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

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.


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



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?



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?



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.



Read

Plan

Solve

Reflect

Step 1

Step 2

Step 3

Step 4


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


Do Now Problem

A

B

1 1

12

1

3 4

3 6 10

4 10 20

1

1



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.



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


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.


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


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


Extension

How many different ways could Melissa get from the YWCA to the YMCA (D)?

A

D


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?



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


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.




Model Problem


B

G

B

G

B

G

B

G

B

G

B

G

B

G

B

GBGBGBGB

GBGBGBG

BBBBBBBBBG

GGGGG

BBBGG

BG

. . . . .

. . . . .

. . . . .

. . . . .


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


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


Model Problem


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


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


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


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?


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


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



4 + 7 + 10 + · · · + · · · · + · · · · · · · · 61 + 64 + 67

71

even # of numbers?

If so, how many pairs?

sums equal 71 for successive outer pairings



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


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?



John shakes Mary’s hand and Mary shakes John’s hand counts only as one handshake, not two.


Draw a Diagram


A C

B

three people A C

DE

four people

A C

DE

F

3 handshakes

6 handshakes

5 people 10 handshakes


Draw a Diagram


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


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?


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

Documents

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