Aim: Problem Solving Course: Math Literacy Do Now: Aim: Why is problem solving sooooo...
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Aim: Problem Solving Course: Math Literacy
Do Now:
Aim: Why is problem solving sooooo problematic??!!
A
Melissa lives at the YWCA (A) and works at Macy’s (B). She walks to work. How many different routes can she take?
B
Sutter St.
Post St.
Geary St.
O’Farrell St.
Stock
ton S
t.
Pow
ell St.
Mason
St.
Taylor S
t.
Aim: Problem Solving Course: Math Literacy
Guidelines for Problem Solving
First You have to understand the problem
Second Devise a plan. Find the connection between the data and the unknown. Look for patterns, related to previously solved problem or a know formula, or simplify the given info to give you an easier problem.
Third Carry out the plan
Fourth Look back and examine the solution obtained
Aim: Problem Solving Course: Math Literacy
Guidelines for Problem Solving
First You have to understand the problem
What facts are given?
What does the problem tell me?
Do I know what all the words and phrase mean?
Did I read the problem carefully?
What is the problem asking me?
Aim: Problem Solving Course: Math Literacy
Guidelines for Problem Solving
Second Devise a plan.
What am I trying to find out?
What facts do I need to know?
Should I make a chart, a table or a diagram?
Can I do the problem in my head or do I need paper and pencil or a calculator?
What steps should I follow?
Aim: Problem Solving Course: Math Literacy
Guidelines for Problem Solving
Third Carry out the plan
Review the steps and formulate/calculate the answer.
Fourth Look back and examine the solution obtained
Reread the problem and ask yourself: Does the answer make sense.
Aim: Problem Solving Course: Math Literacy
Guidelines for Problem Solving
Read
Plan
Solve
Reflect
Step 1
Step 2
Step 3
Step 4
Aim: Problem Solving Course: Math Literacy
Model Problem
A library has 2890 science books. The science books are classified into three categories: life, earth, and physical science. There are 190 more books in the earth category than in each other category. How many books are in each category.
Understand the problemDevise a plan
Carry out the planLook back
Aim: Problem Solving Course: Math Literacy
Do Now Problem
A
B
1 1
12
1
3 4
3 6 10
4 10 20
1
1
Understand the problemDevise a plan
Carry out the planLook back
no backtracking; no cutting thru backyards
one way to arrive at this point
start small
Each # of ways to a point is found by adding the two numbers from the upper/left vertices that connect to that point.
Aim: Problem Solving Course: Math Literacy
Guidelines for Problem Solving
A
B
1 1
12
1
3 4
3 6 10
4 10 20
1
1
Each vertex in the rows is found by adding the two numbers from the above row that connect to it
AB
1
1
1
2
13
4
3
6
10
4
10
20
1
1
Aim: Problem Solving Course: Math Literacy
Pascal’s Triangle
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
The first & last numbers in each row are 1
Every other number in each row is formed by adding the two numbers above the number.
Aim: Problem Solving Course: Math Literacy
Pascal’s Triangle & Expansion of (x + y)n
(x + y)0 = 1
(x + y)1 = 1x + 1y
(x + y)2 = 1x2 + 2xy + 1y2
(x + y)3 = 1x3 + 3x2y + 3xy2 + 1y3
(x + y)4 = 1x4 + 4x3y + 6x2y2 + 4xy3 + 1y4
(x + y)5 = 1x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + 1y5
In each expansion there is n + 1 terms.
In each expansion the x and y have symmetric roles.
The sum of the powers of each term is n.
The coefficients increase & decrease symmetrically.
5 5
44
expansion of (x + y)n
zero row
1st row
Aim: Problem Solving Course: Math Literacy
Melissa’s Trip
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
A
3 blocks down
3 blocks over
Aim: Problem Solving Course: Math Literacy
Extension
How many different ways could Melissa get from the YWCA to the YMCA (D)?
A
D
Aim: Problem Solving Course: Math Literacy
Model Problem
A jokester tells you that he has a group of cows and chickens and that he counted 13 heads and 36 feet. How many cows and chickens does he have?
Understand the problemDevise a plan
Carry out the planLook back
No. of chickens No. of cows No. of Heads No. of Feet
0 13 13 521 12 13 50
2 11 13 48
Look for patterns need 36 feet12 feet gotta goeach additional chicken
reduces No. of feet by 2need additional 6 chickens
8 5 13 36
Aim: Problem Solving Course: Math Literacy
Model Problem
If a family has 5 children, in how many different birth orders could the parents havea 3-boy, 2-girl family?
BBBGG
BBGGB
BGGBB
etc.
Understand the problemDevise a plan
Carry out the planLook back
Aim: Problem Solving Course: Math Literacy
Model Problem
If a family has 5 children, in how many different birth orders could the parents havea 3-boy, 2-girl family?
B
G
B
G
B
G
B
G
B
G
B
G
B
G
B
GBGBGBGB
GBGBGBG
BBBBBBBBBG
GGGGG
BBBGG
BG
. . . . .
. . . . .
. . . . .
. . . . .
Aim: Problem Solving Course: Math Literacy
Model Problem
If a family has 5 children, in how many different birth orders could the parents have
a 3-boy, 2-girl family?
Start with smaller family and look for pattern
1 child: B one wayG one way
2 children: BB one wayBGGBGG one way
two ways
3 children: BBB one wayBBGBGBGBBBBGGBGGGBGGG one way
three ways
three ways
Aim: Problem Solving Course: Math Literacy
Model Problem
If a family has 5 children, in how many different birth orders could the parents have
a 3-boy, 2-girl family?
2 children: 1BB 2 1GGtwo ways for 1 B and 1 G
1 child: 1B 1G
3 children: 1BBB 3 3 1GGG3 ways for 2B & 1G 3 ways for 1B & 2G
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
Pascal’s trianglebinomial expansion
Aim: Problem Solving Course: Math Literacy
Model Problem
If a family has 5 children, in how many different birth orders could the parents havea 3-boy, 2-girl family?
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
5B 4B1G 3B2G 2B3G 1B4G 5G
1 child 2 child
3 child 4 child
Ten different ways
Aim: Problem Solving Course: Math Literacy
Do Now:
Aim: Why is problem solving sooooo problematic??!!
As I was going to St. Ives,
I met a man with seven wives.
Every wife had seven sacks.
Every sack had seven cats.
Every cat had seven kits.
Kits, cats, sacks and wives,
How many were going to St. Ives
Aim: Problem Solving Course: Math Literacy
Gauss
When the famous German mathematician Karl Gauss was a child, his teacher required students to find the sum of the first 100 natural numbers. The teacher expected this problem to keep the class occupied for some time. Gauss answer almost immediately.
1 + 2 + 3 + · · · · + 50 + 51 + · · · · · + 98 + 99 + 100
101 x 50 pairs of numbers
= 5050
Aim: Problem Solving Course: Math Literacy
Model Problem
Find the sum of natural numbers from 1 to 50.
1 + 2 + 3 + · · · · + 25 + 26 + · · · · · + 48 + 49 + 50
51 x 25 pairs of numbers
= 1275
Will this method for finding the sum of the natural numbers always work?
Is there a formula?
Aim: Problem Solving Course: Math Literacy
Model Problem
1 + 2 + 3 + · · · · + · · · · + · · · · · · · (n-2) + (n-1) + n
Will this method for finding the sum of the natural numbers always work?
Is there a formula?
must be an even number2
n
1 for is even2
nS n n
n + 1
n + 1 sum of pairs
Aim: Problem Solving Course: Math Literacy
Examine a Related Problem
Ryan was building matchstick square sequences, as shown below. He use 67 matchsticks to form the last figure in his sequence. How many match sticks did he use for the entire project?
4 7
10
Every added box requires three more matches
Aim: Problem Solving Course: Math Literacy
Examine a Related Problem
4 + 7 + 10 + · · · + · · · · + · · · · · · · · 61 + 64 + 67
71
even # of numbers?
If so, how many pairs?
sums equal 71 for successive outer pairings
Aim: Problem Solving Course: Math Literacy
Examine a Related Problem
4 + 7 + 10 + · · · + · · · · + · · · · · · · · 61 + 64 + 67
71
+3 +3 +3 +3 +3
# of Term Term
1 4
2 7 = 4 + 3
3 10 = 4 + 3 + 3
4 13 = 4 + 3 + 3 + 3
n
2 3 1
3 3 1
4 3 1
3 1n
11 pairs
71 x 11 = 781
1 3 3
n = 223n + 1 = 67
Aim: Problem Solving Course: Math Literacy
Draw a Diagram
On the first day of math class, 20 people are present in the room. To become acquainted with one another, each person shakes hands just once with everyone else. How many handshakes take place?
Understand the problemDevise a plan
Carry out the planLook back
John shakes Mary’s hand and Mary shakes John’s hand counts only as one handshake, not two.
Aim: Problem Solving Course: Math Literacy
Draw a Diagram
On the first day of math class, 20 people are present in the room. To become acquainted with one another, each person shakes hands just once with everyone else. How many handshakes take place?
A C
B
three people A C
DE
four people
A C
DE
F
3 handshakes
6 handshakes
5 people 10 handshakes
Aim: Problem Solving Course: Math Literacy
Draw a Diagram
On the first day of math class, 20 people are present in the room. To become acquainted with one another, each person shakes hands just once with everyone else. How many handshakes take place?
three people four people3 handshakes 6 handshakes
5 people10 handshakes
each member shakes the
hand of two other people
3 x 2 = 6
John shakes Mary’s hand and Mary shakes John’s hand counts only as one handshake, not two.
each member shakes the hand of 3
other people
4 x 3 = 12
each member shakes the hand of 4
other people
5 x 4 = 20
6/2 = 3 12/2 = 6 20/2 = 10
(20 x 19)/2 = 190 total handshakes
Aim: Problem Solving Course: Math Literacy
Model Problems
How much dirt is in a a hole 2 feet long, 3 feet wide and 2 feet deep?
Two US coins have a total value of 55 cents. One coin is not a nickel. What are the two coins?
Walter had a dozen apples in his office. He ate all be 4. How many are left?
Sal owns 20 blue and 20 brown socks, which he keeps in a drawer in complete disorder. What is the minimum number of socks that he must pull out of the drawer on a dark morning to be sure he has a matching pair?
Aim: Problem Solving Course: Math Literacy
Types of Problems
In your own words Asks u to discuss or rephrase main ideas or procedures using your own words.
Level 1 Mechanical and drill
Level 2 require understanding of concepts and related to past examples
Level 3 extension of past problems requiring creative thought
Problem Solving original thinking; not based on past examples