Themes in IMSA Problem Sets
In the problem set for MI 2, MI3, and MI 4 different themes are developed through several sets
or within one set. An example from MI 4 and MI 2 with comments follows.
MI 4: Golden Ratio, extended fractions, and related ideas.
Source:
PS 2, #15, #16
PS 3, #2, #10
PS 4, #3, #15
PS 5, #13 - #17
The limit of the ratio of consecutive terms
Fn 1
is the Golden Ratio. Through this series of
Fn
problems students discover how that is found and the connection to extended fractions. By the
end of this theme students should understand the connection between the recursive representation
of a Lucas sequence (such as the Fibonacci sequence) and this limit. As an extension of this
theme, one could further study Lucas sequences and find their explicit formulas.
Commented Solutions
PS 2:
15) Find the 37th Fibonacci number. Clearly indicate how you found this number.
Comment: Previously in this problem set students were given instructions on how to use the
sequence mode on their calculator to generate sequences defined recursively. This theme will
take full advantage of this. Note that once the sequences are in the calculator, the student can
generate any one term on the home screen without having to look at the previous terms or
examine a table.
u1  u1(n  1)  u1(n  2)
Soln: 
ui 1  {1,1}
u1(37)  24,157,817
16) In this problem we will consider the values of the ratio of consecutive terms of the terms of
F
the Fibonacci sequence, that is, n 1 .
Fn
a) Let Gn 
Fn 1
. State (as decimals) the first 5 terms of G. Give to three decimals.
Fn
b) Find G20 .
Themes in IMSA Problem Sets
c) What appears to be the limit of Gn as n   ? The notation we use in mathematics is
lim Gn and by that we mean what value does Gn approach as n increases without
n 
bound.
Comment: Hopefully some students recognize this number in (c) as the Golden Ratio. It would
be good to discuss this problem at some point to increase interest in what follows. Many students
have only seen the mathematical constant  and also perhaps e at this point. Best not to show
the formula
1 5
, as they will discover it, but don’t be surprised if some have seen it.
2
Soln: a) 1, 2, 1.5, 1.667, 1.6
b)
u1(21)
 1.618
u1(20)
c) Gn is approaching the Golden Ratio   1.618...
PS 3
2)
In problem set #2 you explored the Fibonacci sequence with your calculator. Consider a
similar sequence:
L: 3, 7, 10, 17, 27, . . .
a) State the 50th term of this sequence.
L
b) Let Gn  n 1 , for n  1. Find G20 . Give your answer to 6 decimals.
Ln
c) Explore the sequence Gn further. What appears to be lim Gn ?
n 
Comment: In general a Lucas sequence is one that satisfies the recursive relation
xn  P  xn1  Q  xn2 where P and Q are integers.
In this problem P  Q  1 as it is in the Fibonacci sequence. Hopefully the students will be a bit
surprised that the result in part (c) is again the Golden Ratio. Mathematically they will discover
that the ratio only depends on P and Q.
Soln: a) 68873775271
b) 1.618034
c) 1.6180339 . . .
 
Themes in IMSA Problem Sets
1
Find the value of N such that N = 1 
10)
1
1
1
1
1
1
where . . . indicates that the pattern continues indefinitely. State answer exactly.
(Hint: Let x  1 
1
, so that you can solve x  1   .
1
x
1
1
Comment: P and Q as stated above give rise to the 1’s in the extended fraction The connection
will be made on problem set 5.
Soln: x  1 
1
x
x2  x  1
x2  x 1  0
x
1 5

2
PS 4
3)
2
3
Simplify:
2
3
3
2
3
2
where . . . indicates that the pattern continues indefinitely. Give exact answer and a
six decimal approximation.
Comment: This problem and problem #15 are a pair. In this problem the student uses the
extended fraction technique hinted in problem #10 above. Note the values of P and Q.
Soln: Let x  3 
x 2  3x  2  0
2
x
Themes in IMSA Problem Sets
x
3 98
2
x
3  17
 3.561553
2
a1  1

15) Cosider the sequence: a2  1
a  2a  3a
n2
n 1
 n
a) State the frist 8 terms of the sequence.
a
b) Let Gn  n 1 . Find G5 .
an
c) Give a five decimal appproximation of lim Gn . Hmm, this answer looks familiar.
n 
Comment: P and Q are now 2 and 3 respectively. Besides giving students some sequence
practice, this them motivates the use of technology, specifically some more obscure areas of the
calculator. Empowerring students with technology will hopefully inspire them to explore other
ideas on their own.
Soln: a) 1, 1, 5, 17, 61, 217, 773, 2753
b) 3.55738
c) 3.56155
PS 5
a1  1

13) Consider the sequence: a2  1
a  5a  2a
n2
n 1
 n
d) State the first 8 terms of the sequence.
a
e) Let Gn  n 1 . Find G5 .
an
f) Give a five decimal appproximation of lim Gn .
n 
Comment: For any student that has missed what was happening previously, the problems on this
problem set are back to back.
Themes in IMSA Problem Sets
Soln: a) 1, 1, 7, 19, 73, 241, 847, 2899
b) 3.301
c) 3.44949
14) Evaluate the following extended fraction: 2 
5
.
5
2
2
5
2
5
State your answer exactly and as a five decimal approximation.
Comment: P  5 and Q  2 . The student may begin to wonder what about the other solution to
the quadratic. Good question. It turns out both solutions are used in writing the explicit formula
for the sequence, but that is not explored in this theme.
5
x
2
x  2x  5  0
x  1  6  3.44949
Soln: x  2 
a1  5

15) Consider the sequence: a2  3
a  5a  2a
n2
n 1
 n
a) State the first 8 terms of the sequence.
a
b) Let Gn  n 1 . Find G5 . State your answer to five decimal places.
an
c) Give a five decimal appproximation of lim Gn .
n 
d) State the exact value of lim Gn .
n 
Comment: To do this problem correctly the student must make the connection from the recursive
formula to the extended fraction and the associated quadratic on their own.
Soln: a) 5, 3, 31, 77, 309, 1003
b) 3.24595
c) 3.44949
Themes in IMSA Problem Sets
d) 1  6
a1  4

16)
Consider the sequence: a2  6
a  3a  7a
n2
n 1
 n
an 1
a) Let Gn 
. State an extended fraction that equals the exact value of lim Gn .
n 
an
b) Find the exact value of lim Gn .
n 
Comment: In all these problems P and Q have been positive. A question for further student
exploration might consider what happens when they are not both positive.
Soln: a) x  7 
3
7
3
3
x
2
x  7x  3  0
7  61
x
2
b) x  3 
17) Write the first 10 terms as reduced fractions of the sequence given by:
1, 1+1, 1 
1
1
1
, 1
, 1
,...
1
1
11
1
1
1
11
1
11
Comment: Students can see in this problem that considering the terms of the ratio of consecutive
terms of the Fibonacci sequence is the same as the terms of the ‘extended’ fraction sequence.
3 5 8 13 21 34 55 89
Soln: 1, 2, 1, 2, , , , , , , ,
2 3 5 8 13 21 24 55
Themes in IMSA Problem Sets
MI 2: Transformations and Algebraic Representation
This theme develops the relationships of translations and scale changes between graphs and their
algebraic representaions. One might do a similar theme taking advantage of sliders (called
parameters) in Winplot. If nothing else, demonstrating translations or scale changes in class in a
dynamic way can be a powerful learning experience and we strongly recommend a teacher does
a classroom demonstration of this tool.
PS 1
20) a) Graph and label on the same grid: y  x , y  x  4 , and y  x  3 .
b) Describe, in general terms, how the graph of y  x is transformed to produce the graph
of y  x  h .
Comment: This begins horizontal shifts. If students have access to computer to graph, encourage
them to submit printed solutions. All soltutions given here where generated using Winplot.
a)
b) The graph is shifted h units. If h  0 to the right and if h  0 to the left.
PS 2
7)
On the same grid, sketch the graph of each of the following, noting significant points.
Label each graph.
Themes in IMSA Problem Sets
a) y  x ,
y  x 2,
and
y  x 4
b) Generalize how you use the graph of y  x to graph y  x  k .
Comment: The student begins to explore vertical shifts.
Soln:
b) y  x  k translates the graph up or down k units, up if k  0 , down if k  0 .
12) In problem set #1 you looked at y  x  h and above you considered y  x  k . Now we
would like you to consider y  x  h  k .
a) Explore this problem by experimenting with different values of h and k. On your
answer sheet, give three examples you used, stating both the formula and showing its
sketch.
b) Write a generalization about the graph of y  x  h  k and its relation to y  x .
Explain your reasoning.
Comment: Explore implies the student should look at enough other examples to draw a
conclusion. If they have access to Winplot and use sliders, they should be encourage to use them.
A demo at this point would be helpful if you would like to see them explore similar problems
and take advantage of the software. Modeling inquiry type behavior is always a good use of class
time.
Soln: b) y  x  h  k translates the graph h units horizontally and k units vertically.
Themes in IMSA Problem Sets
PS 4
15) (Continuation of problem set 2) Consider the graph of y  x 2 and transformations of the
graph given by y  ( x  h)2  k .
a) Examine the graph of y  x 2 and y  x 2  k for different values of k. What effect do
different values of k have upon the graph of y  x 2 ? Give an example (both equation and
sketch) in your explanation.
b) Examine the graph of y  x 2 and y  ( x  h)2 for different values of h. What effect do
different values of h have upon the graph of y  x 2 ? Give an example (both equation
and sketch) in your explanation.
c) Generalize your results by what effect both h and k have of the graph of y  x 2 when
y  ( x  h)2  k . Give an example (both equation and sketch) in your explanation.
Comment: The student is given less direction in this problem as they are expected to create there
own examples in the spirit of inquiry. One might extend this activity to consider other functions.
Any function that is new to the students would work well, say a trig function, even though they
have not studied any trig at this point. The idea is to explore the idea of translating any function
and reach a generalization about functions which afterall the point of the problem.
Soln: a) Expect a nice variety of answers. This would be a good opportunity to post several
students solutions in class.
b) For the function y  x 2 , replacing x with ( x  h) and y with ( y  k ) translates the graph of
y  x 2 horizontally h units and vertically k units, i.e.,
y  ( x  h)2  k
translates all points (a, b) to (a  h, b  k )
PS 7
3)
In problem set 6 you graphed the greatest integer function y   x  . You should have found
the graph created ‘steps’, each one unit long and one unit high. Your calculator will graph
this function if you use the INT command, that is, set y  int( x) . Be careful though, the
calculator does not indicate which end of the step has an open circle. In making the graphs
below, clearly indicate which end is open and which is closed. In this problem you will
graph the greatest integer function the involves some multiplication.
Themes in IMSA Problem Sets
a. Graph y  2  x 
1 
b. Graph y   x 
3 
1 
c. Graph y  2  x 
3 
d. Generalize your results and describe, in complete sentences, the effect of a and b has on
the graph of y   x  when we graph y  a bx  . You may need to experiment a bit more
to reach your conclusions.
Comment: If students use Winplot they need to use the floor function. Problem set 6 asked
students to graph the basic floor function so they are familiar with it. This problem gets them to
examine scale change. The floor function is an ideal function for this exercise, as it is easy for
students to see vertical and horizontal stretching.
Soln: a)
b)
Themes in IMSA Problem Sets
c)
d) Multiplying by a multiplies the height of each step by a.
Multiplying by b divedes the length of each step by b.
PS 8
On previous problem sets you examined transformations of functions that then changed
their graphs. For the following, a transformation was performed to one of the functions you
studied and the resulting graph is shown. State the algebraic rule for the function graphed.
(For example, g ( x)  x  3 ). Hint: Examine your old problem sets and when you think you
have the rule, graph it on your calculator or computer to check.
6)
a)
Themes in IMSA Problem Sets
b)
c)
Comment: Students now get to reverse the process and determine then formula for each graph. If
they study their old problems and get this correct, they have the main idea.
Soln:
a) y  x  4  1
1 
b) y  2  x 
2 
2
c) y  x  3
Themes in IMSA Problem Sets
PS 9
7)
The graph of f ( x) is given below. State the piecewise rule for function f.
Comment: This is practice for wiriting linear equations and also for transformations in problems
#8 and #9.
 x
1

Soln: f ( x)   3
5
 2 x  2

2
8)
 4  x 1
1  x  1
1 x  3
3 x 5
Using the function graphed in problem 7, sketch the graph of
a) f ( x  3)
b)
1
f  2x
2
Comment: Students in MI 2 class have been doing these type of problems in class, given a
drawing, applying translations and scale changes, so they are familiar with the notation. You
may need to talk about the notation and demonstrate how a graph changes if students are
unfamiliar with this type of problem.
Soln:
a)
Themes in IMSA Problem Sets
b)
9)
a) State the piecewise rule for the graph you made in 8a.
b) Compare your answers to 7 and 9a. Make a conjecture as to how a translation affects the
rule for a function.
Comment: Hopefully the students will compare their genralizations from previous problem sets
to the answer to (a). This would be a good problem to discus in class as to the two different
approaches students might take, just find the equations from the graph or applying the rules for
the transformations to the algebraic representation. This understanding will come in handy when
students study sinusoidal graphs.
( x  3)
1

Soln: a) g ( x)   3
 2 ( x  3)

2
 7  x  4
 4  x  2
2 x 0
0 x2
b) Replacing x with ( x  3) moves the graph 3 units to the left. In general, replacing x with
( x  h) moves the graph of the function h units.
PS 10
8)
The domain of function f is [-5, 2] and its range is [0, 10]. Find the domain and range of:
a) f ( x  2)  3
b) 2 f (3 x)
c)
f ( x  4)  2
Themes in IMSA Problem Sets
Comment: Part (c) could be a bit of a stretch if students have not done some previous work with
absolute value and their graphs. Parts (a) and (b) give another way of thinking about what was
going on in this theme.
Soln: a) Domain:  7,0 Range:  3,7
 5 2 
b) Domain:  ,  Range: 0, 20
 3 3
c) Domain: 1,6 Range: 0,8
9)
Recall the inter-quartile range is the difference between the third quartile and the first
quartile.
At a small firm a survey of the customer service department found the length of time a
caller was put on hold on one given day was the following:
4
5
5
6
7
4
8
7
6
5
5
6
7
6
6
5
8
9
9
10
7
8
11
5
4
6
5
12
13
6
3
7
8
8
9
9
10
9
8
9
a) Determine the mean, median and inter-quartile range of this data set. Good time to use
Excel.
b) A new phone system is being considered. One company claims its system would reduce
wait times by 3 minutes for each call. Determine the mean, median and inter-quartile if the
new system is installed. Note how Excel reduces the work on this question.
c) A second company claims that their system will reduce wait time by 25%. Determine
the mean, median and inter-quartile range under these conditions.
d) Based on your answers to parts b and c, what which system would you recommend the
firm purchase? Justify your answer.
Comment: There has been a theme of data analysis running through MI 2 problem sets. This
problem examines the effect of a tranlation of data has on measures. An interesting extension
would be to examine scale changes with data.
Themes in IMSA Problem Sets
Soln:
a) Mean: 7.125 Median: 7 IQR: 4
b) Mean: 4.12 Median: 4 IQR: 4
c) Mean: 5.34 Median: 5.25 IQR: 3
d) System in (b) reduces both the median and the mean so total wait time is less.
PS 11
1 
10) The graph y  f ( x) is given below. Sketch the graph of y  3 f  x  .
2 
Comment: Here we are setting the students up for their study of trig which will not begin until
next semester. On the answer sheet we provide we give a grid with this graph reproduced to help
the student make an accurate graph.
Themes in IMSA Problem Sets
12) Given the graph y  f ( x) , sketch the graphs, labeling key points.
a) y  f ( x)  2 .
b) y  f  2 x 
c) y  2 f ( x  4)
Comment: This is typical of the problems students were asked to do earlier in MI 2 during class.
The theme developed in the problem set connects this type of work to algebraic representations
of functions.