This document contains a geometry drill with transformation problems and sequence problems, as well as a lesson on inductive and deductive reasoning. It provides examples of identifying patterns to make conjectures, and using counterexamples to disprove conjectures. The document aims to teach students how to use examples to form generalizations and find exceptions through inductive and deductive reasoning.

1. GT Geometry Drill10/8/13
• For each transformation, or set of
transformations, determine the mapping.
Please be sure to show your work for the
multi-step mappings
1. Reflect over the line x = -3
2. Translate along the vector <-6, 1>
3. Reflect across the y-axis, and then
translate along the vector <3, -4>

2. GT Geometry Drill10/8/13
1. What are the next two terms in the
sequence?
1, 4, 9, 16...
2. Write a counterexample for the following
statement:
For any number m, 3m is odd.

3. Accept the two statements as given
information. State the conclusion based on
the information.
• 1. AB is longer than BC; BC is
longer than CD
• 2. 12 is greater than integer M.
M is greater than 8
• 3. 4x + 6 = 14, then x =?

4. Objectives
Use inductive and deductive reasoning
to identify patterns and make
conjectures.
Find counterexamples to disprove
conjectures.

5. When several examples form a pattern
and you assume the pattern will continue,
you are applying inductive reasoning.
Inductive reasoning is the process of
reasoning that a rule or statement is true
because specific cases are true. You may
use inductive reasoning to draw a
conclusion from a pattern. A statement
you believe to be true based on inductive
reasoning is called a conjecture.

6. Deductive reasoning is the process of using
logic to draw conclusions from given facts,
definitions, and properties.

8. Vocabulary
inductive reasoning
conjecture
counterexample

9. Example 1A: Identifying a Pattern
Find the next item in the pattern.
January, March, May, ...
Alternating months of the year make up the pattern.
The next month is July.

10. Example 1B: Identifying a Pattern
Find the next item in the pattern.
7, 14, 21, 28, …
Multiples of 7 make up the pattern.
The next multiple is 35.

11. Example 1C: Identifying a Pattern
Find the next item in the pattern.
In this pattern, the figure rotates 90° counter-clockwise
each time.
The next figure is .

12. Check It Out! Example 1
Find the next item in the pattern 0.4, 0.04, 0.004, …
When reading the pattern from left to right, the next
item in the pattern has one more zero after the
decimal point.
The next item would have 3 zeros after the decimal
point, or 0.0004.

13. Example 2A: Making a Conjecture
Complete the conjecture.
The sum of two positive numbers is ? .
List some examples and look for a pattern.
1 + 1 = 2 3.14 + 0.01 = 3.15
3,900 + 1,000,017 = 1,003,917
The sum of two positive numbers is positive.

14. Example 2B: Making a Conjecture
Complete the conjecture.
The number of lines formed by 4 points, no
three of which are collinear, is ? .
Draw four points. Make sure no three points are
collinear. Count the number of lines formed:
AB AC AD BC BD CD
The number of lines formed by four points, no
three of which are collinear, is 6.

15. Check It Out! Example 2
Complete the conjecture.
The product of two odd numbers is ? .
List some examples and look for a pattern.
1  1 = 1 3  3 = 9 5  7 = 35
The product of two odd numbers is odd.

16. Example 3: Biology Application
The cloud of water leaving a whale’s blowhole
when it exhales is called its blow. A biologist
observed blue-whale blows of 25 ft, 29 ft, 27 ft,
and 24 ft. Another biologist recorded humpback-whale
blows of 8 ft, 7 ft, 8 ft, and 9 ft. Make a
conjecture based on the data.
Heights of Whale Blows
Height of Blue-whale Blows 25 29 27 24
Height of Humpback-whale
8 7 8 9
Blows

17. Example 3: Biology Application Continued
The smallest blue-whale blow (24 ft) is almost
three times higher than the greatest humpback-whale
blow (9 ft). Possible conjectures:
The height of a blue whale’s blow is about three
times greater than a humpback whale’s blow.
The height of a blue-whale’s blow is greater than
a humpback whale’s blow.

18. Check It Out! Example 3
Make a conjecture about the lengths of male and
female whales based on the data.
Average Whale Lengths
Length of Female (ft) 49 51 50 48 51 47
Length of Male (ft) 47 45 44 46 48 48
In 5 of the 6 pairs of numbers above the female is
longer.
Female whales are longer than male whales.

19. To show that a conjecture is always true, you must
prove it.
To show that a conjecture is false, you have to find
only one example in which the conjecture is not true.
This case is called a counterexample.
A counterexample can be a drawing, a statement, or a
number.

20. Inductive Reasoning
1. Look for a pattern.
2. Make a conjecture.
3. Prove the conjecture or find a
counterexample.

21. Example 4A: Finding a Counterexample
Show that the conjecture is false by finding a
counterexample.
For every integer n, n3 is positive.
Pick integers and substitute them into the expression
to see if the conjecture holds.
Let n = 1. Since n3 = 1 and 1 > 0, the conjecture holds.
Let n = –3. Since n3 = –27 and –27  0, the
conjecture is false.
n = –3 is a counterexample.

22. Example 4B: Finding a Counterexample
Show that the conjecture is false by finding a
counterexample.
Two complementary angles are not congruent.
45° + 45° = 90°
If the two congruent angles both measure 45°, the
conjecture is false.

23. Example 4C: Finding a Counterexample
Show that the conjecture is false by finding a
counterexample.
The monthly high temperature in Abilene is
never below 90°F for two months in a row.
Monthly High Temperatures (ºF) in Abilene, Texas
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
88 89 97 99 107 109 110 107 106 103 92 89
The monthly high temperatures in January and February
were 88°F and 89°F, so the conjecture is false.

24. Check It Out! Example 4a
Show that the conjecture is false by finding a
counterexample.
For any real number x, x2 ≥ x.
1
2
Let x = .
1
2
2 1
1
4
1
4
Since = , ≥ .
2
The conjecture is false.

25. Check It Out! Example 4b
Show that the conjecture is false by finding a
counterexample.
Supplementary angles are adjacent.
23° 157°
The supplementary angles are not adjacent,
so the conjecture is false.

26. Check It Out! Example 4c
Show that the conjecture is false by finding a
counterexample.
The radius of every planet in the solar system is
less than 50,000 km.
Planets’ Diameters (km)
Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune
4880 12,100 12,800 6790 143,000 121,000 51,100 49,500
Since the radius is half the diameter, the radius of
Jupiter is 71,500 km and the radius of Saturn is
60,500 km. The conjecture is false.

27. Lesson Quiz
Find the next item in each pattern.
1. 0.7, 0.07, 0.007, … 2.
0.0007
Determine if each conjecture is true. If false,
give a counterexample.
3. The quotient of two negative numbers is a positive
number.
true
4. Every prime number is odd.
false; 2
5. Two supplementary angles are not congruent.
6. The square of an odd integer is odd.
false; 90° and 90°
true