10/5/2016
Burks Echols
[email protected]
PROBLEM SOLVING
WHAT IS IT?
MUST KNOW WHAT’S ONE THING THAT YOU What’s
IN ORDER TO SOLVE A PROBLEM?
your
problem?
HOW DO YOU KNOW THAT YOU HAVE A PROBLEM?
HOUSTON
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10/5/2016
WE HAVE A PROBLEM!
HOW DO YOU KNOW WHEN THINGS ARE NORMAL?
Context
Past Knowledge
Collaboration
HOW MUCH DO PROBLEMS AFFECT OUR
DECISION MAKING?
Completely?
Not at all?
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10/5/2016
LET’S GET TO KNOW EACH OTHER
Have you ever been to…
Australia
Canada
Brazil
Denmark
OR TRAVELED TO…
England
France
Hungary
Germany
Italy
OR, HAVE YOU VISITED THESE PLACES IN RICHMOND?
Maymont
Park
The VMFA
The State Capitol
The Diamond
Richmond Metropolitan Zoo
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10/5/2016
TIME FOR EVEN MORE AUDIENCE PARTICIPATION
Raise your hand if you mow your lawn
or have ever mowed a lawn.
RAISE YOUR HAND IF YOU…
have ever trimmed trees or hedges.
WHICH ONE IS EASIER?
Mowing Lawns
Trimming Hedges
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10/5/2016
WHY ???
Don’t worry, there’s nothing scientific about this question, and I won’t over‐analyze your answers. But, I do have some theories, and I would like for us to have some informal discussions about this important topic.
HOW MANY ARRIVED IN A CAR TODAY?
WHAT SIDE IS YOUR GAS TANK ON?
Left?
Right?
Could you repeat the question?
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10/5/2016
DESCRIBE HOW TO OPERATE
YOUR WINDSHIELD WIPERS
HOW MANY OF YOU HAVE EVER USED A COMPUTER?
LET’S GET TO KNOW EACH OTHER BETTER (MORE COLLABORATION)
Take the number of brothers and sisters you have
Double this number
Add 4
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10/5/2016
THIS IS COMPLICATED. WRITE DOWN EACH STEP.
Multiply by 5
Add 1
Subtract 10 times the number of brothers and sisters
KEEP WRITING DOWN EACH STEP
Subtract a number between 12 and 19 Multiply by 9
Add the digits of your number together
ARE YOU WRITING EVERYTHING DOWN?
Subtract 5
Have you been checking your math?
Are you sure that everything is correct?
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10/5/2016
STAY WITH ME, YOU’RE ALMOST THERE
Match this number with a letter of the alphabet
Pick a country that starts with this letter
Pick an animal whose name begins with the last letter of the country
YOU’VE MADE IT. I PROMISE. ANY QUESTIONS? Pick a color that starts with the last letter of the animal’s name
Make sure that you have written all of this down
Answer one final question for me
HOW WERE YOU SERVED BY…
Context?
Past Knowledge?
Collaboration?
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10/5/2016
LET’S SOLVE SOME PROBLEMS!
• Does everybody like math?
• Of course not, so we will address a variety of problems. However, it may SEEM like ALL of these problems are about math especially to anyone who suffers from math phobia.
DO YOU NEED A CALENDAR TO SOLVE THIS PROBLEM?
• If the 9th day of a month is on Tuesday, on what day is the 25th ?
HOW WELL DO YOU KNOW YOUR MATHEMATICAL SYMBOLS: +, ‐, =, etc.?
• What math symbol can be placed between the 2 and the 3 in “23” to make a number greater than 2 but less than 3?
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10/5/2016
STRAIGHT LINES AND 9 DOTS…BUT NOT EXACTLY MATH
.
Start as shown. Draw only 4 straight lines to connect all 9 dots.
Do not lift your pencil until all the dots are covered.
Start here ›
THINK OUTSIDE THE BOX
A COMPLEX, MULTI‐STEP PROBLEM
• You have three bottles ‐‐ a 10‐liter, a 4‐liter and a 3‐liter. All of the bottles are unmarked and there is no other supply of water available. The 10‐liter bottle is full. You want to divide the water in such a way as to have one liter of water in the 3‐liter bottle, four liters in the 4‐liter bottle and five liters in the 10‐liter bottle. You can do this by pouring the water from one bottle to another. What is the fewest number of pourings that will achieve this division of the water?
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10/5/2016
5 POURINGS
10 L 4 L 3 L
•
10 0 0 start
1.
6 4 0 after filling the 4‐liter from the 10‐liter
2.
6 1 3 after filling the 3‐liter, from the 4‐liter 3.
9 1 0 after pouring what's in the 3‐liter into the 10‐liter
4.
9 0 1 after pouring what's in the 4‐liter into the 3‐liter
5.
5 4 1 after refilling the 4‐liter from the 10‐liter
HOW WELL DO YOU KNOW THE ALPHABET?
The letters S, T, and U have been left out of the sequence of letters below. Write each in its correct place above or below the line.
A E F H I K L M N V W X Y Z
B C D G J O P Q R MORE ALPHABET SOUP
Fill in the missing letter of the alphabet in this pattern:
M V E M J S U N _____
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10/5/2016
THREE SQUARES HAVE BEEN MADE FROM 12 PENCILS BELOW. SHOW HOW TO MOVE ONLY THREE OF THE PENCILS, AND MAKE FOUR SQUARES THIS SAME SIZE.
DRAW ARROWS TO SHOW HOW TO REARRANGE EXACTLY 2 OF THESE TOOTHPICKS SO THAT YOU WILL HAVE 4 SQUARES INSTEAD OF 5. EACH SQUARE IS TO BE THE SAME SIZE AS THE ONES SHOWN.
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MAKE AN X ON EACH OF FOUR TOOTHPICKS YOU COULD REMOVE SO THAT EXACTLY 7 SQUARES, ALL THE SAME SIZE, WOULD BE LEFT.
JOE KEEPS ALL HIS SOCKS IN ONE DRAWER. HE HAS 7 BLUE SOCKS AND 9 BROWN SOCKS. IF HE REACHES IN THE DRAWER WITHOUT LOOKING, WHAT IS THE LEAST NUMBER OF SOCKS HE CAN TAKE OUT TO BE SURE OF GETTING A PAIR OF THE SAME COLOR?
RUSTY CAN CUT A LOG INTO 3 PIECES IN 20 MINUTES. AT THAT RATE, HOW LONG WILL IT TAKE HIM TO CUT ANOTHER SUCH LOG INTO 6 PIECES?
Here comes the answer
You may not like it
50 minutes
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10/5/2016
A TRAIN THAT IS 1 MILE LONG STARTS THROUGH A TUNNEL THAT IS ALSO 1 MILE LONG. THE TRAIN IS TRAVELING 15 MILES PER HOUR. HOW LONG DOES IT TAKE FOR THE TRAIN TO GET COMPLETELY OUT OF THE TUNNEL?
ANOTHER RATE PROBLEM
A patch of water lilies doubles itself in size each day. From the time the first leaf appeared to the time when the pond was completely covered took 40 days.
How long did it take for the pond to be half covered in lily pads?
JANE WAS HELPING HER FAMILY PICK ORANGES IN THEIR GROVE. SHE TOOK SOME ORANGES HOME TO SHARE WITH THREE FRIENDS. SHE GAVE 3 MORE THAN HALF TO JENNIFER. JANICE GOT HALF OF THE REMAINDER AND 3 MORE. SHE GAVE JOSIE HALF OF THE REMAINDER PLUS 3. WHEN SHE GOT HOME, SHE ONLY HAD 10 ORANGES LEFT. HOW MANY DID JANE HAVE WHEN SHE LEFT THE GROVE?
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10/5/2016
WORK BACKWARDS
10 = ½ of something minus 3 [26]
26 = ½ of something minus 3 [58]
58 = 0.5(C) ‐ 3
[122]
BRANDON COUNTED 13 PEOPLE AHEAD OF HIM IN LINE TO BUY CONCERT TICKETS. HE THEN COUNTED 17 BEHIND HIM IN LINE. FIVE PEOPLE BROKE IN LINE AHEAD OF HIM, AND THEN 2 PEOPLE BEHIND HIM DROPPED OUT. HOW MANY PEOPLE WERE IN THE LINE AT THAT POINT?
answer
=
34
YOU ARE HAVING A POOL PARTY AND INVITE 2 OF YOUR BEST FRIENDS. THESE TWO FRIENDS EACH INVITE 2 OTHER PEOPLE. THESE 2 PEOPLE INVITE 2 PEOPLE THAT HAVE NOT BEEN INVITED. HOW MANY PEOPLE WILL BE INVITED IF THIS PROCESS CONTINUES FOR 4 ROUNDS? 15
10/5/2016
SOMETIMES…
DRAWING A DIAGRAM IS A QUICK SOLUTION
Answer =
30
TAKE A SHEET OF PAPER AND FOLD IT IN HALF, FOLD IT IN HALF AGAIN, FOLD IT IN HALF AGAIN, FOLD IT IN HALF AGAIN, AND THEN FOLD IT IN HALF ONCE MORE. IF YOU OPENED THE PAPER, HOW MANY SECTIONS WOULD THERE BE?
AND…
Now fold the paper in half five more times and determine how many sections have been created?
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10/5/2016
ALRIGHT, LET’S TRY SOMETHING EASIER.
LET’S BUILD A FENCE.
A square garden has five fence posts on each side.
How many fence posts are there around the garden?
answer
=
16
AND NOW LET’S BUILD A BIGGER FENCE.
A RECTANGULAR YARD 80 FEET BY 60 FEET HAS A FENCE WITH A POST ON EVERY CORNER AND ANOTHER POST EVERY FIVE FEET. HOW MANY POSTS ARE NEEDED FOR THE ENTIRE FENCE? Answer
=
56
A MAN WALKING THROUGH THE WOODS HAS A GOOSE, A FOX, AND A BAG OF
CORN WITH HIM. HE COMES TO A RIVER, BUT THERE IS ONLY ONE BOAT AVAILABLE FOR CROSSING. THE BOAT WILL HOLD ONLY THE MAN AND ONE
OTHER THING EACH TIME ACROSS THE RIVER. THE MAN CAN’T LEAVE THE FOX AND GOOSE ALONE ON THE RIVER BANK, BECAUSE THE FOX WILL EAT THE GOOSE. HE CAN’T LEAVE THE GOOSE AND CORN ALONE, BECAUSE THE GOOSE WILL EAT THE CORN. WHAT IS THE FEWEST NUMBER OF CROSSINGS HE CAN MAKE IN THE BOAT TO GET EVERYTHING ON THE OTHER SIDE?
(A CROSSING MEANS GOING FROM ONE SIDE OF THE RIVER TO THE OTHER.)
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10/5/2016
REMEMBER, DIAGRAMS OFTEN HELP.
IF YOU SHOT 3 ARROWS AT A TARGET AND ALL 3 ARROWS HIT THE BULL'S EYE, YOU WOULD SCORE 15 POINTS. IF EXACTLY 3 ARROWS HIT THIS TARGET, HOW MANY DIFFERENT TOTAL SCORES ARE POSSIBLE?
DON’T FORGET…
TO CONSIDER ALL ASPECTS OF THE QUESTION.
 (7 scores) There are 11 possible ways the arrows could land
 (1,1,1); (1,1,3); (1,1,5); (1,3,5); (1,3,3); (1,5,5); (3,3,3); (3,3,5); (3,5,5); and (5,5,5)
 However, these only produce seven DIFFERENT scores: 3, 5, 7, 9, 11, 13, and 15.
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10/5/2016
BOX A HAS 3 BLACK MARBLES AND 2 WHITE MARBLES.
BOX B HAS 2 BLACK MARBLES AND 1 WHITE MARBLE.
IF YOU HAVE TO CLOSE YOUR EYES AND PICK A BLACK
MARBLE TO WIN A PRIZE, WHICH BOX GIVES YOU THE BEST CHANCE OF WINNING? AND…MORE MARBLES
Mark has 3 red marbles, 9 white marbles, and 4 green marbles. He wants to divide all the marbles evenly into two jars, but he only wants two colors in each jar.
How can they be divided?
3 red and 5 white in one jar; 4 white and 4 green in the other
AND FINALLY, SOME GOOD OLD FASHIONED ALGEBRA
• Susan’s age is 3 times Andrea’s age.
• Barbara is twice as old as Andrea.
• The sum of their ages is 30.
• How old is each girl?
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10/5/2016
IN HOW MANY DIFFERENT WAYS CAN 4 BOOKS
BE ARRANGED ON A SHELF? AND… SOME SERIOUS, HEAVY‐LIFTING ALGEBRA!
King Louis XIII must pick a team of 5 musketeers to investigate one of Cardinal Richelieu’s nefarious schemes. If there are 10 musketeers to choose from, what is the probability that four of them (Athos, Porthos, Aramis, and d’Artangan) will be selected?
Remember: the probability of any event is calculated by taking the number of desired outcomes over the number of possible outcomes. (who could ever forget this?)
Since 4 of the musketeers are required, there are only six others who could be selected to complete the 5 man team. (10 total – 4 required for the desired outcome = 6 left to choose) So, there are six desired outcomes.
Now, let’s determine how many possible outcomes there are. How hard could this be?
Would you like a hint?
10!
10! 10! 6!
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