Topic: Measurement



Grade: 5 Circles, Circles Everywhere

Mathematics

Geometry & Spatial Sense; Measurement

12

Topic: Measurement

Standard: Selects appropriate customary and metric units of measure for length (including perimeter and circumference), area, capacity/volume, weight/mass, time, and temperature. Length, Millimeter, Inch, Centimeter, Foot, Meter, Yard, Kilometer, Mile, Capacity, Milliliter, Ounce, Centiliter, Cup, Liter, Pint (Liquid and Dry), Quart (Liquid and Dry), Gallon, Weight/Mass, Milligram, Ounce, Gram, Pound, Kilogram, Time, Second, Week, Minute, Month, Hour, Year, Day, Decade, Century, Temperature, Degree Fahrenheit, Degree Celsius

14

Topic: Measurement

Standard: Determine perimeter, area, and volume of various geometric figures connecting concrete experiences (covering, filling, and counting) to computation and formulas.

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Procedures/Activities

Step:  1 Duration: 5 minutes

The teacher will demonstrate how to construct a circle on the board using a chalkboard compass. First, open the compass to the desired length (this will be the length of the radius). Then, mark the center clearly and construct the circle. The teacher will explain the meaning of a circle to the students (a circle is a shape with all points the same distance from its center). The teacher will have the students construct a circle using a compass on the paper.

Step:  2 Duration: 15 minutes

The teacher will explain the meaning of the radius to the students (the radius is the distance from the center of a circle to a point on the circle). The teacher will construct a radius on the circle that was drawn on the chalkboard in Step 1. The teacher will measure the length of the radius using a ruler. Record this length on the chalkboard. Then, draw and measure another radius on the same circle. The students will see that all the radii of a circle are exactly the same length. The students will construct and measure a radius on the circle drawn on paper. The teacher will then explain the meaning of the diameter of a circle (the distance across a circle through its center). The teacher will draw a diameter on the circle that was drawn on the chalkboard. The teacher will ask the students to compare the lengths of the diameter and the radius of the circle. The students should realize that the length of the diameter is twice the length of the radius. The teacher will state the fact that in every circle, no matter how big or how small, the length of the diameter is always twice the length of the radius of the same circle. Therefore, if the radius is given, the diameter will be calculated by multiplying the radius by 2. If the diameter is given, the radius will be calculated by dividing the diameter by 2. For example, if a circle has a radius of 4 cm, the diameter will be 8 cm (4 x 2 = 8). If the diameter is 14 in, then the radius is 7 in (12/2=7). The teacher will reinforce this concept with the students by having them state the radius of a circle given the diameter, and vice versa. The teacher will say the following: "The radius of a circle is 12 ft. What is the diameter?" The students should write "24 ft" on their slates. The teacher will repeat this activity with the following problems: Radius = 3 cm (Diameter = 6 cm); Diameter = 18 mm (Radius = 9 mm); Diameter = 20 in (Radius = 10 in); Radius = 15 cm (Diameter = 30 cm).

Step:  3 Duration: 10 minutes

The students will construct several circles on paper using a compass. The students will construct and measure the radius and diameter on each circle. The students will record the lengths of the radius and diameter of each circle on the attached chart.

Attachments for Step 3

Title: Circumference Chart FileName: Circumference Chart.rtf

Description: The students will record the lengths of the radius and diameter of the circles constructed on paper using a compass.

Step:  4 Duration: 5 minutes

The teacher will hold up a can of soup or a beverage can to the class. The teacher will ask the students for their suggestions on how to find the distance around the can. The teacher will tell the students that the distance around the can is called the "circumference" of the circle. This is the same thing as the "perimeter" of the circle (distance around the outside of the object). The students should suggest using a tape measure or a string and ruler (wrap the string around the outside of the circle, measure the length of the string needed to get around the whole circle). The teacher will use either method to demonstrate how to find the circumference of the can.

Step:  5 Duration: 10 minutes

The students will determine the circumference of each circle that was constructed in Step 3. The students will wrap a piece of string around the outside of the circle and will measure the length of the string. The students will record the circumference on the chart started in Step 3. The teacher will collect the charts at the end of the lesson for assessment purposes.

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Materials and Equipment

Per Student:

1. compass with a pencil 2. paper 3. string and ruler and/or tape measure

4. slate and chalk or white board and wipe-off marker

Per Teacher:

1. chalkboard compass 2. string 3. one can of soup or a beverage can 4. ruler

Standards (Local and/or National) Total Duration 45 minutes

Topic: Measurement Measuring Bubbles [pic]

Procedures/Activities

Step:  1 Duration: Time Varies

Lesson Preparation: The teacher will fill a cup with bubble solution for each student. The teacher will print out the attached Circumference Chart and will make a copy for each student.

Attachments for Step 1

Title: Circumference Chart FileName: Circumference Chart.html

Description: The teacher will print out the attached Circumference Chart and will make a copy for each student. The students will use this chart to record data in today's lesson.

Step:  2 Duration: 30 minutes

The students will clear off their desks. There should be nothing on the desks at all. Each student will pour about three spoonfuls of bubble solution on his/her desk. Then, the student will blow into the solution with a straw. The student should blow very slowly and gently to form a bubble. The student should continue blowing into the straw until the bubble pops. The popped bubble will form a circle on the desk. The student will measure the diameter of the circle using the ruler and will record it on the Circumference Chart. The student will divide the diameter by two to find the radius and will record this on the chart as well. The student will then wrap the string around the outside of the circle and will measure the length of the string to determine the circumference of the circle, which will also be recorded on the chart. Once the radius, diameter, and circumference of the circle are recorded on the Circumference Chart, the student will wipe off the desk with paper towels and will repeat this activity with new bubbles until the teacher calls time. The teacher may make a contest out of this activity to see who can blow the bubble with the largest circumference. The teacher may reward the winner.

Step:  3 Duration: 15 minutes

The students' desks should all be cleared off and free from bubbles. The students should have their Circumference Charts and a pencil on their desk. The teacher will have the students divide each circumference by its corresponding diameter using a calculator. The students will record these calculations on the Circumference Charts. The teacher will discuss the results with the class. The answers should all be close to 3.14. The teacher will point this fact out to the students. The teacher will tell the students that "3.14" is a very special number and it has a very special name. It is called "pi." Whenever the circumference of a circle is divided by its diameter, the answer will always be 3.14, or pi. The concept of pi will be further explored in tomorrow's lesson (Day 128). The teacher will collect the students' Circumference Charts for assessment purposes.

Materials and Equipment

Per Student:

1. compass with a pencil

2. paper 3. string, ruler and/or tape measure 4. white board and wipe-off marker

Topic: Measurement Doughnut Fun[pic]

Procedures/Activities

Step:  1 Duration: 20 minutes

Each student will have a doughnut placed on top of a paper towel. The teacher will guide the students to complete the following activity: The student will measure the diameter of the doughnut using a ruler (measure from one end of the doughnut to the other end of the doughnut while going through the center). The student will write the measurement of the diameter on paper. The student will divide the diameter by 2 to calculate the radius. The student will write the measurement of the radius on the paper as well. Then, the student will calculate the circumference of the doughnut with the aid of a calculator. The teacher may write the formula for circumference on the board to help the students (Circumference = Diameter x Pi (3.14)). The student will write the calculated circumference on the paper. The student will then determine the area of the circle with the aid of a calculator. The teacher may write the formula for the area of a circle on the board to help the students (Area = pi(3.14)x radius squared). The student will write the calculated area on the paper. Once all the students have determined the circumference and area of the doughnuts, the students will share their results with the class. The students may then consume their doughnuts (yummy!).

Step:  2 Duration: 25 minutes

The student will construct a circle on paper using a compass with a pencil. The student will then measure the radius and the diameter of the circle using a ruler. The student will write these measurements on the back of the paper. The student will then calculate the circumference and area of the circle with the aid of a calculator. The student will write these measurements on the back of the paper as well. Then, the student will switch circles with another student. The student will determine the radius, diameter, circumference, and area of the new circle using a ruler and a calculator and will write the results on a slate. The student will check their answers with the answers on the back of the paper. If the results are not a match, the students will meet to discuss the problem and to determine the solution. Once the results are the same, the student may switch circles with another student who is ready to switch. The student will repeat this activity until the teacher calls time.

Description: The teacher will use the attached checklist to record students' performance in the class activities for assessment purposes.

Materials and Equipment

Per Student:

1. doughnut

2. paper towel to place doughnut on

3. compass with pencil

4. paper 5. ruler 6. calculator 7. pencil

8. slate and chalk or white board and wipe-off marker

Mathematics What is Pi?

Geometry & Spatial Sense; Measurement

12

Topic: Measurement

Standard: Selects appropriate customary and metric units of measure for length (including perimeter and circumference), area, capacity/volume, weight/mass, time, and temperature. Length, Millimeter, Inch, Centimeter, Foot, Meter, Yard, Kilometer, Mile, Capacity, Milliliter, Ounce, Centiliter, Cup, Liter, Pint (Liquid and Dry), Quart (Liquid and Dry), Gallon, Weight/Mass, Milligram, Ounce, Gram, Pound, Kilogram, Time, Second, Week, Minute, Month, Hour, Year, Day, Decade, Century, Temperature, Degree Fahrenheit, Degree Celsius

14

Topic: Measurement

Standard: Determine perimeter, area, and volume of various geometric figures connecting concrete experiences (covering, filling, and counting) to computation and formulas.

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Procedures/Activities

Step:  1 Duration: Time varies

The teacher will print out the attached "What Is Pi?" chart. Each student will need a copy of the chart to use in today's activity.

Attachments for Step 1

Title: What Is Pi? FileName: Round Objec1.doc

Description: The teacher will print out the attached "What Is Pi?" chart. Each student will need a copy of the chart to use in today's activity.

Title: Round Object 1 FileName: Round Object 1.cwk

Description: Claris version

Step:  2 Duration: 25 minutes

The teacher will review the concept of pi with the students. The teacher will explain that when the circumference of a circle (the perimeter of the circle) is divided by its diameter (the length of a segment that goes through the center of the circle to reach both ends of the circle), the answer will always be 3.14, or pi. Then, the teacher will pass out the round objects (paper plate, yo-yo, bracelet, mug, can of food, etc.) to the students to use in today's activity. The students may share these objects. The students will write the name of each object on the chart attached in Step 1. Then, the students will measure the diameter of each circle and will record the measurements on the chart. The students will also measure and record the circumference of each object. One of the following methods may be utilized to determine the circumference: wrap a piece of string around the object and measure the string, wrap a measuring tape around the cylinder near its base, or roll the cylinder 1 revolution on a meter stick. Once the students have recorded the diameter and circumference of each object, they will divide the circumference by the diameter for each object and will record this number on the chart. This number should be around 3.14 for all the objects. The teacher will state the fact that anytime the circumference of a circle is divided by its diameter, the answer will always be 3.14 (pi). The concept of pi may be reinforced using the attached Web resource.

Web Resources for Step 2

Title: Activities for Pi math

URL:

Annotation: The students should work in small groups on the computer. The students will explore the concept of pi by reading the information and viewing the online video and activities.

Step:  3 Duration: 10 minutes

The teacher will state the fact again that when the circumference of a circle is divided by its diameter, the answer will always be 3.14 (pi). In other words, Circumference/Diameter=3.14. The teacher will write this formula on the board. The teacher will then say, "How can we figure out the circumference of a circle without measuring it if we only know the diameter?" The teacher will have the students think about the formula "Circumference/Diameter=3.14" and how it can be altered to figure out the Circumference. The teacher will guide the students to come up with the formula "Diameter x 3.14 (pi) = Circumference." For example, if the diameter is 10, then multiply 10 x 3.14 to calculate the circumference, which is 31.4. The teacher will then say to the students, "What happens if only the radius of the circle is given?" The students should realize that the diameter can be calculated by multiplying the radius by 2. Therefore, "2 x Radius x 3.14 (pi) = Circumference." The teacher will write this formula on the board. For example, if the radius is 3.5, then multiply the radius by 2, which is equal to 7. Then, multiply 7 by 3.14, which is equal to 21.98. The circumference is 21.98. The concept of calculating the circumference of a circle may be reinforced using the attached Web resource.

Web Resources for Step 3

Title: Circumference of A Circle

URL:

Annotation: The students should work with a partner or in small groups on the computer. The concept of calculating the circumference of a circle will be reinforced, and the students will be able to practice this skill in an on-line activity.

Step:  4 Duration: 10 minutes

The students will determine the circumference of several circles given the radius or diameter of each circle. The students will use the formula for determining circumference to complete this activity (Circumference = 2 x radius x 3.14 OR diameter x 3.14). The teacher will remind the students that the radius is the length of the segment that reaches from the center of the circle to one end of the circle. The teacher will write the following problems on the board: 1) Radius = 6 m (6 x 2 x 3.14 = 37.7 m) 2) Diameter = 8 ft (8 x 3.14 = 25.1 ft) 3) Diameter = 15 m (15 x 3.14 = 47.1 m) The teacher will guide the students as they complete the first problem. Then, the students will complete the problems on their slates. The teacher will guide volunteers to demonstrate how to solve the problems on the board. The students may practice calculating the circumference of circles using the attached Web resource.

Web Resources for Step 4

Title: Calculate Circumference

URL:

Annotation: The students will work with a partner or in small groups. The students will practice calculating the circumference of circles.

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Materials and Equipment

Per Student:

1. "What Is Pi?" chart (attached in Step 1)

2. pencil

3. measuring tape

4. meter stick

5. string

6. calculator

7. slate and chalk or white board and wipe-off marker

Per Teacher:

1. variety of round objects for the students to use in today's activity (paper plate, bracelet, cup, yo-yo, ball, can of food, beverage can, mug, jar, etc.)

Standards (Local and/or National)

Total Duration

45 minutes

Technology Connection

Calculator On-line Activities (Computer)

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Assessment

The teacher will collect the "What Is Pi?" charts completed in today's activity. The teacher will check for accuracy and understanding (Circumference/Diameter should be close to 3.14 for all objects).

Extension

Remediation

The student may use a calculator for all computation problems. The student may work with a partner for peer assistance.

Mathematics Pi are squared!

Geometry & Spatial Sense; Measurement

14

Topic: Measurement

Standard: Determine perimeter, area, and volume of various geometric figures connecting concrete experiences (covering, filling, and counting) to computation and formulas.

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Procedures/Activities

Step:  1 Duration: Time Varies

Lesson Preparation: The teacher will print out the attached grid paper. Each student will need a copy of the grid paper for today's activity.

Step:  2 Duration: 10 minutes

The teacher will review the concept of circumference with the students. The teacher will remind the students of the circumference formula: Diameter x 3.14 (pi) = Circumference or 2 x Radius x 3.14 (pi) = Circumference. The teacher will write these formulas on the board. The students will practice calculating the circumference of circles using the formula. They may use a calculator to compute the multiplication. The teacher will write the following problems on the board for the students to solve on their slates: 1. Radius = 4 in, What is the Circumference? (25.12 inches) 2. Diameter = 12 ft, What is the Circumference? (37.68 feet) The teacher will guide student volunteers as they demonstrate how to solve the problems on the board. The students will continue to practice this skill using the attached Web resource.

Web Resources for Step 2

Title: Lesson on Circumference of a Circle

URL:

Annotation: This Web site will further explain the concept of circumference of a circle to the students with the use of examples. The students will be able to practice this skill using an on-line activity.

Step:  3 Duration: 10 minutes

The teacher will tell the students that the circumference of a circle is actually the perimeter of the circle. The perimeter of an object is the outside of the object. The students will now learn how to find the area of the circle, or the inside of the circle. Each student will have a piece of grid paper and a compass with a pencil. The student will construct a circle on the grid paper with the compass and pencil. The student will calculate the area of the circle by counting the number of squares on the inside of the circle (this will be an approximate area since parts of squares may be included). The student will write the area on the inside of the circle. The student will construct another circle on the grid paper and will repeat this activity to determine the area of the circle. The student will write the estimated area on the inside of the circle.

Step:  4 Duration: 15 minutes

The teacher will tell the students that just as there is a formula for the circumference of a circle, there is also a formula for the area of a circle. The teacher will write the formula for the area of a circle on the board: Area = pi(3.14) x radius x radius or Area = pi(3.14) x radius squared (radius x radius = radius squared). The teacher will demonstrate how to compute this formula on the board. Write the following on the board: "The radius is 3 in." Multiply the radius by the radius (3 x 3 = 9). Then, multiply 9 by 3.14(pi), which is equal to 28.26 square inches. Explain to the students that the unit is squared because it was also multiplied by itself (inch x inch = inches squared). Demonstrate how to compute this formula again on the board. Write the following on the board: "The diameter is 10 cm." Since the radius is needed in the formula, determine the radius of the circle by dividing the diameter by 2. Therefore, the radius is 5 cm. Then, multiply radius x radius (5 x 5), which is equal to 25. Then, multiply 25 by 3.14 (pi), which is equal to 78.5 square centimeters. This concept will be reinforced using the attached Web resource.

Web Resources for Step 4

Title: Area of a Circle

URL:

Annotation: The students will work with a partner or a small group on the computer. The concept of determining the area of a circle will be reinforced on this Web site. The students will be able to practice this skill interactively.

Step:  5 Duration: 10 minutes

The teacher will write the following on the board: "The radius is 2 in. Determine the area of the circle." The students may use a calculator as they work this problem out on their slates. The teacher will guide a student volunteer as he/she demonstrates how to compute this on the board (2x2=4, 4x3.14=12.56 square inches). The teacher will write the following on the board: "The diameter is 12 cm. Determine the area of the circle." The students may use a calculator as they compute this problem on their slates. The teacher will guide a student as he/she demonstrates how to solve this problem on the board (The diameter is 12, so the radius is 6. Multiply 6x6, which is equal to 36. Multiply 36 by 3.14, which is equal to 113.04 square centimeters). The students will retrieve the grid paper with the two circles that were constructed in Step 3. The students will measure the diameter of each circle using a ruler and they will divide each diameter by 2 to obtain the radius of each circle. Then, the students will use a calculator to determine the area of each circle. The students may also determine the circumference of each circle as well. The teacher will collect the students' completed work to check for understanding. The students will practice determining the area of circles using the attached Web resource.

Web Resources for Step 5

Title: Lesson on Area of a Circle

URL:

Annotation: The students will work with a partner or in a small group on the computer. The concept of determining the area of a circle will be reinforced, and the students will practice this skill with an on-line activity.

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Materials and Equipment

Per Student:

1. compass with pencil

2. grid paper (attached in Step 1)

3. slate and chalk or white board and wipe-off marker

4. calculator

5. pencil

6. ruler

Standards (Local and/or National)

Total Duration

45 minutes

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Assessment

The teacher will collect the students' completed work in Step 5. The teacher will examine the work to check for understanding of the concept (correct use of the formula).

Extension

Challenge the student to figure out the following: "What happens to the area of a circle when the radius doubles?" (The area will quadruple.)

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