Possible solution to DA of Applications of Derivatives

Instead of a multiple choice, numerical response and a written response exam on "Applications of Derivatives", I gave my students an open-ended assessment.

Here is one of my student's work. As you can see, students still can demonstrate knowledge and understanding without shading in "c" on a scantron sheet.

DA with Applications of Derivatives

Here is how I am assessing Applications of Derivatives:

New final Exam

Here is what I am giving to my students for the Final Assessment. 

Math 31 Final Assessment

This assessment will be in addition to a written component.  The written component and this assessment will both be worth 15% of your final grade.  In this course we covered many outcomes, the outcomes which are important to success in future calculus courses are:

·         Slope at a point, using first principles.

·         Limits

o   when do they exist and not exist

o   one sided limits.

o   Limits as x approaches infinity

·         Derivatives using, chain, product, quotient, and implicit differentiation rules of functions which have:

o   Trigonometry

o   Polynomials

o   Exponential

o   Natural Logs

·         Absolute and relative maximums and minimums.

·         Related Rates.

·         Curve Sketching.

·         Integration rules of functions which have:

o   Trigonometry

o   Polynomials

o   Exponential

o   Natural Logs

·         Determining area between two functions, both graphically and algebraically.

You must demonstrate your knowledge of all the outcomes anyway you want.  Your presentation can take any form(s) you would like, a powerpoint presentation, a prezi, a video, a skit, etc.  The presentation will be as long as it takes to demonstrate your understanding; however 30 minutes is the maximum the presentation should be.  You will be presenting in front of a panel of judges, one of which will NOT be a math teacher.  You must relate most of your knowledge to a real world application and demonstrate how calculus is used outside of the math classroom.  After the presentation, 10 minutes will be allotted for questioning from the judges.  Your mark will be decided by the judges and based on your presentation and your answers from the questions you are asked.

You may work in groups up to 3; however you will each receive a separate mark and will be differentiated by your individual answers of the questions from the judges.

Here is a rubric I will use to assess their knowledge.

	Superior	Adequate	Minimal	Inadequate
Content	The speaker provides a variety of types of content appropriate, such as generalizations, details, examples and various forms of evidence. The speaker adapts the content in a specific way to the listener and situation.	The speaker focuses primarily on relevant content. The speaker sticks to the topic. The speaker adapts the content in a general way to the listener and the situation.	The speaker includes some irrelevant content. The speaker wanders off the topic. The speaker uses words and concepts which are inappropriate for the knowledge and experiences of the listener (e.g., slang, jargon, technical language).	The speaker says practically nothing. The speaker focuses primarily on irrelevant content. The speaker appears to ignore the listener and the situation.
Organization	The message is overtly organized. The speaker helps the listener understand the sequence and relationships of ideas by using organizational aids such as announcing the topic, previewing the organization, using transitions, and summarizing.	The message is organized. The listener has no difficulty understanding the sequence and relationships among the ideas in the message. The ideas in the message can outlined easily.	The organization of the message is mixed up and random. The listener must make some assumptions about the sequence and relationship of ideas.	The message is so disorganized you cannot understand most of the message.
Creativity	Very original presentation of material; captures the audience’s attention.	Some originality apparent; good variety and blending of materials / media.	Little or no variation; material presented with little originality or interpretation.	Repetitive with little or no variety; insufficient use of materials / media.

Angry Birds and Calculus

During my first unit, instead of assessing with a traditional exam I used an open ended project.  One of my students submitted the following video to answer question 3:

As you can see, he used a timer on his phone, a ruler and the game Angry Birds.  Here is his work

DA with Quadratics

Before, I have assessed Quadratic functions in the traditional sense with worksheets, homework checks and quizzes.  This year I wanted to assess my students' knowledge through an open-ended project and allow for differientated assessment.

Here is the task:

Quadratic Graph Project

1.      Take a digital copy of a picture which has a parabolic shape in it.

2.      Log on to www.prezi.com using the following log on:

***I just created a class log on and password***

3.      Select blank template.

4.      Select insert picture, and import your picture onto the prezi.

5.      Using insert à shapes àlines, draw in two axes.  Neither axis may go through the vertex of the parabola.

6.      Using either the pen tool or the circle tool, create a point on the vertex of the parabola

7.      Label the vertex using points which are appropriate.

a.       Vertex:____________

8.      Label another co-ordinate on the graph.

a.       Point:_____________

9.      Using your vertex and co-ordinate you created, determine the value of a, either as a fraction or a whole number, if your parabola was written in standard form .  Show your work on Prezi.

a.      

10.    Label another 4 points on your graph.  Try to space out the co-ordinates evenly across the parabola.

a.       Point 1:_______

b.      Point 2:_______

c.       Point 3:_______

d.      Point 4:_______

e.       Vertex:_______

11.  We will now have the calculator create the function.

a.       On your calculator, push STAT, then EDIT…

b.      In the first column (L1) input all the x-values, and in the second column (L2) input all the y-values of the points.

c.       Click STAT, then the right arrow  (à) to the CALC menu, then scroll down to QUADREG.

d.      Write down the values your calculator gives you, to the nearest hundredth if necessary.

                                 a:__________

                                 b:__________

                                 c:__________

12.  Inputting your values into general formyour function will be_____________________

13.  Change your general form into standard form, by completing the square.  Show your work in prezi.

a.       Your standard form now is ________________
b.      State the vertex, domain, range, direction of opening, and axis of symmetry from the function above.

14.  Write a couple of sentences explaining any differences from the vertex you stated in part 7, and the vertex in part 13.

15.  Using the equation from Part 9, determine the functions x and y intercepts, if the picture was extended such that it intercepts both axes.  Solve this part by graphing.

a.       x-intercept________

b.      y-intercept________

16.  Using the equation from Part 12, determine the functions x and y intercepts, if the picture was extended such that it intercepts both axes.  Solve this part by the quadratic formula.

a.       x-intercept________

b.      y-intercept________

Here is an example of my student working up to Section 14.

Grading with O'Connor

Recently I read through "How to Grade for Learning" from Ken O' Connor, who brings up some GREAT points about grading.  Here is synopsis of my own learning:

First and foremost.... "Learning can (and does) happen in the absent of grades!"...This was the biggest myth I believed for quite a while when I first started teaching.  I truly believed that grades were the ONLY way learning could occur...I was wrong (the first and last time I have ever been wrong :) )

Grant Wiggin even said

"Perhaps the most counterproductive aspect of schooling as we know it is the conventional system of letter grades. The problem with grades is not the use of symbols, but the absence of any DEFENSIBLE plan for coming up with the symbol… most grades reflect what is easy to count and average into a final grade.”

O' Conner brought up his 7 persepectives underlying grading:

• Grading is not essential for learning.

• Grading is complicated.

• Grading is subjective & emotional.

• Grading is inescapable.

• Grading has a limited research base.

• Grading has no single best practice.

• Grading that is faulty damages students and teachers!

I have heard many teachers express their views that grades truly motivate students into learning, however O'Conner says:

If grades are extrinsic motivators, they can destroy intrinsic motivation. Good grades may motivate, but poor grades have no motivational value.

Even further, Gusky has said:

“Low grades push students farther from our cause, they don’t motivate students. Recording a D on a student’s paper won’t light a fire under that student to buckle down and study harder. It actually distances the student further from us and the curriculum, requiring us to build an emotional bridge to bring him or her back to the same level of investment prior to receiving the grade.”

Lastly, Marzano (after researching over 8000 studies) said:

The most powerful single modification that enhances achievement is FEEDBACK. The simplest prescription for improving education must be ‘dollops of feedback’.”

And now the big finale...here are 15 "fixes" you can do TODAY to change your assessment:

#1 - Don’t include student behaviors (effort, participation, polite manners, etc.) in grades; include only achievement.

#2 - Don’t reduce marks on late work; provide support for the learner.

#3- Don’t give points for extra credit work or use bonus points; seek only evidence that more work has  resulted in a higher level of achievement.

#4- Don’t punish academic dishonesty with reduced grades; apply other consequences and reassess to determine actual level of achievement.

#5 - Don’t consider attendance in grade determination; report absences separately.

#6- Don’t include group scores in grades; use only individual achievement evidence.

#7 - Don’t organize information in grading records by assessment methods or simply summarize into a single grade; organize and report evidence by standards/learning goals.

#8 - Don’t assign grades using inappropriate or unclear performance standards; provide clear descriptions of

achievement expectations.

#9 - Don’t assign grades based on student’s achievement compared to other students; compare each student’s performance to preset standards.

#10- Don’t rely on evidence gathered using assessments that fail to meet standards of quality; rely only on quality assessments.

#11 - Don’t rely only on the mean; consider other measures of central tendency and use professional

judgment.

#12- Don’t include zeros in grade determination when evidence is missing or as punishment; use alternatives, such as reassessing to determine real achievement or use “I” for Incomplete or Insufficient Evidence.

#13- Don’t use information from formative assessments and practice to determine grades; use only summative evidence.

#14 - Don’t summarize evidence accumulated over time when learning is developmental and will grow with time and repeated opportunities; in those instances, emphasize more recent achievement.

#15 - Don’t leave students out of the grading process. Involve students; they canand should- play key roles in assessment and grading that promote achievement.

Last, Reeves says

“If you wanted to make JUST ONE change that would immediately reduce student failure rates, then the most effective place to start would be challenging prevailing grading practices.”