Worked solutions using the default input values for both project versions. For teachers only.
For teachers — worked solutions using the default values. Students who change inputs get different but similarly-structured answers. Each phase shows: default inputs used, step-by-step arithmetic, the correct final answer, and the exact quick-check answer. Sample expert responses and rubric guidance are included for scoring.
"My smoothie recipe uses 6 cups of fruit to 4 cups of yogurt. I can write this ratio as 6:4, 6 to 4, or 6/4. Since GCF(6, 4) = 2, the simplified ratio is 3:2, meaning for every 3 cups of fruit I need exactly 2 cups of yogurt to keep the same taste."
4 (Expert): Writes all three forms correctly (6:4, 6 to 4, 6/4), identifies GCF = 2, shows the division step (6÷2, 4÷2), states the simplified ratio 3:2, and explains what it means in context ("for every 3 cups of fruit…").
3 (Proficient): All three forms correct and simplified ratio 3:2 stated; GCF work may be implicit.
2 (Developing): Ratio written but one form is missing, or simplification skipped/incorrect.
| Batch Size | Fruit (cups) | Yogurt (cups) | Ratio |
|---|---|---|---|
| 1× batch | 6 | 4 | 6:4 → 3:2 |
| 2× batch | 12 | 8 | 12:8 → 3:2 |
| 3× batch | 18 | 12 | 18:12 → 3:2 |
"To serve more smoothie customers I scaled my base recipe (6 cups fruit, 4 cups yogurt). A double batch uses 12 cups of fruit and 8 cups of yogurt, and a triple batch uses 18 cups and 12 cups. All three ratios simplify to 3:2, confirming the taste stays the same no matter how many I make."
4 (Expert): All three rows correct (6:4, 12:8, 18:12); explains why each ratio is equivalent (same GCF simplification, or "multiplied both parts by the same factor"); connects back to the recipe context.
3 (Proficient): All table rows numerically correct.
2 (Developing): One or two rows correct; pattern not explained.
Part A — Unit Rate
Part B — Better Buy
"Each smoothie costs $3.00 to make ($9.00 ÷ 3), so I must sell each for more than $3.00 to profit. For yogurt, Pack B ($7.20 for 6 cups = $1.20/cup) beats Pack A ($4.50 for 3 cups = $1.50/cup) by $0.30 per cup, so I will order Pack B even though it costs more in total."
4 (Expert): Unit rate $3.00/smoothie correct with division shown; both unit prices computed ($1.50, $1.20), Pack B identified as better buy with a contextual explanation ("saves $0.30/cup").
3 (Proficient): Unit rate correct; better buy correctly identified.
2 (Developing): Unit rate attempted; better buy missing or incorrect (e.g., chose Pack A because the total price is lower).
"My smoothie recipe uses a ratio of 6 cups of fruit to 4 cups of yogurt, which simplifies to 3:2. To serve more customers I can scale up to 12 cups of fruit and 8 cups of yogurt for a double batch, keeping the same 3:2 ratio. Each smoothie costs $3.00 to make (unit rate: $9.00 ÷ 3), so I priced my smoothies at $4.50 to earn a $1.50 profit each. I chose Pack B yogurt because it costs $1.20 per cup vs. Pack A's $1.50 per cup — saving $0.30 per cup. My featured menu ratio is 3:2 fruit to yogurt because the extra fruit creates a sweeter, fruitier taste that customers enjoy."
4 (Expert): Reflection explicitly cites all four numeric results (ratio in three forms, scaled table values, unit rate, both better-buy unit prices) and gives a clear, reasoned choice for the menu board ratio.
3 (Proficient): Uses most numbers and gives a clear decision.
2 (Developing): Vague or missing key numbers (e.g., only mentions the ratio without citing the unit rate or better buy).
"Jordan scored 84 points in 12 games, giving a unit rate of 7.00 points per game (84 ÷ 12). Riley scored 70 points in 10 games, also 7.00 ppg (70 ÷ 10). On a per-game basis the two players are perfectly tied, so I need additional data — like the equivalent-ratio comparison and season projection — to make a recommendation."
4 (Expert): Both unit rates correct (7.00 ppg each); division work shown for both; interpretation stated ("they are tied — this means..."); notes the tie creates a need for further analysis.
3 (Proficient): Both unit rates numerically correct.
2 (Developing): One rate correct or division set up incorrectly (e.g., games ÷ points instead of points ÷ games).
"To compare Jordan and Riley on equal footing, I scaled both to 60 games. Jordan's ratio 84:12 becomes 420:60 (multiply by 5), and Riley's ratio 70:10 becomes 420:60 (multiply by 6). Since 420 = 420, both players would score identically over 60 games, confirming the tie we saw in the unit-rate analysis."
4 (Expert): Scaling factors shown (×5, ×6), both products correct (420, 420), explains why these are equivalent ratios ("I multiplied both parts by the same number, so the ratio stays the same"), and ties the result back to the comparison.
3 (Proficient): Both players scaled correctly to 60 games (420 pts each).
2 (Developing): One scaling correct; factor computed wrong for the other (e.g., added instead of multiplied).
| Games Played | Jordan (projected pts) | Riley (projected pts) | Calculation |
|---|---|---|---|
| 10 games | 70.0 pts | 70.0 pts | 7 × 10 = 70 |
| 20 games | 140.0 pts | 140.0 pts | 7 × 20 = 140 |
| 30 games | 210.0 pts | 210.0 pts | 7 × 30 = 210 |
"Using each player's unit rate of 7 points per game, my ratio table projects: at 10 games both have 70 points, at 20 games both have 140 points, and at 30 games (full season) both project to 210 points. Every row is an equivalent ratio of 7:1, meaning the per-game rate never changes."
4 (Expert): All six table cells correct (70, 140, 210 for each player); explains the pattern ("multiply unit rate by games played"); connects the tie in projections to the recommendation section.
3 (Proficient): All six cells numerically correct.
2 (Developing): Two or more cells correct; pattern partially shown.
"Coach, I analyzed the stats for Jordan and Riley. Jordan's unit rate is 7.00 points per game (84 ÷ 12) and Riley's is also 7.00 points per game (70 ÷ 10). When I scaled both to 60 games using equivalent ratios, both players had 420 projected points — a tie. By the end of the 30-game season, both project to 210 points. Because the math shows no statistical difference, I recommend starting Jordan based on consistency: Jordan reached 7.00 ppg over 12 games (a larger sample) compared to Riley's 10 games, making Jordan's rate more reliable."
4 (Expert): Scouting report cites all three math tools (unit rate numbers, equivalent-ratio scaling with factors, season projection totals); recommendation is clearly stated with a data-backed justification (even when the data shows a tie, the student reasons through sample size or another factor).
3 (Proficient): Report gives a clear recommendation with most numbers present.
2 (Developing): Vague or missing key data points (e.g., no mention of the equivalent-ratio comparison).