Download Reteach Using Inductive Reasoning to Make Conjectures

Name
Date
Class
Reteach
LESSON
2-1
Using Inductive Reasoning to Make Conjectures
When you make a general rule or conclusion based on a pattern, you are using
inductive reasoning. A conclusion based on a pattern is called a conjecture.
Pattern
Conjecture
8, 3, 2, 7, . . .
Next Two Items
7 5 12
12 5 17
Each term is 5 more than
the previous term.
45°
The measure of each angle
is half the measure of the
previous angle.
Find the next item in each pattern.
3, 1, . . .
1 , __
1, __
1. __
4 2 4
11.25°
22.5°
2. 100, 81, 64, 49, . . .
1
1__
4
36
3.
4.
3
6
10
Complete each conjecture.
5. If the side length of a square is doubled, the perimeter of the square
is
doubled
.
6. The number of nonoverlapping angles formed by n lines intersecting in a point
is
2n
.
Use the figure to complete the conjecture in Exercise 7.
7. The perimeter of a figure that has n of these triangles
is
1
1
n2
.
1
1
03
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1
1
1
1
04
1
1
1
1
1
1
05
1
1
1
1
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Holt Geometry
Name
LESSON
2-1
Date
Class
Reteach
Using Inductive Reasoning to Make Conjectures
continued
Since a conjecture is an educated guess, it may be true or false. It takes only
one example, or counterexample, to prove that a conjecture is false.
Conjecture: For any integer n, n 4n.
n
n 4n
True or False?
3
3 4(3)
3 12
true
0
0 4(0)
00
true
2
2 4(2)
2 8
false
n 2 is a counterexample, so the conjecture is false.
Show that each conjecture is false by finding a counterexample.
8. If three lines lie in the same plane, then they intersect in at least one point.
Possible answer: If the lines are parallel, then they do not intersect.
9. Points A, G, and N are collinear. If AG 7 inches and GN 5 inches, then
AN 12 inches.
Possible answer: If point N is between points A and G, then AN 2 inches.
10. For any real numbers x and y, if x y, then x 2 y 2.
2
2
Sample answer: If x 0 and y 1, then x y .
11. The total number of angles in the figure is 3.
$
Sample answer: ABD, DBE,
"
!
EBC, ABE, DBC
%
#
12. If two angles are acute, then the sum of their measures equals the
measure of an obtuse angle.
Sample answer: m1 25,
m2 20
Determine whether each conjecture is true. If not, write or draw a counterexample.
14. If J is between H and K, then HJ JK.
13. Points Q and R are collinear.
(
true
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+
Holt Geometry
Name
LESSON
2-1
Date
Class
Name
Practice A
2-1
Find the next item in each pattern.
Using Inductive Reasoning to Make Conjectures
Find the next item in each pattern.
2. Z, Y, X, . . .
1. 2, 4, 6, 8, . . .
Class
Practice B
LESSON
Using Inductive Reasoning to Make Conjectures
3. fall, winter, spring, . . .
�
5. A statement you believe to be
3. Alabama, Alaska, Arizona, . . .
.
true
north
Complete each conjecture.
5. The square of any negative number is
For Exercises 6–8, complete each conjecture by looking for a pattern
in the examples.
even
positive
.
6. The number of diagonals that can be drawn from one vertex in a convex polygon
.
n�3
that has n vertices is
n
7. The number of sides of a polygon that has n vertices is
�������
4. west, south, east, . . .
Arkansas
based on inductive
reasoning is called a conjecture.
6. The sum of two odd numbers is
3�5�8
13 � 3 � 16
1�1�2
�
36
4. When several examples form a pattern and you assume the pattern will continue,
inductive reasoning
2.
1. 100, 81, 64, 49, . . .
summer
W
10
you are applying
Date
.
Show that each conjecture is false by finding a counterexample.
.
7. For any integer n, n 3 � 0.
Possible answers: zero, any negative number
8. Each angle in a right triangle
has a different measure.
8. When a tree is cut horizontally, a series of rings is visible in the stump. Make a
conjecture about the number of rings and the age of the tree based on the data
in the table.
Number of Rings
3
15
22
60
Age of Tree (years)
3
15
22
60
���
9. For many years in the United States, each bank printed its own currency. The variety
of different bills led to widespread counterfeiting. By the time of the Civil War, a
significant fraction of the currency in circulation was counterfeit. If one Civil War
soldier had 48 bills, 16 of which were counterfeit, and another soldier had 39 bills,
13 of which were counterfeit, make a conjecture about what fraction of bills were
counterfeit at the time of the Civil War.
The number of rings in a tree is the same as the tree’s age.
9. Assume your conjecture in Exercise 8 is true. Find the
number of rings in an 82-year-old oak tree.
82 rings
false
10. A counterexample shows that a conjecture is
One-third of the bills were counterfeit.
.
Make a conjecture about each pattern. Write the next two items.
Show that each conjecture is false by finding a counterexample.
10. 1, 2, 2, 4, 8, 32, . . .
11. For any number n, 2n � n.
Possible answers: zero, any negative number
12. Two rays having the same endpoint make an acute angle.
(Sketch a counterexample.)
3
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Name
LESSON
2-1
11.
�
Each item, starting with the
Possible answer:
Date
Holt Geometry
Class
The dot skips over one vertex
in a clockwise direction.
4
2-1
Using Inductive Reasoning to Make Conjectures
The pattern is the cubes of the negative integers; �125, �216.
Pattern
Conjecture
Each item describes the item before it (one, one one, two ones, . . .);
312211, 13112221.
3. A, E, F, H, I, . . .
45°
The pattern is the letters of the alphabet that are made only from straight
segments; K, L.
�
36
4.
3
true
6
10
��
false
�
Sample answer:
Complete each conjecture.
n
3
�n �
8. If n is an integer (n � 0), then _1_ � _1_
5. If the side length of a square is doubled, the perimeter of the square
true
7. If a � b and b � c, then a � b � a � c.
is
false
Possible answers: n � 1, n � �1
.
9. Hank finds that a convex polygon with n sides has n � 3 diagonals from any one
vertex. He notices that the diagonals from one vertex divide every polygon into
triangles, and he knows that the sum of the angle measures in any triangle is 180�.
Hank wants to develop a rule about the sum of the angle measures in a convex
polygon with n sides. Find the rule. (Hint: Sketching some polygons may help.)
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.
is
2n
.
Use the figure to complete the conjecture in Exercise 7.
7. The perimeter of a figure that has n of these triangles
10. A regular polygon is a polygon in which all sides have the same length and all
angles have the same measure. Use your answer to Exercise 9 to determine the
measure of one angle in a regular heptagon, a regular nonagon,
and a regular dodecagon. Round to the nearest tenth of a degree. 128.6�; 140�;
5
doubled
6. The number of nonoverlapping angles formed by n lines intersecting in a point
Sum of angle measures � [180(n � 2)]�
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2. 100, 81, 64, 49, . . .
3.
Determine if each conjecture is true. If not, write or draw a counterexample.
6. An image reflected across the x-axis cannot appear identical
to its preimage.
�
11.25°
22.5°
1_1_
4
First rotate the figure 180�. Then reflect the figure across a vertical line. Repeat.
5. Three points that determine a plane also determine a triangle.
7 � 5 � 12
12 � 5 � 17
The measure of each angle
is half the measure of the
previous angle.
Find the next item in each pattern.
1. _1_, _1_, _3_, 1, . . .
4 2 4
,
��������
Next Two Items
Each term is 5 more than
the previous term.
�8, �3, 2, 7, . . .
2. 1, 11, 21, 1211, 111221, . . . (Hint: Try reading the numbers aloud in different ways.)
�
Holt Geometry
Class
When you make a general rule or conclusion based on a pattern, you are using
inductive reasoning. A conclusion based on a pattern is called a conjecture.
1. �1, �8, �27, �64, . . .
�
Date
Reteach
LESSON
Using Inductive Reasoning to Make Conjectures
�������
preceding items; 256, 8192.
Name
Practice C
�
,
third, is the product of the two
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All rights reserved.
Make a conjecture about each pattern. Write the next two items.
4.
���
is
n�2
.
1
1
1
��3
1
1
1
1
��4
1
1
1
1
1
��5
1
1
1
1
1
1
��6
150�
Holt Geometry
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All rights reserved.
59
6
Holt Geometry
Holt Geometry
Name
LESSON
2-1
Date
Class
Name
Reteach
Date
Challenge
LESSON
Using Inductive Reasoning to Make Conjectures
2-1
continued
Since a conjecture is an educated guess, it may be true or false. It takes only
one example, or counterexample, to prove that a conjecture is false.
Class
Discovery Through Patterns
The pattern shown is known as Pascal’s Triangle.
1. If the pattern is extended, find the terms in row 7.
Conjecture: For any integer n, n � 4n.
�
�����
1, 6, 15, 20, 15, 6, 1
�
�����
n
n � 4n
True or False?
3
3 � 4(3)
3 � 12
true
0
0 � 4(0)
0�0
true
number. Each of the other numbers is
�����
�2
�2 � 4(�2)
�2 � �8
false
found by adding the two numbers that
�����
2. Make a conjecture for the pattern.
�
�����
Each row has 1 as the first and last
�
�����
�
�
�
�
�
�
�
�
�
�
��
�
�
��
�
�
�
appear just above it.
n � �2 is a counterexample, so the conjecture is false.
3. Make a conjecture about the sum of the terms in each row.
Show that each conjecture is false by finding a counterexample.
The sum of each row of terms after the first is twice the sum of the terms
in the previous row.
8. If three lines lie in the same plane, then they intersect in at least one point.
Possible answer: If the lines are parallel, then they do not intersect.
Refer to the pattern of figures for Exercises 4 and 5.
9. Points A, G, and N are collinear. If AG � 7 inches and GN � 5 inches, then
AN � 12 inches.
Possible answer: If point N is between points A and G, then AN � 2 inches.
2
Figure 1
2
10. For any real numbers x and y, if x � y, then x � y .
Sample answer: If x � 0 and y � �1, then x 2 � y 2.
11. The total number of angles in the figure is 3.
�
�
27; 81
5. Write an algebraic expression for the number
of black triangles in figure n.
3
2
6. For every integer x, x � 2x � 1
is divisible by 2.
12. If two angles are acute, then the sum of their measures equals the
measure of an obtuse angle.
Sample answer: x � 2
Sample answer: n � �2
2
7. For every integer n, n � n is prime.
a
8. Make a table of values for the expression 4 � 1, where a is a positive integer.
Make a conjecture about the type of number that is generated by the rule.
Sample answer: m�1 � 25�,
m�2 � 20�
13. Points Q and R are collinear.
14. If J is between H and K, then HJ � JK.
�
true
7
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All rights reserved.
Name
�
Date
�
Class
a
a
1
2
3
4
5
Determine whether each conjecture is true. If not, write or draw a counterexample.
2-1
Holt Geometry
Problem Solving
4 �1
3
15
63
255
1023
1. Estimate the length of a green iguana
after 1 year if it was 8 inches long when
it hatched.
Length after
Hatching (in.)
Length after
1 Year (in.)
1
10
36
2
9
34
3
11
35
4
12
35
5
10
37
Sample answer: 33 in.
2. Make a conjecture about the average
growth of a green iguana during the
first year.
2
3
4
5
6
7
�������������������
������������������
����������
�����������
���������
�����������
8
5. The table shows the number of cells
present during three phases of mitosis.
If a sample contained 80 cells during
interphase, which is the best prediction
for the number of cells present during
prophase?
A 18 cells
B 24 cells
Match 4 was 18 minutes long.
Sample
C 40 cells
D 80 cells
0
Number of Cells
Interphase Prophase
86
22
5
2
70
28
3
3
76
32
3
4
91
25
5
5
65
16
4
6
89
34
6
H 306 students
A $20.50
C $24.50
G 204 students
J 333 students
B $23.33
D $25.00
9
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1 2 3 4 5 6 �
2. Consider the two conjectures given to explain the pattern seen on the grid.
Metaphase
1
F 102 students
(4, 4)
(3, 3)
(2, 2)
(1, 1)
1. Observe the pattern of the points on the grid.
Conjecture A
or
To find the next point on the grid,
add 1 to the x-coordinate and
subtract 1 from the y-coordinate.
Conjecture B
To find the next point on the grid,
add 1 to both the x-coordinate and
the y-coordinate.
3. Evaluate each of the conjectures and plot the next two points on the graph according to
each conjecture. Which conjecture is correct?
7. Mara earned $25, $25, $20, and $28
in the last 4 weeks for walking her
neighbor’s dogs. If her earnings continue
in this way, which is the best estimate
for her average weekly earnings for next
month?
6. About 75% of the students at Jackson
High School volunteer to clean up a
half-mile stretch of road every year.
If there are 408 students in the school
this year, about how many are expected
to volunteer for the clean-up?
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�
6
5
4
3
2
1
4. These matches were all longer than
a half hour.
Match 7 lasted 1 hour 3 minutes.
Choose the best answer.
������
���������������
Consider the following example:
0:31 0:56 0:51 0:18 0:50 0:34 1:03 0:36
3. Every one of the first eight matches
lasted less than 1 hour.
Holt Geometry
Identify Relationships
The times for the first eight matches of the Santa Barbara Open women’s
volleyball tournament are shown. Show that each conjecture is false by
finding a counterexample.
1
Class
The steps to making a decision based on inductive reasoning are
presented below.
Sample answer: The average growth of a green iguana during the first
year is about 2 ft.
Time
Date
Reading Strategies
2-1
Iguana
8
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All rights reserved.
LESSON
The table shows the lengths of five green iguanas after birth and then after 1 year.
Sample answer: All values of
a
4 � 1 are divisible by 3.
Name
Using Inductive Reasoning to Make Conjectures
Match
n�1
Find a counterexample for each statement.
�
�EBC, �ABE, �DBC
LESSON
Figure 3
�
Sample answer: �ABD, �DBE,
�
Figure 2
4. If the pattern continues, how many black
triangles will there be in Figure 4? in Figure 5?
For Conjecture A, students should plot (5, 3) and (6, 2).
For Conjecture B, students should plot (5, 5) and (6, 6).
Student should conclude that Conjecture B is correct.
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