Patterns and Conjecture A. Write a conjecture that describes the pattern 2, 4, 12, 48, 240. Then use your conjecture to find the next item in the sequence. Step 1 Look for a pattern. 2 4 12 48 240 ×2 ×3 ×4 ×5 Step 2 Make a conjecture The numbers are multiplied by 2, 3, 4, and 5. The next number will be multiplied by 6. So, it will be 6 ● 240 or 1440. Answer: 1440 Example 1

Patterns and Conjecture B. Write a conjecture that describes the pattern shown. Then use your conjecture to find the next item in the sequence. Step 1 Look for a pattern. 3 9 18 +6 +9 Example 1

Check Draw the next figure to check your conjecture. Patterns and Conjecture Step 2 Make a conjecture. Conjecture: Notice that 6 is 3 × 2 and 9 is 3 × 3. The next figure will increase by 3 × 4 or 12 segments. So, the next figure will have 18 + 12 or 30 segments. Answer: 30 segments Check Draw the next figure to check your conjecture. Example 1

A. Write a conjecture that describes the pattern in the sequence A. Write a conjecture that describes the pattern in the sequence. Then use your conjecture to find the next item in the sequence. A. B. C. D. Example 1

A. The next figure will have 10 circles. B. Write a conjecture that describes the pattern in the sequence. Then use your conjecture to find the next item in the sequence. 1 3 6 10 A. The next figure will have 10 circles. B. The next figure will have 10 + 5 or 15 circles. C. The next figure will have 15 + 5 or 20 circles. D. The next figure will have 15 + 6 or 21 circles. Example 1

Step 1 List some examples. 1 + 2 = 3 1 + 4 = 5 4 + 5 = 9 5 + 6 = 11 Algebraic and Geometric Conjectures A. Make a conjecture about the sum of an odd number and an even number. List some examples that support your conjecture. Step 1 List some examples. 1 + 2 = 3 1 + 4 = 5 4 + 5 = 9 5 + 6 = 11 Step 2 Look for a pattern. Notice that the sums 3, 5, 9, and 11 are all odd numbers. Step 3 Make a conjecture. Answer: The sum of an odd number and an even number is odd. Example 2

Step 2 Examine the figure. Algebraic and Geometric Conjectures B. For points L, M, and N, LM = 20, MN = 6, and LN = 14. Make a conjecture and draw a figure to illustrate your conjecture. Step 1 Draw a figure. Step 2 Examine the figure. Since LN + MN = LM, the points can be collinear with point N between points L and M. Step 3 Make a conjecture. Answer: L, M, and N are collinear. Example 2

A. Make a conjecture about the product of two odd numbers. A. The product is odd. B. The product is even. C. The product is sometimes even, sometimes odd. D. The product is a prime number. Example 2

B. Given: ACE is a right triangle with AC = CE B. Given: ACE is a right triangle with AC = CE. Which figure would illustrate the following conjecture? ΔACE is isosceles, C is a right angle, and is the hypotenuse. A. B. C. D. Example 2

Make a statistical graph that best displays the data. Make Conjectures from Data A. SALES The table shows the total sales for the first three months a store is open. The owner wants to predict the sales for the fourth month. Make a statistical graph that best displays the data. Since you want to look for a pattern over time, use a scatter plot to display the data. Label the horizontal axis with the months and the vertical axis with the amount of sales. Plot each set of data. Example 3

Make Conjectures from Data Answer: Example 3

Look for patterns in the data. The sales triple each month. Make Conjectures from Data B. SALES The table shows the total sales for the first three months a store is open. The owner wants to predict the sales for the fourth month. Make a conjecture about the sales in the fourth month and justify your claim or prediction. Look for patterns in the data. The sales triple each month. Answer: The sales triple each month, so in the fourth month there will be $4500 × 3 or $13,500 in sales. Example 3

A. SCHOOL The table shows the enrollment of incoming freshmen at a high school over the last four years. The school wants to predict the number of freshmen for next year. Make a statistical graph that best displays the data. A. B. C. D. Example 3

B. SCHOOL The table shows the enrollment of incoming freshmen at a high school over the last four years. The school wants to predict the number of freshmen for next year. Make a conjecture about the enrollment for next year. A. Enrollment will increase by about 25 students; 358 students. B. Enrollment will increase by about 50 students; 383 students. C. Enrollment will decrease by about 20 students; 313 students. D. Enrollment will stay about the same; 335 students. Example 3

The unemployment rate is highest in the counties with the most people. Find Counterexamples UNEMPLOYMENT Based on the table showing unemployment rates for various counties in Alabama, find a counterexample for the following statement. The unemployment rate is highest in the counties with the most people. Example 4

Find Counterexamples Answer: Perry has a population of 9,652, and it has a higher rate of unemployment than Butler, which has a population of 203,709. Example 4

C. Wisconsin and West Virginia D. Alabama and West Virginia DRIVING This table shows selected states, the 2000 population of each state, and the number of people per 1000 residents who are licensed drivers in each state. Based on the table, which two states could be used as a counterexample for the following statement? The greater the population of a state, the lower the number of drivers per 1000 residents. A. Texas and California B. Vermont and Texas C. Wisconsin and West Virginia D. Alabama and West Virginia Example 4

Truth Values of Conjunctions Use the following statements to write a compound statement for each conjunction. Then find its truth value. Explain your reasoning. p: One foot is 14 inches. q: September has 30 days. r: A plane is defined by three noncollinear points. A. p and q Example 1A

Truth Values of Conjunctions What is statement p? One foot is 14 inches. What is statement q? September has 30 days. What is the statement for p and q? One foot is 14 inches and September has 30 days. Is statement p true? No Is statement q true? Yes Is statement p and q true? No Answer: One foot is 14 inches, and September has 30 days; Although q is true, p is false. So p and q is false. Example 1A

Truth Values of Conjunctions Use the following statements to write a compound statement for each conjunction. Then find its truth value. Explain your reasoning. p: One foot is 14 inches. q: September has 30 days. r: A plane is defined by three noncollinear points. B. ∼p ⋀ r Example 1B

Truth Values of Conjunctions What is statement ~p? One foot is not 14 inches. What is statement r ? A plane is defined by three noncollinear points. What is the statement for ~p ⋀ r ? One foot is not 14 inches and a plane is defined by three noncollinear points. Is statement ~p true? Yes Is statement r true? Yes Is statement ~p ⋀ r true?? Yes Answer: A foot is not 14 inches, and a plane is defined by three noncollinear points; Both ∼p and r are true, so ∼p ⋀ r is true. Example 1B

Truth Values of Disjunctions Use the following statements to write a compound statement for each disjunction. Then find its truth value. Explain your reasoning. p: is proper notation for “segment AB.” q: Centimeters are metric units. r: 9 is a prime number. AB A. p or q Example 2A

Truth Values of Disjunctions What is statement p? is proper notation for “segment AB.” AB What is statement q? Centimeters are metric units. AB What is the statement for p or q? is proper notation for “segment AB” or centimeters are metric units. Is statement p true? Yes Is statement q true? Yes Is statement p or q true? Yes Answer: is proper notation for “segment AB,” or centimeters are metric units. Both p and q are true, so p or q is true. AB Example 2A

Truth Values of Disjunctions Use the following statements to write a compound statement for each disjunction. Then find its truth value. Explain your reasoning. p: is proper notation for “segment AB.” q: Centimeters are metric units. r: 9 is a prime number. AB B. q ⋁ r Example 2B

Truth Values of Disjunctions What is statement q? Centimeters are metric units. What is statement r? 9 is a prime number. What is the statement for q ⋁ r? Centimeters are metric units or 9 is a prime number. Is statement q true? Yes Is statement r true? No Is statement q ⋁ r true? Yes Answer: Centimeters are metric units, or 9 is a prime number. Because q is true, q ⋁ r is true. Example 2B

Truth Values of Disjunctions Use the following statements to write a compound statement for each disjunction. Then find its truth value. Explain your reasoning. p: is proper notation for “segment AB.” q: Centimeters are metric units. r: 9 is a prime number. AB C. ∼p ⋁ r Example 2C

Answer: is not proper notation for segment AB, or 9 is a prime number. Truth Values of Disjunctions What is statement ~p? is not proper notation for “segment AB.” AB What is statement r? 9 is a prime number. AB What is the statement for ~p ⋁ r? is not proper notation for “segment AB” or 9 is a prime number. Is statement ~p true? No Is statement r true? No Is statement p ⋁ r true? No Answer: is not proper notation for segment AB, or 9 is a prime number. Because both ∼p and r are false, ∼p ⋁ r is false. AB Example 2C

Key Concept

Write a Conditional in If-Then Form Write in if-then form. A. Measured distance is positive. Example 3A

Write a Conditional in If-Then Form What is the hypothesis or p? Distance is measured. What is the conclusion or q? It is positive. What is p q? If distance is measured, then it is positive. Answer: If a distance is measured, then it is positive. Example 3A

Write a Conditional in If-Then Form Write in if-then form. B. A five-sided polygon is a pentagon. Example 3B

Write a Conditional in If-Then Form What is the hypothesis or p? A polygon has five sides. What is the conclusion or q? It is a pentagon. What is p q? If a polygon has five sides, then it is a pentagon. Answer: If a polygon has five sides, then it is a pentagon. Example 3B

Truth Values of Conditionals Determine the truth value of each conditional statement. If true, explain your reasoning. If false, give a counterexample. A. If you subtract a whole number from another whole number, then the result is also a whole number. Example 4A

Truth Values of Conditionals Is p q true? No. Why? Subtracting whole numbers can negative answers. What is a counterexample? 2 – 7 = –5 Answer: false; 2 – 7 = –5 Example 4A

Truth Values of Conditionals Determine the truth value of each conditional statement. If true, explain your reasoning. If false, give a counterexample. B. If last month was February, then this month is March. Example 4B

Truth Values of Conditionals Is p q true? Yes. Why? Because March comes after February. Answer: True; the hypothesis is true, and the conclusion is also true, because March is the month that follows February. Example 4B

Truth Values of Conditionals Determine the truth value of each conditional statement. If true, explain your reasoning. If false, give a counterexample. C. If a rectangle has an obtuse angle, then it is a parallelogram. Example 4C

Truth Values of Conditionals Is hypothesis p true? No Is conclusion q true? Yes Is p  q true? Yes Why? A conditional statement with a false hypothesis is always true. Answer: True; the hypothesis is false, because a rectangle can never have an obtuse angle. A conditional statement with a false hypothesis is always true. Example 4C

Key Concept

Key Concept

Related Conditionals Nature Write the converse, inverse, and contrapositive of the following true statement. Determine the truth value of each statement. If a statement is false, find a counterexample. Bats are animals that can fly. What is p  q? If an animal is a bat, then it can fly. What is q  p? If an animal can fly, then it is a bat. Is q  p true? No What is a counter example? Birds can fly and they are not bats. Real-World Example 5

Related Conditionals What is ~p  ~q? If an animal is not a bat, then it cannot fly. Is ~p  ~q true? No What is a counter example? Birds are not bats and they can fly. What is ~q  ~p? If an animal cannot fly, then it is not a bat. Is ~q  ~p true? Yes Why? If the animal cannot fly then it cannot be a bat. Real-World Example 5

Related Conditionals Answer: Conditional: If an animal is a bat, then it can fly. Converse: If the animal can fly, then it is a bat. False; most birds can fly, and they are animals. Inverse: If the animal is not a bat, then it cannot fly. False; most birds can fly, and they are not bats. Contrapositive: If the animal cannot fly, then it is not a bat. True; because the given conditional is true, the animal cannot be a bat. Real-World Example 5

Key Concept

Write Biconditional Statements Rewrite each statement as a biconditional statement. Then determine whether the biconditional is true or false. A. Two lines that intersect at right angles are perpendicular. Example 6

What is statement p? Two lines intersect at right angles. Write Biconditional Statements What is statement p? Two lines intersect at right angles. What is statement q? The lines are perpendicular. What is the biconditional Two lines intersect at right for p and q? angles if an only if they are perpendicular. Is the biconditional statement Yes true? Answer: Two lines intersect at right angles if and only if they are perpendicular; true. Example 6

Write Biconditional Statements Rewrite each statement as a biconditional statement. Then determine whether the biconditional is true or false. B. Whole numbers are rational numbers. Example 6

What is statement p? A number is a whole number. Write Biconditional Statements What is statement p? A number is a whole number. What is statement q? The number is rational. What is the biconditional The number is a whole number for p and q? if and only if the number is rational. Is the biconditional statement No true? Answer: A number is a whole number if and only if it is a rational number; false. Example 6