Reteaching Using Deductive and Inductive Reasoning

Transcription

1 Name Date Class Reteaching Using Deductive and Inductive Reasoning INV There are two types of basic reasoning in mathematics: deductive reasoning and inductive reasoning. Deductive reasoning bases a conclusion on laws or rules, while inductive reasoning bases a conclusion on an observed pattern. Both types are used to support conclusions. Two statements form an argument. Identify the premise and conclusion. Then classify the reasoning as deductive or inductive. Miguel multiplies several numbers. He then says that 5 5. The premise is the pattern that Miguel sees. The conclusion is 5 5. Miguel uses an observation to support his conclusion. He uses inductive reasoning. When your premise is based on proven laws or rule, you are using deductive reasoning. If you are using the laws or rules properly, the conclusion will always be correct. With inductive reasoning, your conclusion is based on patterns and observations. With this type of reasoning, your conclusion is supported, but may not be correct. Hank says that 5 5a because the identity property of multiplication says that for any number a, a a, The premise is the identity property. The conclusion is 5 5. Hank uses a proven law to support his conclusion. He uses deductive reasoning. Complete the steps to identify the type of reasoning as deductive or inductive.. The temperature has been greater than 70 for the past 5 days, so it will be greater than 70 tomorrow. The premise is based on an observation, so this is inductive reasoning.. The power of a power rule says to simplify a power of a power, multiply the exponents, so The premise is based on a rule, so this is deductive reasoning. Identify the type of reasoning as deductive or inductive.. One pound equals 6 ounces, so 6 ounces equal pounds. deductive Saxon. All rights reserved. 87 Saxon Algebra

2 Reteaching continued INV. Mary s brother is 5 years older than she is, so when Mary is her brother will be 9. deductive 5. The Mustangs have not lost a game all season, so they will win Tuesday s game. inductive A conditional statement is written in if-then form. The hypothesis follows the word if, and the conclusion follows the word then. Conditional statements may be true or false. One way to show that a statement is false is to give a counterexample. Identify the hypothesis and conclusion. Then give a counterexample. If a number is multiplied by a negative number, then the product is negative. Hypothesis: a number is multiplied by a negative number Conclusion: the product is negative This statement is not true because the product of negative numbers is positive. A counterexample is (5)() 0. Complete the steps to identify the hypothesis and conclusion. Then give a counterexample. 6. If a month has exactly 0 days, then it must be September, April, or June. Hypothesis: a month has exactly 0 days Conclusion: It must be September, April, or June. Counterexample: November also has 0 days. Identify the hypothesis and conclusion. Then give a counterexample. 7. If a number is prime, then it is odd. Hypothesis: Conclusion: Counterexample: A number is prime. It is odd. is a prime number and is not odd. 8. If the sum of two numbers rolled on a number cube is 6, then both rolls were. Hypothesis: The sum of numbers rolled on a number cube is 6. Conclusion: Both rolls were. Counterexample: Sample: The rolls could have been and 5, or and. Saxon. All rights reserved. 88 Saxon Algebra

3 Name Date Class Reteaching Finding Rates of Change and Slope You have learned how to simplify expressions. Now you will use this skill to find the rate of change. The rate of change is a ratio that compares the change in one quantity with the change in another. You can find the rate of change from a table of values. A shop charges a fee for renting bikes. What is the rate of change for the rental fees? Times (hours) 6 8 Cost $.50 $.50 $.50 $.50 Step : Choose any two sets of values from the table to find the ratio of change in cost to the change in time. (,.50), (6,.50) Step : Write the change of costs Write the change of times. 6 Step : Write the differences in a ratio and simplify. change in cost change in time The rate of change is $5 per hour. 0 5 Complete the steps to find the rate of change Saxon. All rights reserved. 89 Saxon Algebra Choose any two set of values: (, 7 ), Choose any two set of values: (5, 0 ), (8, 9 ) (7, 6 ) Write a ratio and simplify Find the rate of change. Write a ratio and simplify A shop charges a fee for mopeds. What is the rate of change for the rental fees? Times (hours) 5 Cost $5.50 $6.50 $7.50 $58.50 The rate of change is $ per hour.

4 Reteaching continued The slope of a line is a rate of change. It is equal to the ratio of the vertical change (rise) to the horizontal change (run). slope rise run Find the slope of the line. Find the slope of the line. Step : Find the rise and run on the graph. Step : Find the rise and run on the graph. y y (,0) O - (-,-) - x (0,) O - (,) x - Step : Find the ratio of rise to run. slope rise run units units Step : Find the ratio of rise to run. slope rise run units units Complete the steps to find the slope of each line. 6. y (,) 7. y (-,) O - - x (-,) O (,0) x Write the ratio of rise to run. slope rise run units 5 units 5 Write the ratio of rise to run. slope rise run units units Find the slope of each line. 8. y 9. y O x O x Saxon. All rights reserved. 90 Saxon Algebra

5 Name Date Class Reteaching Solving Percent Problems You have multiplied real numbers. Now you will use this skill to solve problems involving percent. A percent is ratio that compares a number to 00. There are three parts to a percent statement. The whole is the total amount. The percent is a rate that quantifies an amount measured with respect to the whole. The percentage is a number that represents a percent of the whole. 0% of 90 is 7. Percent Whole Percentage The word of means multiplication and the word is means equals. What number is 0% of 50? What number is 0% of 60? Step : Change the percent to its decimal form. 0% 0.0 Step : Write the question as an equation. n Step : Multiply the percent by the whole. n So, 0 is 0% of 50. Step : Change the percent to its decimal form. 0% 0.0 Step : Write the question as an equation. n Step : Multiply the percent by the whole. n So, is 0% of 60. Complete the steps to find the percentage.. What number is 5% of 00?. What number is 5% of 60? 5% 0.5 5% 0.5 n n n n So, 50 is 5% of 00. So, 9 is 5% of 60. Find the percentage using an equation.. What number is 0% of 500?. What number is 75% of 00? Tom saves 0% of his allowance to put towards a new CD. His allowance is $0 per week. What amount does he save each week? If the CD costs $, how many weeks must he save? $; 6 weeks Saxon. All rights reserved. 9 Saxon Algebra

6 Reteaching continued A percent equation can be written as a proportion. One ratio compares the part of a quantity to a whole quantity. The other ratio is the percent written in the form of a ratio. What number is 0% of 80? Step : Change the percent to its fraction form. 0% 0 00 Step : Write the question as a proportion. n Step : Cross multiply. 00n 600 Step : Solve for n. n 6 So, 6 is 0% of 80. What number is % of 60? Step : Change the percent to its fraction form. % 00 Step : Write the question as a proportion. n Step : Cross multiply. 00n 70 Step : Solve for n. n 7. So, 7. is % of 60. Complete the steps to find the percentage. 6. What number is 5% of? 7. What number is 5% of 0? 5 5% 00 n % n n n 50 n 6 n.5 So, 6 is 5% of. So,.5 is 5% of 0. Find the percentage using a proportion. 8. What number is 0% of 50? 9. What number is 80% of 0? 0 0. What number is 60% of 96?. What number is 5% of 90? Mary saves 50% of her allowance for a digital audio player. Her allowance is $5 per week. What amount does she save each week? If the digital audio player costs $90, for how many weeks must she save? $7.50; weeks Saxon. All rights reserved. 9 Saxon Algebra

7 Name Date Class Reteaching Simplifying Rational Expressions You have simplified fractions. Now you will use this skill to simplify rational expressions. Rational expressions are fractions containing variables. Since you cannot divide by 0, the denominator of a rational expression must not equal 0. When 0 is in the denominator, the rational expression is undefined. Determine the values for which the rational expression is undefined. x 5 Step : Identify the denominator. x 5 Step : Set the denominator equal to zero. x 5 0 Step : Solve for x. x 5 0 x 5 Subtract 5 from both sides. So, the expression is undefined when x 5. Determine the values for which the rational expression is undefined. x 7 x 8 Step : Identify the denominator. x 8 Step : Set the denominator equal to zero. x 8 0 Step : Solve for x. x 8 0 x 8 Add 8 to both sides. x Divide both sides by. So, the expression is undefined when x. Complete the steps to determine the values for which the rational expression is undefined.. x x 7. 6x x The denominator is x 7. The denominator is x. x 7 0 x 0 x 7 Add 7 to both sides. x Add to both sides. So, the expression is undefined when x 7. x Divide both sides by. Determine the values for which the rational expression is undefined.. x x 6. So, the expression is undefined when x. x 0 x 6 x 5 Saxon. All rights reserved. 9 Saxon Algebra

8 Reteaching continued Rational expressions can be factored. They can also be simplified if the numerator and denominator contain the same common factor. Factor and simplify the rational expression. x 5x Step : Find the GCF of the numerator. GCF none Step : Find the GCF of the denominator. GCF x Step : Factor. x 5x x(x 5) Factor and simplify the rational expression. x 8x x 6x Step : Find the GCF of the numerator. GCF x Step : Find the GCF of the denominator. GCF x Step : Factor. x 8x x x(x ) 6x x(x ) Step : Simplify. x (x ) x (x ) x x Complete the steps to factor and simplify each rational expression. 5. 5x 8x 6 Find the GCF of the numerator. GCF none Find the GCF of the denominator. Factor. 5x 8x 6 GCF 5x (x ) 6. x x 9x x Find the GCF of the numerator. GCF x Find the GCF of the denominator. GCF x Factor. x x x 9x x (x ) x (x ) Factor and simplify each rational expression. 7. x x 5x 9. x x x 8x x x (x 5) Simplify. 8. 5x 0x 5x 5x x x 0. x x 6 x (x ) (x ) x (x ) (x ) x x Saxon. All rights reserved. 9 Saxon Algebra

9 Name Date Class Reteaching Finding Slope Using the Slope Formula You have found slope using a graph. Now you will find slope using the slope formula. Slope is the measure of the steepness of a line. Two points on a line can be used to find the slope of a line. The slope m of a line containing the points ( x, y ) and ( x, y ) is m y y x x. Find the slope of the line containing the points (, ) and (, 5). Step : Identify the points. ( x, y ) is (, ), and ( x, y ) is (, 5). Step : Substitute the points into the formula. m y y x x 5 Step : Subtract and simplify. m 5 So, the slope of the line containing the points (, ) and (, 5) is. Find the slope of the line containing the points (, ) and (8, ). Step : Identify the points. ( x, y ) is (, ), and ( x, y ) is (8, ). Step : Substitute the points into the formula. m y y x x 8 Step : Subtract and simplify. m 8 5 So, the slope of the line containing the points (, ) and (8, ) is 5. Complete the steps to find the slope of the line containing the given points.. (, ) and (5, ). (, ) and (5, ) ( x, y ) is (, ), and ( x, y ) is (5, ). ( x, y ) is (, ), and ( x, y ) is (5, ). m y y x x 5 m y y x x 5 So, the slope of the line containing the So, the slope of the line containing the points (, ) and (5, ) is. points (, ) and (5, ) is Find the slope of the line containing the given points.. (, ) and (, ). (, ) and (8, ) 5 Saxon. All rights reserved. 95 Saxon Algebra

10 Reteaching continued Two points are not always given to substitute into the slope formula. If you are given a table or graph, choose two points to plug into the slope formula. Determine the slope of the line that contains the given points shown below. x y Step : Identify two points. ( x, y ) is (, ), and ( x, y ) is (6, ). Step : Substitute the points into the formula. m y y x x 6 Step : Subtract and simplify. m slope 6 Determine the slope of the line that contains the given points shown below. x y Step : Identify two points. ( x, y ) is (0, 6), and ( x, y ) is (5, 0). Step : Substitute the points into the formula. m y y 0 (6) x x 5 0 Step : Subtract and simplify. 0 (6) m 5 0 slope 5 Complete the steps to find the slope of the line containing the given points in the table. 5. x y x y 5 6 ( x, y ) is (, 6), and ( x, y ) is (6, 7). ( x, y ) is (, 5), and ( x, y ) is (, ). 7 x x 6 m y y 6 x x 5 m y y So, the slope of the line is. So, the slope of the line is. 7. Find the slope of the line containing the x given points in the table at the right. y 5 5 Saxon. All rights reserved. 96 Saxon Algebra

11 Name Date Class Reteaching Translating Between Words and Inequalities 5 You have learned to translate between words and equations. Now you will use this skill to translate between words and inequalities. An inequality is a mathematical statement comparing quantities that are not equal. Inequality Symbol Meaning less than greater than less than or equal to; at most greater than or equal to; at least Translate the sentence into an inequality. The difference of a number and 6 is greater than 0. Step : Identify the operation: subtraction Step : Identify the inequality: Step : Write the inequality. n 6 0 Translate the sentence into an inequality. The sum of the quotient of a number and and 5 is at most 5. Step : Identify the operations: addition, division Step : Identify the inequality: Step : Write the inequality. n 5 5 Complete the steps to translate the sentence into an inequality.. The product of a number and 7 is less than. The difference of a number and 5 is greater or equal to. than 8. Identify the operation: multiplication Identify the operation: subtraction Identify the inequality: Identify the inequality: Write the inequality: 7n Write the inequality: n 5 8. The quotient of and a number is at least 5. n 5. The difference of a number and 0 is less than. n 0 5. The product of and a number is less than 0. n 0 6. The sum of the product of 5 and a number and is greater than or equal to 80. 5n 80 Saxon. All rights reserved. 97 Saxon Algebra

12 Reteaching continued 5 You can also translate between mathematical symbols and words. Write each inequality as a sentence. n 9 Step : Identify the operation: addition Step : Identify the inequality: greater than Step : Write the sentence. The sum of a number and is greater than 9. Write each inequality as a sentence. 7n 5 80 Step : Identify the operations: multiplication and addition Step : Identify the inequality: less than or equal to Step : Write the sentence. The sum of the product of a number and 7 and 5 is less than or equal to 80. Complete the steps to write the inequalities in words. 7. n 5 8. n 0 Identify the operation: subtraction Identify the operation: division Identify the inequality: greater than or Identify the inequality: less than equal to The difference of a number and is The quotient of a number and 0 is greater than or equal to 5. less than. 9. n 5 7 The sum of a number and 5 is less than or equal to n 70. n 7 5 The product of 6 and a number is greater than 70. The difference of the quotient of a number and and 7 is less than 5.. n 5 0 The sum of twice a number and 5 is greater than or equal to 0.. Mary needs to buy pencils that cost $.0 each. If she can spend at most $5, write an inequality to represent the situation. 0.0p 5 Saxon. All rights reserved. 98 Saxon Algebra

13 Name Date Class Reteaching Simplifying Expressions with Square Roots and Higher-Order Roots 6 You have calculated square roots. Now you will simplify expressions with higher-order roots and fractional exponents. To find higher-order roots like the fourth root of 6, think what number as a factor four times equals 6? Higher-Order Roots If a n b, then the nth root of b is a, or n b a. Simplify ()()()() Simplify 5. 5 (5)(5)(5) (5) 5 Complete the steps to simplify each expression ()( )( )( )( ). 6 (6)(6)(6) 5 5 (6) 6 Simplify each expression , ,000 0 Saxon. All rights reserved. 99 Saxon Algebra

14 Reteaching continued 6 Roots can be written as numbers with fractional exponents. For example, 8 8. Notice that the index from 8 becomes the denominator in the exponent of 8. Fractional Exponents n n a a Simplify Simplify (5). (5) 5 (8) 8 Complete the steps to simplify the expression Simplify each expression. 5. () (6) A gift box in the shape of a cube has a volume of 79 cubic centimeters. What is its side length? 9 cm Saxon. All rights reserved. 00 Saxon Algebra

15 Name Date Class Reteaching Solving Problems Involving the Percent of Change 7 You have found percents of numbers. Now you will find percents of increase and decrease. A percent of change is a ratio comparing an amount of change to an original amount. When the new amount is greater than the original amount, this ratio is called a percent of increase. When the new amount is less than the original amount, the ratio is called a percent of decrease. Percent of Change percent of increase amount of increase original amount percent of decrease amount of decrease original amount Monday s high temperature was 60 F. Tuesday s high temperature was 75 F. What was the percent of change? Tuesday s temperature was greater than Monday s, so the percent of change is an increase. amount of increase original amount , or a 5% increase Complete the steps to find the percent of increase or decrease.. population in 000: 5,000. January sales: $5,000 population in 007: 6,500 February sales: $0,000 amount of increase original amount 500 5,000 amount of decrease 5000 original amount 5,000 0 % increase 0 % decrease Find the percent of increase or decrease.. starting pay rate: $7.50 per hour. time to walk the first mile: 8 min new pay rate: $9 per hour time to walk the second mile: 5 min 0% increase 7% decrease 5. students that graduated in 006: heart rate before exercise: 65 students that graduated in 007: 750 heart rate during exercise: 0 6.5% decrease 00% increase Saxon. All rights reserved. 0 Saxon Algebra

16 Reteaching continued 7 A common application of percent of change is markups and discounts. To find a new price after a markup or discount, first use the percent of change and the original price to find the amount of change. For a markup, add this amount to the original price. For a discount, subtract the amount of change from the original price. A store buys sweatshirts for $0 each and marks up the price by 75%. What is the new price of one sweatshirt? The amount of the markup is 75% of $0: (0.75)(0) $7.50. To find the new price, add the amount of markup to the original price. $0 $7.50 $7.50 The new price of each sweatshirt is $7.50. Complete the steps to find the new price. 7. $50, 90% markup 8. $80, 5% discount amount of markup ( 0.9 )(50) amount of discount (0.5)( 80 ) $ 5 $ new price 50 5 new price 80 Find the new price. $ 95 $ $75, 50% markup 0. $85, 80% markup $6.50 $5. $98, 85% markup. $60, 5% discount $8.0 $95. $5, 0% discount. $, 0% discount $7.50 $ Phil buys a new television that has a regular price of $650. He uses a coupon for a 0% discount. What is the price of the television after the discount? $50 6. A video game store buys games for $0 each. If the store marks up prices 70%, what is the new price of each game? $ Saxon. All rights reserved. 0 Saxon Algebra

17 Name Date Class Reteaching Analyzing Measures of Central Tendency 8 You have used graphs to analyze statistical data. Now you will use measures of central tendency to analyze statistical data. Measures of Central Tendency Mean Median Mode sum of all values in a data set divided by the number of data values middle number in a set of data values when arranged in order value or values that occur most frequently in a data set The range of a data set is the difference between the greatest and least values in the set. The range shows how spread out a data set is. Find the mean, median, mode and range of the values in the data set. 8,,,, 6,, 8,, mean sum of values number of values median Order the data values and find the middle value: 6,,,,, 8, 8,,. mode Find the number that occurs more often than any other number:. range greatest value least value 6 6 Complete the steps to find the mean, median, mode, and range of the values in the data set.. 0,, 7, 5,, 7, 8, 6 mean sum of values number of values median Order the data values and find the middle value. Since there is an even number of data values, the median is the average of the two middle numbers., 5, 6, 0,,,7, 7, 8 0 Saxon. All rights reserved. 0 Saxon Algebra 8 mode Find the number that occurs more often than any other number: 7. range greatest value least value 8 7 Find the mean, median, mode, and range of the values in each data set.., 7,, 5,,, 6,, 8. 6, 7, 7, 76, 6, 75, 7, 67 mean median mean 70 median 7 mode range mode 6 range

18 Reteaching continued 8 An outlier is a value that is much greater or less than the other values in the set. Outliers can affect the measures of central tendency. The following data show points scored by a football team in games. 6,, 7, 0,,, 7, 7, 6, 0, 7 a. Identify any outliers in the data set. Write in order: 0, 0,, 6, 6, 7, 7, 7,,, 7 Observe any patterns: 7 is much greater than the other values in the set. b. Describe how the outlier affects the measures of central tendency. Without the outlier With the outlier mean 8 6. median mode 7 7 The outlier of 7 raised the mean score by almost points and the median by 0.5 point. The following data show quiz scores for 5 students. 8, 8, 0, 7, 8, 0, 9,, 8, 7, 0, 7, 9, 0, 8 Complete the steps.. Identify any outliers in the data set. Write in numerical order:,7,7,7,8,8,8,8,8,9,9,0,0,0,0 Observe any patterns: the outliner value of is much less than the other values in the data set. 5. Describe how any outliers affect the measures of central tendency. With the outlier Without the outlier mean median 8 8 mode 8 8 The outlier of lowered the mean score by 0.5. The following data show Ben s earnings from working at the music store over a -day period. Day Sun. Mon. Tues. Wed. Thu. Fri. Sat. Sun. Mon. Tues. Wed. Thu. Earnings $0 $ $ $8 $6 $ $6 $0 $6 $ $ $0 6. Identify any outliers in the data set. $6 Saxon. All rights reserved. 0 Saxon Algebra

19 Name Date Class Reteaching Writing Equations in Slope-Intercept Form 9 You have graphed the equation of a line by finding the intercepts of the line. Now you will graph the equation of a line by first writing the equation in slope-intercept form. Slope-Intercept Form of an Equation y mx b m is the value of the slope of the line. b is the value of the y-intercept. Determine the slope and y-intercept of an equation. y x 5 Equation is in slope-intercept form. m slope b y-intercept 5 x y 0 Equation is not in slope-intercept form. Isolate y to write the equation in slope-intercept form. x _ x y 0 x _ y x y x y x m slope b y-intercept Complete the steps to determine the slope and y-intercept of the equation.. y x 7 Equation is in slope-intercept form.. 5x y Equation is not in slope-intercept form. Isolate y to write the equation in slope-intercept form. 5x y 5x 5x ()(y) ( 5x)() y 5x Determine the slope and y-intercept of each equation.. y 9x. 5x y 8 m slope b y-intercept 7 m slope 5 b y-intercept Slope 9 y-intercept Slope 5 y-intercept 8 Saxon. All rights reserved. 05 Saxon Algebra

20 Reteaching continued 9 Once the equation of a line is in slope-intercept form, it can be graphed on a coordinate plane. Graph y x. slope m y-intercept b Graph the y-intercept, then use it and the slope to plot another point. The slope is. Start at (0, ), move up units and right units to (, ). Draw a line through the two points. - y (,) O x - - (0,-) - Complete the steps to graph the line. 5. y x. slope m y-intercept b = Graph the y-intercept, then use it and the slope to plot another point. The slope is. Start at (0, ), move up units and right units to (, ). - (0,) O y (,) x Draw a line through the two points. Graph each line using an equation that is in slope-intercept form. 6. y x 5 y 7. x y 0 6 x O - - y x 8. A bowling center charges $ for shoe rental and $ per game for each bowler. Write an equation in slope-intercept form to represent the total cost y given that each bowler bowls x games. Then graph the equation. y x Amount ($) Bowling Costs Number of Games Saxon. All rights reserved. 06 Saxon Algebra

21 Name Date Class Reteaching Graphing Inequalities 50 You have translated sentences into inequalities. Now you will graph inequalities and write inequalities from graphs. A graph on a number line can represent the solution set of an inequality. Graph x. Step : Create a number line that includes the endpoint number,. Step : Draw a circle on the number. Step : Decide whether to fill in the circle. If or, leave empty. If or, fill in. Step : Draw a heavy line and arrow. If or, draw arrow to the left. If or, draw arrow to the right Complete the steps to graph the inequality.. x Step : Create a number line that includes Step : Draw a circle on. Step : Decide whether to fill in the circle. means leave empty. Step : Draw a heavy line and arrow. means draw arrow to the right. Graph each inequality.. m. z k c Saxon. All rights reserved. 07 Saxon Algebra

22 Reteaching continued 50 The graph of an inequality can be translated into symbols. Write an inequality for the graph Step : Write a variable and the number at the endpoint indicated by the circle. Step : Look at the direction of the arrow. If arrow points left, use or. If arrow points right, use or. Step : Look at the circle. If circle is empty, use or. If circle is filled in, use or. x? 6 x or 6 x 6 Complete the steps to write an inequality for the graph Step : Write a variable and the number. x? Step : Look at the direction of the arrow. The arrow points left. Use or. Step : Look at the circle. x or The circle is empty. Use or. x Write an inequality for each graph x 7 x x x 0. The freezing point of a liquid is the temperature at which it changes to a solid. The freezing point of water under normal conditions is F. Write an inequality to represent the temperatures for which water is a solid. Graph the solution set of the inequality. t Saxon. All rights reserved. 08 Saxon Algebra