Ratio and Proportional Reasoning Activity Set 4 Trainer Guide Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development
RATIO AND PROPORTIONAL REASONING Activity Set #4 NGSSS 4.G.5. NGSSS 6.G.4. A Piece of Pi In this activity, participants investigate the meaning of pi and use proportions to solve problems involving pi. Materials Transparency/Page: A Piece of Pi Transparency/Page: A Piece of Pi Tally Sheet Transparency/Page: Pi Problems Transparency/Page: Pi Problems Answer Key blank transparency precut cardboard circles paper clips string pencils scissors calculators ruler measuring tapes (centimeters and inches) masking tape Vocabulary pi ratio proportion Time: 30 minutes Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development
RATIO AND PROPORTIONAL REASONING Activity Set #4 Teaching Tip: Precut random-sized circles in groups of three: small, medium, and large. The range of sizes must be from a minimum of 4" to a maximum of 2" in diameter. You will need one set of circles for each participant group and one set for a model. Label each group of three circles with the same unit of measure: centimeters, inches, or paper clips. For example, label one group of three centimeters ; label a second group paper clips ; label a third group inches. Repeat until all the groups of circles are labeled. Introduce Explain to participants that they will determine the ratio between the circumference of a circle and its diameter. Use one of the precut circles (model set) to trace a circle on the blank transparency. Use the ruler as a straightedge (not for measuring) to draw the diameter across the circle. Cut a piece of string to the length of the diameter. Write diameter on a piece of masking tape and attach it to the string. Measure the diameter string using inches. Write the measurement on the piece of tape and on the transparency with the circle. Cut a piece of string to the length of the circumference of the circle. Write circumference on a piece of masking tape and attach it to the string. Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development 2
McGraw-Hill Professional Development RATIO AND PROPORTIONAL REASONING/25 RATIO AND PROPORTIONAL REASONING Activity Set #4 Measure the circumference string using inches. Write the measurement on the piece of tape and on the transparency with the circle. Be sure that each measurement is appropriately labeled with inches on the transparency. Hang the two pieces of string side-by-side on the wall. Have the group estimate the number of diameter strings it will take to make up the circumference string. Write the estimate on the transparency. a piece of pi Do the following for each circle: Cut a string the same length as the diameter of the circle. Cut a string the length of the circumference of the circle. Measure each string using the assigned unit of measure of your circle. Determine the ratio of the length of the diameter to the length of the circumference. Fill in the chart for the circle. Circle your unit of measure: centimeters inches paper clips Circle Circumference Diameter C d C d 2 3 Transparency: A Piece of Pi Have participants use their calculators to divide the circumference by the diameter to check their estimate. Write the result on the transparency. Display Transparency: A Piece of Pi and have participants take out their matching pages. Explain to participants that they will repeat this activity for three different circles, recording their final measurements on their pages. Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development 3
RATIO AND PROPORTIONAL REASONING Activity Set #4 Discuss and Do Ask participants to work in groups of 4 or 5 so that they can share the precut circles. Have one participant from each group pick up a set of precut circles. Point out that the unit of measure that each group should use is written on its circles. Teaching Tip: Have participants within the groups work in pairs to measure and record their data, with each pair taking one circle and sharing data with the group. Circle Circumference Diameter C d C d 2 3 4 5 6 7 8 9 0 2 3 4 5 6 7 8 9 20 a piece of pi Tally Sheet RATIO AND PROPORTIONAL REASONING AcTIvITy SET 4 Copyright 2002 by the McGraw-Hill Companies McGraw-Hill Professional Development TRANS_K6_RA_04 Transparency: A Piece of Pi Tally Sheet Display Transparency: A Piece of Pi Tally Sheet. Explain to participants that as soon as they have completed and recorded their measurements, they should: send a group representative to the front to record their information (for all three circles) on Transparency: A Piece of Pi Tally Sheet hang, on the wall, their circles with the related pieces of string as was previously modeled Give participants approximately 0 minutes to complete their assigned tasks. Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development 4
RATIO AND PROPORTIONAL REASONING Activity Set #4 Conclude Call the group together. Ask participants if they can make any generalizations about the relationship between the diameter strings and the circumference strings. (The diameter strings are about 3 the length of the circumference strings.) Have participants look at the last column on Transparency: A Piece of Pi Tally Sheet. Point out the range of responses. Point out that all (or most) of the numbers fit in a range that closely approximates pi (3.46 rounded to 4 decimal places, or 3.4 rounded to 2 decimal places or as close as we can get using nonspecific instruments of measurement). Point out any outliers and discuss with participants why they might have appeared. (mismeasurements, inaccurate tools for measuring, or computational errors) Teaching Tip: If time permits, tally all the numbers in the final column and compute the mean. (Sum the column and divide the sum by the number of entries.) Point out that the more data points, the more closely the list will approximate pi. Write, on a blank transparency, the ratio 3.4 (the ratio of C d for a -unit circle). Explain that we can use proportions and the ratio shown ( 3.4 ) to help solve circumference and diameter problems. We can solve for the diameter when the circumference is known, and we can solve for the circumference when the diameter is known. Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development 5
RATIO AND PROPORTIONAL REASONING Activity Set #4 John bought his mother a round music box as a gift. It has a circumference of 2 inches. The only boxes the store has are square. How wide must the square box be to hold the music box? (Round your answer to the nearest inch.) Marilyn swam 4 miles straight across the center of Round Lake, while Tana decided to walk around the lake. How far did Tana walk? Solve for C when d = 4 3.4 McGraw-Hill Professional Development pi problems 3.4 C 4 Solve for d when C = 2 2 d RATIO AND PROPORTIONAL REASONING/29 Display Transparency: Pi Problems and have participants take out their matching pages. Walk through all the steps to each solution with the group. (Steps are shown on Transparency: Pi Problems Answer Key.) Ask participants why was this exercise was called A Piece of Pi. (The problems related to the ratio of C to d pi.) Ask participants if, prior to this activity, they had realized that the number 3.4 was a ratio. Transparency: Pi Problems pi problems Answer Key John bought his mother a round music box as a gift. It has a circumference of 2 inches. The only boxes the store has are square. How wide must the square box be to hold the music box? (Round your answer to the nearest inch.) End of A Piece of Pi Solve for d when C = 2 3.4 2 d Marilyn swam 4 miles straight across the center of Round Lake, while Tana decided to walk around the lake. How far did Tana walk? Solve for C when d = 4 3.4 2 = 3.4d 2 = 3.4d 3.4 3.4 3.82 = d C 4 C = (4)(3.4) C = 2.56 Tana walked 2.56 miles. The box will be 4 inches on a side. McGraw-Hill Professional Development RATIO AND PROPORTIONAL REASONING/52 Transparency: Pi Problems Answer Key Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development 6
Do the following for each circle: Cut a string the same length as the diameter of the circle. Cut a string the length of the circumference of the circle. Measure each string using the assigned unit of measure of your circle. Determine the ratio of the length of the diameter to the length of the circumference. Fill in the chart for the circle. A Piece of Pi Circle your unit of measure: centimeters inches paper clips Circle Circumference Diameter C d C d 2 3 Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development Int_RPR_04_PM
Circle Circumference Diameter C d C d 2 3 4 5 6 7 8 9 0 2 3 4 5 6 7 8 9 20 A Piece of Pi Tally Sheet Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development Int_RPR_04_PM
Pi Problems John bought his mother a round music box as a gift. It has a circumference of 2 inches. The only boxes the store has are square. How wide must the square box be to hold the music box? (Round your answer to the nearest inch.) Solve for d when C = 2 3.4 2 d Marilyn swam 4 miles straight across the center of Round Lake, while Tana decided to walk around the lake. How far did Tana walk? Solve for C when d = 4 3.4 C 4 Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development Int_RPR_04_PM
Pi Problems Answer Key John bought his mother a round music box as a gift. It has a circumference of 2 inches. The only boxes the store has are square. How wide must the square box be to hold the music box? (Round your answer to the nearest inch.) Solve for d when C = 2 3.4 2 d Marilyn swam 4 miles straight across the center of Round Lake, while Tana decided to walk around the lake. How far did Tana walk? Solve for C when d = 4 2 = 3.4d 2 = 3.4d 3.4 3.4 3.82 = d The box will be 4 inches on a side. 3.4 C 4 C = (4)(3.4) C = 2.56 Tana walked 2.56 miles. Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development Int_RPR_04_PM
Glossary Ratio and Proportional Reasoning cross multiplication A method of verifying whether 2 fractions are equivalent. equivalent fractions 2 fractions which, when reduced to lowest terms, are equal. equivalent ratios Ratios that can be expressed by equivalent factions. pi The ratio of the circumference of a circle to its diameter (approximately 3.4). proportion A statement that 2 ratios are equivalent. rate A ratio comparing two different units of measure. ratio A comparison of 2 quantities. sample A subset of items taken at random from a complete set. Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development Int_RPR_04_PM