Students create the problem to solve

Math 31 Assessment

Option 1: The head of NASA has approached you asking for your assistance plotting a course for the International Space Station.  The tracking device can be found at http://spaceflight.nasa.gov/realdata/tracking/index.html .  NASA needs an equation of the path, as well as the 1^st and 2^nd derivative.  The director is also asking for all relevant information about the path to be explained.  Create a potential problem the satellite might encounter and provide the solution to the problem.

Option 2: Ethan Hunt, an IMF point man, is securing top secret files at the bottom a cylindrical tube.  He is being lowered by rope into the room at a constant speed.  Unfortunately, while being lowered, another man walks, at a constant rate, towards the room.  Ethan is then raised back up out of the view of the approaching man.  While suspended in air, sweat is building at a constant rate until it reaches a critical value and drops onto the floor.  Using the video (a clip can be found here http://www.youtube.com/watch?v=k-oVuQpjG3s) use calculus to help out Ethan and determine various velocities, critical values, and timing to assist him in securing the evidence.

Option 3: Taking what you’ve learned this semester you must create a problem to solve. It must be based on a real world example (or movie world) and must have at least one solution. Be sure to submit your proposed solution in a sealed envelope. Five bonus points will be awarded if you are able to stump your teacher.

No matter which option you choose you must create the question and problem in the given scenario

	Calculus Student	Calculus Student Teacher	Calculus Teacher	Calculus Master
Real Life Application: Is the problem worthwhile solving? What are the implications of the result of the problem? Who would benefit with the knowledge of the answer to your problem?	Only students of this course would see the relevance of this problem. The purpose is built on strictly recalling facts. The solution is only needed to complete this assignment.	Problem is created from a plausible issue with major changes. Purpose is unclear and does not go beyond the needs of the course. Students in this course would only understand the consequences of the solution to the problem.	The problem is created from a plausible issue or problem with minor changes. The purpose is clear and slightly exceeds the needs of this course. Few, outside this course, would benefit from the solution to the problem.	The problem is created from a real life issue or problem. The purpose is meaningful beyond the needs of the course. The solution to this problem adds to the experience of the students’ real world knowledge.
Explanation of math. Are you using basic math knowledge? Are you demonstrating most of the knowledge you learned in this course?	Focuses strictly on basic recall and basic knowledge of the mathematical skills.	Requires few sections to apply higher level thinking to solve the problem. Math is still largely focused on recall of knowledge.	All levels of understanding, from basic to higher level thinking are implemented throughout the problem.	Focuses on higher level comprehension, the use of the combination of multiple skills is evident throughout the entire solution.
Communication of your answer: Does your work follow sequentially throughout your solution? Are there gaps in your communication?	Minimal response is given with multiple gaps in the explanation process.	The use of similar explanation techniques is used throughout the solution.  Communication is limited with various gaps in the problem solving.	Most of the project is easily understood, and organization is mostly logical.	The use of different means of demonstration is illustrated throughout the solution.  The mathematics is clearly communicated as well as the meaning of the solution(s)

More on Differentiated Assessment

A couple of weeks ago, I was watching a dad teach his son how to ride a bike.  The son had a helmet on, elbow pads, and training wheels on the bike.  As the dad put his son on the bike, he walked behind his son as the son rode the bike in circles in the parking lot.

Just recently, I witnessed the same father and son in the school’s parking lot and this time the training wheels were off.  The dad continued to walk behind the son and the son completed the same circles.  At one point the son fell over and the dad quickly picked him up and put him back on the bike immediately.  After about 10 minutes, where the son did not fall once, the dad stopped walking behind the child and the child started to do more complex paths on his bike.

This is how assessment should be!

It would be ridiculous to mandate that all fathers must spend exactly 10 minutes of time with their child until they stop walking behind them; as each child will require a different amount of time to learn the skill. 

It would be ludicrous to allow a son to write a multiple choice test where, if he scored over 50% (even though he wrote it is ok to play in traffic on a bicycle), the father would let him ride alone as the son “passed the test”, since the son doesn’t understand all the safety issues of riding a bike.

It would be unfortunate if all fathers were required to purchase the same bicycle since not every child is the same height, or has the lower body lengths.

Yet all of these ideas are allowed in schools, why is that?

I wonder what school would look like if instead of holding teachers accountable with mandated common assessment we instead allowed teachers to teach students “how to ride a bike”?

I believe, students would learn the skills at a deeper level before moving on, they could learn at their own pace, and each “test” would be different for each student.

Still not convinced?  Reflect on this picture as it represents the traditional testing model of students.

Why we should not have set deadlines in school.

I wonder what school would look like if we didn’t have set timelines or completion dates for the assessments of students.

This is the thought I wanted to address this year in one of my classes.  Instead of having set dates for exams, and a set timeline for project dates, I created a learning environment that is conducive for the needs of every single student in my class.

Let’s first look at the problem of having a timeline for when students must demonstrate their knowledge.

Usually, a teacher makes their year plan around the goal of covering all the outcomes of the course.  This teacher must make predictions on how long it will take to cover each individual outcome, which is usually based upon previous years and other students.  Test dates are then inserted strategically throughout the year to determine when it is best for the class to demonstrate their knowledge.  The problem….the teacher is worrying about the class not the individual students.

I have heard teachers say they teach at a pace such that the “average students” can follow, and my assessment dates are around when the “average student” should be able to demonstrate knowledge.  By definition then we are actually pleasing no one!  Half of the students will feel this day comes too late as they have already learned the material and could demonstrate it classes ago, while the other half believes that the pace is too quick, and they will need more classes until they are comfortable demonstrating the material.  Once again, it is very unlikely that we are meeting the needs of any student by trying to meet the needs of the “average student”.

How have I changed this?

I teach on the same timeline and give students an assessment similar to this.  DA with Derivatives , but instead of taking 3 days for the test (1-2 days for review then the 3^rd to administer the exam) I provide the student with 1-2 days to complete the assessment.  Students who understand the material quickly are able to work on the assessment ahead of time and complete it immediately, while students who need more time can use as much time as possible.  There is no set date for completion. 

What if a student gets behind?

My first comment would be “Behind what?”  Some teachers have this notion that the pace of the class is the pace every student should be learning at, but does this make sense?  Remember these unit plans are created before even meeting our students, so how can we make a plan that addresses the individual student?  Saying that, if a student is not demonstrating the material at an acceptable standard at a time which you feel is detrimental to learning other outcomes, then instead of giving a bad mark and moving on I sit down with this student at lunch, or after school, and ensure this student learns the material.  Is it not our job to educate students?  By giving a test, and saying “sorry you haven’t learned everything, but I am moving on anyways” is actually not completing our job. 

As teachers we must remember, our class sizes may be large and diverse but this is due to the fact that many individual students are making up this group and our assessment style should not be created to meet the needs of the “average student” but the “individual student”.

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.

DA with Derivatives

Math 31 Derivative Assessment

Complete a newspaper, newsletter, pamphlet, or any informational item showing how Calculus can be used in real life applications.

Your product must demonstrate your knowledge of:

·         Use of the product rule by taking the derivative of the product of two functions, both which have a minimum of 2 terms and are at least degree 2.

·         Use of the quotient rule by taking the derivative of a quotient of two functions, both which have a minimum of 2 terms and are at least degree 2.

·         Implementing the chain rule while taking the derivative.

·         Taking the derivative of a function which must use the combination of two or more of the following:

o   Chain Rule

o   Product Rule

o   Quotient Rule

·         Taking the derivative of a function which requires implicit differentiation. 

In addition, you must also:

·         Determine the slope at a point of a function.

·         Determine the equation of a tangent line of a function at a point.

·         Determine the second derivative of a function.

The work, determining the derivative and other answers can be supplied separate to your final product, but the solutions MUST make sense in the story, or scenario, you have placed them in.

Examples:

Recently the police has determined the crime rate of Red Deer can be shown by the function, c(d) = d^2, where c(d) is the amount of crimes committed on a day, and d is the day of the year.  This function applies to only the first 5 days of the year, then the function changes.  The rate of change of crime from day to day can then be demonstrated by the function c'(d)=2d, and the exact rate of change on the 3^rd day is 6 more crimes each day.

Sylvan lake was under attack, last night, by a mob equipped with catapults.  The height of one of the arms of a catapult, in meters, could be represented by the function h(t) = -t^2+9, where t is from 3 seconds before the arm reaches its maximum height to 3 seconds after it reaches it maximum height.  If the catapult launches its projectile at t = -2, the slope of the projectile would be 4 m/s and an acceleration of    -2 m/s^2 with an equation of 5(x+1)=y-5