Letters Name Numbers
How is an expression different from an equation?
The story unfolds
The mathematics beneath the story
Read, evaluate, and generate equivalent expressions. Distinguish an expression from an equation and a variable’s role as a general number from its role as an unknown. Solve one-step equations with nonnegative rational numbers and check by substitution.
A learning breakpoint to notice
A letter can be treated as an object label instead of a number. Another breakpoint is assuming that matching at one input proves two expressions equivalent. Use a structural argument; a single mismatch disproves equivalence.
See the structure
Teacher investigation
Compare 3(x + 2) and 3x + 6 using three kit diagrams. Explain why they agree for every allowed x. Is 3x + 2 also equivalent?
Try it before revealing the solution. Name every quantity and unit. Accept several valid representations.
Study a completed solution
Each kit contains x red beads and 2 blue beads. Counting all beads in three kits gives 3(x + 2) = 3x + 6: three copies of the red count plus six blue beads, all measured in beads. For the numerical equation x + 7 = 19, subtract 7 from both sides to obtain x = 12; check 12 + 7 = 19.
Compare approaches
Tiles or repeated groups reveal distribution; the equation records the property. Solving x + 7 = 19 can also use an unknown-addend interpretation: what added to 7 makes 19?
Analyze an incorrect response
“3(x + 2) = 3x + 2.” The added 2 occurs in every copy, giving 6 altogether. Testing x = 0 exposes the mismatch, but the grouping argument explains the valid identity.
Identify the first invalid inference. Ask what evidence would help the learner revise it.
Complete the reasoning
4x = 28. Divide both sides by __ to get x = __. Check by substitution.
Check after trying
4; 7. The same nonzero divisor preserves equality; 4 × 7 = 28.
Transfer independently
A ribbon is 3/4 meter long and consists of an unknown piece plus 1/4 meter. Write and solve an equation for the unknown length, then check.
Compare your reasoning
x + 1/4 = 3/4 gives x = 1/2 meter; 1/2 + 1/4 = 3/4.
Pause and reflect
Which unit must students keep track of? What would an accurate answer conceal? Choose a different context or representation to test whether the idea transfers.
Bring it to your classroom
Grade 6 · Printable student practice and a separate teacher key. Includes an optional “Think Like a Mathematician” extension.