Date______Name ______No. ______

Trig/Math Analysis Portfolio Assignment

Type I: Mathematical Investigation

Introduction:

This work is an inquiry into a particular area of mathematics leading to a general result which was previously unknown to the student. The use of a calculator and/or computer is encouraged in this type of activity.

Purpose:

The purpose of portfolio work is to:

  • Develop students’ personal insight into the nature of mathematics and to develop their ability to ask their own questions about mathematics.
  • Provide opportunities for students to complete extended pieces of work in mathematics without the time constraints of an examination.
  • Enable students to develop individual skills and techniques and to allow them to experience the satisfaction of applying mathematical processes on their own.
  • Provide students with the opportunity to experience for themselves the beauty, power and usefulness of mathematics.
  • Provide students with the opportunity to discover, use and appreciate the power of a calculator/computer as a tool for doing mathematics.
  • Enable students to develop qualities of patience and persistence, and to reflect on the significance of the results they obtain.
  • Provide opportunity for students to show, with confidence, what they know and can do.

Requirements:

  • The response must be individual effort in which each student does his/her own work.
  • Appropriate mathematical symbols and correct spelling/grammar are to be used.
  • Diagrams, graphs and tables are to be included as appropriate.
  • Plain, lined, or graph paper is acceptable (white only).
  • The response must be written in blue or black ink, or word-processed. Graphs may be drawn or sketched in pencil.
  • The response should be readable by an individual not familiar with the initial problem.
  • Calculations are to be accurate to three significant figures.
  • Calculator use is permitted; results taken from graphic display calculators must be so noted.
  • Calculation results must distinguish between equality and approximation.
  • This assignment is due on or before ______.

Grading:

Score / Criterion
Criterion A: Use of Notation and Terminology
0 The student does not use appropriate notation and terminology.
1 The student uses some appropriate notation and/or terminology.
2 The student uses appropriate notation and terminology in a consistent manner and does so throughout the activity.
Criterion B: Communication
0 The student neither provides explanations nor uses appropriate forms of representation (e.g. symbols, tables, graphs, diagrams).
1 The student attempts to provide explanations and uses some appropriate forms of representation.
2 The student provides adequate explanations/arguments, and communicates them using appropriate forms of representation.
3 The student provides complete, coherent explanations/arguments, and communicates them clearly using appropriate forms of representation.
Criterion C: Mathematical Content
0 The student recognizes no mathematical concepts which are relevant to the activity.
1 The student recognizes a mathematical concept or selects a mathematical strategy which is relevant to the activity.
2 The student recognizes a mathematical concept and attempts to use a mathematical strategy which is relevant to the activity and consistent with the level of the course.
3 The student recognizes a mathematical concept and uses a mathematical strategy which is relevant to the activity and consistent with the level of the course, and makes few errors in applying mathematical techniques.
4 The student recognizes a mathematical concept, successfully uses a mathematical strategy which is relevant to the activity and consistent with the level of the course, and applies mathematical techniques correctly throughout the activity.
5 The student displays work distinguished by precision, insight and a sophisticated level of mathematical understanding.
Criterion D: Results or Conclusions
0 The student draws no conclusions or gives unreasonable or irrelevant results.
1 The student draws partial conclusions or demonstrates some consideration of the significance or the reasonableness of results.
2 The student draws adequate conclusions or demonstrates some understanding of the significance and reasonableness of results.
3 The student draws full and relevant conclusions or demonstrates complete understanding of the significance, reasonableness or possible limitations of results.
Optional Criterion E: Making Conjectures
0 The student demonstrates no awareness of patterns or structures.
1 The student recognizes patterns and/or structures and attempts to draw inductive generalizations.
2 The student recognizes patterns and/or structures and successfully draws inductive generalizations.
3 The student recognizes patterns and/or structures and successfully draws inductive generalizations.
4 The student recognizes patterns and/or structures, successfully draws inductive generalizations and rationalizes the generalization by way of an informal explanation.
Optional Criterion F: Use of Technology
0 The student does not use a calculator or computer beyond routine calculations.
1 The student attempts to use a calculator or computer in a manner, which could enhance the development of the activity.
2 The candidate makes limited use of a calculator or computer in a manner which does enhance the development of the activity.
3 The student makes full and resourceful use of a calculator or a computer in a manner which significantly enhances the development of the activity.
TOTAL POINTS

Trig/Math Analysis Type I Assessment Criteria A, B, C, D, E, F

Investigation: Linear Regression

The data pairs give the average speed of an airplane during the first ten minutes of a flight, with x in minutes and y in miles per hour. Approximate and state the equation of the best fitting line for the data:

x / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10
y / 180 / 250 / 290 / 310 / 400 / 420 / 410 / 490 / 520 / 510

1. Carefully draw a scatter plot of the data.

(a.) Explain the general purpose of this graph.

(b.) Describe the correlation (positive, negative, or none) you observe.

2. Estimate the line of best fit.

(a.)Sketch the line that appears to follow most closely the pattern given by the points. There should be as many points above the line as below it.

(b.)Choose two points on the line, and estimate the coordinates of each point. These two points do not have to be original data points.

(c.)Find the equation of the line that passes through these two points.

3. Evaluate the line of best fit.

(a.)What values or distances would need to minimized (or maximized) to assure that you have identified the best fitting line for the data? Give example calculations of these values for the line you sketched in step 2.

(b.)Develop a measure of the accuracy of the equation of the line you sketched in step 2.

(c.)Add the sketch of the line to your graph; is this line a better or worse fit than the line you sketched in step 2? Use your measure of accuracy to justify your answer.

(d.)Add the sketch of the line to your graph; is this line a better or worse fit than the line you sketched in step 2? Use your measure of accuracy to justify your answer.

4. Make a conjecture about this process called linear regression.

(a.)How can technology aid you in finding the equation of the line of best fit?

(b.)Would you get the same equation of a best-fitting line each time you graphed these pairs? Justify your answer.

(c.)If a unique, correct “answer” exists for this problem, how does this compare to the equation of the line you sketched in step 2?