1. Problem Solving; Venn Diagrams; Patterns; Pascal’s Triangle; Sequences
MATH st Exam Review
Problem Solving; Venn Diagrams; Patterns; Pascal’s Triangle; Sequences

2. Problem Solving Polya’s 4 Steps Understand the problem Devise a plan
Carry out the plan
Look back

3. Problem Solving Strategies for Problem Solving Make a chart or table
Draw a picture or diagram
Guess, test, and revise
Form an algebraic model
Look for a pattern
Try a simpler version of the problem
Work backward
Restate the problem
Eliminate impossible situations
Use reasoning

4. Problem Solving How many hand shakes? Playing darts Tetrominos
Who am I?
Triangle puzzle

5. Venn Diagrams Vocabulary Intersection Universe Union Element
Set
Subset
Disjoint
Mutually Exclusive
Finite
Intersection
Union
Compliment
Empty Set
Infinite

6. Venn Diagrams What can you say about A and B? A Ç B = Æ A È B = {A, B}
A and B are mutually exclusive or disjoint A B

7. Venn Diagrams What can you say about A and B? A Ç B = A È B = A’ Ç B =

8. Venn Diagrams What can you say about A, B, and C?
A Ç B  C =? A È B  C =?
(A Ç C)  B =? A Ç (C  B) =?
(A Ç B)  C =? C Ç (A  B) =?
(B Ç C)  A =? B Ç (C  A) =?
(A’ Ç B)  C =? (A’ È B)  C =?
A’ Ç B’  C’ =? A’  B’  C’ =? Etc. B A C

9. Patterns
Triangular Numbers
Etc.
T T T3 T4
Tn = Tn-1 + n

10. Patterns
Square Numbers
Etc.
S1 S2 S S4
Sn = n2

11. Patterns Rectangular Numbers R1 R2 R3 R4 Etc. Rn = n (n + 1)

12. Pascal’s Triangle Expanding a binomial expression: (a + b)0 = 1
(a + b)1 = a + b
(a + b)2 = a2 + 2ab + b2
(a + b)3 = a3 + 3a2b + 3ab2 + b3
(a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4

13. Pascal’s Triangle Vocabulary Expansion – the sum of all of the terms
Coefficient – the number in front of the variable(s) for a particular term
Variable(s) – the letters AND their exponents for a particular term
Term – the coefficient AND the variable(s)

14. Pascal’s Triangle 1
1 1
1 2 1

15. Pascal’s Triangle Magic 11’s 110 1 111 1 1 112 1 2 1 113 1 3 3 1
Fails to work after this…

16. Arithmetic Sequences
The difference between any two consecutive terms is always the same.
Examples:
1, 2, 3, …
1, 3, 5, 7, …
5, 10, 15, 20, …
Non-Examples
1, 4, 9, 16, …
2, 6, 12, 20, …

17. Arithmetic Sequences The nth number in a series: Example
an = a1 + (n – 1) d
Example
Given 2, 5, 8, …; find the 100th term
n = 100; a1 = 2; d = 3
a100 = 2 + (100 – 1) 3
a100 = 2 + (99) 3
a100 =
a100 = 299

18. Arithmetic Sequences Summing or adding up n terms in a sequence:
Example:
Given 2, 5, 8, …; add the first 50 terms
n = 50; a1 = 2; a50 = 2 + (50 – 1) 3 = 149
S50 = (50/2) ( )
S50 = 25 (151)
S50 = 3775

19. Arithmetic Sequences Summing or adding up n terms in a sequence:
Example:
Given 2, 5, 8, …; add the first 51 terms
n = 51; a1 = 2; a2 = 5; a51 = 2 + (51 – 1) 3 = 152
S51 = 2 + ((51-1)/2) ( )
S51 = 2 + (50/2) ( )
S51 = (157)
S51 =
S51 = 3927

20. Geometric Sequences
The ratio between any two consecutive terms is always the same.
Examples:
1, 2, 4, 8, …
1, 3, 9, 27, …
5, 20, 80, 320, …
Non-Examples
1, 4, 9, 16, …
2, 6, 12, 20, …

21. Geometric Sequences The nth number in a series: Example an = a1 r(n-1)
Given 5, 20, 80, 320, …; find the 10th term
n = 10; a1 = 5; r = 20/5 = 4
a10 = 5 (4(10-1))
a10 = 5 (49)
a10 = 5 (262144)
a10 =

22. Geometric Sequences Summing or adding up n terms in a sequence:
Example:
Given 5, 20, 80, 320, …; add the first 7 terms
n = 7; a1 = 5; r= 20/5 = 4
S7 = 5(1 – 47)/(1 – 4)
S7 = 5(1 – 16384)/(– 3) = 5(– 16383)/(– 3)
S7 = (– 81915)/(– 3) = (81915)/(3)
S7 = 27305

23. Fibonacci Sequences 1, 1, 2, 3, … Seen in nature Golden ratio
Pine cone
Sunflower
Snails
Star fish
Golden ratio
(n + 1) term / n term of Fibonacci
Golden ratio ≈ 1.618

24. Test Taking Tips Get a good nights rest before the exam
Prepare materials for exam in advance (scratch paper, pencil, and calculator)
Read questions carefully and ask if you have a question DURING the exam
Remember: If you are prepared, you need not fear