Welcome to IB Math Studies Year 1

Transcription

1 Welcome to IB Math Studies Year 1 Some things to know: 1. Lots of info at 2. HW yup. You know you love it! Be prepared to present. 3. Content: 4. Grading Ultimately, you need to pass the IB exam! Presentations, quizzes, tests (80%), Project draft (20%) 5. Bring: Notebooks ($3!), pencil(s), calculator, and you! 1

2 2

3 Syllabus Overview Topic Hours Topic 1 Number & Algebra 20 Topic 2 Descriptive Statistics 12 Topic 3 Logic, Sets & Probability 20 Topic 4 Statistical Applications 17 Topic 5 Geometry & Trig 18 Topic 6 Mathematical Models 20 Topic 7 Calculus 18 Project 25 3

4 For now, we will focus on the following: Sum Difference Product Quotient Factor Divisor Dividend But what about 7 2? and ( 7) 2? Let's see what the calculator does with them. Hmmm...why are they different? Try these: A number that divides evenly into another number is called a factor of that number. Finding factors, prime factors The Fundamental Theorem of Arithmetic Summary of Factoring Prime Numbers: Are only divisible by 1 and itself Composite Numbers: Are the product of two or more primes in addition to 1 Factorisation: Writing a number as a product of two or more other numbers Prime Factorisation: Writing a number as a product of two or more prime numbers Common factors There are 40 girls and 32 boys who want to participate in 6 th grade intramurals. If each team must have the same number of girls and the same number of boys, what is the greatest number of teams that can participate in intramurals? Common multiples Boxes that are 12 inches tall are being stacked next to boxes that are 18 inches tall. What is the shortest height at which the two stacks will be the same height? Three clocks start chiming at exactly the same instant. One chimes every 3hours, one every 4 hours, and the other every six hours. When will they next chime together? Review 1A: #1 12 (Number properties) 4

5 Be careful, it's kind of like base 60 (in fact, that's where it came from Babylonians) Sometimes it's easy/convenient to convert to decimal time (computers) Number is in the form a x 10k where 1 a < 10 World population: 7,176,347,165 (ish) Size of the nucleus of an atom: cm Number of atoms in 12 grams of pure carbon 12: 602,214,129,000,000,000,000,000 On a calculator... Base units Other units are derived from base units Other allowable units in SI Common prefixes Small Big Round to: 1. 3 decimal places 2. The nearest thousandth 3. The nearest hundredth 4. The nearest unit A comparison of two quantities pay attention to units. In science, accuracy and precision are different! Accuracy: how close a number is to the true value being measured. Precision: the range of certainty within which a measurement falls But... for our purposes, the two are treated more or less synonomously. The measurement 3.4, for example refers to any value between 3.35 and is more accurate (or more precise) in that it refers to any value between and How many students do you think are enrolled at Desert? 195 Calculate your error. Notice that it can be positive or negative. Now calculate your percentage error. This is the ratio of the error to the correct amount, written as a percent. A percentage error is always expressed as a positive number. So conceptually: And using symbols, where VA is the approximate value (often measured) and VE is the exact or correct value. A special case of rates. You must pay attention to units. Suppose $1 = How much is $50 worth in Euros? How many US dollars can I buy with $50 Euros? It helps to keep in mind which currency is worth more. Since $1 is only part of a Euro, $50 will only be worth part of (a smaller number than) 50 Euros. Common sense will help you check your result. 5

6 Give examples of words used in mathematics you don't necessarily have to know what they mean (but make a guess if you don't know!) For now, we will focus on the following: Sum Difference Product Quotient Factor Divisor Dividend 1A: #1 11 (Vocabulary) 6

7 But what about 7 2? and ( 7) 2? Let's see what the calculator does with them. Hmmm...why are they different? Try these: The exponent operates on the symbol that immediately precedes it! Let's look at some other properties of exponents. 7

8 The definition of exponents along with properties of multiplication and addition lead to some patterns that we can use as shortcuts when working with expressions involving exponents. Can your understanding of exponents help you evaluate? 1 +1 means multiply 1 by a, n times means divide 1 by a, n times Using your understanding of exponents, write down an equivalent power for each product. Using your understanding of exponents, write down an equivalent power for each power. Using your understanding of exponents, write down an equivalent power for each power. Using your understanding of exponents, write down an equivalent power for each quotient. Using your understanding of exponents, write down an equivalent power for each power. 8

9 There are a lot of useful properes. Do not memorize them! Understand them! Properes of Exponents Let a and b be real numbers and m and n be integers. Then: For now, we will restrict ourselves to integer exponents. More on that next me... This takes pracce... 1B.1: #1,2 4def,5 8 (Exponents) 1B.2: #1ijkl,2efgh (Exponents Base<0) Handout 3B p SL1 book selected 9

10 Consider a regular deck of cards. How many people can play a game that requires you to deal the same number of cards to each player without any left over cards? Find as many answers as you can. You never know how many of your 5 friends are coming over to your house for the Friday night card game but you want to design a game so that no matter how many arrive, everyone would get the same number of cards when you deal the whole deck. How many cards should you put in the deck you design? A number that divides evenly into another number is called a factor of that number. Numbers that have a factor of 2 are called: Numbers that do not have a factor of 2 are called: Even Numbers Odd Numbers Can you write a general mathematical description of an even number? 2n What about an odd number? 2n + 1 Note: The letter n is often used to describe an arbitrary integer (± with no fraction or decimal) Numbers that have no factors other than 1 and itself are called: Prime Numbers So, let's look at 60: Even or odd? Prime or Composite? Can you find all the ways to write 60 as a product of factors? (This is called factorising) Be systematic! Lets look further at 60 as 3 x 20. Notice that we can also factorise 20 since it is 4 x 5. So we could write 60 as: 60 = 3 x 20 = 3 x 4 x 5 Once again, we see that 4 can be factorised into 2 x 2. So: 60 = 3 x 20 = 3 x 4 x 5 = 3 x 4 x 5 = 3 x 2 x 2 x 5 At this point all the factors we have, 2, 3, and 5, cannot be factored further except using 1 (which is really pointless). We're done! This is called prime factorisation. Usually we write the factors from small to large. 60 = 2 x 2 x 3 x 5 There are several systematic ways to do this. Here are two: Factor Tree Factor Ladder The Fundamental Theorem of Arithmetic Notice that all the factors of a number can be written as products of the prime factors of that number. Take 60 for example: Summary of Factoring Prime Numbers: Are only divisible by 1 and itself Composite Numbers: Are the product of two or more primes in addition to 1 Factorisation: Writing a number as a product of two or more other numbers Prime Factorisation: Writing a number as a product of two or more prime numbers 10

11 There are 40 girls and 32 boys who want to participate in 6 th grade intramurals. If each team must have the same number of girls and the same number of boys, what is the greatest number of teams that can participate in intramurals? Common Factors A number, a, that is a factor of both n and m is called a common factor of n and m. The largest number with this property is called the Greatest Common Factor (GCF) or Highest Common Factor (HCF) of of n and m 1C.1: #1 7 (Factors) 1C.2: #1,2cd,3cde,4 (Factors) 1C.3: #1 2 (Common Factors) 11

12 Boxes that are 12 inches tall are being stacked next to boxes that are 18 inches tall. What is the shortest height at which the two stacks will be the same height? Hot dogs come in packages of 6, buns come in packages of 8. How many packages of each do you need to buy to get the same number of hot dogs and buns? 12

13 13

14 Attachments QuadraticsMaster.gsp