Common Core Math I Homework KEY
Measures of Center: Mean
The Phantom Tollbooth is a children's fantasy adventure novel published in 1961. It was written by Norton Juster and
illustrated by Jules Feiffer. It tells the story of a bored young boy named Milo who unexpectedly receives a magic
miniature tollbooth one afternoon and, having nothing better to do, decides to drive through it in his toy car. The tollbooth
transports him to a land called the Kingdom of Wisdom. There he acquires two faithful companions, has many adventures,
and goes on a quest to rescue the princesses of the kingdom from the castle of air, Princess Rhyme and Princess Reason.
The text is full of puns, and many events, such as Milo's jump to the Island of Conclusions, exemplify literal meanings
of English language idioms.1
1) Read the attached excerpt from The Phantom Tollbooth.
2) Explain what it means to have an average of 2.58 children in each family. It means that when you add up all of the
children and divide by how many families there are, you get 2.58 children per family. This means that families
typically have 2-3 children each. It does NOT mean that families have part of a child!
3) The Phantom Tollbooth was published in 1961. According to US Census data, the average number of children per
family in 2000 was 1.86 and in 2010 was 0.94. What factors have contributed to this decline? People’s awareness of
overpopulation; better family planning; women delaying having children until they establish their careers, etc.
4) What does Milo mean when he says “But averages aren’t real, they’re just imaginary”? Sometimes the mean can be a
value that doesn’t make sense if interpreted literally. Like 2.58 children.
5) How does .58 of a child use the idea of fair shares to explain the mean? He talks about a situation of 5 people, 4 of
whom have $10 each and 1 who has no money. If the $40 is distributed equally among the 5 people, then each would
get a “fair share” (or equal share) of $8. So the mean of 8 tells us how much money to give each person.
6) Is .58 of a child correct in the following explanation? Explain why or why not.
“Well, think how much better off you’d be, just because of averages,” he explained convincingly. “And think of
the poor farmer when it doesn’t rain all year: if there wasn’t an average yearly rainfall of 37 inches in this part
of the country, all his crops would wither and die.” He is not correct. The average yearly rainfall describes the
trend in rainfall over time, but does not guarantee that there will be 37 inches of rain in any given year. Each year
varies – some more than 37 inches, some less. So it’s possible (but unlikely) to have a year with no rain at all.
7) Find an example in the newspaper or online of an average. Describe what the average means in context.
8) Using salary information from the Wacky Widget Company, answer the questions below.
Job
President
Vice President
Supervisor
Sales Representative
Warehouse Worker
Custodian
Clerical Worker
Number of
Employees
1
1
2
4
2
2
3
Annual Salary
$200,000
$50,000
$25,000
$21,000
$15,000
$15,000
$12,000
a) Find the mean salary. $32,000
b) Find the median salary. $21,000
c) Why do you suppose most employees were upset by a recent newspaper headline reporting “Average worker at
Wacky Widget making $32,000”? Only two employee’s make $32,000 or more – the President who makes $200,000
and the Vice President who makes $50,000. The mean is not an appropriate measure of the typical salary. The
median of $21,000 is a better descriptor.
d) Create a newspaper headline that you feel is more appropriate. Median Salary at Wacky Widget $21,000 OR Mean
Salary of $32,000 at Wacky Widget Misleading
10) Annette has a hamster that is three years old. She wonders if her hamster is old compared to other hamsters. On the
internet, she finds out that the median age for a hamster is 2 ½ years.
a) What does the median tell Annette about the life span of a hamster? 50% of hamsters live as long as or longer than 2
½ years.
b) How would knowing how the data vary from the least value to the greatest value help Annette predict the life span of
her hamster? If the typical amount that data varies from the center of 2 ½ years is small (like ¼ or ½ of a year) then her
hamster would seem to be really old. If the typical amount is large (like 1 or 2 years), then it’s likely that her hamster may
still live awhile longer.
Ms. Jackson’s class also collected data on the number of pets each student has at their home.
Use this data to answer the following questions.
Number of Pets
11) Make a histogram of the data. Include the graph or a sketch of the graph
from the calculator.
Pets of Students in Ms. Jackson's Class
6
Frequency
5
4
3
2
Number
0
1
2
3
4
5
6
7
8
9
Frequency
2
2
5
4
1
2
3
0
1
1
1
0
0
1
2
3
4
5
6
7
8
9
Number of Pets
12) Find the median. 3
13) Describe the distribution (shape, center, spread, outliers). The distribution of the number of pets owned
by students in Ms. Jackson’s class is slightly skewed to the right. The median number of pets is 3 and the
number of pets per student varies from 0 to 9. There are two students who own as many as 8 and 9 pets.
14) Do you think the students live in a city, the suburbs, or the country? Explain. I think that these students
must live in the suburbs or the country, since 8 students out of 21 own 4 or more pets. I think that you would
need a lot of space for so many pets to live in.