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Part
1: Exploring Sequences in Middle School through Algebra I
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Introduce the activity by listing several
sequences on the board and having the students determine the
“rules” for generating such a sequence.
For this activity, the allowable rules are: 1) what to start
with, and 2) what to multiply by and add on to each number.
Ex ample: 1, 2, 3,
4, 5, 6, 7, 8, 9, 10, ...
Rule: Start with 1, then multiply by 1 and add on 1 each time.
Ex ample: 1, 3, 5,
7, 9, 11, 13, ...
Rule: Start with 1, then multiply by 1 and add on 2 each time.
Ex ample: 1, 4, 10,
22, 46, 94, ...
Rule: Start with 1, then multiply by 2 and add on 2 each time.
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We can set up a spreadsheet to quickly generate
sequences and their cumulative sequences.
Have the students create a spreadsheet like the following.
Although the formulas are given in the following figure, have
the students use their knowledge of spreadsheets and formulas to
create and make sense of the formulas rather than just inputting those
shown.
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A
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B
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1
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SEQUENCE
GENERATOR
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2
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1
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Starting Number
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3
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1
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Multiplier
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4
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1
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Add On
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5
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6
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Sequence
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Cumulative
sequence
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7
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=A2
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=A7
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8
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=A7*$A$3
+ $A$4
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=B7
+ A8
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9
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10
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Fill down the
formulas in cells A8 and B8 so that the first 20 numbers in the
sequence and cumulative sequence can be calculated. Discuss the roles
of cells A2, A3, and A4 as variables and the use of the dollar sign
($) to indicate “absolute” cell references that do not increment
when the formula is filled down.
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With the triple (1, 1, 1) as the (starting
number, multiplier, add on), the counting numbers appear in column A.
Do you recognize the numbers in column B? Use a drawing or
physical model to illustrate the relationship between the counting
numbers and the numbers in column B (Hint: build a “growing”
staircase or triangular array).
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Explore changing the values in the variables
(starting number, multiplier, add on) and observe the resulting
sequences and cumulative sequences.
Try to generate the following sequences.
Discuss the values and strategies used to generate each.
Are there more than one set of triple values that can be used
to generate the same sequence?
- even numbers
- multiples of five (discuss concept of
multiplication as repeated addition)
- powers of three
(discuss concept of using exponents to represent repeated multiplication)
- numbers that
end in seven (connect with place value concepts)
- powers of 10
(connect with place value concepts)
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Be sure students explore the use of negative
values. Suggest the
following triple (5, -1, 5) and discuss the effects of multiplying by
a negative value. Notice
the “zigzag” and “staircase” appearance in the graph.
Pose the following challenges:
- Can anyone
make a zigzag without also having a staircase?
- Can anyone
make a staircase without also having a zigzag?
- Can anyone make a decreasing staircase?
- Can you generalize
any findings about generating a zigzag or staircase sequence?
Part
2: Exploring Convergence and Divergence in
Precalculus
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Set the starting number to 1 and the add on to 0.
Explore different values for the multiplier (M).
Be sure to include positive, negative, and decimal values.
After allowing time for exploration, use the multipliers (M)
in the table on the next page and record the values to which the
cumulative sequences would converge (C)
as n goes to infinity (n is the number of items in the sequence).
For this exploration, it may help to express the last cell in
the cumulative sequences column (about the 20th term) as a proper
fraction or mixed number.
To
express a value in a cell in fractional form:
- Click in the cell containing the value you want
to express in fractional form.
- Under the Format
menu, choose Cells...
- In the dialog box, click on the
Number tab and choose the Fraction
category. Highlight
the choice # ??/?? .
This will allow the value of that cell to be
expressed as proper fraction or mixed number with a maximum of
two digits in the numerator and denominator.
- Click on OK. (Excel does not allow values
expressed as improper fractions.)
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Multiplier
M
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Fraction
Equivalent of M
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Cumulative
sequence Convergent Value C
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Fraction
Equivalent of C
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0.5
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0.25
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0.75
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0.3333333
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0.4
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With the starting number fixed at 1 and the
multiplier at 0.5, what value should be used as the add on to make the
sequence converge to 10? One
way to approach this problem is to guess and test several values. Excel has a built in feature called Goal Seek that can also be used to solve this problem.
Use the Goal Seek to set the value in the cell of the sequence
(probably in cell A25) to 10 by changing the value of the add on (cell
A3).
To use the Goal Seek:
- Under the Tools
menu select Goal Seek...
- In the dialog box, fill in the boxes for Set
Cell (the cell containing a formula that you want to have a
specific value), To Value
(the goal value for the formula), and By Changing Cell (the cell that
contains the value that will effect the formula referenced
in the Set Cell box).
- Click OK. The
spreadsheet will change the values in the “changing” cell
until the “set” cell reaches the goal value.
Note: The
Set Cell box must
refer to a cell containing a formula that references the cell
entered in the By
Changing Cell box.
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- Use the Goal Seek tool to find the add on value
for each of the situations in the table on the next page.
Before “seeking” each value, make a hypothesis of the value
needed for the add on to make the sequence converge to the intended
value.
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Start
Number
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Multiplier
(M)
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Value
Sequence Converges To (S)
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Hypothesized
Add On Needed
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Actual
Add On (A)
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1
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0.5
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4
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1
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0.5
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8
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1
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0.5
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16
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1
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0.25
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4
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1
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0.25
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8
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1
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0.25
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16
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Determine the relationship between the multiplier
(M), add on (A),
and the value to which the sequence converges (S)?
Express this relationship as a function of M
and A. Are there any
conditions under which this relationship does not hold?
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Conjecture how the starting number would affect
the value to which the sequence converges. To explore this question,
fix the multiplier and add on to specific values and only change the
starting number. Compare your results with your conjecture.
Discuss why and how the starting number affects the value to
which the sequence converges.
Discuss the results
as a class.
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Now create a cumulative sequence that converges to