quick reads
a good idea in a small package
Fig. 1 This table of contents lists all the
Fig. 2 Students receive these puzzle
varieties of puzzles that students have
created.
instructions.
MathMania Puzzles
MathMania:
A Middle School Puzzle Book
Alessandra King
l
Learning math can be made fun—and
therefore much more engaging—
through puzzles. Most students, thrilled
by puzzles, riddles, and brainteasers,
stay on task and focus for a long time
to solve them. This project makes the
most of this penchant through additional incentives: Students create their
own math puzzles, combine them into
an anthology in book format, choose
a title for their collective work, and
then distribute it electronically to all
the students in the class and to their
families. This activity provides a visual
hallmark or a year-end product of their
mathematical journey. Showing their
enthusiasm for the project, one of my
latest classes selected MathMania as the
title of their book. It contained a variety
of puzzle styles, as shown by the listing
in its table of contents (see fig. 1).
Edited by Trena Wilkerson, trena_
[email protected], Baylor University,
Waco, Texas. Readers are encouraged
to submit manuscripts through
http://mtms.msubmit.net.
438
Learning Objectives
My main objective is to foster problem-solving, critical-thinking, and
decision-making skills. Although
puzzles are often used to introduce
new material, reinforce skills, and
Mathematics Teaching in the Middle School
●
Vol. 19, No. 7, March 2014
review concepts previously studied,
this objective is much wider: It is
to simulate real-life applications of
mathematics with tasks framed by a
single, pivotal question that could be
answered in many different ways. This
puzzle activity requires students to
analyze and integrate a large quantity
of information and to deploy many
different skills and concepts.
My second objective is to help students showcase their accomplishments
by producing a year-end keepsake,
one comparable to those typically produced in arts and language courses.
Project Design
I invite students to select any type of
puzzle they wish and choose from any
topic or unit that has been studied thus
far during the school year. Comparable
to a real-life problem, this task can be
considered as a complex system of interconnected parts. Such a system can
display properties that may be different
from those of the parts and depend
on various possible initial conditions.
Moreover, small variations during the
Alex’s Sudoku
5
Lillian’s Cryptogram
6
Charlotte’s PuzzlePac Challenge 7
Skyler’s CrossNumber
8
Samantha’s Word Search
9
Lea’s Sudoku Challenge
10
Mira’s Mind-Bender
11
Aleja’s Amazing and
Awesome KenKen Puzzle
12
Annika’s KenKen Puzzle
12
Mary Claire’s KenKen
13
Olivia’s Puzzle
13
Soleine’s Super Stupendous
KenKen
14
Leah’s Puzzle
15
Joelle’s Sudoku Puzzle
16
Wei Wei’s KenKen
16
Rinnie’s Riddle
17
Victoria’s Sudoku
18
Amreen’s KenKen
18
Annelise’s Math Word Search 19
April’s Kindergram
20
Zoe’s Sudoku
20
Sarah’s KenKen
21
Yasmin’s KenKen
21
Ashvike’s KenKen
22
Amanda’s Math Scramble
22
Ava’s Math Word Search
23
Reagan’s Puzzle
23
Shea’s KenKen Puzzle
24
Solutions25
system development may lead to very
different outcomes. Therefore, just
like any real-life math application, the
students find out that—
• a puzzle may have more than one
solution;
• the puzzle’s initial conditions
(the number and position of the
known entries in a Sudoku™
puzzle, for example) may affect
its difficulty level and require a
lot of fine-tuning to produce a
balanced outcome;
For your graded homework
assignment, you will create an
original puzzle. Please look at
the examples of puzzles in the
attached document and research
other types of puzzles, if you
want. Magic squares, Sudoku,
KenKen, math-related word
search or crossword puzzles,
cryptograms, riddles, and logic
puzzles are all acceptable, as long
as you produce it. Research and
read the suggestions for creating
a math puzzle. Do remember that
your puzzle must be original, and
you must create it yourself (no
downloads from the Internet are
allowed). You must also title your
puzzle, submit an answer key,
and briefly explain your strategy.
Please submit your entry in both
hard-copy and electronic forms by
the due date.
• it is essential to master flexibility
and the ability to respond to
unanticipated outcomes; and
• the same task can be completed in
different, yet equally valuable ways.
In addition, the puzzles need to be
prepared within a specified time to exercise the students’ organizational and
study skills and to teach them to plan
with deadlines in mind. Finally, creating puzzles helps the students practice
the mathematical skills picked up
during the school year, review concepts previously introduced, refine
problem-solving techniques, and focus
on proper vocabulary.
Class Activity
Initially, students familiarize themselves with many different formats
and types, such as word problems,
cross-number puzzles, Sudoku,
KenKen®, SET®, Kakuro™, logic
matrices, math-o-grams, word
Vol. 19, No. 7, March 2014
●
Online Tools for
Creating Puzzles
Tools to Write Puzzles
Microsoft Word®
Excel®Apple® Pages®
Numbers®
Adobe® Photoshop®
Tools to Create Brainteasers
Puzzlemaker (http://www.discovery
education.com/free-puzzlemaker/)
Puzzlefast (http://www.puzzlefast
.com/).
searches, riddles, tangrams, magic
squares, and logic problems culled
from various sources (including
graduate school entrance exams).
They solve representative puzzle
examples during the school year and
discuss alternate strategies. I give
them only some general directions
(see fig. 2). Next, students brainstorm,
research, and debate ideas in class for
the content and format of the puzzles
that they will create themselves.
The research and conversation
in small, flexible groups are engaging and productive, and, above all,
inspiring—leaving the students with
a wealth of information with which
to begin work. Each student then
meets with me in a short, focused
one-on-one conversation (somewhat
like a teacher-student conference for
reading). This allows me to assess
students’ understanding and progress,
as well as gain an idea of their final
product. Then I require them to create rough drafts of their puzzles, and
also solve them before submission, to
make sure the item is well conceived
and structured.
Students can use various tools
to write their puzzles or create
brainteasers. (See the sidebar.)
However, students generally prefer to
create their own puzzles from scratch,
rather than use online tools, because
Mathematics Teaching in the Middle School
439
Fig. 3 These sample puzzles provide an idea of the range of math that students accessed.
Fig. 3 (continued) These sample puzzles provide an idea of the range of math that students accessed.
Alex’s Sudoku
Rinnie’s Riddle
Samantha’s Word Search
Instructions
The puzzle is a 9 × 9 grid of cells in
which you need to put nine sets of the
digits 1–9 subject to the restriction
that each digit must appear once in
each row, once in each column, and
once in each 3 × 3 box.
“Why was the math book so sad?”
To answer this question, first find the solution to the following math problems:
Look for the words described by the following mathematical definitions:
1. Part of a whole.
2. The result of an addition.
3. Comparison of two amounts by division.
4. A parallelogram with four right angles and four equal sides.
5. The ratio of the circumference to the diameter of a circle.
6. Term with no variable.
7. A shape that is not a polygon.
8. A mathematical sentence that states that two quantities are equal.
9. The unit of measurement for length in the metric system.
10. The data item that appears most often.
11. An arithmetical value used in counting.
H
If 4a + 13 = 29, what
is a?
B
Percent decrease of 1,500 to
1,416
T
If 6/2 = 15/x, what is x?
L
True or false: |-x| is positive
S
–|–1 + 3|
E
4 is what % of 39 (rounded to
nearest hundredth)?
A
(–2)3 – 42
M
Percent increase of 4 to 7.5
Y
GCF (63, 48)
O
40 + (64 – 96)
N
LCM (72, 63)
R
49% of 60
P
True or false: a2/b > a/b
always?
I
If 28/7 = x/4, find x.
D
Find the area of the second, similar rectangle.
48
24
(a)
x
8
Charlotte’s PuzzlePac Challenge
Across:
1. Find the value of x: x/43 = 2
2. What percent of 80 is 484?
3. 102 × 7.02
4. The square of the length of the
hypotenuse in a right triangle
with legs of 10 and 24 units.
6. The value of x: 3/30 = 9/x
Down:
1. 104 × 8.367
2. 872
3. What is 20% of 350?
4. 16 × 8 + 88
You should find each of your solutions in one or more of the white boxes
below. In the green box above each of these solutions, write the letter that
was paired with the problem. Your answer to the riddle will be revealed!
16
5
4
768
–2
87.5 –24
504
3
False
Mathematics Teaching in the Middle School
29.4
8
(c)
●
Vol. 19, No. 7, March 2014
5.6%
True
10.26 87.5%
F
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N
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A
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A
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T
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(e)
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B
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(d)
it is easier for them to edit their own
work.
8
!
(b)
440
–24
Your solution:
1. _________________
2. _________________
3. _________________
4. _________________
5. _________________
6. _________________
7. _________________
8. _________________
9. _________________
10. ________________
11. ________________
Aleja’s Amazing and Awesome
KenKen Puzzle
Discussion
It is gratifying that the puzzles submitted typically cover a wide range
of mathematical topics: basic number
theory, relations between fractions,
ratios and percentages, logic, geometry
definitions, calculations of area and
perimeter, equations and inequalities,
proportions, scientific notation, and
word problems. Some of the puzzles
initially submitted are simple, but
others are quite elaborate. Many different formats are also used: puzzles
are handwritten, created on a tablet
computer, composed with common
software such as Microsoft Word, and
created with a mixture of tools.
So that the puzzles can be edited
and printed as a single collection, the
final submission has to be in a common, electronic format. To that end, I
issue students some clear guidelines.
After a final check, I compile them in
a book. Sample puzzles are shown in
figure 3.
Evaluation and
Assessment
Students are evaluated during the
class portion of the project, which
involves reading and discussing the
project directions, researching and
working with others in class, communicating and talking about mathematics, and sharing the puzzles and
solutions with classmates. They are
also assessed on the precision of the
Vol. 19, No. 7, March 2014
●
final copy of the puzzle and their solution, and on their review of others’
puzzles. The evaluation focuses on
students’ understanding of the problem and their search for a solution;
their willingness to appreciate different approaches; their openness to,
and understanding of, other students’
directions and work; their ability to
communicate mathematical concepts
and ideas; their respect for others’
thoughts; and their work ethics.
To assess the work done in class,
I use observations, discussion, smallgroup review, and other oral assessments. In addition, I keep a tool (a
clipboard, a notebook, or an iPad®)
with me at all times to take quick
notes about each student’s work
and contributions. Students are
Mathematics Teaching in the Middle School
441
given a simple rubric to help them
review their own work and their
friends’ puzzles. RubiStar at http://
rubistar.4teachers.org was helpful in
building rubrics for the students.
APPROPRIATE FOR
PREALGEBRA WORK
This activity is particularly appropriate for a prealgebra class, in which
the range of mathematical topics is
wide and varied. This type of content
and format encourages and emphasizes initiative, originality, independent work, extensive reading, and
investigation; all of these features
define a truly enriching project that
is also captivating. The enthusiastic
reception of this project by the students and their families leads me to
conclude that this strong activity can
be used regularly, either as is or to
review and reinforce a particular unit,
set of skills, concepts, or procedures.
Math is abstract and, unlike the
arts and languages, not often thought
of as providing an outlet for creativity. It is therefore thrilling for
students and their families to acquire
a memento of the collective achievement of the class as tangible and
original as an anthology of puzzles.
Note: Readers are encouraged to contact the author, Alessandra King, at
[email protected], for
specific information about implementing this activity, including the complete
instructions given to students and
evaluation rubrics used.
REFERENCES
ALTEC. “Rubistar: Create Rubrics
for Your Project-Based Learning
Activities.” Lawrence: University of
Kansas. http://rubistar.4teachers.org
Discovery Education. 2013. “Puzzlemaker.” http://www.discovery
education.com/free-puzzlemaker
PuzzleMaker. “PuzzleFast.” http://
www.puzzlefast.com
Alessandra King, alessandra.king@
holton-arms.edu, studies mathematics
with her students at the Holton-Arms
School in Bethesda, Maryland. She has
taught mathematics and physics at the
middle school and high school level in
Europe, Asia, and America, and is interested in creative problem solving, critical
thinking, and quantitative reasoning.
Coming April 2014 from NCTM
Principles to Actions: Ensuring Mathematical Success for All
What it will take to turn the opportunity
of the Common Core into reality in
every classroom, school, and district.
Continuing its tradition of mathematics
education leadership, NCTM has undertaken a
major initiative to define and describe the
principles and actions, including specific teaching
practices, that are essential for a high-quality
mathematics education for all students.
This landmark new publication offers guidance to
teachers, mathematics coaches, administrators,
parents, and policymakers:
• Provides a research-based description of eight
essential Mathematics Teaching Practices
• Builds on NCTM’s Principles and Standards for
School Mathematics and supports
implementation of the Common Core State
Standards for Mathematics to attain much
higher levels of mathematics achievement for
all students
• Identifies obstacles, unproductive and
productive beliefs, and key actions that must
be understood, acknowledged, and addressed
by all stakeholders
• Encourages teachers of mathematics to
engage students in mathematical thinking,
reasoning, and sense making to significantly
strengthen teaching and learning
• Describes the conditions, structures, and policies
©April 2014, Stock #14861
www.nctm.org/PrinciplestoActions
that must support the Teaching Practices
For more information or to place an order,
please call 800.235.7566 or visit www.nctm.org/catalog.
442
Mathematics Teaching in the Middle School
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Vol. 19, No. 7, March 2014