Objective To introduce a formula to calculate the area. Family Letters. Assessment Management


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1 Area of a Circle Objective To introduce a formula to calculate the area of a circle. epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment Management Common Core State Standards Curriculum Focal Points Interactive Teacher s Lesson Guide Teaching the Lesson Ongoing Learning & Practice Differentiation Options Key Concepts and Skills Find the median of a data set. [Data and Chance Goal 2] Investigate and apply a formula for finding the area of a circle. [Measurement and Reference Frames Goal 2] Use ratios to describe the relationship between radius and area. [Measurement and Reference Frames Goal 2] Use patterns in a table to define the relationship between radius and area. [Patterns, Functions, and Algebra Goal 1] Key Activities Students draw circles by tracing round objects on centimeter grids. They measure the areas and radii and find that the ratio of a circle s area to the square of its radius is close to the value of π. They use the formula to calculate the areas of the circles. Ongoing Assessment: Informing Instruction See page 833. Materials Math Journal 2, pp B Study Link 10 8 Math Masters, p. 436 transparencies of Math Masters, pp. 314 and 436 slate collection of round objects from Lesson 10 8 calculator metric ruler Converting Units of Measure Math Journal 2, pp. 366A and 366B Students convert units of measure. Playing First to 100 Student Reference Book, p. 308 Math Masters, pp per partnership: 2 sixsided dice, calculator Students practice solving open number sentences. Ongoing Assessment: Recognizing Student Achievement Use Math Masters, page 458. [Patterns, Functions, and Algebra Goal 2] Math Boxes Math Journal 2, p. 367 Students practice and maintain skills through Math Box problems. Study Link Math Masters, p. 315 Students practice and maintain skills through Study Link activities. ENRICHMENT 2 Modeling πr Math Masters, pp. 316 and 317 scissors colored pencil or marker construction paper glue or tape Students apply their understanding of area formulas to verify the formula for the area of a circle. EXTRA PRACTICE Calculating the Circumferences and Areas of Circles Math Masters, p. 318 Geometry Template calculator Students use formulas to solve problems involving circumferences and areas of circles. Advance Preparation For Part 1, copy the table on journal page 365 onto the board. Make transparencies of Math Masters, pages 314 and 436. Teacher s Reference Manual, Grades 4 6 pp , 185, 186, 221 Lesson 831
2 Getting Started Mental Math and Reflexes Have students record an appropriate unit of measure for given situations. Suggestions: Amount of carpet needed to carpet a bedroom ft 2, yd 2, or m 2 Distance from where you live to a summer camp on a lake Miles, or kilometers Length of a dollar bill Inches, or centimeters Amount of juice the average person drinks in a week Cups, gallons, or liters Volume of cubes that could be stacked in a desk drawer, filling every space cm 3 or in 3 Math Message Solve Problems 1 4 on journal page 364. Study Link 10 8 FollowUp Have partners compare answers and resolve differences. 1 Teaching the Lesson Math Message FollowUp (Math Journal 2, p. 364; Math Masters, p. 314) ELL WHOLECLASS DISCUSSION NOTE All measurement is inexact, but finding the area of a circle by counting square centimeters is especially so. Area, unlike length, volume, or mass, is difficult to measure directly. Area is usually found by measuring lengths and applying a formula. Nevertheless, if students are particularly precise, measuring area by counting square units can be an accurate and practical technique. Verify that students were able to determine the diameter, radius, and circumference of the circle. Then ask students to share their answers to Problem 4. List their measurements in order on the board or a transparency, find the median of the measurements, and have students record the median in Problem 5 on the journal page. Point out the wide variation in the area measurements. Discuss why it is difficult to measure the area of a circle by counting squares. The pieces are irregular. Then use the transparency of Math Masters, page 314 to demonstrate the following method for counting squares Math Message Measuring the Area of a Circle Use the circle at the right to solve Problems The diameter of the circle is about 8 centimeters. 2. The radius of the circle is about 4 centimeters. 3. a. Write the open number sentence you would use to find the circumference of the circle. C π 8, or C b. The circumference of the circle is about 25 centimeters. 4. Find the area of this circle by counting squares. About 50 cm 2 Answers vary. 5. What is the median of all the area measurements in your class? cm 2 1cm 2 1. Make a check mark in each whole centimeter square. 2. Mark each centimeter square that is nearly a whole square with an X. 3. Find combinations of partial squares that are about equivalent to a whole square. Number each set, using the same number in each partial square. 4. Count the approximate total number of squares. In the circle in Problem 1, the area of the circle is about 52 square centimeters. 6. Pi is the ratio of the circumference to the diameter of a circle. It is also the ratio of the area of a circle to the square of its radius. Write the formulas to find the circumference and the diameter of a circle that use these ratios. The formula for the circumference of a circle is C π d, or C πd. The formula for the area of a circle is A π r 2, or A πr 2. Ask the class to suggest situations in which one might need to find the area of a circle. To support English language learners, list students suggestions on the board. Explain that students will count squares and use a formula to find the area of circles. Math Journal 2, p Unit 10 Using Data; Algebra Concepts and Skills
3 Exploring the Relationship between Radius and Area (Math Journal 2, pp. 364 and 365; Math Masters, p. 436) PROBLEM SOLVING WHOLECLASS Algebraic Thinking This exploration requires students to measure the radius of circles where the center is not given and to find the approximate areas of these circles. Demonstrate what students are to do using a round object and the transparency of Math Masters, page 436: 1. Make a circle on the transparency by tracing the object. 2. Find the approximate area of the circle by counting squares. Record the name of the object and its approximate area in the first and second columns of the table on the board. 3. Use a rightangled corner of a piece of paper to find a diameter of the circle. Position the right angle on the circle as shown below. Mark the points where the sides of the angle intersect the circle. These are the endpoints of a diameter of the circle. Measure the distance between endpoints Areas of Circles Work with a partner. Use the same objects, but make separate measurements so you can check each other s work. Answers vary. 1. Trace several round objects onto the grid on Math Masters, page Count square centimeters to find the area of each circle. 3. Use a ruler to find the radius of each object. (Reminder: The radius is half the diameter.) Record your data in the first three columns of the table below. Object Ratio of Area to Radius Squared Area Radius A (sq cm) (cm) as a Fraction r 2 as a Decimal 4. Find the ratio of the area to the square of the radius for each circle. Write the ratio as a fraction in the fourth column of the table. Then use a calculator to compute the ratio as a decimal. Round your answer to two decimal places, and write it in the last column. 5. Find the median of the ratios in the last column. Math Journal 2, p. 365 diameter 4. Ask students how to find the radius when the diameter is known. Divide by 2. Record the radius in the third column of the table. Ongoing Assessment: Informing Instruction Watch for students who struggle with measuring accurately. Have them trace and measure larger objects. Measurement errors are less significant with longer lengths. Have partners trace several round objects on a centimeter grid using Math Masters, page 436. Then have students measure the radius and the area of each of the tracings using the techniques just demonstrated. Partners use the same objects but measure independently and check each other s work. They record their results in the first three columns of the table on the journal page. When all groups have traced and measured at least three objects, bring the class together to demonstrate how to complete the last two columns in the table on the journal page. Continue to use the circle you traced on the transparency earlier. Lesson 833
4 A Formula for the Area of a Circle Your class just measured the area and the radius of many circles and found that the ratio of the area to the square of the radius is about 3. This was no coincidence. Mathematicians proved long ago that the ratio of the area of a circle to the square of its radius is always equal to π. This can be written as: A_ r = 2 π Usually this fact is written in a slightly different form, as a formula for the area of a circle. The formula for the area of a circle is A = π r 2 where A is the area of a circle and r is its radius. 1. What is the radius of the circle in the Math Message on journal page 364? 4 cm 2. Use the formula above to calculate the area of that circle cm 2 3. Is the area you found by counting square centimeters more or less than the area you found by using the formula? How much more or less? Answers vary. Answers vary. Sample answers: cm cm cm 2 4. Use the formula to find the areas of the circles you traced on Math Masters, page Which do you think is a more accurate way to find the area of a circle, by counting squares or by measuring the radius and using the formula? Explain. The formula is more accurate because it tells exactly how the area and radius are related. Counting squares is difficult and less accurate because partial squares are irregular. Math Journal 2, p _EMCS_S_G5_MJ2_U10_ indd 366 2/22/11 5:22 PM In the fourth column of the table on the board, write the ratio of the circle s area to the radius squared as a fraction. Ask volunteers for another way to express the meaning of radius squared. radius radius Use a calculator to convert the fraction to a decimal, rounded to two decimal places. Write the resulting decimal in the fifth column of the table. Have students complete the last two columns of the table on the journal page, find the median of the ratios in the last column, and record it as the answer to Problem 5. When students have finished, ask them to share the values they found and record them on the board. Most median values should be close to 3, though some might be far off because of various errors using the diameter instead of the radius, for example, or measuring the radius in inches rather than centimeters. Ask what number the ratios are close to. π Explain that this is no coincidence: The ratio of the area of a circle to the square of its radius is always equal to π. Help students recognize how remarkable this is the same number, π, is the ratio of the circumference to the diameter and the ratio of the area to the radius squared. These ratios are the bases for the formulas that can be used to find the circumference and the area of a circle. Have students write the formulas in Problem 6 on journal page 364. Using a Formula to Find the Area of a Circle (Math Journal 2, pp ; Math Masters, p. 436) WHOLECLASS Converting Units of Measure Customary System Length Weight Liquid Capacity Algebraic Thinking Have students read journal page 366 and use the formula to calculate the areas of the Math Message circle and the circles they traced on Math Masters, page 436. They compare the areas they found by counting square centimeters with the areas from the formula. 1 foot (ft) = 12 inches (in.) 1 pound (lb) = 16 ounces (oz) 1 pint (pt) = 2 cups (c) 1 yard (yd) = 3 feet (ft) 1 ton (T) = 2,000 pounds (lb) 1 quart (qt) = 2 pints (pt) 1 mile (mi) = 5,280 feet (ft) 1 gallon (gal) = 4 quarts (qt) Metric System Length Mass Liquid Capacity 1 centimeter (cm) = 1 gram (g) = 1 liter (L) = 10 millimeters (mm) 1,000 milligrams (mg) 1,000 milliliters (ml) 1 meter (m) = 1 kilogram (kg) = 1 kiloliter (kl) = 100 centimeters (cm) 1,000 grams (g) 1,000 liters (L) 1 kilometer (km) = 1,000 meters (m) 1. Tell if you should multiply or divide. a. To convert from a larger unit to a b. To convert from a smaller unit to a smaller unit (such as from ft to in.), larger unit (such as from m to km), you multiply. you divide. 2. Find the equivalent customary measurement. a. 24 in. = 2 ft b. 3 yd = 108 in. c. 12 qt = 3 gal d. 3.5 gal = 14 qt e. 1.5 mi = 7,920ft = 2,640yd f. 2.5 qt = pt = c Ongoing Learning & Practice Converting Units of Measure (Math Journal 2, pp. 366A and 366B) Students convert differentsize units of measure, compare units of measure, and solve problems involving measurement conversions. 3. Find the equivalent metric measurement. 1, , , a. 10 m = cm b. 50 mm = cm c. 250 g = kg d. 2.5 kl = L e. 7.5 m = cm = mm f. 100 mg = g = kg Math Journal 2, p. 366A 366A366B_EMCS_S_MJ2_G5_U10_ indd 366A 3/22/11 12:43 PM 834 Unit 10 Using Data; Algebra Concepts and Skills
5 Playing First to 100 (Student Reference Book, p. 308; Math Masters, pp ) PARTNER Algebraic Thinking Students play First to 100 to practice solving open number sentences. This game was introduced in Lesson 47. For detailed instructions, see Student Reference Book, page 308. Ongoing Assessment: Recognizing Student Achievement Math Masters Page 458 Use the First to 100 Record Sheet (Math Masters, page 458) to assess students facilities with replacing variables and solving problems. Students are making adequate progress if their number sentences and solutions are correct. [Patterns, Functions, and Algebra Goal 2] Converting Units of Measure continued 4. Which is less? a. 1.5 gallons or 20 cups b L or 1,300 ml 20 cups 1.25 L 5. Which is more? a. 1 3_ 4 lb or 28 oz b. 1,299 g or 1.3 kg They are equal. 6. Arrange each set of measurements in order from least to greatest. a. 9 oz, 1_ 2 lb, 0.75 lb b m, 800 mm, 85 cm 1_ 2 lb, 9 oz, 0.75 lb 7. Arrange each set of measurements in order from greatest to least. a yd, 1.5 ft, 39 in. b. 2,500 g, 100,000 mg, 2.55 kg 39 in., 0.75 yd, 1.5 ft 8. The standard length of a marathon is 26 miles 385 yards. How many yards is that in all? 46,145 yards 9. You can estimate the adult height of a 2yearold child by doubling the child s height. Suppose a 2yearold child is 35 1_ 2 inches tall. Estimate what the child s adult height will be in feet and inches. 5 feet 11 inches 10. Mt. McKinley in Alaska is km tall. This is 705 m greater than the height of Mt. St. Elias in Alaska. How tall is Mt. St. Elias in kilometers? km 1.3 kg 0.75 mm, 800 mm, 85 cm 2.55 kg, 2,500 g, 100,000 mg Math Boxes (Math Journal 2, p. 367) INDEPENDENT 11. A recipe for chicken soup calls for 12 cups of chicken broth. _ 12 a. How many gallons is that equivalent to? 16, 3_ 4 b. How many gallons are needed if the recipe is doubled? 1 1_ 2 Math Journal 2, p. 366B, or 0.75 gallon, or 1.5 gallons Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lessons 105 and The skill in Problem 5 previews Unit 11 content. Writing/Reasoning Have students write a response to the following: Exchange the exponent and base for each of the numbers in Problem 2, for example, Write the standard notation for each new number. 289; 279,936; 1,000,000; 1,000; 59,049 Which number is larger than its corresponding original number? Explain why. Sample answer: The original number 7 6 in standard notation is 117,649; 6 7 or 279,936 is larger than the original number because 6 used as a factor 7 times is greater than 7 used as a factor 6 times. Study Link (Math Masters, p. 315) Home Connection Students identify the best measurement to find in specific situations. They solve a set of problems using the formulas for area and circumference of circles. INDEPENDENT 366A366B_EMCS_S_MJ2_G5_U10_ indd 366B Math Boxes 1. Monica is y inches tall. Write an algebraic expression for the height of each person below. a. Tyrone is 8 inches taller than Monica. Tyrone s height: b. Isabel is 1 1_ 2 times as tall as Monica. Isabel s height: c. Chaska is 3 inches shorter than Monica. Chaska s height: d. Josh is 10 1_ 2 inches taller than Monica. Josh s height: e. If Monica is 48 inches tall, who is the tallest person listed above? Isabel How tall is that person? 72 in. 2. Use a calculator to rename each of the following in standard notation. a = 131,072 b. 7 6 = 117,649 c = 60,466,176 d = 59,049 e. 5 9 = 1,953,125 6 y _ 2 y, or 1 1_ 2 y y  y _ Solve. Solution a d = 14 b. 28 e = 2 c. b + 18 = 24 d. 14 = f 7 e. 12 = 16 + g d = 26 e = 30 b = 42 f = 7 g = /6/11 11:02 AM 4. Complete the What s My Rule? table and state the rule. Rule: Subtract 12 from in, in out or in  12 Math Journal 2, p Find the volume of the cube. Volume = length * width * height Volume = 6 units cube 216 units _EMCS_S_G5_MJ2_U10_ indd 367 2/22/11 5:22 PM Lesson 835
6 Study Link Master Name STUDY LINK 10 9 Area and Circumference Circle the best measurement for each situation described below. 1. What size hat to buy (Hint: The hat has to fit around a head.) Differentiation Options 2. How much frosting covers the top of a round birthday cake 3. The amount of yard that will be covered by a circular inflatable swimming pool 4. The length of a can label when you pull it off the can Fill in the oval next to the measurement that best completes each statement. 5. The radius of a circle is about 4 cm. The area of the circle is about 12 cm 2 39 cm 2 50 cm 2 25 cm 2 6. The area of a circle is about 28 square inches. The diameter of the circle is about 3 in. 6 in. 9 in. 18 in. 7. The circumference of a circle is about 31.4 meters. The radius of the circle is about 3 m 5 m 10 m 15 m 8. Explain how you found your answer for Problem 7. Sample answer: The circumference is about 31.4 meters, and this equals π º d or about 3.14 º d. Because 3.14 º , the diameter is about 10 meters. The radius is half the diameter or about 5 meters. Math Masters, p. 315 Area of a circle: A π º r 2 Circumference of a circle: C π º d 315 ENRICHMENT PARTNER Modeling πr Min (Math Masters, pp. 316 and 317) Algebraic Thinking To apply students understanding of area formulas, have them cut a circle into pieces and arrange them in the shape of a parallelogram. They draw and label the parts of the parallelogram to compare them to the parts of the circle. Partners follow the directions on Math Masters, page 316 to verify the formula for the area of a circle. When students have finished, discuss any difficulties they encountered. EXTRA PRACTICE INDEPENDENT Calculating the Circumferences and Areas of Circles (Math Masters, p. 318) 5 15 Min Algebraic Thinking Students use formulas to solve problems involving the circumferences and areas of circles. They draw the circles before finding the areas and circumferences. Remind students to use the fix function on their calculators to round the calculations to the nearest hundredth. Teaching Master Name 10 9 More Area and Circumference Problems Circle Formulas Circumference: C = π º d Area: A = π º r 2 where C is the circumference of a circle, A is its area, d is its diameter, and r is its radius. Measure the diameter of the circle at the right to the nearest centimeter. 4 cm 2 cm cm cm 2 Sample answer: The circumference is the perimeter of the circle. 1. The diameter of the circle is. 2. The radius of the circle is. 3. The circumference of the circle is. 4. The area of the circle is. 5. Explain the meaning of the word circumference. 6. a. Use your Geometry Template to draw a circle that has a diameter of 2 centimeters. b. Find the circumference of your circle. c. Find the area of your circle. 7. a. Use your Geometry Template to draw a circle that has a radius of inches. b. Find the circumference of your circle in. c. Find the area of your circle in cm 3.14 cm 2 Math Masters, p Unit 10 Using Data; Algebra Concepts and Skills
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