1. EXAMPLE 3
Make a conjecture
Given five collinear points, make a conjecture about the number of ways to connect different pairs of the points.
SOLUTION
Make a table and look for a pattern. Notice the pattern in how the number of connections increases. You can use the pattern to make a conjecture.

2. EXAMPLE 3
Make a conjecture
Conjecture: You can connect five collinear points , or 10 different ways.
ANSWER

3. GUIDED PRACTICE
for Examples 3 and 4
Suppose you are given seven collinear points. Make a conjecture about the number of ways to connect different pairs of the points. 3.
Conjecture: You can connect seven collinear points , or 21 different ways.
ANSWER

4. EXAMPLE 1
Describe a visual pattern
Describe how to sketch the fourth figure in the pattern. Then sketch the fourth figure.
SOLUTION
Each circle is divided into twice as many equal regions as the figure number. Sketch the fourth figure by dividing a circle into eighths. Shade the section just above the horizontal segment at the left.

5. EXAMPLE 2
Describe a number pattern
Describe the pattern in the numbers –7, –21, –63, –189,… and write the next three numbers in the pattern.
Notice that each number in the pattern is three times the previous number.
Continue the pattern. The next three numbers are –567, –1701, and –5103.
ANSWER

6. GUIDED PRACTICE
for Examples 1 and 2
Describe the pattern in the numbers 5.01, 5.03, 5.05, 5.07,… Write the next three numbers in the pattern. 2.
5.13
Notice that each number in the pattern is increasing by 0.02.
5.11
+0.02
5.09
5.07
5.05
5.03
5.01
Continue the pattern. The next three numbers are , 5.11 and 5.13
ANSWER

7. GUIDED PRACTICE
for Examples 3 and 4 4.
Make and test a conjecture about the sign of the product of any three negative integers.
Test:
Test conjecture using the negative integer –2, –5 and –4
–2 –5 –4
= –40
Conjecture: The result of the product of three negative number is a negative number.
ANSWER

8. EXAMPLE 5
Find a counterexample
A student makes the following conjecture about the sum of two numbers. Find a counterexample to disprove the student’s conjecture.
Conjecture: The sum of two numbers is always greater than the larger number.
SOLUTION
To find a counterexample, you need to find a sum that is less than the larger number.

9. EXAMPLE 5
Find a counterexample
–2 + –3
= –5
–5 > –2
Because a counterexample exists, the conjecture is false.
ANSWER

10. EXAMPLE 6
Standardized Test Practice X

11. EXAMPLE 6
Standardized Test Practice
SOLUTION
Choices A and C can be eliminated because they refer to facts not presented by the graph. Choice B is a reasonable conjecture because the graph shows an increase over time. Choice D is a statement that the graph shows is false.
The correct answer is B.
ANSWER

12. ( )2 GUIDED PRACTICE for Examples 5 and 6 5.
Find a counterexample to show that the following conjecture is false.
Conjecture: The value of x2 is always greater than the value of x. 12
( )2 = 14 14
> 12
Because a counterexample exist, the conjecture is false
ANSWER

13. Daily Homework Quiz
Describe a pattern in the numbers. Write the next number in the pattern. 20, 22, 25, 29, 34, . . . 1.
ANSWER
Start by adding 2 to 22, then add numbers that successively increase by 1; 40. 2.
Find a counterexample for the following conjecture: If the sum of two numbers is positive, then the two numbers must be positive.
ANSWER
Sample: 20 + (– 10) = 10

14. Daily Homework Quiz
The scatter plot shows the average number of hours of homework done per week by a student during the first 10 weeks of a school term. Make a conjecture that could be true. Explain your reasoning. 3.
ANSWER
Sample answer: The student will do about 11 hours of homework in week 11. The number of hours of homework per week increased steadily during the first 10 weeks.