Unit 6 · Teacher Resource

Unit 6 Projects — Teacher Answer Key

Worked solutions for both project versions using each calculator's default input values. Covers all four phases, quick-check answers, sample expert responses, and rubric scoring notes.

Version A — Game Studio Scoring Engine

Version A · 6.EE.1 · 6.EE.2

Game Studio Scoring Engine — Worked Solutions

Phase 1 · Powers & Exponents · 6.EE.1

Level-Up Score Multiplier

Base score = 3 Level (exponent) = 4

Formula: score = base^level = base × base × … (exponent times)
Expanded form: 3 × 3 × 3 × 3
Step-by-step: 3 × 3 = 9 → 9 × 3 = 27 → 27 × 3 = 81
3⁴ = 81 points

81 points

The expanded form shows 4 factors of 3 — students should count them. A common error is computing 3 × 4 = 12 instead of 3⁴ = 81. The exponent counts how many times the base appears as a factor, not how many times to multiply by the exponent itself.

Phase 2 · Write an Algebraic Expression · 6.EE.2a

Coin Bonus Formula

Coins collected (c) = 12 Expression: 3c + 50

Expression: 3c + 50 | Substitute c = 12
3 × 12 + 50
= 36 + 50
= 86 points

86 points

Correct written expression: 3c + 50 (also accepted: 50 + 3c)
Coefficient = 3 (rate per coin), variable = c (coins), constant = 50 (flat bonus).

Students enter the expression in a text box; the checker accepts "3c+50" or "50+3c" (spaces stripped, lowercase). Common errors: writing "3 + c + 50" (missing coefficient as multiplier) or "3c × 50" (using multiplication instead of addition for the bonus).

Phase 3 · Evaluate an Expression · 6.EE.2c

Power Boost Calculator: 2x² + 5

Power level x = 3 Expression: 2x² + 5

Order of operations: exponent first, then multiply, then add
Step 1 — exponent: x² = 3² = 9
Step 2 — multiply: 2 × 9 = 18
Step 3 — add: 18 + 5 = 23

23 points

Critical order-of-operations sequence: squaring before multiplying by 2, multiplying before adding 5. The most common error is computing (2 × 3)² + 5 = 36 + 5 = 41 instead of 2 × (3²) + 5 = 23. Emphasize that the exponent applies only to x, not to the coefficient 2.

Phase 4 · Compare Formulas & Quick Check · 6.EE.2c · 6.EE.1

Which Formula is Fairer?

Player level x = 4 Formula A: x² + 10 Formula B: 4x + 6

Evaluate both formulas at x = 4
Formula A: 4² + 10 = 16 + 10 = 26
Formula B: 4 × 4 + 6 = 16 + 6 = 22
Formula A is higher at x = 4. (At low x values, B may lead; they cross around x = 2.)

Formula A: 26 | Formula B: 22

At x = 1: A = 11, B = 10 (A slightly higher). At x = 2: A = 14, B = 14 (tie). At x = 3: A = 19, B = 18 (A ahead). At x = 4 and beyond, Formula A grows faster because it is quadratic (x²) while Formula B is linear (4x). Students who try multiple x values earn the "Expert" rating.

Quick-Check Answer: Evaluate 5².
5² = 5 × 5 = 25
Hard-coded correct answer: 25 (see checkPow())

Sample Expert Response — Design Document (4 / Expert) In our Quest Blaster scoring engine, the level-up score at base 3 raised to level 4 is 81 points (3 × 3 × 3 × 3). The coin bonus expression is 3c + 50, which gives 86 points for 12 coins collected (3 × 12 + 50). The elite formula 2x² + 5 evaluates to 23 at power level x = 3, following order of operations: 3² = 9, then 2 × 9 = 18, then + 5. I recommend Formula A (x² + 10) because it scales faster than Formula B (4x + 6) as players advance — at level 4, Formula A gives 26 points versus 22, rewarding skilled players more generously at higher levels.

Sample 4 / 3 / 2 Rubric Scoring Note 4 — Expert: Expanded form shown for exponent; expression written correctly; 2x² + 5 evaluated with all three steps in order; comparison tested at multiple x values; design doc uses every computed number with clear reasoning.
3 — Proficient: All answers correct; design doc uses most numbers; order-of-operations steps may be condensed.
2 — Developing: Exponent treated as multiplication (3 × 4 = 12), or order-of-operations error in 2x² + 5 (e.g., (2×3)²), or expression has wrong coefficient.

Version B — App Pricing Engine

Version B · 6.EE.2 · 6.EE.3 · 6.EE.4

App Pricing Engine — Worked Solutions

Phase 1 · Equivalent Expressions · 6.EE.4

Are These Pricing Expressions Equal?

x = 5 Expression 1: 2(x + 3) Expression 2: 2x + 6

Test at x = 5 (default value)
2(x + 3) at x = 5: 2 × (5 + 3) = 2 × 8 = 16
2x + 6 at x = 5: 2 × 5 + 6 = 10 + 6 = 16
Both equal 16 — expressions are equivalent.

Why they are always equal: 2(x + 3) = 2·x + 2·3 = 2x + 6 by the distributive property.

Both = 16 at x = 5 ✓

Students are encouraged to try multiple x values — they will always match. For extra credit: ask students to verify at x = 0 (both give 6) and x = 10 (both give 26). The algebraic proof is the distributive property itself (6.EE.3).

Phase 2 · Distributive Property · 6.EE.3

Expand the Discount Expression: a(b + c)

a = 3 b = 8 c = 4

Formula: a(b + c) = a·b + a·c
Factored form: 3(8 + 4) = 3 × 12 = 36
Expanded form: 3·8 + 3·4 = 24 + 12 = 36
Both give 36 — distributive property confirmed.

Students should show both paths: (1) add inside parentheses then multiply, and (2) distribute then add. A common error is multiplying a by only b (forgetting to distribute to c), giving 3 × 8 + 4 = 28 instead of 36.

Phase 3 · Evaluate a Cost Expression · 6.EE.2c

Calculate the Real Cost: 9.99n + 4.50

n = 6 months Expression: 9.99n + 4.50

Order of operations: multiply before adding (multiplication before addition)
Step 1 — multiply: 9.99 × 6 = $59.94
Step 2 — add setup fee: $59.94 + $4.50 = $64.44

$64.44

The order-of-operations teaching point: multiplication (9.99 × n) is performed before addition (+ 4.50). A student who adds first would compute 9.99 × (6 + 4.50) = 9.99 × 10.50 = $104.90 — an incorrect application of order of operations.

Phase 4 · Compare Plans & Quick Check · 6.EE.2c · 6.EE.4

Which Plan Costs Less?

n = 6 months StreamPro: 9.99n + 4.50 StreamBasic: 7.50n + 12.00

Evaluate both expressions at n = 6
StreamPro: 9.99 × 6 + 4.50 = 59.94 + 4.50 = $64.44
StreamBasic: 7.50 × 6 + 12.00 = 45.00 + 12.00 = $57.00
StreamBasic is cheaper by $7.44 at n = 6 months.

Note: The plans cross over around n = 3. At n = 3: StreamPro = $34.47, StreamBasic = $34.50 (nearly tied). For n ≤ 3, StreamPro is slightly cheaper due to the lower setup fee; for n ≥ 4, StreamBasic is cheaper due to the lower monthly rate.

StreamPro: $64.44 | StreamBasic: $57.00

Quick-Check Answer: Evaluate 3(4 + 2) using the distributive property.
Method 1 (inside first): 3 × (4 + 2) = 3 × 6 = 18
Method 2 (distribute): 3·4 + 3·2 = 12 + 6 = 18
Hard-coded correct answer: 18 (see checkDist())

Sample Expert Response — Pricing Memo (4 / Expert) Manager, I verified that 2(x + 3) and 2x + 6 are equivalent expressions — for example, when x = 5, both equal 16, and this holds for any value of x because the distributive property guarantees 2(x + 3) = 2x + 6. Using the distributive property, 3(8 + 4) expands to 3·8 + 3·4 = 24 + 12 = 36, confirming both forms give the same result. StreamPro costs $64.44 for 6 months (9.99 × 6 + 4.50), while StreamBasic costs $57.00 (7.50 × 6 + 12.00). I recommend StreamBasic for customers planning to subscribe 4 or more months, as it saves $7.44 at the 6-month mark and the gap grows with longer subscriptions.

Sample 4 / 3 / 2 Rubric Scoring Note 4 — Expert: Equivalence verified for multiple x values and explained via distributive property; distribution shown both ways (factored and expanded); cost evaluated with order-of-operations steps shown; plan comparison with crossover insight; memo uses all computed values and gives a justified recommendation.
3 — Proficient: All calculations correct; memo uses most computed values; recommendation is stated but brief.
2 — Developing: Equivalence tested for only one x with no explanation; distribution error (missing one term); or cost expression evaluated with wrong operation order.