3-85 (10 minutes)
“Sketch the slope function for each function below.” -
Students will be allowed to choose which slope function they will sketch based upon their level of comfort. Students will then share their responses with their group and one student will be selected to be the “Traveling Salesman” to rotate to their neighboring group and compare and revise their sketches.
Coaching Prior to the traveling salesman activity, the teacher will conduct a huddle to meet with the travellers to provide guided feedback on the shape of their slope functions and presenting it to others.
6 - Lesson 3.3.1 Part 5: 3-86 (10 minutes)
CURVE CONSTRUCTOR, Part One
“Computers often have drawing programs that allow users to create pictures and diagrams. Your firm is designing software that will allow users to construct curves so they do not have to draw them by hand.
As the main software designer, you need to provide the user the option of drawing all different kinds of curves.
Your boss has asked you to design a set of four screen icons that will create arcs by clicking and dragging. The direction and size of the arc depends on where you click and where you drag the mouse. The user can use this button to create each part of a long, interesting curve.
Consider all the ways that you can sketch a curve from one corner of a square to the opposite corner. Sketch these on sticky notes. Are four icons enough, or do you need more? Explain.
Get creative! Without rotating the sticky notes, connect them head to tail to create as many different known functions as you can. Can you build a parabola, a cubic, an exponential function? What else can you create?
Your first assignment as a graphic designer is to use these four icons (the four curves of calculus) to sketch f(x) = sin(x) over the interval [0, 2π]. Which icons will you select, and in which order? Work with your team to build the sine wave by connecting the sticky notes. Then sketch it on your paper—be sure to identify the points where one arc begins and another ends.
Under each curve of the sine wave, answer these questions about f(x) = sin(x) on that domain.
Is f(x) positive or negative? Increasing or decreasing? Concave up or concave down?
Is f ′(x) positive or negative? Increasing or decreasing?
Is f ″(x) positive or negative?”
Students complete this problem with their study teams. Teams will create their composite sine wave for part (c) on a piece of cardstock to share to add a sense of permanence and have a reminder for the second curve constructor activity which will take place later in the unit.
7 - Closure: (10 minutes)
“The graph at right will be presented to students and the following questions will be answered in the student’s preferred format:
Over what intervals is the function increasing? Is f’(x) positive or negative?
Over what intervals is the function decreasing? Is f’(x) positive or negative?
Where is f’(x)=0?
Sketch y=f’(x) from this information. “
Students will complete the closure problem silently and individual and will be prompted, “Complete the closure individually, it is important for me to see where you are at currently. Don’t worry where your level of understanding is right now. The principles from today’s lesson will be repeated in forthcoming lessons.” This phrasing is designed to prevent discouragement for students that do you yet currently understand
8 - Closure Reflection and Rubric Self-Assessment (10 minutes)
After completing the closure activity students will take out their Chapter 3 Overview document, which includes the rubric above and self-evaluate their progress towards the learning goal of Sketching Derivatives. Students will then update their long term summative goal if needed. All students will then record their self-reported scores on a sticky note to be collected and analyzed by the teacher.
Stage 4 – Feedback Strategies, including Timeliness (Module 5)
The “Algebra Sandwich” model (Jackson 2014) and the Closure self-assessment allow students to be reflective on the progress and have an immediate sense of their progress.
Homework quizzes are grading daily and returned to students on the following day, so that they also receive a small portion of teacher feedback on a consistent basis.
Students are urged to follow metacognitive strategies and the class is working towards student self- and peer-assessment is the primary driver of growth, as teacher decentralization will increase student mastery and long term self-sustainability.