[PDF] Unit 09 More About Variables Formulas and Graphs.pdf | Plain Text

STUDY LINK 9 1 Name Date Time Multiplying Sums 285 Copyright © Wright Group/McGraw-Hill 248 249 1. For each expression in the top row, find one or more equivalent expressions below it. Fill in the oval next to each equivalent expression. a. (8 7) 4 b. (6 5) (6 8) c. 3 (9 4) (8 4) (7 º 4) (8 6) (5 6) (9 4) (3 4) 4 (7 8) 6 (5 8) 9 (3 4) (8 4) 7 (8 5) 6 (4 9) 3 (8 4) (7 4) (6 5) (6 8) (9 3) (4 3) 2. The area of Rectangle M is 45 square units. a. What is the value of b? b. Write 2 different number sentences to describe the area of the unshaded part of Rectangle M. () ()  ()  3. Each of the following expressions describes the area of one of the rectangles below. Write the letter of the rectangle next to its expression. a. (3 2) 7 b. (2 3) (7 3) c. (7 2) 3 d. (3 7) (2 7) e. 2 (7 3) f. 3 (2 7) 4. Sandra wants to buy envelopes and stamps to send cards to 8 friends. Envelopes cost $0.10 and stamps cost $0.39. How much will she spend? Write a number model to show how you solved the problem. 5 3b Rectangle M 7 2 3Rectangle N 2 7 3Rectangle P 3 27Rectangle O

STUDY LINK 9 2 Name Date Time Using the Distributive Property 286 Copyright © Wright Group/McGraw-Hill 248 249 1. Use the distributive property to rewrite each expression. a. 7 (3 4) (º ) () b. 7 (3 π) (º ) () c. 7 (3 y) (º ) () d. 7 (3 (2 4)) () ((2 4)) e. 7 (3 (2 π)) () ((2 )) f. 7 (3 (2 y)) () (()) 2. Use the distributive property to solve each problem. Study the first one. a. 7 (11025) b. 20 (42 19)  c. (32 50) 40  d. (90 8) 11  e. 9 (15 25)  3. Circle the statements that are examples of the distributive property. a. (80 5) (120 5) (80 120) 5 b. 6 (3 0.5) (6 3) 0.5 c. 12(dt) 12d12t d. (ac) nancn e. (16 4m) 9.7 16 (4m9.7) f. (9 º 1 2) ( 1 3º 1 2) (9  1 3) º 1 2 (7 110)+(7 25) 770 + 175 945 Write each quotient in lowest terms. 4. 1 5 11 5 5. 3 7 16 1 6. 111 97 1 2 PracticeReminder:aº (xy) (aº x) (aº y) aº (xy) (aº x) (aº y)

LESSON 9 2 Name Date Time Applying the Distributive Property 287 Copyright © Wright Group/McGraw-Hill 1. Cheng and 5 of his friends are buying lunch. Each person gets a hamburger and a soda. How much money will they spend in all? Write a number model to show how you solved the problem. Answer Explain how the distributive property can help you solve Problem 1. 2. Minowa signed her new book at a local bookstore. In the morning she signed 36 books, and in the afternoon she signed 51 books. It took her 5 minutes to sign each. How much time did she spend signing books? Write a number model to show how you solved the problem. Answer 3. Ms. Hays bought fabric for the school musical chorus. She bought 4 yards each of one kind for 30 group costumes and 4 yards each of another kind for 6 soloists. How many yards did she buy in all? Write a number model to show how you solved the problem. Answer 4. Mr. Katz gave a party because all the students got 100% on their math test. He had budgeted $1.15 per student. It turned out that he saved $0.25 per student. If there are 30 students, how much did he spend? Write a number model to show how you solved the problem. Answer Fill in the missing numbers according to the distributive property. 5. 28 6 () 6 6. (6) (6) (20  8) 6 $.90 $1.10

STUDY LINK 9 3 Name Date Time Combining Like Terms 288 Copyright © Wright Group/McGraw-Hill 252 Simplify each expression by rewriting it as a single term. 1. 3x12x 2. (1 3 5)y(1 13 0)y 3. (5t) 6t 4. 4d(3d)  Complete each equation. 5. 15k(9 )k 6. 3.6pp0.4p 7. (8 ) m5m 8. j4.5j3.8j Simplify each expression by combining like terms. Check your answers by substituting the given values for the variables. Show your work on the back of this sheet. Example:18 6m2m26 Combine the mterms. 6m2m8m Combine the number, or constant, terms. 18 26 44 So, 18 6m2m26 8m44. Check: Substitute 5 for m. 18 (6 5) (2 5) 26 (8 5) 44 18 30 10 26 40 44 84 84 9. 8b9 4b3b(2b) (5)  Check for: b 6 10. 1 2a 3 4t 2 3a( 1 2t)  Check for: a2 and t2 Practice 11. 117 64  12. 9 (32)  13. 12  (11)  14. 57 3 

LESSON 9 3 Name Date Time Simplifying and Evaluating Expressions 289 Copyright © Wright Group/McGraw-Hill Each expression on the left of the equal sign can be simplified to the expression on the right. Fill in the missing variable or constant terms. 1. 9x10 7x13 2. 4m8n6 4m32n6 3. 2t(2v) 2w4t2v4w 4. 3c2c6d4d10 2d2 5. 8f4g13 fg10 Simplify each expression below. Then evaluate the expression for x2, y3, and z4. 6. 2y6z4x2z8y12x 7. 3x10 4x9y6x4z 8. 2x9 6z3y6z15 9. 3z5x9yxy5z 10. 1 2x 2 3y 3 4z 3 2x

LESSON 9 4 Name Date Time Area and Variables 290 Copyright © Wright Group/McGraw-Hill Copyright © Wright Group/McGraw-Hill Math Message Write an expression to show how you could find the area of each rectangle. Then find the area by substituting the given value for the variable. Expression Evaluate for y3. If y3, then the area of Rectangle A is units 2. 5 2y4y Rectangle A 2m 5m7 Rectangle B LESSON 9 4 Name Date Time Area and Variables Math Message Write an expression to show how you could find the area of each rectangle. Then find the area by substituting the given value for the variable. Expression Evaluate for y3. If y3, then the area of Rectangle A is units 2. 5 2y4y Rectangle A 2m 5m7 Rectangle B Expression Evaluate for m5. If m5, then the area of Rectangle B is units 2. 248 249 Expression Evaluate for m5. If m5, then the area of Rectangle B is units 2. 248 249

STUDY LINK 9 4 Name Date Time Simplifying Expressions 291 Copyright © Wright Group/McGraw-Hill 248 249 251 252 Simplify each expression by removing parentheses and combining like terms. Check by substituting the given values for the variables. Show your work. 1. 7(7 5f) (f6)10 Check: Substitute  1 5for f. 2. 3(4 5m) 12 (3m) Check: Substitute 1 3for m. 3. (12 3 5k)6 4k2(k5) Check: Substitute 0.5 for k. 4. 5(yb) 3b6y4(6 b) Check: Substitute 1 for yand 2 3for b. Practice Find each product or quotient. 5. 0.658 10 5 6. 234.8 10 3 7. 5,163 10 4 8. 7.96 10 2

LESSON 9 4 Name Date Time Variable-Term Tiles 292 Copyright © Wright Group/McGraw-Hill

LESSON 9 4 Name Date Time Constant-Term Tiles 293 Copyright © Wright Group/McGraw-Hill

LESSON 9 4 Name Date Time Modeling and Simplifying Algebraic Expressions 294 Copyright © Wright Group/McGraw-Hill For each expression, use the tiles to represent the variable terms and the tiles to represent the constant terms. Example:Simplify 3(2x3). Step 13(2x3) means 3 of(2x3). Model (2x3) three times. Step 2Combine like terms. Combine variable terms. Combine constant terms. Step 3Simplify. 3(2x3) 6x9 Use tiles or pictures to model each expression. Combine like terms and simplify. 1. 2(y4)  2. 4(2k3)  3. 5(w2) 5  4. 6(1 x) 3x 5. p2(p3) 2p 6. 2(n3n) n1 

STUDY LINK 9 5 Name Date Time Equivalent Equations Copyright © Wright Group/McGraw-Hill Each equation in Column 2 is equivalent to an equation in Column 1. Solve each equation in Column 1. Write Any number if all numbers are solutions of the equation. Match each equation in Column 1 with an equivalent equation in Column 2. Write the letter label of the equation in Column 1 next to the equivalent equation in Column 2. Column 1 Column 2 A4x2 6 Solution B3s6 Solution C3y2yy Solution D5a7a Solution 6j8 8  6j 2c1 3 6w12 2 2h h1 3 3q6 4 3(r4) 18 2(5x1) 10x2 5x5(2 x) 2(x7) s0 5b3 2b6b3 4t3 2 1 2 6z12 2a(4 7)a A Write each product or quotient in exponential notation. 1. 222 3 2. 3. 525 2 4. 4342 10 4 10 2 Practice 295 251 252

LESSON 9 5 Name Date Time Revisiting Pan Balances 296 Copyright © Wright Group/McGraw-Hill Solve the equations. For each step, record the operation you use and the equation that results. Check your solution by substituting it for the variable in the original equation. 1. Original equation Operation    2. Original equation Operation    3. Original equation Operation    4. Original equation Operation   

LESSON 9 5 Name Date Time Writing and Solving Equations 297 Copyright © Wright Group/McGraw-Hill Sometimes you need to translate words into algebraic expressions to solve problems. Example:The second of two numbers is 4 times the first. Their sum is 50. Find the numbers. If nthe first number, then 4nthe second number, and n4n50. Because 5n50, n10. The first number is 10 and the second number is 4(10), or 40. For each problem, translate the words into algebraic expressions. Then write an equation and solve it. 1. The larger of two numbers is 12 more than the smaller. Their sum is 84. Find the numbers. Equation Smaller number Larger number 2. Mr. Zock’s sixth-grade class of 29 students has 9 more boys than girls. How many girls are in the class? Equation Number of girls Sometimes it helps to label a diagram when you are translating words into algebraic expressions. 3. The base (b) of a parallelogram is 3 times as long as an adjacent side (s). The perimeter of the parallelogram is 64 m. What is the length of the base? Label the diagram at the right. Then write an equation and solve it. Equation Length of the base units b

LESSON 9 5 Name Date Time More Simplifying and Solving of Equations 298 Copyright © Wright Group/McGraw-Hill Simplify each equation. Then solve it. Show your work. 1. 4(5t7) 10t2 2. 18(m6) 15m6 Solution Solution 3. 4(12 8w) w18 4. 3g8(2g6) 2 14g Solution Solution 5. 7(1 4y) 13(2y3) 6. 4n5(7n3) 9(n5) Solution Solution 7. 2(6v 3) 18 3(16 3v) 8. 5 (15d1) 2(7d16) d Solution Solution

STUDY LINK 9 6 Name Date Time Expressions and Equations 299 Copyright © Wright Group/McGraw-Hill Solve. 1. 3x9 30x 2. 73  1 2(108 f)f 3. 55 (9 d) 11d 4. (m15) (m6) 42m Simplify these expressions by combining like terms. 5. 8y27 6y(4) 6. 7b17 9b15 7. 3f80 25 10k 8. 240 5g3(10g5) Circle all expressions that are equivalent to the original. There may be more than one. Check your answer by substituting values for the variable. 9. Original: 3r17 2r6 5r23 23 rr23 13 r 10. Original: 8(9 b) 4b 89 3b72 3b4b72 72 (4b) 11. The top mobile is in balance. The fulcrum is at the center of the rod. A mobile will balance when WD wd. Look at the bottom mobile. What is the weight of the object on the left? Write and solve an equation to answer the question. WDwd Equation Solution The weight of the object on the left is units. Try This fulcrum(at center of rod) d D Ww 5b 15 15 12 4 30 12. 81 32 7 8 13. 35 6º 24 14. 25 4 3 8 Practice

300 LESSON 9 6 Name Date Time Challenge: Balancing a Mobile Copyright © Wright Group/McGraw-Hill In the mobile shown below, each rod is suspended at its center. 1. Is each rod in perfect balance? If not, which of the rods is not balanced? Explain how you found the answer. 2. If you found a rod in the mobile that was not balanced, how would you move exactly one of the suspending wires so the rod would balance? 20 20 17 10 10 10 8.5 3 15.31 5 6 10 10 10 7.6 8 9.5 rod weight = 6 rod weight = 3 rod weight = 3 3

301 LESSON 9 7 Name Date Time Evaluating Expressions Copyright © Wright Group/McGraw-Hill Name Math Message Evaluate the expression (b 2º 4) k for the following values: 1. b3 and k5 2. b 1 2and k 3 4 3. b2 and k10 4. b5 and k115 Name Math Message Evaluate the expression (b 2º 4) k for the following values: 1. b3 and k5 2. b 1 2and k 3 4 3. b2 and k10 4. b5 and k115 Name Math Message Evaluate the expression (b 2º 4) k for the following values: 1. b3 and k5 2. b 1 2and k 3 4 3. b2 and k10 4. b5 and k115 Name Math Message Evaluate the expression (b 2º 4) k for the following values: 1. b3 and k5 2. b 1 2and k 3 4 3. b2 and k10 4. b5 and k115

LESSON 9 7 Name Date Time Sample Spreadsheet 302 Copyright © Wright Group/McGraw-Hill totals pencils graph paper ruler book Subtotal tax 7% Totalunit price 0.29 1.19 0.50 5.95quantity 6 2 1 1display bar address boxcancel accept ABCD item name 1.74 2.38 0.50 5.95 10.57 0.74 $11.31 Supplies ($) 1 2 3 4 5 6 7 8 9 10D3 x✓  B3 ºC3

STUDY LINK 9 7 Name Date Time Circumferences and Areas of Circles 303 Copyright © Wright Group/McGraw-Hill 1. Complete the spreadsheet at the left. For each radius, calculate the circumference and area of a circle having that radius. Round your answers to tenths. 2. Use the data in the spreadsheet to graph the number pairs for radius and circumference on the first grid below. Then graph the number pairs for radius and area on the second grid below. Connect the plotted points. A 2 7 8 9 B C Circles circumferences and areas of circles radius (ft) circumference (ft) area (ft 2) r2rr 2 0.5 1.0 1.5 2.0 2.5 3.09.4 12.67.1 12.6 1 3 4 5 6 0123 0 10 20 30 Radius (ft) Circumference (ft) 0123 0 10 20 30 Radius (ft) Area (ft 2) 3. A circular tabletop has an area of 23 square feet. Use the second line graph to estimate the radius of the tabletop. Radius: About 140 213218 (unit)

LESSON 9 7 Name Date Time Stopping Distance for an Automobile 304 Copyright © Wright Group/McGraw-Hill Drivers sometimes need to stop quickly. The time it takes to stop depends on the car’s speed. A driver takes about 3 4second to react before actually stepping on the brake pedal. After the brake has been depressed, additional time passes before the car comes to a complete stop. The spreadsheet below shows the minimum stopping distances for various speeds. 1. The spreadsheet is not completely filled in. Calculate and record the numbers for the cells in rows 9, 10, and 11. (Hint:Use the formulas in cells B4, C4, and D4.) 2. Circle any cell that contains labels. 3. Circle any cell that contains numbers used in calculations but not in formulas. 4. Circle any cell in which formulas are stored. 5. Write the formula stored in each cell. B7 D11 6. If you change the number in cell A7 to 35, will the numbers in any other cells change? If so, which cells? D4 B10 C6 A3 B4 A5 D5 C10 D9 B5 A11 C4 minimum stopping distance on a dry, level, concrete surface speed (mph) reaction-time distance (ft) distance  1.1 º speedbraking distance (ft) distance  0.06 º speed 2 total stopping distance (ft) distance  1.1 º speed + 0.06 º speed 2 A 1 2 3 4 5 6 7 8 11 B C D Stopping Distances 10 20 30 40 50 60 7011 22 33 446 24 54 9617 46 87 140 9 10

LESSON 9 7 Name Date Time Stopping Distance for an Automobile cont. 305 Copyright © Wright Group/McGraw-Hill 7. Use the data in the spreadsheet on Math Masters,page 304. a. Graph the number pairs for speed and reaction-time distance on the first grid below. Make a line graph by connecting the plotted points. b. Graph the number pairs for speed and braking distance on the second grid below. Use a curved line to connect the plotted points. 8. How are the two graphs different? 9. Complete the statement. At speeds of 50 miles per hour or more, 0 10203040506070 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 Speed (mph) Reaction-time distance (ft) Reaction-Time Distance 0 10203040506070 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 Speed (mph) Braking distance (ft) Braking Distance

STUDY LINK 9 8 Name Date Time Area Problems 306 Copyright © Wright Group/McGraw-Hill 1. parallelogram Area 4. triangle Area 2. rectangle Area 5. triangle Area 3. parallelogram Area 6. trapezoid Area Calculate the area of each figure in Problems 1–6. Remember to include the unit in each answer. In Problems 7 and 8, all dimensions are given as variables. Write a true statement in terms of the variables to express the area of each figure. Example: Area 7. Area 8. Area m n xy a a b c d ab c 1 2 ºc ºd Try This Practice 9. x5.3 12 x 10. 3.1 31ww 7 in. 8 in. 16 in. 1 ft 30 in. 12 cm 9 cm 20 cm 13 mm 10 mm 7 mm 10 ft 11 ft 14.9 ft 30 m10 m 18 m 30 m 24 m 26 m 215– 217

LESSON 9 8 Name Date Time Areas of Parallelograms 307 Copyright © Wright Group/McGraw-Hill 1. Do not cut out the shapes on this page. Instead cut out Parallelogram A on Math Masters,page 309 and follow the directions there. Parallelogram A Tape your rectangle in the space below. base cm length cm height cm width cm Area of parallelogram cm 2 Area of rectangle cm 2 2. Do the same with Parallelogram B on Math Masters,page 309. Parallelogram B Tape your rectangle in the space below. base cm length cm height cm width cm Area of parallelogram cm 2 Area of rectangle cm 2 3. Write a formula for the area of a parallelogram.

LESSON 9 8 Name Date Time Areas of Triangles 308 Copyright © Wright Group/McGraw-Hill 1. Do not cut out the triangle below. Instead cut out Triangles C and D from Math Masters,page 309 and follow the directions there. Triangle CTape your parallelogram in the space below. base cm length cm height cm width cm Area of triangle cm 2 Area of parallelogram cm 2 2. Do the same with Triangles E and F. Triangle E Tape your parallelogram in the space below. base cm length cm height cm width cm Area of triangle cm 2 Area of parallelogram cm 2 3. Write a formula for the area of a triangle.

LESSON 9 8 Name Date Time Parallelograms 309 Copyright © Wright Group/McGraw-Hill Cut out Parallelogram A. (Use the second Parallelogram A if you make a mistake.) Cut it into 2 pieces and tape the pieces together to make a rectangle. Tape the rectangle onto Math Masters,page 307. Do the same with Parallelogram B. Triangles Cut out Triangles C and D. Tape them together at the shaded corners to form a parallelogram. Tape the parallelogram onto the space next to Triangle C on Math Masters,page 308. Do the same with Triangles E and F. A A B B C D E F

LESSON 9 9 Name Date Time Calculating the Volume of the Human Body 310 Copyright © Wright Group/McGraw-Hill head (sphere) neck (cylinder) torso (rectangular prism) 2 arms (cylinders) 2 upper legs (cylinders) 2 lower legs (cylinders) torso (rectangular prism) (side view) scale is 1 mm:1 cm

STUDY LINK 9 9 Name Date Time Area and Volume Problems 311 Copyright © Wright Group/McGraw-Hill Area formulas Rectangle:Abº h Parallelogram:Abº h Triangle:A 1 2º bº h Volume formulas Cylinder:VBº hπ º r 2º h Rectangular prism:VBº hlº wº h Sphere:V 4 3º π º r 3 Circumference formulaC2πr Calculate the area or volume of each figure. Pay close attention to the units. 1. 2. Volume Volume 3. 4. Area Volume 5. 6. Area Volume 47 yd 100 yd 23 yd 30 yd 1.6 m 6.5 m 2 2.1" 5.7" 5"4"6" Aarea Vvolume Barea of base Ccircumference blength of base hheight llength wwidth rradius (unit) (unit) (unit)(unit) (unit) (unit) Practice 7. 0.95 m cm 8. 378 mm cm 9. 1.4 m mm Try This diameter 12" Use 3.14 for π. 221 222 224 Use 3.14 for π. 4.1 ft ft π Use 3.14 for π.

BoxArea of Number of Total number base (B)layers (h) of cubes (V) Cube cm 2 cm cm 3 Rectangular prism cm 2 cm cm 3 312 Copyright © Wright Group/McGraw-Hill LESSON 9 9 Name Date Time Exploring Volume of Rectangular Prisms Use what you know about area to help you find the volume of solid figures. 1. Find the area of each base (B) for the partial nets on Math Masters,page 313. Record each base’s area in the space provided on each net. 2. Cut out the nets. Fold along the lines and tape the sides together to make an open box from each net. 3. Fill each box by carefully layering centimeter cubes. Count the number of layers and total number of cubes needed to fill each box. Record the results in the table below. 4. Compare the area of the base (B) and the number of layers (h) with the total number of cubes (V). Explain any patterns you notice. 5. Use the pattern(s) from Problem 4 to find the volume of the rectangular prisms described below. a. B40 cm 2; h4 cmVcm 3 b. B50 cm 2; h8 cmVcm 3 c. l5 cm; w9 cm; h6 cmVcm 3 6. Suppose you know the volume of a rectangular prism is 135 cm 3and its square base has a side measuring 3 cm. Explain how to find the height of this prism.

Name Date Time 313 Copyright © Wright Group/McGraw-Hill LESSON 9 9 Partial Nets TAB fold fold fold fold TAB TAB TAB TAB Area of base (B) = cm 2 fold fold fold fold TAB TAB TAB Area of base (B) = cm 2

Name Date Time 314 Copyright © Wright Group/McGraw-Hill LESSON 9 9 Comparing Capacities Using Formulas Compare the capacities of the cylinders shown at the right. The formula for the volume of a cylinder can be used to find its capacity. To find the capacity, first find the area of the circular base (Aπr 2) and multiply by the cylinder’s height. Because the circumference is given, use it to find the radius; then find the area of the base. Finally, find the capacity. For the taller cylinder: 1. Use the formula for circumference to find the radius. Cπ º 2r (Circumference π º 2 º radius) Use 3.14 for π. 2. Substitute the radius in the formula Aπr 2to find the area of the base. Round this area to the nearest hundredth. 3. Multiply the area of the base by the cylinder’s height to find the capacity. For the shorter cylinder: 4. Use the formula for circumference to find the radius. Use 3.14 for π. 5. Substitute the radius in the formula Aπr 2to find the area of the base. Round this area to the nearest hundredth. 6. Multiply the area of the base by the cylinder’s height to find the capacity. 7. Which cylinder holds more?  πº 2r A π º π º2r A π º height  8.5 in. circumference  11 in. circumference  8.5 in. height  11 in.

STUDY LINK 9 10 Solving Equations by Trial and Error 315 241–243 Name Date Time Copyright © Wright Group/McGraw-Hill Find numbers that are close to the solution of each equation. Use the suggested test numbers to get started. 1. Equation: r 2r15 My closest solution ________________________ 2. Equation: x 22x23 rr 2 r2rCompare r 2rto 15. 39 12 15 416 20 15 3.5 12.25 15.75 15 xx 2 2x x 22xCompare x 22x to 23. 636 12 24 23 525 10 15 23 5.5 30.25 11 19.25 23 My closest solution ________________________ 3. 56 42.52  4. 23.5 5.88  Practice

LESSON 9 10 Name Date Time A Box Problem 316 Copyright © Wright Group/McGraw-Hill Suppose you have a square piece of cardboard that measures 8 inches along each side. To construct an open box out of the cardboard, you can cut same-size squares from the 4 corners of the cardboard and then turn up and tape the sides. 1. José cut out small square corners to make his box. Amy cut out large square corners to make her box. a. Whose box was taller? b. Whose box had a greater area of the base? The volume of the box depends on the size of the squares cut from the corners. 2. Find the dimensions of the box with the greatest possible volume. Use trial and error to solve the problem. Keep a record of your results in the spreadsheet below. a. Three test values for h(the height of the box) are listed in Column A. Complete rows 4, 5, and 6. (8  2h)" (8  2h)" h" cutout corner Box made by cutting out square corners and folding up sides 8" h" h" 8"original sheet A 1 2 3 4 5 6 7 8 9 B C Boxes Problem: Find the length that maximizes the box volume. box height (in.) box length, width (in.)base area of box (in. 2) h 8  2h(8  2h) 2 1 2 3636 D volume of box (in. 3) (8 – 2h) 2 º h b. Use the spreadsheet results to select new test values for hthat are likely to give a box of greater volume. Complete the statement below. The box that I found with the greatest volume has a height of inches and a volume of cubic inches.

STUDY LINK 9 11 Using Formulas 317 Name Date Time Copyright © Wright Group/McGraw-Hill 245 246 Each problem below states a formula and gives the values of all but one of the variables in the formula. Substitute the known values for the variables in the formula and then solve the equation. 1. The formula C 5 9º (F 32) may be used to convert between Fahrenheit and Celsius temperatures. a. Convert 77°F to degrees C. b. Convert 50°C to degrees F. Equation Equation Solve. Solve. 77°F °C 50°C °F 2. The formula for the area of a trapezoid is A 1 2º (a b) º h. a. Find the area (A) of a trapezoid if a7 cm, b10 cm, and h 5 cm. Equation Equation Solve. Solve. Area Height 3. The formula for the volume of a cone is V 1 3º π º r 2º h. a. Find the volume (V) of a cone if b. Find the height (h) of a cone if r2 inches and h9 inches.r3 cm and V94.2 cm 3. Equation Equation Solve. Solve. Volume Height a h b r h (unit) (unit) (unit) (unit) Use 3.14 for π. b. Find the height (h) of a trapezoid ifa6.5 inches, b5.5 inches, and A90 inches 2.

LESSON 9 11 Name Date Time Perimeter and Area Problems 318 Copyright © Wright Group/McGraw-Hill Study the example. Then solve the problems. Example:The area of triangle CBAis 21 square inches. What is the length of side BA? Solution 1. Write the formula for the area of a triangle.A 1 2º bº h 2. Substitute the dimensions in the formula. 21  1 2º 6 º (3x1) 3. Solve the equation. 21 3 º (3x1) 21 9x3 18 9x,so 2 x. 4. Answer the question. 3x1 (3 º 2) 1, or 7 Side ABis 7 inches long. 5. Check the answer: Area  1 2º 6 º 7  1 2º 42 21 in. 2 1. The area of rectangle RPQTis 14 ft 2. Find the length of side RP. Formula: Area  Substitute  Solve Length of R P Check 2. The area of parallelogram FLOWis 15 in. 2Find the length of side FL. Formula: Area  Substitute  Solve Length of F L Check 3. The perimeter of triangle MONis 29 cm. Find the length of each side. Formula: Perimeter  Substitute  Solve Length of NM ON  MO Check CA B 63x  1 2 4y  3 R P T Q 3 2x  1 FW O L y 2y  4 3y  5 M N O

STUDY LINK 9 12 Pythagorean Theorem 319 Name Date Time Copyright © Wright Group/McGraw-Hill Mentally find the positive square root of each number. 1. 144   2. 200 2   3. 900   4. 0.16   5. 12 25 1  6. 10,00  0  Use a calculator to find each square root. Round to the nearest hundredth. 7. 12   8. 51   9. 63   Use the Pythagorean theorem to find each missing length. Round your answer to the nearest tenth. 10. 11. cb (unit) (unit) 12. a 13. Find the distance (d) from home plate to second base. dft 7 m 24 mc b 5 ft 11 ft 120 yd 122 yd a 2nd base 3rd base1st base Home 90 ft 90 ft 90 ft 90 ft d Simplify. 14. 2[9(6 5)]  15. 5  3 4 8 2 7  Practice (unit) 167 285 286

LESSON 9 12 Name Date Time Pythagorean Triples 320 Copyright © Wright Group/McGraw-Hill Sets of positive integers that are solutions of the equation a 2b 2c 2are Pythagorean triples. The smallest Pythagorean triple is 3, 4, 5. You can find Pythagorean triples by choosing any 2 positive integers, xand y,where x y, and using the formulas ax 2 y 2, b2xy,and cx 2 y 2. 1. Study the example in the table below. Then use the formulas to complete the table. Use the Pythagorean theorem to make sure each triple works. a = 2  xy 2 c = x 2  y 2 b = 2xy 2. Use any patterns you notice in the table above and a trial-and-error strategy to help you find the values of xand ythat generate each triple. a. 21, 20, 29xy b. 27, 36, 45xy Formulas Leg aLeg bLeg c Pythagorean triple xy x 2y 2 2xy x 2y 2 212 2 - 1 2 = 3 2 (2) (1) = 4 2 2 + 1 2 = 5 3, 4, 5 31 32 41 43 54 65

STUDY LINK 9 13 Unit 9 Review 321 Name Date Time Copyright © Wright Group/McGraw-Hill 1. Simplify the following expressions by combining like terms. a. 4x3x b. 3x7 x c. 4 º (x2) 2x6  d. (x3) º 2 2x 2. Liani simplified the expression 8(x10) as (8 º x) 10. What did she do wrong? Explain her mistake and show the correct way to solve the problem. 3. Solve each equation. Show your work on the back of this sheet. a. 3x4 4x6 b. 5 º (2 6) 4g c. 3(2y3) 15 d. (2x 31)  9 4. The perimeter of triangle ABCis 18 inches. What is the length of each side? AB BC AC 5. The perimeter of right triangle GLD is 12 centimeters. What is the area of the triangle? 6. Toshi often walks to school along Main Street and Elm Street. If he were to take Pythagoras Avenue instead, how many fewer blocks would he walk? A B Cx  1x  1 2x G LD x  2 x  1x  3 school home 5 blocks 12 blocks Main St. Elm St. Pythagoras Ave. 7. 28 4 2  8. 161  92  9. 200 1 20  Practice 251 252

LESSON 9 13 Name Date Time An Indirect Measurement Problem 322 Copyright © Wright Group/McGraw-Hill Work with 3 other students. Your teacher has taped a target on the wall. You will use an indirect method to determine the height of the target above the floor. Study the diagram shown below. Each student has a special job. Observer:Sit on the floor and face the target. Sit about 4.5 to 6 meters from the target. Supporter:You and the observer hold a meterstick so it is at the observer’s eye level. Make sure the meterstick is parallel to the floor. Pointer:Take a second meterstick and place the 0 end on top of the end of the meterstick that the supporter is holding. The supporter holds the ends of the sticks together. Make sure to hold the meterstick vertically so angle ACBis approximately a right angle (90°). Observer:Hold the end of the meterstick (point A) near your eye and look at the target (point D). Instruct the pointer to slide a finger up or down the vertical meterstick until the finger appears to point to the target (point D). Record the length of B C. Measurer:Measure the height above the floor of the observer’s meterstick (the height of ACabove the floor). Also measure the distance from the observer’s eye to the wall (the length of AE). A CBD E measurer pointer observer supportertarget

LESSON 9 13 Name Date Time An Indirect Measurement Problem cont. 323 Copyright © Wright Group/McGraw-Hill 1. Record your measurements. AC100 cm AEcm (distance from observer’s eye to wall) BCcm Distance from observer’s eye to floor cm 2. Draw sketches of triangles ACBand AEDthat include your measurement information. 3. Triangles ACBand AEDare similar figures. What is the size-change factor for these figures? Use the size-change factor to calculate the length of D E. DEcm 4. What is the height of the target above the floor?

LESSON 9 13 Name Date Time Proportions and Indirect Measurement 324 Copyright © Wright Group/McGraw-Hill One way to solve an indirect measurement problem is to write a proportion. Study the example below. Example: A road sign casts a shadow that is 4.6 meters long. A stop sign near the road sign casts a shadow that is 3 meters long. The road sign and its shadow form 2 legs of a right triangle that are similar to the 2 legs of a right triangle formed by the stop sign and its shadow. Find the height of the road sign. Use the diagram to write a proportion involving the corresponding sides of the triangles. → → 1 h.8  43 .6 ← ← Use cross products to write an equation. 3h(1.8)(4.6) Solve. 3 3h 8. 328 h2.76 The height of the road sign is 2.76 meters. The triangles in Problems 1 and 2 are similar. Solve each problem by writing and solving a proportion. 1. 2. Height of birdhouse Height of flagpole  length of stop sign’s shadowlength of road sign’s shadow stop sign’s height road sign’s height STOP Ratio Rd. 3 m 1.6 m h 1.8 m 3 m 7 m 6 m h h 5 ft 4 in. 12 ft 27 ft (unit) (unit)

STUDY LINK 9 14 Unit 10: Family Letter 325 Name Date Time Copyright © Wright Group/McGraw-Hill Geometry Topics Unit 10 includes a variety of activities involving some of the more recreational, artistic, and lesser-known aspects of geometry. In Fifth Grade Everyday Mathematics, students explored same-tile tessellations.A tessellation is an arrangement of closed shapes that covers a surface completely, without gaps or overlaps. Your kitchen or bathroom floor may be an example of a tessellation. A regular tessellation involves only one kind of regular polygon. Three examples are shown at the right. In Unit 10 of Sixth Grade Everyday Mathematics, your child will explore semiregular tessellations. A semiregular tessellationis made from two or more kinds of regular polygons. For example, a semiregular tessellation can be made from equilateral triangles and squares as shown below. The angles around every vertex point in a semiregular tessellation must be congruent to the angles around every other vertex point. Notice that at each vertex point in the tessellation above, there are the vertices of three equilateral triangles and two squares, always in the same order. The artist M. C. Escher used transformation geometry—translations, reflections, and rotations of geometric figures—to create intriguing tessellation art. Ask your child to show you the translation tessellation that students created in the style of Escher. Your child will also explore topology. Topology, sometimes called rubber-sheet geometry,is a modern branch of geometry that deals with, among other topics, properties of geometric objects that do not change when the objects’ shapes are changed. Ask your child to share with you some ideas from topology, such as Möbius strips. Please keep this Family Letter for reference as your child works through Unit 10. vertex

regular polygon A polygon in which all sides are the same length and all angles have the same measure. regular tessellation A tessellation of one regular polygon.The three regular tessellations are shown below. rotation symmetry A figure has rotation symmetry if it is the rotation image of itself after less than a full turn around a center or axis of rotation. genus Intopology,the number of holes in a geometric shape. Shapes with the same genus are topologically equivalent. For example, a doughnut and a coffee cup are equivalent because both are genus 1. Möbius strip (Möbius band) A 3-dimensional figure with only one side and one edge, named for the German mathematician August Ferdinand Möbius (1790 –1868). order of rotation symmetry The number of times a rotation image of a figure coincides with the figure before completing a 360° rotation. Copyright © Wright Group/McGraw-Hill 326 Unit 10: Family Letter cont. STUDY LINK 914 Math Tools Your child will use the Geometry Templateto explore and design tessellations. This tool includes a greater variety of shapes than the pattern-block template from previous grades. It might more specifically be called a geometry-and-measurement template. The measuring devices include inch and centimeter scales, a Percent Circle useful for making circle graphs, and a full-circle and a half-circle protractor. Vocabulary Important terms in Unit 10: Genus 0 Genus 1 A figure with order 5 rotation symmetry Sha pes with rotation s ymmetr y The three regular tessellations Regular polygons Möbius strip

topological transformation A transformation that pairs a figure with its image after shrinking, stretching, twisting, bending, or turning inside out. Tearing, breaking, and sticking together are not allowed. Shapes that can be changed into one another by a topological transformation are called “topologically equivalent shapes.” For example, a doughnut is topologically equivalent to a coffee cup. translation tessellation Atessellationmade of a tile in which one or more sides is a translation image of the opposite side(s). Dutch artist M. C. Escher(1898 –1972) created many beautiful and elaborate translation tessellations. vertex point A point where the corners of tessellation tiles meet. 327 Copyright © Wright Group/McGraw-Hill Do-Anytime Activities To work with your child on the concepts taught in this unit, try these interesting and rewarding activities: 1.Familiarize yourself with the definition of regular tessellation(p. 326). Encourage your child to find tessellations in your home, such as floor tile patterns, wallpaper patterns, and wall tile patterns. Have your child identify the shapes that make up the pattern. 2.Encourage your child to use the local library or the Internet to find examples of M. C. Escher’s artwork. 3.If you have art software for your home computer, allow your child time to experiment with computer graphic tessellations. Encourage him or her to share the creations with the class. Unit 10: Family Letter cont. STUDY LINK 914 In Unit 10, your child will reinforce skills and concepts learned throughout the year by playing the following games: Angle TangleSeeStudent Reference Book,page 306 Two players will need a protractor, straightedge, and blank paper to play Angle Tangle.Skills practiced include estimating angle measures as well as measuring angles. Name That NumberSeeStudent Reference Book,page 329 This game provides your child with practice in writing number sentences using order of operations. Two or three players need 1 complete deck of number cards to play Name That Number. Building Skills through Games A translation tessellation

Copyright © Wright Group/McGraw-Hill 328 Study Link 10 1 1.rotation 2.translation 3. Answers vary. 4.Answers vary. 5. 114.534 6.35.488 7. 0.0338 8.31.7025 Study Link 10 2 1. 2. 3. 4. 0.8 5.1.6 6.8.9 7.5.1 Study Link 10 3 1.2 2.1 3.4 4. 6 5.2 6.infinite 7. 2, 3, 5, 6, 9, 10 8.2, 3, 6, Study Link 10 5 Sample answers: 1. The paper clips are linked to one another. 2. The paper clips and the rubberband are linked. 3. All the paper clips are linked. 4. 60 5.50 6.63 7.493 1 2 4 3 5 6 7 8 9 10 0 12345678 910 0 image N  L K  O  M  1 2 4 3 5 6 7 8 9 10 0 12345678 910 0 image G  I J H  F E  1 2 4 3 5 6 7 8 9 10 0 1 2345678 910 0 image C  A B D  Unit 10: Family Letter cont. STUDY LINK 914 As You Help Your Child with Homework As your child brings assignments home, you may want to go over the instr\ uctions together, clarifying them as necessary. The answers listed below will guide you through some of the Unit 10 St\ udy Links.