Mathematics - Seminar

1. GARRY C. CACHUELA, PhD
West Visayas State University- Lambunao
Campus

2. In mathematics, a conjecture is…
”I believe that this is
TRUE, but I have NO
PROOF yet"

3. In mathematics,
a conjecture is…
”throw
together…"
Throwing
together
many
observations
into one idea.

4. Conjecture as part of Math
Investigation (Palomo, E.)
-Exploring systematically should permit
some patterns or relationships to begin to
emerge
-Making general statements about patterns
or relationships observed in the cases
considered.

5. Stages in Mathematical Investigation
(Arellano, E.)
- preceded by sample data from which the
conjecture was generated
- stated in concise mathematical statements
(in symbols if appropriate)
- clearly generated from the data, that is, data
are sufficient, appropriate, well
organized, and logically presented
- patterns are noted

6. Looking for Patterns

7. Variables That Change Predictably
linear, quadratic, exponential, etc.
n 1 2 3 4 5 6
F(n) -3 -1 1 3 5 7
Linear
2 2 2 2 2

8. When the first differences are
constant, the pattern represents
a linear equation. In this case,
the equation is, y= 2x-5.
The constant difference is the
coefficient of x in the equation.

9. Variables That Change Predictably (Linear)

10. Situation:
A karenderia has only rectangular
tables that can seat 6 people each, as
shown below. If n tables must be
joined to form a single long table,
investigate…

11. 6 people 6 + 4 = 10
Problem:
How many people can sit in nth
table?
Data Gathering using Illustration…

12. 6 + 4 + 4 + 4= 18
6 + 4 + 4 = 14

13. Summarizing the data in the table, we get
number of tables (n) Number of people (P)
1 6 = 6 + 4(0) = 6
2 6+4 = 6 + 4(1) = 10
3 6+ 4+4= 6 + 4 (2) = 14
4 6 + 4 + 4 + 4 = 6 + 4(3) = 18
... ...
n
=6 + 4 (n-1)
= 6 + 4n -4
=4n + 2 or
=2 (2n + 1)

14. Using differences method
and linear equation
no of tables
(n)
1 2 3 4 5
Number of
People (B)
6 10 14 18 22
4 4 4 4
first differences

15. Since the common difference occur at the
first differences the equation is in the form
of a linear equation An + B=0
If n = 1, f (n) = 6
A(1) + B = 6
A + B = 6 equation 1
If n = 2, f (n) = 10
A (2) + B = 10
2A + B = 10 equation 2

16. Equation 1-Equation 2
A + B = 6  2A + 2B = 12
2A + B = 10  - (2A + B = 10)
B = 2
Substitute B = 2 to equation 1 to get A
A + 2 = 6  A = 4

17. Conjecture:
The number of people (P) for
number of tables (n) when
tables are joined side by side is
P = 2 ( 2n + 1) = 2+4n  n Є N
Since A = 4, B = 2, then An + B = 0
is 4n + 2 = 0 or 2 (2n + 1)=0

18. Equation of the line can also be used
y - y1 = m ( x - x1)
Solving for slope m, we get the points (1, 6) and (2, 10)
𝑚 =
(10−6)
(2−1)
= 4
By substitution, using m=4 and (1,6), we get
y-6 = 4 (x-1)
y-6 = 4x - 4
y = 4x + 2
y = 2 (2x + 1)

19. Variables That Change Predictably
linear, quadratic, exponential, etc.
n 1 2 3 4 5 6
F(n) 2 5 10 17 26 37
Quadratic
3 5 7 9 11
2 2 2 2

20. When the second differences are
constant, the pattern represents
a quadratic equation. In this
case, the equation is, y = x2 + 1.
The constant difference divided
by 2, gives the coefficient of x2 in
the equation.

21. Variables That Change Predictably (Quadratic)

22. Situation:
Angela wants to make a stairway
using blocks as shown below. Continue
the pattern and investigate.
1 3 6

23. 1 1+2=3 1+2+3=6 1+2+3+ =10
Pattern Finding and Conjecture
Problem:
How many blocks would she need
to make an n-step stairway?

24. Summarizing the data in the table:
Number of
Layers
Total number of blocks (B)
1 1
2 1+2=3
3 1+2+3=6
4
.
.
n
1+2+3+4=10
.
.
1+2+3+...+ (n-2) + (n-1)+n

25. Using number pattern:
1+2+3+4 = 5 x 2=10
5
5
1+2+3+4+5= 5x2 + 5
5
5
= 5(2+1)
= 5(3)
=15

26. 1+2+3+4+5+6 = 7 x 3=21
7
7
1+2+3+4+5+6+7= 7x3 + 7
7
7
= 7(3+1)
= 7(4)
=28
7
7

27. Patterns
Found
Odd
Layers
Even
Layers

28. For even layers
(1+2+3+4)
(first + last) x number of pairs
=(1+n)(n/2)

29. For odd layers
(1+2+3+4+5)
(first + (last-1)) x (last-1)/2 + last
=(1+(n-1)) x (n-1)/2 + n
=n x (n-1)/2 + n
=(n2-n)/2+ n
=[(n2-n) + 2n]/2
=n(n+1)/2
=(1+n)(n/2)

30. Conjecture:
For any n layers of a
stair, the total number of
blocks (B) is
B = (n+1)(n/2),  n Є N

31. Using differences method and quadratic
equation
no of layers
(n)
1 2 3 4 5
Number of
blocks (B)
1 3 6 10 15
2 3 4 5
1 1 1
first differences
second differences

32. Since the common difference occur at the second differences,
the equation is quadratic of the form:
An2 + Bn + C=0
If n = 1, f(n)= 1
A(1)2 + B(1) + C = 1
A + B + C = 1 equation 1
If n = 2, f(n) = 3
A (2)2 + B(2) + C = 3
4A + 2B + C = 3 equation 2
If n = 3, f(n) = 6
A ( 3)2 + B(3) + C = 6
9 A + 3B + C = 6 equation 3

33. Using elimination method to get A, B, and C
Equation 1 - Equation 2
A + B + C = 1  4 A + 4B + 4C = 4
4A + 2B + C = 3  - ( 4A + 2B + C = 3)
2B + 3C=1 equation 4
Equation 1 - Equation 3
A + B + C = 1  9A + 9B + 9C = 9
9 A + 3B + C = 6  - (9A + 3B + C = 6)
6B + 8C = 3 equation 5

34. 2B + 3C = 1  6 B + 9C = 3
6B + 8C = 3  - (6B + 8C = 3)
C = 0
Substitute C=0 to equation 4
2B + 3(0) = 1
2B =1
B = 1/2
Substitute C=0, and B= 1/2 to equation 1
A + B + C = 1
A + 1/2 + 0 = 1
A = 1/2
Equation 4 – Equation 5

35. Since A=1/2, B=1/2 and C=0, then
An2 + Bn + C = 0
1/2n2 + 1/2 n + 0 = 0
(1/2 n) ( n + 1) =0
Thus, The number of Blocks (B)
for n layers is
B=(1/2n)(n+1)

37. References
• Abrams, Joshua (2001). Mathematical modeling.
Retrieved at www.meaningfulmath.org/modeling.
• Arellano, E. Lecture Notes in Mathematics
Investigations
• Jinon, S. (2017). 8th MTAP-IC Seminar Workshop and
Convention
• Libutaque, M. Raising the Bar of Mathematics
Education through Mathematical Investigation.
• Palomo, E. Work text in Mathematical Investigations
with Modelling
• Education Development Center, Inc. (2002). Retrieved
at
http://www2.edc.org/makingmath/handbook/teacher/
conjectures/Conjectures.pdf