Warm-Up Exercises

1. Find the length of a segment with endpoints A(1, –3) and B(–2, –7).

ANSWER (0, –4)

2. If M(4, –3) is the midpoint of RS, and the coordinates of R are (8, –2), find the coordinates of S.

ANSWER 5

Warm-Up Exercises

ANSWER 45º

3. A and B are supplementary. If the measure of A is three times the measure of B, find the

measure of B.

Warm-Up ExercisesEXAMPLE 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.

Warm-Up ExercisesEXAMPLE 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

Warm-Up ExercisesGUIDED PRACTICE for Examples 1 and 2

1. Sketch the fifth figure in the pattern in example 1.

ANSWER

Warm-Up ExercisesGUIDED 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

+0.02

5.07

+0.02

5.05

+0.02

5.03

+0.02

5.01

+0.02

Continue the pattern. The next three numbers are 5.09, 5.11 and 5.13

ANSWER

Warm-Up ExercisesEXAMPLE 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.

Warm-Up ExercisesEXAMPLE 3 Make a conjecture

Conjecture: You can connect five collinear points 6 + 4, or 10 different ways.

ANSWER

Warm-Up ExercisesEXAMPLE 4 Make and test a conjecture

Numbers such as 3, 4, and 5 are called consecutive integers. Make and test a conjecture about the sum of any three consecutive numbers.

SOLUTION

STEP 1Find a pattern using a few groups of small numbers.

3 + 4 + 5 7 + 8 + 910 + 11 + 12 16 + 17 + 18

= 12= 4 3 = 12= 8 3= 33= 11 3 = 51= 17 3

Conjecture: The sum of any three consecutive integers is three times the second number.

ANSWER

Warm-Up ExercisesEXAMPLE 4 Make and test a conjecture

STEP 1Test your conjecture using other numbers. For example, test that it works with the groups –1, 0, 1 and 100, 101, 102.

–1 + 0 + 1 100 + 101 + 102= 0= 0 3 = 303= 101 3

Warm-Up ExercisesGUIDED 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 15 + 6, or 21 different ways.

ANSWER

Warm-Up ExercisesGUIDED 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

Warm-Up ExercisesEXAMPLE 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.

Warm-Up ExercisesEXAMPLE 5 Find a counterexample

–2 + –3

–5 > –2

= –5

Because a counterexample exists, the conjecture is false.

ANSWER

Warm-Up ExercisesEXAMPLE 6 Standardized Test Practice

X

X

Warm-Up ExercisesEXAMPLE 6 Standardized Test Practice

SOLUTION

The correct answer is B.

ANSWER

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.

Warm-Up ExercisesGUIDED 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

Warm-Up ExercisesGUIDED PRACTICE for Examples 5 and 6

6. Use the graph in Example 6 to make a conjecture that could be true. Give an explanation that supports your reasoning.

Conjecture: The number of girls playing soccer will increase; the number of girls playing soccer has increased every year for the past 10 years.

ANSWER

Warm-Up ExercisesDaily 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

Warm-Up ExercisesDaily Homework Quiz

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.

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.