6-1The Set of Rational Numbers

Transcription

1 6-1The Set of Rational Numbers Equivalent or Equal Fractions Simplifying Fractions Equality of Fractions Ordering Rational Numbers Denseness of Rational Numbers Rational numbers The set of numbers Q such that Numerator, Denominator Uses of Rational Numbers Rational Number Models 6.1 The set of Rational Numbers Page 1

2 Meaning of a Fraction To understand the meaning of any fraction, using the parts-to-whole model, we must consider each of the following: 1. The whole being considered. 2. The number b of equal-size parts into which the whole is divided. 3. The number a of parts of the whole that we are selecting. Number Line Model What numbers are represented on the number line? Proper fraction Examples Improper fraction Examples Equivalent or Equal Fractions Equivalent fractions are numbers that represent the same point on a number line. 6.1 The set of Rational Numbers Page 2

3 Fraction Strips Fundamental Law of Fractions Example 1 Find a value for x such that Simplifying Fractions A rational number is in simplest form if a and b have no common factor greater than 1, i.e. they are relatively prime. Example 2 Write each of the following fractions in simplest form if they are not already so: a. b. c. d. 6.1 The set of Rational Numbers Page 3

4 Equality of Fractions Show that Ordering Rational Numbers If a, b, and c are integers and b > 0, then if and only if a > c. If a, b, c, and d are integers and b > 0, d > 0, then ad > bc. if and only if Denseness of Rational Numbers Given two rational numbers a and c, there is another rational b d number between these two numbers. Example 3 a. Find two fractions between b. Show the sequence is increasing. 6.1 The set of Rational Numbers Page 4

5 Homework 1. A student asks if 0 is in its simplest form. How do you respond? 6 2. A student writes 15 < 1 15 because 3 15< Another student writes = 1. Where is the fallacy? 3. A student claims that there are no numbers between 999 together. What is your response? 1000 and 1 because they are so close 6.1 The set of Rational Numbers Page 5

6 6-2 Addition, Subtraction, and Estimation with Rational Numbers Mixed Numbers Properties of Addition for Rational Numbers Subtraction of Rational Numbers Definition of Greater Than and Less Than in Terms of Subtraction Estimation with Rational Numbers Addition of Rational Numbers Area model Number-line model Addition of Rational Numbers with Like Denominators Addition of Rational Numbers with Unlike Denominators Example 1 Find each of the following sums: 6.2 Addition, Subtraction, and Estimation with Rational Numbers Page 1

7 Mixed Numbers numbers that are made up of an integer and a fractional part of an integer A mixed number is a rational number, and therefore, it can always be written in the form of Properties of Addition for Rational Numbers Additive Inverse Property Properties of the additive inverse for rational numbers are analogous to those of the additive inverse for integers. Addition Property of Equality 6.2 Addition, Subtraction, and Estimation with Rational Numbers Page 2

8 Subtraction of Rational Numbers Definition of Greater Than and Less Than in Terms of Subtraction Estimation with Rational Numbers Many of the estimation and mental math techniques that are used with whole numbers also work with rational numbers Example: A student added and 4 2 obtained4. How would you use 6 estimation to show that this answer could not be correct? Example: Estimate each of the following: a b Addition, Subtraction, and Estimation with Rational Numbers Page 3

9 Homework 1. Suppose a large pizza is divided into 3 equal-size pieces and a small pizza is divided into 4 equal-size pieces and you get 1 piece from each pizza. Does represent 3 4 the amount that you received? Explain why or why not? 2. When we add two fractions with unlike denominators and convert them to fractions with the same denominator, must we use the least common denominator? What are the advantages of using the least common denominator? 3. When the least common denominator is used in adding or subtracting fractions, is the result always a fraction in simplest form? 4. Kara spent 1 of her allowance on Saturday and 1 of what she had left on Sunday. 2 3 Can this situation be modeled as 1 1? Explain why or why not? Addition, Subtraction, and Estimation with Rational Numbers Page 4

10 6-3 Multiplication and Division of Rational Numbers Multiplication of Rational Numbers Properties of Multiplication of Rational Numbers Multiplication with Mixed Numbers Division of Rational Numbers Estimation and Mental Math with Rational Numbers Extending the Notion of Exponents Multiplication of Rational Numbers Multiplication as repeated addition Multiplication as part of an area Example 1 If 5 of the population of a certain city are college graduates and of the city s college graduates are female, what fraction of the population of that city is female college graduates? Properties of Multiplication of Rational Numbers Multiplicative Identity The number 1 is the unique number such that for every rational number a b, then 6.3Multiplication and Division of Rational Numbers Page 1

11 Multiplicative Inverse (Reciprocal) For any nonzero rational number, the number 1 is the unique rational number such that Distributive Property of Multiplication Over Addition Multiplication Property of Equality Multiplication Property of Inequality Multiplication Property of Zero If a b is any rational number, then Example 2 A bicycle is on sale at 3 of its original price. If the sale price is $330, what was 4 the original price? Multiplication with Mixed Numbers Using Improper Fractions Using the Distributive Property 6.3Multiplication and Division of Rational Numbers Page 2

12 Example 3 Solve for x: Division of Rational Numbers The bar of length 3 4 is made up of 6 equal-size pieces of length. There is at least one length of 3 in If we put another bar of length 3 on the number line, we see there is 1 more of the 6 4 equal-length segments needed to make 7. Therefore, the answer is1 1 or Definition of Division of Rational Numbers 6.3Multiplication and Division of Rational Numbers Page 3

13 When two fractions with the same denominator are divided, the result can be obtained by dividing the numerator of the first fraction by the numerator of the second. To divide fractions with different denominators, we rename the fractions so that the denominators are equal. Example 4 A radio station provides 36 min for public service announcements for every 24 hr of broadcasting. a. What part of the broadcasting day is allotted to public service announcements? b. How many 3 min. public service announcements can be allowed in the 36 minutes? 4 Example 5 There are 35 1 yards of material available to make towels. Each towel 2 requires 3 yards of material. 8 a. How many towels can be made? b. How much material will be left over? 6.3Multiplication and Division of Rational Numbers Page 4

14 Example 6 If 3 of a jump rope is 5 yards longs, what is the length of the entire rope? 4 Estimation and Mental Math with Rational Numbers Estimation and mental math strategies that were developed with whole numbers can also be used with rational numbers. Example 7 Estimate each of the following: a b Extending the Notion of Exponents Multiplication and Division of Rational Numbers Page 5

15 8. 9. Example 8 Write each of the following in simplest form using positive exponents in the final answer: a. b. c. d. Homework 1. Amy says that dividing a number by 1 is the same as taking half of a number. How do 2 you respond? 2. Noah says that dividing a number by 2 is the same as multiplying it by 1.He wants to 2 know if he is right and if so, why. How do you respond? 6.3Multiplication and Division of Rational Numbers Page 6

16 6.4Ratios, Proportions, and Proportional Reasoning Proportions Scale Drawings Ratios A ratio is used to compare quantities. usually written or. used to represent part-to-part, part-to-whole, or whole-to-part comparisons. Example 1 There were 7 males and 12 females in the Dew Drop Inn on Monday evening. In the game room next door were 14 males and 24 females. a. Express the number of males to females at the inn as a ratio (partto-part). b. Express the number of males to females at the game room as a ratio (part-to-part). c. Express the number of males in the game room to the number of people in the game room as a ratio (part-to-whole). Proportions A proportion is a statement that two given ratios are equal. If a, b, c, and d are all real numbers, and b 0 and d 0, then the proportion a b = c d is true if and only if. 6.4 Ratios, Proportions, and Proportional Reasoning Page 1

17 Example 2 Allie and Ben type at the same speed. Allie started typing first. When Allie had typed 8 pages, Ben had typed 4 pages. When Ben has typed 10 pages, how many has Allie typed? This is an example of an relationship. Students should reason that since the two people type at the same speed, when Ben has typed an additional 6 pages, Allie should have also typed an additional 6 pages, so she should have typed 8 + 6, or 14, pages. Carl can type 8 pages for every 4 pages that Dan can type. If Dan has typed 12 pages, how many pages has Carl typed? This is an example of a relationship. Since Carl types twice as fast as Dan he will type twice as many pages as Dan. Therefore, when Dan has typed 12 pages, Carl has typed 24 pages. Example 3 If there are 3 cars for every 8 students at a high school, how many cars are there for 1200 students? Constant of Proportionality 6.4 Ratios, Proportions, and Proportional Reasoning Page 2

18 A central idea in proportional reasoning is that a relationship between two quantities is such that the ratio of one quantity to the other remains unchanged as the numerical values of both quantities change. Cross-Multiplication Method Scaling Strategy Use the scaling strategy to determine which is the better buy 12 tickets for $15 or 20 tickets for $23? The scaling strategy for solving the problem would involve finding the cost for a common number of tickets, the LCM. LCM(12, 20) = In the first plan, since 12 tickets cost $15, then 60 tickets cost $. In the second plan, since 20 tickets cost $23, then 60 tickets cost $. The second plan is the better buy. Unit-Rate Strategy Use the unit-rate strategy to determine which is the better buy 12 tickets for $15 or 20 tickets for $23? The unit-rate strategy for solving this problem involves finding the cost of one ticket under each plan and then comparing the unit cost. In the first plan, since 12 tickets cost $15, then 1 ticket costs $. In the second plan, since 20 tickets cost $23, then 1 ticket costs $. The plan is the better buy. 6.4 Ratios, Proportions, and Proportional Reasoning Page 3

19 Example 4 Kai, Paulus, and Judy made $2520 for painting a house. Kai worked 30 hr., Paulus worked 50 hr., and Judy worked 60 hr. They divided the money in proportion to the number of hours worked. If they all earn the same rate of pay, how much did each earn? Scale Drawings Ratio and proportions are used in scale drawings. The scale is the ratio of the size of the drawing to the size of the object. Example 5 The floor plan of the main floor of a house in the figure is drawn in the scale of 1:300. Find the dimensions in meters of the living room. 3.7 cm 2.7 cm Homework 1. Mary is working with measurements and writes the following proportion: 12 in/1ft = 5ft/60 in How would you help her? 2. Nora said she can use division to decide whether two ratios form a proportion; for example, 32:8 and 40:10 form a proportion because 32 8 = 4 and = 4. Is she correct? Why or why not? 3. Mark s friend told him the ratio of girls to boys in his new class is 5:6. Mark was very surprised to think his friend s class had only 11 students. What do you tell him? 6.4 Ratios, Proportions, and Proportional Reasoning Page 4