Reasoning Activities
Using 0.999... = 1 in
Developmental
Mathematics
Chris L. Yuen, CLYUEN@buffalo.edu
SUNY University at Buffalo
November 13, 2014
Cross Examination
O Inherent the Wind (1960)
https://www.youtube.com/watch?v=l5Kdc0
LLSW8
O Witness for the Prosecution (1957)
https://www.youtube.com/watch?v=Fq3UK
04pNrY
O Legally Blonde (2001) Start the clip at 3:14
https://www.youtube.com/watch?v=ytWGiO
uzpe4
.999… as an Object of Interest
O Why use .999…? Yopp et al. (2011) studied
in-service teachers’ perception toward
.999…
Another Finding…
More findings from Yopp
1
3
O Some teachers believe that .333 … ≠ .
O Some teachers believe “small” quantities
are irrelevant.
O Teachers’ (mis)understanding of real
numbers can be tied to their teaching as
well as perceptions of their students
learning.
And More…
O Some teachers modify “truth” to fit their
subjective beliefs.
O Some teachers dismiss conflict as
unimportant or uninteresting as it relates to
their teaching.
O Some teachers teach that approximations
are good enough and “small” does not
matter.
Developmental Math
O Much instruction directed to development
math is pedagogically delivered differently
from higher math.
O Emphasis of procedures
O Emphasis of formulas
O Emphasis of performing well on traditional
tests and exams
Proving .999… = 1
Proofs requiring addition:
2
= .222 …
9
7
= .777 …
9
9
= .999 … = 1
9
A Multiplication Proof
1
= .333 …
3
1
× 3 = .333 … × 3
3
1 = .999 …
An algebraic proof
Let 𝑥 = .999 …
10𝑥 = 9.999 …
10𝑥 − 𝑥 = 9.999 … − .999 …
9𝑥 = 9
𝑥=1
Proof by Limit
Define 𝑘𝑛 =. 999 … 99
𝑛
Find the limit:
. 999 … = lim 𝑘𝑛 = lim . 999 … 99
𝑛→∞
𝑛
= lim
𝑛→∞
𝑘=1
𝑛→∞
𝑛
9
10𝑘
1
1
lim 1 − 𝑛 = 1 − lim 𝑛 = 1 − 0 = 1
𝑛→∞
𝑛→∞ 10
10
How to Engage Students?
O The proofs alone are not going to motivate
students.
O Developmental students often do not know
much about the concepts of “limits.”
O One can show an application of proof in
legal arguments – Reductio ad absurdum
O So, involving a trial for the students to craft
arguments may be a productive alternative.
What is
Reductio ad absurdum?
O Big Bang Theory
https://www.youtube.com/watch?v=ytWGiOuz
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Reductio ad absurdum
O Reduction to absurdity
O To demonstrate a statement is true by
showing that a false, untenable, or absurd
result follows from its denial
O This sounds very much like “Proof by
Contradiction
The Trial
Is .999 < 1?
Mathematics Zealot vs. Point Nine Repeat
Analysis of the Activity
O The trial encompasses all the proofs that a
developmental mathematics student could
comprehend.
O The trial co-create mathematical discourse
with students in a social manner, as
opposed to proving a mathematical fact in a
solitary setting.
O Mathematics can be system/postulate
specific, and the court room setting is
analogous to a specific system.
Application
O This activity allows students to see how
mathematical argumentation and logic be
applied in an unexpected setting.
O Other form of argumentation can be used in
this activity, such as “Proof by Contradiction”
(e.g. 2 is irrational).
O The overall effect of this activity is to show
that argumentation is indeed applied, and in
this case, there is a link between math and
the legal field.
Questions/Comments
Contact Information:
Chris L. Yuen, Ed.D., CLYUEN@buffalo.edu
EOC Assistant Professor of Mathematics
University at Buffalo
Educational Opportunity Center
555 Ellicott Street
Buffalo, NY 14203