Table of Contents Number and Operation Lesson 1 Goldbach s Conjecture........................ 5 Prime Factorization Lesson 2 Micro Mites.................................... 11 Division with Decimals Lesson 3 The Stock Market............................. 17 Multiply and Divide Integers Lesson 4 Let s Sleep on It.............................. 23 Percents Lesson 5 shopping at Home........................... 29 Working with Percents Lesson 6 Famous Faces................................ 35 Divisibility Rules Lesson 7 Deep Blue.................................... 41 Square Roots Geometry Lesson 8 Tremendous Triangles........................ 47 Pythagorean Theorem Lesson 9 twins......................................... 53 Missing Sides of Similar Figures Lesson 10 Alone Around the World..................... 59 Angle Pairs Measurement Lesson 11 a Capital Idea................................ 65 Four-Quadrant Grid Lesson 12 the Titanic................................... 71 Scale Drawings Lesson 13 Papermaking................................. 77 Surface Area Lesson 14 Let s Get Cereal!............................. 83 Volume of a Pyramid
Algebra Lesson 15 Rock Stars.................................... 89 Evaluate Expressions Lesson 16 The Drive-In.................................. 95 Two-Step Equations Lesson 17 Go Figure.................................... 101 Inequalities Lesson 18 Cable Cars................................... 107 Distributive Property Lesson 19 on the Move................................. 113 Graph Linear Functions Data Analysis and Probability Lesson 20 a Helping Hand............................. 119 Histograms Lesson 21 Living Giants................................ 125 Box-and-Whisker Plots Lesson 22 The Earl of Sandwich........................ 131 Math Tools Combinations Glossary....................................................... 137 Order of Operations............................................ 140 Divisibility Rules................................................ 141 Perimeter, Area, and Volume..................................... 142 Angle Pairs.....................................................143 Fractions, Decimals, and Percents................................ 144
prime factorization 1 Goldbach s Conjecture In mathematics, a conjecture is a belief that something is right even though it hasn t been proven. One of the most famous conjectures in all of mathematics is called Goldbach s Conjecture. In 1742, a mathematics professor named Christian Goldbach wrote to another mathematician, Leonard Euler, that he believed that every even integer greater than 2 could be expressed as the sum of two prime numbers. Think about the even number 8. It can be expressed as the sum of the prime numbers 3 and 5. To this day, Golbach s Conjecture has not been proven, but no one has been able to disprove it either! Get Started Ms. Cardona s class was exploring Goldbach s Conjecture. Michael chose the even numbers below to see if the conjecture works. Complete Michael s chart. The first one has been done for you. even number sum of two greater than 2 primes 12 5 7 14 18 24 36 40 44 56 Remember The number 1 is neither prime nor composite. Name two odd numbers that can also be written as the sum of two primes: and Lesson 1: Prime Factorization 5
Working with Prime Factorization You can write any whole number greater than 1 as a product of different factors. For example, you can write 12 as 2 6 or 3 4 or 2 2 3. Writing a number this way is called factoring a number. You can factor the number 24 in many different ways: 24 24 24 2 12 4 6 3 8 2 2 6 2 2 2 3 3 2 4 2 2 2 3 3 2 2 2 When you write a composite number, like 24, as a product of only prime factors, you are writing the prime factorization of that number. Look at the diagram above. The expression 2 2 2 3, or 3 2 2 2, is the prime factorization of the number 24. Each shows the number 24 written as a product of only prime factors. prime factorization the expression of a number as a product of prime 1. Why is the expression 3 2 4 not a prime factorization of 24? 2. Can you write a prime number, like 19, as the product of only prime numbers? Why or why not? 6 Level G
Practice Write the prime factorization for each number below. Show your work. 3. 18 4. 32 5. 15 6. 36 7. 20 8. 25 9. 39 10. 80 11. 28 Solve a Problem 12. Look at Problem 10 above. Show how you could use exponents to write the prime factorization of 80. It s a Fact! In 2000, a British publisher offered a prize of one million dollars to anyone who could submit a correct proof of Goldbach s conjecture by April 2002. The prize was never claimed. Lesson 1: Prime Factorization 7
Writing Prime Factorizations You can use a factor tree to help you write the prime factorization of a number. Follow the steps below to write the prime factorization of 56. STEP 1 Write the number you are factoring. Then write this number as the product of any two factors. Draw lines to each factor as shown. 56 7 8 STEP 2 Write each composite number as the product of two factors and draw lines to each factor. Bring down any prime factors. 56 7 8 7 2 4 STEP 3 Repeat Step 2 until you are left with only prime factors. The prime factorization of the number is shown on the bottom row of your factor tree. 56 7 8 7 2 4 7 2 2 2 STEP 4 Write the prime factors in order from least to greatest: I Then use exponents to write the prime factorization: 56 3 Explain how you can check your answer. 8 Level G
Show What You Know Use the steps you learned to write the prime factorization of each number below. Use exponents to write each prime factorization. Remember, the factors in a prime factorization are written in order from least to greatest. 1. 64 2. 42 64 42 3. 90 4. 108 90 108 On Your Own Look at Problem 3 above. How could you use the prime factorization of 90 to write the prime factorization of 180? Lesson 1: Prime Factorization 9
1 Test Yourself 1. When has the prime factorization of a number been found? when all factors are composite numbers when all factors are even numbers when all factors are prime numbers when all factors are less than 5 4. The expression 2 3 5 2 is the prime factorization of which number? 10 40 100 200 2. Which represents the prime factorization of 48? 2 24 5. Find the prime factorization of 60 by making a factor tree. Show your work below. 2 2 12 2 2 2 6 2 2 2 2 3 3. Which shows the prime factorization of 50? 5 10 2 52 2 3 5 2 2 32 60 6. Think Back Write the formula for the area of a circle: Then find the area of a circle with a radius of 8 feet. Use π = 3.14. 10 Level G