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Chapter 3

Chapter 3. Rational Numbers. 3-1-A Explore: The Number Line. You have already graphed integers and positive fractions on a number line. Today, you will graph negative fractions. Let’s graph - on a number line

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Chapter 3

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  1. Chapter 3 Rational Numbers

  2. 3-1-A Explore: The Number Line You have already graphed integers and positive fractions on a number line. Today, you will graph negative fractions. Let’s graph - on a number line • Draw a number line. Place a zero on the right side an a -1 on the left. Divide the line into fourths. • Starting from the right, label the line with -1/4, -2/4, and -3/4. • Draw a dot on the number line on the -3/4 mark. -1 0 -½ -¼ -¾

  3. Graph the pair of numbers on a number line. Then write which number is less. Remember! The denominator of the fraction determines the number of sections to be marked on the number line between two integers! • Remember the steps! • Draw a number line. Place a zero on the right side an a -2 on the left. Divide the line into the appropriate parts. • Starting from the right, label the line with the fractions. • Draw a dot on the number line to mark the values. Self-Assessment: Try pg. 127 # 1-8 on your own. Then, check answers with a partner.

  4. 3-1-B Terminating & Repeating Decimals The table shows the winning speeds for a 10-year period at the Daytona 500. • What fraction of the speeds are between 130 and 145 miles per hour? • Express this fraction using words and then as a decimal. • What fraction of the speeds are between 145 and 165 miles per hour? Express this fraction using words and decimals.

  5. Fractions to Decimals: Mental Math! A goal for today is to change fractions to decimals. Try these! • 7/20 • Think: 35/100 so 0.35 • 5 ¾ • Think: 75/100 so 5.75 • 3/25 • Think: 12/100 so 0.12 • -6 ½ • Think: 50/100 so -6.5 TIP You should use MENTAL MATH whenever possible when writing fractions as decimals. Think about if the denominator is a factor of 10, 100, or 1,000.

  6. Fractions to Decimals: Division Any fraction can be written as a decimal by dividing its numerator by its denominator! You should get -0.025. Remember to keep the negative sign! You should get 0.375! You should get -0.875 2.125 7.45

  7. Not all fractions are TERMINATING DECIMALS. Remember, a TERMINATING DECIMAL is a decimal with digits that end. REPEATING DECIMALS have a pattern in their digit (s) that repeats forever! Consider 1/3. When you divide 1 by 3, you get 0.3333... Use BAR NOTATION to indicate a that a number pattern repeats indefinitely. A bar is written over only the digit (s) that repeat.

  8. PRACTICE: • Write each as a decimal. • 7/9 • 2/3 • -3/11 • 8 1/3 • ---------------------------------------------- • Use the table to find what fraction of the fish in an aquarium are goldfish. Write in simplest form. • Determine the fraction of the aquarium made up by each fish. Write the answer in simplest form! • molly • guppy • angelfish Self-Assessment: Try pg. 131 # 1-12 on your own. Then, check answers with a partner.

  9. 3-1-C Compare & Order Rational Numbers The batting average of a softball player is found by comparing the number of hits to the number of times at bat. Melissa had 50 hits in 175 at bats. Harmony had 42 hits in 160 at bats. Write the two batting averages as fractions. Which girl had the better batting average? Explain. Describe two methods you could use to compare the batting averages.

  10. RATIONAL NUMBERS: numbers that can be expressed as a ratio of two integers expressed as a fraction (in which the denominator is not zero). Includes common fractions, terminating and repeating decimals, percents, and all integers. Rational Numbers 0.8 20% 2.2 ½ 1 2/3 -1.44 Integers -1 -3 Whole Numbers 2 1

  11. Today, your goal is to be able to compare and order RATIONAL NUMBERS (fractions, mixed numbers, and decimals). Graph each rational number on a number line. Mark off equal size increments of 1/6 between -2 and -1.

  12. You won’t always be comparing rational numbers that have common denominators. A COMMON DENOMINATOR is a common multiple of the denominators of two or more fractions. The LEAST COMMON DENOMINATOR or LCD is the LCM of the denominators. The LCD is used to compare fractions! What is the least common denominator? What does that make your numerators?

  13. In Mr. Reed’s math class, 20% of the students own Sperry shoes. In Mrs. Crowe’s math class, 5 out of 29 students own Sperry. In which math class does a greater fraction of students own Sperry? Express each number as a decimal and then compare. 20% = 0.2 5/29 = .1724 Since 0.2 > 0.1724, 20% > 5/29 Therefore, a greater fraction of students in Mr. Reed’s class own Sperry shoes. In a second period class, 37.5% of students like to bowl. In a fifth period class, 12 out of 29 students like to bowl. In which class does a greater fraction of the students like to bowl?

  14. 3.44 3.1415926… 3.14 3.4444444444 Remember to line up the decimal points and compare using place value! Self-Assessment: Try pg. 136 # 1-7 on your own. Then, check answers with a partner.

  15. Add & Subtract Positive Fractions Sean surveyed ten classmates to find out which type of tennis shoe they like to wear! • What fraction liked cross trainers? • What fraction liked high tops? • What fraction liked either cross trainers OR high tops? Fractions that have the same denominator are called LIKE FRACTIONS Fractions that do not have the same denominator are called UNLIKE FRACTIONS.

  16. You can use FRACTION TILES as a model to help solve problems that require addition and subtraction of fractions. With your “elbow partner” , complete Fraction Discovery #1. In it, you will be asked to do three things: Draw a model to represent the problem and use that model to find a solution (no numbers allowed) Draw a model to represent the problem and AT THE SAME TIME, write an expression using numbers. Find a solution using both methods. Write a numerical expression only to solve the problem. By 7th grade, you should already know fraction addition & subtraction rules! But your CHALLENGE is to complete some of the problems without those rules

  17. Key Concepts Review Add and Subtract Like Fractions To add or subtract like fractions, add or subtract the numerators and write the result over the denominator.

  18. Key Concepts Review • Add and Subtract Unlike Fractions • To add or subtract like fractions with different denominators • Rename the fractions using the least common denominator (LCD) • Add or subtract as with like fractions • If necessary, simplify the sum or difference

  19. Add & Subtract Negative Fractions • Can fractions be negative? • YES! • Although we may not think about it much, you use negative fractions when you: • Give part of something away • Eat a part of something • Lose part of something • Pour out part of something • Go part of the way backwards • Go part of the way down With your “elbow partner”, complete Fraction Discovery #2. Today, you will need PINK fractions for NEGATIVE numbers and YELLOW fractions POSITIVE. Use what you already know about INTEGER RULES and FRACTION OPERATIONS to help you!

  20. Key Concepts Review When you have like denominators, keep the denominator and use your INTEGER RULES to find the sum or difference in the numerator! When you have unlike denominators, first, find a COMMON DENOMINATOR! Then, you can just use the INTEGER RULES to find the sum or difference in the numerator!

  21. Practice adding and subtracting with fraction tiles.

  22. Practice Without Tiles! Answers Questions

  23. Self-Assessment: Try pg. 148 # 1-10 on your own. Then, check answers with a partner.

  24. 3-2-D Add & Subtract Mixed Numbers 1. Write an expression to find how much more Stephen weighs than Nora. 2. Rename the fractions using the LCD. 3. Find the difference of the fractional parts and then the difference of the whole numbers. To add or subtract mixed numbers, first add or subtract the fractions. If necessary, rename them using the LCD. Then add or subtract the whole numbers and simplify if necessary.

  25. Add and write in simplest form. For these problems, you can add the whole numbers and the fractions separately. Subtract. Write in simplest form. For these problems, you can subtract the whole numbers and the fractions separately.

  26. Many times, it is not possible to subtract the whole numbers and fractions separately. In this case, it is often best to convert to IMPROPER FRACTIONS IMPROPER FRACTION: Has a numerator that is greater than or equal to the denominator

  27. Real World Problems! Self-Assessment: Try pg. 154 # 1-9 on your own. Then, check answers with a partner.

  28. 3-3-A Explore: Fraction Discovery • With a partner, complete Fraction Discover #3 • You will use rectangular models to find the answer to fraction problems. • Your challenge is to find an answer WITHOUT using rules you have learned in the past!

  29. 3-3-B Multiply Fractions For each the first problem, create a sketch or model to solve. Represent these two situations with equations. Are the equations the same or different?

  30. 3-3-D Divide Fractions KEY CONCEPT: Words: To divide a fraction, multiply by its multiplicative inverse, or reciprocal

  31. Practice Dividing by Fractions

  32. Practice Dividing by Mixed Numbers To divide by a mixed number, first rename it as an improper fraction. Estimation a great way to check your solution! Ms. Holloway has 8 ¼ cups of coffee. If she divides the coffee into ¾ cup servings, how many servings will she have? Mrs. Bybee bought 4 ½ gallons of ice cream to serve at her birthday party. If a pint is 1/8 of a gallon, how many pint-sized servings can be made? Self-Assessment: Try pg. 170 # 1-10 on your own. Then, check answers with a partner.

  33. 3-4-A Multiply & Divide Monomials Examine the exponents of the powers in the last column. What do you observe? Write a rule for determining the exponent of the product when you multiply powers with the same base. For each increase on the Richter scale, an earthquake’s vibrations, or seismic waves, are 10 times greater! So, an earthquake of magnitude 4 has seismic waves that are 10 times greater than that of a magnitude 3 earthquake.

  34. REMEMBER:Exponents are used to show repeated multiplication. Use the definition of an exponent to find a rule for multiplying powers with the SAME BASE. 23x24 = (2 x 2 x 2) x (2 x 2 x 2 x 2) = 27 PRODUCT OF POWERS Words: To multiply powers with the same base, add their exponents Symbols: am x an = am+n Example: 32x 34 = 32+4= 36

  35. Practice Multiplying Powers! 73 x 71 53 x 54 (0.5)2 x (0.5)9 8 x 85 Common Mistake: When multiplying powers, do not multiply (evaluate) the bases that are the same! MONOMIAL A number, variable, or product of a number and one or more variables. Monomials can also be multiplied using the rule for the product of powers. x5 (x2) (-4n3)(6n2) -3m(-8m4) 52x2y4 (53xy4)

  36. If we get the PRODUCT OF POWERS using ADDITION, we should get the QUOTIENT OF POWERS using…… QUOTIENT OF POWERS Words: To divide powers with the same base, subtract their exponents Symbols: am ÷ an = am-n Example: 34÷ 32 = 34-2= 32

  37. The table compares the processing speeds of a specific type of computer in 1999 and in 2008. Find how many times faster the computer was in 2008 than in 1999. The number of fish in a school of fish is 43. If the number of fish in the school increased by 42times the original number of fish, how many fish are now in the school? Evaluate the power. Self-Assessment: Try pg. 179 # 1-10 on your own. Then, check answers with a partner.

  38. 3-4-B Negative Exponents • Describe the pattern of the powers in the first column. Continue the pattern by writing the next two values in the table. • Describe the pattern of values in the second column. Then complete the second column. • Determine how 3-1 should be defined.

  39. KEY CONCEPT: NEGATIVE EXPONENET Words: Any nonzero number to the negative n power is the multiplicative inverse of its nth power. • PRACTICE! • Write each expression using a positive exponent. • 6-2 • x-5 • 5-6 • t-4

  40. When given a fraction with a positive exponent or square, you can rewrite it using a negative exponent. PRACTICE! Write each expression using a negative exponentother than -1.

  41. Perform Operations with Exponents Simplify x3 (x-5)

  42. Perform Operations with Exponents Nanometers are often used to measure wavelengths. 1 nanometer= 0.000000001 meter. Write the decimal as a power of 10 A unit of measure called a micron equals 0.001 millimeter. Write this number using a negative exponent. Self-Assessment: Try pg. 183 # 1-13 on your own. Then, check answers with a partner.

  43. 3-4-C Scientific Notation More than 425 million pounds of gold have been discovered in the world. If all this gold were in one place, It would form a cube seven stories on each side! • Write 425 million in standard form 425,000,000 • Complete: 4.25 x _________________ 100,000,000 When you deal with very large numbers like 425,000,000, it can be difficult to keep track of the zeros! You can express numbers such as this in SCIENTIFIC NOTATION by writing the number as the product of a factor and a power of 10.

  44. Words: A number is expressed in scientific notation when it is written as the product of a factor and a power of 10. The factor must be greater than or equal to 1 and less than 10. Symbols: a x 10n, where 1≤ a < 10 and n is an integer Example: 425,000,000 = 4.25 x 108 Express Large Numbers in Standard Form: 2.16 x 105 2.16 x 100,000 = 216,000 (move the decimal point 5 places) 7.6 x 106 7,600,000 (move the decimal point 6 places) 3.201 x 104 32,010 (only move the decimal point 4 places) FOCUS: On moving the decimal rather than adding the zeros!

  45. SMALL NUMBERS TOO! Scientific notation can also be used to express very small numbers. Study the pattern of products at the right. Notice that multiplying by a NEGATIVEPOWERof 10 moves the decimal point to the LEFT the same number of places as the absolute value of the exponent. 1.25 x 102 = 125 1.25 x 101= 12.5 1.25 x 100= 1.25 1.25 x 10-1=0.125 1.25 x 10-2=0.0125 1.25 x 10-3= 0.00125 EXPRESS IN SCIENTIFIC NOTATION 1,457,000 1.457 x 106 0.00063 6.3 x 10-4 35,000 3.5 x 104 0.00722 7.22 x 10-3 EXPRESS SMALL NUMBERS IN STANDARD FORM 5.8 x 10-3 = 0.0058 (move the decimal 3 places left) 4.7 x 10-5= 0.000047 9 x 10-4= 0.0009

  46. The Atlantic Ocean has an area of 3.18 x 107square miles. The Pacific Ocean has an area of 6.4 x 107square miles. Which ocean has a greater area? Since the exponents are the same and 3.18 < 6.4, the Pacific Ocean has a greater area. Earth has an average radius of 6.38 x 103 kilometers. Mercury has an average radius of 2.44 x 103 kilometers. Which planet has the greater average radius? Compare using <, >, or = 4.13 x 10-2_____ 5.0 x 10-3 0.00701_____7.1 x 10-3 5.2 x 102_____ 5,000 Self-Assessment: Try pp. 187 # 1-12 on your own. Then, check answers with a partner.

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