Foundations and Pre-Calculus 10 – Ski Hill Project

Learning Outcomes Addressed:

3.1 Determine the slope of a line segment by measuring or calculating the rise and run.

3.4 Explain why the slope of a line can be determined by using any two points on that line.

3.6 Draw a line, given its slope and a point on the line.

3.7 Determine another point on the line, given the slope and point on the line

3.9 Solve a contextual problem involving slope.

 Purpose:

The purpose of the project is to take what you have learned about slope and apply it to a real world scenario in which slopes are used.

 

Scenario:

An indoor ski hill recently opened in Dubai in the United Arab Emirates.  You and your two partners have been hired by a retail developer to propose a design for an indoor ski hill to be placed inside a mall in Southern California.  You will start with a short warm up to give you an idea of where to start; then you will be responsible for producing your own profile and a 3-D mock up of what the hill may look like in reality.

 Warm-up:

1)      Graph the following co-ordinates on grid paper. Join the points to create a simplified elevation profile of a ski hill.

Horizontal Position (ft)	Vertical Position (ft)
0	100
50	70
65	65
100	60
125	35
145	30
165	10
170	3
180	2
190	0
200	0

2)      Calculate the slope between the following pairs of points:

a)      (0, 100) to (50, 70)                                                                          m =                 

b)      (65, 65) to (100, 60)                                                                       m =                 

c)      (145, 30) to (165, 10)                                                                     m =                 

3)      Which part of the run has the steepest slope?                                                                

4)      Find the slope for the last 10 feet of the run.                                           m =                 

5)      Find the average slope from beginning to end.                                        m =                 

6)      Calculate the length of this ski run.                                                          l =                   

Your Proposal:

Your proposal is to include three aspects.

1. Your ski hill elevation profile is to meet the following specifications:

·         The total height of the ski hill should be 100 feet.

·         The total horizontal distance should be 120 feet.

·         The total length of the run should be no more than 300 feet.

·         You should include at least 3 different slopes and a more gentle finish to allow the skier to slow down and stop.

·         Your drawing should be on grid paper with an appropriate scale given.

2. You must also include a paragraph description of the ski hill with the following information:

·         The total length of the ski run.

·         The slope of the steepest section.

·         The average slope of the ski run.

3. Your 3-D Model is to meet the following specifications:

·         Must be done to scale, ideally the scale you used for your elevation profile.

·         Must include your scale on your model.

·         Must be able to see the 3 different slope sections and the gentle finish.

·         Must look like it would in reality, i.e. make it look like you would see it in the mall (add snow, the lift, skiers, etc…, be creative)

Timeline:

·         You will have one class period to complete the warm up and discuss what your proposal will look like.

·         You will have an additional 2 class periods to work as a group to develop your profile, written description and model.

·         If additional time is needed the classroom will be open after school for one additional week so you may complete your work at that time if necessary.

Extensions:

If you wish to go above and beyond the project specifications here are some ideas for you; you may choose one or you may choose to do them all.

• Research the Dubai project – cost to build, timeline from start to finish, slopes involved, cost to keep hill operational.
• Research the practicality (cost to build, marketing, cost to run yearly, etc…) of developing an indoor ski hill in a Western Canadian mall.
• Research how outdoor ski hills are run and how they may use slopes to classify the different runs.
• What is the connection between man-made ski hills (not necessarily indoor) and polygons?

 Rubric for the ski hill project:

Knowledge and Understanding - Calculations show complete understanding of the mathematical concepts used to complete the assignment. All calculations are done correctly; co-ordinates are graphed accurately on grid paper.         /10

Completeness of Assignment - Contains all the required parts: graph, slopes, steepest slope, slope for the last 10 feet, average slope and total length of the ski run. The proposal for the ski hill elevation profile meets all required specifications. Includes a brief description with all required information .     /15                                           
Neatness and organization - The work is presented in a neat, clear, organized fashion.      /5 
                                                                                              Total      /30

Review of a math 9 polyhedra project created by Susan Gerofsky Experience:

With Deborah and Nadine

Researching the four topics was very interesting although obtaining information for part one was difficult. Also the mathematics involved for some parts is too complicated for a grade nine to understand. No guidance as to the weighting of parts a, b, and c is given which could lead to some groups putting emphasis on the biography and others on the mathematical ideas. Making the origami model was fun for those of us who enjoy art projects; however this task is frustrating for those who do not.

Overall we found this project interesting but dated. There was no relationship with today’s world and we question the value this project has beyond providing interesting facts.

Benefits:

* Students will learn interesting facts through research and from peers rather than their teacher
* History is incorporated into the classroom, enriching the learning experience
* Students will see how mathematical ideas can extend into the arts and architecture, making mathematics seem less isolated as a subject
* Learning about mathematicians, connecting math to real people can spark interest and provide comfort to those who may see the subject as alien
* Allowing students to create and thus touch and experience in real life the Platonic solids allows them to realise their simplicity


Weaknesses:

* It is unclear as to why some people work with partners and some alone to write the paper
* There are no specifics on whether each person writes about a different part or all contribute to every part, writing an essay with someone else is not easy
* Each person has to make an origami model, time consuming & some students may struggle with this, not beneficial, there needs to be more of a point to making them
* Other than introducing polyhedra and some history the intended learning outcomes of this project are not obvious
* No specifics for what the presentation should look like
* Students may not connect with mathematicians from the past
* The math involved in some sections is too complex for the grade level
* No connections to the lives of the students
* Group work, especially of this nature, is very time consuming



Uses:

* Introducing the class to polyhedra, good start to 3D geometry
* Can be incorporated into volume, or surface area units
* Allowing the class to experience group work and obtain skills in researching
* Incorporating history into the math classroom



Modifications & Extensions:

* Only one student per group makes an origami model, the teacher can bring in the fifth one. They show the template to the other group members before gluing so that all students can see how it is constructed and benefit
* All students should be involved in the research, then two can work together to produce the written work, one can present and one can make the model so that everyone contributes, if they want to change that they can but they must inform the teacher of their roles
* Find modern day uses for polyhedral, compare with historical
* Research more than one of the areas or people then groups can compare
* Presentation only, no essay
* Using paper templates students can cut out all kinds of regular polygons with the same side length, then try to create all kinds of regular 3D objects by gluing the regular polygons together
* Find patterns in the construction of the solids
* Use Geomag for constructing the solids (stick magnets with iron balls, easier and more fun to use than paper and glue)

Response to "Creativity, flexibility, adaptivity, and strategy use in mathematics"

Creativity, flexibility and adaptability are all necessary attributes of problem solving. I think the aim of teaching mathematics should be creative problem solving, to encourage students to seek out solutions, exploring patterns, finding new strategies- not just memorizing procedures, formulas and doing exercises. In order to achieve creative thinking in the classroom, students should be encouraged to use different problem solving strategies.

This article ties in well with what we have explored earlier this term about relational and instrumental learning. Instrumental learning provides a narrow curriculum based on mastering facts and procedures which does not allow students to become creative thinkers. Relational understanding, on the other hand, trains students to think creatively and develops their problem solving ability. Mathematically powerful students are capable of interpreting large amount of data and this allows them the flexibility and adaptability to come up with new strategies or to choose the most appropriate strategy to solve a problem.

Ferit's compensation strategy, adding 100 first and then subtracting 1, worked for 127+99, but he did not get the right answer for numbers such 133. His creativity enabled him to come up with a strategy that was new to him or at least unfamiliar to him. The lack of a solid mathematical background prevented him to switch between different strategies, he used the compensation strategy for all of the addition problems he encountered getting the wrong answer most of the time.

Two Column Problem Solving

The Restless Fly from "The Moscow Puzzles" by Boris Kordemsky

Two cyclists began a training run simultaneously, one starting from Moscow, the other from Simferopol.
When the riders were 180 miles apart, a fly took an interest. Starting on one cyclist's shoulder, the fly flew ahead to meet the other cyclist. On reaching the latter, the fly at once turned back.
The restless fly continued to shuttle back and forth until the pair met; then it settled on the nose of one of the cyclists.
The fly's speed was 30 miles per hour. Each cyclist's speed was 15 miles per hour.
How many miles did the fly travel?

The problem looks more complicated that it actually is. At first, students might say there is not enough information given, since we don't know the distance between the two cities. On the other hand, although it is highly unlikely that a fly would travel from biker to biker until the two meet, students will probably find it funny and will try to figure out the solution. It is not a practical problem, but probably it is both memorable and strange because of the travelling fly. I still remember some physics problems where we had to determine the speed and angle of a brick thrown out the window of a tall building so it would land on the pedestrian's head who was walking by. It was a weird problem but interestig enough for us to figure out the solution to it.
We could extend the problem by having different speeds for the cyclists and maybe having another fly travelling from the opposite direction with a different speed.

If the two cyclists had the same speed, they meet halfway, both travelling 90 miles. Since the fly's speed was twice as much as the cyclists' speed, and flew without stopping, it had to travel twice as much, a total of 180 miles.
If we look at the time spent until the cyclists meet, t=90/15, we get 6 hours. The fly travels between the two cyclists until they meet, for the same amount of time. This means that the distance covered by the fly will be d=30x6, d=180 miles.

Short Practicum Stories

#1

My short practicum started with an unexpectedly warm welcome from the school administration followed by the tour of the scool. We (together with two other UBC teacher candidates) were introduced to almost everybody in the school and spent about an hour getting to know the school policies and procedures in the board office with all three vice principals and principal in attendance. I really appreaciate their effort to make our short practicum a positive experience. All the teachers of the Math department heard about me coming to observe and teach in my SA's class. Two of them came over to see me teach when they had a free block. I got a lot of advice from so many of them, I fell really thankful.

#2

I wasn't aware that the day my FA was coming is actually going to be a modified day, so I prepared a nice long lesson on the quadratic formula and the proof of the formula. I had to adjust the lesson to be able to finish it in 50 min insted of 70 min but wasn't able to go over everyting I have planned.
During the lesson one of the student's desk he was sitting on fell apart and the whole class laughed for about 5 minutes, which shortened my already short time. At almost the end of the period the announcement went on eating another 5 minutes not leaving me time to wrap up the lesson. It was really frustrating but I learned that I have to be more flexible as a teacher because situations like these happen and I have to be able to deal with the unexpected all the time.

LESSON PLAN ON AREAS OF RECTANGLES,CIRCLES AND SQUARES with Feda and Namrat

	What	How long	Materials used
Bridge	Review the definitions of rectangles, squares and circles. Take several responses to get an idea of what the students know.	2 min
Learning Objective	Students will calculate the area of a rectangle, a square and a circle. Students will solve problems involving the areas of rectangles, squares or circles.
Teaching Objective	Students will be able to solve problems individually and cooperatively. Students will be able to clearly and logically communicate a solution to a problem and the process used to solve it.
Pre-test	Ask the students about the formula of areas of rectangles, squares and circles.	2 min
Participatory learning	Ask the Students to solve a problem that involves the calculations of the area of a rectangle and a circle combined. Ask students to describe in words the steps that they will need to take to obtain the answer. Students will work in small groups to find the areas by using the formulas for the area of a rectangle and of a circle.	5 min	Rectangle shaped cardboards with a cut-out circle in the middle.
Post-test	Ask students to solve a similar problem involving many circles marked on a rectangle. Compare results with other groups.	5 min	Another cardboard with many circles marked on a rectangle
Summary	Give overview	1 min