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Section 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
−8, −3, 2, 7, . . .
Conjecture
Each term is 5 more than the
previous term.
Next Two Items
7 + 5 = 12
12 + 5 = 17
The measure of each angle is
half the measure of the
previous angle.
Find the pattern in each of the sequences, then write the next three numbers in the sequence.
1) 1000, 500, 250, _________, _________,_________.
2) 1,1,2,3,5,8,13, _________, _________,_________.
3)
1 2 1 4
, , , , _________, _________, _________.
8 7 2 5
4) 11.2, 9.2, 7.2, _________, _________,_________.
5) 6, 12, 24, _________, _________,_________.
6) 101, 102, 105, 110, 117, _________, _________,_________.
7) -1, -5, -9, -13_________, _____________, __________
8) 17, 15, 12, 8, _________, _____________, __________
9) 3, 5, 9, 15, _________, _____________, __________
10)
1, 4, 10, 19, _________, _____________, __________
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11)
1 3 5 7
, , , , _________, _____________, __________
2 4 6 8
12) Draw the next figure in the pattern:
13) 4, 8, 12, 16, _______, _______
14) 400, 200, 100, 50, 25, _______, _______
15) 1, 3, 9, 27, 81, ________, _______
16) 1, 5, 14, 30, 55, _______, _______
17) Use inductive reasoning to predict the number of dots in the 8th figure for the problem
below.
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Section 2.1
Definition: A counterexample is an example that justifies that a statement is false.
Example 1: The numbers of days in every month is 31. A counterexample is February, since it
only has 28 days. Another counter example is April, it only has 30 days.
Example 2: For any integer n, n < 4n. A counterexample is n = -3, since -3 is not greater than 12.
Show that each conjecture is false by finding a counterexample.
1)
The absolute value of every real number is positive.
2) All odd numbers are prime.
3)
Multiplying any number by -1 produces a product that is less than -1.
4)
If J is between H and K, then HJ = JK.
5. If three lines lie in the same plane, then they intersect in at least one point.
________________________________________________________________________________________
6. Points A, G, and N are collinear. If AG = 7 inches and GN = 5 inches, then
AN = 12 inches.
________________________________________________________________________________________
7. For any real numbers x and y, if x > y, then x 2 > y 2.
________________________________________________________________________________________
8. The total number of angles in the figure is 3.
________________________________________
________________________________________
12. If two angles are acute, then the sum of their measures equals the
measure of an obtuse angle.
________________________________________
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A conclusion based on a pattern is called a conjecture.
Determine the pattern, then make a conjecture.
Cube Problem 1: The first three objects are show. How many blocks are in the sixth figure?
Cube Problem 2: The first three objects are show. How many blocks are in the sixth figure?
Triangle Problem:
Draw the next figure, determine the perimeter.
The perimeter of a figure that has n of these triangles is _____________?
Square Problem: If the side length of a square is doubled, the perimeter of the square
is __________________________________ .
(hint, try a few problems)
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Find the next item in each pattern.
1.
1 1 3
, , , 1, . . .
4 2 4
2. 100, 81, 64, 49, . . .
________________________________________
3.
________________________________________
4.
________________________________________
________________________________________
5) 1000, 500, 250, _________, _________,_________.
Pattern:_______________________________________________________________
6) 1,1,2,3,5,8,13, _________, _________,_________.
Pattern:_______________________________________________________________
7)
1 2 1 4
, , , , _________, _________, _________.
8 7 2 5
Pattern:_______________________________________________________________
8) 11.2, 9.2, 7.2, _________, _________,_________.
Pattern:_______________________________________________________________
9) 6, 12, 24, _________, _________,_________.
Pattern:_______________________________________________________________
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5. If the side length of a square is doubled, the perimeter of the square
is __________________________________ .
Use the figure to complete the conjecture in Exercise 7.
7. The perimeter of a figure that has n of these triangles
is __________________________________ .
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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)
0≤0
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.
________________________________________________________________________________________
9. Points A, G, and N are collinear. If AG = 7 inches and GN = 5 inches, then
AN = 12 inches.
________________________________________________________________________________________
10. For any real numbers x and y, if x > y, then x 2 > y 2.
________________________________________________________________________________________
11. The total number of angles in the figure is 3.
________________________________________
________________________________________
12. If two angles are acute, then the sum of their measures equals the
measure of an obtuse angle.
________________________________________
________________________________________
Determine whether each conjecture is true. If not, write or draw a counterexample.
13. Points Q and R are collinear.
________________________________________
14. If J is between H and K, then HJ = JK.
________________________________________
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Answers for the chapter Geometric Reasoning
USING INDUCTIVE REASONING TO
MAKE CONJECTURES
Practice A
1. 10
2. W
3. summer
3. The pattern is the letters of the alphabet
that are made only from straight
segments; K, L.
4. First rotate the figure 180°. Then reflect
the figure across a vertical line. Repeat.
5. true
6. false
4. inductive reasoning
5. true
6. even
Sample answer:
7. n
8. The number of rings in a tree is the
same as the tree’s age.
9. 82 rings
7. true
8. false
Possible answers: n = 1, n = −1
10. false
11. Possible answers: zero, any negative
number
9. Sum of angle measures = [180(n − 2)]°
10.128.6°; 140°; 150°
Reteach
1
1. 1
4
12. Possible answer:
Practice B
1. 36
2.
3. Arkansas
4. north
5. positive
6. n − 3
7. Possible answers: zero, any negative
number
8.
9. One-third of the bills were counterfeit.
10. Each item, starting with the third, is the
product of the two preceding items;
256, 8192.
11. The dot skips over one vertex in a
clockwise direction.
Practice C
1. The pattern is the cubes of the negative
integers; −125, −216.
2. Each item describes the item before it
(one, one one, two ones, . . .); 312211,
13112221.
2. 36
3.
4.
5. doubled
6. 2n
7. n + 2
8. Possible answer: If the lines are
parallel, then they do not intersect.
9. Possible answer: If point N is between
points A and G, then AN = 2 inches.
10. Sample answer: If x = 0 and y = −1,
then x 2 < y 2.
11. Sample answer: ∠ABD, ∠DBE, ∠EBC,
∠ABE, ∠DBC
12. Sample answer: m∠1 = 25°, m∠2 = 20°
13. true
14.
Challenge
1. 1, 6, 15, 20, 15, 6, 1
2.Each row has 1 as the first and last
number. Each of the other numbers is
found by adding the two numbers that
appear just above it.
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