Download ch 5 finding a pattern notes

Finding a Pattern Notes - Sequences
-The study of ________________ is often called the study of _________________.
-As a problem-solving strategy, _____________ patterns enable you to ____________ a complex problem to a
_________________ and then use the pattern to find a _____________. Often the key to finding a pattern is
_______________ information.
A) A __________________ is an ordered string of numbers tied together by a ___________ ___________, or set
of rules, that _________________ the next term in the sequence.
-A ________________ is an individual member of a sequence.
Ex) Find the pattern and predict the next four terms. Then write a sentence to explain the pattern.
1. 1, 2, 4, _______, _______, _______, ________
a. _______________ each term to find the next term in the sequence.
b. Start by _________________ one to the first term, then _________ one greater number to each
successive term.
2. 1, 3, 5, 7, _______, _______, ________, ________
a. ________ _________ to each term to find the next.
3. 1, 6, 11, 16, ________, ________, _________, ________
a. _________ ________ to each term.
4. 1, 3, 6, 10, ________, _______, ________, ________
a. Start by _________ _______ to the first term, then _______ ________ greater number to each
successive term.
Having trouble with sequences? Try the following process:
1) _______________ down the sequence you’re working with:
3,4,6,9,13,18,24,…
2) Write down each __________ of the sequence and its _______________ in the sequence, as show in the
table below.
Position
Term
1
3
2
4
3
6
4
9
5
13
6
18
7
24
6
18
7
24
3) Write down the _________________ between the terms of the sequence:
Position
Term
1
3
2
4
3
6
4
9
5
13
4) ________________ for a way to ___________ the terms to their positions. Look for a way to relate the
___________________ between terms to the ___________________.
Position
Term
1
3
2
4
3
6
4
9
5
13
6
18
7
24
5) If figuring the differences between terms doesn’t work, try using other operations. (mult., div,..)
One __________ type of sequence is called the _____________ Sequence. This sequence has __________ to
start it off and usually it’s the first ______ numbers, sometimes even the first _________. This sequence
_________ the two or three __________ terms to find the next.
Find the next two terms
Ex) 2, 2, 4, 6, 10, 16, 26, ______, ______
Ex) 1, 2, 3, 6, 11, 20, 37, ______, ______
Patterns can be identified on different level of ___________________. That is, sometimes it lies in the
differences between the ______________, sometimes it lies in the differences between the ________________
and so on.
Finding Patterns Example
Dodger Stadium
Radio broadcasters joke about the number of people who start leaving Dodger Stadium during the 7th
inning of baseball games. One evening, during a particularly boring baseball game in which the Dodgers
were trailing by six runs after six innings, the fans began to leave at a record pace. After the first out in the
top of the seventh inning, 100 fans left. After the 2nd out, 150 fans left. After the third out, 200 fans left.
The pattern continued in this way, with 50 more fans leaving after each out than had left after the
previous out. The ridiculous thing was, the Dodgers tied up the game in the bottom of the 9th inning, and
people still kept leaving early. The game lasted 10 innings, and the pattern continued through the bottom
of the 10th inning. How many fans left early?
Out #
1
2
3
4
5
Base Fans
Leaving
Additional
Fans
Leaving
Additional
Fans
Leaving
Additional
Fans
Leaving
6
7
…
Total
…
…
…
…
Pattern:
2nd Paragraph
I solved this problem by ___________ a ________________. First, I made a ______________ with the outs
and total labeled on top and the number of __________ leaving on the left. Next, I knew the base number
of fans leaving every inning so I filled in 100 in every column of that row. Then, I added another ________
fans every inning of play ______________ the columns of outs that was before the out number I was on.
After that, I saw a _____________ in the total number column after the first 2400 so I did not make the
chart for the total 24 outs nor did I continue with the additional fans leaving rows. Lastly, I followed the
pattern and added the _____________ number column to find the total number of fans leaving.
The total number of fans leaving was ________________ people.
Finding Pattern Notes - Functions
Sequences taken from algebra are featured next. These problems are models of ________________. A function
assigns an ___________ to each _____________. For a function there is a consistent rule that uses the input
number to generate the output number.
Ex) Table of values – Determine the rule for each function shown below
In
x
0
1
2
3
4
5
895
Out
?
5
6
7
8
9
?
?
To find each successive terms, you’d simply add 1 to each previous term.
However looking at this example in the form of a table of values allows
you to find a _____________ that you can apply to a certain term to
find every other term. Let x represent that certain term, or input. The
table shows that when x is 2, the output is _____. When x is 4, the
output is ______. So the rule of __________ _____ to the input seems
to work for the inputs 2 and 4. Experimenting further shows you that
this rule works for all the inputs. Now that we know that x+5 is the
function rule to apply to each input, determining the output for an input
of 895 is easy: 895+5=900.
*If you can’t figure out what the rules are, simply treat each problem as if it was a sequence
written _________________.
Clearly there is a big difference between finding patterns in tables of values and finding patterns in
simple sequences.
*When finding patterns in _________________, you attempt to go through the entire sequence, always
try the next term in the sequence.
*When looking for a pattern in a ____________ of ______________ , you try to find a relationship
between the input number and the output number.
-Finding this _______________ allows you to jump way ahead in the sequence and find the _____________ number (such as 895) for a ___________ number _________________ having to extend
the sequence all the way to the 895th term.