MS133 - Jacksonville State University | Where You're Going.

Transcript MS133 - Jacksonville State University | Where You're Going.

PROBLEM SOLVING

Chapter 1

       My web page A Fable Information Sheet Syllabus Grade Sheet Chapter 1 Answers Myths about Mathematics  History of Math Education

Math Anxiety

 Lets go to Taco Bell    Burrito $1.79

$2 Change: $18.21

Math Anxiety

 Timed Tests   Overcoming Math Anxiety by Sheila Tobias MATH – An American Phobia by Marilyn Burns

Three approaches to problem solving:  Experimentation  Intuition  Deduction

Try to remember as little as possible! Try to figure out as much as possible !

Real learning takes place when we connect social knowledge (facts that have to be told to you) to what’s inside the learner.

In the Math classroom we will be creating situations that cause you to interact with ideas and by that, connect something new to what you already know.

Water Jug Problem

You are stranded on a desert island. You have an unlimited amount of water but only 2 containers: a 5 gallon jug and a 3 gallon jug. You need exactly 4 gallons and you will be miraculously transported off the island. Both jugs are irregularly shaped and no markings can be made on them. How can you use these 2 containers and your unlimited source of water to obtain exactly 4 gallons of water?

Marilyn Burns Video

DAY 2

Cannibals and Missionaries

In a far away place there are 3 cannibals and 3 missionaries who wish to cross a river. They have learned to live together peacefully without any threat to their lives as long as they abide by one rule. The cannibals can never outnumber the missionaries. As long as there are more missionaries than cannibals or an equal number of each, everyone is safe. If the There is a boat that will hold 2 people that will be used to get all six across the river. No one swims. The boat cannot travel unmanned. If there is a missionary on one side by himself and 2 cannibals come to that side, there is no rule that keeps one in the boat. He will get out and eat that missionary.

Problem Solving is the Cornerstone of School Mathematics.

The NCTM Principles and Standards page 2

You have a tremendous responsibility and a great privilege!

Page 375

 SPOT  Fixation  The Book of Think by Marilyn Burns

     Pick a number between 1 and 100 Double it Add 10 to your answer Take half of what you have now Subtract 5 from that

What did you get?

      Pick a number between 1 and 100 Add 10 to it Double the result Add 100 Take half of the result Subtract the number you started with

What did you get?

      Choose any two odd numbers or any two even numbers.

Add the two numbers together.

Divide the sum by 2.

Subtract the smaller of the two original numbers from the larger.

Divide the difference by 2.

Add the two quotients.

What did you get?

        Pick a 2 digit number Add 11 Multiply by 6 Subtract 3 Divide by 3 Subtract 6 less than the original number Subtract one more than the original number Divide by 2

What did you get?

Three digit fun!

       Write down the number of the month you were born in.

Double it.

Add 5.

Multiply by 50 Add 1756 Subtract the year of your birth Circle the last 2 digits.

What do you notice about the result?

Looking for Patterns

1 x 9 = 21 x 9 = 321 x 9 = 4321 x 9 = 54321 x 9 = 87654321 x 9 =

67 x 67 = 667 x 667 = 6667 x 6667 = 6,666,667 x 6,666,667 =

9 x 1 = 9 x 2 = 9 x 3 = 9 x 4 = 9 x 5 = 9 x 6 = 9 x 7 = 9 x 8 = 9 x 9 =

11 x 4 = 11 x 26 = 11 x 89 = 11 x 9 = 11 x 72 = 11 x 2345 =

“Fifteen”

Fill in the box below with the Counting Numbers 1-9. Use each number only once and make each row, column, and diagonal have a sum of 15.

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45 45 ÷ 3 = 15 1 + 5 + 9 = 15 1 + 6 + 8 = 15 2 + 4 + 9 = 15 2 + 5 + 8 = 15 2 + 6 + 7 = 15 3 + 4 + 8 = 15 3 + 5 + 7 = 15 4 + 5 + 6 = 15

Standard 3 X 3 Magic Square

2 X 2 Magic Square

Old McDonald had a total of 37 chickens and pigs. All together they had 98 feet. How many chickens were there and how many pigs were there?

Old McDonald had a total of 37 chickens and pigs. All together they had 98 feet. How many chickens and how many pigs?

Old McDonald had a total of 37 chickens and pigs on his farm. All together they had 98 feet. How many chickens were there and how many pigs?

Toni is thinking of a number.

If you double the number and add 11, the result is 39.

What number is Toni thinking of?

Guess and Check

Place the digits 1,2,3,4, and 5 in these circles so that the sums across and vertically are the same.

10 Commandments for Teachers

Introduction to Problem Solving

 Two engineers were standing on a street corner. The first engineer was the second engineer’s father but the second engineer was not the first engineer’s son. How could this be?

 If an electric train is traveling 40 miles an hour due west and a wind of 30 miles per hour is blowing due east, which way is the smoke from the train blowing?

Introduction to Problem Solving

 At noon a rope ladder with rungs 1 foot apart is hanging over the side of a ship, and the twelfth rung down is even with the water surface. Later, after the tide has risen 3 feet, which rung of the ladder is just even with the surface of the water?

DAY 3

Homework Questions Page 6

Introduction to Problem Solving Answers

Census Taker Problem

During a recent census, a census taker went to a man's house who had three children. The father of the children told the census taker that the product of his children's' ages is 72. The census taker waited for more information. The man then told him that the sum of their ages is the same as his house number. The census taker looked at the house number and then replied "I still don't know how old they are." The father said "Oh, I forgot to tell you that my oldest child likes chocolate pudding!" The census taker was then able to write down the children's ages and go on to the next house. How old are the children. How do you know?

Polya’s Problem Solving Principles page 9

 Understand the problem  Devise a plan  Carry out the plan  Look back

The process is more important than the answer!

Guess and Check

16 8 14 2 6 12 18 11 15

8 14 18 22 12 13 17 13 9

Make an Orderly List/Table

How many different scores could you make if you hit the dartboard with three darts?

1 5 10

10’s 5’s 1’s Score

Explain how you could determine the number of ways you can make change for a dollar using quarters, nickels, dimes and pennies.

242

ways to make change for $1

Draw a Diagram Example1.5 page 16 In a stock car race the first five finishers in some order were a Ford, a Pontiac, a Chevrolet, a Buick, and a Dodge.

Problem Solving Strategies

   Guess and Check Make an orderly list Draw a diagram

Polygon Perimeter Lab

DAY 4

Homework Questions Page 18

Polygon Perimeter Lab

M R DUCKS M R NOT DUCKS O S M R DUCKS L I B I C M R DUCKS!

YY U R YY U B I C U R YY 4 ME!

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0 PhD BS MD

______ Program

deathlife

T O U C H

KNEE LIGHTS

HEHIMSELF

MAN BOARD

WEAR LONG

R O R O A D S D S

T O W N

CYCLE CYCLE CYCLE

chair

Dice Dice

GROUND FEET FEET FEET FEET FEET FEET

MIND MATTER

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Number Sequence Lab

Number Sequence Example

1st 2nd 3rd 4th 5th 4 7 10 13 16 6th 19

20 th

Figure

• Three horizontal rows of dots. The top row has 20 dots, the middle row has 20 dots, and the bottom row has 21 dots.

th Figure

• Three horizontal rows of dots. The top row has n dots, the middle row has n dots, and the bottom row has n + 1 dots.

DAY 5

 Word Picture Answers

1st

Number Sequence Lab

2nd 3rd 4th 5th

1st

Number Sequence Lab

2nd 3rd

1st

Number Sequence Lab

2nd 3rd

Number Sequence Lab

1st 2nd 3rd

Continue these numerical sequences.

 15, 30, 45, 60, ___, ___, ___ 75, 90, 105, . . .

Arithmetic Sequence common difference = 15  15, 30, 45, 75, ___, ___, ___, . . .

120, 195, 315, . . .

Fibonacci “type” sequence

 1, 1, 2, 3, 5, ___, ___, ___, . . .

8, 13, 21, . . .

The Fibonacci sequence



Leonardo of Pisa AKA Fibonacci

Fibonacci Fun, Fascinating Activities with Intriguing Numbers by Trudi Hammel Garland  Fascinating Fibonaccis, Mystery and Magic in Numbers by Trudi Hammel Garland  The Famous Rabbit Problem

 15, 30, 60, 120, ___, ___, ___, . . .

240, 480, 960, . . .

Geometric sequence common ratio = 2  15, 30, 90, 360, ___, ___, ___, . . .

1,800, 10,800, 75,600, . . .

 15, 30, 120, 960, ___, ___, ___, . . .

15,360, 491,520, 31,457,280, . . .

 15, 20, 26, 33, ___, ___, ___, . . .

41, 50, 60, . . .

 1, 4, 7, 10, 13, ___, ___, ___, . . .

16, 19, 22, . . .

Arithmetic sequence common difference = 3  19, 20, 22, 25, 29, ___, ___, ___, . . .

34, 40, 47, . . .

• 1, 4, 9, 16, 25, ___, ___, ___, . . .

• 1, 8, 27, 64, 125, ___, ___, ___, . . .

• A, E, F, H, I, K, ___, ___, ___, . . .

• 8, 5, 4, 9, 1, ___, ___, ___, . . .

Carl Gauss

  Page 27 Fractal, Googols, and Other Mathematical Tales by Theoni Pappas 1 + 2 + 3 + 4 + . . . + 98 + 99 + 100 = 5050

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 =

1 + 2 + 3 + 4 + . . . + 98 + 99 + 100 =

34, 39, 44, 49, . . .

 What are the most likely choices for the next two numbers in the sequence?

 What number will be the 46 th term?

(Make a table – establish a pattern)

34, 39, 44, 49, . . .

• What number will be the 46 th term?

34, 39, 44, 49, . . .

• What number will be the 46 th term?

Term # 1 2 3 4 Term 34 = 34 + 5(0) 39 = 34 + 5 = 34 + 5(1) 44 = 34 + 5 + 5 = 34 + 5(2) 49 = 34 + 5 + 5 + 5 = 34 + 5(3)

34, 39, 44, 49, . . .

• What number will be the n th term?

Term # 1 2 3 4 Term 34 = 34 + 5( 0 ) 39 = 34 + 5( 1 ) 44 = 34 + 5( 2 ) 49 = 34 + 5( 3 ) 46 259 = 34 + 5( 45 )

34, 39, 44, 49, . . .

• Which term is 359?

Term # 1 2 3 4 Term 34 = 34 + 5( 0 ) 39 = 34 + 5( 1 ) 44 = 34 + 5( 2 ) 49 = 34 + 5( 3 ) n 34 + 5( n – 1 ) = 5n + 29

34, 39, 44, 49, . . .

• Which term is 359?

Term # 1 2 3 4 Term 34 = 34 + 5( 0 ) 39 = 34 + 5( 1 ) 44 = 34 + 5( 2 ) 49 = 34 + 5( 3 ) 66 359 = 34 + 5( 65 )

Find the sum (S).

34 + 39 + 44 + 49 + . . . + 359 = S

Find the sum (S).

34 + 39 + 44 + 49 + . . . + 359 = S 359 + 354 + . . . . . . . . + 34 = S 393 + 393 + 393 + . . . . . + 393 = 2S 393 shows up in this list 66 times.

(66)(393) = 25,938 If two of the sums = 25,938 The sum is 29,938 ÷ 2 = 12,969

27, 30, 33, 36, . . . , 363

• Find the next two missing terms.

27, 30, 33, 36, . . . , 363

• Find the 84 th term.

Term # Term

27, 30, 33, 36, . . . , 363

• How many terms are there?

Term # Term 1 27 = 27 + 3( 0 ) 2 3 4 30 33 36 = 27 + 3( = 27 + 3( = 27 + 3( 1 2 3 ) ) ) 84 276 = 27 + 3( 83 )

27, 30, 33, 36, . . . , 363

• How many terms are there?

Term # Term 1 27 = 27 + 3( 0 ) 2 3 4 30 33 36 = 27 + 3( = 27 + 3( = 27 + 3( 1 2 3 ) ) ) 84 113 276 = 27 + 3( 83 ) 363 = 27 + 3( 112 )

Find the sum (S).

27 + 30 + 33 + . . . + 363 = S

Find the sum (S).

27 + 30 + 33 + . . . + 363 = S 363 + 360 + . . . . . + 27 = S 390 + 390 + 390 + . . .+ 390 = 2S (113)(390) = 44,070 44070 ÷ 2 = 22,035

52, 59, 66, 73, . . . ,815

• Find the 48 th term.

52, 59, 66, 73, . . . ,815

• How many terms are there?

Term # 1 2 3 4 Term 52 = 52 + 7( 0 ) 59 = 52 + 7( 1 ) 66 = 52 + 7( 2 ) 73 = 52 + 7( 3 ) 48 381 = 52 + 7( 47 )

52, 59, 66, 73, . . . ,815

• How many terms are there?

Term # 1 2 3 4 Term 52 = 52 + 7( 0 ) 59 = 52 + 7( 1 ) 66 = 52 + 7( 2 ) 73 = 52 + 7( 3 ) 110 815 = 52 + 7( 109 )

Find the sum (S).

52 + 59 + 66 + 73 + . . . + 815 = S

Find the sum (S).

52 + 59 + 66 + . . . + 815 = S 815 + 808 + . . . . . . . + 52 = S 867 + 867 + . . . . . . + 867 = 2S (110)(867) = 95,370 95,370 ÷ 2 = 47,685

Pascal’s Triangle Example 1.10 Page 31

1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1

Problem Solving Strategies

     Look for a pattern Make a Table Use a variable Solve an equivalent problem Solve an easier, similar problem

Did You Know?

Page 35

A lecture hall has 40 rows of seats. There are 10 seats in the first row, 12 in the second row, and so on, with two more seats in each row than the previous row. How many seats are in the lecture hall?

DAY 6

NIM (The Gold Coin Game)

Place 15 gold coins (or toothpicks) on a desktop. The players play in turn and, on each play, can remove one, two, or three coins from the desktop. The player who takes the last coin wins the game.

Work backwards to come up with a strategy to win!

 You pick from 1, 2 or 3 coins – pick all and win!

 You go first, pick 3 – AND WIN!

 He picks from 12 (take 3 - leave 9; take 2 - leave 10; take 1 leave 11)  You pick from 9 (take 1 - leave 8) or 10 (take 2 – leave 8) or 11 (take 3 – leave 8)  He 7) picks from 8 (take 3 - leave 5; take 2 - leave 6; take 1 - leave  You pick from 5 (take 1 - leave 4) or 6(take 2 – leave 4) or 7 (take 3 – leave 4)  He 3) picks from 4 (take 3 - leave 1; take 2 - leave 2; take 1 - leave  You pick from 1, 2 or 3 coins – pick all and win!

Pigeon Hole Principle

If you have more pigeons than pigeon holes, at least one hole has more than one pigeon in it.

3 pigeons, 2 holes

The sock problem

John owns 20 blue and 20 black socks which he keeps in complete disorder. What is the minimum number of socks that he must pull from the drawer on a dark morning to be sure he has a matching pair?

   8 Red 12 Yellow 16 Blue

Jar of Marbles

What is the minimum to be drawn to be sure you have at least 2 of the same color?

Make sure you have more pigeons than pigeon holes.

    8 Red 12 Yellow 16 Blue 20 Green

Jar of Marbles

What is the minimum to be drawn to be sure you have at least 2 of the same color?

    8 Red 12 Yellow 16 Blue 20 Green

Jar of Marbles

What is the minimum to be drawn to be sure you have at least 3 of the same color?

Jar – student names

   24 students 14 girls 10 boys How many must you draw to insure 2 of the same sex?

Jar – student names

   24 students 14 girls 10 boys How many must you draw to insure 3 of the same sex?

Jar – student names

   24 students 14 girls 10 boys How many must you draw to insure 5 of the same sex?

Jar – student names

   24 students 14 girls 10 boys How many must you draw to insure 2 of opposite sex?

Eliminating Possibilities

   Either Jim, John or Yuri are in the shower.

You can recognize John’s voice and Yuri’s voice.

You do not recognize the voice of the person singing in the shower.

Who is in the shower?

(a) (b) Beth, Jane and Mitzi play on the basketball team. Their positions are forward, center and guard. Given the following information, determine who plays each position.

Beth and the guard bought a milkshake for Mitzi.

Beth is not a forward.

forward center guard Beth Jane Mitzi

   Beth plays center Jane plays guard Mitzi plays forward

Classic Handshake Problem

If everyone in class today were to shake hands with each person in the class today, how many handshakes would take place?

*Make an easier problem, Use a model, Make a table, Look for a pattern.

Number of People Number of Handshakes

Problem Solving Strategies

   Working backwards Pigeon Hole Principle Eliminating Possibilities

Homework Questions Page 35

#7 1 = 1 1 + 2 + 1 = 4 1 + 2 + 3 + 2 + 1 = 9 1 + 2 + 3 + 4 + 3 + 2 + 1 = 16 1+2+3+…+ 99 + 100 + 99 +…+3+2+1= 1+2+3+…+ (n-1) + n + (n-1) +…+3+2+1=

2 n n = 1 n = 2 n 2 – n + 2 n = 1 n = 2

Page 35 #9

n = 3 n = 3 n = 4 n = 4 n 3 – 5n 2 + 10n - 4 n = 1 n = 2 n = 3 n = 4

Page 35, #13

40 th row?

10, 12, 14, 16, . . .

40 th row?

1 2 3

10, 12, 14, 16, . . .

10 = 10 + 2( 0 ) 12 = 10 + 2( 1 ) 14 = 10 + 2( 2 ) 40 88 = 10 + 2( 39 )

10, 12, 14, 16, . . . , 88

10 + 12 + 14 + … + 88 = S

10, 12, 14, 16, . . . , 88

10 + 12 + 14 + … + 88 = S 88 + 86 + . . . . + 10 = S 98 + 98 + 98 + . . .+ 98 = 2S 40 rows, (40)(98) = 3920 Divide by 2 1960

BAGELS – no digit is correct PICO – digit is correct, wrong place FERME – correct digit, correct place

1.5 Reasoning Mathematically

 We use inductive reasoning to draw a general conclusion based on information obtained from specific examples.

 Think about all the bears you have seen (or seen pictures of). You could draw the conclusion that all bears are black, brown or white.

Now, think about all the square whole numbers: 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, . . Notice that they are all either multiples of 4 or one number larger than a multiple of 4.

Inductive reasoning leads us to generalize that the square of any whole number is either a multiple of 4 or 1 more than a multiple of 4.

Inductive Reasoning is a powerful way to create and organize information. It leads us to what seems to be true. In order to state the observation for fact we would have to prove our conjecture.

*There is in fact a rare type of bear in Alaska whose fur in dark blue, called the blue bear.

Example 1.14

Page 52

Look at several examples of 3 consecutive integers.

8, 9, 10 25, 26, 27 33, 34, 35 100, 101, 102 121, 122, 123 600, 601, 602 It appears that exactly one of the three will always be a multiple of 3.

, 1, 2,

, 4, 5,

, 7, 8,

, 10, 11,

, 13, 14,

• Place n dots on a circle.

• Join each pair of dots with a line segment making sure that no more than 2 line segments intersect in a point.

• Observe the number of regions inside each circle.

Use Inductive Reasoning

   Observe a property that holds in several examples.

Check to see if the property holds in more examples. Try to find an example in which it does not hold.

If it holds in every example, state a generalization that the property is probably true.

Conjecture

 A generalization that seems to be true but has not yet been proven.

Theorem

 A conjecture that has been proved.

Representational Reasoning

A representation is an object that captures the essential information needed for understanding and communicating the properties and relationships.

Often it is displayed visually, as in a diagram, a graph, a map, or a table.

NCTM Standard page 54

Page 36 Homework problem #7 1=1 1+2+1=4 1+2+3+2+1=9 1+2+3+4+3+2+1=16 1+2+3+4+5+4+3+2+1=25

1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1 = 25

Test – one week from today!

    Have ALL homework completed before next class period.

I will answer questions on sections 4 and 5 first then any left over questions you might have from previous sections.

Print a copy of the review list from my web page.

Elimination lab next time.

DAY 7

Homework questions Page 48

Page 48, #3

• Jumps • $ Doubles • Pays $32 • Jumps • $ Doubles • Pays $32 • Jumps • $ Doubles • Pays $32

• 1 st • 2 nd • 3 rd • 4 th card card card card • 5 th • 6 th • 7 th • 8 th card card card card • 9 th • 10 th card card

Page 48, #5

0 1 2 3 4 5 6 7 8 9

# 19 - 20 people, at least 2 have same number of friends at party. Friendship is mutual.

• Case 1: Everyone has at least one friend.

• Case 2: Exactly one person has 0 friends.

• Case 3: 2 or more people have 0 friends.

Homework questions Page 60

Review List

Canoe Problem (frog problem)

Elimination Lab

1. The Browns were giving a small dinner party and couples.

2. Don called to say that he would be late but his wife would come with the Carters.

Abe, Ann, and Betty.

4. When Mr. Jones arrived, Bill and his host greeted him at the door.

5. Abe’s wife and Ann helped Candy get the dinner were planning the up-coming weekend of golf.

Abe Bill Carl Don Adams Brown Carter Jones Ann Betty Candy Doris Adams Brown Carter Jones

5 2 7 1 6 2 5 3 4 6 4 9

SUDOKU

5 7 4 8 3 1 1 9 4 2 5 8 7 2 5 8 9 6 1 8 9 3 1 4