Problems vs. Exercises
Deborah Moore-Russo
What is the difference
in math problems and exercises?
PROBLEMS vs. Exercises
Problems vs. EXERCISES
Example of an exercise
31 8423761
Example of an exercise
31 9423761
3
93
.
.
.
12
Example of an exercise
31 9423761
30
93
.
.
.
12
00
Difficult because it is tedious and easy to make a small mistake, but you should know how to proceed immediately.
Problem or exercise?
According to a news report, a certain private school in Washington, recently was faced with a unique situation. A number of 12-year-old girls were beginning to use lipstick and would put it on in the bathroom. That was fine, but after they put on their lipstick, they would press their lips to “kiss” the mirror leaving dozens of little lip prints. Every night the maintenance man would remove them, and the next day the girls would put them back.
Finally, the principal decided that something had to be done.
Recognizing Appropriate Problems
Situations that pose a challenge and that require effort and thought to solve
Pólya noted the difference between what he called:
He argued that the motives and procedures at arriving at a solution are similar.
George Pólya
Pólya’s 4 steps for solving problems
Pólya’s 4 steps for solving problems
Pólya’s 4 steps for solving problems
etc.
Pólya’s 4 steps for solving problems
Pólya’s 4 steps for solving problems
Revisit a practical problem
According to a news report, a certain private school in Washington, recently was faced with a unique situation. A number of 12-year-old girls were beginning to use lipstick and would put it on in the bathroom. That was fine, but after they put on their lipstick, they would press their lips to “kiss” the mirror leaving dozens of little lip prints. Every night the maintenance man would remove them, and the next day the girls would put them back.
Finally, the principal decided that something had to be done.
Solution
The principal called all the girls to the bathroom and met them there with the maintenance man. She explained that all these lip prints were causing a major problem for the custodian who had to clean the mirrors every night. To demonstrate how difficult it had been to clean the mirrors, she asked the maintenance man to show the girls how much effort was required. He took out a long-handled squeegee, dipped it in the toilet, and cleaned the mirror with it.
Since then, there have been no lip prints on the mirror.
Some solutions are better or
more “elegant” than others.
Pólya’s 4 steps for solving problems
Train Task
A model train is set up on a circular track. Six telephone poles are spaced evenly around the track. The engine takes 10 seconds to go from the first pole to the third pole. How long would it take the engine to go all the way around the track assuming it maintains a constant speed?
Worm Task
A worm is at the bottom of a 12-foot wall. Every day he crawls up 3 feet, but at night he slips down 2 feet. How many days does it take the worm to get to the top of the wall?
Coin Task: Apply Pólya’s 4 Steps
Bob has 10 pockets and 44 coins. He wants to put his coins into his pockets so distributed that each pocket contains a different number of coins. Can he do so?
High-Five Task
Two soccer teams each with 11 players finished a match. Each winning team player high-fived each player on the losing team. Each player on the winning team also high-fived each other player on the winning team. How many high fives were given?
Elevator Task
An elevator’s capacity is either 15 adults, who weigh the same, or 20 children, who also weigh the same. If 12 children are in the elevator, how many adults can still get in with them?
Farm Task
Farmer McDonald has only ducks and cows. One day he counted that the animals had 12 heads and 32 feet. How many of the animals were ducks and how many were cows?
2-Digit Number Task
If you take a certain 2-digit number, reverse its digits to make a second 2-digit number, and add these two numbers together, their sum is 121. What was the original number?
Need information: Pythagorean Theorem
Ant Task
An ant is going to walk from corner A to corent B in the room as shown below. The room is a 3cm x 3cm x 3cm cube. (Ant rooms are small!) What is the shortest route?
A
B
Labels
Near wall
Distant wall
Left wall
Right wall
Ceiling
Floor
“Open up” the room
Near wall - N
Distance wall - D
Left wall - L
Right wall - R
Ceiling - C
Floor - F
L F R
N
C
D
Place points
Floor, Ceiling, and 4 walls:
Near, Distant, Left, Right
L F R
N
C
D
A
B
Labels
Floor, Ceiling, and 4 walls:
Near, Distant, Left, Right
L F R
N
C
D
B
A
B
Labels
Floor, Ceiling, and 4 walls:
Near, Distant, Left, Right
L F R
N
C
D
A
B
A
B
Labels
Floor, Ceiling, and 4 walls:
Near, Distant, Left, Right
L F R
N
C
D
A
B
A
B
Labels
Floor, Ceiling, and 4 walls:
Near, Distant, Left, Right
L F R
N
C
D
A
B
A
B
3 cm
6 cm
Labels
Floor, Ceiling, and 4 walls:
Near, Distant, Left, Right
L F R
N
C
D
A
B
A
B
3 cm
6 cm
Use the Pythagorean Theorem to find the distance.
If you would like to see a slightly different solution, see:
The best problems are like the best kinds of gum
They don’t lose their flavor easily;
so, you keep chewing on them longer.
Other ways to “open up” the room.
Would one of them give a shorter path?
N
C
B
Fly Task: Another good problem to “chew” on
A housefly is going to fly from point A to point B in a cubic room that is 9 ft x 9 ft x 9 ft. What is the shortest route if the fly started on the near top left corner and then flew to the distant bottom right corner?
A
B
Fly Task: Another good problem to “chew” on
A housefly is going to fly from point A to point B in the room shown. The cubic room is 9 ft by 9 ft by 9 ft. What is the shortest route if the fly started at point A and flew to point B?
Beetle Task: Yet another good problem to “chew” on
A beetle is going to walk from point A to point B in the room shown below. The room is shaped like a cube with a 9 ft by 9 ft square floor and a 9 ft ceiling. What is the shortest route?
To revisit this talk: