Proof

1
Proof
2
Proof
Mathematics grows through
Improvement of guesses by speculation and
criticism
By logic of proof
Need proofs in the study of mathematics
Without them, we may arrive at incorrect
conclusions

3
Proof
Proof convinces that
Conclusions are factual
Argument can be replicated
Discussion does not contradict a know truth

4
Proof
Which is the real triangle ABC?
Why SSA does not prove 2 triangles congruent.

5
Proof
Prove ?DEF ?DFE at least 2 different ways.

6
Proof
DEF is a triangle Given
DG ? EF Construction
?DGF rt. ? Dfn. ? lines
?DGF is rt. Dfn. rt. ?
?DGE rt. ? Dfn. ? lines
?DGE is rt. Dfn. rt. ?
7
Proof
8
Proof
Proof requires
Understanding definitions and logic
Depends on insight into
How
Why
Before students prove, must become
Independent thinkers
Understand need for
Precise language
Definition
Expression
9
Proof
What is the probability that 2 people in a group
of 23 people share the same birthday (month and
day)?
Likely or not
Each person give estimate
0.5073
10
Proof
Birthday table
11
Proof
36 in 1 yd
9 in 0.25 yd (divide both sides by 4)
3 in 0.5 yd (positive square root of both
sides)
Is it true that 3 inches equals a half a yard?
What is wrong with this proof?

12
Proof
Seeing is not always believing
Individual fire walks on bed of coals
Mind over matter excludes pain
Spectators convinced
Fact - - dehydrated wood
Has extremely low heat content
Poor conductor of heat
Person walking quickly feels little heat
Investigation will yield a proof.
13
Proof
President Garfield proof of the Pythagorean
theorem
Good connection to Social Studies
14
Proof
What is the divisibility rule for 2?
Show why the 2 rule is true
7,358 (735)(10) 8
(735)(5)(2) (4)(2)
2(735)(5) 4
23675 4
23679 Unless the ones digit is a 0, 2, 4,
6, or 8, a 2 could not be factored out of the
sum and the original number would not be
divisible by 2

15
Proof
What is the divisibility rule for 5?
Show why the 5 rule is true
Solution 3465 (346)(10) 5
(346)(5)(2) 5
5(346)(2) 1
5692 1
5693 Unless the ones digit is a 0 or 5, a
5 could not be factored out of the sum and the
original number would not be divisible by 5
16
Proof
What is the divisibility rule for 4?
5,732 57(100) 32
57 (4)(25) 4 (8)
457(25) 8
17
Proof
What is the divisibility rule for 8?
Show why the 8 rule is true
18
Proof
What is the divisibility rule for 3?
Prove the 3 rule using X, Y, and Z for digits
XYZ (X)(100) (Y)(10) Z
(X)(99 1) (Y)(9 1) Z
(X)(99) (X)(1) (Y)(9) (Y)(1) Z
99X 9Y X Y Z
9 and 99 are divisible by 3, so if X Y Z is
divisible by 3 the original number must also be
divisible by 3.

19
Proof
What is the divisibility rule for 9?
20
Proof
Why does the divisibility for 4 Work?
21
Proof
Divisibility rule for 11
If the number is UVWXYZ, subtract (U W Y)
from (V X Z) (or vice versa)
If the missing addend is a multiple of 11, the
original number is a multiple of 11
If the missing addend is not a multiple of 11,
the original number is not divisible by 11
22
Proof
Proof of divisibility rule for 11
UVWXYZ rewritten as
U(10)5 V(10)4 W(10)3 X(10)2 Y(10)1
Z(10)0 , which is
U(11 - 1)5 V(11 - 1)4 W(11 - 1)3 X(11 -
1)2 Y(11 - 1)1 Z(11 - 1)0
Each term in expansion divisible by 11
Except -U, V, -W, X, -Y, and Z
Notice subtraction of sums of alternate digits
23
Proof
Prove x0 1 (n ? 0, ?)
x0 1 by transitive property of equality
24
Proof
25
Proof
Fast adding
We will add five 2-digit addends
Do not repeat digits (33, 77, etc.) within
addend
The sum will be 258
You pick 2 numbers
I pick 3 numbers
You go first
26
References
Brumbaugh, D. K., Ortiz, E., Gresham, G.
(2006) Teaching Middle School Mathematics Mahwah,
NJ Lawrence Erlbaum Associates Brumbaugh, D.,
Rock, D. (2006 (3rd Ed.) Teaching Secondary
Mathematics Mahwah, NJ Lawrence Erlbaum
Associates Brumbaugh, D., Rock, D.
(2001) Scratch Your Brain C1 Pacific Grove, CA
Critical Thinking Books and Software.