TABLE OF CONTENTS Click on a title to go directly to the handout. Handout 2: Estimating Challenge. Handout 3: Din-O-Rama Exploration

Transcription

1 SCALE CITY The Road to Proportional Reasoning: Dinosaur World Handouts TABLE OF CONTENTS Click on a title to go directly to the handout. Handout 1: Review: Fractions, Decimals, and Percents Problems assessing student understanding of fractions, decimals, and percents Handout 2: Estimating Challenge Practice with estimating using fractions, decimals, and percents Handout 3: Din-O-Rama Exploration Using to-scale cutouts to determine unknown heights Handout 4: Measuring and Comparing with Fractions, Decimals, and Percents Proportional reasoning problems related to measurement Handout 5: Scale and Proportion Proportional reasoning problems requiring students to scale objects

2 DINOSAUR WORLD: HANDOUT 1 Review: Fractions, Decimals, and Percents Name: Date: 1. Model 60% in the 10 by 10 grid. 2. Model 2.5% in the 10 by 10 grid. 3. Draw a model or picture that represents the fraction 2/3. 4. Draw a model or picture that represents 4/5.

3 Fractions, Decimals, and Percents 5. Show the following on the number line: A. 2/3 B. 3/4 C. 2 and 5/8 t u 0 1/2 6. Show the following on the number line: A B. 1.2 C. 0.5 D. Give an approximate value for the point indicated by the arrow. t u Draw a picture that represents the following statement: There was 0.7 of the pie left.

4 Fractions, Decimals, and Percents 8. Represent the following statements by shading in a portion of the 10 by 10 grid. a of the poster was wet. b. Her locker was 0.9 full. 9. Write other ways to express the following information. Write the fractions in simplest form. 67% /4 Decimal Fraction Percent Fraction Percent Decimal Solve the following problems. Show your work, and circle your answer. 10. The doctor thinks the girl is 75% of the height she will be as an adult. The girl is 48 inches tall. How tall does the doctor think the girl will be?

5 Fractions, Decimals, and Percents 11. One-third of the students are eating lunch in the cafeteria. If there are 450 students in the school, how many students are eating in the cafeteria? 12. The first year of Chess Club, four students joined. The next year, membership increased by 125%. How many students joined the next year? 13. Juan is 0.85 of his brother s height. If Juan s brother is 72 inches tall, how tall is Juan? 14. Sota is 66 inches tall. How would that be expressed in feet using a decimal? 15. The bone was 1.2 meters long. How many centimeters is that?

6 KEY: DINOSAUR WORLD: HANDOUT 1 Fractions, Decimals, and Percents 1. Model 60% in the 10 by 10 grid. 2. Model 2.5% in the 10 by 10 grid. The student s grids should clearly shade the specified area, as in the sample answers illustrated above. 3. Draw a model or picture that represents the fraction 2/3. 4. Draw a model or picture that represents 4/5. The student s models should show clear understanding of fractions. For example, she might draw a rectangle for question 3, divide it into three equal sections, and shade two sections. For question 4, she might draw four stick figures close together and a fifth figure standing apart.

7 5. Show the following on the number line: A. 2/3 B. 3/4 C. 2 and 5/8 KEY: Fractions, Decimals, and Percents t 0 1/2 2/3 3/4 2 and 5/8 u 6. Show the following on the number line: A B. 1.2 C. 0.5 D. Give an approximate value for the point indicated by the arrow. Accept value from 0.41 to 0.43 t u 7. Draw a picture that represents the following statement: There was 0.7 of the pie left. The student s picture should clearly represent the decimal value. For example, the student might draw a circle divided into 10 approximately equal segments, with three of the segments shaded out. Or he might draw two circles, one divided into 10 segments and the other with three segments missing and seven segments remaining. Or he even might draw 10 plates, seven of which hold a slice of pie and three of which are empty.

8 KEY: Fractions, Decimals, and Percents 8. Represent the following statements by shading in a portion of the 10 by 10 grid. a of the poster was wet. b. Her locker was 0.9 full. The student s grids should clearly represent the specified decimals, as in the sample answers illustrated above. 9. Write other ways to express the following information. Write the fractions in simplest form. Note: Students should not be penalized for failing to put a 0 before the decimal point. 67% /4 Decimal 0.67 Fraction 67/100 Percent 45% Fraction 9/20 Percent 75% Decimal 0.75 Solve the following problems. Show your work, and circle your answer. 10. The doctor thinks the girl is 75% of the height she will be as an adult. The girl is 48 inches tall. How tall does the doctor think the girl will be? 64 inches tall 11. One-third of the students are eating lunch in the cafeteria. If there are 450 students in the school, how many students are eating in the cafeteria? 150 students 12. The first year of Chess Club, four students joined. The next year, membership increased by 125%. How many students joined the next year? 5 students

9 KEY: Fractions, Decimals, and Percents 13. Juan is 0.85 of his brother s height. If Juan s brother is 72 inches tall, how tall is Juan? 61.2 inches tall 14. Sota is 66 inches tall. How would that be expressed in feet using a decimal? 5.5 feet 15. The bone was 1.2 meters long. How many centimeters is that? 120 centimeters

10 DINOSAUR WORLD: HANDOUT 2 Estimating Challenge Name: Date: Estimate the answer to these questions. You will not be given enough time to calculate your answers. For multiple choice questions 1-6, circle the answer that is closest to your estimate. Follow the directions in question 7, and then write your estimates for questions /8 + 8/9 = A. 1 B. 15 C. 2 D = A. 2 B. 0.2 C D x 0.22 = A B C. 0.8 D = A. 10 B. 1 C D = A. 0.6 B C D Circle the greatest number. A. 1.9 or 0.23 B or 1.4 C. 0.19, 0.036, 0.195, or Find the sum. A = B = 9. Find the least common denominator. A. 7/15 + 4/9 B. 2/3 + 5/6 C. 1 1/ /3 10. Arrange from the least to the greatest. 5/8, 3/10, 3/5, 1/4, 2/ = A. 50 B. 500 C D. 1250

11 KEY: DINOSAUR WORLD: HANDOUT 2 Estimating Challenge 1. C, 2. D, 3. B, 4. D, 5. B, 6. A 7. Circle the greatest number. A. 1.9 B. 1.4 C Find the sum. A. 1.3 (accept a range from 1 to 1.5) B (accept a range from 49 to 50) 9. Find the least common denominator A. 45 B. 6 C Arrange from the least to the greatest. 1/4, 3/10, 3/5, 5/8, 2/3

12 DINOSAUR WORLD: HANDOUT 3 Din-O-Rama Exploration Name: Date: Let s play paper dolls! Cut out the figures of Matt, Emilie, Chi kah, Baby Styra, T. rex, and Iguanodon and use them to answer these questions. Measure from lowest to highest point of the figures and round to the nearest 1/2 inch. 1. Compare Chi kah and the Tyrannosaurus rex. How many Chi kahs tall is the T. rex? 2. Estimate what percent of the T. rex s height Chi kah s height represents. 3. Use a ruler to measure Chi kah and the T. rex. Round to the nearest 1/2 inch. A. What is Chi kah s height in inches? B. What is the T. rex s height in inches? 4. In reality, Chi kah is five feet tall. What is the T. rex s real height in feet? 5. Compare Chi kah and the Iguanodon. How many Chi kahs tall is this dinosaur? (Hint: You can use a fraction of the paper cutout of Chi kah to answer the question.) 6. Use a ruler to measure the Iguanodon. What is its height in inches? Round to the nearest 1/2 inch. 7. Based on the information above, what is the Iguanodon s real height in feet? (Hint: Remember, Chi kah is five feet tall.) 8. How tall is Baby Styra in reality? Compare his cutout to the other figures and use what you ve determined about Chi kah and the big dinosaurs to estimate Baby Styra s height. No fair using a ruler! 9. How about Matt and Emilie? They re the same height, but what is it? No rulers!

13 Din-O-Rama Exploration EMILIE CHI KAH MATT BABY STYRA

14 Din-O-Rama Exploration T. REX

15 Din-O-Rama Exploration IGUANODON

16 KEY: DINOSAUR WORLD: HANDOUT 3 Din-O-Rama Exploration Let s play paper dolls! Cut out the figures of Matt, Emilie, Chi kah, Baby Styra, T. rex, and Iguanodon and use them to answer these questions. Measure from lowest to highest point of the figures and round to the nearest 1/2 inch. 1. Compare Chi kah and the Tyrannosaurus Rex. How many Chi kahs tall is the T. rex? 3 Chi kahs 2. Estimate what percent of the T. rex s height Chi kah s height represents. 33 percent 3. Use a ruler to measure Chi kah and the T. rex. Round to the nearest 1/2 inch. A. What is Chi kah s height in inches? 2 1/2 inches B. What is the T. rex s height in inches? 7 1/2 inches 4. In reality, Chi kah is five feet tall. What is the T. rex s real height in feet? 15 feet 5. Compare Chi kah and the Iguanodon. How many Chi kahs tall is this dinosaur? (Hint: You can use a fraction of the paper cutout of Chi kah to answer the question.) 2 3/5 Chi kahs (accept answers ranging from 2 1/2 to 2 3/4) 6. Use a ruler to measure the Iguanodon. What is its height in inches? Round to the nearest 1/2 inch. 6 1/2 inches 7. Based on the information above, what is the Iguanodon s real height in feet? (Hint: Remember, Chi kah is five feet tall.) 13 feet 8. How tall is Baby Styra in reality? Compare his cutout to the other figures and use what you ve determined about Chi kah and the big dinosaurs to estimate Baby Styra s height. No fair using a ruler! 3 feet 9. How about Matt and Emilie? They re the same height, but what is it? No rulers! 6 feet

17 DINOSAUR WORLD: HANDOUT 4 Measuring and Comparing with Fractions, Decimals, and Percents Name: Date: Draw a picture to demonstrate the concept and then calculate. Circle your answers. 1. Samson is 6 feet tall. He estimates that the tree outside the window is five times his height. About how tall is the tree? 2. Jen s little sister is 4 feet tall. Jen s father is 1.5 times her sister s height. How tall is the girls father?

18 Measuring and Comparing with Fractions, Decimals, and Percents 3. Cho s tree house stands at about 2/3 the height of his house. If the house is 30 feet tall, about how tall is the tree house? 4. Henry is 58 inches tall. His uncle is 70 inches tall. What percent of his uncle s height is Henry s height? Calculate: percent of percent of 62

19 Measuring and Comparing with Fractions, Decimals, and Percents percent of percent of percent of A child is standing by the giraffe exhibit at the zoo. The giraffe is 16 feet tall, and the child is 48 inches tall. What percent of the giraffe s height is the child? 11. If a boy is 0.8 as tall as a door, and the door is 80 inches tall, how tall is the boy? 12. The salesperson says a doghouse door should be 3/4 the height of the dog. The boy s dog is half his sister s height, and she is 40 inches tall. How tall should the doghouse door be?

20 Measuring and Comparing with Fractions, Decimals, and Percents 13. The girl is 62 inches tall. The mounted dinosaur skeleton in the museum is three times her height. How tall is the skeleton? 14. Your science teacher wants you to make a model of a T. rex that is 40 percent of its actual height. If the T. rex measures 13 feet tall, how tall would the model be? 15. A girl is 32% of the height of an Iguanodon sculpture. The girl is 5 feet tall. How tall would the sculpture be?

21 KEY: DINOSAUR WORLD: HANDOUT 4 Measuring and Comparing with Fractions, Decimals, and Percents Note to teacher: These problems are designed to complement the activities involving relative heights presented in Size-O-Rama, the first part of the interactive. In Session 2, after you introduce Din-O-Rama (the second part of the interactive) and the concept of scale, you might return to a few of the word problems in this handout and ask your students to discuss what scale they represent. Draw a picture to demonstrate the concept and then calculate. Circle your answers. 1. Samson is 6 feet tall. He estimates that the tree outside the window is five times his height. About how tall is the tree? 30 feet 2. Jen s little sister is 4 feet tall. Jen s father is 1.5 times her sister s height. How tall is the girls father? 6 feet tall 3. Cho s tree house stands at about 2/3 the height of his house. If the house is 30 feet tall, about how tall is the tree house? 20 feet 4. Henry is 58 inches tall. His uncle is 70 inches tall. What percent of his uncle s height is Henry s height? 83% Calculate: percent of percent of percent of percent of percent of

22 KEY: Measuring and Comparing with Fractions, Decimals, and Percents 10. A child is standing by the giraffe exhibit at the zoo. The giraffe is 16 feet tall, and the child is 48 inches tall. What percent of the giraffe s height is the child? 25% 11. If a boy is 0.8 as tall as a door, and the door is 80 inches tall, how tall is the boy? 64 inches 12. The salesperson says a doghouse door should be 3/4 the height of the dog. The boy s dog is half his sister s height, and she is 40 inches tall. How tall should the doghouse door be? 15 inches 13. The girl is 62 inches tall. The mounted dinosaur skeleton in the museum is three times her height. How tall is the skeleton? 186 inches or 15.5 feet 14. Your science teacher wants you to make a model of a T. rex that is 40 percent of its actual height. If the T. rex measures 13 feet tall, how tall would the model be? 5.2 feet 15. A girl is 32% of the height of an Iguanodon sculpture. The girl is 5 feet tall. How tall would the sculpture be? feet

23 DINOSAUR WORLD: HANDOUT 5 Scale and Proportion Name: Date: 1. Jen has a polar bear figurine that is 2 inches tall. A. What does she need to know to determine the scale of the figurine? B. How would she determine the scale? 2. As a present to his grandparents, Caleb wants to make a scale drawing of the family farm. Here s what he knows: The tractor is 1/4 as tall as the barn. The tree is 75% the height of the barn. The fence is 1/8 the height of the barn. The house is 0.7 the height of the barn. If Caleb draws the barn 8 inches tall on his paper, how tall will he draw the following? A. the tractor B. the tree C. the fence D. the house 3. Kip doesn t know how big to make a banner for the pep club. The banner will hang from the stands at the ballgame. He has a picture of himself by last year s banner. How can he use the picture to determine the height of the banner this year?

24 Scale and Proportion 4. The model scene was 1/72 the size of the original. What would you need to do to determine the height of an image for the background drawing? 5. An architect who once attended the school is coming on Career Day. As a token of appreciation, the class is creating a model of the school. The trees for the model are 3 inches tall. The actual trees are now 0.5 as tall as the school. How tall would the model school be? 6. The blue whale is the largest animal in the world. At about 80 feet long, it is thought to be the largest animal to have ever lived on earth. To demonstrate the length of a blue whale, you d like to create a scale drawing comparing the blue whale to the length of your school cafeteria, gymnasium, and school bus. What would you need to do create this model? 7. Emily s little brother was given an inflatable goal post that is 6 feet tall. A goal post in the NFL is 20 feet tall. A. What is the scale of the inflatable goal post height to the NFL goal post height? B. If an NFL player is 74 inches tall, how tall would a NFL action figure to match the scale of the inflatable goal posts be? 8. The Iguanodon was about 10 meters long. Some pet iguanas are about 1 meter long. A. What is the ratio of the length of a pet iguana to the length of the Iguanodon? B. If you are creating a model comparing the size of a pet iguana to the Iguanodon and your model Iguanodon is 15 centimeters long, how long should the model pet iguana be?

25 Scale and Proportion 9. The class visited Dinosaur World where the models of dinosaurs are actual size. The teacher took a picture of each student by the same dinosaur. The next week in math class, the class was told to use the pictures to determine the percent of the student s height to the dinosaur s height, the scale of the photograph compared to the actual size, and the height of the dinosaur. A. What would the students do to determine what percent of the dinosaur s height the student s height is? B. What would the students do to determine the scale of the photograph compared to the actual size? C. Describe one way the students could determine the actual height of the dinosaur using their other computations. 10. The model of a house was 1.5 feet high, and the actual house was 30 feet tall. An actual tree standing in front of the house was 20 feet. How tall should the model tree be?

26 KEY: DINOSAUR WORLD: HANDOUT 5 Scale and Proportion Note to teacher: Many of these questions are open-ended. It would be good to accept a variety of possible answers and methods and then to ask students to explain and justify their answers. 1. Jen has a polar bear figurine that is 2 inches tall. A. What does she need to know to determine the scale of the figurine? She needs to know the actual size of the polar bear. B. How would she determine the scale? Students should come up with answers that indicate that they understand that scale represents a proportional relationship between a model and a real object, in this case, the height of the figurine and the actual height of a polar bear. One way in which a student might explain this is to say that Jen would set up a ratio comparing the figurine s height with the polar bear s height (a to b, a:b or a/b). Or a student might suggest creating a fraction with the figurine height of 2 inches as the numerator and the actual polar bear height as the denominator. Dividing the figurine s height by the polar bear s actual height (in inches) would allow Jen to determine a percent or decimal value for the figurine s scale. 2. As a present to his grandparents, Caleb wants to make a scale drawing of the family farm. Here s what he knows: The tractor is 1/4 as tall as the barn. The tree is 75% the height of the barn. The fence is 1/8 the height of the barn. The house is 0.7 the height of the barn. If Caleb draws the barn 8 inches tall on his paper, how tall will he draw the following? A. the tractor 2 inches B. the tree 6 inches C. the fence 1 inch D. the house 5.6 inches 3. Kip doesn t know how big to make a banner for the pep club. The banner will hang from the stands at the ballgame. He has a picture of himself by last year s banner. How can he use the picture to determine the height of the banner this year? Students answers should indicate the understanding that Kip s height and the banner s height in the photograph are proportional to their actual heights. One possible answer students might give is that Kip can measure himself and the banner in the picture. Using these values he can determine what percent of his photo height the banner photo height is (dividing the banner photo height by his photo height). He could then multiply his actual height by this percent.

27 KEY: Scale and Proportion 4. The model scene was 1/72 the size of the original. What would you need to do to determine the height of an image for the background drawing? Find out the actual height of the object that will be depicted in the drawing. Multiply this height by 1/72 or divide it by 72 to determine its height in the drawing. 5. An architect who once attended the school is coming on Career Day. As a token of appreciation, the class is creating a model of the school. The trees for the model are 3 inches tall. The actual trees are now 0.5 as tall as the school. How tall would the model school be? 6 inches 6. The blue whale is the largest animal in the world. At about eighty feet long, it is thought to be the largest animal to have ever lived on earth. To demonstrate the length of a blue whale, you d like to create a scale drawing comparing the blue whale to the length of your school cafeteria, gymnasium, and school bus. What would you need to do create this model? Find out the lengths of the cafeteria, gymnasium, and school bus. Determine a reasonable scale size for the drawing depending on the size of the paper. Multiply actual lengths by the scale to determine the lengths for the drawing. To communicate about the concept of a reasonable scale, some students might provide or need an example. For instance, if the paper is 11 inches by 17 inches, a person might decide that it would be best to line the drawings up one above the other down the length of the paper from shortest to longest. If the gym at 90 feet is the longest of the four lengths being compared, a scale of 1 inch to 10 feet (120 inches) might make sense. That way, the drawing of the gym would be 9 inches long, with room on the paper for a margin on either side. 7. Emily s little brother was given an inflatable goal post that is 6 feet tall. A goal post in the NFL is 20 feet tall. A. What is the scale of the inflatable goal post height to the NFL goal post height? 6 feet to 20 feet or 3 feet to 10 feet. The value of this ratio may be expressed as 0.3 or 30%. B. If an NFL player is 74 inches tall, how tall would a NFL action figure to match the scale of the inflatable goal posts be? 22.2 inches 8. The Iguanodon was about 10 meters long. Some pet iguanas are about 1 meter long. A. What is the ratio of the length of a pet iguana to the length of the Iguanodon? 1:10 or 1/10 or 1 to 10 B. If you are creating a model comparing the size of a pet iguana to the Iguanodon and your model Iguanodon is 15 centimeters long, how long should the model pet iguana be? 1.5 centimeters

28 KEY: Scale and Proportion 9. The class visited Dinosaur World where the models of dinosaurs are actual size. The teacher took a picture of each student by the same dinosaur. The next week in math class, the class was told to use the pictures to determine the percent of the student s height to the dinosaur s height, the scale of the photograph compared to the actual size, and the height of the dinosaur. A. What would the students do to determine what percent of the dinosaur s height the student s height is? Use a ruler to measure the photographic height of the student and the dinosaur. Divide the photographic height of the student by the photographic height of the dinosaur to determine what percent the height the student is of the dinosaur s height. B. What would the students do to determine the scale of the photograph compared to the actual size? Create a fraction with the photographic height of the student as the numerator and the actual height of the student as the denominator. C. Describe one way the students could determine the actual height of the dinosaur using their other computations. Option 1: Students could divide the actual height of the student by the percent of the student s height to the dinosaur s height. Option 2: Divide the dinosaur s photographic height by the scale of the person s photographic size compared to the actual person s size. Option 3: Students could set the problem up as two equivalent ratios with the photographic size as the numerators and the actual size as the denominators. Students could then cross multiply and divide to find the unknown quantity. 10. The model of a house was 1.5 feet high, and the actual house was 30 feet tall. An actual tree standing in front of the house was 20 feet. How tall should the model tree be? 1 foot