The Expression Studio
The studio reads, writes, and remixes expressions: exponents as repeated multiplication, letters standing for numbers, and the properties of operations proving that 3(2 + x) and 6 + 3x are the same song in different arrangements — while 2x and x² only rhyme by accident.
Common Core standards covered
- 6.EE.A.1Write and evaluate numerical expressions involving whole-number exponents.
- 6.EE.A.2Write, read, and evaluate expressions in which letters stand for numbers.
- 6.EE.A.2.aWrite expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation “Subtract y from 5” as 5 − y.
- 6.EE.A.2.bIdentify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression 2(8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.
- 6.EE.A.2.cEvaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s³ and A = 6s² to find the volume and surface area of a cube with sides of length s = 1/2.
- 6.EE.A.3Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3(2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6(4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.
- 6.EE.A.4Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.
© Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
Teacher-facing overview
Big idea: Write and evaluate expressions with whole-number exponents and variables, apply properties of operations to generate equivalent expressions, and identify when two expressions are equivalent.
Student target: “I can evaluate expressions with exponents in conventional order, write expressions from verbal descriptions, generate equivalents with properties, and judge equivalence by property or counterexample.”
Success criteria: I can evaluate powers and formulas, order of operations intact. · I can write 'subtract y from 5' as 5 − y and name an expression's parts. · I can distribute, factor, and combine like terms with the property named. · I can prove equivalence with properties and disprove it with one counterexample.
Essential question
Two expressions can LOOK different and BE the same — or look similar and be strangers. What counts as proof either way?
Learning target: I can evaluate expressions with exponents in conventional order, write expressions from verbal descriptions, generate equivalents with properties, and judge equivalence by property or counterexample.
Success criteria
- I can evaluate powers and formulas, order of operations intact.
- I can write 'subtract y from 5' as 5 − y and name an expression's parts.
- I can distribute, factor, and combine like terms with the property named.
- I can prove equivalence with properties and disprove it with one counterexample.
Why this matters
6.EE.A is algebra's grammar school: exponents (A.1), the discipline of letters-for-numbers (A.2), and the two-sided coin the strand turns on — properties GENERATE equivalents (A.3) and define what equivalent MEANS (A.4). The prove-or-kill epistemology installed here (properties prove; one counterexample kills; agreement at a point proves nothing) is exactly what Grade 7's Border Crew and every later algebra course assume on day one.
Students should already be able to
- The distributive identity on numbers — The Snack Pack Factory.
- Order of operations with whole numbers.
- Fraction multiplication — The Serving Scoop.
Vocabulary to define explicitly
- Exponent (power)
Repeated multiplication, counted: sⁿ is n copies of s multiplied.
Example: 3⁴ = 3·3·3·3 = 81.
Not an example: 3⁴ = 12 — that's 3 × 4, addition's cousin, not a power.
🔒 3 more terms with examples and non-examples — Parts of an expression, Equivalent expressions, Counterexample — are defined in the full lesson. See plans →
Where this shows up outside the classroom
Assessment plan
Formative — for feedback during the lesson
- Warm-up expansions.
- Booth boards during floor work.
- The faulty-identity diagnosis.
Summative — the graded evidence: The exit ticket (power, phrase, remix, verdict), scored with the four-criterion rubric.
The task students work on
The studio has four booths. POWER BOOTH: evaluate 3⁴, 2³ + 4², and the cube formulas V = s³ and A = 6s² for a die with s = 1/2 (the standard's own job). WRITING BOOTH: turn 'subtract y from 5,' 'the product of 4 and one more than x,' and 'y added to itself three times' into symbols, and describe 2(8 + 7) in structure words. REMIX BOOTH: distribute 3(2 + x), factor 24x + 18y, collapse y + y + y. VERDICT BOOTH: are 3(2 + x) and 6 + 3x equivalent? Are 2x and x² — they matched at x = 2!
Studio board — property or counterexample cited on every verdict.
| Booth | Job | Work | Result |
|---|---|---|---|
| Power | 3⁴; 2³ + 4²; V and A at s = 1/2 | ||
| Writing | the three phrases; describe 2(8 + 7) | ||
| Remix | 3(2 + x); 24x + 18y; y + y + y | ||
| Verdict | 3(2 + x) vs 6 + 3x; 2x vs x² |
Anticipated student strategies
Expansion grounds exponents, narration guards the writing, and the table-then-property sequence teaches what evidence can and cannot conclude.
- Repeated-multiplication expansion. The student unrolls 3⁴ into 3·3·3·3 before computing.
Keeps powers from collapsing into products — the 81-vs-12 fork. Share first.
🔒 2 more anticipated strategies with select-and-sequence guidance are in the full lesson. See plans →
Lesson script
3·3·3·3 = 81 vs 4·4·4 = 64: the smaller base wins on repetition. The near-universal error surfaced on purpose: 3⁴ ≠ 12. Then order of operations: 2³ + 4² = 8 + 16 = 24 — powers before sums.
Listen for: The exponent counts multiplications, and an extra multiplication by 3 beats a bigger base — position matters more than size here.
Structure language drilled on 2(8 + 7): a product of two factors; the second factor is itself a sum of two terms — one object at two zoom levels.
- Direction words (from, less than) reverse the order.
- Chunks in parentheses are single entities until opened.
Listen for: The 5 is the starting pile; y leaves it. From-language names the start — order in subtraction is meaning, not style.
🔒 The remaining 4 steps — fully scripted with teacher-says lines, discourse questions, look-fors, and interventions — are in the full lesson. See plans →
Misconceptions and how to respond
- “3⁴ = 12.” The exponent counts FACTORS, not a multiplier: 3·3·3·3 = 81. Unroll before you compute.
- “'Subtract y from 5' is y − 5.” — the teacher response is in the full lesson.
- “They matched at x = 2, so they're equivalent.” — the teacher response is in the full lesson.
- “4(x + 1) = 4x + 1.” — the teacher response is in the full lesson.
🔒 Every misconception ships with a tested classroom response. See plans →
Scaffolded practice
The workbook follows gradual release rather than a page of blank prompts: 1 fully worked, 1 partially completed, then 4 independent.
- I doRun the power booth: 3⁴, 2³ + 4², and the die formulas V = s³, A = 6s² at s = 1/2.
🔒 5 more prompts across all three release tiers are in the full lesson. See plans →
Worked solutions and scaffolds for every prompt are in the downloadable kit, included with membership. See plans and pricing.
Exit ticket
- Evaluate 5³, and 3 + 2⁴.
🔒 2 more exit-ticket questions are in the full lesson. See plans →
Worked answers, the completed answer tables, and the scoring rubric are in the downloadable kit, included with membership. See plans and pricing.
Printable handouts in the kit
All handouts are in Printable_Handouts.pdf inside the kit download.
Materials and classroom resources
Student experience
Students enter through a low-floor prompt, build or sketch a model, test a strategy, revise their thinking, collaborate, and present a final solution. The task remains accessible while allowing for sophisticated reasoning.
Detailed facilitation notes
- Step 1: Evaluate exponent expressions, formulas included. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 2: Write expressions from words; name their parts. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 3: Remix with the distributive property and combine like terms. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 4: Test equivalence: properties prove, counterexamples kill. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
Differentiation and inclusion
Support: Substitution tables scaffold every verdict; extend with the fraction-edge cube and the 2x-vs-x² trap.
Language support: “I started by…”, “My model shows…”, and “I know my answer is reasonable because…”.
Extension: Require a second solution, alter a condition, or have students write a challenge for another group.
Accessibility: Manipulatives, enlarged visuals, oral response choices, partner reading, and reduced-copy student pages.
Assessment plan
Formative checks: Strategy choice, accuracy, representation, collaboration, and explanation.
End evidence: One evaluated formula, one remix with the property named, and one equivalence verdict with evidence.
Mastery: The student completes the task accurately and explains the strategy with limited prompting.