Math and Super Mario Brothers

Usually I would hand out a worksheet on calculating when a function is closest to a point and have students complete 10-12 questions.  This year, I took the a different approach by bringing Super Mario Brothers in my class.

Using this picture,

I informed my class about Mario Brothers; when you jump, with Yoshi, you can jump again.  I then posed the question, "When would be the best time to jump off Yoshi if you want to get to the top level?"

Using Calculus, and geogebra you can calculate the path of Yoshi and the co-ordinate of the top level to get:

From here we calculated the equation of the parabola, and a distance function based on any point (x, f(x)) on the function.  Ultimately, we calculated the closest distance Yoshi comes to the point, and when to double jump to get the coin.
Students enjoyed this more than completing the 10 questions on the worksheet.  Feel free to use and fix as you see fit.

Rate of change of beaker.

Here is a lesson you can do with your students on related rates of change:

Before this, ensure your students understand at least similar triangles and have seen some related rates questions before.

1) Bring students to a chemistry lab with different beakers, or bring different beakers into your class.  **You will need a water tap**

2) Divide the students into groups of 3 or 4 and provide each group with one of each of the following beakers, shown in the picture below.

 3) Provide the following question to the students:


 Hypothesis:  Will the height change at a constant rate or will it change throughout, if we were fill these beakers up with water? Explain.  If the rate of change of the height of the water changes, when will the rate be the largest, and when will it be lowest.



Procedure: Turn the tap on and record the time it takes to fill the beaker up to the last marking on the glass, using a constant stream of water.  Record this three times and average your times.  What does this represent?



Fill up both beakers, using this stream of water.  Was your hypotheis correct?  If not, what did you notice?



Calculations:  If you were to fill up the cylindrical beaker with the stream of water how fast would the height change when it was half full? When it was entirely full?



If you were to fill up the conical beaker with the stream of water how fast would the height change when it was half full?  When it was entirely full?



I have used this before with my students and watched how they had to determine not only WHAT to measure but HOW to measure it.  Feel free to use, change, tweak as you see fit.

Here is video one student made:

**SOON TO BE UPDATED**

And his Prezi:

Copy of Dante Bencivenga's Unit 3 Assessment on Prezi

Cut the Rope in math class.

I provided my students with a laptop, which had Geogabra on it, and the following picture:

I then showed, on my iPad, how the rockets travels around the point.  When the rope is cut, the rocket will fly tangent to the curve.

I used this in my calculus class and asked "Determine the exact location the rocket should be when the rope is cut"

Students determined the equation of the circle, found the derivative and made it equal to the slope between any point (x,y) on the circle and the co-ordinates of the mouth of the monster (or goblin).
**WARNING, this is a hard question**

After, I was thinking this could be used in lower classes, by changing to one the questions below:

•  allow students to use the tangent button on geogabra, and ask for the reference angle.
• Determine the amount of time which should pass before cutting the rope.
• There are always two tangent lines on a circle through an exterior point, are there two times you could cut the rope in the picture?

There are more I could think of.

My students enjoyed the activity, and feel free to change and adapt as you see necessary. 

Angry Birds and calculus:

First provide the students with a laptop with Geogebra, and the following photo.

Introduce the meaning of Piecewise functions, and how the yellow bird, when clicked, shoots off at a tangent to the curve.

Ask the students to describe what are the two functions that create this curve (Parabola and a line).

Using prior knowledge have students graph the maximum point on the parabola, and use the dot where the yellow bird took off at a tangent (B), to create the equation of the parabola.  Have students graph the parabola in geogabra, overtop the picture, to ensure all calculations have been done correctly.

Next, ask students how to determine the equation of line.

We will need either a) two points or b) a slope and a point.  Both of which is impossible, without the use of the tangent button in Geogabra.  I explained, we can calculate this without that button!

Have students pick another “point on” the parabola (c), and to calculate the slope between B and C.  Ask how do we make this slope more accurate to the slope of the tangent…

Move C closer to B..

Have students move C closer to B, while still staying on the parabola and calculate the new slope.  Eventually move C as close as possible to B.  The slope should be -0.5  Below is a picture of one of the students’ work

Next using the point B and the slope you can create the equation of line.

From here your choice for extensions: I had students graph the piecewise functions on their calculator and got the following image

Lastly, here is a video on how to determine the slope at a point:

Learning can occur on a final exam

Historically, I would say that the least amount of learning occurs during exam week.  This year, however, I have tried to make this statement false through giving my students a new final exam.  When the first group presented, not only did I see, hear and was taught about unique examples of how to use calculus, I can actually say the group still LEARNED through the assessment. 

Below is a video of their PowerPoint presentation, which they used as well as supplemented with dialogue. 

After the presentation they opened the floor to me by giving me 10 minutes of questioning.  Most of the questions they answered swift and correctly, until I asked “Do all functions, on a closed interval, have an absolute maximum?” . Their answer was “Yes”.  Of course this is incorrect.

Now here is how the learning occurred… First I will address the traditional way of assessment:

If this was a traditional final exam, I would have marked this question wrong and moved on to the next question and continued marking.  These students would never have received any feedback, as in the past, I have yet to see many students come back to see WHAT they did incorrectly on a final exam.  These students would have then gone on to university/college with this false knowledge.

How it has changed this year…

I didn’t let this false information continue.  I then asked, “What if I told you, I could draw a function on a closed interval without an absolute maximum?”  The girls looked at each other with confused eyes, and pondered the idea.  After some passing minutes, one replied accusing me of a liar.  I wanted to ensure they didn't continue on with this false information, so I then asked, “Is there any kind of function that continues upward on forever?”.  One quickly answered, “WAIT! A function could have a vertical asymptote and therefore have no absolute maximum.  I guess you weren’t lying”, the other girl smiled and agreed. 

After this, I wanted to ensure they had a true understanding and therefore I asked, “Can a function, without any asymptotes, on a closed interval, not have an absolute maximum?”  The enlightenment has occurred!  Both girls whispered quietly, and then turned and replied “If the curve has an open point at the highest point, then it would not have an absolute maximum”.

Learning had occurred, and yet it was a final exam.

I continued with my questioning, which they answered correctly, and I am happy to say that this experience has been a success with the first group!

Calculus and Kobe Bryant

I used this video in my calculus class to show how math can either support or disprove a movie.

I showed this movie to my students and asked if this looked possible.  One jumped up and said "No way!" while others thought it could happen.  What was interesting was when I asked "Do you have proof?", as the class went silent. 

The math in this movie is incredible. 

Where I thought the students would go was completely the opposite of what happened.  My students timed how long Kobe was in the air, we measured his height in the movie and compared it to his real height to create a scale.  We used integration using gravitational pull to be -9.81m/s^2, to create a velocity and distance function.  The calculus was amazing. 

Whether we proved or disproved the reality is a secret I keep with me as I challenge you to give this to your students and see what they do.  Just watch the movie then ask "Any questions?"  I bet you will get lots.  The secret is then to let them "play" with the math and the movie.

Learning is road that they must travel down themselves and we should only be guiding them not pulling them along by the hand.

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)

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.

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

DA with first principles and limits

Here is an example of how I have changed the definition of an exam.  Before, when assessing first principles and limits in my calculus class, I would have each student write an exam.  This year, I am giving students the option of writing an exam or completing the following to demonstrate their learning.

Unit 1 Assessment Project

1.      Create a formula for the distance of an object at any time t.  The formula must be at least degree 2 with a minimum of 2 terms.  Using your formula you must:

·         Determine average velocity of the object for the first 10 seconds.

·         Determine, with an explanation, the instantaneous velocity after 10 seconds.

·         Determine, with an explanation, the instantaneous velocity for any time t.

·         Determine if the object is ever traveling 50 m/s.

2.      Create a piecewise function over the domain of the real numbers. The function must contain at least 3 different pieces and may not be continuous and may not have a limit at at least one point.  Using your function you must:

·         Provide a graph of your function.

·         Explain why your function is not continuous at the specific point.

·         Explain why a limit does not exist at the point above. Explain how you could change your function such that there is a limit.  Calculate the left and right side limits at this point.

3.      Demonstrate, through a real life application, how you can calculate the instantaneous rate of change at a specific data point.  You may use data from a chart, create a video, etc., but the data may not follow a specific function.  The point at which you calculate the rate is entirely up to you, but must have an explanation.

4.      Create three functions, one which:

·          approaches infinite,

·         one which approaches 0,

·         and another which approaches a line which is not 0

as you extend these functions to infinite.  Explain how you can determine the limit as x approaches infinite, for each of your functions.

5.      Create a real life example of a convergent series, and calculate the sum of this series.

Intro to Tangents

Here is an activity I used in my Math Calculus class to introduce tangents at a point.

I gave each student a copy of the video below.

Then, in groups of 2, students had to answer the following questions (which is part of the comments in the full YouTube video)

1)  Watch the video

2)  Determine the average speed for the first 5 minutes.

3)  Determine the average speed for the entire trip.

4)  At the end of the video, the speed of the vehicle is 109  km/h, explain any discrepancies from your calculation in part 3.

5)  Determine the average speed of the vehicle after 15 min of driving.

6)  Determine, with the least amount of error, the speed of the vehicle  after 15 min of driving.

7)  Explain, if any, the difference from part 5 and part 6.

Feel free to use, change as you seem necessary.

For the parts that is hard to see:

The time at each break in the bottom right corner is the total driving time to that point.
The distances at each break are -7.6 km/16.7 km/23.3km/32.1km

I do apologize the video is quite horrible with YouTube.  The original is much better.  If you require a copy of the original, email me and I can get you a copy.

Samples of work for Integration Project

Below are some samples of the work completed by various students for the Integration Project

Part 1:

Part 2:

Video for Part 3:

Work



Open ended project on Integration

In the past, in my Pre-Calc Class, I have used traditional exams to assess the knowledge of integration techniques.  This year I plan on using the following open ended project.

Integration Project

1.      Create an equation for the velocity of a particle at any time t, stating the initial position, which cannot equal 0.  The equation must include all of:

·         A polynomial function

·         Rational function –Chain rule must be applied

·         Trigonometric function

a.       Determine what must be true about a function so that it is able to be integrated.

b.      Determine the distance of the particle at any time t. – must use u substitution for the rational term.

c.       Create a velocity-time graph, as well as a distance-time graph on the same grid. 

d.      Determine, using appropriate sums of rectangles, an over and under estimation of the displacement of the particle in the first 10 seconds.

                                                              i.      Explain how this estimation could be made more exact.

e.       Determine the exact displacement of the particle for the first 10 seconds, and then determine the exact location of the particle after 10 seconds.

f.       Determine the average acceleration of the particle from 0.  Illustrate how your answer could have been determined by the graphs.

2.      Create two functions, one representing the revenue of a company while the second will represent the costs of a company over a 12 month period.  The curves must intersect and cross each other.  Determine the area between the two curves, for the 12 months and interpret your answer.

3.      Knowing the gravitational pull on Earth is 9.81m/s², create a video an object in free-fall.  You must throw your object, and calculate the velocity of the object when it left your hand.  Appropriate measurements must be included.

Other ideas can be found here:

At http://courses.ncssm.edu/math/POW/POWindex.htm

2010-11 Problem 7 The Lanczos derivative (which is defined as an improper integral)

Problem 8 What is the average value of all possible averages?

2009-10 Problem 7 Non-Fundamental functions (just a curiosity)

2008-09 Problem 10 Introduction to Fourier series

2007-08 Problem 8 Chebychev polynomials

A MISSION POSSIBLE…

Your mission, should you choose to accept it …actually you don’t have much choice…will consist of mathematically explaining everything you know about an irregularly shaped symmetrical object such as a bottle or a vase. Include detailed drawings and calculations.

Thank you to the Calculus Discussion Board for the ideas!