This lesson will help students learn that recognizing and using patterns is a valuable problem solving tool.
- The Kings Chessboard, by David Birch; ISBN 0-14-054880-7
- One Grain of Rice: A Mathematical Folktale, by Hitz Demi;
- Navigating Through Algebra in Grades 3-5, by Gilbert J. Cuevas and
Karol Yeatts; ISBN 0-87353-500-7
Background for Teachers
Finding patterns is the underlying theme of mathematics.
Recognizing and using patterns is a valuable problem-solving tool. By
exploring, discovering, and analyzing patterns, students can begin to
make sense of things in mathematics. Searching for patterns begins with
concrete activities, but moves to discovery, application, and a greater
sense of understanding.
Intended Learning Outcomes
2. Become mathematical problem solvers.
3. Reason mathematically.
Invitation to Learn
Read The Kings Chessboard to the point where the wise man makes
his request and leaves the hall.
Ask the students how much rice they think the wise man will receive
before the chessboard is full. Was it a good request for his reward? Why
were the counselors and nobles laughing at the wise mans request?
The class will see if the wise mans request was truly wise after the
- Pass out copies of the
Which Salary is the Best? worksheet and
discuss the scenario:
You want to buy a go-cart for $1,000 and need to find a job to
raise the money. You found job openings to mow lawns with two
different companies that have different pay scales. One company,
Lawns Are Us, will double your salary each day. You will earn
$1 the first day, $2 the second day, $4 the third day, $8 the fourth day,
and so on. The second company, Smith Lawn Care will
increase your salary by $4 each day. You will make $4 the first
day, $8 the second day, $12 the third day, $16 the fourth day, and
so on. Which company will help you reach the $1,000 needed to
buy a go-cart the fastest?
- After reading the scenario, ask students to
predict which company
would enable them to reach their $1,000 goal the fastest and
explain their reasoning in their math journals.
- Students create a table
for each of the two companies and
complete the tables until day 5. Which company pays the most at
this point? Write a short paragraph in their journals about which
company they would choose at day 5 and why.
- Students complete the chart
until $1,000 is made by both
companies. Which company was the best choice? Why? Have
students write a paragraph in their journals explaining what
happened with the salaries.
- As a class, write a function to find out which
salary would pay
more on the nth day.
- Create a multiple line graph on the centimeter
graph paper. The
x-axis should be Total Earnings and the y-axis should be Number
of Days. Have them graph the total earnings of each company in
- Discuss the graph. For what days does Lawns Are Us yield
total earnings? For what days does Smith Lawn Care yield better
- Does the chart or graph illustrate the information more
effectively? Why? Have students record their thoughts in their
- Finish reading The Kings Chessboard and discuss.
- Exponential growth is a number
pattern that occurs in mitosis, or
cell division. An e-coli cell is one of the fastest growing bacteria
cells. It can reproduce itself in 15 minutes. Have students create
a table showing mitosis of an e-coli cell in one hour. Have them
find a pattern in the growth rate.
- Using Excel, have students create a double
bar graph comparing
the salary of the two different jobs.
- Students poll their family members
about which job they would
choose. Would they rather get $1 the first day, $2 the second day,
$4 the third day, $8 the fourth day, and so on? Or would they
rather get $4 the first day, $8 the second day, $12 the third day,
$16 the fourth day, and so on? After they get their families
opinions, students explain which job is better and why.
- Students read The
Kings Chessboard or One Grain of Rice with
- Observation of students creating their tables and
- Class discussion.
- Which Salary is the Best? worksheet and graph.
Brenner, M.E. (1995) The Role of Multiple Representations in Learning Algebra.
http://eric.ed.gov ERIC #
Prealgebra students learned about functions by representing problems
in multiple formats. Students learning that was anchored by a
meaningful thematic context had gains in translating word problems into
equations, tables, and graphs. The same results were found in lower achieving
students and language-minority students.
Capraro, R.M., Kulm, G. & Caprano,
M.M. (2002) Investigating the Complexity of Middle
Grade Students Understanding of Mathematical Constructs: An example from Graphic
Representation. http://eric.ed.gov ERIC # ED465799
This study examined four components of prior understanding required
for graphic representation: coordinate relationships, graphs showing a
variety of relationships, reading simple tables, and graphic displays.