SCALE CITY
The Road to Propor tional Reasonin
g:
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
DINOSAUR WORLD: Introducing Proportion and Scale © KET, 2009
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Review: Fractions, Decimals, and Percents
Name:
1. Model 60% in the 10 by 10 grid.
Date:
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.
DINOSAUR WORLD: Introducing Proportion and Scale © KET, 2009
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. 0.45
B. 1.2
C. 0.5
D. Give an approximate value for the point indicated by the arrow. ______________
t
u
0
0.2
7. Draw a picture that represents the following statement: There was 0.7 of the pie left.
DINOSAUR WORLD: Introducing Proportion and Scale © KET, 2009
Fractions, Decimals, and Percents
8. Represent the following statements by shading in a portion of the 10 by 10 grid.
a. 0.05 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%
0.45
3/4
Decimal __________________
Percent __________________
Percent __________________
Fraction __________________
Fraction __________________
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?
DINOSAUR WORLD: Introducing Proportion and Scale © KET, 2009
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?
DINOSAUR WORLD: Introducing Proportion and Scale © KET, 2009
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KEY: DIN
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.
DINOSAUR WORLD: Introducing Proportion and Scale © KET, 2009
KEY: Fractions, Decimals, and Percents
5. Show the following on the number line:
A. 2/3
B. 3/4
C. 2 and 5/8
t
0
2/3 3/4
2 and 5/8
↓↓
↓
u
1/2
€€
€
6. Show the following on the number line:
A. 0.45
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_____
0.45 0.5
t
0
1.2
↓ ↓
↓
u
0.2
€ €
€
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.
DINOSAUR WORLD: Introducing Proportion and Scale © KET, 2009
KEY: Fractions, Decimals, and Percents
8. Represent the following statements by shading in a portion of the 10 by 10 grid.
a. 0.05 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%
0.45
3/4
Decimal
0.67
Percent
45%
Percent
75%
Fraction
67/100
Fraction
9/20
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
DINOSAUR WORLD: Introducing Proportion and Scale © KET, 2009
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
DINOSAUR WORLD: Introducing Proportion and Scale © KET, 2009
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DINOS
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 8-10.
1. 7/8 + 8/9 =
6. 0 .239 ÷ 0.4 =
A. 1
B. 15
C. 2
D. 17
A. 0.6
B. 0.06
C. 0.006
D. 6
2. 0.02 + 0.0002 + 0.000002 =
A. 2
B. 0.2
C. 0.00022
D. 0.02
7. Circle the greatest number.
A. 1.9 or 0.23
B. 1.15 or 1.4
C. 0.19, 0.036, 0.195, or 0.2
8. Find the sum.
3. 0.3827 x 0.22 =
A. 0.7 + 0.4 + 0.2 =
B. 7.1 + 0.23 + 42.123 =
A. 0.008
B. 0.08
C. 0.8
D. 8
9. Find the least common denominator.
4. 30 ÷ 317 =
A. 10
B. 1
C. 0.01
D. 0.1
A. 7/15 + 4/9 _______
B. 2/3 + 5/6
_______
C. 1 1/4 + 2 2/3 _______
10. Arrange from the least to the greatest.
5/8, 3/10, 3/5, 1/4, 2/3
5. 250 ÷ 0.5 =
A. 50
B. 500
C. 1000
D. 1250
DINOSAUR WORLD: Introducing Proportion and Scale © KET, 2009
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KEY: D
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. 0.2
8. Find the sum.
A. 1.3 (accept a range from 1 to 1.5)
B. 49.453 (accept a range from 49 to 50)
9. Find the least common denominator
A. 45
B. 6
C. 12
10. Arrange from the least to the greatest.
1/4, 3/10, 3/5, 5/8, 2/3
DINOSAUR WORLD: Introducing Proportion and Scale © KET, 2009
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“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!
DINOSAUR WORLD: Introducing Proportion and Scale © KET, 2009
Din-O-Rama Exploration
EMILIE
MATT
DINOSAUR WORLD: Introducing Proportion and Scale © KET, 2009
CHI’KAH
BABY STYRA
Din-O-Rama Exploration
T. REX
DINOSAUR WORLD: Introducing Proportion and Scale © KET, 2009
Din-O-Rama Exploration
IGUANODON
DINOSAUR WORLD: Introducing Proportion and Scale © KET, 2009
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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
DINOSAUR WORLD: Introducing Proportion and Scale © KET, 2009
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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?
DINOSAUR WORLD: Introducing Proportion and Scale © KET, 2009
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:
5. 85 percent of 16
6. 223 percent of 62
DINOSAUR WORLD: Introducing Proportion and Scale © KET, 2009
Measuring and Comparing with Fractions, Decimals, and Percents
7. 52 percent of 35
8. 63 percent of 182
9. 119 percent of 85
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?
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?
DINOSAUR WORLD: Introducing Proportion and Scale © KET, 2009
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?
DINOSAUR WORLD: Introducing Proportion and Scale © KET, 2009
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KEY: DIN
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:
5. 85 percent of 16
13.6
6. 223 percent of 60
133.8
7. 52 percent of 35
18.2
8. 63 percent of 182
114.66
9. 119 percent of 85
101.15
DINOSAUR WORLD: Introducing Proportion and Scale © KET, 2009
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?
15.625 feet
DINOSAUR WORLD: Introducing Proportion and Scale © KET, 2009
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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. A
s 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?
DINOSAUR WORLD: Introducing Proportion and Scale © KET, 2009
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. T
he 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?
DINOSAUR WORLD: Introducing Proportion and Scale © KET, 2009
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?
DINOSAUR WORLD: Introducing Proportion and Scale © KET, 2009
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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. A
s 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.
DINOSAUR WORLD: Introducing Proportion and Scale © KET, 2009
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. E
mily’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%.
. If an NFL player is 74 inches tall, how tall would a NFL action figure to match the scale of the inflatable goal
B
posts be?
22.2 inches
8. T
he 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
DINOSAUR WORLD: Introducing Proportion and Scale © KET, 2009
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
DINOSAUR WORLD: Introducing Proportion and Scale © KET, 2009