Lesson 5.03

You will learn to solve complex mathematical and real-world problems by "Changing Focus"—starting from a known end result and working in reverse to find the initial state.
In most situations, we think "forwards" (Action A leads to Result B). However, many problems are more efficiently solved by starting at the goal and reversing the steps

Real-Life Examples:

Scheduling

To catch a 12:40 p.m. flight, you don’t just leave your house whenever. You start at the flight time and subtract the time needed for security, the drive to the airport, and returning a rental car to determine your departure time.

Investigations

Accident investigators start with the wreckage (the end state) and work back to figure out the cause.

Budgeting

People often determine how much they can spend today by working backwards from the total money they have available for the month.

To successfully work backwards, you must apply two specific changes to every step of the problem:

Reverse the Order: Start with the last thing that happened and move toward the first.
Reverse the Action (Inverse Operations): Do the opposite of what the problem describes.
- Addition becomes Subtraction.
- Multiplication becomes Division.
- Taking a Right Turn becomes Taking a Left Turn.

You already use this strategy in Algebra! When you solve for a variable, you are isolating it by performing inverse operations in reverse order.

Example: "I’m thinking of a number. If I multiply it by 3 and add 5, I get 32. What is my number?".

Forward Path: x→(*3)→(+5)=32.
Reverse Path: 32→(−5)→(÷3)=9.
Verification: Always check your answer by running the number through the original "forward" problem: (9×3)+5=32.

The most difficult part of working backwards often involves fractions.

If Barry eats 3/5 of Willie’s cookies and Willie has 8 left, you cannot simply "add back" 3/5 of 8.

Instead use a diagram. If 3/5 are gone, the 8 cookies left represent the remaining 2/5. Since 2 "parts" of a diagram equal 8, then each part is 4. Since there were 5 parts total, the original number was 4×5=20.

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