Semantics and Logic Honors

1. Semantics and Logic Honors Mr. Torres

2. Bell ringer

3. Homework Review • Which questions from the homework gave you the most trouble? Why? • Remember that the SAT Math and Writing sections are presented in order of increasing difficulty, placing the easier questions at the beginning of the section and the harder questions at the end. • If you find yourself having trouble with easy problems, skip them to tackle more difficult ones, but make sure to answer as many questions CORRECTLY as possible. You lose ¼ point for every wrong answer.

4. The sat math section: key vocabulary • Factor • Multiple • Remainder • Greatest Common Factor (GCF) • Least Common Multiple (LCM) • Prime Number and Prime Factorization • Ratio • Proportion • Average (arithmetic mean) • Percent • Cross-Multiplication • Sequences • Geometric • Arithmetic • Set Notation • Median, Mode, Arithmetic Mean • Elements • Union of Sets • Intersection of Sets • Fundamental Counting Principle • Factorial Notation • Permutation • Combination

5. sat math: elementary number theory • Factors are positive integers that can be evenly divided into another number WITHOUT a remainder. • 2 is a factor of every even number, because it can divide evenly into each one. • E.g. – What are the factors of 24? • Divisible by means that a number can be evenly divided by another without a remainder. • E.g. – 15 is divisible by 5 because 15 divided by 5 yields no remainder. 15 is not divisible by 10 because the remainder would be 5. • When numbers have common factors, they have one or more factors that divide evenly into both. The largest factor that two numbers share is called the Greatest Common Factor, or GCF. • E.g. – the Greatest Common Factor of 24 and 40 is 8. • Multiples of any given number are those numbers that can be divided by that number WITHOUT a remainder. • 10 is a multiple of 5, but 5 is not a multiple of 10 because it cannot be divided evenly (without remainder) by 10. • What are the first seven multiples of 3? • The multiples of any number will also be multiples of any of its factors. • E.g. – since 12 is a multiple of 2, 3, 4, and 6, all of its multiples (12, 24, 36, 48, 60, etc…) will be multiples of each of its factors! • When numbers have common multiples, they have one or more multiples that those numbers all divide into evenly. The smallest multiple that is common to two (or more) numbers is known as the Least Common Multiple, or LCM. • E.g. – the Least Common Multiple of 6 and 8 is 24.

6. sat math: elementary number theory examples Example 1: Which is the least positive integer divisible by the numbers 2, 3, 4, and 5? Example 2: Which of the following could be the remainders when four consecutive positive integers are each divided by 3? • 1, 2, 3, 1 • 1, 2, 3, 4 • 0, 1, 2, 3 • 0, 1, 2, 0 • 0, 2, 3, 0

7. SAT Math: Elementary Number Theory (cont.) Prime Numbers • Prime Numbers are positive integers greater than 1 whose only two factors are 1 and itself. • What are the first 10 prime numbers? • Is there anything that all prime numbers have in common besides the number of factors they have? • Prime Factorization is a process by which you identify all of the prime factors of a number. It is also called a Factor Tree. • Breaking down any number into its prime factors allows you to find square roots easily. • Be wary of sneaky composite numbers that look prime. Make sure a number is prime before you stop factoring. • A composite number is any number that is not prime. It is known as the product of two distinct integers.

8. Sat math: ratios, Percents, and averages Ratios: • A ratio expresses the relationship between two amounts. Specifically, a ratio is a quotient of those quantities. • Ratios can be written in the form , , or . • E.g. – there are 4 roses and 6 daisies in a bouquet. What is the ratio of roses to daisies? • Ratios only give you the relationship between the two quantities, not necessarily the actual amount. If you have one of the actual values, you can determine the other value and the total number of items in a ratio. • E.g. – the ratio of cats to dogs in a pet store is 3:5. There are 24 cats in the pet store. How many dogs are there? Total animals? • If you know the ratio, but not either amount, you can still determine what the total amount must be a multiple of. • E.g. – the ratio of girls to boys in a class is 2:1. How many people are in the class? • If you have two ratios that share a common term and you want to compare them, find a common term between them. • E.g. – the ratio of a to b is 3:4. The ratio of b to c is 3:5. What is the ratio of a to c? • A proportion is simply two ratios set equal to one another. You can solve for an unknown in a proportion by cross-multiplication.

9. Sat math: ratios, Percents, and averages Percents: • A percent is a ratio in which the second quantity is 100. It can be expressed as or . • Percent questions are great for choosing numbers if you do not know the original value. What number should you pick? • Percents have a three-part formula: . This formula can be adopted in different forms. • X% of Y  Averages • The average, or arithmetic mean, of a group of numbers can be expressed in a three-part formula as well. • If you know the average and EITHER the sum or number of terms, you can solve for the remaining unknown. • E.g. – The average test score in Mr. Torres’ 2nd period class was 89. If the total number of points scored on his test was 2314, how many students are in Mr. Torres’ 2nd period class?

10. Sat math: sequences Sequences are ordered lists of numbers. There are two kinds of sequences. Arithmetic Sequences Geometric Sequences Geometric Sequences are formed by multiplying each previous term by some constant. This constant is know as the ratio between terms. The formula for solving geometric sequences is where is the number of the desired term, is the first term, and is the ratio between terms. E.g. – what is the next number in the sequence ? • Arithmetic Sequences are formed by adding some constant to the previous term. This constant is known as the difference between terms. • The formula for solving arithmetic sequences is where is the number of the desired term, is the first term, and is the different between terms. • E.g. – what is the next number in the sequence ?

11. Sat math: sets • A set is a collection of things, and those things are called elements. • Set Notation – • Median – the middle number of a set when it has been ordered in ascending order. What is the median of the set above? • Mode – the number that appears the most in a set. Not all sets will have a mode. What is the mode of the set above? • Arithmetic Mean – see averages. What is the arithmetic mean of the set above? • A union of sets (represented by the symbol ) is the set consisting of all elements that are in either OR both of the two sets being united. Numbers that appear in both sets ARE NOT REPEATED in their union. • An intersection of sets (represented by the symbol ) is the set consisting of elements that are common to BOTH of the intersecting sets. Set A: Set B: What is the union of sets A and B? What is their intersection?

12. Sat math: fundamental counting principle • The Fundamental Counting Principle dictates that “if one event can happen Yways, and another event can happen Z ways, then there are YZways that both events can happen together.” • E.g. – On a restaurant menu, there are three appetizers and four main courses. How many different dinners can be ordered if each dinner consists of one appetizer and one main course? • Factorial Notation – represented by a symbol after a number. Factorial notation means that the integer is multiplied by every integer between itself and zero.

13. SAT Math: permutations and Combinations Permutations Combinations A combination is a group in which order does not matter. E.g. – There are 12 students in the school theater class. Two students will be responsible for finding the props needed for the class skit. How many different pairs of students can be chosen to find the props? Formula for combinations: where is the total number of items to choose from and is the number of desired items. • A permutation is a sequence in which order matters. • E.g. – A security system uses a four-letter password, but no letter can be used more than once. How many possible passwords are there? • Formula for Permutations: • where is the total number of items to choose from and is the number of desired items.

14. SAT math: classwork • Open your textbooks to page 423. • You will have the rest of the class period to work on this section. Please show your work for each problem on a separate sheet of paper with your name, the date, and your period number on it. • When you have finished, please turn it in and sit quietly until you are dismissed. • Remember to check the school website for Mr. Torres for your homework.