Summative Assessment

Patterns: Retirement Riches

Answer Key/Scoring Guide

You have landed your dream job and expect to retire from this company after 30 years of employment.  You will earn an excellent salary with a choice of retirement plans.  You must choose either the Penny Plan or the Grand Plan.  Your employer gives you the following information about the plans.

v      Penny Plan:  Your employer will deposit one cent into your retirement plan at the end of your first year of employment and double that amount each year there after.  That is, the first year your account will receive one cent; the second year your account will receive two cents; the third year your account will receive four cents, continuing in this same pattern year after year.

v      Grand Plan:  Your retirement account will have $1000 deposited at the end of each year of employment.  For example, the first year your account will receive $1000; the second year you will receive $1000; the third year your account will receive $1000, continuing in this same pattern year after year.

Which plan would you choose and why?

(Turn in this page to the instructor when complete.)

Exemplary response

I would choose the Grand Plan since $1000.00 is far greater than $.01.  Even though the Penny Plan doubles each time, it would take a long time for it to be greater than $1000.00

                                                          OR

I would choose the Penny Plan since it doubles each time.  By thirty years, it will be far greater than the Grand Plan.

Scoring guide

1 point             Gives an opinion and supports the opinion with a reason.

0 points           Opinion with no support or other answer.

You have landed your dream job and expect to retire from this company after 30 years of employment.  You will earn an excellent salary with a choice of retirement plans.  You must choose either the Penny Plan, the Grand Plan, or the Grand Plus Plan.  Your employer gives you the following information about the plans.

v      Penny Plan:  Your employer will deposit one cent into your retirement plan at the end of your first year of employment and double that amount each year there after.  That is, the first year your account will receive one cent; the second year your account will receive two cents; the third year your account will receive four cents, continuing in this same pattern year after year.

v      Grand Plan:  Your retirement account will have $1000 deposited at the end of each year of employment.  For example, the first year your account will receive $1000; the second year you will receive $1000; the third year your account will receive $1000, continuing in this same pattern year after year.

v      Grand Plus Plan:  Your retirement plan will have an additional $1000 deposited into your retirement account each year.  This means your account will receive $1000 in the first year, $2000 in the second year, and $3000 in the third year, continuing in this same pattern year after year.

(Use this page to answer questions 1 through 3.)

1.     How much will be deposited into your retirement account during the 20^th year of employment for each plan?  Provide evidence to support your answers using both function rules and tables or graphs.

Exemplar Response

 Penny Plan:

 $5,242.88

      AND

A(x) =   OR    a_n = 2a_n-1; a₁ = .01   OR    NEXT = NOW · 2; START = .01

      AND

      AND

Grand Plan: 

$1000.00

     AND

A(x) = 1000   OR     a_n = a_n-1; a₁ = 1000    OR     NEXT = NOW, START = 1000

     AND

     AND

Grand Plus Plan:

$20,000

     AND

A(x) = 1000n  OR   a_n = a_n-1+ 1000; a1 = 1000 OR NEXT = NOW + 1000, START = 1000

     AND

         OR

Holistic Scoring Guide

2.     Compare and contrast the properties of the graphs/function rules for each retirement plan.  Include a description of each type of function.

Exemplar Response

The Penny Plan is an exponential function with a base of 2.  The Grand Plan is a constant function (at 1000).  The Grand Plus Plan is a linear function.

      OR

The graph of the Penny Plan rises at a varying rate.  The Grand Plan has a graph that is a horizontal line.  The Grand Plus Plan has a graph that is a line with a slope of 1000 and a y-intercept of 0. 

Scoring guide

2 points           Exemplary response

1 point             Correctly states the type of function 2 of three plans are.

                                                     .                      

0 points           Other

3.     Employee A, employee B, and employee C want to compare how much money they will have in their retirement plans after 30 years.  Employee A chose the Penny Plan.  Employee B chose the Grand Plan.  Employee C chose the Grand Plus Plan.

A.     What is the total amount of money that each employee will have in their retirement account at the end of thirty years employment?  Support your answers with function rules and tables or graphs.

B.    Based on your findings, which employee chose the best plan and why?

Exemplary Response

A. 

Penny Plan – Employee A

$10,737,418.23

     AND

 where A is the amount invested and n is the year.

     OR

Grand Plan – Employee B

$30,000

     AND

 where A is the amount invested and n is the year.

     AND

     OR

3.  Grand Plus Plan – Employee C

$465,000

     AND

  since there are 15 sets of 31.  In general this is  where A is the amount invested and n is the year.

     AND

      OR

B.  Based on our findings of Employee A earning $10,737,418.23, Employee B earning $30,000 and Employee C earning $465,000, the best choice of plans is Employee A with the Penny Plan because that one earned the most money.

Holistic Scoring Guide

Constructed response:

4.     Now that you have a steady job, you can indulge in your favorite hobby:  skeet shooting.  At your first tournament, skeet is shot into the air at 82 ft/s and an angle of 20 degrees.  Its height is a function of time and is described by the equation h(t) = -16t² + 28t.  This results in the skeet reaching a maximum height of 12.25 feet after 0.875 seconds.  In the next shot, the skeet is released at 88 ft/s and an angle of 20 degrees.  Describe the effect the resulting equation of h(t) = -16t² + 30t will have on the maximum height reached by the skeet.  Support your answer with a graph, algebra, or a table.

Exemplary response

The increase in initial velocity will cause the maximum height to rise to 14.0625 feet after 0.9375 seconds.  This is an increase of 1.8125 feet.

AND

t = - ^b/_2a = - ³⁰/_[2(-16)] = ¹⁵/₁₆ OR 0.9375

h(¹⁵/₁₆) = -16(¹⁵/₁₆)² + 30(¹⁵/₁₆) = ²²⁵/₁₆ OR 14.0625

14.0625 – 12.25 = 1.8125

OR

OR

Scoring guide

2 points           Correct answer and supporting work.

1 point             Correct answer with errors in work.

0 points           States increase or decrease with no supporting work.

5.      A young couple just had their first baby.  They want to help provide financial security for their child’s retirement.  They have $1,000.00 to invest for their child’s future.  The investment they have chosen has an expected return ranging from six to eight percent annually. 

a.      How much will the account be worth in 65 years (estimated retirement age) if the account has a 6% compounded annually rate of return?  Show the work necessary to obtain your answer.

Exemplar Response

b.      How much will the account be worth in 65 years if the account has an 8% compounded annually rate of return?  Show the work necessary to obtain your answer.

Exemplar Response

Scoring Guide

2 points                       Exemplary response (2 answers with supporting work)

1 point                         Correct answers without work

                                                  or

                                    1 correct answer with supporting work

0 points                       Other