1. The given pattern must represent a relationship of the form mx + b (linear), x 2 (simple quadratic), simple arithmetic progression, or simple geometric progression with all exponents being positive.
   

Indicator 1.1.4: The student will describe the graph of a non-linear function and discuss its appearance in terms of the basic concepts of maxima and minima, zeros (roots), rate of change, domain and range, and continuity.

Assessment Limits:

1. A coordinate graph will be given with easily read coordinates.
   
2. “Zeros” refers to the x-intercepts of a graph, “roots” refers to the solution of an equation in the form p(x) = 0.
   
3. Problems will not involve a real-world context.

Expectation 1.2: The student will model and interpret real-world situations using the language of mathematics and appropriate technology.

Indicator 1.2.4: The student will describe how the graphical model of a non-linear function represents a given problem and will estimate the solution.

Assessment Limits:

1. The problem is to be in a real-world context.
   
2. The function will be represented by a graph.
   
3. The equation of the function may be given.
   
4. The features of the graph may include maxima/minima, zeros (roots), rate of change over a given interval (increasing/decreasing), continuity, or domain and range.
   
5. “Zeros” refers to the x-intercepts of a graph, “roots” refers to the solution of an equation in the form p(x) = 0.
6. Functions may include step, absolute value, or piece-wise functions

Back to top

See Real World Setting below.

Back to top

Back to top

Level I: Acquiring Data - Data students will acquire in this standards-based task:

Vocabulary: exponential equation, exponential growth, exponential decay

Level III: Applying Knowledge - Visualized information that is applied knowledge in this standards-based task:

Solving problems: Solving exponential problems such as money investments and population growth and radioactive decay.

Back to top

Use of Resources - The students will provide classroom materials such as pencils, paper, notebooks.

Customer for Student Work - The students will present their work as evidence of task completion to the teacher and peers.

Assessment of Student Work - The following people will be involved in assessing student work generated to complete the task: the student's teacher.

Assessment of Student Work - The following forms of assessment will be used to determine progress and results: 5 classwork BCR type problems, 5 homework BCR type problems, 2 multiple choice test items on the quiz the next day, and 2 BCR type problems on the unit test.

Timeline - The estimated time needed to plan, teach, and score this task is two planning periods and one class meeting.

Back to top

20-60-20 Teaching Model

Segment One: Mini-Lesson - Estimated Time: 20% of allocated time

During this segment of the lesson:

1. Objective: The student will know exponential growth, exponential decay, applications, and how to use the number e.
   
   
2. For a prior knowledge assessment of my students, I taught almost all of these students Precalculus last semester. Thus, this is basically a review lesson.

Drill:

1. Age(weeks) 2 3 4
   Weight (lb.) 4 7 11
   Find the linear regression and use it to estimate the weight at 6 weeks. Round it to the nearest pound.
   
   
2. y = 2x^5 + x^3
   Is this funtion odd, even, or neither? Why?
   
   
3. Describe how you will model and verbalize assumptions underlying the lesson

Technology for this segment: TI-83 graphing calculators
Student product or performance for this segment: 5 BCR's for classwork problems and 5 BCR's for homework
Scoring tool for this segment: BCR rubric

Segment Two: Practice - Estimated Time: 60% of allocated time

Describe for this segment of the lesson how you will:

1. Acquire a knowledge of student strengths and weaknesses: By grading and reviewing the homework due today by having students write all the work and answers on the board and explain it orally with follow-up questions by the teacher and other students, I will both formally and informally assess students' strengths and weaknesses. In addition, part of the homework was to read, study, and copy the new examples for today out of section 1.3 of Calculus by Finney, Demana, Waits, and Kennedy. I will ask almost every student, there are only 7 students in the class, to fully explain an example to further determine their strengths and weaknesses. Of course, I give them hints and clues when necessary, and I try to get other students to repeat and expand on the explanations. I also ask students to use the proper vocabulary for this section which is exponential functions, exponential growth, and exponential decay. I emphasize the algebraic, graphical, and numerical approahes for all of the problems and especially for the four examples on pages 20-23. The exploration was assined as extra credit and/or a project idea
   
   
2. Clearly delineate expectations set for each student: Each student's drill and homework is graded, and then they are explained again and again for extra credit. Alot of partial credit is given for corrections.
   
   
3. Clearly delineate instructions:
   • First, students know that homework may be put on the board first thing for extra credit. So, most try to come to class as early as possible.
   • Second, I circulate, and grade the homework while they work on the drill.
   • Third, students are encouraged to put up the drill work on the board
   • Fourth, later students explain it for extra credit.
   • Fifth, I circulate and grade the drill.
   • Sixth, students explain their drill work.
   • Seventh, students explain their homework, the new examples, and at least the first part of the investigations listed in the textbook. This is where I emphasize multiple approaches as in algebraically, graphically and numerically to solve the problem. In this lesson, the textbook emphasizes the graphical approach. However, because this is basically a review lesson, I emphasized and the students suggested several variations on all three of these methods.
   • Eighth, students ask any additional questions and start the classwork.
   • Ninth, I circulate giving help if students have already asked their group members for help.
   • Tenth, students put up the classwork for extra credit.
   • Eleventh, I quickly circulate, and give classwork grades.
   • Twelfth, students explain their classwork that is on the board to each other. Alot of questions are asked usually from one student to another.
   • Thirteenth, I sometimes challenge them with a classwork extra credit. For this lesson, I did not have time to do this; so, I assined problem 42 as homework extra credit and/or a project idea.
   • Fourteenth, I summarize, or get a student to and assign the new homework and extra credit that is already written on the board. At several points, especially when a student has a question about a similar problem ("what if ... ?"), I often model the problem solving process in solving that new problem.
   • Finally, the next day I adjust my grades for drill, classwork, and homework if they have made corrections while I check the new homework. This is the procedure I followed for this lesson, and the procedure that I use most of the time.

Technology for this segment: Graphing calculators
Materials for this segment: Calculus textbook
Student product or performance for this segment: Written and oral presentation of the examples p20-23
Scoring tool for this segment: BCR rubric

Segment Three: Feedback - Estimated Time 20% of allocated time

Describe for this segment of the lesson how you will:

1. Administer an assessment of student work: Classwork pages 24-25:4,12,20,28,36
   
   
2. Provide feedback between the student and the teacher: I circulate giving help if students have already asked their group members for help. Then students put up the classwork for extra credit. I summarize , or get a student to, and assign the new homework pages 24-25: 8,16,24,32,40 and study for the quiz tomorrow and extra credit that is already written on the board page 26:42.

Technology for this segment: Graphing calculators
Materials for this segment: Calculus textbook
Student product or performance for this segment: completion of the classwork pages 24-25:4,12,20,28,36
Scoring tool for this segment: BCR rubric

Back to top

Assessment Benchmarking Example: Classwork pages 24-26:4,12,20,28,36

4. The graph of y = -.5^-x or equivalently y = -2^x is the reflection about the x-axis of the graph in Exercise 1(C).
12. 16^(3x) = (2^4)^(3x) = 2^(12x)
20. From y = 1 to -2 the change in y is -3, from y= -2 to -5 the change in y is -3, and from y = -5 to -8 the change in y is -3.
28. A = amount of initial investment
t = number of years
Solve: A(1 + .0625/12)^(12t) = 2A
(1 + .0625/12)^(12t) = 2
Graphically y1 = (1 + .0625/12)^(12t) and y2 = 2 t = 11.119 years
36. (A) When t = 0, B = 100e^0 = 100 bacteria initially
(B) When t = 6, B = 100e^(.693*6) = 6394.351 bacteria
(C) Solving 100e^(.693t) = 200 graphically t = 1.000. The population will be 200 after 1 hour. The doubling time is 1 hour.

Assessment Benchmarking Example:Homework pages 24-26:8,16,24,32,40

8. Domain All real numbers, Range (3,infinity), x-intercept None, y-intercept 4
16. Intersection x = 1.3862944 in window (-6,6) by (-2,6)
24. (a) The population is given by P(t) = 6250(1.0275)^t where t is the number of years after 1890 Population in 1915 P(25) = 12315, Population in 1940 P(50) = 24265 (b) Solving P(t) = 50000 graphically I find that t = 76.651. The population reached 50000 about 77 years after 1890 in 1967.
32. A = initial investment, t = number of years Ae^(.0575t) = 3A e^(.0575t) = 3 Graphically t = 19.106 years
40. (a) In the window of (-5,5) by (-2,10) they intersect twice although a third crossing off screen sems likely. (b) It happens by x = 4. (c) Graphically, x = -.7667, x = 2, x = 4 (d) The solution set is about (-.7667,2) union (4,infinity)

Assessment Benchmarking Example: Quiz problems 6 and 7
6. 25^(4x) = (5^2)^(4x) = 5^(8x) choice B
7. Use a graph to solve 3^(-x) = 7 y1 = 3^(-x) y2 = 7 x= -1.77 choice B

Assessment Benchmarking Example: Practice Unit Test problems pertaining to this lesson
6. State the domain, range, and intercepts of the function y = 2^(-x) - 1
7. Use a graph to solve the equation 4 - 3^x = 0
8. Suppose that in any given year, the population of a certain endangered species is reduced by 25%. If the population is now 7500, in how many years will the population

Assessment Benchmarking Example: Unit Test problems pertaining to this lesson
6. State the domain, range, and intercepts of the function y = 9 - 3^x
7. Use a graph to solve the equation 2^(-x) - 6 = 0
8. Suppose that in any given year, the value of a certain investment is increased by 15%. If the value is now $15000, in how many years will the value be $21000 ? These problems are from Calculus Assessment by Scott Foresman-Addison Wesley copyright 1999.

Back to top

Maryland High School Mathematics Rubric: Brief Constructed Response Items

LEVEL 3
The response indicates application of a reasonable strategy that leads to a correct solution in the context of the problem. The representations are essentially correct. The explanation and/or justification is logically sound, clearly presented, fully developed, supports the solution, and does not contain significant mathematical errors. The response demonstrates a complete understanding and analysis of the problem.

LEVEL 2
The response indicates application of a reasonable strategy that may be incomplete or undeveloped. It may or may not lead to a correct solution. The representations are fundamentally correct. The explanation and/or
justification supports the solution and is plausible, although it may not be well developed or complete. The response demonstrates a conceptual understanding and analysis of the problem.

LEVEL 1
The response indicates little or no attempt to apply a reasonable strategy or applies an inappropriate strategy. It may or may not have the correct answer. The representations are incomplete or missing. The explanation and/or justification reveals serious flaws in reasoning. The explanation and/or justification may be incomplete or missing. The response demonstrates a minimal understanding and analysis of the problem.

LEVEL 0
The response is completely incorrect or irrelevant. There may be no response, or the response may state, "I don't know."

Explanation refers to the student using the language of mathematics to communicate how the student arrived at the solution.

Justification refers to the student using mathematical principles to support the reasoning used to solve the problem or to demonstrate that the solution is correct. This could include the appropriate definitions, postulates and theorems.

Essentially correct representations may contain a few minor errors such as missing labels, reversed axes, or scales that are not uniform.

Fundamentally correct representations may contain several minor errors such as missing labels, reversed axes, or scales that are not uniform.

Source:
http://www.mdk12.org/mspp/high_school/structure/algebra/index.html

Back to top

Cognitive Information: Items to keep in mind for students:

• I will ask the students to explain how they solved the goal, problem, or issue in this task.
  

Back to top

Analysis: The classwork grades of the seven students from this lesson all were threes as were the homework grades on the BCR rubric. I converted these to 90% and above based on details and extra credit. Every student got the two multiple choice items on the quiz pertaining to this section correct. That, of course, exceeds our school goal of a 10 percentage point increase in previous passing percent. In addition, four of the seven students showed another method to solve problem 7 on the quiz that pertained to this lesson. The only disappointment was the other extra credit that ranges from sectons 1.2 to 1.6 on graphs, domain, range, zeros, even or odd, symmetry, periodic or not, one-to-one or not, and transformations was not well done by students although six out of the seven attempted it. Of course, we had only covered up to and including section 1.3, but really this entire chapter is review. Obviously, there is a bigger loss of long term memory from Precalculus than I anticipated. On the actual test on this unit, the BCR items that pertained to this lesson were items 6,7, and 8. Five out of eight (62.5%) got a 2 or 3 on item 6, eight out eight (100%) got a 2 or 3 on item 7, and 6 out of 8 (75%) got a 2 or 3 on item 8. Again, this obviously exceeds our school goal of a 10 percentage point gain.