Circling 2.1
Geometry: Circling the Bases - Level 2
Lesson 1 – Pre-Visit Ballpark Figures – Part 1 Objective: Students will be able to: • Estimate, measure, and calculate length, perimeter, and area of various rectangles. Time Requirement: 1 class period, longer for activity Materials Needed: - Pencils - Paper (regular and graph paper) - Rulers - Calculators - Copies of the “Stuff My Locker!” worksheet – 1 for each student Vocabulary: Angle - The figure formed by two lines extending from the same point Congruent - Having the same size and shape Length - The measured distance from one end to the other of the longer side of an object Polygon – A closed figure made up of line segments Rectangle - A quadrilateral with two pairs of congruent sides and four right angles Right Angle - An angle measuring exactly 90 degrees Parallel – Lines moving in the same direction but always the same distance apart Perpendicular – Lines that intersect at 90 degree angles Perimeter - The distance around the outside of a polygon Width - The measured distance from one end to the other of the shorter side of an object
4
Geometry: Circling the Bases - Level 2
Applicable Common Core State Standards: CCSS.Math.Content.6.G.A.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems. CCSS.Math.Content.7.G.A.1 Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. CCSS.Math.Content.7.G.A.2 Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. CCSS.Math.Content.7.G.B.6 Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. CCSS.Math.Content.6.EE.A.2 Write, read, and evaluate expressions in which letters stand for numbers. • CCSS.Math.Content.6.EE.A.2a Write expressions that record operations with numbers and with letters standing for numbers. CCSS.Math.Content.7.EE.B.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
5
Geometry: Circling the Bases - Level 2
Lesson
1. Begin the lesson by reviewing that geometry is a branch of mathematics that deals with points, lines, angles, and shapes. 2. Review the term polygon. Explain that a polygon is a plane shape (two-dimensional or “flat”) with straight sides that connect. The points at which the sides connect are angles. In fact, the word polygon actually means “many-angles.” 3. Polygons are often named after the number of sides they have. Draw several different types of polygons on the board, and then label each, demonstrating how the number of sides gives each polygon its name. - Triangle: A three-sided polygon. - Quadrilateral: A four-sided polygon. - Pentagon: A five-sided polygon. - Hexagon: A six-sided polygon. 4. Briefly discuss squares and rectangles. Both shapes are quadrilaterals. Squares have four congruent sides and four right angles. Rectangles have two congruent sides and four right angles. 5. Explain that today you will be reviewing different ways to measure polygon s. Review the following definitions: - Length - The measured distance from one end to the other of the longer side of an object - Width - The measured distance from one end to the other of the shorter side of an object 6. Explain (or review) that perimeter is the measure of the distance around the outside of a polygon. It is found by adding the lengths of all sides of a figure. Sometimes the lengths of each side are given; sometimes the lengths will need to be measured.
6
Geometry: Circling the Bases - Level 2
7. On the board, draw a rectangle labeled with a length of 4 feet and width of 3 feet. Then draw a right triangle with a base of 4 feet, height of 3 feet, and a hypotenuse of 5 feet. Demonstrate that to measure the perimeter of any polygon, the lengths of each side are added together. 8. Provide students with the formula to find the perimeter of a rectangle: Perimeter = 2 x (length + width) 9. Remind students that rectangles have two pairs of parallel sides. Opposite sides are equal. Squares have four equal sides, so if the length of one side is known, the lengths of all other sides are known as well. 10. Draw a diagram of a baseball field as shown below. Discuss that the infield is commonly known as the “baseball diamond.” Home plate, first, second, and third bases make up the four points of the diamond. 11. The baseball “diamond” is a square. Review that a square is a quadrilateral with all sides of equal length and all angles measuring 90 degrees. Point out that each side of a Major League baseball diamond measures 90 feet. Have students determine the perimeter of the baseball diamond.
90 ft.
90 ft.
90 ft.
90 ft.
7
Geometry: Circling the Bases - Level 2
12. Review that 90’ + 90’ + 90’ + 90’ = 360’. 13. Explain that the area of a figure measures the amount of space inside it. Area is measured using square units. For example, if inches are used to measure length, then the area will be measured in square inches. 14. Show students that square units are indicated with a superscript 2 following the units of measure. For example, 90’². 15. Provide students with the formula to find an object’s area: Area = length x width 16. Have students determine the area of the infield drawn previously. 17. Review that 90’x 90’ = 8100’². 18. Introduce the activity.
8
Geometry: Circling the Bases - Level 2
Activity
1. Introduce the activity by explaining that perimeter and area are used all the time by baseball groundskeepers and stadium architects. In this multi-day activity, students will have the chance to act as both groundskeepers and architects as they put area and perimeter formulas into practice. 2. Challenge students with the following problem: Your hometown has just decided to build a brand new baseball field for your Little League team. The city council has set aside $100,000 to build this new field. A rectangular piece of land (275 feet x 600 feet) has been donated to the city for the purpose of building this field. In order to build the field, it first must be determined if the new field will fit on the donated land, and also if the city has budgeted enough money to build it. 3. Before proceeding, ensure that each student has a pencil, a calculator, regular paper, graph paper, and a ruler. 4. Ask students, “What is the area of the donated land?” (Answer: 165,000 ft2) 5. Have students use graph paper to draw a scale outline of the donated lot. *Note* Students may want to connect multiple pieces of graph paper in order to create this model. 6. Next, review the required measurements of a Little League baseball field. Write the following on the board or on a sheet of chart paper for reference. Base to base
60 feet
Pitching rubber to home plate
46 feet
Backstop to home plate
25 feet
Pitchers mound to the grass line of the infield
50 feet
Foul lines
180 feet
9
Geometry: Circling the Bases - Level 2
7. Have students use the given measurements to draw an outline of the proposed baseball field within the scale outlines they created earlier. 8. Ask students, “Based on the dimensions of the Little League baseball field, will there be enough room to build the Little League baseball field on the donated land?” (Yes) 9. Ask, “Is there anything not shown on the diagram of the baseball field that needs to be accounted for that is found at a baseball field?” (Answer: Stands, Dugouts, Outfield Fences) 10. As a class, discuss how much space you want to allow for stands and team dugouts at this ballpark. *Note* All outfield fences must be at least 180 feet from home plate. 11. Have students calculate the amount of fencing that will be required to encircle the baseball field. (This figure will vary depending on your outfield distance choices.) 12. Have students calculate the area of the infield and the area of the outfield. (Infield area = 8100’²; Outfield area will vary based on your class’ choice in step #10) 13. Collect students’ scale drawings for use in Lesson 2 of this unit.
Conclusion: To conclude this lesson and check for understanding, provide students with “Stuff My Locker!” worksheet (included), and have students work independently to find the area and perimeter of the objects given.
10
Geometry: Circling the Bases - Level 2
Stuff My Locker!
Name: __________________
Date: ______________
Instructions: You want your locker at school to reflect that you’re a HUGE baseball fan. Your locker measures 12” wide, 12” deep, and 30” tall. Counting the inside of the door, you have four surfaces that you can cover with baseball gear. First, calculate the total surface area of the walls of your locker. Then, calculate the perimeter and area of each of the following posters. Will they fit? Show your work alongside each problem, or on a separate sheet of paper. 1. Determine the area of each wall of your locker.
2. Determine the total area of the 4 walls of your locker.
11
Calculate the perimeter and/or area of each of the following posters: 3. B
A
Side A = 10” Side B = 20” Perimeter: _______________ Will it fit? Circle one: YES
NO
4. A
B
C Side A = 6” Side B = 10” Side C = 3.5” Perimeter: _______________ Will it fit? Circle one: YES
NO
12
5. A
B
Side A = 9.5” Side B= 12” Perimeter: _______________ Area: _________________ Will it fit? Circle one: YES
6.
NO
A
Side A = 12” Perimeter: _______________ Area: _________________ Will it fit? Circle one: YES
NO
13
7. A
B
Side A = 8” Side B= 32” Perimeter: _______________ Area: _________________ Will it fit? Circle one: YES
8.
NO
A
B
Side A = 5” Side B= 4” Perimeter: _______________ Area: _________________ Will it fit? Circle one: YES
NO
14
Geometry: Circling the Bases - Level 2
Stuff My Locker! – Answer Key Your locker measures 12” wide, 12” deep, and 30” tall. 1. Determine the area of each wall of your locker.
12” x 30” = 360”2
2. Determine the total area of the 4 walls of your locker.
360” 2 x 4 = 1440” 2
3. Side A = 10”, Side B = 20”, *Side C = 20” (Not given, students must determine) Perimeter: 50” Will it fit? YES (If hung vertically) 4. Side A = 6”, Side B = 10”, Side C = 3.5”, *Side D = 3.5”, *Side E = 10” Perimeter: 33” Will it fit? YES 5. Side A = 9.5”, Side B= 12”, *Side C = 9.5”, *Side D = 12” Perimeter: 43” Area: 114” 2 Will it fit? YES 6. Side A = 12”, *Sides B, C, D = 12” Perimeter: 48” Area: 144” 2 Will it fit? YES 7. Side A = 8”, Side B= 32”, *Side C = 8”, *Side D = 32” Perimeter: 80” Area: 256” 2 Will it fit? NO 8. Side A = 5”, Side B = 4”, *Side C = 5”, *Side D = 4” Perimeter: 18” Area: 20” 2 Will it fit? YES
15
Lesson 1 – Pre-Visit Ballpark Figures – Part 1 Objective: Students will be able to: • Estimate, measure, and calculate length, perimeter, and area of various rectangles. Time Requirement: 1 class period, longer for activity Materials Needed: - Pencils - Paper (regular and graph paper) - Rulers - Calculators - Copies of the “Stuff My Locker!” worksheet – 1 for each student Vocabulary: Angle - The figure formed by two lines extending from the same point Congruent - Having the same size and shape Length - The measured distance from one end to the other of the longer side of an object Polygon – A closed figure made up of line segments Rectangle - A quadrilateral with two pairs of congruent sides and four right angles Right Angle - An angle measuring exactly 90 degrees Parallel – Lines moving in the same direction but always the same distance apart Perpendicular – Lines that intersect at 90 degree angles Perimeter - The distance around the outside of a polygon Width - The measured distance from one end to the other of the shorter side of an object
4
Geometry: Circling the Bases - Level 2
Applicable Common Core State Standards: CCSS.Math.Content.6.G.A.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems. CCSS.Math.Content.7.G.A.1 Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. CCSS.Math.Content.7.G.A.2 Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. CCSS.Math.Content.7.G.B.6 Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. CCSS.Math.Content.6.EE.A.2 Write, read, and evaluate expressions in which letters stand for numbers. • CCSS.Math.Content.6.EE.A.2a Write expressions that record operations with numbers and with letters standing for numbers. CCSS.Math.Content.7.EE.B.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
5
Geometry: Circling the Bases - Level 2
Lesson
1. Begin the lesson by reviewing that geometry is a branch of mathematics that deals with points, lines, angles, and shapes. 2. Review the term polygon. Explain that a polygon is a plane shape (two-dimensional or “flat”) with straight sides that connect. The points at which the sides connect are angles. In fact, the word polygon actually means “many-angles.” 3. Polygons are often named after the number of sides they have. Draw several different types of polygons on the board, and then label each, demonstrating how the number of sides gives each polygon its name. - Triangle: A three-sided polygon. - Quadrilateral: A four-sided polygon. - Pentagon: A five-sided polygon. - Hexagon: A six-sided polygon. 4. Briefly discuss squares and rectangles. Both shapes are quadrilaterals. Squares have four congruent sides and four right angles. Rectangles have two congruent sides and four right angles. 5. Explain that today you will be reviewing different ways to measure polygon s. Review the following definitions: - Length - The measured distance from one end to the other of the longer side of an object - Width - The measured distance from one end to the other of the shorter side of an object 6. Explain (or review) that perimeter is the measure of the distance around the outside of a polygon. It is found by adding the lengths of all sides of a figure. Sometimes the lengths of each side are given; sometimes the lengths will need to be measured.
6
Geometry: Circling the Bases - Level 2
7. On the board, draw a rectangle labeled with a length of 4 feet and width of 3 feet. Then draw a right triangle with a base of 4 feet, height of 3 feet, and a hypotenuse of 5 feet. Demonstrate that to measure the perimeter of any polygon, the lengths of each side are added together. 8. Provide students with the formula to find the perimeter of a rectangle: Perimeter = 2 x (length + width) 9. Remind students that rectangles have two pairs of parallel sides. Opposite sides are equal. Squares have four equal sides, so if the length of one side is known, the lengths of all other sides are known as well. 10. Draw a diagram of a baseball field as shown below. Discuss that the infield is commonly known as the “baseball diamond.” Home plate, first, second, and third bases make up the four points of the diamond. 11. The baseball “diamond” is a square. Review that a square is a quadrilateral with all sides of equal length and all angles measuring 90 degrees. Point out that each side of a Major League baseball diamond measures 90 feet. Have students determine the perimeter of the baseball diamond.
90 ft.
90 ft.
90 ft.
90 ft.
7
Geometry: Circling the Bases - Level 2
12. Review that 90’ + 90’ + 90’ + 90’ = 360’. 13. Explain that the area of a figure measures the amount of space inside it. Area is measured using square units. For example, if inches are used to measure length, then the area will be measured in square inches. 14. Show students that square units are indicated with a superscript 2 following the units of measure. For example, 90’². 15. Provide students with the formula to find an object’s area: Area = length x width 16. Have students determine the area of the infield drawn previously. 17. Review that 90’x 90’ = 8100’². 18. Introduce the activity.
8
Geometry: Circling the Bases - Level 2
Activity
1. Introduce the activity by explaining that perimeter and area are used all the time by baseball groundskeepers and stadium architects. In this multi-day activity, students will have the chance to act as both groundskeepers and architects as they put area and perimeter formulas into practice. 2. Challenge students with the following problem: Your hometown has just decided to build a brand new baseball field for your Little League team. The city council has set aside $100,000 to build this new field. A rectangular piece of land (275 feet x 600 feet) has been donated to the city for the purpose of building this field. In order to build the field, it first must be determined if the new field will fit on the donated land, and also if the city has budgeted enough money to build it. 3. Before proceeding, ensure that each student has a pencil, a calculator, regular paper, graph paper, and a ruler. 4. Ask students, “What is the area of the donated land?” (Answer: 165,000 ft2) 5. Have students use graph paper to draw a scale outline of the donated lot. *Note* Students may want to connect multiple pieces of graph paper in order to create this model. 6. Next, review the required measurements of a Little League baseball field. Write the following on the board or on a sheet of chart paper for reference. Base to base
60 feet
Pitching rubber to home plate
46 feet
Backstop to home plate
25 feet
Pitchers mound to the grass line of the infield
50 feet
Foul lines
180 feet
9
Geometry: Circling the Bases - Level 2
7. Have students use the given measurements to draw an outline of the proposed baseball field within the scale outlines they created earlier. 8. Ask students, “Based on the dimensions of the Little League baseball field, will there be enough room to build the Little League baseball field on the donated land?” (Yes) 9. Ask, “Is there anything not shown on the diagram of the baseball field that needs to be accounted for that is found at a baseball field?” (Answer: Stands, Dugouts, Outfield Fences) 10. As a class, discuss how much space you want to allow for stands and team dugouts at this ballpark. *Note* All outfield fences must be at least 180 feet from home plate. 11. Have students calculate the amount of fencing that will be required to encircle the baseball field. (This figure will vary depending on your outfield distance choices.) 12. Have students calculate the area of the infield and the area of the outfield. (Infield area = 8100’²; Outfield area will vary based on your class’ choice in step #10) 13. Collect students’ scale drawings for use in Lesson 2 of this unit.
Conclusion: To conclude this lesson and check for understanding, provide students with “Stuff My Locker!” worksheet (included), and have students work independently to find the area and perimeter of the objects given.
10
Geometry: Circling the Bases - Level 2
Stuff My Locker!
Name: __________________
Date: ______________
Instructions: You want your locker at school to reflect that you’re a HUGE baseball fan. Your locker measures 12” wide, 12” deep, and 30” tall. Counting the inside of the door, you have four surfaces that you can cover with baseball gear. First, calculate the total surface area of the walls of your locker. Then, calculate the perimeter and area of each of the following posters. Will they fit? Show your work alongside each problem, or on a separate sheet of paper. 1. Determine the area of each wall of your locker.
2. Determine the total area of the 4 walls of your locker.
11
Calculate the perimeter and/or area of each of the following posters: 3. B
A
Side A = 10” Side B = 20” Perimeter: _______________ Will it fit? Circle one: YES
NO
4. A
B
C Side A = 6” Side B = 10” Side C = 3.5” Perimeter: _______________ Will it fit? Circle one: YES
NO
12
5. A
B
Side A = 9.5” Side B= 12” Perimeter: _______________ Area: _________________ Will it fit? Circle one: YES
6.
NO
A
Side A = 12” Perimeter: _______________ Area: _________________ Will it fit? Circle one: YES
NO
13
7. A
B
Side A = 8” Side B= 32” Perimeter: _______________ Area: _________________ Will it fit? Circle one: YES
8.
NO
A
B
Side A = 5” Side B= 4” Perimeter: _______________ Area: _________________ Will it fit? Circle one: YES
NO
14
Geometry: Circling the Bases - Level 2
Stuff My Locker! – Answer Key Your locker measures 12” wide, 12” deep, and 30” tall. 1. Determine the area of each wall of your locker.
12” x 30” = 360”2
2. Determine the total area of the 4 walls of your locker.
360” 2 x 4 = 1440” 2
3. Side A = 10”, Side B = 20”, *Side C = 20” (Not given, students must determine) Perimeter: 50” Will it fit? YES (If hung vertically) 4. Side A = 6”, Side B = 10”, Side C = 3.5”, *Side D = 3.5”, *Side E = 10” Perimeter: 33” Will it fit? YES 5. Side A = 9.5”, Side B= 12”, *Side C = 9.5”, *Side D = 12” Perimeter: 43” Area: 114” 2 Will it fit? YES 6. Side A = 12”, *Sides B, C, D = 12” Perimeter: 48” Area: 144” 2 Will it fit? YES 7. Side A = 8”, Side B= 32”, *Side C = 8”, *Side D = 32” Perimeter: 80” Area: 256” 2 Will it fit? NO 8. Side A = 5”, Side B = 4”, *Side C = 5”, *Side D = 4” Perimeter: 18” Area: 20” 2 Will it fit? YES
15
