Strategy 3, Part B: Connecting 'Working Backwards' to Solving Equations
Working backwards to solve equations connects arithmetic reasoning to formal algebra in Grade 8 math (Yoshiwara Core Math). When a result is known but the starting value is unknown, reverse each operation in reverse order. For example, if a number is multiplied by 3 then 7 is added to get 22: working backwards means 22 − 7 = 15, then 15 ÷ 3 = 5. This matches solving 3x + 7 = 22 algebraically. Understanding this link builds intuition for why inverse operations isolate variables.
Key Concepts
Property To solve an equation: 1. List the operations performed on the variable in order. 2. Undo those operations in reverse order.
Examples Solve $\frac{4a 5}{3} = 9$. The operations on $a$ are: multiply by 4, subtract 5, divide by 3. Undo in reverse: multiply by 3 to get $4a 5 = 27$, add 5 to get $4a = 32$, and divide by 4 to get $a = 8$. Solve $\frac{b + 7}{2} 3 = 8$. Undo in reverse: add 3 to get $\frac{b+7}{2} = 11$, multiply by 2 to get $b+7 = 22$, and subtract 7 to get $b = 15$. Solve $10 + \frac{c}{5} = 14$. The operations on $c$ are: divide by 5, then add 10. Undo in reverse: subtract 10 to get $\frac{c}{5} = 4$, then multiply by 5 to get $c = 20$.
Explanation Solving an equation is like unwrapping a gift. The variable is the gift inside, and the operations are the wrapping paper and box. To get to the gift, you must undo each layer in the reverse order it was applied.
Common Questions
What does 'working backwards' mean?
Start from the known result and apply inverse operations in reverse order to find the unknown starting value.
How does working backwards connect to solving equations?
Both use inverse operations. Working backwards is informal; formal equation solving uses algebraic notation for the same logic.
How do you solve 'doubled then decreased by 5 gives 11' working backwards?
Start at 11, add 5 → 16, divide by 2 → 8. The number is 8.
What is the reverse order rule?
Apply inverse operations in the opposite order from the original. If first multiplied then added, reverse by first subtracting then dividing.
Why is working backwards useful?
It provides a concrete step-by-step method to find an unknown given the final result, before setting up a formal equation.