Transcript Using Differences to Identify Patterns

Using Differences to Identify
Patterns
Section 1.1
number _________
sequence is a string of
• A _______
numbers, or terms, in a certain order.
• If the difference from one term to the next
in a number sequence is always the
same the difference is called a
______
constant ___________.
difference
_________
Example 1:
• Find the next three terms of each
sequence by using constant differences.
A. 1, 3, 5, 7, 9, …
11
1 3 5 7 9 ___
+2
+2
+2
+2
+2
13 ___
15
___
+2
+2
Example 1
B. 80, 73, 66, 59, 52, …
•
80 73 66 59 52 ___
45
-7
-7
-7
-7
-7
___
38 ___
31
-7
-7
Try these…
C. 1, 4, 7, 10, 13, …
1 4 7 10 13 ___
___ ___
16 19
22
+3
+3
+3
+3
+3
+3
+3
Try these…
D. 30, 25, 20, 15, 10, …
•
0 ___
30 25 20 15 10 ___
5 ___
-5
-5
-5
-5
-5
-5
-5
-5
Example 2
•
Find the next three terms of each
sequence by using constant differences.
E. 1, 4, 9,16, 25, …
1 4 9 16 25 ___
36 ___
49 ___
64
+3
+5
+2
+7
+2
+9
+2
+11
+2
+13
+2
+15
+2
First differences
Second differences
Example 2
• F. 37, 41, 48, 58, 71, …
•
87 106 128
37 41 48 58 71
+4
+7
+3
+10 +13
+3
+3
+16
+3
+19
+3
+22
+3
First differences
Second differences
Try these…
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G. Find the next three terms of each
sequence by using constant differences.
2, 6, 12, 20, 30, …
42 ___
56 72
2 6 12 20 30 ___
___
+4
+6
+2
+8
+2
+10
+2
+12
+2
+14
+2
+16
+2
First differences
Second differences
Try these…
•
•
•
H. Find the next three terms of each
sequence by using constant differences.
8, 20, 30, 38, 44, …
8 20 30 38 44 ___
48 ___
50 ___
50
+12 +10
-2
+8
-2
+6
-2
+4
-2
+2
-2
+0
-2
First differences
Second differences
• A___________
conjecture is a statement about
observations that is believed to be true.
• Mathematicians try to prove or disprove
conjectures.
• Let’s observe the next relationship and
see if a conjecture can be made.
Example 3
• The table below shows the relationship between
temperatures in Celsius and temperatures in Fahrenheit.
Use the method of constant differences to find the
Fahrenheit temperatures that correspond to the Celsius
temperatures of 50, 60, and 70.
Celsius
0
10 20 30
40
Fahrenheit 32 50 68 86 104
+18
+18 +18
+18
50 60 70
122
140
158
+18 +18 +18
• What conjecture can you make about this
relationship?
For every 10 degrees that Celsius increases, the
Fahrenheit increases 18 degrees.
• Some sequences can also be studied with
diagrams.
For example, the sequence 2, 6, 12, 20, 30, … is found by counting the
number of dots in the pattern below.
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12
2
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23
6
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34
12
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6  7  42
The next three terms are _________________,
7  8  56
8  9  72
_________________,
and ____________________.
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56
30
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Problem solving strategies can include:
•
•
•
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Drawing a diagram
Solving a simpler problem
Making a table or chart
Looking for a pattern
Example 4
• Suppose that 10 friends have just returned
to school. Each friend has exactly one
conversation with each of the other friends
to talk about what they did during summer
break. Use problem-solving strategies to
determine how many conversations there
will be.
one
person

two
people
three
people
four
people

 
 
 
 
Arrange the information from the simpler problems in a table. Look for a pattern.
People
1 2 3 4 5 6 7 8 9 10
Conversations 0 1 3 6 10 15 21 28 36 45
1
2
3
4
5
6
7
8
9
Use differences to determine
how the number of conversations
45 conversations
With 10 friends, it takes _______
for each person to have exactly
is increasing.
Then extend
the pattern
to 10 people.
one conversation
with each
other person.