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Lesson 53
TAKS Grade 9 Objective 10
(8.16)(A)
Making Conjectures
A conjecture, or statement predicting what is true in certain cases, is often a
part of mathematical problems. To make conjectures, you will need
information that can be confirmed as either true or false.
New Vocabulary
• conjecture
• sequences
Using Patterns to Make Conjectures
Patterns that contain repeating sequences of numbers, shapes, or colors
can be used for conjectures. Note what features in the sequence are
repeated, and how often they are repeated. From this information, you
can predict where these repetitions will occur at a point in the sequence
that is not given explicitly.
Sometimes the repetition is not in the form of a recurring number or figure,
but in a procedure. In the sequence 2, 4, 6, 8, 10, … , no number is repeated.
Instead, the adding of 2 to each number to produce the next is repeated.
This repetition allows you to make the conjecture that each number in the
sequence will be 2 larger than the previous number.
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EXAMPLE 1
What conjecture can you make about the following sequence
of numbers?
1, 1, 2, 3, 5, 8, 13, 21, 34, …
Step 1 Determine the pattern in the given sequence.
The third number, 2, equals the sum of the first two
numbers, 1 and 1. The fourth number, 3, equals the sum
of the second and third numbers, 1 and 2. The fifth
number, 5, equals the sum of the third and fourth
numbers, 2 and 3.
Step 2 Make your conjecture based on the pattern.
After the first two numbers in the sequence, each
number in the sequence equals the sum of the previous
two numbers.
Step 3 Test your conjecture using the given information.
The seventh number is 13. The eighth number is 21, The
ninth number, 34, should equal 13 21. This is the case,
which supports the conjecture.
Be sure that you have
recognized the pattern
that is taking place in
the sequence, so that
you can accurately
predict where the
repetitions will occur
at a later point in the
sequence.
Quick Check 1
1a. What conjecture can you make about the
following sequence of numbers?
1, 5, 9, 13, 17, 21, …
TAKS Review and Preparation Workbook
1b. What conjecture can you make about the
following sequence of numbers?
1, 4, 2, 4, 3, 4, …
LESSON 53
■
Making Conjectures
157
TAKS Objective 10 (8.16)(A)
LESSON 53
Using Examples or Non-Examples to Make Conjectures
Sometimes sets of examples have a common property about which you
can make a conjecture. Sometimes a conjecture can even be made about a
property that is not a part of the examples. These “non-examples” indicate
what the conjecture is. Start by looking at all of the examples to see what
they have in common. These are factors you can ignore. Look for patterns
in how you get from one example to the next. Use these patterns to make
a conjecture about the next example. In the event that there is something
the examples are all missing, a conjecture can still be made from this set
of non-examples.
If there is enough
information to make a
conjecture, then there
is enough information
to check it.
EXAMPLE 2
A group of students were given 20 problems to solve in different amounts of time. The
number of mistakes made by students for each block of time is shown in the scatterplot
below. What conjecture can you make about the data?
6
5
4
3
2
1
0
5
10 15 20 25
Time (minutes)
30
Step 1 Examine the information for common features or patterns.
The number of mistakes decreases as the time for solving the problems increases.
Step 2 Make your conjecture.
The greater the time spent on each problem, the fewer mistakes that are made.
Step 3 Test your conjecture using the given information.
The line that follows the trend of the data has a negative slope. Therefore, as the
number in the domain (amount of time per problem) increases, the number in the
range (number of mistakes made) decreases.
Quick Check 2
2a. What conjecture would you make if the data in Example 2 formed
a horizontal line?
2b. What conjecture would you make if the data in Example 2 followed
no particular trend?
158
LESSON 53
■
Making Conjectures
TAKS Review and Preparation Workbook
Copyright © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved.
Number of Mistakes
Number of Mistakes
vs. Time
Name__________________________Class____________Date________
1 The figures below have a repeating pattern.
...
1
2
3
4
5
6
7
8 ...
Which of the following will appear as the
18th figure in the pattern?
A
C
B
D
4 Five balls are chosen such that the radius
of each ball is as follows: 2 centimeters,
4 centimeters, 8 centimeters, 16 centimeters,
and 32 centimeters. Assuming that each
ball is a perfect sphere, which of the
following statements about the four
largest balls is correct?
F The volume of any ball is 8 times greater
than the volume of the next smallest ball.
G The volume of any ball is 4 times greater
than the volume of the next smallest ball.
H The volume of any ball is 3 times greater
than the volume of the next smallest ball.
J The volume of any ball is 2 times greater
than the volume of the next smallest ball.
2 Which statement is a correct conjecture
about the following numerical sequence?
1, 1, 1,…
1, 14, 19, 16
25 36
Copyright © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved.
F Each number in the sequence equals
one half the previous number (12n).
5 Angela is comparing the prices of different
shoes. She notes the prices for two sizes and
two styles for each of three brands. Angela’s
findings are listed in the table below.
G Each number in the sequence equals
one third the previous number (13n).
H Each number in the sequence equals
the reciprocal of the square of the
number 12 .
(n )
J Each number in the sequence equals
the reciprocal of the cube of the
number 13 .
(n )
3 The arrangement of colored lights along the
edge of a carnival booth is as follows: blue,
green, yellow, and purple. If the sequence
starts with a blue light, what color will the
32nd light in the sequence be?
Size 6
dress
Size 8
dress
Size 6
loafers
Size 8
loafers
Brand A
$22
$22
$25
$25
Brand B
$16
$16
$11
$11
Brand C
$19
$19
$14
$14
Which conjecture follows from the data in
the table?
A The cost of all shoes increases with
shoe size.
B The cost of shoes does not depend on
the brand.
C The cost of shoes is always highest for
brand A and lowest for brand B.
D The cost of loafers is always less than
the cost of dress shoes.
A blue
B green
C yellow
D purple
TAKS Review and Preparation Workbook
LESSON 53
■
Making Conjectures
159