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?
These assessments, following Activities 5.4 and 5.8, will give you an opportunity to demonstrate your ability to fi nd the perimeter and area of composite fi gures made up of triangles, circles, and quadrilaterals and to classify triangles and quadrilaterals, to perform transformations on a coordinate grid, and to fi nd the volume of a solid.
Embedded Assessment 1
Area and Perimeter p. 271
Embedded Assessment 2
Polygons, Trans formations, and Geometry p. 319
Unit OverviewIn 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 VocabularyAs you study this unit, add these terms to your vocabulary notebook.
5.1Area and PerimeterThe Dot GameSUGGESTED LEARNING STRATEGIES: Summarize/Paraphrase/Retell, Think/Pair/Share, Use Manipulatives
Have you ever played Th e Dot Game? Here is your chance to play a game during math class. Have fun.
Game BoardTh e game is played on a rectangular grid of dots.
Object of the GameEach 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 fi rst 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. Th e game board will be fi lled with squares, each containing the initial of the player who completed it.
Winning the GameCount the squares for each player. Th e 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 My Notes 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?
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?
8. Look at Game Board 1 from the fi rst two games and identify the largest shape made with your initialed adjacent squares. Draw that shape in the My Notes 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 My Notes space, but do not draw any interior line segments.
Adjacent squares must have at least one side and two vertices in common. Vertices is the plural of vertex.
1 2. How many one-unit line segments are there in the perimeter of the fi gure 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?
Now play the game again. Th e rules for playing this time are the same, but the winner will be decided diff erently. Instead of the winner completing the greatest number of squares, the winner in this game will have completed adjacent squares that form a fi gure that has the greatest perimeter.
14. Betty and Andy played Th e Dot Game on a 4 cm × 5 cm grid. Th eir results are shown below. Each segment drawn is 1 cm in length and each square has an area of 1 cm 2 .
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.
16. How did you determine the perimeter of your fi gure? Show your work.
SUGGESTED LEARNING STRATEGIES: Quickwrite, Close Reading, Create Representations
READING MATH
1 cm 2 is read as one square centimeter. Sometimes you will see 1 sq cm used for 1 cm 2 .
246 SpringBoard® Mathematics with MeaningTM Level 1
Area and PerimeterACTIVITY 5.1continued The Dot GameThe Dot Game
17. For Betty and Andy’s Dot Game, draw and describe the diff erent units of measure used to decide who won the fi rst round of Th e Dot Game and who won the second round.
18. Make a fi gure that has the same area as Betty’s fi gure, but with a diff erent perimeter. Use the grid in the My Notes space. You may also use square tiles to form a fi gure and then make a drawing of it in the My Notes space.
19. Use the grid in the My Notes space to draw the rectangles.
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 diff erent rectangle with the same area but a diff erent perimeter. Draw a diagram in the My Notes space.
c. Can you make a diff erent rectangle with the same perimeter as the rectangle in Part a, but with a diff erent area? If so, draw a diagram in the My Notes 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.
SUGGESTED LEARNING STRATEGIES: Quickwrite, Group Presentation, Create Representations, Think/Pair/Share
Recall that opposite sides of a rectangle have the same measures.
5.2Investigating πGoing 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 fi nd the distance around the garden, but is not sure which measuring tool she will need. In this activity, you will investigate a method for fi nding 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. Th e distance around a circle is called the circumference.
1. Draw and label a diameter and a radius in the circle.
2. What is the relationship between the length of a diameter and the length of a radius of a circle?
Th ere 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. Th en calculate the ratios.
First Circle
Second Circle
Th ird Circle
Fourth Circle
Fift h Circle
Circumference
Diameter
Ratio of circumference to diameter
(as a fraction)
Ratio of circumference to diameter
(as a decimal)
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.
MATH TERMS
A plane can be thought of as a fl at surface that extends in all directions. A parallelogram is usually used to model a plane.
250 SpringBoard® Mathematics with MeaningTM Level 1
My Notes
Investigating πACTIVITY 5.2continued Going In Circles Going In Circles
SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Look for a Pattern, Create Representations, Group Presentation, Identify a Subtask
Th e 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.
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.
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 fi gure below represents.
1. Look at the shape of each fi gure, and write the name of the playground equipment next to each letter.
SUGGESTED LEARNING STRATEGIES: Use Manipulatives, Look for a Pattern, Self Revision/Peer Revision, Debriefing
ACTIVITY 5.3continued
Area of Polygons and Circles Play AreaPlay Area
6. Cut out one of the two congruent parallelograms on page 255. Th en 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 fi nd its area.
b. Sketch the rectangle and record your measurements in the My Notes 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. Th e hexagon in the aerial view of the playground is made up of triangles and pentagons. List some characteristics of each fi gure.
a. hexagon
b. triangle
c. pentagon
The height of a fi gure is always drawn perpendicular to its base. Perpendicular lines (⊥) meet to form right angles.
ACADEMIC VOCABULARY
Two or more fi gures that have the same shape and size are congruent.
258 SpringBoard® Mathematics with MeaningTM Level 1
SUGGESTED LEARNING STRATEGIES: Use Manipulatives, Create Representations, Quickwrite
Area of Polygons and Circles ACTIVITY 5.3continued Play AreaPlay Area
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 fi nding the area of a triangle.
Another shape seen in the aerial view of the playground looks like the fi gure at right. Th is fi gure is called a trapezoid.
13. Find the congruent trapezoids on page 255. • Cut out the two congruent trapezoids.• On the inside of each fi gure label the
bases as b1 and b2 as shown at right. • 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. Th e long legs of the trapezoids will be adjacent.
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.
MATH TERMS
READING MATH
Sometimes subscripts are used to label segments.
b1 is read as “b sub 1” and represents one base of the trapezoid.
b2 is read as “b sub 2” and represents a second base of the trapezoid.
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, Group Presentation, Self Revision/Peer Revision, Quickwrite
ACTIVITY 5.3continued
Area of Polygons and Circles Play AreaPlay Area
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 play-ground. Use what you have learned about fi nding the area of rectangles, triangles, parallelograms, and trapezoids to describe how to fi nd the area of this pentagon.
Th e last shape found in the aerial view of the playground on the fi rst 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.
260 SpringBoard® Mathematics with MeaningTM Level 1
SUGGESTED LEARNING STRATEGIES: Look for a Pattern, Guess and Check, Group Presentation, Debriefing
Area of Polygons and Circles ACTIVITY 5.3continued Play AreaPlay Area
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 fi gure. What part of the circle does the height represent?
22. What other part of the circle is about the length needed to fi nd the area of the shape you named in Question 21?
23. Using words, symbols, or both, describe how you can now fi nd the area of the circle.
24. Th e dimensions of some of the pieces of playground equipment are shown with their drawings below. Find the area of each fi gure. Explain how you found each area.
a. Figure E
b. Figure G
10 feet
2 feet
4 feet
2 feet
Right angles are often identifi ed with a small square in the corner of the angle.
Th is 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 22 ___ 7 .
262 SpringBoard® Mathematics with MeaningTM Level 1
SUGGESTED LEARNING STRATEGIES: Think/Pair/Share
Area of Polygons and Circles ACTIVITY 5.3continued Play AreaPlay Area
25. Based on the dimensions given for the other fi gures 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 fi gure. Remember to label your answer.
1.
2.
3. Draw the fi gure, and then fi nd 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.
5. Use π ≈ 22 ___ 7 to fi nd the area of the circle.
6. Find the area.
7. Mikel is going to build a doghouse for his new puppy. Th e fl oor’s shape is shown below. Find the area of the doghouse fl oor.
8. Draw a circle with a radius of 2.3 cm, and then fi nd its area.
5.4Area and Perimeter of Composite Figures Putting Back the Pieces
According to Chinese legend, a man dropped a porcelain tile. It broke into the seven pieces you see below. Th ey are called tangram pieces. While he was trying to reassemble the seven pieces into a square, he found that he could make hundreds of diff erent shapes.
1. Cut out the seven pieces or use a tangram set. Assemble them into the original square tile.
266 Springboard® Mathematics with Meaning™ Level 1
SUGGESTED LEARNING STRATEGIES: Identify a Subtask, Create Representations, Group Presentation
Area and Perimeter of Composite Figures ACTIVITY 5.4continued Putting Back the PiecesPutting Back the Pieces
5. Divide the composite fi gure below into shapes whose area you know how to fi nd. Use the grid lines as a guide.
a. Determine the area of each shape you made. Show your work.
b. Give the total area of the composite fi gure.
6. Compare and contrast your solution with your classmates.
CONNECT TO APAP
In AP Calculus, an important problem is fi nding the area under a curve. You can count squares on a coordinate plane to do this.
2
1
1 2
y
x3 4 5
4
3
5
You will also learn other procedures that result in much more precise or even exact calculations of the area of irregular fi gures when you take calculus.
SUGGESTED LEARNING STRATEGIES: Marking the Text, Quickwrite, Use Manipulatives
ACTIVITY 5.4continued
Area and Perimeter of Composite Figures Putting Back the PiecesPutting Back the Pieces
Tyrone loves to go fi shing with his mother. Th ey oft en fi sh at Big Trout Pond in a nearby state park. One day aft er Tyrone caught several fi sh, he was concerned that the pond would run out of fi sh. His mother explained that the park rangers routinely restock the pond.
Tyrone wanted to know how the rangers know how many fi sh to add to the pond. Tyrone did some research and learned that the number of fi sh 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 fi gure into shapes whose area you know how to fi nd? Explain why or why not.
b. Carefully cut out the scale drawing of the pond. Th en trace it on the grid on page 269.
SUGGESTED LEARNING STRATEGIES: Quickwrite, Simplify the Problem, Think/Pair/Share, Debriefi ng, Group Presentation
ACTIVITY 5.4continued
Area and Perimeter of Composite Figures Putting Back the PiecesPutting Back the Pieces
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 fi gure?
e. Shade every square that contains some piece of the perimeter of the fi gure. 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 fi gure?
8. Estimate the area of the fi gure that represents the pond. Describe how you arrived at your answer
Write your answers on notebook paper. Show your work.
Th e students at Bailey Middle School participate in a community service project every year. Th is 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. Th e two favorite designs are shown below.
Th e fl oor plan for Design 1 includes a regular octagon with sides measuring four meters and four congruent triangles. Th e fl oor plan for Design 2 includes a rectangle, a trapezoid, a parallelogram, and a semicircle.
1. Find the areas of the fl oors for both designs. Explain your thinking by giving formulas and showing your work.
2. Find the perimeters of the fl oors 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.
Mr. Javarra asked his students to make up some math games involving facts about triangles. Katie and Allie suggested the following game.
Triangle Trivia RulesProperties of Triangles—Perimeter Variation
Players: Th ree to four studentsMaterials: Th ree 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.
Directions: 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. Th e value of the perimeter of any triangle that can be formed is added to that player’s score. Th e fi rst 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 fi nd 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.
1. 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.
LEARNING STRATEGIES: Close Reading, Think Aloud, Marking the Text, Summarize/Paraphrase/Retell, Look for Pattern, Use Manipulatives
Player 1 Player 2 Player 3 Player 4Numbers Score Numbers Score Numbers Score Numbers Score
274 SpringBoard® Mathematics with MeaningTM Level 1
Properties of TrianglesACTIVITY 5.5continued Triangle TriviaTriangle 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.
Properties of TrianglesTriangle TriviaTriangle Trivia
LEARNING STRATEGIES: Close Reading, Think Aloud, Marking the Text, Summarize/Paraphrase/Retell
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.
Triangle Trivia Rules—Name My Sides VariationNumber of Players: Th ree to four studentsEquipment needed: Th ree 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.
276 SpringBoard® Mathematics with MeaningTM Level 1
Properties of TrianglesACTIVITY 5.5continued Triangle TriviaTriangle 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.
Properties of TrianglesTriangle TriviaTriangle Trivia
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°. Th e angle measure of an acute angle is less than 90° and the angle measure of an obtuse angle is greater than 90°.
8. Identify each of these angles as right, acute, or obtuse.
LEARNING STRATEGIES: Create Representations, Use Manipulatives
Player 1 Player 2 Player 3 Player 4Numbers Score Numbers Score Numbers Score Numbers Score
278 SpringBoard® Mathematics with MeaningTM Level 1
Properties of TrianglesACTIVITY 5.5continued Triangle TriviaTriangle Trivia
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.
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.
LEARNING STRATEGIES: Look for a Pattern, Quickwrite, Create Representations
Properties of TrianglesTriangle TriviaTriangle Trivia
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.
LEARNING STRATEGIES: Look for a Pattern, Quickwrite, Create Representations
Scalene, right Isosceles, right Equilateral, right
280 SpringBoard® Mathematics with MeaningTM Level 1
Properties of TrianglesACTIVITY 5.5continued Triangle TriviaTriangle Trivia
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 fi rst 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.
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. Th e acute angles of an isosceles triangle are complementary.
b. Th e 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. Th e largest angle of a scalene triangle can be opposite the smallest side.
LEARNING STRATEGIES: Look for a Pattern, Quickwrite
A conjecture is a statement that seems to be true but has not been proven to be either true or false
Properties of TrianglesTriangle TriviaTriangle Trivia
Amir used what he learned about angle relationships in triangles to write a variation of Triangle Trivia. He called it the Th ird Angle Variation. Use his directions to answer Question 13.
Triangle Trivia Properties of Triangles—Th ird Angle Variation
Directions
Shuffl e the cards and place them facedown. Draw two cards. Th e 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.
Th e fi rst player to reach 300 points wins.
13. When playing Amir’s Th ird Angle variation of Triangle Trivia, suppose you drew cards with the following numbers on diff erent 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
LEARNING STRATEGIES: Close Reading, Think Aloud, Marking the Text, Summarize/Paraphrase/Retell, Quickwrite
282 SpringBoard® Mathematics with MeaningTM Level 1
Properties of TrianglesACTIVITY 5.5continued Triangle TriviaTriangle Trivia
Another student in Mr. Javarra’s class invented this variation. Play this game in groups of three or four.
Triangle Trivia RulesProperties of Triangles—Triangle Trio Game
Players: Th ree to four students
Materials: 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.
Goal: Be the fi rst player to make two sets. A set is three cards whose triangles have the same classifi cation 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.
Directions: • 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 . Th e 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.
LEARNING STRATEGIES: Close Reading, Think Aloud, Marking the Text, Summarize/Paraphrase/Retell, Quickwrite
Properties of TrianglesTriangle TriviaTriangle Trivia
Write your answers on notebook paper. Show your work.
CHECK YOUR UNDERSTANDING
Write your answers on notebook paper. Show your work.
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 = 9b. a = 2 b = 2 c = 5c. a = 6 b = 3 c = 8d. 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, obtuseb. Acute, isoscelesc. Obtuse, equilateral
3. You are given two of the angles of a triangle. Find the third angle and use as many of the following words as possible to describe the triangle.(scalene, isosceles, equilateral, acute, obtuse, right)a. 32°; 58°
b. 162°; 9°
c. 60°; 60°
4. Read the following conjecture and determine whether it is always true, sometimes true, or never true. Explain your reasoning.Th e side of an isosceles triangle between two equal sides is longer than the other two sides.
5. MATHEMATICAL R E F L E C T I O N
Th e games in this activity were designed to help
you better understand the relationship between the sides and angles of diff erent kinds of triangles. Explain how triangles are classifi ed using angle measures and side lengths, and give two examples.
Properties of QuadrilateralsThe Sagging GateSUGGESTED 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. Th e fence included a large rectangular gate.
Th e climate where Gabrielle lives is very rainy in the spring and cold and snowy in the winter. One year aft er 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.
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?
Understanding angle relationships is important. It is easier to work with angles if you know the diff erent 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.
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 1.
294 SpringBoard® Mathematics with MeaningTM Level 1
Properties of QuadrilateralsACTIVITY 5.6continued The Sagging GateThe Sagging Gate
Meanwhile, Gabrielle asked her father to help her fi x 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!
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 fi gures below and measure them to the nearest 0.1 cm. Fill in the measurements.
SUGGESTED LEARNING STRATEGIES: Use Manipulatives
Quadrilateral 1 Quadrilateral 2
MR: _________ CL: _________
AE: _________ TO: _________
A diagonal is a line segment joining two nonadjacent vertices of a polygon.
Properties of QuadrilateralsThe Sagging GateThe Sagging Gate
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.
b. Do the diagonals of Quadrilateral 2 bisect each other? Explain.
22. Th e diagonals of six quadrilaterals are drawn below. Identify which quadrilaterals are rectangles and draw in the sides to check your answer.
296 SpringBoard® Mathematics with MeaningTM Level 1
My Notes
Properties of QuadrilateralsACTIVITY 5.6continued The Sagging GateThe Sagging Gate
SUGGESTED LEARNING STRATEGIES: Use Manipulatives, Create Representations, Look for a Pattern, Think/Pair/Share, Quickwrite, Self Revision/Peer Revision, Group Presentation
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.
298 SpringBoard® Mathematics with MeaningTM Level 1
Properties of QuadrilateralsACTIVITY 5.6continued The Sagging GateThe 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.
Write your answers on notebook paper. Show your work.
CHECK YOUR UNDERSTANDING
Write your answers on notebook paper. Show your work.
1. PONY is a parallelogram
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 PY = 5 inches and PO = 7 inches,
what are the lengths of ___
ON and ___
YN ?f. If m∠P = 112°, what are the measures
of ∠O, ∠N, and ∠Y? 2. PINT is a rectangle.
Name the 5 pairs of equal segments in the fi gure and explain why they are equal.
3. Write all names that apply to a quadrilateral with the given properties. Draw each fi gure.a. A quadrilateral with diagonals that
bisect each other.b. A quadrilateral with both pairs of
opposite sides having the same length and the diagonals having the same length.
c. A quadrilateral with diagonals that have the same length.
d. A quadrilateral with diagonals that are perpendicular and consecutive sides that are perpendicular.
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 R E F L E C T I O N
Naming a quadrilateral 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.
5.7Symmetry and TransformationsTracking the Migration
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.
Th ese examples of simple geometric designs have lines of symmetry.
1. Th e fi gure below is a square and has four lines of symmetry. Draw the lines of symmetry.
2. Draw the lines of symmetry for each fi gure below.
CONNECT TO SOCIAL STUDIESSOCIAL 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.
SUGGESTED LEARNING STRATEGIES: Close Reading, Marking the Text, Create Representations, Think/Pair/Share
Symmetry and TransformationsTracking the MigrationTracking the Migration
SUGGESTED LEARNING STRATEGIES: Create Representations, Group Presentation
Many of the standard fi gures in geometry can be defi ned 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 fi gure 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
302 SpringBoard® Mathematics with MeaningTM Level 1
My NotesMy Notes
Symmetry and TransformationsACTIVITY 5.7continued Tracking the MigrationTracking the Migration
FigureBest Name for the
FigureValue of n for
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 fi gure. A fi gure has rotational symmetry if it can be rotated about its center for less than 360° and fi t exactly over its original shape.
For example, if you rotate a square 90° about its center, it will fi t exactly over its original shape. If you rotate it another 90°, it will again fi t over its original shape. Aft er 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.
A fi gure that fi ts exactly over its original shape n times as it is rotated 360° about its center has n-fold symmetry.
MATH TERMS
8. For each fi gure, mark the center of the fi gure, then name the fi gure, and fi nd the value of n for n-fold symmetry.
Symmetry and TransformationsTracking the MigrationTracking the Migration
SUGGESTED LEARNING STRATEGIES: Close Reading, Marking the Text, Use Manipulatives, Create Representations, Look for a Pattern
Band patterns can be horizontal, vertical, or any other position.
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 fi gure, the pattern has translation symmetry.
Nine stars in a pattern are drawn below. Th e pattern continues on forever in both directions.
Th e 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. Th e 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 diff erent locations were related.
a. Circle the band pattern below that has 2-fold rotational symmetry but no vertical line symmetry.
CONNECT TO SOCIAL STUDIESSOCIAL STUDIES
Artifacts are objects manufac-tured, used, or modifi ed by humans, such as tools, utensils, and art.
304 SpringBoard® Mathematics with MeaningTM Level 1
Symmetry and TransformationsACTIVITY 5.7continued Tracking the MigrationTracking the Migration
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. Th e pottery piece below shows many characteristics of early traditional Native American artwork. Th e border near the top of the pot is an example of band symmetry. List all of the diff erent types of symmetry that can be found in the band.
SUGGESTED LEARNING STRATEGIES: Look for a Pattern, Think/Pair/Share
Symmetry and TransformationsTracking the MigrationTracking the Migration
A band pattern can be created by fi rst graphing a fi gure that is the basic design element and then using transformations to make congruent images of the fi gure.
12. What transformations do you already know?
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)
d. Plot (-4, 5) and (1, 5) and fi nd the distance between the points.
e. Plot (6, 2) and (6, 5) and fi nd the distance between the points.
8
10
6
4
2
–8–10 –6 –4 –2 2 4 6 8 10–2
–4
–6
–8
–10
y
x
MATH TERMS
Quadrant 2 Quadrant 1
Quadrant 3 Quadrant 4
x
8
10
6
4
2
–8–10 –6 –4 –2 2 4 6 8 10–2
–4
–6
–8
–10
y
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.
ACADEMIC VOCABULARY
A movement of a fi gure on a plane is a transformation.
CONNECT TO ALGEBRAALGEBRA
The distance between points that have different x-coordinates but the same y-coordinate is the abso-lute 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 fi nd the distance between two points that have dif-ferent y-coordinates but the same x-coordinate?
17. �A'B'C' is the image of �ABC aft er 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.
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'.
308 SpringBoard® Mathematics with MeaningTM Level 1
My Notes
Symmetry and TransformationsACTIVITY 5.7continued Tracking the MigrationTracking 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 fi gure on a coordinate grid by working with the coordinates of its vertices.
You have fl ipped fi gures over lines in earlier grades. Th is kind of transformation is a refl ection. Now you are going to refl ect a shape using the x-axis or the y-axis as a line of refl ection.
21. Graph points G (4, -1), H (2, -2), and J (3, -5) on the grid. Connect the points to form �GHJ.
Use the grid in Question 21 for Questions 22 and 23.
22. Draw a refl ection 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'.
Symmetry and TransformationsTracking the MigrationTracking the Migration
READING MATH
Remember, A" is read as “A double prime.”
SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Create Representations, Look for a Pattern
23. Draw a refl ection 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".
In earlier grades, you learned what happened when you turned or rotated fi gures. Now you will explore a rotation of a shape about a point on the coordinate plane.
24. 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'.
25. �K'L'M' is the image of �KLM aft er 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'.
310 SpringBoard® Mathematics with MeaningTM Level 1
Symmetry and TransformationsACTIVITY 5.7continued Tracking the MigrationTracking the Migration
CHECK YOUR UNDERSTANDING
Write your answers on notebook paper. Show your work.
1. Sketch each fi gure. Th en draw any lines of symmetry for each fi gure.a.
b.
c.
2. Look at the letters below.ABCDEFGHIJKLMNOPQRSTUVWXYZ
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?
3. List all of the diff erent types of symmetry that can be found in each of the band patterns below.
a.
b.
c.
d.
4. Without graphing fi nd the coordinates of A', B', and C' aft er 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 refl ection of �DEF over the x-axis. Label the points D', E', and F'.
b. Draw the refl ection of �DEF over the y-axis. Label the points D", E", and F".
6. MATHEMATICAL R E F L E C T I O N
Describe the relationship between the coordinates of
a point and the coordinates of its image aft er a refl ection across the x-axis or the y-axis.
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.
2. 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 fi gure such as a cube, a ball, or a cylinder.
3. How does this game relate to volume?
4. How would knowing how to fi nd volume of a fi gure make it easier to make a reasonable prediction? Would you be able to fi nd the exact number in the jar every time? Explain.
See if you can discover a formula for fi nding the volume of your group’s popcorn container.
5. Th e 2-dimensional drawing at right, representing the popcorn container, is a solid. Label each dimension on the drawing.
SUGGESTED LEARNING STRATEGIES: Guess and Check, Quickwrite, Debrief, Vocabulary Organizer
ACADEMIC VOCABULARY
Volume is the amount of space occupied by a three-dimensional fi gure. It is measured in cubic units, such as cubic inches (in . 3 ).
ACADEMIC VOCABULARY
A solid is a 3-dimensional geometric fi gure with dimensions of length, width, and height.
312 SpringBoard® Mathematics with MeaningTM Level 1
My NotesMy Notes
Exploring VolumeACTIVITY 5.8continued Guess How ManyGuess How Many
Th e popcorn container is a type of solid called a prism. Look at the fi gures in the table to help remember what a prism is.
Attributes are characteristics or qualities of something.
MATH TERMS
SUGGESTED LEARNING STRATEGIES: Group Presentation, Look for a Pattern, Vocabulary Organizer, Use Manipulatives
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.
To fi nd the amount of 3-dimensional space that is fi lled 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 specifi c name and explain your reasoning.
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 fi ll a 3-dimensional space.
Since you do not have enough cubic units to fi ll the popcorn container as a way of fi nding 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 fi ndings.
Length Width Height Volume
Figure 1 2 3 1
Figure 2 2 3 2
Figure 3 2 3 3
Figure 4 2 3 4
12. Use the data in the table to describe a pattern that can be used for fi nding 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 fi nd 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.
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 diff erent.
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 fi lled. Th e drawings below relate to this container.
3-D View (not drawn to scale) Top View
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.
5 in.18
2 in.58
1 in.34
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 fi gure 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.
316 SpringBoard® Mathematics with MeaningTM Level 1
My Notes
Exploring VolumeACTIVITY 5.8continued Guess How ManyGuess How Many
SUGGESTED LEARNING STRATEGIES: Look for a Pattern, Create Representations, Use Manipulatives, Debriefing, Vocabulary Organizer, Think/Pair/Share
22. Determine the formula for fi nding the volume of a rectangular prism and explain how you could use it to develop a formula for fi nding 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?
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 fi nding the volume of a prism to develop a formula for fi nding the volume of a cylinder. Explain your reasoning.
CONNECT TO APAP
In calculus, you will learn how to compute the volume of an irregular solid like the vase shown below.
318 SpringBoard® Mathematics with MeaningTM Level 1
Exploring VolumeACTIVITY 5.8continued Guess How ManyGuess How Many
CHECK YOUR UNDERSTANDING
Write your answers on notebook paper. Show your work.
1. What is the diff erence 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.
4. How much can the following container hold in cubic inches?
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?
6. A glass company sells vases in 3 diff erent styles:
a. Th e dimensions of the rectangular prism are shown below. Find the volume.
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.
base = radius = 2.7 in.altitude = 4 in. height = height = 10 in.
7. MATHEMATICAL R E F L E C T I O N
Compare and contrast the formulas for fi nding
volume of rectangular prisms, triangular prisms, and cylinders.
Polygons, Transformations, and VolumeGRAPHIC GEOMETRY
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. Refl ect �ABC across the y-axis. Label the refl ection 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:
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 diff erent kinds of fi gures 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.
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.
5. Find the area and perimeter of the following fi gure:
ACTIVITY 5.2
Answer each problem, and then use estimation to see if your answer is reasonable.
6. Find the circumference.
7. Find the circumference.
8. Th e diameter of a circular stone is 15 cm. What is the stone’s circumference?
9. Evelyn wants to glue a ribbon around a circular fl ower planter. Th e radius of the planter is 7 inches. How much ribbon (not including a bow) does she need?
322 SpringBoard® Mathematics with MeaningTM Level 1
10. Find and correct the mistake(s) that Myriam made when fi nding the circumference of a circle with a radius of 4 meters.
C = 4 × π C ≈ 4 × 3.14 C ≈ 12.56 m 2 11. 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.
13.
14.
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 fi gure.
16.
17.
18. Find the area and circumference.
19. Janice works at the local park and tends a fl ower garden that is in the shape of a circle with a diameter of 22 feet. Find the area of the garden.
20. Th e top and bottom of Monty’s kaleidoscope is a hexagon, as shown below. Find its area.
ACTIVITY 5.4
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 21.
23. Determine the perimeter of shape you made in Question 21.
24. Th e 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 fi nd the approximate area and perimeter of your state.
ACTIVITY 5.5
26. 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
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 5.6
29. FOAL is a rectangle.
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,
324 SpringBoard® Mathematics with MeaningTM Level 1
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 fi gure.
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 5.7
32. Copy each fi gure. Draw any lines of symmetry you see.
a.
b.
c.
33. List all the diff erent types of symmetry that can be found in each band pattern.
a.
b.
34. Draw a coordinate grid. Graph each point on the grid. Th en 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 aft er 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?
326 SpringBoard® Mathematics with MeaningTM Level 1
An important aspect of growing as a learner is to take the time to refl ect 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 specifi c 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:
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
Is Your Understanding Strong (S) or Weak (W)?
Concept 1
Concept 2
Concept 3
a. What will you do to address each weakness?
b. What strategies or class activities were particularly helpful in learning the concepts you identifi ed 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?
1. A botanist wants to study the plant life in a circular zone of the Florida Everglades. Th e 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. 150.7 square miles C. 1808.64 square miles
B. 7234.6 square miles D. 301.4 square miles
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?
8.5 cm
3 cm
3. Th e formula for the circumference of a circle can also be used for the circumference of a sphere. Th e planets in our solar system are spherical. Some measures are listed in this table.
Planet Diameter Circumference
Earth 24,888 milesMercury 3032 milesVenus 23,616 milesJupiter 88,846 miles
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. Th en double the diameter of Mercury and calculate what the new diameter would be. Verify your prediction.
![mlit.go.jp · 2019. 2. 1. · [235] [235) 123 [24.2] [240] [240] [24.3] [242 [242 [242] [242) [245 43] [242 (242 [242] [24.2] [ú.2] [242] [242 [240] [242] 27 087 087 [24.6] [24.6]](https://static.documents.pub/doc/80x56/613019b41ecc51586943e0fb/mlitgojp-2019-2-1-235-235-123-242-240-240-243-242-242-242.jpg)
