Math Fundamentals PoW Packet
Dragons and Tigers and Hares, Oh My!
Problem 4627
Welcome
Welcome!
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This packet contains a copy of the problem, the “answer check,” our solutions, some teaching
suggestions, and samples of the student work we received in February 2008. The text of the problem
is included below. A print-friendly version is available using the “Print” link on the problem page.
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Standards
In Dragons and Tigers and Hares, Oh My! students are asked to use clues about the Chinese
calendar to determine when Lin’s great-grandfather was born and how old he is. The key concepts
are addition, subtraction, division, multiples, and an understanding of how leap years work.
If your state has adopted the Common Core State Standards, this alignment might be helpful.
Grade 4: Operations & Algebraic Thinking
Use the four operations with whole numbers to solve problems.
Grade 4: Measurement & Data
Solve problems involving measurement and conversion of measurements.
Grade 5: Operations & Algebraic Thinking
Analyze patterns and relationships.
Mathematical Practices
1.
2.
4.
7.
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Model with mathematics.
Look for and make use of structure.
Additional alignment information can be found through the Write Math with the Math Forum service,
where teachers can browse by NCTM and individual state standards, as well as popular textbook
chapters, to find related problems.
The Problem
Dragons and Tigers and Hares, Oh My!
The Chinese calendar counts years in a 12-year cycle. Each year in the cycle is represented by an
animal. The year 2008 was called the Year of the Rat, as was 1996. Here are all 12 animals and two
cycles of the years they symbolize:
Rat
Ox
Tiger
Hare
Dragon
Snake
1996
1997
1998
1999
2000
2001
2008
2009
2010
2011
2012
2013
Horse
Sheep
Monkey
Rooster
Dog
Boar
2002
2003
2004
2005
2006
2007
2014
2015
2016
2017
2018
2019
Lin Chen lives in San Francisco. Her ancestors came to America from Shanghai over 100 years ago.
One day in December 2007 she asked her great-grandfather Xiang how old he was. Here is what Xiang
told her:
“I was born in this city in April during a leap year. I have lived to see seven years of the Boar. I will also
tell you that the year of my first birthday was a multiple of seven, just as the year 2002 was. Now child,
can you tell me how old I am?”
1. In what year was Great-grandfather born? What was its animal symbol?
2. How old was he in December 2007?
Explain how you solved the problem.
Extra: Lin’s mother was born in July during a year of the Snake. How likely is it that she was born in a
leap year? Explain how you know.
Answer Check
After students submit their solution, they can choose to “check” their work by looking at the answer
that we provide. Along with the answer itself (which never explains how to actually get the answer) we
provide hints and tips for those whose answer doesn’t agree with ours, as well as for those whose
answer does. You might use these as prompts in the classroom to help students who are stuck and
also to encourage those who are correct to improve their explanation.
1. Great-grandfather was born in 1924, a Year of the Rat.
If your answer doesn’t match ours,
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did you understand that a Year of the Boar occurs every 12 years?
did you understand that a leap year occurs every 4 years? The year 2004 was a leap year.
did you understand that the year after Great-grandfather’s birth year was a multiple of 7?
did you check your arithmetic?
If any of those ideas help you, you might revise your answer, and then leave a comment that tells
us what you did. If you’re still stuck, leave a comment that tells us where you think you need help.
If your answer does match ours,
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is your explanation clear and complete?
did you answer Question 2?
did you try the Extra questions?
did you verify that your answers meet all of Great-grandfather’s clues?
Revise your work if you have any ideas to add. Otherwise leave us a comment that tells us how
you think you did—you might answer one or more of the questions above.
Our Solutions
Method 1: Noticing and Wondering
Our group read the scenario our teacher gave us and we started to list our noticings.
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© 2013 Drexel University
A Chinese calendar has a 12-year cycle
The number of years from 1996 (Rat) to 2008 (Rat) is 12.
There are 12 different animals on the calendar.
If we count 1997, 1998, …, 2007, 2008 there will be 12 years or if we count 1996, 1997, …,
2006, 2007 there will be 12 years.
We can see that works for the Rat or the Ox or any of the different years named for an animal.
(Our group talked a lot about this so that we’d really understand how the calendar works.)
Lin Chen talked to her great-grandfather Xiang.
They live in San Francisco.
Their ancestors came from Shanghai.
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The ancestors came to America over 100 years ago.
In December 2007 Lin asked Xiang how old he was.
Great-grandfather said he was born in “this city” or San Francisco.
He said he was born in April.
He was born in a leap year.
Since Lin asked him how old he was in December but he was born in April, he’s already had his
birthday for 2007.
Great-grandfather has lived “to see seven years of the Boar.”
His first birthday year was a multiple of seven.
2002 is a multiple of seven. (We checked and 2002 ÷ 7 = 286!)
2002 was the Year of the Horse.
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As we talked we had a few wonderings.
• Was Great-grandfather over 100 years old?
• Was Great-grandfather born in America or in Shanghai? (As we kept reading we saw he said he
was born in San Francisco.)
• What language (Chinese or English) does Lin speak with her great-grandfather?
We decided we liked the idea of thinking about the “seven years of the Boar” and we made this chart.
Boar Year
Year
0
1st
2nd
3rd
4th
5th
6th
7th
1923
1935
1947
1959
1971
1983
1995
2007
We included the “0” column to think about which Boar year happened before he was born thinking that
might give us a clue. We decided that we had narrowed down the years that he was born to be after
1923 but before 1935. We made a list of those and then checked to see which years were a multiple of
7 or, in other words, which numbers could be divided by 7 with no remainder.
Year
1924
1925
1926
1927
1928
1929
1930
1931
1932
1933
1934
÷7
evenly?
no
yes
(=275)
no
no
no
no
no
no
yes
(=276)
no
no
Either 1925 or 1932 could have been Great-grandfather’s first birthday. We now had to decide
between 1924 and 1931 as the year he was born. We looked through the clues we had noticed to see
if there was one more that we could use to decide our final answer. We saw that Great-grandfather
was born in a leap year. We asked ourselves, how do we find out if 1924 or 1931 was a leap year? We
searched Ask Dr. Math and found this page:
http://mathforum.org/dr.math/faq/faq.calendar.html
We decided 1924 was a leap year but 1931 was not (since it’s odd).
Now we had to decide what animal that was. From our first table we knew that 1923 was a “Year of
the Boar” (and before he was born) and so 1924 would be the next animal in the cycle. It was a “Year
of the Rat.”
If Great-grandfather was born in April 1924, we can find his age in December 2007 by subtracting 1924
from 2007. We got 83.
1.
2.
Great-grandfather was born in 1924. He was born in the Year of the Rat.
He was 83 years old in December of 2007.
Method 2: Working Backwards
Since Years of the Boar come every 12 years, I started in 2007 and subtracted 12s to find the last 7
Years of the Boar that Xiang has seen:
2007, 1995, 1983, 1971, 1959, 1947, 1935
This tells me he had to be born in 1935 or earlier. Leap years occur in years that are multiples of 4, so
1935 was not a leap year. The leap years between 1935 and 1923, the last Year of the Boar before
1935, were:
1932, 1928, 1924
Since the year of Xiang’s first birthday was a multiple of 7, I tried dividing 1933, 1929 and 1925 each
by 7. Only one of them, 1925, had no remainder. Xiang was born in 1924. I added seven 12s, or 84 to
1924 to find out what his symbol was. 1924 + 84 = 2008, the year of the Rat.
© 2013 Drexel University
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2. To find Xiang’s age in December 2007 (after his 2007 birthday) I subtracted the year he was born
from 2007.
2007 - 1924 = 83
Xiang is 83 years old.
Extra: Lin’s mother could not have been born in a leap year. Years of the Snake must always be odd,
since 2001 is odd. Subtracting groups of 12 from any odd number will always result in an odd number.
Leap years are always even numbers, since they are always multiples of 4.
Method 2 – Subtracting Six Cycles of 12
1. Xiang has lived to see 7 Years of the Boar. That means he must have lived through six complete
cycles of 12, or 72 years.
6 • 12 = 72 years
2007 - 72 = 1935
the first Year of the Boar during his lifetime
Leap years come in years that are multiples of 4. The closest leap year before 1935 was 1932. The
year of Xiang’s first birthday was a multiple of 7.
1933 ÷ 7 = 276 r1 so 1932 is a multiple of 7. The multiple of 7 just before that was 1925 because
1932 - 7 = 1925.
If Xiang’s first birthday was in 1925, he was born in 1924, which was also a leap year.
I added 12s to 1924 until I found a year on the chart.
1924, 1936, 1948, 1960, 1972, 1984, 1996
1996 was a Year of the Rat, so 1924 was, too.
2. From 1924 to 2008, another Year of the Rat, is 7 cycles of 12, or 84 years. In December 2007 Xiang
would have been 83 years old.
Teaching
Suggestions
We found that the most common source of errors in this problem was in reading and interpreting the
clues. The prompts in the answer check may help you develop students’ understanding of the key
ideas or using the Notice/Wonder activity to get things started. Also, since organization is a challenge
in this problem, the Make a Table strategy could provide students with a way to keep track of all the
numbers they are thinking about.
This problem offers an opportunity to integrate social studies and literacy with math. There are some
excellent children’s books that address Chinese culture and the history of the Chinese in America:
The Last Dragon, Susan Miho Nunes, illus. Chris K Soentpiet, Clarion Books
Big Jimmy’s Kum Kau Chinese Take Out, Ted Lewin, Harper Collins
Coolies, Yin, illus. Chris K Soentpiet, Philomel Books
Brothers, Yin, illus. Chris K Soentpiet, Philomel Books
Dragonwings, Lawrence Yep, Harper Collins
Our Understand the Problem strategy may help students get started. We hope, however, that you
resist the urge to give direct instructions on a specific approach. This problem also provides a good
chance to practice the Work Backwards or Make a Table strategy. You’ll find everything you need from
the Problem Solving Activities link in the left menu bar.
The questions in the Answer Check, above, might serve as useful prompts to help students make
progress. Encourage students to use a strategy that works for them. As you can see from many
methods we used in Our Solutions, above, your students might approach this problem in a number of
different ways.
If you would like a calendar of the Current Problems, consider bookmarking this page:
http://mathforum.org/pow/support/
© 2013 Drexel University
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Sample
Student
Solutions
focus on
Strategy
Joshua
Neal
age 11
In the solutions below, we’ve provided the scores the students would have received in the Strategy
category of our scoring rubric. Our comments focus on what we feel is the area in which they need the
most improvement.
Novice
Has no ideas
that will lead
them toward a
successful
solution or
shows no
evidence of
strategy.
Apprentice
Practitioner
Uses a strategy
Uses a strategy that relies on skill,
that uses luck
not luck, which might include:
instead of skill, or
• thorough noticing and
doesn’t provide
wondering
enough detail to
• working backwards
determine
• logical reasoning
whether it was
• making a table or chart or list
luck or skill.
her grandpa is 86 years old.
i multiplyed 12 and7 and got 84 i add 2.
Strategy
Clarity
Novice
Expert
Does one or more of these:
Uses two different
strategies.
Uses a good Extra strategy.
Uses an unusual or
sophisticated strategy, e.g.,
effective and appropriate
use of technology
I notice that Joshua
mentions 12 and 7, both
numbers referred to in the
problem. He may be using
the common “strategy” of
“find numbers and use an
operation to quickly get an
answer.”
I wonder why he decided to
multiply those two numbers.
I also wonder if he noticed
any of the other clues from
the problem. A through
Notice/Wonder activity
might help him think more
about each of the
conditions.
Aryan
Neal
age
age11
9
Strategy
Clarity
Novice
answer 1 he was born in 1924 his symbol is the rat.answer2 was 83.
for the first one.
2007-82=1925
1925-1=1924
then keep adding 12 from 1924 until you get to 1996
then add from 1924 to 2007
Aryan may not be using a
random “pick some
numbers and operate”
strategy but with the limited
explanation provided it’s
difficult to really know the
thinking behind the solution.
I would ask Aryan for some
details, including
Why subtract 82?
Why subtract 1?
Why change from
subtraction to addition?
© 2013 Drexel University
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Neal
Jill
age
age11
9
Strategy
Clarity
Apprentice
Novice
Kelley
Neal
age 11
Strategy
Clarity
Apprentice
Novice
Neal
Ben
age
age11
8
Strategy
Clarity
Practitioner
Novice
Grandfather was born in 1917 in the year of the Boar and is 90 years old.
For the 7 Boar cycles I subtracted 12x7 = 84 from 2007. Since he
went through only 7 boar cycles, his birthday was between 1911 and
1923. I tested the numbers that were divisible by 7. 1918 was the
only number didvisible by 7. Since that was his first birthday, he
was born in 1917. I then subtracted 1917 from 2007.
Lin’s great grandfather is 79 years old, was born in the, in 1928, and was
born in the animal of the Snake.
I looked for the Horse and went back to the Snake because the horse
was his first birthday not the year he was born.
Next, Grandpa said that he’d lived through 7 years of the Boar. So
I put tallies on all the animals until I had 7 tallies on the Boar.
Then I stopped so I wouldn’t go over the 7 Boars other wise it
wouldn’t be accurate.
Fourth, I counted all of the tallies to get his age for this year.
I got an age of 79.
After that was finished I subtracted 79 from 2007 to get the year
of his birth. That makes him born the year of 1928. Grandpa’s animal
symbol is a Snake. In December he will still be 79.
Extra: Lin’s mom has a 25% chance of being born on a Leap Year.
Leap Years only happen every 4 years.
Lin’s grandpa is 79 years old and was born in the years of the
Snake in 1928.
1.Great-grandfather was born in 1924.His animal symbol was the rat.
2.Great-grandfather was 83 years old in December 2007.
Jill, like Joshua, multiplies 12
x 7 assuming the result of
84 will help her find the year
Great-grandfather was born.
While it could be a starting
point, I would suggest to Jill
that she list those “Boar”
years. If it includes 2007,
how many more will there
be?
I notice that Kelley focuses
the most on the clue that
Great-grandfather “lived to
see seven years of the
Boar.” Without more
explanation (or looking at her
paper with tallies) it’s difficult
to know how effective her
strategy is.
I would suggest that she
consider some of the other
clues from the problem.
Perhaps she can adapt her
tally system to include the
other clues as she reflects
and revises.
Leap years: 2008 2004 2000 1996 1992 1988 1984 1980 1976 1972 1968
1964 1960 1956 1952 1948 1944 1940 1936 1932 1928 1924 1920 1916
1912 1908 1904 1900
Ben’s “listing the years”
strategy for each of the main
clues clearly shows what he
thought about and how he
made his decisions.
Seven years of the boar: 2007 1995 1983 1971 1959 1947 1935 (1923)
My only suggestion would
be to try the Extra.
Multiples of seven: 2002 1995 1988 1981 1974 1967 1960 1953 1946
1939 1932 1925 1918
First, I looked at the leap years in between 1935-1923 because he
had 7 years of the boar so it had to be between those years. I
found these numbers: 1932 1928 and 1924. 1925 was a multiple of
seven and the year of his first birthday so I went back one year and
found 1924. To find his animal symbol, I looked at the ones that
were leap years which were rat, dragon and monkey. I knew it had to
be one of those symbols. I started at 1935 and counted back to
1924. I figured out that his animal symbol was the rat. To get
to how old he was in 2007 I added 3 too 1924 and I was at 1927 so I
added 80 to that and it made it 2007.So he was 83 in 2007.
© 2013 Drexel University
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Neal
Erin
age 11
Strategy
Clarity
Practitioner
Novice
Connor
Neal
age
age11
9
Strategy
Clarity
Novice
Expert
Great Granfather was born in 1924, during the year of the rat. In
December of 2007 he was 83 years old.
To solve this problem I first went over what I knew:
He was born on a leap year
he had seen seven years of the boar.
and his first birthday was a multiple of 7.
I started with the fact that he had seen 7 years of the boar. I kept
subtracting 12 from 2007, the last year of thge boar.
2007-12=1995
1995-12=1983
1983-12=1971
1971-12=1959
1959-12=1947
1947-12=1935
That was the seven years of the boars. I knew that it had to be
between 1935 and 1923 because that would be eight years of the
boars. So then I found the leap years between these 2 years.
They were between 1932, 1928, and 1924.
I tried the numbers right above them to see if they were a multiple of 7.
1933/7=276.14285
1929/7=275.57142
1925/7=275
Since 1925 was the only one without a decimal I new that that was
the year of his first birthday. His birthday was 1924. In December
2007 he was 83 years old.
I found out that he was born in the year of the rat because from
before I knew that the year of the boar was in 1923. And the year of
the rat comes right after the year of the boar.
1. Great grandfather was born in 1924. His animal symbol is a rat.
2. In December 2007, he was 83 and 8 months old.
Extra: There is no chance that Lin’s mother was born in a leap year.
1. How long ago is 7 boar years?
2007 = 7th boar year
6 more boar cycles - 6x12=72 years
2007-72=1935
1-Therefore Great grandfather was baorn before 1935 but after 8 boar
cycles in 1923.
The leap years it could be are 1932, 1928, 1924
Note: all leap years are divisible by 4 except for the years on the
hundreds.
I notice Erin lists the clues
that she knew before she
started using any of them.
Her strategy of listing what
she thought about as she
considered each of the
clues, works well to help her
find answers to the three
questions.
All that’s left to suggest is
that she consider the Extra.
Connor has done a great job
with this problem. His
strategy of figuring the Year
of the Rat from the Year of
the Boar within a specific
time period works out
particularly well.
When is his first birthday?
(leap year+1) divide by 7
(1924+1) divide by 7= 275 no remainder
(1928+1) divide by 7= 274 4 remainder
(1932+1) divide by 7= 276 1 remainder
His first birthday was in 1925, so he was born in 1924
1935 was a year of the boar
1935-12=1923 year of the boar
One more year is the year of the rat
Since his birthday is in April, he is 83 years old and 8 months.
© 2013 Drexel University
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EXTRA Lin’s Mom couldn't be born in a leap year because all snake
years are uneven numbers and all leap years have to be end in an even
number. Leap years are divisible by 4.
Scoring Rubric
A problem-specific rubric can be found linked from the problem to help in assessing student
solutions. We consider each category separately when evaluating the students’ work, thereby
providing more focused information regarding the strengths and weaknesses in the work. A generic
student-friendly rubric can be downloaded from the Teaching with PoWs link in the left menu (when
you are logged in). We encourage you to share it with your students to help them understand our
criteria for good problem solving and communication.
We hope these packets are useful in helping you make the most of Math Fundamentals Problems of
the Week. Please let me know if you have ideas for making them more useful.
~ Suzanne
© 2013 Drexel University
<[email protected]> and
Claire
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