Rational Numbers. 3-1-A Explore: The Number Line Previously, you have graphed integers and positive fractions on a number line. Today, you will graph

Rational Numbers

3-1-A Explore: The Number Line Previously, you have graphed integers and positive fractions on a number line. Today, you will graph negative fractions. 0 - - -

Remember! The denominator of the fraction determines the number of sections to be marked on the number line between two integers! Graph the pair of numbers on a number line. Then identify which number is less. Remember the steps! 1. Draw a number line. Place a zero on the right side an a -2 on the left. Divide the line into the appropriate parts. 2. Starting from the right, label the line with the fractions. 3. Draw a dot on the number line to mark the values. Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare your work with your partners. Are there any differences?

3-1-B Terminating & Repeating Decimals The table shows the winning speeds for a 10-year period at the Daytona 500. 1. What fraction of the speeds are between 130 and 145 miles per hour? 2. Express this fraction using words and then as a decimal. 3. What fraction of the speeds are between 145 and 165 miles per hour? Express this fraction using words and decimals. YearWinner Speed (mph) 1999J. Gordon148.295 2000D. Jarrett155.669 2001M. Waltrip161.783 2002W. Burton142.971 2003M. Waltrip133.870 2004 D. Earnhardt Jr. 156.345 2005J. Gordon135.173 2006J. Johnson142.667 2007K. Harvick149.335 2008R. Newman152.672

Look at the following example: 7/20 We can easily change this to have a denominator of 100 by multiplying the numerator and denominator by 5. This would make the fraction 35/100 or 0.35. Now you try! 5 Think: 75/100so5.75 3/25 Think: 12/100so0.12 -6 Think: 50/100so-6.5 Mental Math! Converting Fractions to Decimals Because our decimal system is based on powers of 10 such as 10, 100, and 1,000, we can use mental math to convert fractions to decimals. If the denominator of a fraction is a power or multiple of ten, then you can use place value to write the fraction as a decimal.

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

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.

FishAmount Guppy0.25 Angelfish0.4 Goldfish0.15 Molly0.2 Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare your work with your partners. Are there any differences?

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. 1.Write the two batting averages as fractions. 2.Which girl had the better batting average? Be ready to explain how you found your answer. 3.Describe two different methods you could use to compare the batting averages.

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 Integers Whole Numbers 0.8 20% 2.2 2/3 -1.44 -3-1 2 1

What is the least common denominator? What does that make your numerators? Be careful! Negative numbers may look bigger because they have a larger absolute value. However, the larger the negative, the smaller the numbers actual value! You wont 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!

In Mr. Reeds math class, 20% of the students own Sperry shoes. In Mrs. Crowes 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. Reeds class own Sperry shoes. Now you try! 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?

Remember to line up the decimal points and compare using place value! 3.44 3.1415926 3.14 3.4444444444 Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare your work with your partners. Are there any differences?

Add & Subtract Positive Fractions Sean surveyed ten classmates to find out which type of tennis shoe they like to wear! 1. What fraction liked cross trainers? 2. What fraction liked high tops? 3. 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. Shoe Type Number Cross Trainer 5 Running3 High Top2

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: 1.Draw a model to represent the problem and use that model to find a solution (no numbers allowed) 2.Draw a model to represent the problem and AT THE SAME TIME, write an expression using numbers. Find a solution using both methods. 3.Write a numerical expression only to solve the problem. By 7 th grade, you should already know fraction addition & subtraction rules! But your CHALLENGE is to complete some of the problems without those rules

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

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!

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!

Practice adding and subtracting with fraction tiles.

Questions Answers Practice Without Tiles!

Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare your work with your partners. Are there any differences?

3-2-D Add & Subtract Mixed Numbers BabyBirth Weight Adelaide Stephen Micah Nora 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.

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.

Many times, it is not possible to subtract the whole numbers and fractions separately. In this case, two different strategies could be used: 1.Convert mixed numbers to improper fractions OR 2.Borrow from the whole number and add 1 to the fraction. IMPROPER FRACTION: Has a numerator that is greater than or equal to the denominator.

Real World Problems! Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare your work with your partners. Are there any differences?

Fraction Discovery #3 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!

3-3-BMultiply Fractions For the problem, create a sketch or model to solve. Represent these two situations with equations. Are the equations the same or different? Two-thirds of the students chose pizza at lunch. One-half of those students chose pepperoni pizza.

Key Concepts 1 2 15 When multiplying with mixed numbers, you MUST change the mixed numbers to improper fractions BEFORE you multiply.

Evaluate each verbal expression: a)One-half of five-eighths b)Four-sevenths of two-thirds c)Nine-tenths of one-fourth d)One-third of eleven-sixteenths

Fraction Discovery #4 With a partner, complete Fraction Discover #4 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!

3-3-D Divide Fractions KEY CONCEPT: To divide a fraction, multiply by its multiplicative inverse, or reciprocal. PAY ATTENTION ! The divisor (or second fraction) is the ONLY fraction that is flipped during this process. DO NOT FLIP THE FIRST FRACTION.

Practice Dividing by Fractions Show your work in your notes. Simplify when necessary.

Practice Dividing by Mixed Numbers To divide by a mixed number, first rename it as an improper fraction. Self-Assessment: Try pg. 170 # 1-10 on your own. When signaled, compare your work with your partners. Are there any differences?

3-4-A Multiply & Divide Monomials For each increase on the Richter scale, an earthquakes 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. 1.Examine the exponents of the powers in the last column. What do you observe? 2.Write a rule for determining the exponent of the product when you multiply powers with the same base. Richter Scale Times Greater than Magnitude 3 Earthquake Written using Powers 410 x 1 = 1010 1 510 x 10 = 10010 1 x 10 1 = 10 2 610 x 100 = 1,00010 1 x 10 2 = 10 3 710 x 1,000 = 10, 00010 1 x 10 3 = 10 4 810 x 10,000 = 100,00010 1 x 10 4 = 10 5

REMEMBER: Exponents are used to show repeated multiplication. We can use the definition of an exponent to find a rule for multiplying powers with the SAME BASE. 2 3 x 2 4 =(2 x 2 x 2)x(2 x 2 x 2 x 2)=2 7 PRODUCT OF POWERS Words : To multiply powers with the same base, add their exponents Symbols : a m x a n = a m+n Example : 3 2 x 3 4 = 3 2+4 = 3 6

Practice Multiplying Powers! 1.7 3 x 7 1 2.5 3 x 5 4 3.(0.5) 2 x (0.5) 9 4.8 x 8 5 Common Mistake: When multiplying powers, do not multiply (evaluate) the bases that are the same! Example: 3 3 x 3 5 9 8 3 3 x 3 5 = 3 8 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. 1.x 5 (x 2 ) 2.(-4n 3 )(6n 2 ) 3.-3m(-8m 4 ) 4.5 2 x 2 y 4 (5 3 xy 4 ) STUCK? Remember that the coefficients are multiplied!

QUOTIENT OF POWERS Words: To divide powers with the same base, subtract their exponents. Symbols: a m a n = a m-n Example: 3 4 3 2 = 3 4-2 = 3 2 If we get the PRODUCT OF POWERS using ADDITION, we should get the QUOTIENT OF POWERS using

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. Year Processing Speed (instructions per second) 199910 3 200810 9 The number of fish in a school of fish is 4 3. If the number of fish in the school increased by 4 2 times 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. When signaled, compare your work with your partners. Are there any differences?

3-4-B Negative Exponents 1. Describe the pattern of the powers in the first column. Continue the pattern by writing the next two values in the table. 2. Describe the pattern of values in the second column. Then complete the second column. 3. Using what you observed in the table, determine how 3 -1 should be defined. PowerValue 2626 64 2525 32 2424 16 2323 8 2 4 2121 2 2020 2 -1 Take a look at the table:

PRACTICE! Write each expression using a positive exponent. 6 -2 x -5 5 -6 t -4

When given a fraction with a positive exponent or square, you can rewrite it using a negative exponent.

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. When signaled, compare your work with your partners. Are there any differences?

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! 1. Write 425 million in standard form 425,000,000 2. Complete: 4.25 x = 425,000,000 100,000,000 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.

Check it out! Express the following numbers in standard form: 2.16 x 10 5 2.16 x 100,000 2.16 _ _ _ 2.16 0 0 0 216,000 (move the decimal point 5 places) You try! 7.6 x 10 6 7,600,000 (move the decimal point 6 places) 3.201 x 10 4 32,010 (move the decimal point 4 places ) WATCH OUT! Common mistake: Make sure you are counting decimal places rather than just adding zeroes.

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 NEGATIVE POWER of 10 moves the decimal point to the LEFT the same number of places as the absolute value of the exponent. Practice: Express each number 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 Practice: Express each number in scientific notation. 1,457,000 1.457 x 10 6 0.00063 6.3 x 10 -4 35,000 3.5 x 10 4 0.00722 7.22 x 10 -3

The Atlantic Ocean has an area of 3.18 x 10 7 square miles. The Pacific Ocean has an area of 6.4 x 10 7 square 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 10 3 kilometers. Mercury has an average radius of 2.44 x 10 3 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 10 2 _____ 5,000 Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare your work with your partners. Are there any differences?

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Rational Numbers. 3-1-A Explore: The Number Line Previously, you have graphed integers and positive fractions on a number line. Today, you will graph