Algebra Strand
M6A1: Students will understand the concept of ratio and use it to represent
quantitative relationships.
M6A2: Students will consider relationships between varying quantities
b. Use manipulative or draw pictures to solve problems involving
proportional relationships.
M6P5: Students will represent mathematics in multiple ways
a. Create and use representations to organize, record, and
communicate mathematical ideas.
b. Select, apply, and translate among mathematical representations to
solve problems.
c. Use representations to model and interpret physical, social and
mathematical phenomena.
Lesson One:
The complete article on this activity can be found in “Miniature Toys Introduce Ratio
and Proportion with a Real-world connection” Mathematics Teaching in the Middle,
Sept 2001.
Background: The instruction begins on a concrete level by using miniature toys to
model a set with two obvious subsets. Because the use of ratios depends on natural
relationships, the two sub-groups are chosen to represent objects that would normally be
considered related in the real world. For each modeled set, multiple relationships exist
that can be expressed as ratios using the terms and numbers that represent the entire set
and each subset. Guide students to write or state ratios expressions that relate the set to
each of the subsets and that relate the two subsets. For any set with only two obvious
subsets, six ratio expressions can be found- three direct relationships and the reciprocal of
each.
Example: Start with a set of musical instruments- 3 saxophones and 1 French horn.
The following ratios can be written.
A. The ratio of musical instruments to saxophones is 4 to 3. The ratio represents the
relationship of the set to subset 1.
B. The ratio of saxophones to musical instruments is 3 to 4. This ratio represents the
relationship of subset 1 to the set and is the reciprocal of ratio expression A.
C. The ratio of musical instruments to French horns is 4 to 1. This ratio represents
the relationship of the set to subset 2.
D. The ratio of French horn to musical instruments is 1 to 4. This ratio represents the
relationship of subset 2 to the set and is the reciprocal of ratio C.
E. The ratio of saxophones to French horns is 3 to 1. This ratio represents the
relationship of subset 1 to subset 2.
F. The ratio of French horns to saxophones is 1 to 3. This ratio represents the
relationship of subset 2 to subset 1, and is the reciprocal of ratio expression E.
Have students practice creating and writing examples like the one above.
Concrete modeling of proportions involves a given set and requires creating a second set.
In creating the second set, a consistent factor must be applied.
Example: Start with a set of insects and subsets of flies and ants. The top set represents
the original set and that each set below it was created to be proportional to the original.
Notice that two of the proportional sets are larger than the original set and that one is
smaller. Tell the students that the simplest way to create proportional models is to
concentrate on the two subsets. Larger proportional sets can be created by applying the
same factor to create multiples of each of the original subsets. Any factor greater than 1,
including mixed numbers, can be used. It is usually easiest to make the first subset 2
times the size of the corresponding subset in the original set. (Note line 2) The second
subset (line 2) was created by a factor of 1 1/3 times the size of the original set. Smaller
proportional sets can also be created by applying the same factor to each of the original
subsets. For smaller sets, however, the factors must be less than 1. The third set was
created by applying the factor 2/3 to each of the subsets in the original set; changing each
subset tin the new set to 2/3 the size of the corresponding subset in the original set.
Now allow the students to work with creating their own sets, subsets and telling what
factor they used to create the new proportional sets and subsets.
The following website has pictures that allow the same concepts to be taught.
http://math.rice.edu/~lanius/proportions/index.html
Students complete worksheet in groups/ independently.
Examples for lesson 1
Original set
Set increased by a factor of two
Set increased by a factor of 1 1/3
Set decreased by a factor of 2/3
Worksheet for lesson 1
Write six ratios expressions for the set of tools.
Write six ratio expressions for the set of school supplies
Create a larger proportional set. What factor did you use?
Create a smaller proportion set. What factor did you use?
Complete the second set to be proportional to the original set.
Use some of the miniature toys provided and create 2 of your own sets and proportions.
Lesson for later in the unit.
Types of problems that can be solved using proportions.
Discuss with students that one type of proportion problem is known as the part-partwhole. Which of the problems worked on so far would be examples of this. (Saxophones
to instruments in lesson 1 works here) another type of proportion problem is know as
the Associated sets, which types of problems have we done that would be examples of
this type of proportion. (The saxophones to French horns example works here) The third
type of problem is well known measures. (This is where you use the words per minute
typed, miles per gallon, etc) The final type of proportion problem is known as the growth
(stretching and shrinking) problems. This is where you use examples of enlarging or
shrinking a picture or model. Give students the worksheet on the four types of problems.
Have them work groups/individually to solve and write about the strategies that they used
to solve. When finished, have a class discussion on these strategies and how they fit into
the four types of proportion problems. Now that students have an understanding of
proportion problems some may want to use these strategies for all problem solving. Use
the back of the worksheet to have a discussion on when it makes sense to use proportions
to solve problems and when it does not. Have students complete the next set of
problems independently.
Work these problems and identify the problems as: part-part-whole; Associated sets; measures;
Growth.
Dr. Day drove 156 miles and used 5 gallons of gasoline. At this rate, can he drive 561 miles on a full
tank of 21 gallons of gasoline?
Ellen, Jim, and Steve bought 3 helium-filled balloons and paid $2 for all 3 balloons. They decided to
go back to the store and buy enough balloons for everyone in the class. How much did they pay for 24
balloons?
Mrs. Jones put her students into groups of 5. Each group had 3 girls. If she has 25 student, how many
girls and how many boys does she have in her class?
A 6”x8’ photograph was enlarged so that the width changed from 8” to 12”. What is the height of the
new photograph?
Do the following statements make sense? What distinguishes those that make sense from those that do
not?
1. If one girl can walk to school in 10 minutes, two girls can walk to school in 20
minutes.
2. If one box of cereal cost $2.80, two boxes of cereal cost $5.60.
3. If one boy makes one model car in 2 hours, then he can make three models in 6 hours.
4. If Huck can paint the fence in 2 days, then Huck Tom, and a third boy can paint the
fence in 6 days.
5. If one girl has 2 cats, then four girls have 8 cats.
6. During mathematics class, the sixth grade students were grouped at three tables with
2 girls and 4 boys at each table. During science class, they were arranged into two
groups with 3 girls and 6 boys in each group. What has changed? What has not
changed?
7. When does it make sense to use a ratio?
Solve the following problems and state the strategy used.
1. A player’s batting average in baseball is the number of hits he would make in
1,000 times at bat. If a player makes 2 hits in 5 times at bat, how many hits can
he expect to make in 20 times at bat?
2. An original 5 by 8 inch photo must be reduced to 1 ¼ by 2 inches to fit in the
yearbook. What is the scale of the reduced photo to the original in simplest form?
3. Alonso drove the first 300 miles of his trip in 6 hours. How long will it take him
to make the 800 miles trip?
4. There are 2 teachers for every 57 students at Elmwood Middle School. If there
are 855 students, how many teachers are at the school?
Tell if each of the following is correct and if it can be solved using a proportion. Justify
your answer.
1. If 17 out of 110 people said they liked canoeing, then 15 people out of 100 would
like canoeing.
2. If one person can paint the floor of a 12’x 12’ porch in 2 hours, then it would take
him 4 hours to paint a porch floor that is 24’x 24’.
Suggested middle school reading that has proportional concepts included:
All of the Borrower’s books by Norton
Honey I Shrunk the Kids
The Little cartoons
Harry Potter and the Soccer’s Stone
The Fellowship of the Ring
The Perfect Storm
Below are pictures that I use for a comparison shopping activity. A similar activity can
be found on the web
http://students.cs.byu.edu/~tim/Page11.html)
46 ounces of Tide for $6.87
56 ounces of Cheer for $8.96
24 ounce box of Cheerios for $4.79
22 ounce box of Fruit Loops for $4. 43
12 ounce can of Coke for $.75
16 ounce bottle of Coke for $1.00
1 pkg (3 boxes- 6 oz each) of Hi-C drink for $1.29
1 pkg (4 pouches 4.5 oz each ) of Capri Sun drink for $1.98
4 socks for $ 6.88
5 socks for $7.29
12 ounces of L’oreal for $2.67
14 ounces of Johnson’s for $2.84
16 ounce bottle of Curel for $ 3.79
15 ounce bottle of Apreive for $ 3. 54
12 ounce bag of Baked Lay’s for $2.89
10 ounce bag of Gibbles’ for $2.51
1 box ( 4 CDs- 650 MB each) for $7.98
1 Box (4 CDs- 700 MB each for $ 9.59
10 pencils for $2.25
12 pencils for $2.75
1 box of 250 paper clips for $.89
1 box of 525 paper clips for $1.79
1 gallon of milk for $2.98
1 quart of milk for $.79
Complete each of the following tables and explain the relationships that you find. How
can these be represented on a graph? What is the constant of proportionality in each
problem?
Inches
Miles
1
5
2
10
Students
Adults
36
24
72
48
Tablespoons
of Cocoa
3
Ounces of
Milk
8
Stickers
Dollars
4
20
18
12
6
3
108
9
21
32
50
$10.00
Cost of Dollars
Apples
3
6
Oranges
Dollars
3
$0.49
40
48
25
1
$2.00
6
12
$1.00
1
2
$0.40
1
9
$0.98
$1.96
Name_________________
Reading in the content Area
Main Idea
Subject Matter
Supporting Details
Conclusion
Clarifying Details
Vocabulary
1. Mark
the main idea with an M.
Mark the statement that it too broad with a B.
Mark the statement that is too narrow with an N.
x 3
______to solve 
multiply 6 by 3 and then divide by 9
6 9
______You can solve proportions by using cross products.
______Proportions can be used to solve real-life problems.
2.
This lesson is mainly about how to ________
a. Find equivalent fractions.
b. Solve problem with a variable.
5 3
c. Solve problems like x
7 13
24
s

d. Solve a proportion like
32 500
5 25

you should_______
7 m
a. Multiply the numerators and the denominators.
b. Find the cross products.
c. Divide both sides by 7
d. Multiply both sides by 7
1
9
4. To determine whether
and
form a proportion, you
3
27
a. add the fractions
b. multiply the fractions
c. find if the cross products are equal
d. find a common factor for 3 and 27
3. To
solve the proportion
5. The Key concepts shows that proportions are
equations__________
a. where tow ratios are equivalent
b. where two ratios are unequal
c. that have many properties
d. where tow ratios are comparison
6. A ratio means___________
a. Any two numbers whose product is one.
b. The difference of two numbers.
c. A comparison of two numbers by division
d. The distance from the center of a circle to any point on
the circle.
(The complete article from which the performance task was taken can be found in
Mathematics Teaching in the Middle; January 2004).
Movies such as Harry Potter and the Sorcerer’s Stone, Lord of the Rings, and The
Perfect Storm were all filmed using sets to make a person or object look exceptionally
tall or large. In addition, since many movie scenes are enhanced through computer
graphics, film directors need to consider making objects appear in the proper proportions.
When preparing a scene in such a movie, set designers need to determine the ratio
between the objects they want to film together and adjust the set or computer generated
scene accordingly.
Choose one of the books, write a 3 paragraph book review, and complete the tasks about
that book and movie.
Harry Potter and the Sorcerer’s Stone
In Harry Potter and the Sorcerer’s Stone, Harry is a young boy who survives the deaths
of his parents and has a miserable childhood living with relatives. After coming to live at
the Hogwarts school of Witchcraft and Wizardry, ten-year-old Harry and his friends
experience numerous adventures as the forces of good and evil parry.
Hagrid, the groundskeeper at Hogwarts, befriends Harry. In the book, Hagrid is twice the
height of an average man and three times as wide. In the movie, Hagrid is very large,
head and shoulders taller than the other men, although not twice their height. In a
number of scenes, we see that Hagrid is more than just a large man. He appears to be at
least 4 feet taller than Harry and his friends. In one scene, he appears to be 1 ½ feet taller
than a door that he removes from its hinges. He is nearly as tall as the Hogwarts Express
Train. In addition, although the items in Hagrid’s hut are normal sized for Hagrid, they
are very large for Harry and his friends. The audience is led to believe that wizards can
magically conjure up the items that they need in the necessary sizes. However, laws of
reality govern set designers. Thus, the items that Hagrid used need to be in proportion to
him, just as the sizes of the items used by the students and teachers need to be in
proportion to them.
It is not possible for the actor who played Hagrid to be nearly 12 feet tall as suggested in
the book, so let’s look at how a set can be built to make Hagrid appear twice the size of a
normal man.
How large are Hagrid’s table and chairs to make them appear in the same
proportion as those of an average man?
How large would a fork be for Hagrid, if Harry and friends use normal-sized
forks?
How large would Hagrid’s teacup be? What human sized object would be of
similar size? If we are to give the impression in the movie that Hagrid is
twice the size of a normal man, how can that be done?
Now let us look at the set itself. Consider the scene in which Harry meets Hagrid
for the first time. Harry and his relatives are hiding in a primitive cabin on a
remote island. Hagrid arrives in the middle of the night, lifts the door of the cabin
from its hinges, and crouches to enter the room.
What are the dimensions of the door so that it appears to be 6 feet 8 inches
tall next to a 12-foot man?
If normal ceilings are 8 feet high, how high must the cabin ceiling be to make
Hagrid look 12 feet tall?
The movie producer decided to make Hagrid look unusually large but not twice
the height of a normal man. How do the solutions to each of these questions
change if Hagrid is to be portrayed as 8 feet? 10 feet? Rather than 12 feet.
Completing the table will help answer the above questions
Item
Adult Height
Table Height
Fork length
Teacup height
Chair height from floor
Chair height from seat to
top
Length and width of chair
seat
Average for Adult Males
For Hagrid
The Lord of the Rings
The Lord of the Rings, Part I: The Fellowship of the Ring is a movie in which the
set designer used proportional reasoning. In part I, while Gandalf is visiting the
hobbits, he notices that a ring Bilbo found while on another adventure is making
him act strangely. Gandalf realizes that the dark forces that have misplaced the
ring are gathering strength. If they find the ring, then all of Middle Earth,
including dwarves, elves, hobbits, and men, is doomed. Gandalf encourages
Bilbo to entrust the ring to his nephew, Frodo, who must leave Bag End to ensure
the ring’s safekeeping. The story revolves around the fellowship of elves,
dwarves; hobbits, men, and Gandalf who have sworn to protect Frodo as he
delivers the ring to the fire in which it was made, thereby neutralizing its power
and saving Middle Earth.
In the book, hobbits are between 2 feet and 4 feet tall with most reaching a height
of about 3 feet. Frodo and Bilbo are about 4 feet- tall for hobbits. Gandalf, a
wizard who befriended Bilbo and Frodo, is 7 feet tall. In real life, the actors who
play Bilbo and Frodo are both about 5 feet 5 inches tall. The actor who plays
Gandalf is about 6 feet tall. Yet when filmed together, the hobbits appear as they
should (about 4 ft) and Gandalf appear to be at least 3 feet taller than the hobbits!
How did the producers make actors whose heights are only 7 inches apart
look so different on film?
The hobbits, Bilbo Baggins and his nephew Frodo, live in Bag End, a dwelling
built into the side of a hill. When Bilbo and Frodo are walking in Bag End, the
ceiling appears to be nearly twice their height. However, in the same setting,
Gandalf can barely stand erect and keeps hitting his head on the semicircular
archways between rooms. He also has to dodge a chandelier so that it does not hit
him in the face. Frodo and Bilbo pay no attention to the chandelier, easily
walking under it.
What dimensions for ceiling heights would make 5 foot 5 inch actor
appear to be 4 feet all and a 6 foot actor look 7 feet tall? Could both of these
sizes be accomplished in a single set? Explain your findings in detail.
What dimensions would make the 6-foot actor who plays Gandalf
appear as thought eh is sitting at a hobbit-sized table?
What dimensions were used for objects in the hobbit-focus set so that
when consecutive scenes were viewed in the movie, it appeared that Gandalf
and the hobbits were among the same objects?
Complete the table to help you answer the questions
Item
Average Male
Hobbit size
Gandalf Size
average height
Ceiling height
Height of chandelier
from floor
Table height
Chair height
Archway height
The Perfect Storm
The Perfect Storm dramatizes the last days of the crew of the 72-foot swordfish
boat, the Andrea Gail, during the storm of October 1991. The Andrea Gail,
sailing form Gloucester, Massachusetts, got caught in the convergence of three
major weather systems: energy from the Great Lakes, a nor’easter weakening in
the North Atlantic, and an old hurricane from the South Atlantic. The
Midwestern storm regenerated the other two, causing the worst storm to hit the
area in over 100 years. The crew of the Andrea Gail was lost at sea during this
perfect storm.
The movie showed the actor on the ship struggling with the forces of nature in 50to 100 foot seas. How were these scenes filmed? To recreate the scenes, the
movies company expanded its soundstage tank to 95ft/100 ft/ 22 ft deep (formerly
8ft deep) and used an exact replica of the ship to shoot all scenes involving the
actors on the boat. What were the dimensions of the original tank? The tank
was surrounded by blue screen so that the large waves could be added through
computer generation after the scenes were shot. A model of the ship was imposed
on actual footage of large waves. Set designers had to determine how large to
make the model of the ship so that the waves would appear proportionally 75 feet
to 100 feet tall. What was the size of the model ship? In the scene where the
ship was driving up the face of a 100 foot wave and was seen backsliding, burying
its stern in the trough of the wave as the wave began to break, quickly flipping the
boat end to end. To make this scene realistic, what were the proportions
between the wave and the boat? (Note that a 45 foot wave has an angled face of
60-70 feet) Assuming the same steepness, what is the angled face of a 100 ft
wave? According to oceanographers the maximum theoretical height for winddriven waves is 198 ft. How would the boat appear against the wave in this
case? Waves are periodic. Instead of coming every 15 seconds as in a normal
storm, waves were breaking every 8-9 second during the perfect storm. At one
point, the captain decided he could no longer run with the “following seas”;
waves were breaking over the stern or back of the boat. He turned the boat to run
directly into the waves. A boat has a much better chance of flipping over
broadside in seas. It takes 30 second to turn a boat around. How many waves hit
the boat broadside? Assuming that the captain started his turn either during a
trough or peak of a wave, in a normal storm only 1 wave would hit the boat
broadside. In the perfect storm how many waves hit the boat broadside?
The Andrea Gail was a ‘longliner swordfisher’. It contained enough fishing line
to lay 40 miles of line with 100 hooks. Graph the relationship of line to hooks.
While using the sound stage tank for filming, the animated swordfish had to also
be proportional. What were the sizes of the swordfish used in the tank? The
crew of the Andrea Gail could land 6,000-7,000 pounds of swordfish in a good
day. While on the expedition, the captain wanted to catch at least 40,000 pounds
before heading back to port; how many good fishing days would he need?