FOUR-STEP PLAN FOR PROBLEM SOLVING

Transcription

1 - FOUR-STEP PLAN FOR PROBLEM SOLVING. Explore Read the problem carefully. Ask yourself questions like, What facts do I know? and What do I need to find out?. Plan See how the facts relate to each other. Make a plan for solving the problem. Estimate the answer. 3. Solve Use your plan to solve the problem. If your plan does not work, revise it or make a new plan. 4. Examine Reread the problem. Ask, Is my answer close to my estimate? Ask, Does my answer make sense for the problem? If not, solve another way. Example How many buses are needed to transport all 38 seventhgrade students at Oak Heights Middle School to the state museum? Each bus seats 56 students. Explore You know how many seventh-graders there are. You also know how many students each bus will hold. Plan To find the number of buses needed, divide 38 by 56. Solve The answer is a little more than 4. Since you can t have part of a bus, 5 buses are needed. Examine With 4 buses, there would be room for 4 56 or only 4 students. With 5 buses, there would be room for 5 56 or 80 students. This is enough room.

2 -4 AN ALGEBRAIC EXPRESSION INCLUDES AT LEAST ONE OF EACH OF THE FOLLOWING: numbers variables (such as a, n, or x) operations (,,, or ) To evaluate an expression, replace all of the variables with numbers and find the value. Evaluate 7 n if n 4. 7 n 7 4 Replace n with 4. 3 Subtract. Evaluate 7x y if x 3 and y 5. 7x y 7(3) (5) Replace x with 3 and y with 5. 0 Multiply. Subtract. 3 Evaluate cd if c and d 3. cd cd means c d. Replace 3 6 c with and d with 3. Multiply. 6 means 6. Divide.

3 3-5 The statements at the right are equations. An equation is a mathematical statement that contains an equals sign y 97 5 b 4 7 You solve an equation when you replace the variable with a number that makes the equation true. That number is called a solution. Which of the numbers, 3, or 4 is a solution of 4x 4? Replace x with. Replace x with 3. Replace x with 4. 4x 4 4x 4 4x 4 4()? 4 4(3)? 4 4(4)? false 4 4 true 56 4 false The solution is 3. Solve 45 a 30 mentally. 45 a You know that The solution is 5.

4 4-8 CHANGING METRIC UNITS km m cm mm kg g cg mg kl L cl ml, DIVIDE Complete. 0.6 cm? mm To change from centimeters to millimeters, multiply by 0 since cm 0 mm cm 6 mm 74 ml? L To change from milliliters to liters, divide by,000 since L,000 ml. 74, ml 0.74 L 3 59 mg? g 4 3. kg? g To change from milligrams to grams, divide by,000 since g,000 mg. To change from kilograms to grams, multiply by,000 since kg,000 g. 59, mg 0.59 g 3.,000 3,0 3. kg 3,0 g

5 To change from a larger unit to a smaller unit, multiply. MULTIPLY, overlay

6 To change from a smaller unit to a larger unit, divide., DIVIDE -8 overlay

7 5 - Example MAKING A FREQUENCY TABLE. Draw a table with three columns.. In the first column, list the items or intervals in the set of data. 3. In the second column, mark the tallies. 4. In the third column, write the frequency or number of tallies. Find the range for the set of data below. Choose an appropriate scale and interval. Then make a frequency table. 0.,.7, 0.8, 0.9,,.3, 0.,.5 The range is the difference between the greatest number and the least number in the set of data..3 greatest number 0. least number. range Since the data set includes numbers from 0. to.3, you might use a scale of 0. to 3 with an interval of. Therefore, the categories in the first column are 0.,., and. 3. In the second column, tally the data. In the third column, write the number of tallies. INTERVALS TALLY FREQUENCY

8 6 - You can use line graphs, bar graphs, and scatter plots to predict future events. T-Shirt Sales The graph shows the sales of T-shirts designed by the East Middle School Math Club. Use the graph to predict whether the club will reach its goal of selling 00 T-shirts in 0 days. The line graph flattens out around 0 T-shirts sold. Therefore, it does not seem likely that the club will reach its goal. Number Sold O Day The graph shows the favorite subjects of students ages 0 7. Use the graph to predict the top three subjects if another survey is taken of students in the same age range. Favorite Subjects Math % Science 6% Art History/Social Studies 5% English 3% Foreign Language 5% 8% Source: National Science Foundation The graph shows that math, science, and art, respectively, were the students favorite subjects. You could predict that those three subjects would be favorites in a larger survey as well.

9 7-3 A line plot is a picture of information on a number line. MAKING A LINE PLOT. Draw a number line.. Choose a scale and interval. Remember that the line plot does not need to start at Place an above or close to the number that represents each item of data. Example The table shows the average price for several types of exercise equipment. Make a line plot of these data. Round all prices to the nearest fifty dollars. TYPE PRICE treadmill $699 home gym 40 ski machine 394 free weights 350 stair climber 349 aerobic rider 8 stationary bike 9 Step Draw a number line. Since the lowest price is $9 and the highest price is $699, you can use a scale of $00 to $700 and an interval of $50. $00 $50 $300 $350 $400 $450 $500 $550 $600 $650 $700 Step Put an above the number that represents each price. $00 $50 $300 $350 $400 $450 $500 $550 $600 $650 $700

10 8-5 MAKING A STEM-AND-LEAF PLOT. Find the least and greatest data values.. Identify the stems and list them on the left side of the plot. 3. Identify the leaves and write them in order on the right side of the plot across from the corresponding stems. 4. Include a key to the data. Example Make a back-to-back stem-and-leaf plot for the data. Temperatures ( F) in Miami and Los Angeles, August 0 DATE MIAMI L.A. DATE MIAMI L.A. 8/ / / / / / / / / / Since the tens place is the greatest place value of the data, write the digits in the tens places as the stems. List the stems in increasing order. The numbers on either side of the stems are the leaves. Use the digits in the ones place for the leaves. List the leaves in increasing order. Stem F Always include a key to the data.

11 Miami Stem Los Angeles overlay

12 Los Angeles overlay

13 9 3-4 Use counters to solve b 5. Place 5 negative counters on the mat. Then place positive counters on the mat. Remove all of the zero pairs. b 5 b 3 Use a number line to solve x 5 (7) Both addends are negative, so the sum is negative. Start at 0 and move 5 to the left. Then move 7 more units to the left. x 5 (7) x 3 3 Evaluate h (9) if h 7. h (9) 7 (9) Replace h with , so the sum is negative. The difference of 7 and 9 is. So, h (9).

14 0 3-5 Subtracting Integers Words: To subtract an integer, add its additive inverse. Symbols: Arithmetic Algebra (5) a b a (b) Solve y 4 8. y 4 8 y 4 (8) y To subtract 8, add 8. Solve t 3 (). t 3 () t 3 To subtract, add. t 5 3 Evaluate b a if a and b. b a () Replace b with and a with. To subtract, add. 0

15 3-6 Multiplying Integers with Different Signs The product of two integers with different signs is negative. Multiplying Integers with the Same Sign The product of two integers with the same sign is positive. Solve each equation. 3(4) x 8(6) t The two integers have the same signs. The product will be positive. 3(4) x x 3 y (5) 4 g 3(7) The two integers have different signs. The product will be negative. 8(6) t 48 t The two integers have different signs. The product will be negative. y (5) y 60 The two integers have the same signs. The product will be positive. g 3(7) g 5 Evaluate the expression 3a if a 5. 3a 3(5) 3(5 5) 3(5) 75 Replace a with 5.

16 3-7 DIVIDING INTEGERS The quotient of two integers with the same sign is positive. The quotient of two integers with different signs is negative. Solve k 36 (9). k 36 (9) The signs are the same. k 4 The quotient is positive. Solve 65 5 m m The signs are different. 3 m The quotient is negative. 3 Solve d 8 (9). d 8 (9) The signs are different. d 9 The quotient is negative. 4 Evaluate a b if a 5 and b 3. a b 5 3 Replace a with 5 and b with 3. 7 Since the signs are different, the quotient is negative.

17 3 4- Subtraction Property of Equality Words: If you subtract the same number from each side of an equation, then the two sides remain equal. Symbols: Arithmetic Algebra 5 5 a b 5 5 a c b c 3 3 Solve x Check your solution. x Check: x x ? 395 x The solution is 335. Addition Property of Equality Words: If you add the same number to each side of an equation, then the two sides remain equal. Symbols: Arithmetic Algebra a b 6 6 a c b c 8 8 Solve 355 y 40. Check your solution. 355 y 40 Check: 355 y y ? y The solution is 495.

18 4 4-3 Division Property of Equality Words: If you divide each side of an equation by the same nonzero number, then the two sides remain equal. Symbols: Arithmetic Algebra 5 5 a b a c b c, c Solve each equation. Check your solution. 7c 7 7c c 7 7 Divide each side of the equation by 7. c 6 Check: 7c 7 7 6? 7 Replace c with a a a Divide each side of the equation by a Check: a 88? Replace a with

19 5 4-4 Solve each equation. Check your solution. y y y y 0 Subtract from each side. y 0 Divide each side by. y 5 Check: y 5? Replace y with 5. 0? 0 4r 0 4r 0 4r 8 4r r r Add to each side. Divide each side by 4. Check: 0 4r 0? 4 Replace r with. 0? 8 0 0

20 6 4-5 An inequality is a mathematical sentence that contains one of the symbols at the right. greater than less than greater than or equal to less than or equal to GRAPHING A SOLUTION ON A NUMBER LINE. Draw a circle at the number that all other numbers in the solution are compared to.. Fill in the circle if the number is included in the solution set. 3. Draw a thick arrow to the right or the left to show the numbers that are solutions. Example Solve n 5. Check your solution. Then graph the solution. n 5 n 5 n 7 Add to each side. Check: Try 6, a number less than 7. n 5 6? 5 Replace n with

21 4-5 overlay

22 7 5- Method Make a list. List the factors of each number. Identify the common factors. The greatest of the common factors is the GCF. FINDING THE GCF Method Use prime factorization. Write the prime factorization of each number. Identify all common prime factors. The product of the common prime factors is the GCF. Example Find the GCF of 44 and 0. Method List the factors. factors of 44:,, 4,,, 44 factors of 0:,, 3, 4, 5, 6, 8, 0,, 5, 0, 4, 30, 40, 60, 0 common factors:,, 4 Thus, the GCF of 44 and 0 is 4. Method 44 Write the prime factorization The common prime factors are and. The GCF is or 4.

23 8 5-4 In a terminating decimal, the division ends or terminates when the remainder is zero. In a repeating decimal, there is a pattern in the digits that repeats forever Method Use paper and pencil to divide the numerator by the denominator. WRITING DECIMALS AS FRACTIONS Express 4 as a decimal. Express 3 9 as a decimal. Method Method Use paper and pencil. Use paper and pencil The digit will repeat since will continue to be the remainder So, or 3. Method Use a calculator. Method Use a calculator. 4 = = 3. So, Use bar notation to indicate that the digit repeats. Method Use a calculator to divide the numerator by the denominator.

24 9 5-6 Percent Fraction Decimal 37% TO WRITE A DECIMAL AS A PERCENT: Multiply the decimal by 00 and add the percent symbol. TO WRITE A PERCENT AS A DECIMAL: Divide the percent by 00 and remove the percent symbol. Express each decimal as a percent Write as a fraction. 30% , Write as a fraction. 875, Divide to make the denominator or 87.5% Express each percent as a decimal. 3 67% 4.5% 67% % So, 67% So,.5% 0.5.

25 0 5-7 Method Make a list. List several multiples of each number. Identify the common multiples. The least of the common multiples is the LCM. FINDING THE LCM Method Use prime factorization. Write the prime factorization of each number. Identify all common prime factors. Then find the product of the prime factors using each common prime factor only once and any remaining factors. This product is the LCM. Example Find the LCM of 6, 9, and. Method Make a list. multiples of 6: 6,, 8, 4, 30, 36,... multiples of 9: 9, 8, 7, 36, 45, 54,... multiples of :, 4, 36, 48, 60, 7,... The LCM of 6, 9, and is 36. Method Use prime factorization or 3 3 or 3 The LCM of 6, 9, and is 3 3 or 36.

26 6- ADDING AND SUBTRACTING FRACTIONS With Like Denominators. Add or subtract the numerators.. Use the same denominator in the sum or difference. With Unlike Denominators. Rename the fractions with a common denominator.. Add or subtract the numerators. 3. Simplify. Add or subtract. Write each sum or difference in simplest form Estimate: 3 5 LCD: or 4 5 So, is close to the estimate, Estimate: 7 8 LCD: or 6 So, is close to the estimate,.

27 6-4 Multiplying Fractions Words: To multiply fractions, multiply the numerators and then multiply the denominators. Symbols: Arithmetic Algebra b a c d b ac d, b, d 0 Multiply. Write each product in simplest form Estimate: The product is close to the estimate, Estimate: or 4 0 The product is close to the estimate, Estimate: The product is close to the estimate, 7.

28 3 6-7 CUSTOMARY UNITS OF CAPACITY cup (c) 8 fluid ounces (fl oz) pint (pt) cups quart (qt) pints gallon (gal) 4 quarts CUSTOMARY UNITS OF WEIGHT pound (lb) 6 ounces (oz) ton (T),000 pounds Complete. 64 qt? gal smaller unit larger unit Since 4 qt gal, divide by qt 6 gal 300 T? lb larger unit smaller unit Since T,000 lb, multiply by, , , T 600,000 lb 3 6 pt? fl oz 4 30 c? qt larger unit smaller unit smaller unit larger unit Since c 8 fl oz and pt c, multiply by 6. Since c pt and pt qt, divide by pt 96 fl oz 30 c 7 qt

29 4 6-9 A circle is a set of points in a plane, all of which are the same distance from a fixed point in the plane called the center. circumference (C) diameter (d ) center radius (r) Circumference of a Circle Words: The circumference of a circle is equal to times its diameter or times twice its radius. Symbols: C d Model: or C r d r C The decimal 3.4 and the fraction 7 are used as approximations for. Example Find the circumference of a circle with a radius of 9.5 feet. 9.5 ft C r Replace with 3.4 and r with The circumference is about.46 feet.

30 5 7- A ratio is a comparison of two numbers by division. At the right are four ways to express the ratio that compares 3 to 5. 3 to 5 3:5 3 out of Express each ratio as a fraction in simplest form. 8: The GCF of 8 and 30 is 6. 5 books out of 90 books The GCF of 5 5 and 90 is The ratio in simplest form is 3 5 or 3:5. The ratio in simplest form is 5 8 or 5 out of 8. 3 Tell whether 6:40 and 48:0 are equivalent ratios. Express each ratio as a fraction in simplest form. The GCF of 6 and 40 is The GCF of 48 and 0 is The ratios in simplest form are equal. So, 6:40 and 48:0 are equivalent ratios.

31 6 7-3 Proportion Words: A proportion is an equation that shows that two ratios are equivalent. Symbols: Arithmetic Algebra b a c d (b 0, d 0) Property of Proportions Words: The cross products of a proportion are equal. Symbols: If a b d c, then ad bc. (b 0, d 0) Solve each proportion. 3 7 n 6 r n 6 r n 7 6 Cross products 3n n 4 3 Divide. n 4 r Cross products 7r r Divide. The solution is 4.

32 3 7 n 6 r n 7 6 Cross products 3n n 4 3 Divide. n 4 The solution is overlay r Cross products 7r r Divide. r 0 The solution is 0.

33 7 7-6 A percent greater than 00% represents a number greater than. A percent less than % represents a number less than 0.0 or 00. Express each percent as a decimal. 50% 0.08% 50% % So, 50% So, 0.08% Express each number as a percent % So, %. 680% So, %.

34 8 7-8 Percent Proportion The percent proportion is B P 0 r 0, where P represents the percentage, B represents the base, and r represents the number per hundred. Find each number. Round to the nearest tenth if necessary. What number is 55% of 60? B P 0 r 0 Write the percent proportion. P Replace B with 60 and r with 55. P Find the cross products. 00P 8, P 8, Divide each side by 00. P is 55% of 60. What percent of 65 is 54? B P 0 r 0 Write the percent proportion r 0 Replace P with 54 and B with r Find the cross products r = r 54 is about 83.% of 65.

35 9 8- What number is 53% of 60? Estimate: of P R B P Replace R with 0.53 and B with 60. P is 53% of is close to the estimate of 300. is what percent of 67? Estimate: or 0%. P R B R 67 Replace P with and B with R Divide each side by = R Round to the nearest hundredth. is about 8% of 67. 8% is close to the estimate of 0%. 3 9 is 6% of what number? Estimate: 9 is 5% or 0 of 80. P R B B Replace P with 9 and B with B Divide each side by = 50 9 is 6% of is close to the estimate of 80.

36 Example Dreamy Yogurt expects to sell 5,000 yogurt cups at the first annual Taste of the Town food fair. To determine how much of each flavor to have on hand, Dreamy Yogurt conducted a poll in which 5 yogurt lovers were FLAVOR NUMBER OF PEOPLE vanilla 47 chocolate 34 strawberry 8 other 6 asked which kind of yogurt they preferred. Use the results given in the table to predict how many people will prefer chocolate and vanilla yogurt if the people who prefer other will settle for vanilla. You can use the percent proportion to find the number who prefer chocolate n 5, ,000 5n , n = 360 You can predict that,360 people will prefer chocolate.

37 8-3 overlay Use the percent proportion to find the number who prefer vanilla, including those who prefer other n 5, ,000 5n , n = 50 You can predict that about,50 people will prefer vanilla, including those who prefer other.

38 3 8-5 The price of a round-trip ticket between New York and Chicago on an airline is normally $355 when the ticket is purchased days in advance. For a limited time, the airline is offering a 7% discount on all advance purchase round-trip tickets. How much would you pay for this ticket with the discount? Method First, find the amount of the discount. 7% of $355 d = The discount is $ Method First, subtract the percent of discount from 00%. 00% 7% 73% The discounted price is 73% of the normal price. Then, subtract to find the discounted price. $355 $95.85 $59.5. Then, multiply to find the discounted price = 59.5 The discounted price is $59.5. Find the total cost of a shirt that costs $9.95 with 6% sales tax. Method 0.06 of $9.95 s Method 00% 6% 06% =.797 The price with sales tax is The sales tax is $ % of the normal price. $9.95 $.80 $ = The total cost is $3.75.

39 3 8-6 Simple interest is the amount paid for the use of money. The formula for simple interest is I prt, where I is the interest, p is the principal, r is the annual interest rate, and t is the time in years. Mr. Jones deposited $800 in his savings account. His account earned 4.5% interest annually. He did not deposit or withdraw any money for 8 months. How much interest did he earn? I prt I p $800, r 4.5%, t 8 mos or yr = 54 I 54 The interest earned on $800 in 8 months was $54. Find the interest to the nearest cent on $750 on a credit card at 4.5% interest for years. I prt I p $750, r 4.5%, t years = 7.5 I 7.50 The interest paid on a $750 credit card balance over years was $7.50.

40 33 9- The probability of an event is the ratio of the number of ways an event can occur to the number of possible outcomes. What is the probability that the spinner will land on a vowel? H A B C G D F E P(spinning a vowel) 8 number of ways to spin a vowel number of ways to spin a letter There are vowels on the spinner, A and E. There are 8 letters on the spinner. 4 The probability is 4 or 5%. In a survey of 00 people, 36 said they talked on the telephone in the kitchen the most, 5 said the living room, 3 said the bedroom, and 6 said other rooms. If a person is chosen at random, what is the probability that he or she talks on the telephone in the kitchen the most? P(kitchen) number of people who talk in the kitchen the most total number of people people talk in the kitchen the most. There are 00 people in the survey. 9 5 The probability is 9 5 or 36%.

41 The Counting Principle If event M can occur in m ways and is followed by event N that can occur in n ways, then the event M followed by N can occur in m n ways. Use the Counting Principle to find the total number of outcomes in each situation. choosing a pair of pants if the pants come cuffed or uncuffed, regular fit or relaxed fit, and in the following colors: stone, khaki, wheat, olive, and black choices choices choices total number for cuffs for fit for color of choices } } 5 0 spinning the spinners shown } } } choices for choices for choices for total number first spinner second spinner third spinner of choices } } }

42 A combination is an arrangement of objects in which order is not important. In how many ways can the colors of paint shown at the right be mixed two colors at a time? You can make a list. The order is not important. So, there is no difference between and. Cross out each package that is the same as another package. You can also use the formula. C 4 3 or 6 There are 4 3 permutations of two colors chosen from four. There are! or colors being mixed. There are six different ways to mix these colors. Does this problem represent a permutation or a combination? In how many ways can six people pose for a picture if they all stand in a row? Since order does matter, this is a permutation of 6 people. 6! or 70 There are 70 ways for six people to pose for the picture.

43 The order is not important. So, there is no difference between and. Cross out each package that is the same as another package. 9-5 overlay

44 Two or more events in which the outcome of one event does not affect the outcome of the other event(s) are called independent events. Probability of Independent Events The probability of two independent events can be found by multiplying the probability of one event by the probability of the second event. If the results of one event does affect the result of a second event, the events are called dependent events. Example Find the probability of selecting a circle card followed by a star card with replacement of the first card. Since the first card is being replaced, the events are independent. P(circle, then star) P(circle) P(star) number of specified cards total number of cards

45 37 0- Example According to a recent survey of 500 parents, 33% of the parents said their kids do homework in the bedroom, 5% said the kitchen, % said the family or living room, and 0% said the dining room. Make a circle graph to represent the data. Step Find the number of degrees for each part. Use P R B. bedroom: 33% of kitchen: 5% of family or living room: % of dining room: 0% of Step Draw a circle. Then draw a radius. Step 3 Use a protractor to draw an angle of 8.8. Step 4 Repeat for the other sections. Where Kids Do Their Homework Step 5 Label each section of the graph with the category and percent. Give the graph a title. Kitchen: 5% Family/Living Room: % Bedroom: 33% Dining Room: 0%

46 Similar Polygons Words: Symbols: Model: Two polygons are similar if their corresponding angles are congruent and their corresponding sides are in proportion. ABCD WXYZ The symbol means is similar to. A B W X D C Z Y Example If ABC DEF, find the length of ED. E x ft 6 ft 3 ft B 4 ft A 5 ft C D F 0 ft BA and ED are corresponding sides. AC and DF are corresponding sides. Let x represent the missing measure. B ED A 3 x 5 0 A D C F 3(0) x 5 Find the cross products. 60 5x x Divide each side by 5. The length of ED is feet. Write a proportion using the measures of corresponding sides.

47 A polygon is a simple closed figure formed by three or more line segments. triangle (3 sides) quadrilateral (4 sides) pentagon (5 sides) hexagon (6 sides) heptagon (7 sides) octagon (8 sides) nonagon (9 sides) decagon (0 sides) Any polygon with all sides congruent and all angles congruent is called a regular polygon. Determine which figures are polygons. If the figure is a polygon, name it and tell whether it is a regular polygon. If the figure is not a polygon, explain why. 3 quadrilateral not a polygon; no sides are segments triangle; regular

48 Figures that match exactly when folded in half have line symmetry. Each fold line is called a line of symmetry. Determine which figures have line symmetry. Draw all of the lines of symmetry. 3 A reflection is a mirror image of a figure across a line of symmetry. 4 Complete an Escher-like drawing using the change shown at the right. Complete the first pattern unit by drawing the reflection of the design on the other side of the triangle. Now add another triangle. Reflect the new pattern in the second triangle and complete the tessellation.

49 0-9 overlay no symmetry

50 4 - square square root or because 9 8 A radical sign, 00, represents a nonnegative square root. Evaluate 8. Evaluate x 55 3 Find Find 34. Since 3 69, nd 34 [0] 8 5 Juan is flying to Europe with his parents for vacation. Since it is a clear day, he can use the formula d.4 h to estimate the distance in miles that he can see from the airplane. In this formula, d is the distance from the object in miles, and h is the height in feet Juan s eyes are above the surface. If Juan s plane is flying at 3,400 feet, how far can he see? d.4 h d.4 3,400 Replace h with 3,400. d ,400 nd [0] 80 d 5 Juan can see about 5 miles.

51 4 - Estimate 8. Since 8 is not a perfect square, estimate 8 by finding the two perfect squares closest to 8. List some perfect squares., 4, 9, 6,... } 8 is between 4 and Find the square root of each number. 8 3 This means that 8 is between and 3. So, 8 is between and 3. Since 8 is closer to 9 than to 4, 8 is closer to 3 than to. The best whole number estimate for 8 is 3. Estimate 30., 4, 9, 6, 5, 36, 49, 64, 8, 00,, 44, 69,... } 30 is between and Find the square root of each number is between and. So, 30 is between and. Since 30 is closer to than to 44, 30 is closer to than. The best whole number estimate for 30 is.

52 43-3 Pythagorean Theorem Words: In a right triangle, the sum of the squares of the lengths of the legs (a and b) is equal to the square of the length of the hypotenuse (c). Symbols: Arithmetic Algebra a b c Model: c a b If one side of a triangle measures 6 centimeters and the other side measures 3 centimeters, find the length of the hypotenuse to the nearest tenth. a b c Pythagorean Theorem 6 3 c Replace a with 6 and b with c 05 c 05 c Definition of square root 4.3 c 05 nd [0] The hypotenuse measures about 4.3 centimeters. If 5 inches, inches, and 3 inches are the lengths of the sides of a triangle, is the triangle a right triangle? a b c Pythagorean Theorem 5? 3 Replace a with 5, b with, and c with ? It is a right triangle.

53 44-5 Area of a Triangle Words: The area (A) of a triangle is equal to half the product of its base (b) and height (h). Symbols: A bh Model: h b Find the area of a triangle with a base of 3 inches and a height of 4 inches. A bh Formula for the area of a triangle A 3 4 Replace b with 3 and h with 4. A or 6 The area is 6 square inches. Area of a Trapezoid Words: The area (A) of a trapezoid is equal to half the product of the height (h) and the sum of the bases (a b). Symbols: A h(a b) Model: a h b Find the area of a trapezoid with bases of 6 meters and 8 meters and a height of 4 meters. A h(a b) Formula for the area of a trapezoid A (4)(6 8) Replace h with 4, a with 6, and b with 8. A (4) or 8 The area is 8 square meters.

54 45 - rectangular prism triangular prism pyramid cone cylinder Make a perspective drawing of a figure by using the top, side, and front views of the figure below. top side front First, sketch a rectangle for the top. Then, add the front and side views. Finally, add dashed lines to show hidden edges. Draw a top, a side, and a front view of the figure.

55 - overlay

56 top side front - overlay

57 46-3 A cylinder is a solid figure that has two congruent, parallel circles as its bases. Volume of a Cylinder Words: The volume (V ) of a cylinder is found by multiplying the area of the base (r ) by the height (h). Symbols: V r h Model: h Find the volume of a cylinder with a radius of 6 feet and a height of 4 feet. Use 3.4 for. Estimate: V r h Formula for volume of cylinder V Replace r with 6 and h with 4. V 45.6 The cylinder has a volume of about 45 cubic feet. Find the volume of the cylinder. Estimate: 3 6 0,080 V r h V 3.4 (5.5) V, m m The cylinder has a volume of about,045 cubic meters.

58 47-4 Surface Area of a Rectangular Prism Words: Symbols: Model: The surface area of a rectangular prism equals the sum of the areas of the faces. surface area w h wh h w Find the surface area of the rectangular prism. surface area w h wh 6 in Multiply first. Then add. 88 The surface area of the rectangular prism is 88 square inches. in. 4 in. Find the surface area of a rectangular prism with a length of meters, a width of 7 meters, and a height of 5 meters. surface area w h wh Multiply first. Then add. 358 The surface area of the rectangular prism is 358 square meters.

59 48-5 Surface Area of a Cylinder Words: Symbols: Model: The surface area of a cylinder equals the sum of the areas of the circular bases (r ) and the area of the curved surface (rh). surface area r rh r h Find the surface area of the cylinder. ft ft surface area r rh π x + π = The surface area of the cylinder is about 38 square feet. Find the surface area of a cylinder with a diameter of centimeters and a height of 8 centimeters. The diameter of the cylinder is centimeters. Therefore, the radius is 5.5 centimeters. surface area r rh (5.5) π 5.5 x + π = The surface area of the cylinder is about 8 square centimeters.