
STRATEGY 6: START AT THE END
Sometimes in order to accomplish something you have to start at the end. Athletes see themselves winning even before the competition begins. It is called visualizing success. Engineers make drawings of finished products even before they know how to build them. Stephen Covey in his famous book 7 Habits of Highly Effective People says that highly effective people “start with the end in mind.” In Understanding by Design, a book about teaching and learning, Grant Wiggins and Jay McTighe describe a method called “backwards design”: you start by asking what you will ask your students to do to show that they understand . . . and then you plan to teach them how to do it. Very often, the road to success starts at the end and not at the beginning.
So it is with problem solving. To solve some problems, you start at the end and work backwards. However, the directions for going backwards are not exactly the same as the directions for going forwards. Imagine leaving the school to go to the Post Office and then returning to the school.
FORWARDS to go to the Post Office:
 Turn left out of the school (Independence Avenue)
 Take the 3rd right turn (National Avenue)
 Take the 2nd left turn (Station Street)
 Cross two streets on Station Street
 Turn left into the Post Office

BACKWARDS to the school
 Turn right out of the Post Office (Station Street)
 Take the 3rd right on to National Avenue
 Take the 2rd left turn on to Independence Avenue
 Cross two streets on Independence Avenue
 Turn right into the school

Here is a wellknown problem that can be solved by starting at the end.
THE MANGOES PROBLEM
One night the King could not sleep. He went to the royal kitchen, where he found a bowl full of mangoes. Being hungry, he took 1/6 of the mangoes in the bowl.
Later that same night, the Queen could not sleep, and she was hungry. She found the mangoes and took 1/5 of what the King had left in the bowl.
Still later, the youngest Prince awoke, went to the kitchen, and ate 1/4 of the remaining mangoes.
Even later, the second Prince ate 1/3 of what his younger brother had left.
Finally, the third Prince, the heir to the throne, ate 1/2 of what his younger brothers had left, and then there were only three mangors left in the bowl.
How many mangoes were in the bowl when the King found them?
This problem and its solution can be found on the Illuminations website of the National Council of Teachers of Mathematics http://illuminations.nctm.org/LessonDetail.aspx?ID=L264 
To solve the Mangoes Problem, start with the 3 mangoes left in the bowl after the King, the Queen, and the three Princes have all eaten their share—and work BACKWARDS from there.
The third Prince ate 1/2 of the mangoes he found in the bowl and left 3. So he must have found 6 mangoes in the bowl.
The second Prince ate 1/3 of the mangoes that he found in the bowl and left 6. Therefore 6 = 2/3 of the mangoes he found, and 1/3 = 3. The second Prince must have found 9 mangoes in the bowl.
The youngest Prince ate 1/4 of the mangoes he found in the bowl, leaving 3/4. Therefore 3/4 = 9, and 1/4 = 3. The youngest Prince must have found 12 mangoes.
That means the Queen left 12. Since she ate 1/5, 4/5 = 12 and 1/5 = 3. Therefore the Queen found 15 mangoes in the bowl.
Since the King left 15 mangoes after eating 1/6, 5/6 = 15 and 1/6 = 3. Therefore there were 18 mangoes in the bowl when the King found them.
You started at the end and worked BACKWARDS to the beginning!

Problems:
Problem 1: How Much Was Dinner?
Problem 2: When Should We Leave?
Problem 3: Town Planning
Problem 1: How Much Was Dinner?
Mr. and Mrs. Atkins had friends from Canada come to visit them. They decided to take their friends to their favorite restaurant for dinner. In addition to the cost of the dinner, Mr. Atkins had to pay some extra expenses. He paid $12 for parking, $18 for tax, and he left a tip of $30 for the waiters. When they got home, Mrs. Akins asked Mr. Atkins how much the dinner had cost. “Well,” he said, looking in his wallet. “I know I started with $300, and now I have $15.”
What will he tell Mrs. Atkins? How much was the dinner?
Solution >
Problem 2: When Should We Leave?
Later today, you mother will take you to the doctor’s office for a checkup. “When do you think we should leave?” she asks. “Help me decide.”
Since she always has errands to run, so you ask her, “What do we have to do on the way?”
She answers, “I’d like to go to the drycleaners in the mall, and then let’s have lunch at the restaurant in the mall. Then we can get dogfood at the pet shop and money at the bank. And then we can go see the doctor.”
“OK,” you tell her, “let’s say that it takes 20 minutes to drive to the mall and park, and ten minutes to get the dry cleaning, then 45 minutes for lunch, 10 minutes to get dog food and 10 minutes to get money at the bank. After that we’ll need 20 minutes to drive to the doctor’s office. What time is our appointment.”
“Our appointment is at 2:00 p.m., so when should we leave?”
Solution >
Problem 3:
Town Planning Problem
This is a street map of New Town. The Town Planning Commission wants to know how many different ways you can drive a car from A to B, going only North and/or East.
Solution >
