Geometry. Unit. Essential Questions

Transcription

1 Geometry Unit 5 Unit Overview In this unit you will learn about the perimeter and area of quadrilaterals, circles, and triangles and discover new ideas about the relationships of angles and sides of triangles and quadrilaterals. You will investigate transformations in a new way on a coordinate plane with four quadrants. You will explore the concept of volume and learn ways to calculate the volume of some solids. Academic Vocabulary As you study this unit, add these terms to your vocabulary notebook. altitude bisect congruent equilateral perimeter regular polygon solid transformation volume?? Essential Questions What characteristics do various quadrilaterals share, and why is it possible to determine perimeter and area of quadrilaterals using related formulas? In what ways is symmetry important in real-world situations? EMBEDDED ASSESSMENTS These assessments, following Activities 5.4 and 5.8, will give you an opportunity to demonstrate your ability to find the perimeter and area of composite figures made up of triangles, circles, and quadrilaterals and to classify triangles and quadrilaterals, to perform transformations on a coordinate grid, and to find the volume of a solid. Embedded Assessment 1 Area and Perimeter p. 271 Embedded Assessment 2 Polygons, Trans formations, and Geometry p

2 UNIT 5 Getting Ready Write your answers on notebook paper. Show your work. 1. Name each shape. a. 5. Copy each figure. Draw any lines of symmetry. Tell how many lines of symmetry. a. b. b. c. c. 2. Compare and contrast the three shapes in Question Explain what it means to slide a figure to the right. 7. Draw this figure after it has made a quarter turn counterclockwise. 3. Define or describe an angle. 4. Use a protractor to find the measure of each angle. a. b. c. 8. Write the letter of the figure that shows this figure after a flip from left to right. a. b. c. d. 242 SpringBoard Mathematics with Meaning TM Level 1

3 Area and Perimeter The Dot Game SUGGESTED LEARNING STRATEGIES: Summarize/ Paraphrase/Retell, Think/Pair/Share, Use Manipulatives Have you ever played The Dot Game? Here is your chance to play a game during math class. Have fun. Game Board The game is played on a rectangular grid of dots. Object of the Game Each player tries to create as many squares as possible that have sides one unit long by drawing line segments that connect the dots on the game board. Playing the Game Two players are needed. (Find a partner in your class.) Decide who will have the first turn. Players take turns connecting two dots with a horizontal or vertical line segment that is one unit long. If a player completes a square, that player places his or her initial in the square and continues to play by drawing another line segment. A player s turn ends when he or she draws a line segment that does not complete a square. Play continues until all the dots are connected by line segments. The game board will be filled with squares, each containing the initial of the player who completed it. Game Board 1 Game Board 2 ACTIVITY 5.1 Winning the Game Count the squares for each player. The player who has completed the greatest number of squares is the winner. 1. Play the game two times with your partner. Use the game boards in the space. Tell who wins each game. 2. In each game, how many squares did you mark? Game 1 Game 2 3. In each game, how many squares did your partner mark? Game 1 Game 2 Unit 5 Geometry 243

4 ACTIVITY 5.1 Area and Perimeter The Dot Game SUGGESTED LEARNING STRATEGIES: Guess and Check, Quickwrite, Create Representations, Self Revision/Peer Revision 4. What is the combined number of squares for both players in each game? Game 1 Game 2 5. How many small squares make up the large square on a game board? 6. What is the area of a small square? 7. What is the area of the large square? MATH TERMS Adjacent squares must have at least one side and two vertices in common. Vertices is the plural of vertex. Game Board 8. Look at Game Board 1 from the first two games and identify the largest shape made with your initialed adjacent squares. Draw that shape in the space. Be sure to draw all the small squares. 9. What is the area of the shape above? How did you determine the area of the shape? 10. Rewrite the part of the directions labeled Winning the Game using the concept of area. 11. Look at the shape you drew for Question 8. Draw the shape again on the game board in the space, but do not draw any interior line segments. 244 SpringBoard Mathematics with Meaning TM Level 1

5 Area and Perimeter The Dot Game ACTIVITY 5.1 SUGGESTED LEARNING STRATEGIES: Quickwrite, Close Reading, Create Representations 1 2. How many one-unit line segments are there in the perimeter of the figure that you drew for Question 11? 13. Compare your answer for Question 1 2 to your partner s answer. Did the person who had the most squares on game board 1 also have the most line segments in Question 11? ACADEMIC VOCABULARY Perimeter is the distance around a figure. Game Board 1 Now play the game again. The rules for playing this time are the same, but the winner will be decided differently. Instead of the winner completing the greatest number of squares, the winner in this game will have completed adjacent squares that form a figure that has the greatest perimeter. 14. Betty and Andy played The Dot Game on a 4 cm 5 cm grid. Their results are shown below. Each segment drawn is 1 cm in length and each square has an area of 1 cm 2. A A A A A B B B B A B B B B A READING MATH 1 cm 2 is read as one square centimeter. Sometimes you will see 1 sq cm used for 1 cm 2. B B B B A Betty (B) claims she won, while Andy (A) claims he won. How can both be correct? Justify your decision. 15. Play the game twice with your partner. Remember, the player whose squares make up a shape made of adjacent squares that has the greatest perimeter is the winner. Game Board How did you determine the perimeter of your figure? Show your work. Unit 5 Geometry 245

6 ACTIVITY 5.1 Area and Perimeter The Dot Game SUGGESTED LEARNING STRATEGIES: Quickwrite, Group Presentation, Create Representations, Think/Pair/Share 17. For Betty and Andy s Dot Game, draw and describe the different units of measure used to decide who won the first round of The Dot Game and who won the second round. 18. Make a figure that has the same area as Betty s figure, but with a different perimeter. Use the grid in the space. You may also use square tiles to form a figure and then make a drawing of it in the space. 19. Use the grid in the space to draw the rectangles. Recall that opposite sides of a rectangle have the same measures. a. Draw a rectangle that is formed with 16 squares. What are the perimeter and the area of the rectangle? Include units in your answer and draw a diagram of your rectangle. b. Make a different rectangle with the same area but a different perimeter. Draw a diagram in the space. c. Can you make a different rectangle with the same perimeter as the rectangle in Part a, but with a different area? If so, draw a diagram in the space. 20. Write a rule, in words, to determine the area of a rectangle. 21. If a rectangle has length l, width w, and area A, write an equation that relates all three variables. 246 SpringBoard Mathematics with Meaning TM Level 1

7 Area and Perimeter The Dot Game ACTIVITY 5.1 SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Group Presentation 2 2. Write a rule, in words, to determine the perimeter of a rectangle. 23. If a rectangle has length l, width w, and perimeter P, write an equation that relates all three variables. 24. The drawing below is a composite figure. 2.3 cm 5.2 cm MATH TERMS A composite figure is made up of two or more figures. 3.9 cm 4.1 cm a. Find the measure of each side that is not labeled. b. Find the area of the composite figure. Show your work. c. Find the perimeter of the composite figure. Show your work. 25. Find the area and perimeter of the composite figure below. Show your work. 3 ft 5 ft 6 ft 9 ft 3 ft 2 ft Unit 5 Geometry 247

8 ACTIVITY 5.1 Area and Perimeter The Dot Game CHECK YOUR UNDERSTANDING Write your answers on notebook paper. Show your work. 1. Colleen and David played The Dot Game I. Their game card is shown below. Who won? Explain how you know. C D D C D D C C D D D D C C D C C D D C 4. Make a rectangle that is formed using 24 squares and has a perimeter of Find the area and perimeter of a rectangle that has a length of 23 meters and a width of 17 meters. Show your calculations. 6. Find the area and perimeter of the figure below Sammi and John played the Dot Game II. Their game card is shown below. Who won? Explain how you know S J J J J J 8 S J J S S S S S S J 3. Marcy found the area and perimeter of this rectangle. Label the area and perimeter with the appropriate units. 3 m 4 m A = 12 P = 14 J J 7. MATHEMATICAL REFLECTION What are the dimensions of a rectangle that has a perimeter of 16 cm and has the greatest possible area? 248 SpringBoard Mathematics with Meaning TM Level 1

9 Investigating π Going In Circles SUGGESTED LEARNING STRATEGIES: Marking the Text, Visualize, Quickwrite, Use Manipulatives Maria has a circular garden that she wants to enclose with a decorative fence. She knows that she must find the distance around the garden, but is not sure which measuring tool she will need. In this activity, you will investigate a method for finding the distance around Maria s garden. A circle is the set of points in the same plane that are an equal distance from a given point, called the center. The distance around a circle is called the circumference. 1. Draw and label a diameter and a radius in the circle. ACTIVITY MATH TERMS A plane can be thought of as a flat surface that extends in all directions. A parallelogram is usually used to model a plane What is the relationship between the length of a diameter and the length of a radius of a circle? There is also a relationship between the circumference and the diameter of a circle. Your teacher will give you some material to measure the circumference and diameter of several circles. 3. Use the table to record your data. Then calculate the ratios. Circumference Diameter First Circle Second Circle Third Circle Fourth Circle Fifth Circle MATH TERMS A line segment through the center of a circle with both endpoints on the circle is called the diameter. A line segment with one endpoint on the center and the other on the circle is called the radius. Ratio of circumference to diameter (as a fraction) Ratio of circumference to diameter (as a decimal) Unit 5 Geometry 249

10 ACTIVITY 5.2 Investigating π Going In Circles SUGGESTED LEARNING STRATEGIES: Think/Pair/ Share, Look for a Pattern, Create Representations, Group Presentation, Identify a Subtask The ratio of the circumference to the diameter of a circle is called pi. We use the Greek letter π to represent pi. 4. Use your table of approximations to give a good estimate of the number π. Describe the method you used. READING MATH The symbol means approximately equal to. 5. π 3.14 is a commonly used approximation for pi. Which measurement tools used by your class gave the most accurate approximation of pi? Why do you think this is true? 6. Write the equation that relates π to the circumference and diameter of a circle and relates the circumference, C, to the diameter, d, and the number π. 7. Write an equation that relates the circumference, C, to the radius, r, and the number π. 8. Should the circumference of a circle be labeled with units or square units? Explain your decision. 9. Find the amount of decorative fencing that Maria needs to enclose her garden that has a diameter of 6 feet. Show your work. 250 SpringBoard Mathematics with Meaning TM Level 1

11 Investigating π Going In Circles ACTIVITY 5.2 SUGGESTED LEARNING STRATEGIES: Simplify the Problem, Create Representations, Think/Pair/Share, Work Backward 10. Decorative fencing is sold in packages of 12-foot sections. How many packages must Maria buy? Sometimes 22 is used as an approximation of π Why is this fraction a good approximation of pi? 12. Find the circumference of a circular dog pen that has a radius of 35 meters. Use π Find the circumference of each circle. Use estimation to check whether your answer is reasonable. a. b. 14 cm 3 in. 14. These steps show how to find the diameter of a circle that has a circumference of 40π cm. Explain each step. Step Explanation a. C = π d b. 40 π cm = π d c. 40π cm π = πd π d. 40 cm = d Unit 5 Geometry 251

12 ACTIVITY 5.2 Investigating π Going In Circles SUGGESTED LEARNING STRATEGIES: Simplify the Problem, Create Representations, Think/Pair/Share, Work Backward 15. Find the diameter of a circle if C = 32π in. 16. Find the radius of a circle if C = 17π cm. CHECK YOUR UNDERSTANDING Write your answers on notebook paper. Show your work. Answer each problem, and then use estimation to see if your answer is reasonable. Find the circumference in. 6 mm 3. The diameter of a pizza is 14 inches. What is the circumference of the pizza? 4. The radius of a circular mirror is 4 cm. What is the circumference of the mirror? 5. The radius of a circular garden is 28 ft. What is the circumference? 6. Find the diameter of a circle if C = 25 π ft. 7. Find the radius of a circle if C = 38 π m. 8. MATHEMATICAL REFLECTION Does the size of the diameter of a circle determine the accuracy of π? 252 SpringBoard Mathematics with Meaning TM Level 1

13 Area of Polygons and Circles Play Area SUGGESTED LEARNING STRATEGIES: Think/Pair/Share ACTIVITY 5.3 Pictured below is an aerial view of a playground. An aerial view is the view from above something. Decide what piece of playground equipment each figure below represents. A B C D E G 1. Look at the shape of each figure, and write the name of the playground equipment next to each letter. F A. B. C. D. E. F. G. Unit 5 Geometry 253

14 ACTIVITY 5.3 Area of Polygons and Circles Play Area SUGGESTED LEARNING STRATEGIES: Quickwrite, Self Revision/Peer Revision, Group Presentation To plan the layout of a playground, a designer must know how much area each piece of playground equipment takes up. 2. The aerial view of the playground contains many polygons. a. What is a polygon? b. Is a circle a polygon? Explain your reasoning. 3. Complete the table by listing all the geometric shapes you can identify in each figure in the aerial view of the playground. Figure A B C D E F G Geometric Shape(s) 4. Explain how you would find the area of Figure E. 5. Now consider the parallelogram in the aerial view of the playground that is not also a rectangle. List some characteristics of a parallelogram. Page 255 contains shapes you will work with in this activity. Cut each one out as you start the question that uses that shape. 254 SpringBoard Mathematics with Meaning TM Level 1

15 Area of Polygons and Circles Play Area ACTIVITY 5.3 Two Congruent Parallelograms (Cut these out when you start Question 6.) Two Congruent Triangles (Cut these out when you start Question 10.) Two Congruent Trapezoids (Cut these out when you start Question 13.) Circle (Cut this out when you start Question 19.) Unit 5 Geometry 255

16 This page is blank.

17 Area of Polygons and Circles Play Area ACTIVITY 5.3 SUGGESTED LEARNING STRATEGIES: Use Manipulatives, Look for a Pattern, Self Revision/Peer Revision, Debriefing 6. Cut out one of the two congruent parallelograms on page 255. Then cut that parallelogram once in such a way that the two pieces can be put together to form a rectangle. a. Use a ruler to measure the rectangle and find its area. ACADEMIC VOCABULARY Two or more figures that have the same shape and size are congruent. b. Sketch the rectangle and record your measurements in the space. 7. Explain how the lengths of the base and height of the rectangle you formed relate to those of the original parallelogram. (Use the second parallelogram to compare.) 8. Find a relationship between the base, height, and area of a parallelogram. Describe that relationship using words, symbols, or both. 9. The hexagon in the aerial view of the playground is made up of triangles and pentagons. List some characteristics of each figure. a. hexagon The height of a figure is always drawn perpendicular to its base. Perpendicular lines ( ) meet to form right angles. b. triangle c. pentagon Unit 5 Geometry 257

18 ACTIVITY 5.3 Area of Polygons and Circles Play Area ACADEMIC VOCABULARY The altitude of a triangle is a perpendicular line segment from a vertex to the line containing the opposite side. The measure of an altitude is height. SUGGESTED LEARNING STRATEGIES: Use Manipulatives, Create Representations, Quickwrite 10. Find the congruent triangles on page 255. Cut out one of the two triangles. Label one of its sides b. Draw the altitude of the triangle by drawing a line segment perpendicular to side b. Label the segment h. Cut out the second triangle. Place the two triangles together to form a parallelogram whose base is the side you labeled b. 11. How does the area of each triangle compare to the area of the parallelogram from Question 10? Explain below. 12. Using words, symbols, or both, describe a method for finding the area of a triangle. MATH TERMS A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides of a trapezoid are called the bases. The two sides that are not parallel are called the legs. READING MATH Sometimes subscripts are used to label segments. b 1 is read as b sub 1 and represents one base of the trapezoid. b 2 is read as b sub 2 and represents a second base of the trapezoid. Another shape seen in the aerial view of the playground looks like the figure at right. This figure is called a trapezoid. 13. Find the congruent trapezoids on page 255. Cut out the two congruent trapezoids. b On the inside of each figure label the 1 bases as b 1 and b 2 as shown at right. b 2 Draw in the height of the trapezoid and label it h. Form a parallelogram by turning one of the trapezoids so that its short base lines up with the long base of the other trapezoid. The long legs of the trapezoids will be adjacent. 258 SpringBoard Mathematics with Meaning TM Level 1

19 Area of Polygons and Circles Play Area ACTIVITY 5.3 SUGGESTED LEARNING STRATEGIES: Use Manipulatives, Group Presentation, Self Revision/Peer Revision, Quickwrite 14. What is the height of the parallelogram? How does it compare to the height of the original trapezoid? 15. What is the length of the base of the parallelogram? How does it compare to the base of the trapezoid? 16. What is the area of the parallelogram? 17. What is the area of one of the trapezoids used to form the parallelogram? 18. A pentagon is another polygon in the aerial view of the playground. Use what you have learned about finding the area of rectangles, triangles, parallelograms, and trapezoids to describe how to find the area of this pentagon. The last shape found in the aerial view of the playground on the first page of this activity is a circle. Use the circle on page 255 to complete the following questions. 19. Cut the circle into eight congruent pie-shaped pieces. Arrange your eight pieces using the alternating pattern shown at right. Unit 5 Geometry 259

20 ACTIVITY 5.3 Area of Polygons and Circles Play Area SUGGESTED LEARNING STRATEGIES: Look for a Pattern, Guess and Check, Group Presentation, Debriefing 20. Sketch the shape you just made with the circle pieces. What shape does it resemble? 21. In your sketch, draw and label the height of the figure. What part of the circle does the height represent? 22. What other part of the circle is about the length needed to find the area of the shape you named in Question 21? 23. Using words, symbols, or both, describe how you can now find the area of the circle. Right angles are often identified with a small square in the corner of the angle. 24. The dimensions of some of the pieces of playground equipment are shown with their drawings below. Find the area of each figure. Explain how you found each area. a. Figure E 2 feet 10 feet b. Figure G 2 feet 4 feet 260 SpringBoard Mathematics with Meaning TM Level 1

21 Area of Polygons and Circles Play Area ACTIVITY 5.3 SUGGESTED LEARNING STRATEGIES: Group Presentation c. Figure A 1 ft 0.5 ft 1 ft 1 ft 2 ft 2 ft 8 ft d. Figure F 3.46 ft 1 ft 2 ft 2 ft 3 ft 2 ft 2 ft 3.46 ft This aerial view is composed of three triangles and three pentagons. Each of the outside segments measures 3.46 feet while each of the inside segments measures 2 feet. e. Figure B Recall that we can approximate π as either 3.14 or ft Unit 5 Geometry 261

22 ACTIVITY 5.3 Area of Polygons and Circles Play Area SUGGESTED LEARNING STRATEGIES: Think/Pair/Share 25. Based on the dimensions given for the other figures and the location of Figures C and D on the playground, make an estimate of the area of Figures C and D. Explain how you arrived at your estimate. CHECK YOUR UNDERSTANDING Write your answers on notebook paper. Show your work. Find the area of each figure. Remember to label your answer in. 5. Use π 22 to find the area of the circle Find the area. 14 in. 7 in in. 7.8 in. 15 cm 12 cm 3. Draw the figure, and then find the area of a triangle with a base that measures 8.3 cm and a height that measures 7.2 cm. 4. Find the area. 12 cm 8 cm 15.4 cm 7. Mikel is going to build a doghouse for his new puppy. The floor s shape is shown below. Find the area of the doghouse floor. 1.5 m 8. Draw a circle with a radius of 2.3 cm, and then find its area. 9. MATHEMATICAL REFLECTION 14.3 in. 2 m 1.7 m How does knowing the area of a rectangle help you find the areas of other figures? Explain. 262 SpringBoard Mathematics with Meaning TM Level 1

23 Area and Perimeter of Composite Figures Putting Back the Pieces SUGGESTED LEARNING STRATEGIES: Use Manipulatives ACTIVITY 5.4 According to Chinese legend, a man dropped a porcelain tile. It broke into the seven pieces you see below. They are called tangram pieces. While he was trying to reassemble the seven pieces into a square, he found that he could make hundreds of different shapes. 1. Cut out the seven pieces or use a tangram set. Assemble them into the original square tile. Unit 5 Geometry 263

24 This page is blank. 264 Springboard Mathematics with Meaning Level 1

25 Area and Perimeter of Composite Figures Putting Back the Pieces ACTIVITY 5.4 SUGGESTED LEARNING STRATEGIES: Quickwrite, Group Presentation, Think/Pair/Share, Look for a Pattern 2. Use a ruler to measure the square you made from the seven tangram pieces. a. What is the area of the square? Use the space to show your calculations. b. What is the perimeter of the square? Show your calculations. 3. At right is another shape that can be created using all seven tangram pieces. The shape is not drawn to scale. a. Reassemble your pieces into this shape, or create a shape of your own. b. What is the area of the shape you just created? Compare the area to the area you found for the square in Question 2. c. What is the perimeter of the shape you just created? Compare the perimeter to the perimeter you found for the square in Question What can be said about the areas and perimeters of different shapes that can be formed using all seven tangram pieces? Unit 5 Geometry 265

26 ACTIVITY 5.4 Area and Perimeter of Composite Figures Putting Back the Pieces SUGGESTED LEARNING STRATEGIES: Identify a Subtask, Create Representations, Group Presentation 5. Divide the composite figure below into shapes whose area you know how to find. Use the grid lines as a guide. CONNECT TO AP In AP Calculus, an important problem is finding the area under a curve. You can count squares on a coordinate plane to do this. y 5 a. Determine the area of each shape you made. Show your work You will also learn other procedures that result in much more precise or even exact calculations of the area of irregular figures when you take calculus. x b. Give the total area of the composite figure. 6. Compare and contrast your solution with your classmates. 266 Springboard Mathematics with Meaning Level 1

27 Area and Perimeter of Composite Figures Putting Back the Pieces ACTIVITY 5.4 SUGGESTED LEARNING STRATEGIES: Marking the Text, Quickwrite, Use Manipulatives Tyrone loves to go fishing with his mother. They often fish at Big Trout Pond in a nearby state park. One day after Tyrone caught several fish, he was concerned that the pond would run out of fish. His mother explained that the park rangers routinely restock the pond. Tyrone wanted to know how the rangers know how many fish to add to the pond. Tyrone did some research and learned that the number of fish that a pond can support depends in part on the surface area of the pond. 7. Look at the scale drawing of the pond below. a. Are you able to divide the figure into shapes whose area you know how to find? Explain why or why not. b. Carefully cut out the scale drawing of the pond. Then trace it on the grid on page 269. Big Trout Pond Unit 5 Geometry 267

28 This page is blank. 268 Springboard Mathematics with Meaning Level 1

29 Area and Perimeter of Composite Figures Putting Back the Pieces ACTIVITY 5.4 SUGGESTED LEARNING STRATEGIES: Quickwrite, Simplify the Problem, Think/Pair/Share, Debriefing, Group Presentation c. Shade every square that is completely within the perimeter of the tracing of the scale drawing of Big Trout Pond. Count the shaded squares and record the number. d. How does the area of the shaded squares compare to the area of the figure? e. Shade every square that contains some piece of the perimeter of the figure. How many squares did you shade in this part? Add the number of squares from Part c and this part and record the number. f. How does the total area of the shaded squares now compare to the area of the figure? 8. Estimate the area of the figure that represents the pond. Describe how you arrived at your answer Unit 5 Geometry 269

30 ACTIVITY 5.4 Area and Perimeter of Composite Figures Putting Back the Pieces SUGGESTED LEARNING STRATEGIES: Group Presentation, Discussion Group, Self Revision/Peer Revision 9. Let each square of the graph paper represent 100 square meters. Estimate the area of the pond. Show your calculations. 10. How can you make a more accurate estimate of the size of the pond? 11. Devise a method to find the perimeter of the pond. Use your method to estimate the perimeter of the pond. CHECK YOUR UNDERSTANDING Write your answers on notebook paper. Show your work. 1. Create a shape using 3 or 4 of your tangrams. Draw its outline and label its dimensions. 2. What is the area of the shape? Show your work. 3. What is the perimeter of the shape? Show your work. 4. Divide the figure below into shapes whose area you know how to find. Determine the area of the figure. Show your work. 5. Find the area and perimeter of the following figure. 8 ft 2 ft 12 ft 4 ft 3 ft 7 ft 6. Find the approximate area of the shape. Each square on the graph paper represents 2 ft MATHEMATICAL REFLECTION When might you need to know how to find the area of irregular shapes? Explain and give examples. 270 Springboard Mathematics with Meaning Level 1

31 Area and Perimeter DESIGNING A CLUBHOUSE Embedded Assessment 1 Use after Activity 5.4. Write your answers on notebook paper. Show your work. The students at Bailey Middle School participate in a community service project every year. This year they have decided to build a clubhouse to serve as a meeting place at the local elementary school. Interested students were asked to submit designs for the clubhouse. The two favorite designs are shown below. The floor plan for Design 1 includes a regular octagon with sides measuring four meters and four congruent triangles. The floor plan for Design 2 includes a rectangle, a trapezoid, a parallelogram, and a semicircle. 6 m 10 m 4 m 7.28 m 4 m 5 m 7 m 6 m 3 m 8 m 7 m 6 m 7 m 8 m 3 m 1. Find the areas of the floors for both designs. Explain your thinking by giving formulas and showing your work. 2. Find the perimeters of the floors for both designs. Explain your thinking by giving formulas and showing your work. 3. Compare the area and perimeter of the two designs. a. Which design has the greater area? Explain. b. Which design has the greater perimeter? Explain. 4. Which design would you recommend the students use for the clubhouse? Use mathematical reasons to support your decision. Unit 5 Geometry 271

32 Embedded Assessment 1 Use after Activity 5.4. Area and Perimeter DESIGNING A CLUBHOUSE Exemplary Proficient Emerging Math Knowledge #1, 2, 3 Communication #1, 2, 4 The student: Calculates the areas (1) and perimeters (2) of both designs correctly. Compares the total areas and total perimeters accurately (3). The student: Gives formulas or explains thinking for calculating area (1) and perimeter (2), and shows the calculations involved. Chooses a design and gives a justification to support the decision (4). The student: Uses accurate methods to calculate areas and perimeters, but one calculation may contain minor errors Compares the total perimeters and total areas found accurately. The student: Shows calculations for area and perimeter but does not explain the thinking used or provide formulas OR Provides formulas for area and perimeter or explanations of process but shows no work. Chooses a design and gives no justification. Student calculations are inaccurate and comparisons may or may not be accurate. The student attempts all three questions but only one response is complete and correct. 272 SpringBoard Mathematics with Meaning TM Level 1

33 Properties of Triangles Triangle Trivia LEARNING STRATEGIES: Close Reading, Think Aloud, Marking the Text, Summarize/Paraphrase/Retell, Look for Pattern, Use Manipulatives Mr. Javarra asked his students to make up some math games involving facts about triangles. Katie and Allie suggested the following game. Players: Materials: Directions: Triangle Trivia Rules Properties of Triangles Perimeter Variation Three to four students Three number cubes and a segment pieces set of three each of the following lengths: 1 inch, 2 inches, 3 inches, 4 inches, 5 inches, and 6 inches. Take turns. Roll the three number cubes. Find a segment piece to match each number rolled. See whether a triangle can be formed from those segment pieces. The value of the perimeter of any triangle that can be formed is added to that player s score. The first player to reach 50 points wins. Amir said he thought that Katie and Allie s game had nothing to do with triangles because all they did was find the sum of the numbers rolled and then add that to their score. Katie and Allie told Amir that there was more to their game than he thought. ACTIVITY Play Katie and Allie s game to see if what they told Amir is true. Follow the rules above. Record your results in the table. Player 1 Player 2 Player 3 Player 4 Numbers Score Numbers Score Numbers Score Numbers Score Unit 5 Geometry 273

34 ACTIVITY 5.5 Properties of Triangles Triangle Trivia LEARNING STRATEGIES: Quickwrite, Think/Pair/Share, Look for a Pattern 2. Is there more to the game than adding the three numbers on the cubes and then adding that total to your score? Explain below. Amir found that he did not actually need to use the segment pieces to tell whether a triangle could be formed. 3. Explain how Amir could determine whether a triangle can be formed from three given lengths. 4. Katie and Allie s game illustrates what is known as the Triangle Inequality Property. Use this property to determine whether a triangle can be formed with the given side lengths listed in inches. Show your work or explain. a. a = 8, b = 6, c = 4 b. a = 3, b = 4, c = 7 c. a = 5, b = 5, c = 5 d. a = 3, b = 3, c = 7 e. a = 7, b = 4, c = 4 f. a = 8, b = 4, c = 5 g. a = 1, b = 2, c = 8 Amir had an idea that he thought would make Katie and Allie s game more interesting. He calls his idea the Name My Sides variation. See the rules at the top of the next page. 274 SpringBoard Mathematics with Meaning TM Level 1

35 Properties of Triangles Triangle Trivia ACTIVITY 5.5 LEARNING STRATEGIES: Close Reading, Think Aloud, Marking the Text, Summarize/Paraphrase/Retell Triangle Trivia Rules Name My Sides Variation Number of Players: Three to four students Equipment needed: Three number cubes. Directions: Take turns rolling three number cubes. If you can, form a scalene triangle...add 5 points an isoceles triangle...add 10 points an equilateral triangle...add 15 points no triangle...add 0 points If you are caught making a mistake, deduct 10 points from your last correct score. The first player to reach 25 points wins. ACADEMIC VOCABULARY Equilateral means that all sides of a figure are equal in length. Katie said she was not sure if she knew what scalene, isosceles, and equilateral meant. Amir showed her the following examples of each type of triangle. The sets of matching tick marks, such as and, show that the marked sides are congruent. Scalene Triangles Isosceles Triangles Equilateral Triangles Unit 5 Geometry 275

36 ACTIVITY 5.5 Properties of Triangles Triangle Trivia LEARNING STRATEGIES: Quickwrite 5. Based on the examples that Amir showed Katie, describe each type of triangle. a. Scalene triangle b. Isosceles triangle c. Equilateral triangle 6. When playing Amir s Name My Sides variation of Triangle Trivia, suppose that your cubes landed on the following numbers. Tell how many points you would add to your score and why. a. 5, 5, 5 b. 1, 6, 4 c. 3, 2, 4 d. 6, 6, 4 e. 1, 4, 1 7. Play the Name My Sides variation of Triangle Trivia. Use the table at the top of the next page to record your results. 276 SpringBoard Mathematics with Meaning TM Level 1

37 Properties of Triangles Triangle Trivia ACTIVITY 5.5 LEARNING STRATEGIES: Create Representations, Use Manipulatives Player 1 Player 2 Player 3 Player 4 Numbers Score Numbers Score Numbers Score Numbers Score Another way to classify triangles is by their angles. To do this, you need to know whether an angle is acute, obtuse, or right. A right angle has an angle measure of 90. The angle measure of an acute angle is less than 90 and the angle measure of an obtuse angle is greater than Identify each of these angles as right, acute, or obtuse. Unit 5 Geometry 277

38 ACTIVITY 5.5 Properties of Triangles Triangle Trivia LEARNING STRATEGIES: Look for a Pattern, Quickwrite, Create Representations Now that you can identify angles as acute, right, and obtuse, you can classify triangles by their angles. Look at these examples and think about how each kind of triangle is related to its angles. Acute Triangles Obtuse Triangles Right Triangles 9. Describe each type of triangle. a. Acute triangle b. Obtuse triangle. c. Right triangle. 10. A triangle that has been labeled as acute, obtuse, or right can also be labeled as scalene, isosceles, or equilateral. a. Label each triangle at the top of this page as scalene, isosceles, or equilateral. b. Choose a triangle from the table and explain how the two labels provide a better description of the triangle than either one alone. 278 SpringBoard Mathematics with Meaning TM Level 1

39 Properties of Triangles Triangle Trivia ACTIVITY 5.5 LEARNING STRATEGIES: Look for a Pattern, Quickwrite, Create Representations 11. Sketch a triangle described by each pair of words below or state that it is not possible. Use tick marks and right angle symbols where appropriate. If it is not possible to sketch a triangle, explain why not. Scalene, right Isosceles, right Equilateral, right Scalene acute Isosceles, acute Equilateral, acute Scalene, obtuse Isosceles, obtuse Equilateral, obtuse Unit 5 Geometry 279

40 ACTIVITY 5.5 Properties of Triangles Triangle Trivia LEARNING STRATEGIES: Look for a Pattern, Quickwrite Amir wondered if he could design a variation of Triangle Trivia based on the measures of the angles of a triangle. He decided he would first investigate the sum of the measures of the angles of a triangle. He measured the angles of some scalene, isosceles, and equilateral triangles and recorded his results. Scalene Triangles Isosceles Triangles Equilateral Triangles MATH TERMS A conjecture is a statement that seems to be true but has not been proven to be either true or false 12. Amir made some conjectures about triangles. Determine whether each conjecture below is always true, sometimes true, or never true. Explain why you chose each answer. a. The acute angles of an isosceles triangle are complementary. b. The three angles of any triangle have a sum of 180 degrees. c. An isosceles triangle can have three equal sides. d. An equilateral triangle can have a right angle. e. The largest angle of a scalene triangle can be opposite the smallest side. 280 SpringBoard Mathematics with Meaning TM Level 1

41 Properties of Triangles Triangle Trivia ACTIVITY 5.5 LEARNING STRATEGIES: Close Reading, Think Aloud, Marking the Text, Summarize/Paraphrase/Retell, Quickwrite Amir used what he learned about angle relationships in triangles to write a variation of Triangle Trivia. He called it the Third Angle Variation. Use his directions to answer Question 13. Triangle Trivia Properties of Triangles Third Angle Variation Directions Shuffle the cards and place them facedown. Draw two cards. The number on each card is the measure of an angle of a triangle. Find the third angle measure. If it is equal to: each of the other two, add 3 times the third angle measure to your score. one of the other two angles, add 2 times the third angle measure to your score. a right angle, subtract 90 from your score. neither of the other two and is not a right angle, add the third angle measure to your score. The first player to reach 300 points wins. 13. When playing Amir s Third Angle variation of Triangle Trivia, suppose you drew cards with the following numbers on different turns. Tell how many points you would add to your score each time and why. a. 43, 94 b. 38, 52 c. 57, 39 d. 140, 12 e. 60, 60 Unit 5 Geometry 281

42 ACTIVITY 5.5 Properties of Triangles Triangle Trivia LEARNING STRATEGIES: Close Reading, Think Aloud, Marking the Text, Summarize/Paraphrase/Retell, Quickwrite Another student in Mr. Javarra s class invented this variation. Play this game in groups of three or four. Players: Materials: Goal: Directions: Triangle Trivia Rules Properties of Triangles Triangle Trio Game Three to four students One set of Triangle Trio cards (24 cards on pages 285 and 287). All sides of equal length and all right angles are marked on the cards. Be the first player to make two sets. A set is three cards whose triangles have the same classification either by sides or angles. For example, three acute triangles form a set or three equilateral triangles form a set. A card may be used only once to form a set. Deal all the cards face down so that each player has an equal number of cards. Players pick up their cards. If any player can make two sets of three cards, that player wins the round. If not, each player chooses one of their cards to pass to the player on their left. The players continue to try to make two sets of three cards to win the round. Play continues in this manner until someone wins the round. Use the answer sheet to verify that the winner has two correct sets. 14. Draw at least two examples of possible winning sets. 15. Explain what strategy you used to try to win the game. 282 SpringBoard Mathematics with Meaning TM Level 1

43 Properties of Triangles Triangle Trivia ACTIVITY 5.5 CHECK YOUR UNDERSTANDING Write your answers on notebook paper. Show your work. 3. You are given two of the angles of a Show your work. triangle. Find the third angle and use as many of the following words as possible to describe the triangle. 1. Use the Triangle Inequality Property to determine whether a triangle can be formed with the given length sides in centimeters. Show your work or explain. a. a = 4 b = 5 c = 9 b. a = 2 b = 2 c = 5 c. a = 6 b = 3 c = 8 d. a = 3 b = 5 c = 5 2. Draw a triangle described by each pair of words below or state that it is not possible. If it is not possible, explain why not. a. Scalene, obtuse b. Acute, isosceles c. Obtuse, equilateral (scalene, isosceles, equilateral, acute, obtuse, right) a. 32 ; 58 b. 162 ; 9 c. 60 ; Read the following conjecture and determine whether it is always true, sometimes true, or never true. Explain your reasoning. The side of an isosceles triangle between two equal sides is longer than the other two sides. 5. MATHEMATICAL REFLECTION The games in this activity were designed to help you better understand the relationship between the sides and angles of different kinds of triangles. Explain how triangles are classified using angle measures and side lengths, and give two examples. Unit 5 Geometry 283

44 ACTIVITY 5.5 Properties of Triangles Triangle Trivia Triangle Trio Game Card Descriptions Card Triangle Classification 1 equilateral acute 2 equilateral acute 3 equilateral acute 4 equilateral acute 5 isosceles acute 6 isosceles acute 7 scalene right 8 scalene right 9 isosceles right 10 isosceles obtuse 11 isosceles obtuse 12 isosceles right 13 scalene obtuse 14 scalene acute 15 scalene acute 16 scalene obtuse 17 scalene acute 18 scalene obtuse 19 scalene obtuse 20 isosceles acute 21 scalene obtuse 22 isosceles obtuse 23 scalene right 24 scalene obtuse 284 SpringBoard Mathematics with Meaning TM Level 1

45 Triangle Trio Cards, Set 1 Carefully cut out these cards for the Triangle Trio game Unit 5 Geometry 285

46 This page is blank. 286 SpringBoard Mathematics with Meaning TM Level 1

47 Triangle Trio Cards, Set 2 Carefully cut out these cards for the Triangle Trio game Unit 5 Geometry 287

48 This page is blank. 288 SpringBoard Mathematics with Meaning TM Level 1

49 Properties of Quadrilaterals The Sagging Gate SUGGESTED LEARNING STRATEGIES: Close Reading, Think Aloud Gabrielle always wanted a horse. When she was old enough to take care of a horse, her father gave her one. Her father was very good with tools, so with Gabrielle s help he designed and built the stable and fence. The fence included a large rectangular gate. ACTIVITY 5.6 The climate where Gabrielle lives is very rainy in the spring and cold and snowy in the winter. One year after Gabrielle and her father built the stable and fenced in the paddock, they noticed that the gate was sagging. Gabrielle realized that the shape of the gate had changed. She drew diagrams to represent the gate. Quadrilateral 1 represents the gate when it was new, and Quadrilateral 2 represents the gate one year later. M A C O E R T Quadrilateral 1 Quadrilateral 2 L Unit 5 Geometry 289

50 ACTIVITY 5.6 Properties of Quadrilaterals The Sagging Gate SUGGESTED LEARNING STRATEGIES: Use Manipulatives 1. Find the length of each side of Quadrilateral 1. Measure to the nearest 1 4 in. M A E Quadrilateral 1 R READING MATH MA is read as line segment MA. MA stands for the measure of MA. MA = AR = ER = ME = 2. In a quadrilateral, consecutive sides intersect. ME and ER form one pair of consecutive sides. List three additional pairs of consecutive sides. 3. In a quadrilateral, the line segments forming opposite sides do not intersect. List the two pairs of opposite sides in Quadrilateral 1. READING MATH MA ER is read as line segment MA is parallel to line segment ER. 4. Circle the letter of each statement that is true for the sides of Quadrilateral 1. a. Consecutive sides are equal in length. b. Opposite sides are equal in length. 5. MA and ER are parallel. You can write MA ER. a. What does that tell you about MA and ER? b. Are both pairs of opposite sides in Quadrilateral 1 parallel? 290 SpringBoard Mathematics with Meaning TM Level 1

51 Properties of Quadrilaterals The Sagging Gate ACTIVITY 5.6 SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Look for a Pattern, Use Manipulatives, Think/Pair/Share 6. Use a protractor to find the measure of each angle to the nearest degree in Quadrilateral 1. m M = m A = m R = m E = 7. Right angles are formed by perpendicular lines. You can write AR RE. Circle the letter of each statement that is true for Quadrilateral 1. a. Consecutive sides are perpendicular. b. Opposite sides are perpendicular READING MATH m A is read as the measure of angle A. AR RE is read as line segment AR is perpendicular to line segment RE. 8. What is the best name for this quadrilateral? 9. Find the length of each side of Quadrilateral 2. Measure to the nearest 1 4 in. C O T CO = OL = TL = CT = 10. Use a protractor to find the measure of each angle to the nearest degree in Quadrilateral 2. m C = m O = m L = m T = 11. What properties do Quadrilateral 1 and Quadrilateral 2 have in common? How are they different? L Unit 5 Geometry 291

52 ACTIVITY 5.6 Properties of Quadrilaterals The Sagging Gate SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge While working with Quadrilateral 1, you found opposite and consecutive sides. Now work with Quadrilateral 2 to see that the angles of a quadrilateral can also be opposite or consecutive. 12. In Quadrilateral 2, C and L are opposite angles and T and C are consecutive angles. a. List the other pair of opposite angles. b. Find the angle sum of each pair of opposite angles. c. List the three other pairs of consecutive angles. d. Find the angle sum of each consecutive angle pair. 13. What is the best name for Quadrilateral 2? Figure 1 N O Understanding angle relationships is important. It is easier to work with angles if you know the different ways that angles can be related. You know that angles are formed when lines intersect. In Figure 1 to the left, AL, NE, and SO all intersect at point I. A I L S E 14. Vertical angles are two angles that share a common vertex and whose sides form two lines. AIN and EIL are vertical angles. Angles that have the same vertex, a common side, but no common interior are adjacent angles. AIS and SIE are adjacent angles. Two angles are complementary angles if their measures have a sum of 90. AIS and SIE are complementary angles. Two angles are supplementary angles if their measures have a sum of 180. AIE and EIL are supplementary angles. a. List three pairs each of vertical and adjacent angles in Figure 1. b. List two pairs each of complementary and supplementary angles in Figure SpringBoard Mathematics with Meaning TM Level 1

53 Properties of Quadrilaterals The Sagging Gate ACTIVITY 5.6 SUGGESTED LEARNING STRATEGIES: Create Representations, Quickwrite, Discussion Group, Debriefing 15. Circle the letter of each statement about Quadrilateral 2 that is true. a. Opposite angles are supplementary. b. Opposite angles have the same measure. c. Consecutive sides have the same length. d. Consecutive sides are perpendicular. e. Consecutive angles are supplementary. f. Consecutive angles have the same measure. g. Opposite sides have the same length. h. Opposite sides are perpendicular. 16. Gabrielle decided to organize her findings in a table. For each property, place an X under the quadrilateral to which it applies. Property Parallelogram Rectangle Both pairs of opposite sides are parallel. Both pairs of opposite sides have the same length. Both pairs of opposite angles have the same measure. Consecutive angles are supplementary. All angles are right angles. Consecutive sides are perpendicular. 17. Is every rectangle a parallelogram? Explain. 18. Is every parallelogram a rectangle? Explain. Unit 5 Geometry 293

54 ACTIVITY 5.6 Properties of Quadrilaterals The Sagging Gate SUGGESTED LEARNING STRATEGIES: Use Manipulatives Meanwhile, Gabrielle asked her father to help her fix the gate. He attached a strong wire to the upper left and lower right corners and included a device called a turnbuckle, which allowed him to tighten the wire. As he tightened the wire, the right side of the gate rose to its original position! MATH TERMS A diagonal is a line segment joining two nonadjacent vertices of a polygon. Gabrielle suspected that her father s repair involved a diagonal of the gate. Quadrilateral 1 (MARE) represents the gate when it was new and again now that it has been repaired, and Quadrilateral 2 (COLT) represents the gate when it was sagging. 19. Draw the diagonals in the figures below and measure them to the nearest 0.1 cm. Fill in the measurements. M A C O E R T Quadrilateral 1 Quadrilateral 2 L Quadrilateral 1 Quadrilateral 2 MR: CL: AE: TO: 294 SpringBoard Mathematics with Meaning TM Level 1

55 Properties of Quadrilaterals The Sagging Gate ACTIVITY 5.6 SUGGESTED LEARNING STRATEGIES: Quickwrite, Create Representations, Work Backward 20. If Quadrilateral 2 represents the sagging gate and Quadrilateral 1 represents the repaired gate, explain how Gabrielle s father repaired the gate by adjusting the diagonals. Which measurements changed, and which remained the same? 21. If a line segment is bisected, it is divided into two equal halves. a. Do the diagonals of Quadrilateral 1 bisect each other? Explain. ACADEMIC VOCABULARY Bisect means to divide into two equal parts. b. Do the diagonals of Quadrilateral 2 bisect each other? Explain. 22. The diagonals of six quadrilaterals are drawn below. Identify which quadrilaterals are rectangles and draw in the sides to check your answer. Quadrilateral A Quadrilateral B Quadrilateral C Quadrilateral D Quadrilateral E Quadrilateral F Unit 5 Geometry 295

56 ACTIVITY 5.6 Properties of Quadrilaterals The Sagging Gate SUGGESTED LEARNING STRATEGIES: Use Manipulatives, Create Representations, Look for a Pattern, Think/Pair/Share, Quickwrite, Self Revision/Peer Revision, Group Presentation Rhombus Square 23. List all the properties of a rhombus and of a square. Measure the angles of the rhombus and the sides of both quadrilaterals. Begin each list with those properties of a parallelogram or rectangle, if they apply. Remember to include the properties of the diagonals. Rhombus Square 24. Is every rhombus a square? Explain 25. Is every square a rhombus? Explain. 296 SpringBoard Mathematics with Meaning TM Level 1

57 Properties of Quadrilaterals The Sagging Gate ACTIVITY 5.6 SUGGESTED LEARNING STRATEGIES: Create Representations, Visualize, Group Presentation 26. Write the name of each figure in the Venn diagram. Parallelogram Rectangle Rhombus Square Quadrilaterals 27. Use a ruler and protractor to draw each quadrilateral as described. Then write all names that apply to each quadrilateral. a. A quadrilateral with diagonals that are perpendicular. b. A quadrilateral with opposites sides parallel and diagonals that are perpendicular. c. A quadrilateral with consecutive supplementary angles and diagonals that are equal in length. Unit 5 Geometry 297

58 ACTIVITY 5.6 Properties of Quadrilaterals The Sagging Gate SUGGESTED LEARNING STRATEGIES: Visualize, Group Presentation d. A quadrilateral whose diagonals bisect each other and at least one angle is obtuse. e. A quadrilateral with opposite sides of equal length and four right angles. CHECK YOUR UNDERSTANDING Write your answers on notebook paper. Show your work. 3. Write all names that apply to a Show your work. quadrilateral with the given properties. 1. PONY is a parallelogram Draw each figure. P O a. A quadrilateral with diagonals that bisect each other. Y N b. A quadrilateral with both pairs of a. Name a pair of opposite sides. opposite sides having the same length and the diagonals having the same length. b. Name a pair of consecutive sides. c. A quadrilateral with diagonals that have c. Name a pair of opposite angles. the same length. d. Name a pair of consecutive angles. d. A quadrilateral with diagonals that are e. If PY = 5 inches and PO = 7 inches, what are the lengths of ON and perpendicular and consecutive sides YN? that are perpendicular. f. If m P = 112, what are the measures of O, N, and Y? 2. PINT is a rectangle. P T O Name the 5 pairs of equal segments in the figure and explain why they are equal. I N e. A quadrilateral with both pairs of opposite angles having the same measure and with diagonals that are perpendicular. f. A quadrilateral with consecutive angles that are supplementary angles. 4. MATHEMATICAL Naming a quadrilateral REFLECTION means to give it the one name that best describes it. Out of the many names that would describe a quadrilateral, how do you pick the one that describes it best? Explain your reasoning. 298 SpringBoard Mathematics with Meaning TM Level 1

59 Symmetry and Transformations Tracking the Migration SUGGESTED LEARNING STRATEGIES: Close Reading, Marking the Text, Create Representations, Think/Pair/Share Some anthropologists study how early human groups migrated throughout North America by studying their art. To understand the patterns found in the art of early human groups, you must understand some properties of symmetry. If you have ever made a valentine by folding a piece of paper to cut out a heart shape or if you have ever made a picture by mixing paints on one side of a piece of paper and then folding the paper to transfer the paint to the other side, you have some understanding of symmetry. Both of these activities illustrate a type of symmetry known as line symmetry. These examples of simple geometric designs have lines of symmetry. A B C D E ACTIVITY 5.7 CONNECT TO SOCIAL STUDIES An anthropologist is a person who studies the ways that humans have lived throughout history. Anthropologists have observed that even if the themes in the art of a group of people changed when the group migrated to new locations, the patterns of symmetry used by the group would remain much the same. Sudden changes in patterns of symmetry are thought to suggest that two groups of people merged and developed a new art form. 1. The figure below is a square and has four lines of symmetry. Draw the lines of symmetry. 2. Draw the lines of symmetry for each figure below. Unit 5 Geometry 299

60 ACTIVITY 5.7 Symmetry and Transformations Tracking the Migration ACADEMIC VOCABULARY In a regular polygon, the length of each side is the same and the measures of the angles are equal. SUGGESTED LEARNING STRATEGIES: Create Representations, Think/Pair/Share 3. How many lines of symmetry does the regular pentagon below have? Draw the lines of symmetry. 4. Symmetry often occurs in nature. How many lines of symmetry does each picture have? Draw the lines of symmetry for each picture. Flower Starfish Snowflake 5. Demonstrate an understanding of line symmetry by completing each design so that the dotted line is a line of symmetry. 300 SpringBoard Mathematics with Meaning TM Level 1

61 Symmetry and Transformations Tracking the Migration ACTIVITY 5.7 SUGGESTED LEARNING STRATEGIES: Create Representations, Group Presentation Many of the standard figures in geometry can be defined using only the concept of line symmetry. For example, an isosceles triangle is a triangle with exactly one line of symmetry and a square is a quadrilateral with four lines of symmetry. 6. Draw an isosceles triangle and its line of symmetry. 7. Draw each figure and its lines of symmetry. Find as many lines of symmetry as you can. a. A parallelogram that is not a rectangle or a rhombus b. A rectangle that is not a square c. A rhombus that is not a square d. A circle Unit 5 Geometry 301

62 ACTIVITY 5.7 Symmetry and Transformations Tracking the Migration MATH TERMS A figure that fits exactly over its original shape n times as it is rotated 360 about its center has n-fold symmetry. SUGGESTED LEARNING STRATEGIES: Close Reading, Marking the Text, Use Manipulatives, Create Representations, Group Presentation Another type of symmetry called rotational symmetry can be used to describe a figure. A figure has rotational symmetry if it can be rotated about its center for less than 360 and fit exactly over its original shape. For example, if you rotate a square 90 about its center, it will fit exactly over its original shape. If you rotate it another 90, it will again fit over its original shape. After rotating 90 four times, the square will return to its original position. Because it takes 4 turns to return a square to its original position, a square is said to have 4-fold symmetry. One side of the square below is darkened so you can see how it rotates. 8. For each figure, mark the center of the figure, then name the figure, and find the value of n for n-fold symmetry. Figure Best Name for the Figure Value of n for n-fold symmetry 302 SpringBoard Mathematics with Meaning TM Level 1

63 Symmetry and Transformations Tracking the Migration ACTIVITY 5.7 SUGGESTED LEARNING STRATEGIES: Close Reading, Marking the Text, Use Manipulatives, Create Representations, Look for a Pattern Beside line and rotational symmetries, anthropologists studying human migration look at a symmetry that involves translations. If you can slide (or translate) a copy of a pattern, part by part, along the original pattern and all parts of the copy lie on top of parts of the original figure, the pattern has translation symmetry. Nine stars in a pattern are drawn below. The pattern continues on forever in both directions. The row of stars has translation symmetry because when you place a copy of the pattern over the star pattern so that the stars match up and slide the copy one star to the right or left, the stars in the copy lie exactly on top of the stars underneath. 9. Draw lines of symmetry in the band of arrows below. Anthropologists study patterns on artifacts. One such pattern is called a band pattern. The band of stars you just looked at is an example of a band pattern. When analyzing any band pattern, you can assume that the pattern goes on forever. 10. By comparing symmetries of band patterns in fabrics and pottery, anthropologists can determine if groups of people in different locations were related. a. Circle the band pattern below that has 2-fold rotational symmetry but no vertical line symmetry. CONNECT TO SOCIAL STUDIES Artifacts are objects manufactured, used, or modified by humans, such as tools, utensils, and art. Band patterns can be horizontal, vertical, or any other position. Unit 5 Geometry 303

64 ACTIVITY 5.7 Symmetry and Transformations Tracking the Migration SUGGESTED LEARNING STRATEGIES: Look for a Pattern, Think/Pair/Share b. Circle the band pattern below that has vertical line symmetry but no rotational symmetry. c. Circle the band pattern below that has translation symmetry but not rotational symmetry or line symmetry. 11. The pottery piece below shows many characteristics of early traditional Native American artwork. The border near the top of the pot is an example of band symmetry. List all of the different types of symmetry that can be found in the band. 304 SpringBoard Mathematics with Meaning TM Level 1

65 Symmetry and Transformations Tracking the Migration ACTIVITY 5.7 SUGGESTED LEARNING STRATEGIES: Quickwrite, Self Revision/ Peer Revision, Think/Pair/Share, Create Representations A band pattern can be created by first graphing a figure that is the basic design element and then using transformations to make congruent images of the figure. 12. What transformations do you already know? ACADEMIC VOCABULARY A movement of a figure on a plane is a transformation. In an earlier activity, you graphed ordered pairs in which both the x-coordinate and the y-coordinate were positive. Now you will graph ordered pairs on a coordinate plane that includes negative coordinates. As you do this, you will continue the study of band patterns. 13. Starting each time at the origin (0, 0), describe how to locate each of the following points, plot the point, and name the quadrant in which the point is located. a. (3, 4) b. (-2, 1) c. (-4, -2) y d. Plot (-4, 5) and (1, 5) and find the distance between the points. e. Plot (6, 2) and (6, 5) and find the distance between the points. x MATH TERMS y Quadrant 2 4 Quadrant 1 2 x Quadrant 3 6 Quadrant In a coordinate plane, the x-axis and the y-axis are perpendicular to each other, intersect at a point called the origin, and divide the coordinate plane into four regions called quadrants. CONNECT TO ALGEBRA The distance between points that have different x-coordinates but the same y-coordinate is the absolute value of the difference of the x-coordinates. For example, the distance between (-4, -2) and (1, -2) on the coordinate plane is -4-1 = -5 = 5 units. How would you find the distance between two points that have different y-coordinates but the same x-coordinate? Unit 5 Geometry 305

66 ACTIVITY 5.7 Symmetry and Transformations Tracking the Migration SUGGESTED LEARNING STRATEGIES: Create Representations, Quickwrite, Self Revision/Peer Revision, Group Presentation 14. Now write a generalization for locating the point represented by the coordinates (a, b). 15. Graph and label each point on the grid. Then connect the points to form ABC. A(-3, -1), B(-1, -2), C(-2, -5) y x Add 5 to the x-coordinate and add 0 to the y-coordinate of each ordered pair to produce three new ordered pairs, A', B', and C'. a. Fill in the coordinates below. READING MATH A' is read as A prime. Changing the coordinates of point A gives a new point, A' A(, ) A'(, ) B(, ) B'(, ) C(, ) C'(, ) b. Graph A', B', and C' on the grid in Question 15 and connect the points to form A'B'C'. 306 SpringBoard Mathematics with Meaning TM Level 1

67 Symmetry and Transformations Tracking the Migration ACTIVITY 5.7 SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Quickwrite, Self Revision/Peer Revision, Create Representations 17. A'B'C' is the image of ABC after a translation. a. What do you notice about ABC and its image A'B'C'? b. Describe the horizontal and the vertical translations needed to go from point A to point A'. c. How do the translations you found in part b compare to the horizontal and vertical translations needed to go from point B to point B' and from point C to point C'? 18. Graph points D (-3, 4), E (-1, 3), and F (-2, 0) on the grid. Connect the points to form DEF y x 19. Translate DEF 4 units right and 3 units down to form D'E'F'. Graph D'E'F' on the same grid as DEF. Show how the coordinates of ABC are transformed to the coordinates of D'E'F'. D(, ) D'(, ) E(, ) E'(, ) F(, ) F'(, ) Unit 5 Geometry 307

68 ACTIVITY 5.7 Symmetry and Transformations Tracking the Migration SUGGESTED LEARNING STRATEGIES: Quickwrite, Self Revision/Peer Revision, Group Presentation, Activating Prior Knowledge, Create Representations, Look for a Pattern 20. Explain how to translate any figure on a coordinate grid by working with the coordinates of its vertices. You have flipped figures over lines in earlier grades. This kind of transformation is a reflection. Now you are going to reflect a shape using the x-axis or the y-axis as a line of reflection. 21. Graph points G (4, -1), H (2, -2), and J (3, -5) on the grid. Connect the points to form GHJ y x Use the grid in Question 21 for Questions 22 and Draw a reflection of GHJ over the x-axis. a. Label the coordinates of the new points G', H', and J'. b. Describe any patterns that you see in the coordinates of G and G', H and H', and J and J'. 308 SpringBoard Mathematics with Meaning TM Level 1

69 Symmetry and Transformations Tracking the Migration ACTIVITY 5.7 SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Create Representations, Look for a Pattern 23. Draw a reflection of GHJ over the y-axis. a. Label the coordinates of the new points G", H", and J". b. Describe any patterns that you see in the coordinates of G and G", H and H", and J and J". READING MATH Remember, A" is read as A double prime. In earlier grades, you learned what happened when you turned or rotated figures. Now you will explore a rotation of a shape about a point on the coordinate plane y x Use the coordinate grid above. a. Graph points K(-6, 2), L(-6, 4), and M(-2, 2) on the grid. Connect the points to form KLM. b. Graph points K'(6, -2), L'(6, -4), and M'(2, -2) on the grid. Connect the points to form K'L'M'. c. Find the length of sides KL, K'L', MK, and M'K' K'L'M' is the image of KLM after a rotation. a. What is the center of the rotation? b. How many degrees are there in the rotation? c. Find a pattern in the coordinates of K and K', L and L', and M and M'. Unit 5 Geometry 309

70 ACTIVITY 5.7 Symmetry and Transformations Tracking the Migration CHECK YOUR UNDERSTANDING Write your answers on notebook paper. Show your work. 1. Sketch each figure. Then draw any lines of symmetry for each figure. a. 3. List all of the different types of symmetry that can be found in each of the band patterns below. a. b. c. b. d. c. 2. Look at the letters below. ABCDEFGHI JKLMNOPQR STUVWXYZ a. Which of the letters have horizontal line symmetry? b. Which of the letters have vertical line symmetry? c. Which of the letters have both horizontal and vertical line symmetry? d. Which of the letters have line symmetry that is neither horizontal nor vertical? 4. Without graphing find the coordinates of A', B', and C' after translating ABC two units left and 6 units up. A(-4, -3) A'(, ) B(-2, 0) B'(, ) C(-1, -5) C'(, ) 5. Graph the following points on grid paper and then connect the points to form DEF. D(1, -3); E(3, -2); F(5, -5) a. Draw the reflection of DEF over the x-axis. Label the points D', E', and F'. b. Draw the reflection of DEF over the y-axis. Label the points D", E", and F". 6. MATHEMATICAL REFLECTION Describe the relationship between the coordinates of a point and the coordinates of its image after a reflection across the x-axis or the y-axis. 310 SpringBoard Mathematics with Meaning TM Level 1

71 Exploring Volume Guess How Many SUGGESTED LEARNING STRATEGIES: Guess and Check, Quickwrite, Debrief, Vocabulary Organizer Look at the container of popcorn that your teacher has given your group. Guess how many pieces of popcorn it holds without touching it. 1. Record your guess below and explain the strategy you used. ACTIVITY Containers like these are commonly called estimation jars. a. Have you ever played a game like this before? If so, describe where it was and the purpose? b. Is there any way of knowing the exact answer? Explain. Volume is a measure of the space inside a figure such as a cube, a ball, or a cylinder. 3. How does this game relate to volume? 4. How would knowing how to find volume of a figure make it easier to make a reasonable prediction? Would you be able to find the exact number in the jar every time? Explain. See if you can discover a formula for finding the volume of your group s popcorn container. ACADEMIC VOCABULARY Volume is the amount of space occupied by a threedimensional figure. It is measured in cubic units, such as cubic inches (in. 3 ). ACADEMIC VOCABULARY A solid is a 3-dimensional geometric figure with dimensions of length, width, and height. 5. The 2-dimensional drawing at right, representing the popcorn container, is a solid. Label each dimension on the drawing. Unit 5 Geometry 311

72 ACTIVITY 5.8 Exploring Volume Guess How Many MATH TERMS Attributes are characteristics or qualities of something. SUGGESTED LEARNING STRATEGIES: Group Presentation, Look for a Pattern, Vocabulary Organizer, Use Manipulatives The popcorn container is a type of solid called a prism. Look at the figures in the table to help remember what a prism is. Prisms Not Prisms 6. In your own words, describe the attributes of a prism. 7. Why is the popcorn container called a rectangular prism? Circle all rectangular prisms in the chart above Question 6. WRITING MATH Cubic units can be written as units 3. To find the amount of 3-dimensional space that is filled with popcorn, you need to measure the volume of the popcorn container. You use cubic units to measure volume. Look at the blocks your teacher has given you. 8. Why is one of these blocks called a cubic unit? 9. Measure the dimensions of the block. Now give the cubic unit a more specific name and explain your reasoning. 312 SpringBoard Mathematics with Meaning TM Level 1

73 Exploring Volume Guess How Many ACTIVITY 5.8 SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Create Representations, Use Manipulatives, Look for a Pattern, Group Presentation 10. Name at least two other cubic units that could be used to fill a 3-dimensional space. Since you do not have enough cubic units to fill the popcorn container as a way of finding its volume, you can build smaller prisms and look for a pattern. 11. Use the cubic-unit blocks to build rectangular prisms with the dimensions given in the table. Count the blocks to determine the volume of each prism, and record your findings. Length Width Height Volume Figure Figure Figure Figure Use the data in the table to describe a pattern that can be used for finding volume of any rectangular prism. 13. You can write formulas to represent these patterns. a. Write a formula for volume, V, to represent the pattern you have found. Use l for length, w for width, and h for height. b. Write a formula for volume, V, relating the area of the base, B, to the height, h. Compare this formula to the one you wrote in Part a. 14. Use both formulas from Question 13 to find the volume of a rectangular prism with a length of 4 units, width of 5 units, and a height of 2 units. Use blocks to check your answer. Unit 5 Geometry 313

74 ACTIVITY 5.8 Exploring Volume Guess How Many SUGGESTED LEARNING STRATEGIES: Create Representations, Debriefing, Use Manipulatives, 15. It is possible to make different rectangular prisms with a total of 12 cubic units. a. Make as many different prisms as you can. Make a table below to record your results. b. Use either formula from Questions 13 and 14 to confirm that your dimensions are accurate. 16. Use a ruler to measure. a. Find the volume of your popcorn container in cubic centimeters. b. Will this tell you the number of popcorn pieces? Explain why or why not. 17. This time find the volume of the popcorn container using popcorn as the unit. Your teacher will provide you with a handful of popcorn. l = w = h = V = 18. How does volume in cubic centimeters relate to volume in popcorn units? 314 SpringBoard Mathematics with Meaning TM Level 1

75 Exploring Volume Guess How Many ACTIVITY 5.8 SUGGESTED LEARNING STRATEGIES: Discussion Group, Look for a Pattern, Vocabulary Organizer, Quickwrite 19. Your teacher will tell you how much popcorn is in each container. Compare the actual number to your answer to Question 17? Explain why they are the same or different. 20. Refer to your original guess. How close were you to the actual amount and by how much? Who had the closest guess in your group? Now play the estimation jar game again. 21. Look at the second popcorn container your teacher has filled. The drawings below relate to this container. 3-D View (not drawn to scale) Top View In a triangle, a height is the distance from a vertex to the line containing the opposite side. This distance is the length of the perpendicular line segment from the vertex to the line containing the opposite side. In a prism, or other three-dimensional figure with parallel bases, the height is the distance between the parallel bases. This distance is the length of the line segment perpendicular to both bases and with endpoints on those bases in in in. a. What is the name of this container? Justify your thinking. b. Guess how many pieces of popcorn are in the container. Write your guess and your name on a sticky note and post it when your teacher asks you to do so. c. Describe your estimation strategy. Unit 5 Geometry 315

76 ACTIVITY 5.8 Exploring Volume Guess How Many SUGGESTED LEARNING STRATEGIES: Look for a Pattern, Create Representations, Use Manipulatives, Debriefing, Vocabulary Organizer, Think/Pair/Share 22. Determine the formula for finding the volume of a rectangular prism and explain how you could use it to develop a formula for finding the volume of the second popcorn container. 23. Find the volume of the second popcorn container in each unit. a. popcorn units b. cubic centimeters c. cubic inches 24. Your teacher will tell you the actual volumes. How close were your answers? CONNECT TO AP In calculus, you will learn how to compute the volume of an irregular solid like the vase shown below. 25. An estimation game can be played with a glass jar in the shape of a cylinder. a. Compare and contrast cylinders and prisms. b. Is a cylinder a type of prism? Explain. 26. Use what you know about finding the volume of a prism to develop a formula for finding the volume of a cylinder. Explain your reasoning. 316 SpringBoard Mathematics with Meaning TM Level 1

77 Exploring Volume Guess How Many ACTIVITY 5.8 SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Quickwrite, Self Revision/Peer Revision 27. Apply your formula to find the missing variable in each problem below. Figures are not drawn to scale. a. b. c. r = 2.3 cm d = 200 in. r = 15 m h = 7.6 cm h = 90 in. h = V = V = V = 19,792 m Look at the estimation jar your teacher has displayed. Consider what you have learned about finding volume. a. Make an educated estimate of the number of items in the estimation jar: b. Describe your estimation strategy. c. Write your estimate and your name on a sticky note and post it when your teacher asks you to do so. 29. How does knowing the formulas for volume help you to make better estimates when playing the estimation jar game? Unit 5 Geometry 317

78 ACTIVITY 5.8 Exploring Volume Guess How Many CHECK YOUR UNDERSTANDING Write your answers on notebook paper. Show your work. 1. What is the difference in unit measures when calculating area and volume? 2. Find the volume of a rectangular prism with a height of 5 cm, length of 7 cm, and width of 8 cm. 3. Find the missing dimension of each rectangular prism below. a. b. 6.5 ft h 3.2 ft 173 mm V = ft 3 V = 6,099,980 m m 3 4. How much can the following container hold in cubic inches? w 215 mm 6. A glass company sells vases in 3 different styles: a. The dimensions of the rectangular prism are shown below. Find the volume. h = 10 in. w = 4 in. l = 6 in. b. What do the dimensions of the other vases need to be so that they hold about the same amount of water as the rectangular prism vase? Show your work. 3 in. 8.3 in. 2.7 in. 5. A can of breadcrumbs has a diameter of 4 in. and a height of 10.5 in. What is the volume of the can? base = radius = 2.7 in. altitude = 4 in. height = height = 10 in. 7. MATHEMATICAL REFLECTION Compare and contrast the formulas for finding volume of rectangular prisms, triangular prisms, and cylinders. 318 SpringBoard Mathematics with Meaning TM Level 1

79 Polygons, Transformations, and Volume GRAPHIC GEOMETRY Embedded Assessment 2 Use after Activity 5.8. Write your answers on notebook paper or grid paper. Show your work. 1. Classify each triangle in the margin by its sides and by its angles. Explain your reasoning. 2. Use a coordinate grid. a. Graph each point: A(-10, 4), B(-7, 8) and C(-4, 6). b. Connect the points to form ABC. c. Reflect ABC across the y-axis. Label the reflection of A, B, and C with A, B, and C and name its coordinates. 3. Use your coordinate grid from Question 2. a. Graph each point: F G H P(6, -1), Q(7, -3), R(6, -5), and S(5, -3). b. Connect the points to form quadrilateral PQRS. c. Translate quadrilateral PQRS 4 units up and 2 units right. Label the translation of P, Q, R, and S with P, Q R, and S. What are the coordinates of P,Q R, and S? 4. What is the best name for quadrilateral PQRS? Is that also the best name for quadrilateral P Q R S? Justify your answer. 5. Copy this band pattern. Assume it continues in both directions. a. What is the best name for each of the three different kinds of figures in the band pattern? Explain why. b. Draw any lines of symmetry. Name the other type of symmetry in the band pattern. 6. A solid has two circular bases each with a diameter of 14 inches. Its height is 10 inches. a. What is the name of the solid? b. What is the volume of the solid? Unit 5 Geometry 319

80 Embedded Assessment 2 Use after Activity 5.8. Polygons, Transformations, and Volume GRAPHIC GEOMETRY Exemplary Proficient Emerging Math Knowledge #1, 4, 5, 6 The student: Classifies triangle by sides and angles correctly (1) and names the quadrilaterals (4), Names all three figures in the band pattern (5a), Names the type of symmetry used (5b), Names and calculates volume of the solid (6a, 6b). Student correctly and completely answers four of these six items. Student correctly and completely answers at least two of these items. Representation #2, 3, 5 The student: Plots, connects and labels the points accurately(2,3) Reflects the figure and labels and names its coordinates correctly(2) Translates the figure and labels and names its coordinates correctly (3), Draws all lines of symmetry correctly (5). Student attempts all five items completing four of them accurately. Student attempts at least three of the items and completes only two of the accurately. Communication #1, 4, 5 The student: Classifies the triangles using both methods with explanations(1), Justifies the naming of both quadrilaterals (4), Explains why chosen names are assigned to each figure in the band pattern (5). Student gives explanations for all three questions, but only two of the explanations are complete and correct. Student attempts at least two explanations, but only one is complete and correct. 320 SpringBoard Mathematics with Meaning TM Level 1

81 Practice UNIT 5 ACTIVITY Give the area of each figure. a. 5. Find the area and perimeter of the following figure: 4.3 cm 5.1 cm 13.5 cm b. 8.3 cm 2. Find the perimeter of each figure. a. ACTIVITY 5.2 Answer each problem, and then use estimation to see if your answer is reasonable. 6. Find the circumference. 13 mm b. 3. Make a rectangle that has an area formed by 42 squares. a. What is its perimeter? b. Can you draw a different rectangle with an area of 42 square units? 4. Find the area and perimeter of the rectangle below. 7.3 ft 4.2 ft 7. Find the circumference. 9 ft 8. The diameter of a circular stone is 15 cm. What is the stone s circumference? 9. Evelyn wants to glue a ribbon around a circular flower planter. The radius of the planter is 7 inches. How much ribbon (not including a bow) does she need? Unit 5 Geometry 321

82 UNIT 5 Practice 10. Find and correct the mistake(s) that Myriam made when finding the circumference of a circle with a radius of 4 meters. C = 4 π C C m Find the diameter of a circle if the circumference is 8π yd. 12. Find the radius of a circle if the circumference is 11π in. ACTIVITY 5.3 Find the area. Remember to label your answer in in cm 11 cm 5 cm 4 cm 18. Find the area and circumference. 19. Janice works at the local park and tends a flower garden that is in the shape of a circle with a diameter of 22 feet. Find the area of the garden. 20. The top and bottom of Monty s kaleidoscope is a hexagon, as shown below. Find its area. 3 cm 5 cm in. 5 cm 2.5 cm in. 15. Find the area of a rectangle with a length that measures 12 cm and a height that measures 8 cm. Label your answer. Find the area of each figure m 11 m 15 m ACTIVITY cm 21. Use your tangram set to create a shape. Draw the shape. Measure and label the parts you need to determine the area and perimeter of your shape. 22. Determine the area of the shape you made in Question Determine the perimeter of shape you made in Question SpringBoard Mathematics with Meaning TM Level 1

83 Practice UNIT The following shape is used for windows and is called a Norman Window. Find its area and perimeter. 25. Trace the outline of your state on grid paper, then find the approximate area and perimeter of your state. ACTIVITY Use the Triangle Inequality Property to determine whether a triangle can be formed with each set of side lengths. Show your work or explain your reasoning. a. a = 4 cm b = 4 cm c = 4 cm b. a = 4 cm b = 4 cm c = 5 cm c. a = 4 cm b = 4 cm c = 8 cm 4 ft 9 ft 27. Draw a triangle described by each pair of words below or state that it is not possible. If it is not possible, explain why not. a. Obtuse, isosceles b. Scalene, right 28. Two angles of each triangle are given. Find the third angle and use as many of the following words as possible to describe each triangle: scalene, isosceles, equilateral, acute, obtuse, right. a. 52 ; 64 b. 45 ; 90 c. 24 ; 15 ACTIVITY FOAL is a rectangle. F L a. Name a pair of opposite sides. b. Name a pair of consecutive sides. c. Name a pair of opposite angles. d. Name a pair of consecutive angles. e. If FO = 8 inches and OA = 5 inches, what are the lengths of FL and AL? f. What are the measures of F, O, A, and L? O A Unit 5 Geometry 323

84 UNIT 5 Practice 30. HERS is a parallelogram. Name the 4 pairs of equal line segments in the drawing. Explain why they are equal. 31. Write all names that apply to a quadrilateral with the given features. Draw each figure. a. A quadrilateral with opposite angles having the same measure. b. A quadrilateral with perpendicular diagonals. c. A quadrilateral with all the sides having the same length and with diagonals having the same length. d. A quadrilateral with diagonals that bisect each other and are also perpendicular to each other. e. A quadrilateral with each pair of opposite sides having the same length and with diagonals having the same length. ACTIVITY Copy each figure. Draw any lines of symmetry you see. a. b. S H O R E 33. List all the different types of symmetry that can be found in each band pattern. a. b. 34. Draw a coordinate grid. Graph each point on the grid. Then connect the points to form quadrilateral DEFG. D(1, 2); E(3, 2); F(4, 3); G(2, 5) a. Reflect quadrilateral DEFG over the y-axis. Write the coordinates of your new points D'E'F'G'. b. Reflect quadrilateral D'E'F'G' over the x-axis. Write the coordinates of your new points D"E"F"G". c. What transformation would move quadrilateral DEFG directly to quadrilateral D"E"F"G"? 35. Triangle A' B'C' is the image of triangle ABC after it was translated up 4 units and right 2 units. If the coordinates of A'B'C' are A'(-2, 1); B'(1, 2); C'(1, -1), what are the coordinates of A, B, and C? c. 324 SpringBoard Mathematics with Meaning TM Level 1

85 Practice UNIT 5 ACTIVITY Explain the relationship between area and volume, and their units. 37. How many cubic inches of cereal can a box with the following dimensions hold? 41. Find the missing dimension of each figure. Sketch each figure and label its measurements, if that will help you. a. 38. What is the length of a rectangular prism with h = 32.5 yd, w = 19 yd, and V = 14,943.5 yd 3? 39. How many cubic decimeters of water can this fish tank hold? Draw the figure and label each dimension. base = 5.4 dm 9 in. altitude = 3.7 dm height = 7.1 dm 12 in. 3 in. 40. How many cubic centimeters of water will fill this glass? 3 cm b. V = 96.2 ft 3 r = 2.5 ft h = V = m 3 base = 2 m altitude = height = 7.7 m 20 cm Unit 5 Geometry 325

86 UNIT 5 Reflection An important aspect of growing as a learner is to take the time to reflect on your learning. It is important to think about where you started, what you have accomplished, what helped you learn, and how you will apply your new knowledge in the future. Use notebook paper to record your thinking on the following topics and to identify evidence of your learning. Essential Questions 1. Review the mathematical concepts and your work in this unit before you write thoughtful responses to the questions below. Support your responses with specific examples from concepts and activities in the unit. What characteristics do various quadrilaterals share, and why is it possible to determine perimeter and area of quadrilaterals using related formulas? In what ways is symmetry important in real-world situations? Academic Vocabulary 2. Look at the following academic vocabulary words: altitude bisect congruent equilateral perimeter regular polygon solid transformation volume Choose three words and explain your understanding of each word and why each is important in your study of math. Self-Evaluation 3. Look through the activities and Embedded Assessments in this unit. Use a table similar to the one below to list three major concepts in this unit and to rate your understanding of each. Unit Concepts Concept 1 Concept 2 Concept 3 Is Your Understanding Strong (S) or Weak (W)? a. What will you do to address each weakness? b. What strategies or class activities were particularly helpful in learning the concepts you identified as strengths? Give examples to explain. 4. How do the concepts you learned in this unit relate to other math concepts and to the use of mathematics in the real world? 326 SpringBoard Mathematics with Meaning TM Level 1

87 Math Standards Review Unit 5 1. A botanist wants to study the plant life in a circular zone of the Florida Everglades. The zone has a diameter of 48 miles. Which of the following is closest to the area of the zone that the botanist wants to study? A square miles C square miles B square miles D square miles 1. Read Solve Explain 2. Mr. Patel is building a display case in the shape of a rectangular prism for an exhibit that needs to have a total volume of at least 56 square centimeters. To the nearest tenth of a centimeter, what should the height be? 3 cm 8.5 cm 3. The formula for the circumference of a circle can also be used for the circumference of a sphere. The planets in our solar system are spherical. Some measures are listed in this table. Planet Diameter Circumference Earth Mercury 3032 miles Venus Jupiter 88,846 miles 24,888 miles 23,616 miles 2. Part A: What formula can be used to calculate the missing measures? Use it to complete the table. Part B: Predict how the circumference of a planet would change if its diameter were doubled. Then double the diameter of Mercury and calculate what the new diameter would be. Verify your prediction. Solve and Explain Unit 5 Geometry 327

88 Math Standards Review Unit 5 () Read Solve Explain 4. Use the grid to find the measures of parts of this composite figure. A H I B G L M J C F K D E Part A: Find these areas. ACEG BDFH IJKL Unshaded area Part B: Identify a pair of supplementary angles and a pair of complementary angles. Support your choices. Solve and Explain Part C: Identify two similar figures in the composite figure. Explain why they are similar. Solve and Explain 328 SpringBoard Mathematics with Meaning TM Level 1