Texas Mathematics Teacher Fall/Winter 2021 - 22

We've Got Your Number! Fun with Operations
Vince Schielack
Teachers are constantly on the lookout for ways to keep
students engaged and motivated. One source of activities
is mathematical puzzles and games, especially those
that have strong connections to the usual curriculum.
The Texas Essential Knowledge and Skills (TEKS) for
Mathematics expect middle-school students to be fluent
in the use of the integer arithmetic operations (addition,
subtraction, multiplication, and division) and in the
order of operations, and this is reiterated in the TEKS for
Algebra I (Texas Education Agency, 2012). This article
describes a game, We've Got Your Number, that requires
extensive use of these operations, giving students a
somewhat non-traditional platform for honing their
knowledge regarding them.
An easily-accessible, often-used problem of uncertain
origin is to use a set of digits (for example, four 4's) and
basic mathematical operations (usually unspecified)
to construct expressions that equal some subset of the
positive integers (for example, 1 through 100). Martin
Gardner, in The Magic Numbers of Dr. Matrix, traced
the history of the four 4's problem back to 1881 (Gardner,
1985, p. 50). Constance Reid, in From Zero to Infinity,
dedicated the last page of Chapter 4 to this problem, even
including the claim that, " it is possible, if we do not limit
ourselves as to notations, to represent all numbers by four
4's " (Reid, 1964, p. 70). An excellent review of the 50th
Anniversary Edition of From Zero to Infinity discusses
at length the four 4's problem and how it motivated the
reviewer mathematically when he was a child (Reznick,
2007). More recently, the problem has been discussed in
the letters section of NCTM's Mathematics Teaching in
the Middle School (Cohen, 2001; Poetsch, 2001; Threlkeld,
2001). Reid (1964) gave the following instructions for the
four 4's problem:
All four 4's must be used for every number, but
various mathematical notations may also be used, as
in the four examples below.
We've Got Your Number: The Contest
In 2004, the author codified a set of rules to be used for
a contest based on this type of problem, called We've
Got Your Number (WGYN). This contest became part
of the Texas Science Olympiad, an annual science and
mathematics competition whose winning schools qualify
for the national Science Olympiad competition. Science
Olympiad has teams at both the middle school (or B
Division) and high school (or C Division) levels, and the
WGYN competition is conducted at both levels.
Try now to find similar representations for five
through twelve in the terms of four 4's. (p.70)
Figure 1 displays the rules for the high school division.
(The rules for the middle school division differ only in
that factorials and logarithms are not allowed.) Teams
consist of two players, and they are given two different
lists of four digits, some of which may repeated on the list.
They may collaborate as desired, and are not allowed to
use calculators. They are given an answer sheet for each
set, upon which they write their expressions for as many
of the integers from 1 through 100 as they can. These rules
also contain scoring rules, plus tips and advice for those
administering and grading the contest.
22 | Fall/Winter 2021
Texas Mathematics Teacher
Texas Mathematics Teacher Fall/Winter 2021

Table of Contents for the Digital Edition of Texas Mathematics Teacher Fall/Winter 2021
