BlendEd Best Practices - Applications of Differentiation
Unit Title: Applications of Differentiation
Title and Description
This unit is titled Applications of Differentiation
In this unit students will analyze and manipulate functions in order to apply them to real-life optimization (minimum/maximum) situations.
The
resource was submitted by Melinda Stelling, Pierce Jr./Sr. High, as part of the Nebraska ESUCC Special
Project BlendEd Best Practices
Content Area Skill:
Add Skill(s)
Analyze, Manipulate, and Apply Functions using Derivatives
Digital Age Skill(s):
Add Skill(s)
Empowered Learner
Knowledge Constructor
Computational Thinker
Creative Communicator
Duration of Unit:
Approximately 12 days
Overview of Unit:
In this unit students will analyze and manipulate functions in order to apply them to real-life optimization (minimum/maximum) situations.
Empower Learners:
Content Area Skills (NE and ISTE Standards):
MA12.3.2.a (AT) Identify features of a function (e.g., local and global maxima and minima, concavity, approximate locations of points of inflection and vertical and horizontal asymptotes) from its graph.
MA12.2.1.a (AT) Analyze and graph non-linear functions (e.g., quadratic, trigonometric, square root, logarithmic, rational, higher-order polynomials, exponential, absolute value, piecewise, and sinusoidal).
MA11.3.3 Students will perform and compare measurements and apply formulas.
MA11.2.1.d Identify domain and range of functions represented in either algebraic or graphical form.
MA11.2.3.a Analyze, model, and solve real-world problems using various representations (graphs, tables, linear equations and inequalities, systems of linear equations, quadratic, exponential, square root, and absolute value functions).
MA11.2.2.f Solve an equation involving several variables for one variable in terms of the others.
Student Friendly Objectives:
MA12.3.2.a I can identify important features of a function such as extrema, concavity, points of inflection, and asymptotes.
MA12.2.1.a I can analyze and graph non-linear functions.
MA11.3.3 I can apply formulas to compare measurements.
MA11.2.1.d I can identify the domain of a function.
MA11.2.3.a I can analyze, model, and solve real-world problems using graphs and functions.
MA11.2.2.f I can solve an equation involving several variables for one variable in terms of the others.
Empower Learner Activity:
Detailed Description:
Students will take a Pre-Test using the “Applying Derivatives to Analyze Functions” Unit Test on Khan Academy. - Students will set goals for the unit based on the learning targets and pre-assessment data, on www.sowntogrow.com.
- Students will self assess each student-friendly learning target using a likert scale on www.sowntogrow.com
Knowledge Application:
The purpose of Knowledge Application is to promote deeper-level learning through student creation of learning artifacts demonstrating both Content Area and Digital Age Skills. A key goal is to provide some degree of control over how students will show what they know.
Artifact Profile:
Title of the Artifact:
Container Optimization Investigation
Detailed Description:
Students will choose three cylindrical containers to analyze. After finding the necessary measurements, calculations will be performed to determine the most “volume optimized” container based on the surface area.
Content Area Skills Addressed:
MA12.2.1.a
MA12.3.2.a
MA11.3.3
MA11.2.1.d
MA11.2.3.a
MA11.2.2.f
Digital Age Skills
Knowledge Constructor
Computational Thinker
Creative Communicator
Link to Rubric
Knowledge Deepening:
Detailed Description:
During Knowledge Deepening, the Content Area Skills required of the Learning Artifact are broken down into bite-sized tasks. A key goal is to provide some degree of choice over how students practice the Content Area Skills.
Tasks
Task 1: Understand and find critical numbers and extrema and understand and use The Extreme Value Theorem.
Must Do’s: Watch Extreme Value Theorem, global vs. local extrema, and critical points videos(see choice board) OR read textbook Section 3.1. Do assigned textbook exercises. |
May Do’s: Watch more videos for clarification, do practice exercises on Khan Academy, read Khan Academy article, read textbook, meet with teacher as group or individually. |
Resources: |
Task 2: Understand and use Rolle’s Theorem and the Mean Value Theorem.
Must Do’s: Watch Rolle’s Theorem video and at least first 3 videos on Mean Value Theorem(see choice board) OR read textbook Section 3.2. Do assigned textbook exercises. |
May Do’s: Watch more videos for clarification, do practice exercises on Khan Academy, read Khan Academy articles, read textbook, meet with teacher as group or individually. |
Resources: |
Task 3: Determine intervals on which a function is increasing or decreasing and apply the first derivative test to find relative extrema of a function.
Must Do’s: Watch 1st five Increasing/Decreasing Functions and 1st Derivative Test videos on either Khan Academy or Hippocampus(see choice board) OR read textbook Section 3.3. Do assigned textbook exercises. |
May Do’s: Watch more videos for clarification, do practice exercises on Khan Academy, read Khan Academy articles, read textbook, meet with teacher as group or individually. |
Resources: |
Task 4: Determine intervals on which a function is concave upward or downward, find points of inflection of a graph of a function, and apply the second derivative test to find relative extrema of a function.
Must Do’s: Watch 1st six plus Second Derivative Test videos on Khan Academy or all six Hippocampus videos (see choice board) OR read textbook Section 3.4. Do assigned textbook exercises. |
May Do’s: Watch more videos for clarification, do practice exercises on Khan Academy, read Khan Academy articles, read textbook, meet with teacher as group or individually. |
Resources: |
Task 5: Determine limits at infinity and horizontal asymptotes.
Must Do’s: Watch four Hippocampus videos. (see choice board) OR read textbook Section 3.5. Do assigned textbook exercises. |
May Do’s: Watch more videos for clarification, do practice exercises on Khan Academy, read textbook, meet with teacher as group or individually. |
Resources: |
Task 6: Analyze a function to sketch its graph.
Must Do’s: Watch either the Khan Academy or Hippocampus Curve Sketching videos.(see choice board) OR read textbook Section 3.6. Do assigned textbook exercises. |
May Do’s: Watch more videos for clarification, read textbook, or meet with teacher as group or individually. |
Resources: |
Task 7: Solve optimization(minimum/maximum) problems.
Must Do’s: Watch two Khan Academy and one Hippocampus Optimization (min/max) videos. (see choice board) OR read textbook Section 3.7. Do assigned textbook exercises. |
May Do’s: Watch more videos for clarification, do practice exercises on Khan Academy, read textbook, meet with teacher as group or individually. |
Resources: |
Task 8: Find differentials of functions.
Must Do’s: Watch Differentials video. (see choice board) OR read textbook section 3.9. Do assigned textbook exercises. |
May Do’s: Read textbook, meet with teacher as group or individually. |
Resources: |
Direct Instruction:
Detailed Description
Direct Instruction scaffolds the learning process for all students. A key goal is to incorporate BlendEd strategies to effectively differentiate instruction in order to maximize learning.
Learning Path:
| Day # | Description | BlendEd Model | Resources |
| 1 | Pre-Test: Applying Derivatives to Analyze Functions Unit Test on Khan Academy Task 1: Extreme on an Interval |
Flipped |
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| 2 | Task 1 continued Task 2: Rolle’s Theorem and the Mean Value Theorem |
Flex and Flipped |
|
| 3 |
Task 2 continued Task 3: Increasing and Decreasing Functions and 1st Derivative Test |
Flex and Flipped | |
| 4 | QUIZ over Tasks 1&2(not included due to copyright) Task 3 continued Task 4: Concavity, Points of Inflection, and the 2nd Derivative Test |
Flex and Flipped | |
| 5 | Task 4 continued Task 5: Limits at Infinity |
Flex and Flipped |
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| 6 | QUIZ over Tasks 3&4(not included due to copyright) Task 5 continued Task 6: Sketching Curves of Functions |
Flex and Flipped |
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| 7 | Task 6 continued Task 7: Optimization(Min/Max) Problems |
Flex and Flipped |
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| 8 | QUIZ over Tasks 5&6 (not included due to copyright) Task 7 continued Assign Optimization project Task 8: Differentials |
Flex and Flipped |
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| 9 | Task 8 continued Optimization project continued |
Flex | |
| 10 | Finish Optimization Project Post-Test: Applying Derivatives to Analyze Functions Unit Test on Khan Academy |
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| 11-12 | Present Optimization Project |
Resources:
Provide a list of resources