Unit 2 · Teacher Resource
Unit 2 Projects — Teacher Answer Key
Worked solutions for both versions of the Unit 2 culminating project (Fraction Division). Covers Version A: Recipe Remix Bakery and Version B: Maker Workshop Cut List.
🥁 Recipe Remix Bakery
Standard: 6.NS.1 • Fraction division in a bakery context
1 Phase 1 — Portion Control: Mini-Loaf Pans (Fraction ÷ Fraction · 6.NS.1)
| Input | Default | Worked Solution | Final Answer |
|---|---|---|---|
| Dough numerator (a) | 3 |
Rule: (a/b) ÷ (c/d) = (a×d) / (b×c)
(3/4) ÷ (1/8) = (3×8) / (4×1) = 24 / 4 = 6 Simplify: GCF(24, 4) = 4 24/4 = 6 (whole number) Decimal: 24 ÷ 4 = 6.0 Check: 6 × (1/8) = 6/8 = 3/4 ✓ |
6 mini-loaves (whole number, no dough left over) |
| Dough denominator (b) | 4 | ||
| Per loaf numerator (c) | 1 | ||
| Per loaf denominator (d) | 8 |
2 Phase 2 — Scaling Down: How Many Servings? (Whole Number ÷ Unit Fraction · 6.NS.1)
| Input | Default | Worked Solution | Final Answer |
|---|---|---|---|
| Total cups of frosting | 4 |
Rule: whole ÷ (1/d) = whole × d
4 ÷ (1/3) = 4 × 3 = 12 Interpretation: each cup holds 3 servings of 1/3-cup each; 4 cups × 3 = 12 cupcakes Check: 12 × (1/3) = 12/3 = 4 cups ✓ |
12 cupcakes |
| Each cupcake uses 1/(this) cup | 3 |
3 Phase 3 — Roll Cuts: Mixed-Number Dough (Mixed Number ÷ Fraction · 6.NS.1)
| Input | Default | Worked Solution | Final Answer |
|---|---|---|---|
| Whole part | 2 |
Step 1 — Convert mixed number:
2 and 1/2 = (2×2 + 1)/2 = 5/2 Step 2 — Divide fractions: (5/2) ÷ (3/4) = (5×4) / (2×3) = 20/6 Simplify: GCF(20, 6) = 2 20/6 = 10/3 As mixed number: 10 ÷ 3 = 3 remainder 1 = 3 and 1/3 Full pieces: 3 Leftover fraction of a piece: 1/3 Leftover in feet: (1/3) piece × (3/4) ft/piece = 3/(3×4) = 3/12 = 1/4 ft left over |
3 full pieces (10/3 = 3⅓ pieces; 1/4 ft left over) |
| Fraction numerator | 1 | ||
| Fraction denominator | 2 | ||
| Cut size — numerator | 3 | ||
| Cut size — denominator | 4 |
4 Phase 4 — Best Deal Decision (Real Decision · 6.NS.1)
| Input | Default | Worked Solution | Final Answer |
|---|---|---|---|
| Total cups (Supplier A) | 6 |
6 ÷ (2/3)
= 6 × (3/2) = 18/2 = 9 Check: 9 × (2/3) = 18/3 = 6 cups ✓ Every cup of batter is used (whole number result). |
9 muffins |
| Each muffin uses 2/(this) cup | 3 |
Check: 9 × (2/3) = 18/3 = 6 ✓
Note: This is the same problem as Phase 4 with the default inputs, so the calculator and quick-check give the same answer.
Sample Expert Written Menu Card (Final Deliverable)
"At the Recipe Remix Bakery, I can bake 6 mini-loaves from 3/4 of a pan of dough, because (3/4) ÷ (1/8) = (3×8)/(4×1) = 24/4 = 6 — a perfect whole number with no dough left over. With 4 cups of frosting I can frost 12 cupcakes because dividing by 1/3 is the same as multiplying by 3: 4 × 3 = 12. My 2½-foot dough roll cuts into 3 full pieces of 3/4 ft each (5/2 ÷ 3/4 = 20/6 = 10/3 = 3⅓ pieces; I get 3 full pieces with 1/4 ft left over). From 6 cups of batter I can make 9 muffins because 6 ÷ (2/3) = 6 × (3/2) = 18/2 = 9."
Sample 4 / 3 / 2 Rubric Scoring Notes
Expert (4) responses demonstrate: (Fraction ÷ Fraction) correct answer 6, simplified fraction shown (24/4 = 6), and decimal or check shown; (Whole ÷ Unit Fraction) correct count of 12 with explicit explanation of why multiplying by the reciprocal works (e.g., "each cup holds 3 servings of 1/3"); (Mixed Numbers) mixed number correctly converted to 5/2 first, division result 10/3 simplified, full pieces (3) AND remainder (1/4 ft) both identified; menu card names every calculated value with correct units and connects each to the division operation used.
Proficient (3): All four numerical answers correct; menu card uses most numbers but may omit the remainder interpretation or skip showing the reciprocal step.
Developing (2): At least two phases attempted; error in the mixed-number conversion or a sign/simplification error; menu card present but incomplete.
🧒 Maker Workshop Cut List
Standard: 6.NS.1 • Fraction division in a workshop context
1 Phase 1 — Shelf Boards: How Many Shelves? (Whole Number ÷ Fraction · 6.NS.1)
| Input | Default | Worked Solution | Final Answer |
|---|---|---|---|
| Board length (feet) | 8 |
Each shelf is 2/3 ft (numerator fixed at 2)
Rule: L ÷ (2/d) = L × (d/2) 8 ÷ (2/3) = 8 × (3/2) = 24/2 = 12 Full shelves: 12 Leftover: 24 mod 2 = 0 pieces remainder → no scrap — perfect cut! Check: 12 × (2/3) = 24/3 = 8 ft ✓ |
12 shelves (no scrap — perfect cut) |
| Each shelf is 2/(denominator) ft | 3 |
2 Phase 2 — Ribbon Roll: Smaller Pieces (Fraction ÷ Fraction · 6.NS.1)
| Input | Default | Worked Solution | Final Answer |
|---|---|---|---|
| Ribbon numerator (a) | 5 |
Rule: (a/b) ÷ (c/d) = (a×d) / (b×c)
(5/6) ÷ (1/4) = (5×4) / (6×1) = 20/6 Simplify: GCF(20, 6) = 2 20/6 = 10/3 Decimal: 10 ÷ 3 ≈ 3.333... Full pieces: 3 Remainder: 20 − (3×6) = 20 − 18 = 2 (in sixths-of-quarter units) Leftover in yards: remPiecesNum×c / (rDen×d) = 2×1 / (6×4) = 2/24 = 1/12 yd |
3 full pieces (10/3 ≈ 3.33; 1/12 yd left over) |
| Ribbon denominator (b) | 6 | ||
| Per piece numerator (c) | 1 | ||
| Per piece denominator (d) | 4 |
3 Phase 3 — Pipe Sections: Mixed-Number Length (Mixed Number ÷ Fraction · 6.NS.1)
| Input | Default | Worked Solution | Final Answer |
|---|---|---|---|
| Whole part (W) | 3 |
Cut size is fixed at 1/4 m
Step 1 — Convert mixed number: 3 and 1/2 = (3×2 + 1)/2 = 7/2 m Step 2 — Divide by 1/4: (7/2) ÷ (1/4) = (7/2) × 4 = 28/2 = 14 Full pieces: 14 Remainder: 28 mod 2 = 0 → no scrap — perfect cut! Check: 14 × (1/4) = 14/4 = 7/2 = 3.5 m ✓ |
14 pipe pieces (each 1/4 m; no scrap) |
| Numerator (n) | 1 | ||
| Denominator (d) | 2 |
4 Phase 4 — Which Cut Wastes Less? (Decision + Quick Check · 6.NS.1)
Both options use a 7-foot board; inputs are hard-coded in this phase.
| Option | Calculation | Full Pieces | Leftover (ft) |
|---|---|---|---|
| Option A 3/4-ft cuts |
7 ÷ (3/4) = 7 × (4/3) = 28/3
28 ÷ 3 = 9 remainder 1 Leftover: (1 piece-rem) × (3/4 ft/piece) = 1×3 / (3×4) = 3/12 = 1/4 ft |
9 pieces | 1/4 ft (= 0.25 ft) |
| Option B 2/3-ft cuts |
7 ÷ (2/3) = 7 × (3/2) = 21/2
21 ÷ 2 = 10 remainder 1 Leftover: (1 piece-rem) × (2/3 ft/piece) = 1×2 / (2×3) = 2/6 = 1/3 ft |
10 pieces | 1/3 ft (= 0.333... ft) |
|
Winner: Option A (3/4-ft cuts) wastes less — only 1/4 ft of scrap vs. 1/3 ft. Comparison: 1/4 < 1/3 because 3 < 4 (same numerator, larger denominator = smaller fraction). |
|||
Check: 10 × (1/2) = 5 ft ✓
Sample Expert Written Cut-List Memo (Final Deliverable)
"Workshop team: The 8-ft board yields 12 shelves at 2/3 ft each, with no scrap at all (8 ÷ 2/3 = 8 × 3/2 = 12 — a perfect whole-number result). The 5/6-yd ribbon yields 3 full decoration pieces of 1/4 yd each (5/6 ÷ 1/4 = 20/6 = 10/3 ≈ 3.33), with 1/12 yd of ribbon left over. The 3½-meter pipe gives 14 quarter-meter pieces (7/2 ÷ 1/4 = 7/2 × 4 = 14 — again no waste). For the 7-ft board, Option A (3/4-ft cuts) wastes less, leaving only 1/4 ft of scrap compared to 1/3 ft for Option B, because 1/4 < 1/3."
Sample 4 / 3 / 2 Rubric Scoring Notes
Expert (4) responses demonstrate: (Whole ÷ Fraction) correct shelf count of 12, leftover stated as zero (or "no scrap"), and multiplication check shown; (Fraction ÷ Fraction) correct result 10/3, full piece count 3, simplified fraction shown, AND leftover in yards (1/12 yd) identified; (Mixed Numbers) correct conversion 3½ → 7/2 written explicitly, division 14 shown, no-scrap conclusion stated; (Decision) both leftover amounts computed and compared using fraction comparison reasoning (1/4 < 1/3); memo names the better option (Option A) with a mathematical justification, not just a guess.
Proficient (3): Correct answer for each phase; memo identifies Option A but may not explain the fraction comparison; leftover amounts present but may lack units.
Developing (2): At least two phases correct; error in the mixed-number conversion or leftover calculation; memo present but missing the cut-option comparison.