FOUNDATIONS Outline Sec. 3-1 Gallo Name: Date: Review Natural Numbers: Whole Numbers: Integers: Rational Numbers: Comparing Rational Numbers Fractions: A way of representing a division of a whole into The is the number of parts chosen. The is the total number of parts. Numerator Denominator A fraction is a comparison of two numbers by
Comparing Fractions Using Fraction Bars: Which is larger, 4 9 or 5 6? Use a number line to compare 3 5 and 5 8. 3 5 5 8
Comparing Fractions Using Equivalent Fractions: Another way of comparing fractions is to convert them to fractions that have the same denominator. Then simply compare the. The fraction with the numerator is now the larger fraction. Equivalent Fractions: Created by or the numerator and denominator by the SAME number Example: Compare 5 6 to 4 5 by creating equivalent fractions Step 1- Both 6 and 5 a factor of what number? Step 2 Multiply the numerator and denominator of 5 6 by denominator. 5 6 Step 3 Multiply the numerator and denominator of 4 5 by denominator. to get 30 in the to get 30 in the 4 5 Step 4 Which numerator is larger? Therefore, is the larger fraction.
Comparing Fractions Using Cross-Products: Another way of comparing fractions is to and the denominators. the numerators Then simply compare the. The side with the cross-product is now the larger fraction. ALWAYS START FROM THE BOTTOM!!!! Example: Compare 5 7 to 9 11 by cross-products. Step 1- Multiply 7 and 9 and place it on the right. Step 2 Multiply 11 and 5 and place it on the left. 5 9 7 11 Step 3 Which cross-product is larger? Therefore, is the larger fraction. Homework : Section 3.1 pages 134-135 # s 5-33
FOUNDATIONS Outline Sec. 3-2 Gallo Name: Date: Using Equivalent Fractions Factor: A number that another number evenly. The factors of 36 are: A fraction is in denominator is 1. terms when the only factor common to both the numerator and 3 6 9 12 15 60,,,,, and 5 10 15 20 25 100 are equivalent. Only is in lowest terms To express a fraction in lowest terms find the of the numerator and denominator and divide them both by the GCF The result is a fraction in lowest terms. Examples: Express each fraction in lowest terms: a) 5 15 b) 14 21 c) 75 100 GCF: DIV: LT: fractions can be used to compare one fraction with another. When fractions have the same you only need to look at the numerators to see which is the greater fraction.
For example, the least common multiple of 4 and 6 is: This will be the LEAST COMMON DENOMINATOR (LCD) of the two fractions Find the LCD of the following fractions: 1 3 a) and 3 16 2 4 b) and 9 15 Compare 3 4 to 5 6 by creating equivalent fractions 3 4 5 6 3 5 4 6 Homework : Section 3.2 pages 139-140 # s 1-32
FOUNDATIONS Outline Sec. 3-3 Gallo Name: Date: Exploring Decimals PLACE VALUE The value of a digit in a decimal is determined by its or place. For example, given the decimal 0.35, the 3 is in the place and the 5 is in the place. 1, 2 3 4. 5 6 7 8 What is this number? thousand, hundred AND thousand, hundred ten thousandths
MORE FRACTIONS A 5.45 is a number written with both a whole number and a fraction. Write the following numbers as a decimal and a mixed number with the fraction in lowest terms. Decimal Mixed Number a) one thousand and one hundred-thousandth b) twenty-two and two thousand, five hundred ten-thousandths TERMINATING AND REPEATING DECIMALS A decimal is called a terminating decimal if the decimal number terminates, or. A decimal is called a repeating decimal if the decimal number does not terminate. It repeats a. Bar Notation: A bar indicates what number(s) in the decimals will repeat: 0.6666666 = Rewrite the repeating decimals using bar notation A.) 56.11111 B.) -4.356356356
Examples: Use division to determine whether each of the following fractions terminate or repeat. A.) 4 5 B.) 1 3 C.) 5 8 D.) 41 333 Comparing Decimal Numerals: 1.) each decimal with the number of decimal places. 2) up the decimal points. 3) Write the numerals in. 4) Change the decimals to the form. Order the decimals from least to greatest A.) 5.32, 5.2, 4.97, 5.037, 5.3 Most number of decimal places? B.) -7.11, -7.011, -7.105, -7.01, -7.1 Most number of decimal places?
Write each decimal as a fraction or mixed number in lowest terms Fraction Lowest Terms a).23 b) 4.015 c) 45.45 d) 81.225 Order the numbers from least to greatest; 3 5 1 a) 7.3, 7, 7.045, 7, 7 8 6 3 b.) 2 3 1 0.6,, 0.75,, 3 8 3 Homework : Section 3.3 pages 146-148 # s 6-47, 54-69
FOUNDATIONS Outline Sec. 3-4 Gallo Name: Date: Addition and Subtraction of Rational Numbers Changing an improper fraction to a mixed numeral A fraction such as 12 7, where the numerator is than the denominator is called an Process: Divide: denominator into numerator: How many times does 7 divide 12 evenly? What is the remainder? The numerator of the fraction part of the mixed number is the remainder. The denominator is the number you by. What is the fraction component? Rewrite as a mixed number: Examples: Rewrite as a mixed number: a) 25 6 b) 37 c) 14 156 36
Fractions With Like Denominators: When adding or subtracting fractions with like denominators, simply add or subtract the and place the total over the denominator. 1 3 2 5 Example - = 8 8 8 8 Your Turn: Find the sum of the following fractions: 3 6 1 5 a) b) 3 4 c) 8 8 6 6 3 2 2 3 7 7 Fractions with Unlike Denominators: 1. Find a denominator 2. Rewrite the fractions. 3. Add the. 4. Reduce the fraction to terms. Add: 3 3 = = 4 7 Your Turn: Find the sum of the following fractions: a) 2 6 3 b) 3 7 1 5 5 2 6
c) 2 7 1 1 8 3 8 2 d) 4 7 6 9 5 8 Find two mixed numbers that have the given sum: a) 3 6 8 b) 4 2 9 Find two mixed numbers that have the given difference: a) 1 4 3 b) 1 1 2 Homework : Section 3.4 pages 152-153 # s 6-56 even
FOUNDATIONS Outline Sec. 3-5 Day 1 Gallo Name: Date: Multiplying Proper Fractions Multiplication of Rational Numbers To multiply proper fractions, simply find the of the numerators and place it over the of the denominators. Then the fraction to its terms. Examples: A.) 1 2 3 5 = B.) 2 3 4 = 3 8 5 Cross Simplification Since multiplication is a operation, i.e., we can change the order in which we multiply the factors, we can reduce before we. If any numerator has a common factor with denominator, we can divide both of these numbers by the common factor. Take Example B.) from above: 2 3 4 = 3 8 5 3 in the numerator shares a factor of in the denominator. 8 in the denominator shares factors of and in the numerator. 2 3 4 = 3 8 5
Changing a mixed numeral improper fraction to an improper fraction A multiplication problem such as mixed numeral to an 2 1 1 3 3 4 cannot be determined until we convert each fraction. Then, we follow the procedure above. Process: Multiply: denominator times the whole number: 3 times 1? Add the numerator to the product 3 + 2? The numerator of the improper fraction is this answer. The denominator stays the. What is the first improper fraction?. What is the second improper fraction?. What is the product?. Examples: Rewrite as an improper fraction: a) 1 4 6 b) 9 2 14
Multiplying Mixed Numerals To multiply mixed numerals, change the mixed numerals to. Then, the fractions and. Examples: A.) 1 2 2 1 = 3 5 3 5 3 5 15 15 B.) 2 1 4 1 = 3 8 3 8 Area: To find the area of a rectangle, the length by the. Example Find the area of the following rectangles: A.) 2 B.) 1 C.) 2 3 4 1 4 3 1 8 1 7 2
Reciprocals Reciprocals are pairs of numbers whose is. The reciprocal of a number is over the number, or the number. Examples of reciprocals: and and and Why? Homework : Section 3.5 pages 159-161 # s9-32
FOUNDATIONS Outline Sec. 3-5 Day 2 Gallo Name: Date: Division of Rational Numbers Money, Money, Money!!!! A $0.50 cent coin is equivalent to a of a dollar. A quarter or $0.25 is equivalent to a of a dollar. A dime or $0.10 is equivalent to a of a dollar. A nickel or $0.05 is equivalent to a of a dollar. Question: How many 1 -dollars are there in $3.00? 2 1 $3 2 This is the same as what multiplication statement? How are 1 2 and 2 related? Question: How many nickels are there in $5.00? 1 $5 20 This is the same as what multiplication statement? How are 1 20 and 20 related?
Division Dividing by a number is the same as multiplying by its When dividing fractions, first change all mixed numerals to fractions. Then multiply the first fraction by the of the second. Examples: A.) 1 1 1 = 2 4 2 B.) 1 4 1 = 3 5 3 C.) 2 1 2 1 = 5 3 Homework : Section 3.5 pages 160-161 # s 33-69
FOUNDATIONS Outline Sec. 3-6 Gallo Date: Name: Ratios and Proportions Ratios A is a of two like quantities using. Ratios can be written written different ways. For example, 2 compared to 7 as a ratio can be A number is any number that can be written as in which the denominator is not their. We always want to express our ratios in terms. However, if 1 is in the denominator, we leave it there. Example: Joey made 5 shots for every 8 he attempted in the last game. A.) What was Joey s ratio of shots made to shots attempted? B.) What was Joey s ratio of shots made to shots missed? C.) What was Joey s ratio of shots missed to shots attempted? D.) How many shots would Joey likely make if he attempts 24 shots?
Proportions A proportion is an stating two or more ratios are. In a proportion, the - are equal. Example: 2 10 is a proportion and =. 3 15 2 and 15 are called the. 3 and 10 are called the. Property of Proportions: Two ratios are equal if and only if the cross products are equal. (The product of the MEANS = the product of the EXTREMES) a b c d if and only if ad = bc ( b 0 and d 0 ) Complete each proportion: A.) 4 n 15 30 B.) 8 16 n C.) 36 4 64 15 n Homework : Section 3.6 pages 166-167 # s5-21, 25-27, 31-37
FOUNDATIONS Outline Sec. 3-7 Gallo Name: Date: Percent Review Since we already studied percents, decimals, and fractions on ACELLUS, we will quickly review the procedures surrounding the conversion Changing A Percent to A Fraction Step 1 - Place the percent over. Step 2 - the fraction to a proper fraction or mixed numeral in lowest terms. Example 1 - Change 20% to a fraction 20% = 100 Example 2 - Change 125% to a fraction 125% = 100 Changing A Fraction to A Percent Step 1 - Change the to a. (Divide the numerator by the denominator) Step 2 - Move the decimal point places to the. Example 3 - Change 7 12 to a percent. 7 12 =. %
Changing A Decimal to A Percent Step 1 - Move the decimal point places to the. Example 4 - Change 0.815 to a percent 0.815 = % Changing A Percent to A Decimal Step 1 - Move the decimal point places to the. (If there is no decimal point, it goes at the right of the last number in the percent) Example 5 - Change 315% to a decimal 315% = Solving Simple Percent Problems EVERY percent problem has an, an, and a percent in it. Since percent means PER HUNDRED we can solve every percent problem using the ratio IS % OF 100 Example 6 What is 30% of 50. 100 100 x 30(50) 100 x 1500 100x 1500 100 100 x 15
YOUR TURN: 1.) 10 is what percent of 18? 2.) 5 is 20% of what number? IS % OF 100 IS % OF 100 100 100 3.) Find 75% of 25. 4.) 9 is 45% of what number? IS % OF 100 IS % OF 100 100 100 Homework : Section 3.7 pages 174-175 # s 7-45 odd
FOUNDATIONS Outline Sec. 3-8 Gallo Name: Date: Experimental Probability ACTIVITY: Your job is to flip a coin, record the result (H or T), roll a die, and record that result 20 times. Fill in the chart and answer the questions that follow. TRY NUMBER COIN DIE COMBINED 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1.) How many times did you get a HEAD on the coin flip? 2.) Therefore, the PROBABILITY of a HEAD was 3.) How many times did you get a TAIL on the coin flip? 4.) Therefore, the PROBABILITY of a TAIL was 5.) How many times did you get a 1 on the die? 6.) Therefore, the PROBABILITY of rolling a 1 was 7.) How many times did you get a 4 on the die? 8.) Therefore, the PROBABILITY of rolling a 4 was 9.) How many times did you get an EVEN # on the die? 10.) Therefore, the PROBABILITY of rolling an EVEN was 11.) How many times did you get a PRIME # on the die? 12.) Therefore, the PROBABILITY of rolling a PRIME was 13.) How many times did you get a HEAD on the flip AND Roll an ODD number on the die? 14.) Therefore, the PROBABILITY of getting a HEAD on the flip AND rolling an ODD number on the die was
FOUNDATIONS Outline Sec. 3-9 Gallo Name: Date: Theoretical Probability Each time an experiment such as one toss of a coin, one roll of a dice, one spin on a spinner etc. is performed, the result is called an Theoretical Probability is a measure of what you to occur. A for that experiment. for an experiment is the set of possible outcomes Example 1: Create a sample space for the following situation: Mr. and Mrs. Sanderson are expecting triplets. Assume there is an equally likely chance that the Sandersons will have a boy or girl. Total Number of Outcomes: Theoretical Probability = Number of FAVORABLE outcomes in the sample space Number of TOTAL outcomes in the sample space PE ( ) Notation: The probability of a certain event occurring is notated by Where P stands for probability and E is the event occurring.
Example 2 - Using the sample space above, if the couple has 3 children, what is the probability of having 2 boys and 1 girl? P( of 2 boys) = P( of 3 girls) = P( of 1 boys) = P( of 2 girls) = Example 3 - Out of 100 families with 3 children how many would you expect to have all girls? INDEPENDENT EVENTS: The above situation is an example of an event because the outcome of one event does not affect the probability of the other events occurring. Example 4 - Radcliff is playing a game where he spins the spinner below and tosses and coin right after. Create a sample space for all possible outcomes, and then answer the questions below. 1 2 4 3 Spinner Coin What is the P(1)? What is P(Tails)? What is P(1 and Tails)? What is P(Even and Heads)? What is P(5 and Tails)? What is P(Odd or Heads)?
Mathematically- AND means we can MULTIPLY each individual probability. OR means we can ADD the probabilities. (But don t count an event twice!) P(Even and Heads) = P(E) x P(H) = = P(Odd or Heads) = P(O) + P(H) - P(O and H) = - = Homework : Section 3.9 pages 193-194 # s 6-29
FOUNDATIONS Outline Sec. 3.9 Day 2 Gallo Name: Date: Theoretical Probability (Continued) WITH or WITHOUT REPLACEMENT Some probability events require the act of an item back before choosing another item. These events are called events Other probability events require the act of an item before choosing another item. These events are called events. Example 1 - Suppose a bag contains 12 marbles: 6 red (R), 4 Green (G), and 2 yellow (Y). Two marbles are randomly drawn. Use a grid to find the following probabilities: Sample space: First marble returned First marble not returned (independent event) (dependent event) P(R, then R) P(R, then G) P(R, then Y) P(G, then R) P(G, then G) P(G, then Y) P(Y, then R) P(Y, then G) P(Y, then Y)
CARDS A standard deck of playing cards consist of cards. There are colors; RED and BLACK. of each. There are suits; HEARTS, DIAMONDS, CLUBS, and SPADES. of each. Each suit consist of the cards 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King, and Ace. of each. Example 2 - Suppose you are going to pull two cards from a standard deck of 52, one right after the other WITHOUT replacing the first card. Find the following probabilities: 1.) P(A red and then a black) = 2.) P(Spade and then a Heart) = 3.) P(Jack and then an ACE) = 4.) P(2 Reds) = 5.) P(2 Kings) = Homework : Section 3.9 pages 193-194 # s 30-45