Describe the pattern in the numbers 5.01, 5.03, 5.05, 5.07,… Write the next three numbers in the pattern. Notice that each number in the pattern is increasing by 0.02. +0.02 5.07 5.05 5.03 5.01 5.09, 5.11 5.13

Describe a visual pattern Describe how to sketch the fourth figure in the pattern. Then sketch the fourth figure. 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.

  A conjecture is an educated guess/conclusion based on patterns. Conjectures may be True or False. True: only if all cases can be proven true False: only need 1 counterexample to prove false Counterexample: an example that goes against or disproves the statement.

Find a counterexample Conjecture: The sum of two numbers is always greater than the larger number. To find a counterexample, you need to find a sum that is less than the larger number. –2 + –3 = –5 –5 > –2 Because a counterexample exists, the conjecture is false.

Hints for observing patterns: Figures- rotation, shading, additional division of regions, increased number of rows or columns Numbers- Getting larger- multiply or add Getting smaller- divide or subtract *could be using consecutive numbers, square numbers, or prime numbers *could include more than 1 operation

2.2 Analyze Conditional Statements Conditional Statement: Compound sentence made up of 2 parts that can be expressed in “if-then” form. It implies that a condition be met in order to achieve a result.

Conditional Statements can be True or False True: if you are able to prove the conclusion is true every time the hypothesis is true. False: if you are able to show the conclusion is false when the hypothesis is true. Example: Conjecture: Mrs. Duplechien wears flats on Tuesdays. Conditional statement. If Mrs. Duplechien is wearing flats, then it is Tuesday. Hypothesis: Mrs. Duplechien is wearing flats Conclusion: It is Tuesday. Is it possible that the conclusion would be false? Maybe I decided to wear flats on Thursday.

Rewrite a statement in if-then form Rewrite the conditional statement in if-then form. All birds have feathers. a. SOLUTION First, identify the hypothesis and the conclusion. When you rewrite the statement in if-then form, you may need to reword the hypothesis or conclusion. a. All birds have feathers. If an animal is a bird, then it has feathers.

Rewrite a statement in if-then form b. Two angles are supplementary if they are a linear pair. Two angles are supplementary if they are a linear pair. If two angles are a linear pair, then they are supplementary.

Rewrite a statement in if-then form 1. Tourists at the Alamo are in Texas. ANSWER If tourists are at the Alamo, then they are in Texas. 2. 2x + 7 = 1, because x = –3 ANSWER If x = –3, then 2x + 7 = 1

Negation- the opposite of a statement. I can jump. ----- Negation I cannot fly. ----- Negation ----- I cannot jump. ----- I can fly.

Related Conditionals: Each of these new conditionals is formed by rearranging an original conditional statement. Converse: formed by interchanging the hypothesis and conclusion of the original statement. In other words, the parts of the sentence change places; however, the words "if" and "then" do not move. Inverse: formed by negating the hypothesis and the conclusion of the original statement. Most of time this is simply done by adding the word "not.” But be careful, if the original statement is negative (already has “not”) then negating it would make it a positive statement. Contrapositive: formed by negating both the hypothesis and the conclusion, and then interchanging the resulting negations. It does BOTH the jobs of the INVERSE and the CONVERSE.

Write the converse, the inverse, and the contrapositive of the conditional statement. Tell whether each statement is true or false. Great Danes are large dogs. Conditional: If a dog is a Great Dane, then it is large. True ANSWER Converse: If the dog is large, then it is a Great Dane, False Inverse: If dog is not a Great Dane, then it is not large, False Contrapositive: If a dog is not large, then it is not a Great Dane, True

  Statement True or False Counterexample Conditional If you live in the US, then you live in Texas Converse Inverse Contrapositive F You live in Georgia If you live in Texas, then you live in the US T If you do NOT live in the US, then you do NOT live in Texas T If you do NOT live in Texas, then you do NOT live in the US You live in Alabama F Equivalent Statements have the same Truth value. They are either both true or both false.

Bi-conditional statements: Contain the phrase “if and only if” and can be rewritten as a conditional statement and its converse. True Biconditional statement = True conditional statement + True converse   ALL TRUE DEFINITIONS ARE BICONDITIONAL STATEMENTS. Therefore definition can be stated in 2 ways.

Perpendicular Lines: Two lines that intersect to form a right angle are perpendicular lines. Conditional: If two lines intersect to form a right angle, then they are perpendicular lines. Converse: If two lines are perpendicular, then the intersect to form a right angle. Rewritten as biconditional: Two lines intersect to form a right angle “if and only if” they are perpendicular.

Use definitions Decide whether each statement about the diagram is true. Explain your answer using the definitions you have learned. a. AC BD b. AEB and CEB are a linear pair. c. EA and EB are opposite rays.

Use the diagram shown. Decide whether each statement is true Use the diagram shown. Decide whether each statement is true. Explain your answer using the definitions you have learned. JMF and FMG are supplementary. Use the diagram shown. Decide whether each statement is true. Explain your answer using the definitions you have learned. Point M is the midpoint of FH .