Shape-Variable Equation Puzzles: Solve by Substitution

Algebra and Function Practice

Shape-variable puzzles use pictures instead of letters. Each shape stands for one unknown number, and the same shape always means the same number. You solve the system by substitution: find a shape whose value you can determine, replace that shape everywhere it appears, and repeat until every shape has a value. This guide gives a clear convention, worked reasoning, common mistakes, and four self-contained practice tasks. All arithmetic uses whole numbers and consistent units.

Practice materials: Printable downloads are not available for this lesson yet. Use the instructions and four exercises below.

The substitution convention

Treat each distinct shape as a variable. A triangle is not a number until you assign it one. Write a legend such as triangle equals t, circle equals c, and square equals s. The legend is your working dictionary.

Substitution means replacing every occurrence of a shape with its known value, then simplifying. If two shapes appear in one equation, substitute only the shape you know and keep the other as a symbol until you can solve it.

Use one equation to isolate one shape, then insert that expression into another equation. This is the same substitution method used with letters, but the pictures make the pattern visible.

1. Copy the system and write a legend assigning one letter to each shape.
2. Find an equation containing only one shape and solve it.
3. Substitute that value into every other equation where the shape appears.
4. Solve the simplified equation for the next shape.
5. Repeat until all shapes have values, then check each original equation.

Reading and checking shape equations

Read a shape equation left to right as a statement of equality. Two triangles plus a circle equals eleven means 2t + c = 11. A shape written twice means two copies of the same unknown, not two different unknowns.

After solving, substitute all values back into every original equation. A correct solution makes each equation true. If one equation fails, the error is usually in the substitution step, not in the arithmetic.

Keep units consistent. If a puzzle uses counters, state that each counter is one unit. Do not mix counters and centimetres in the same system.

1. Translate each picture equation into a letter equation.
2. Solve the single-shape equation first.
3. Substitute and simplify one equation at a time.
4. Check every original equation with the final values.

Handwriting and diagram convention

When you draw your own puzzle, use a clear convention: draw each shape at least one centimetre tall, keep the same shape identical in size and orientation, and label the legend beside the system. X-height means the distance from baseline to midline; letter styles vary, so do not judge handwriting by a single ideal shape.

For the printable plan, each puzzle will have a substitution workspace with one line per substitution step. The answer key will show the letter equation, the substituted line, and the final value for each shape.

Valid alternatives are allowed. A student may solve by isolating a different shape first, or may use a table of trials. The rubric rewards correct substitution and checking, not a single route.

1. Draw each shape clearly and keep repeated shapes identical.
2. Write the legend before solving.
3. Show one substitution per line in the workspace.
4. Check by substituting all values into every original equation.

Worked example

A triangle plus a circle equals 9. Two triangles plus a circle equals 13. Find the value of each shape.

Four practice exercises

Common mistakes

• Treating two different shapes as the same unknown. Assign a separate letter to each distinct shape and keep the legend visible.
• Substituting a value into only one side of an equation. Replace every occurrence of that shape on both sides before simplifying.
• Skipping the final check. Substitute all values into each original equation and confirm equality.
• Changing a shape's value midway through the puzzle. Once a shape is solved, keep that value fixed for the whole system.

Self-check rubric

• Legend assigns one letter to each distinct shape.
• Each picture equation is translated correctly.
• Substitution replaces every occurrence of the known shape.
• Arithmetic is correct and units are consistent.
• Final values are checked in all original equations.
• Workspace shows one substitution per line.
• Alternative valid solution routes are accepted if checking is complete.

Questions and answers