Move between objects, pictures, and symbols
When a new idea is difficult to picture, build it with counters, blocks, measuring tools, or fraction strips. Draw what the objects show. Then write an equation or use mathematical language to describe the same relationship.
- Concrete: Make or measure the quantities.
- Pictorial: Draw a number line, array, diagram, or bar model.
- Abstract: Record the relationship with numbers and symbols.
You may begin with symbols when you already understand the idea. Be ready to explain what the numbers represent.
Use a model to show relationships
A diagram is useful when it helps you see how quantities compare or combine. Label each part, show what is known, and mark what you need to find. The model should match the problem; it is not decoration.
For example, if a whole is 12 and one part is 7, a part–whole drawing shows that the missing part is 12 − 7 = 5. Check by combining the parts: 7 + 5 = 12.
A routine for solving problems
- Read the problem and identify the question.
- List or mark the information you know.
- Choose a picture, model, table, or equation that fits.
- Solve carefully and label your answer.
- Check whether the result makes sense in the situation.
- Explain why your method works.
There can be more than one sound method. If your method is mathematically correct, clearly shown, and answers the question, it deserves consideration even when it differs from a worked example.