Work Backward: A Problem Solving Strategy Kids Can Use Starting in Grade 2

Mathematics is usually taught as a forward process: you start with what you know, apply an operation, and arrive at an unknown result. This forward direction is natural and comfortable because it mirrors the physical experience of taking action and observing an outcome.

But some mathematical problems are structured in the opposite direction. You know the outcome. You need to find what you started with, or what happened in the middle. These problems cannot be solved efficiently by guessing a starting point and working forward until you either hit the known outcome or give up and try another starting point.

They can be solved very efficiently by starting at the known outcome and working backward through the problem, undoing each step in reverse order.

This is the work backward strategy, and it is one of the most elegant, versatile, and consistently underused problem solving tools available to elementary school children. It is accessible to children as young as seven when introduced with concrete, familiar contexts. And it builds a form of mathematical flexibility, the ability to approach a problem from its end rather than its beginning, that transfers directly to algebra and to a wide range of real world reasoning situations.

What the Work Backward Strategy Is

Working backward means starting from the known result of a problem and reversing each operation in the problem's sequence until you reach the unknown starting point.

Every addition can be undone by subtraction. Every multiplication can be undone by division. Every step forward can be reversed. Working backward means applying these inverse operations in reverse sequence, starting from the end.

The simplest possible example: "I am thinking of a number. I added 5 to it and got 12. What was my number?" A child who works forward would guess numbers until one works. A child who works backward starts at 12 and subtracts 5 to arrive at 7.

This is trivial in a single step problem, but the strategy becomes increasingly powerful as problems involve more steps, because working backward navigates complex sequences efficiently without any guessing.

Why This Strategy Is Underused

The work backward strategy is standard content in most mathematics problem solving curricula, but it is often introduced and practiced so briefly that it does not become part of a child's active repertoire. A single worksheet, a single lesson, and the strategy is considered covered.

The reason it needs more than this is that it requires a specific cognitive shift that is not intuitive: starting at the end rather than the beginning feels wrong to most children, and this wrongness needs to be overcome through deliberate practice before the strategy becomes available automatically when a problem calls for it.

The resistance to starting at the end is related to a broader tendency in human cognition to address problems in the direction of their natural narrative flow. A problem that describes a sequence of events invites the reader to follow the narrative: first this happened, then this, then this. Working backward requires deliberately resisting this pull and starting at the conclusion rather than the beginning.

Children who have practiced this resistance, who have built the habit of asking "could I start from the end here?" when approaching an unfamiliar problem, have a tool that many of their peers lack.

When to Use It

The work backward strategy is most effective for problems with these characteristics.

The end result is known, but a starting quantity or an intermediate quantity is unknown. Any time a problem tells you where something ended up and asks where it began, working backward is likely the right approach.

The problem involves a sequence of operations performed in order. The more steps involved, the more the work backward strategy outperforms guessing and checking or working forward from an assumed starting point.

Inverse operations are straightforward to apply. Problems involving standard arithmetic operations are ideal candidates. Problems involving more complex relationships may require additional thought about what the inverse operation is.

How to Introduce It to Young Children

For children in second and third grade, the most effective introduction uses familiar, concrete contexts that make the backward movement feel natural rather than contrived.

The reverse trip. "You drove to the store, then to the library, and ended up at Grandma's house. Grandma's house is three miles from the library. The library is five miles from the store. How far from your house is the store?"

This problem can be solved by working backward from Grandma's house through each leg of the journey. Children who act this out physically, walking backward through the route, experience the strategy in the most concrete possible way.

The mystery number. "I am thinking of a number. I multiplied it by 3. Then I added 7. The result was 22. What was my number?" Working backward: start at 22, undo the addition by subtracting 7 to get 15, undo the multiplication by dividing by 3 to get 5.

This type of problem is genuinely engaging to many children because it has the structure of a puzzle or a mystery. The unknown number feels like something to be discovered rather than calculated.

The backwards recipe. "We made cookies and used all our flour. We started the day with some flour, used two cups for breakfast muffins, then used three more cups for the cookies. We had one cup left before making the cookies. How much flour did we start with?" Working backward: we had one cup before the cookies, we used three cups for the cookies, so before the muffins we had four cups, we used two cups for the muffins, so we started with six cups.

The cooking context makes the sequence of events narratively familiar, and acting it out backward, adding flour back to the container at each step, makes the reversal concrete.

The Connection to Algebra

Working backward is not just an elementary school strategy. It is the foundational reasoning behind solving algebraic equations.

When a student solves the equation 2x + 5 = 17, they are working backward. They start from the known result, 17. They undo the addition of 5 by subtracting 5, arriving at 2x = 12. They undo the multiplication by 2 by dividing by 2, arriving at x = 6.

Every step in solving a linear equation is an application of the work backward strategy: identifying what was done last, undoing it, and repeating until the unknown value is isolated. Students who have developed genuine comfort with this strategy in arithmetic, who have built the cognitive habit of starting at the end and reversing operations, find algebraic equation solving considerably more natural than students who encounter the formal algebraic version without this prior experience.

The strategic continuity between working backward in arithmetic and solving equations in algebra is one of the most direct and most underappreciated connections in mathematics education.

A Practice Structure That Works

For parents and homeschooling educators who want to build this strategy into a child's repertoire, the most effective practice structure is a brief, regular encounter with problems specifically designed for backward work.

Three or four problems per week, mixed with other problem types so the child cannot identify in advance which strategy applies, practiced over several months, is sufficient to make the strategy genuinely available when needed.

The problems should increase in complexity as the strategy becomes familiar. Begin with single step reverse problems. Progress to two step, then three step sequences. Eventually include problems where identifying that working backward is the appropriate strategy is itself part of the challenge.

And alongside the practice, make the strategy explicit. Name it. Explain when it works. Ask the child to explain why starting at the end is useful for this type of problem. The metacognitive awareness of having a strategy and knowing when to apply it is as important as the strategy itself.