Table of Contents. Number and Operation. Geometry. Measurement. Lesson 1 Goldbach s Conjecture Lesson 2 Micro Mites... 11
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2 Table of Contents Number and Operation Lesson 1 Goldbach s Conjecture Prime Factorization Lesson 2 Micro Mites Division with Decimals Lesson 3 The Stock Market Multiply and Divide Integers Lesson 4 Let s Sleep on It Percents Lesson 5 shopping at Home Working with Percents Lesson 6 Famous Faces Divisibility Rules Lesson 7 Deep Blue Square Roots Geometry Lesson 8 Tremendous Triangles Pythagorean Theorem Lesson 9 twins Missing Sides of Similar Figures Lesson 10 Alone Around the World Angle Pairs Measurement Lesson 11 a Capital Idea Four-Quadrant Grid Lesson 12 the Titanic Scale Drawings Lesson 13 Papermaking Surface Area Lesson 14 Let s Get Cereal! Volume of a Pyramid
3 Algebra Lesson 15 Rock Stars Evaluate Expressions Lesson 16 The Drive-In Two-Step Equations Lesson 17 Go Figure Inequalities Lesson 18 Cable Cars Distributive Property Lesson 19 on the Move Graph Linear Functions Data Analysis and Probability Lesson 20 a Helping Hand Histograms Lesson 21 Living Giants Box-and-Whisker Plots Lesson 22 The Earl of Sandwich Math Tools Combinations Glossary Order of Operations Divisibility Rules Perimeter, Area, and Volume Angle Pairs Fractions, Decimals, and Percents
4 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 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
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 Writing a number this way is called factoring a number. You can factor the number 24 in many different ways: 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 , or , 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 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
6 Practice Write the prime factorization for each number below. Show your work 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 The prize was never claimed. Lesson 1: Prime Factorization 7
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 STEP 2 Write each composite number as the product of two factors and draw lines to each factor. Bring down any prime factors 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 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
8 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 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
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 is the prime factorization of which number? Which represents the prime factorization of 48? Find the prime factorization of 60 by making a factor tree. Show your work below Which shows the prime factorization of 50? 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 = Level G
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