Problem Solving Strategy ~ Work Backwards ~

Content

 Process Problem 1 Process Problem 2 Process Problem 3 Problem Worksheet

Process Problem 1

Phil was given his allowance on Monday. On Tuesday he spent \$1.50 at the fruit stand. On Wednesday, Jed paid Phil the \$1.00 he owed him. If Phil now has \$2.00, how much is his allowance? (Hint: Using the facts given, start with the amount Phil has now and work backwards.)

Understanding the Problem

· How much money did Phil have after Wednesday? (\$2.00)
· Do you know how much Phil's allowance is? (no)
· How much did Phil spend at the fruit stand? (\$1.50)
· Was Phil given any money after he got his allowance? (Yes, Jed gave him \$1.00.)

Planning a Solution

· Did Phil have Jed's \$1.00 on Tuesday night? (no)
· How much money did Phil have at the end of Tuesday? (\$2.00 - \$1.00 = \$1 00)
· Did Phil spend money on Tuesday? (yes, \$1.50) How much money did Phil have before he spent the \$1.50? (\$1.00 + \$1.50 = \$2.50)

Work Backwards

Phil's allowance is \$2.50.

Problem Extension

On Phil's birthday his father increased his allowance. Phil was so happy he went to the store and bought 2 cans of spray paint for his model airplanes. The paint cost \$1.50. After Phil bought the paint, he had \$3.50 left. How much of an increase did Phil get in his allowance? (\$2.50, twice as much).

Process Problem 2

Jacob, Jesse, and James uncovered a strongbox containing some gold nuggets. They buried half of the nuggets in the Grand Canyon and divided the remaining nuggets evenly among themselves. Jesse received 2,000 gold nuggets. How many nuggets were in the strongbox? (Hint: Start with the number of nuggets Jesse received and work backwards.)

Understanding the Problem

· Do you know how many gold nuggets were in the strongbox? (no)
· Do you know how many nuggets were left after they buried half in the Grand Canyon? (no) After they divided the remaining nuggets evenly among themselves? (yes, 2,000 each)

Planning a Solution

· If Jesse received 2,000 nuggets, how many did James and Jacob receive? (2,000 each)
· Together, how many nuggets did the men have? (6,000)
· The money the men had is how much of what was in the strongbox? (Half, because half of the money was buried in the Grand Canyon.)

Work Backwards

 | | v Start with 2,000 nuggets. Multiply by 3, the number of men—3 x 2,005 = 6,000. Multiply by 2 for the half they buried—2 x 6,000 = 12,000.

There were 12,000 gold nuggets in the strongbox.

Problem Extension

The men decided to place half of the nuggets in the bank first, bury half of the remaining nuggets, and then divide the nuggets that were left evenly among themselves. Again, each man received 2,000 nuggets. How many nuggets were in the strongbox? (24,000)

Process Problem 3

Leon wanted to know the age of a black bear at the zoo. The zoo keeper told Leon that if he added 10 years to the age of the bear and then doubled it, the bear would be 90 years old. How old is the bear? (Hint: Using the facts given, start with 90 years and work backwards.)

Understanding the Problem

· Did the zoo keeper tell Leon the bear's age? (no)
· What was the last thing the zoo keeper did to the bear's age? (He doubled it.)
· What was the first thing the zoo keeper did to the bear's age? (He added 10.)

Planning a Solution

· If you double a number and get 10, what number did you double? (5) What operation did you use to get 5? (division—10 ÷ 2 = 5)
· The zoo keeper doubled a number and got 90. What operation could you use to get the number he doubled? (division—divide by 2)
· Is the bear 45 years old? (no)
· What did the zoo keeper do before he doubled the bear's age? (He added 10 to the bear's age.)
· Which operation would you use to find out how old the bear is? (subtraction—45 - 10 = 35)

Work Backwards

Divide by 2 to get the number that was doubled—90 ÷ 2 = 45.
Subtract 10 to get the age of the bear before 10 years was added—45 - 10 = 35

The bear was 35 years old.

Problem Extension

Leon also wanted to know the age of a turtle he saw at the zoo. The zoo keeper said that if he added 14 years to the age of the turtle and then doubled it, the turtle would be 200 years old. How old was the turtle? (86)

Process Worksheet: Work Backwards

1. Rabbits multiply at an amazing rate. In year 1 there are X rabbits. The rabbit population doubles each year. The forest is crowded in year 7 when there are 3200 rabbits. How many rabbits were there in year one if the population doubles each year?

2. I bought a bag of apples. I kept half of them for myself. I gave the rest to 3 friends. Each friend got 2 apples. How many apples did I buy?

3. What is the starting number?

 ? | v Add 12 ----> Subtract 6 | v Multiply by 2 ----> 20

 4. ? | v Divide by 7 | v Multiply by 2 | v Add 2 | v 20