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Math 1150Summer 2012
Problem Solving and Number Patterns
How do you solve a problem?
Polya’s 4-Step problem-solving process (pg. 4)
1) Understand the problem2) Devise a plan
• Look for a pattern• Examine related problems• Look at a simpler case• Make a table or list• Draw a picture• Write an equation• Guess and Check• Work backward• Use direct or indirect reasoning
3) Carry out the plan4) Look back
• Does your answer make sense?• Did you answer the question that was
asked?• Is there another way to find the solution?
How do you solve a math problem?
4th grade - 1959
4th grade - 2008
1. A pen at the zoo holds giraffes and ostriches. Altogether, here are 30 eyes and 44 feet on the animals. How many of each type of animal are there?
How would you solve this problem?
Let x = the number of giraffesLet y = the number of ostriches
2x + 2y = 304x + 2y = 44
4x + 4y = 60–4x – 2y = –44
2y = 16 y = 8 x = 7
1. A pen at the zoo holds giraffes and ostriches. Altogether, here are 30 eyes and 44 feet on the animals. How many of each type of animal are there?
How could a 2nd grader do this problem?
1. A pen at the zoo holds giraffes and ostriches. Altogether, here are 30 eyes and 44 feet on the animals. How many of each type of animal are there?
Each animal has two eyes 15230 Each animal has at least two feet
We still need 14 more feet (44 – 30)
7 giraffes8 ostriches
1. A pen at the zoo holds giraffes and ostriches. Altogether, here are 30 eyes and 44 feet on the animals. How many of each type of animal are there?
How did a 7th grader do this problem?
1. A pen at the zoo holds giraffes and ostriches. Altogether, here are 30 eyes and 44 feet on the animals. How many of each type of animal are there?
2. Find the sum: 1 + 2 + 3 + … + 48 + 49 + 50
2. Find the sum: 1 + 2 + 3 + … + 48 + 49 + 50
Gauss’ Method
5151
Sum of each pair = 51
Number of pairs = Number of numbers / 2 = 25
Sum of numbers = (Number of pairs)(Sum of each pair)= (25 )(51)= 1275
3. If ten people are in a room, how many handshakes can they exchange?
People 1 2 3 4 5 6 7 8 9 10
Handshakes 0 1 3 6 10 15 21 28 36 45
+ 1
+ 2
+3
+ 4
+ 5
+ 6+ 7+ 8
+ 945 handshakes
4. Mark and Bob began reading the same novel on the same day. Mark reads 6 pages a day, and Bob reads 5 pages a day. If Mark is on page 78, what page is Bob on?
Mark: 78 / 6 = 13 days of reading
Bob: 5 (13) = page 65
4. Mark and Bob began reading the same novel on the same day. Mark reads 6 pages a day, and Bob reads 5 pages a day. If Mark is on page 78, what page is Bob on?
5. If a bag of potato chips and a Snickers bar together cost $3.00, and the chips cost $1.50 more than the candy bar, how much does each item cost separately?
5. If a bag of potato chips and a Snickers bar together cost $3.00, and the chips cost $1.50 more than the candy bar, how much does each item cost separately?
Chips$1.50$2.50$2.25
Guess-and-check must show at least one incorrect guess and the correct guess
Candy$1.50$ .50$ .75
Chips and candy cost the sameChips cost $2.00 moreChips cost $1.50 more
Chips: $2.25Candy: $0.75
6. Billy spent 2/3 of his money on baseball cards and 1/6 of his money on a candy bar, and after that he still had 50 cents left in his pocket. How much money did he start with?
Baseball cards Candy bar
50₵50₵ 50₵ 50₵ 50₵ 50₵
6 x 50₵ = $3.00
Billy’s whole box of money
6. Billy spent 2/3 of his money on baseball cards and 1/6 of his money on a candy bar, and after that he still had 50 cents left in his pocket. How much money did he start with?
Number Patterns
The student council plans to build a flower box with several sections in front of the school cafeteria. Each section will consist of squares made with railroad ties of equal length. Due to money constraints, the box will be built in several phases. Below is the plan for the first three phases. (Each side of the square represents one tie.)
From MODESE Model Curriculum Unit (6th grade)
Phase 1 Phase 2 Phase 3
Create a table to show how many ties will be needed for each of the first six phases. Explain how you know whether the pattern represents a linear or a nonlinear function.
Phase 1 Phase 2 Phase 3
Phase 1 2 3 4 5 6
Ties 4 7 10 13 16 19
+3
+3
+3
+3
+3
Linearbecause the phase number increases at a constant rate of 1, and the ties increase at a constant rate of 3
Phase 1 2 3 4 5 6
Ties 4 7 10 13 16 19
A sequence is a pattern of numbers or symbols.
A term is a number in a sequence.
In an arithmetic sequence, we add a constant number (called the common difference) to find the next term in the sequence.
For this sequence, d = 3
Term 1 2 3 4 5 6
Ties 4 7 10 13 16 19
A sequence is a pattern of numbers or symbols.
A term is a number in a sequence.
Term a1
a2
a3
a4
a5
a6
Ties 4 7 10 13 16 19
A sequence is a pattern of numbers or symbols.
A term is a number in a sequence.
Term a1
a2
a3
a4
a5
a6
Value 4 7 10 13 16 19
A sequence is a pattern of numbers or symbols.
A term is a number in a sequence.
Term a1
a2
a3
a4
a5
a6
an
Value 4 7 10 13 16 19
A sequence is a pattern of numbers or symbols.
A term is a number in a sequence.
Pattern 4 4 + 1(3) 4 + 2(3) 4 + 3(3) 4 + 4(3) 4 + 5(3) 4 + (n-1)(3)
+3
+3
+3
+3+3
Term a1
a2
a3
a4
a5
a6
an
Value 4 7 10 13 16 19
What is the 100th term in the sequence?
4 + (100 – 1)(3) = 4 + 297 = 301
Pattern 4 4 + 1(3) 4 + 2(3) 4 + 3(3) 4 + 4(3) 4 + 5(3) 4 + (n-1)(3)
+3
+3
+3
+3+3
Term a1
a2
a3
a4
a5
a6
an
Pattern 4 4 + 1(3) 4 + 2(3) 4 + 3(3) 4 + 4(3) 4 + 5(3) 4 + (n-1)(3)
nth term formula for an arithmetic sequence:
an = a1 + (n – 1)d
a1 = 4
d = 3
an = a1 + (n – 1)d
5, 9, 13, 17, …
Find the next three terms
d = 4
21, 25, 29
Find an nth term formula for the sequence
a1 = 5 d = 4
an = 5 + (n – 1)4
an = 5 + 4n – 4
an = 4n + 1
Find the 100th term of the sequence
a100 = 4(100) + 1 = 401
5, 9, 13, 17, …
What position in the sequence is the number 609?
an = 4n + 1
609 = 4n + 1-1 -1
608 = 4n 4 4
152 = n 609 is the 152nd term
DESE sample MAP item, 3rd grade
4, 12, 36, 108, …
X 3 X 3 X 3
In a geometric sequence, we multiply by a constant (called the common ratio) to find the next term in the sequence.
For this sequence, r = 3
Find the common ratio for the sequence128, 64, 32, 16, …
1632 =
12
Term a1
a2
a3
a4
an
Value 4 12 36 108
Pattern 4 4 x (3) 4 x (3)2
4 x (3)3
4 x (3)n - 1
x3
x3
x3
4, 12, 36, 108, …
What is the 100th term of the sequence?
a100 = 4 x 3100 – 1 = 4 x 399
Term a1
a2
a3
a4
an
Pattern 4 4 x (3) 4 x (3)2
4 x (3)3
4 x (3)n - 1
a1 = 4 r = 3
nth term formula for a geometric sequencean = a1 · rn - 1
What are the next three terms in this sequence:
1, 3, 4, 7, 11, 4729, 18,
In a Fibonacci sequence, we add two consecutive terms to find the next term.
The first two terms of a sequence are 1 and 5. Find the next two terms if the sequence is
a) Arithmetic
b) Geometric
c) Fibonacci
1, 5,
1, 5,
1, 5,
9, 13
+ 4 + 4 + 4
x 5 x 5 x 5
25, 125
6, 11
A finite sequence has a limited number of terms.
One way to count the number of terms in a sequence is to find a one-to-one correspondence with the counting numbers.
How many terms are in the sequence?
A) 4, 8, 12, …, 40divide each term by 4
1, 2, 3, …, 10
The sequence has 10 terms
How many terms are in the sequence?
B) 4, 9, 14, …, 59add 1 to each term
5, 10, 15, …, 60divide each term by 5
1, 2, 3, …, 12
The sequence has 12 terms
How many terms are in the sequence?
C) 62 , 63 , 64, …, 620
subtract 1 from each exponent 61 , 62 , 63, …, 619
look at exponents1, 2, 3, …, 19
The sequence has 19 terms
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