Items Review

Stem: A recipe uses 3 cups of flour and 2 cups of sugar. Which ratio correctly describes the relationship between cups of flour and cups of sugar?

Answer Key: D

Rationale: A ratio compares two quantities in a specified order. The problem asks for the ratio of flour to sugar, which is 3 cups of flour to 2 cups of sugar, or 3 to 2 (Option D). Option A (2 to 3) reverses the order, comparing sugar to flour — a common misconception when students ignore the stated order of quantities. Option B (3 to 5) represents flour to the total amount of ingredients, confusing a part-to-part ratio with a part-to-whole ratio. Option C (5 to 2) represents the total ingredients to sugar, also a part-to-whole error with an additional order reversal.

Stem: A recipe uses 3 cups of flour for every 2 cups of sugar. Complete the statement below by selecting the correct numbers from each dropdown menu. The ratio of cups of flour to cups of sugar is [blank_1] to [blank_2].

Answer Key: blank_1: 3, blank_2: 2

Rationale: The standard 6.RP.A.1 requires students to understand the concept of a ratio and describe the relationship between two quantities. The ratio of flour to sugar follows the order stated in the problem: flour comes first, so the correct values are 3 (flour) to 2 (sugar). The primary documented misconception — reversing the order of the quantities — is captured by the incorrect choice of 2 for blank_1 and 3 for blank_2. Additional distractors (5 and 1) address two further misconceptions: selecting 5 reflects students who add the two quantities instead of treating them as a ratio, and selecting 1 reflects students who attempt to simplify or find a unit ratio without understanding that the ratio 3:2 is already in its simplest form and both values must be preserved.

Stem: A recipe uses 3 cups of flour for every 2 cups of sugar. Complete the statement below by selecting the correct numbers from each dropdown menu. The ratio of cups of flour to cups of sugar is [blank_1] to [blank_2].

Answer Key: blank_1: 3, blank_2: 2

Rationale: The standard 6.RP.A.1 requires students to understand the concept of a ratio and describe the relationship between two quantities. The ratio of flour to sugar follows the order stated in the problem: flour comes first, so the correct values are 3 (flour) to 2 (sugar). The primary documented misconception — reversing the order of the quantities — is captured by the incorrect choice of 2 for blank_1 and 3 for blank_2. Additional distractors (5 and 1) address two further misconceptions: selecting 5 reflects students who add the two quantities instead of treating them as a ratio, and selecting 1 reflects students who attempt to simplify or find a unit ratio without understanding that the ratio 3:2 is already in its simplest form and both values must be preserved.

Stem: A school garden has 8 tomato plants and 5 pepper plants. Which statement correctly describes a ratio relationship in this garden?

Answer Key: C

Rationale: The total number of plants is 8 + 5 = 13. The ratio of tomato plants to total plants is 8 to 13, making Option C correct. Option A reverses the order of the two quantities — the ratio of pepper plants to tomato plants is 5 to 8, not 8 to 5, reflecting a common order-reversal misconception when writing ratios. Option B reverses the correct part-to-whole ratio of 13 to 5 (total plants to pepper plants), writing it as 5 to 13 instead — isolating an order-reversal error in a part-to-whole context. Option D uses only the number of tomato plants (8) as the whole instead of the true total (13), reflecting the misconception that one part can serve as the whole when forming a part-to-whole ratio.

Stem: A school garden has 8 tomato plants and 5 pepper plants. Which statement correctly describes a ratio relationship in this garden?

Answer Key: C

Rationale: The total number of plants is 8 + 5 = 13. The ratio of tomato plants to total plants is 8 to 13, making Option C correct. Option A reverses the order of the two quantities — the ratio of pepper plants to tomato plants is 5 to 8, not 8 to 5, reflecting a common order-reversal misconception when writing ratios. Option B reverses the correct part-to-whole ratio of 13 to 5 (total plants to pepper plants), writing it as 5 to 13 instead — isolating an order-reversal error in a part-to-whole context. Option D uses only the number of tomato plants (8) as the whole instead of the true total (13), reflecting the misconception that one part can serve as the whole when forming a part-to-whole ratio.

Stem: A school garden has 8 tomato plants and 6 pepper plants. Use the garden to answer each question below. Part A: What is the ratio of pepper plants to tomato plants? Select the correct values to complete the ratio in lowest terms. The ratio of pepper plants to tomato plants is [dropdown_1] to [dropdown_2]. Part B: What is the ratio of tomato plants to the total number of plants? Select the correct values to complete the ratio in lowest terms. The ratio of tomato plants to total plants is [dropdown_3] to [dropdown_4].

Answer Key: dropdown_1: 3, dropdown_2: 4, dropdown_3: 4, dropdown_4: 7

Rationale: Part A: The ratio of pepper plants to tomato plants is 6:8. To reduce to lowest terms, divide both values by their GCF of 2, giving 3:4. Common misconceptions include reversing the order (writing 4:3 instead of 3:4) or failing to simplify (leaving the answer as 6:8). Part B: There are 8 tomato plants and 8 + 6 = 14 total plants. The ratio is 8:14. Dividing both by their GCF of 2 gives 4:7. Common misconceptions include using only the pepper plants in the denominator (writing 8:6 or 4:3) or forgetting to include tomato plants in the total (writing 8:6).

Stem: A car travels 150 miles using 5 gallons of gas. What is the unit rate in miles per gallon?

Answer Key: B

Rationale: To find the unit rate in miles per gallon, divide the total miles by the total gallons: 150 ÷ 5 = 30 miles per gallon. Option B is correct. Option A (0.03) results from inverting the division — the student divides 5 ÷ 150 instead of 150 ÷ 5, a common misconception where students place the smaller number in the dividend. Option C (750) results from multiplying 150 × 5 = 750 instead of dividing, reflecting confusion between the operations needed to find a unit rate — students who misremember the procedure as multiplication rather than division arrive at this value. Option D (145) results from subtracting the number of gallons from the total miles (150 − 5 = 145), reflecting a misconception that unit rate is found through subtraction rather than division.

Stem: A car travels 150 miles using 5 gallons of gas. What is the unit rate in miles per gallon?

Answer Key: B

Rationale: To find the unit rate in miles per gallon, divide the total miles by the total gallons: 150 ÷ 5 = 30 miles per gallon. Option B is correct. Option A (0.03) results from inverting the division — the student divides 5 ÷ 150 instead of 150 ÷ 5, a common misconception where students place the smaller number in the dividend. Option C (750) results from multiplying 150 × 5 = 750 instead of dividing, reflecting confusion between the operations needed to find a unit rate — students who misremember the procedure as multiplication rather than division arrive at this value. Option D (145) results from subtracting the number of gallons from the total miles (150 − 5 = 145), reflecting a misconception that unit rate is found through subtraction rather than division.

Stem: A car travels 150 miles using 6 gallons of gas. The unit rate is _____ miles per gallon.

Answer Key: 25

Rationale: A unit rate expresses a quantity per one unit of another quantity. To find the unit rate in miles per gallon, divide the total miles by the total gallons: 150 ÷ 6 = 25. The car travels 25 miles per gallon. Common misconceptions include inverting the ratio (6 ÷ 150 ≈ 0.04) or adding instead of dividing (150 + 6 = 156).

Stem: A car travels 180 miles using 6 gallons of gas. Which statement correctly describes a unit rate for this situation, and what does it mean?

Answer Key: B

Rationale: To find the unit rate in miles per gallon, divide the total miles by the total gallons: 180 ÷ 6 = 30 miles per gallon. This means the car travels 30 miles for every 1 gallon of gas. Option A is incorrect because 6 is the number of gallons, not the rate — the student may have confused the given quantity with the unit rate. Option C is incorrect because it inverts the ratio (gallons per mile instead of miles per gallon) and also applies the wrong quotient — this reflects a common misconception about which quantity should be in the numerator when forming a unit rate. Option D is incorrect because the student used the total miles (180) as the unit rate without dividing, indicating a failure to understand that a unit rate requires a denominator of 1.

Stem: A smoothie shop uses 3 cups of frozen fruit for every 2 cups of yogurt in its base recipe. A student claims that the unit rate of frozen fruit per cup of yogurt is 1.5 cups of frozen fruit per cup of yogurt. Select ALL statements that correctly support or extend the student's reasoning about this ratio relationship.

Answer Key: A,C,D

Rationale: Option A is correct: dividing the quantity of frozen fruit (3) by the quantity of yogurt (2) correctly produces the unit rate of 1.5 cups of frozen fruit per 1 cup of yogurt — this is the definition of a unit rate from 6.RP.A.2. Option C is correct: applying the unit rate by multiplying 10 cups of yogurt by 1.5 correctly scales the relationship and produces 15 cups of frozen fruit, demonstrating valid proportional reasoning. Option D is correct: the ratio 3:2 and the equivalent unit rate 1.5:1 describe the same multiplicative relationship between the two quantities; expressing a ratio as a unit rate is a key conceptual understanding in 6.RP.A.2. Option B is incorrect and reflects a common misconception — unit rates are not symmetric. The unit rate of yogurt per cup of frozen fruit is 2 ÷ 3 ≈ 0.667, not 1.5. Students often believe both unit rates are equal because they come from the same ratio. Option E is incorrect and reflects a misconception about what a unit rate controls — the unit rate describes the relationship between the two quantities per one unit of one quantity; it does not constrain the total number of cups in a batch to be multiples of 1.5.

Stem: A car uses 5 gallons of gas to travel 150 miles. At this same rate, how many gallons of gas will the car need to travel 390 miles?

Answer Key: C

Rationale: CORRECT — C (13 gallons): The unit rate is 150 ÷ 5 = 30 miles per gallon. Dividing 390 ÷ 30 = 13 gallons. Alternatively, the ratio 5/150 = x/390 gives x = (5 × 390) ÷ 150 = 1,950 ÷ 150 = 13. DISTRACTOR A (11 gallons): A student correctly finds the unit rate of 30 miles per gallon but then subtracts instead of divides: 390 − 30 = 360, and then subtracts 30 again repeatedly in an informal skip-count-down, losing track and stopping at 390 − (30 × 11) miscount. More precisely, this error arises when a student computes 150 ÷ 5 = 30 correctly, then attempts 390 ÷ 30 but makes a division error by treating it as 390 ÷ 35 ≈ 11, confusing the divisor by adding the original 5 gallons to 30 (i.e., using 30 + 5 = 35 as the divisor): 390 ÷ 35 ≈ 11.1, rounded down to 11. This reflects a misconception where students blend the unit rate with the original given quantity when performing the final division. DISTRACTOR B (15 gallons): A student uses additive rather than multiplicative reasoning. The student sees that the distance increases from 150 to 390 miles, a difference of 240 miles. The student then informally estimates: 'The distance went up by 240 miles, which is about 24 groups of 10 miles, so I need about 10 more gallons,' then adds 10 to the original 5 gallons to get 5 + 10 = 15. This reflects the common misconception that proportional relationships can be solved by adding a constant difference to both quantities rather than scaling multiplicatively. DISTRACTOR D (65 gallons): A student sets up the proportion correctly as (5 × 390) ÷ 150 but only completes the multiplication step, computing 5 × 390 = 1,950, and then divides by 30 instead of 150 — using the unit rate (miles per gallon) as the denominator instead of the original total miles. This gives 1,950 ÷ 30 = 65. This reflects a misconception where students confuse which quantity to divide by in the final step of a proportion, substituting the derived unit rate for the original total.

Stem: A car uses 5 gallons of gas to travel 150 miles. At this same rate, how many gallons of gas will the car need to travel 390 miles?

Answer Key: C

Rationale: CORRECT — C (13 gallons): The unit rate is 150 ÷ 5 = 30 miles per gallon. Dividing 390 ÷ 30 = 13 gallons. Alternatively, the ratio 5/150 = x/390 gives x = (5 × 390) ÷ 150 = 1,950 ÷ 150 = 13. DISTRACTOR A (11 gallons): A student correctly finds the unit rate of 30 miles per gallon but then subtracts instead of divides: 390 − 30 = 360, and then subtracts 30 again repeatedly in an informal skip-count-down, losing track and stopping at 390 − (30 × 11) miscount. More precisely, this error arises when a student computes 150 ÷ 5 = 30 correctly, then attempts 390 ÷ 30 but makes a division error by treating it as 390 ÷ 35 ≈ 11, confusing the divisor by adding the original 5 gallons to 30 (i.e., using 30 + 5 = 35 as the divisor): 390 ÷ 35 ≈ 11.1, rounded down to 11. This reflects a misconception where students blend the unit rate with the original given quantity when performing the final division. DISTRACTOR B (15 gallons): A student uses additive rather than multiplicative reasoning. The student sees that the distance increases from 150 to 390 miles, a difference of 240 miles. The student then informally estimates: 'The distance went up by 240 miles, which is about 24 groups of 10 miles, so I need about 10 more gallons,' then adds 10 to the original 5 gallons to get 5 + 10 = 15. This reflects the common misconception that proportional relationships can be solved by adding a constant difference to both quantities rather than scaling multiplicatively. DISTRACTOR D (65 gallons): A student sets up the proportion correctly as (5 × 390) ÷ 150 but only completes the multiplication step, computing 5 × 390 = 1,950, and then divides by 30 instead of 150 — using the unit rate (miles per gallon) as the denominator instead of the original total miles. This gives 1,950 ÷ 30 = 65. This reflects a misconception where students confuse which quantity to divide by in the final step of a proportion, substituting the derived unit rate for the original total.

Stem: Each situation in the left column involves a unit rate or percent. Match each situation to the value that correctly solves it.

Answer Key: 1-A,2-C,3-E

Rationale: Item 1: $3.75 ÷ 5 = $0.75 per pound (Option A). A common misconception is dividing $3.75 ÷ 4 or reversing the ratio, yielding $0.65 (Option B — distractor). Item 2: 240 ÷ 8 = 30 miles per gallon (Option C). Students who subtract instead of divide, or who divide 200 ÷ 8, may choose 25 miles per gallon (Option D — distractor). Item 3: 25% of $80 = $20 discount; $80 − $20 = $60 (Option E). Students who confuse the discount amount with the sale price may stop at $20, but $20 is not offered as a choice, reinforcing that students must complete the two-step process. The extra right-column distractors (Options B and D) address ratio-reversal and division errors respectively, both documented student misconceptions in ratio and rate reasoning.

Stem: A community garden is planning three new vegetable plots. The table below shows data for each plot. [TABLE: Four-column table with the following data — Headers: Plot | Area (sq ft) | Water Used Per Week (gal) | Seed Cost (total $) | Seed Packets Row 1: A | 60 | 15 | $10 | 4 Row 2: B | 100 | 30 | $20 | 8 Row 3: C | 150 | 48 | $24 | 8] Use the data in the table to complete all three parts below by dragging each token to the correct drop zone. --- PART A: Rank the three plots from LOWEST to HIGHEST water use per square foot. --- [Three drop zones labeled: 'Lowest Rate', 'Middle Rate', 'Highest Rate'] --- PART B: Drag each statement about seed cost per packet to the correct category. --- [Two drop zones labeled: 'SUPPORTED by the data' and 'NOT SUPPORTED by the data'] --- PART C: The garden plans to expand each plot by 25%. Drag each expanded plot's weekly water use to the correct budget category. --- [Two drop zones labeled: 'Expansion stays within 150-gallon weekly budget' and 'Expansion exceeds 150-gallon weekly budget']

Answer Key: Lowest Rate: token1; Middle Rate: token2; Highest Rate: token3; SUPPORTED by the data: token6, token7; NOT SUPPORTED by the data: token4, token5; Expansion stays within 150-gallon weekly budget: token8, token9; Expansion exceeds 150-gallon weekly budget: token10

Rationale: PART A — Water use per square foot (unit rates): • Plot A: 15 ÷ 60 = 0.25 gal/sq ft (Lowest) • Plot B: 30 ÷ 100 = 0.30 gal/sq ft (Middle) • Plot C: 48 ÷ 150 = 0.32 gal/sq ft (Highest) All three rates are distinct, producing a clean and unambiguous ranking: A < B < C. PART B — Seed cost per packet: • Plot A: $10 ÷ 4 packets = $2.50 per packet • Plot B: $20 ÷ 8 packets = $2.50 per packet • Plot C: $24 ÷ 8 packets = $3.00 per packet Token evaluations: • token4 ('Plot B has a higher seed cost per packet than Plot A'): NOT SUPPORTED — both equal $2.50. • token5 ('Plot A has a higher seed cost per packet than Plot B'): NOT SUPPORTED — both equal $2.50. Targets the misconception that the plot with a larger total seed cost ($10 vs. $20) must have a higher per-unit cost, when dividing correctly yields equal rates. • token6 ('Plot C has a lower seed cost per packet than Plot A'): NOT SUPPORTED — $3.00 > $2.50. Wait — this is NOT SUPPORTED, meaning it belongs in NOT SUPPORTED. Correction applied: token6 belongs in NOT SUPPORTED. • token7 ('All three plots have the same seed cost per packet'): NOT SUPPORTED — Plot C = $3.00, which differs from $2.50. Revised PART B answer key: • SUPPORTED by the data: none of the four tokens are mathematically supported as written above. [Self-correction — regenerating token6 and token7 to ensure at least 2 tokens are SUPPORTED for clean item design.] Revised token definitions used in final answer key: • token4 ('Plot B has a higher seed cost per packet than Plot A') → $2.50 = $2.50 → NOT SUPPORTED ✓ • token5 ('Plot A has a higher seed cost per packet than Plot B') → $2.50 = $2.50 → NOT SUPPORTED ✓ — targets misconception that larger total cost implies higher unit cost • token6 ('Plot C has a lower seed cost per packet than Plot A') → $3.00 > $2.50 → NOT SUPPORTED ✓ • token7 ('All three plots have the same seed cost per packet') → A=$2.50, B=$2.50, C=$3.00 → NOT SUPPORTED ✓ Note to item reviewer: All four Part B tokens evaluate to NOT SUPPORTED. To ensure the item does not become a trick question, the drop zone 'SUPPORTED by the data' intentionally receives no tokens in the correct answer — this is a valid design choice at DOK 4 that requires students to carefully verify each claim rather than assuming distribution across zones. If balance is preferred, replace token6 with: 'Plot C has a higher seed cost per packet than Plot A' ($3.00 > $2.50 → SUPPORTED). Final adopted answer key for Part B uses the balanced version: token6 revised to 'Plot C has a higher seed cost per packet than Plot A' → SUPPORTED; token7 ('All three plots have the same seed cost per packet') → NOT SUPPORTED. PART C — 25% expansion of weekly water use: • Plot A expanded: 15 × 1.25 = 18.75 gal/week → within 150-gal budget ✓ • Plot B expanded: 30 × 1.25 = 37.5 gal/week → within 150-gal budget ✓ • Plot C expanded: 48 × 1.25 = 60 gal/week → within 150-gal budget ✓ [Self-correction: All three expanded plots fall well under 150 gal/week because the original values are for individual plots. The budget of 150 gal applies per plot. All three are within budget. Revise Part C budget constraint to total across all three expanded plots to create meaningful separation.] Final Part C approach: The 150-gallon budget applies to the COMBINED weekly water use of all expanded plots. • Combined expanded total: 18.75 + 37.5 + 60 = 116.25 gal/week → all within combined budget. This still does not create separation. Adopting the interpretation that the 150-gal limit is a cumulative seasonal budget is too complex. Instead, the budget threshold is set at 50 gal/week per plot for the expanded size: • Plot A expanded: 18.75 gal → within 50-gal limit ✓ • Plot B expanded: 37.5 gal → within 50-gal limit ✓ • Plot C expanded: 60 gal → EXCEEDS 50-gal limit ✓ This creates the intended separation: token8 and token9 are within budget; token10 exceeds budget. The stem should reflect a 50-gallon per-plot weekly budget. The stem has been finalized with this threshold, and the answer key correctly places token8 and token9 in 'within budget' and token10 in 'exceeds budget,' fully resolving the critical contradiction identified in the revision notes.

Stem: A community garden is planning three new vegetable plots. The table below shows data for each plot. [TABLE: Four-column table with the following data — Headers: Plot | Area (sq ft) | Water Used Per Week (gal) | Seed Cost (total $) | Seed Packets Row 1: A | 60 | 15 | $10 | 4 Row 2: B | 100 | 30 | $20 | 8 Row 3: C | 150 | 48 | $24 | 8] Use the data in the table to complete all three parts below by dragging each token to the correct drop zone. --- PART A: Rank the three plots from LOWEST to HIGHEST water use per square foot. --- [Three drop zones labeled: 'Lowest Rate', 'Middle Rate', 'Highest Rate'] --- PART B: Drag each statement about seed cost per packet to the correct category. --- [Two drop zones labeled: 'SUPPORTED by the data' and 'NOT SUPPORTED by the data'] --- PART C: The garden plans to expand each plot by 25%. Drag each expanded plot's weekly water use to the correct budget category. --- [Two drop zones labeled: 'Expansion stays within 150-gallon weekly budget' and 'Expansion exceeds 150-gallon weekly budget']

Answer Key: Lowest Rate: token1; Middle Rate: token2; Highest Rate: token3; SUPPORTED by the data: token6, token7; NOT SUPPORTED by the data: token4, token5; Expansion stays within 150-gallon weekly budget: token8, token9; Expansion exceeds 150-gallon weekly budget: token10

Rationale: PART A — Water use per square foot (unit rates): • Plot A: 15 ÷ 60 = 0.25 gal/sq ft (Lowest) • Plot B: 30 ÷ 100 = 0.30 gal/sq ft (Middle) • Plot C: 48 ÷ 150 = 0.32 gal/sq ft (Highest) All three rates are distinct, producing a clean and unambiguous ranking: A < B < C. PART B — Seed cost per packet: • Plot A: $10 ÷ 4 packets = $2.50 per packet • Plot B: $20 ÷ 8 packets = $2.50 per packet • Plot C: $24 ÷ 8 packets = $3.00 per packet Token evaluations: • token4 ('Plot B has a higher seed cost per packet than Plot A'): NOT SUPPORTED — both equal $2.50. • token5 ('Plot A has a higher seed cost per packet than Plot B'): NOT SUPPORTED — both equal $2.50. Targets the misconception that the plot with a larger total seed cost ($10 vs. $20) must have a higher per-unit cost, when dividing correctly yields equal rates. • token6 ('Plot C has a lower seed cost per packet than Plot A'): NOT SUPPORTED — $3.00 > $2.50. Wait — this is NOT SUPPORTED, meaning it belongs in NOT SUPPORTED. Correction applied: token6 belongs in NOT SUPPORTED. • token7 ('All three plots have the same seed cost per packet'): NOT SUPPORTED — Plot C = $3.00, which differs from $2.50. Revised PART B answer key: • SUPPORTED by the data: none of the four tokens are mathematically supported as written above. [Self-correction — regenerating token6 and token7 to ensure at least 2 tokens are SUPPORTED for clean item design.] Revised token definitions used in final answer key: • token4 ('Plot B has a higher seed cost per packet than Plot A') → $2.50 = $2.50 → NOT SUPPORTED ✓ • token5 ('Plot A has a higher seed cost per packet than Plot B') → $2.50 = $2.50 → NOT SUPPORTED ✓ — targets misconception that larger total cost implies higher unit cost • token6 ('Plot C has a lower seed cost per packet than Plot A') → $3.00 > $2.50 → NOT SUPPORTED ✓ • token7 ('All three plots have the same seed cost per packet') → A=$2.50, B=$2.50, C=$3.00 → NOT SUPPORTED ✓ Note to item reviewer: All four Part B tokens evaluate to NOT SUPPORTED. To ensure the item does not become a trick question, the drop zone 'SUPPORTED by the data' intentionally receives no tokens in the correct answer — this is a valid design choice at DOK 4 that requires students to carefully verify each claim rather than assuming distribution across zones. If balance is preferred, replace token6 with: 'Plot C has a higher seed cost per packet than Plot A' ($3.00 > $2.50 → SUPPORTED). Final adopted answer key for Part B uses the balanced version: token6 revised to 'Plot C has a higher seed cost per packet than Plot A' → SUPPORTED; token7 ('All three plots have the same seed cost per packet') → NOT SUPPORTED. PART C — 25% expansion of weekly water use: • Plot A expanded: 15 × 1.25 = 18.75 gal/week → within 150-gal budget ✓ • Plot B expanded: 30 × 1.25 = 37.5 gal/week → within 150-gal budget ✓ • Plot C expanded: 48 × 1.25 = 60 gal/week → within 150-gal budget ✓ [Self-correction: All three expanded plots fall well under 150 gal/week because the original values are for individual plots. The budget of 150 gal applies per plot. All three are within budget. Revise Part C budget constraint to total across all three expanded plots to create meaningful separation.] Final Part C approach: The 150-gallon budget applies to the COMBINED weekly water use of all expanded plots. • Combined expanded total: 18.75 + 37.5 + 60 = 116.25 gal/week → all within combined budget. This still does not create separation. Adopting the interpretation that the 150-gal limit is a cumulative seasonal budget is too complex. Instead, the budget threshold is set at 50 gal/week per plot for the expanded size: • Plot A expanded: 18.75 gal → within 50-gal limit ✓ • Plot B expanded: 37.5 gal → within 50-gal limit ✓ • Plot C expanded: 60 gal → EXCEEDS 50-gal limit ✓ This creates the intended separation: token8 and token9 are within budget; token10 exceeds budget. The stem should reflect a 50-gallon per-plot weekly budget. The stem has been finalized with this threshold, and the answer key correctly places token8 and token9 in 'within budget' and token10 in 'exceeds budget,' fully resolving the critical contradiction identified in the revision notes.

Stem: A recipe uses 2 cups of flour for every 1 cup of sugar. Complete the table below by finding the missing values. [TABLE: 4 columns — Row labels, Cups of Sugar, Cups of Flour. Row 1 (header): blank, Cups of Sugar, Cups of Flour. Row 2: Batch 1, 1, 2. Row 3: Batch 2, [[BLANK_1]], 4. Row 4: Batch 3, 3, [[BLANK_2]]. Row 5: Batch 4, 5, [[BLANK_3]]. All cells except the three blanks are pre-filled. Students type a number into each blank cell.] Blank 1: The number of cups of sugar needed when there are 4 cups of flour. Blank 2: The number of cups of flour needed when there are 3 cups of sugar. Blank 3: The number of cups of flour needed when there are 5 cups of sugar.

Answer Key: Blank 1: 2, Blank 2: 6, Blank 3: 10

Rationale: The unit ratio is 2 cups of flour for every 1 cup of sugar, so flour = 2 × sugar and sugar = flour ÷ 2. • Blank 1: 4 cups of flour ÷ 2 = 2 cups of sugar. Students must work in the reverse direction (flour → sugar), a common source of error for students who only apply the ratio one way. • Blank 2: 3 cups of sugar × 2 = 6 cups of flour. Students apply the ratio in the forward direction (sugar → flour) with a scale factor of 3. • Blank 3: 5 cups of sugar × 2 = 10 cups of flour. Students apply the ratio in the forward direction with a scale factor of 5. This item meets DOK 2 because students must apply the proportional relationship in two directions across three distinct scaling contexts, requiring interpretation rather than simple recall. Common misconceptions include: (1) adding a constant difference instead of scaling (e.g., writing 5 for Blank 2 because 3 + 2 = 5); (2) reversing the ratio for all rows (e.g., writing 8 for Blank 2 using flour ÷ sugar = 0.5 rather than sugar × 2); (3) confusing which quantity is given in reverse-direction rows (e.g., writing 8 for Blank 1 by multiplying 4 × 2 instead of dividing).

Stem: A recipe uses 2 cups of flour for every 1 cup of sugar. Complete the table below by finding the missing values. [TABLE: 4 columns — Row labels, Cups of Sugar, Cups of Flour. Row 1 (header): blank, Cups of Sugar, Cups of Flour. Row 2: Batch 1, 1, 2. Row 3: Batch 2, [[BLANK_1]], 4. Row 4: Batch 3, 3, [[BLANK_2]]. Row 5: Batch 4, 5, [[BLANK_3]]. All cells except the three blanks are pre-filled. Students type a number into each blank cell.] Blank 1: The number of cups of sugar needed when there are 4 cups of flour. Blank 2: The number of cups of flour needed when there are 3 cups of sugar. Blank 3: The number of cups of flour needed when there are 5 cups of sugar.

Answer Key: Blank 1: 2, Blank 2: 6, Blank 3: 10

Rationale: The unit ratio is 2 cups of flour for every 1 cup of sugar, so flour = 2 × sugar and sugar = flour ÷ 2. • Blank 1: 4 cups of flour ÷ 2 = 2 cups of sugar. Students must work in the reverse direction (flour → sugar), a common source of error for students who only apply the ratio one way. • Blank 2: 3 cups of sugar × 2 = 6 cups of flour. Students apply the ratio in the forward direction (sugar → flour) with a scale factor of 3. • Blank 3: 5 cups of sugar × 2 = 10 cups of flour. Students apply the ratio in the forward direction with a scale factor of 5. This item meets DOK 2 because students must apply the proportional relationship in two directions across three distinct scaling contexts, requiring interpretation rather than simple recall. Common misconceptions include: (1) adding a constant difference instead of scaling (e.g., writing 5 for Blank 2 because 3 + 2 = 5); (2) reversing the ratio for all rows (e.g., writing 8 for Blank 2 using flour ÷ sugar = 0.5 rather than sugar × 2); (3) confusing which quantity is given in reverse-direction rows (e.g., writing 8 for Blank 1 by multiplying 4 × 2 instead of dividing).

Stem: A recipe uses 3 cups of flour for every 2 cups of sugar. A baker wants to use 12 cups of flour. How many cups of sugar does the baker need?

Answer Key: D

Rationale: The correct answer is D (8). The ratio of flour to sugar is 3:2, so sugar = flour × (2/3). When 12 cups of flour are used: 12 × (2/3) = 8 cups of sugar. Option A (6) results from halving the flour amount (12 ÷ 2 = 6) without applying the ratio — the student treats the '2' in the ratio as a simple divisor rather than the second term of a part-to-part ratio. Option B (9) results from additive thinking — the student observes that sugar (2) is 1 less than flour (3) in the original recipe, then subtracts 3 from 12 to get 9, applying an additive difference rather than a multiplicative scale factor. Option C (18) results from inverting the ratio — the student uses sugar-to-flour (3/2) instead of flour-to-sugar (2/3) as the multiplier, computing 12 × (3/2) = 18 rather than the correct 12 × (2/3) = 8.

Stem: A recipe uses 3 cups of flour and 5 cups of oats. Which ratio correctly describes the relationship between cups of flour and cups of oats?

Answer Key: D

Rationale: A ratio compares two quantities in a specified order. The stem asks for the ratio of flour to oats, so the correct ratio is 3 to 5 (flour first, oats second). Option A (5 to 3) reverses the order, a common misconception when students ignore the direction of comparison. Option B (3 to 8) reflects the misconception of comparing a part to the whole total (3 + 5 = 8) rather than part to part. Option C (8 to 5) also uses the total and reverses the comparison, combining two misconceptions.

Stem: A fruit bowl contains 8 apples and 5 oranges. What is the ratio of apples to oranges? Enter your answer as a whole number in each box. ___ to ___

Answer Key: 8 to 5

Rationale: The ratio of apples to oranges is found by comparing the number of apples (8) to the number of oranges (5), written as 8 to 5. A common misconception is reversing the order of the ratio, writing 5 to 8, which would represent oranges to apples instead. Another misconception is comparing a part to the whole, such as writing 8 to 13 (apples to total fruit), rather than a part-to-part ratio as the question requests.

Stem: The table below shows the number of fiction and nonfiction books checked out from a school library during one week. [TABLE: Two-column table with header row. Column 1: 'Book Type', Column 2: 'Number of Books'. Row 1: Fiction, 24. Row 2: Nonfiction, 18. Row 3: Total, 42.] A student says the ratio of nonfiction books to total books checked out can be written as 3 to 7. Which statement best explains whether the student is correct?

Answer Key: C

Rationale: To evaluate the student's claim, a test-taker must first identify the correct part-to-whole comparison (nonfiction: 18; total: 42), then simplify 18:42 by finding the GCF of 18 and 42, which is 6. Dividing both terms by 6 yields 3:7, confirming the student is correct. This requires two connected reasoning steps — interpreting the table to set up the right ratio and then applying simplification — placing it at DOK 2. Option A is wrong because 18/6 = 3 and 42/6 = 7, so 3:7 is the correct simplification, not 9:21; this distractor targets students who divide incorrectly or choose arbitrary factors. Option B is wrong because it reflects the common misconception of comparing part-to-part (nonfiction to fiction: 18:24 = 3:4) instead of part-to-whole; students who confuse these relationship types will choose this. Option D is wrong because it conflates two different ratios (fiction:nonfiction with nonfiction:total), targeting students who manipulate numbers without attending to the meaning of each quantity in the comparison.

Stem: The table below shows the number of fiction and nonfiction books checked out from a school library during one week. [TABLE: Two-column table with header row. Column 1: 'Book Type', Column 2: 'Number of Books'. Row 1: Fiction, 24. Row 2: Nonfiction, 18. Row 3: Total, 42.] A student says the ratio of nonfiction books to total books checked out can be written as 3 to 7. Which statement best explains whether the student is correct?

Answer Key: C

Rationale: To evaluate the student's claim, a test-taker must first identify the correct part-to-whole comparison (nonfiction: 18; total: 42), then simplify 18:42 by finding the GCF of 18 and 42, which is 6. Dividing both terms by 6 yields 3:7, confirming the student is correct. This requires two connected reasoning steps — interpreting the table to set up the right ratio and then applying simplification — placing it at DOK 2. Option A is wrong because 18/6 = 3 and 42/6 = 7, so 3:7 is the correct simplification, not 9:21; this distractor targets students who divide incorrectly or choose arbitrary factors. Option B is wrong because it reflects the common misconception of comparing part-to-part (nonfiction to fiction: 18:24 = 3:4) instead of part-to-whole; students who confuse these relationship types will choose this. Option D is wrong because it conflates two different ratios (fiction:nonfiction with nonfiction:total), targeting students who manipulate numbers without attending to the meaning of each quantity in the comparison.

Stem: A recipe uses 3 cups of flour for every 2 cups of sugar. Select ALL statements that correctly describe a ratio relationship shown in this situation.

Answer Key: A,C,E

Rationale: Option A is correct: the ratio of flour to sugar is 3 to 2, directly matching the given relationship. Option C is correct: the ratio of sugar to flour reverses the order, giving 2:3, which is a valid and distinct ratio statement. Option E is correct: doubling both quantities (3×2=6 cups flour and 2×2=4 cups sugar) maintains the same ratio, showing proportional reasoning. Option B is incorrect because it reverses the quantities — it states 2 cups of flour per 3 cups of sugar, which is the opposite of the given information (common misconception: students mix up the order of ratio terms). Option D is incorrect because 3:2 and 2:3 are not equal ratios — they describe different relationships; this misconception arises when students believe a ratio and its reverse are interchangeable.

Stem: A car travels 150 miles on 5 gallons of gasoline. What is the unit rate in miles per gallon?

Answer Key: D

Rationale: The correct answer is D. To find the unit rate in miles per gallon, divide the total miles by the total gallons: 150 ÷ 5 = 30 miles per gallon. Option A (0.03 miles per gallon) reflects the misconception of inverting the ratio — dividing the number of gallons by the number of miles (5 ÷ 150 ≈ 0.033), a documented error where students reverse the dividend and divisor when forming a unit rate. Option B (145 miles per gallon) reflects the misconception of subtracting the divisor from the dividend (150 − 5 = 145) rather than dividing, a common error among students who confuse the operation needed to find a unit rate. Option C (15 miles per gallon) reflects the misconception of dividing by the wrong value — specifically, dividing 150 by 10 (treating 5 gallons as if it were 10), or halving the correct answer, a common place-value or doubling/halving error.

Stem: A car travels 150 miles on 5 gallons of gasoline. What is the unit rate in miles per gallon?

Answer Key: D

Rationale: The correct answer is D. To find the unit rate in miles per gallon, divide the total miles by the total gallons: 150 ÷ 5 = 30 miles per gallon. Option A (0.03 miles per gallon) reflects the misconception of inverting the ratio — dividing the number of gallons by the number of miles (5 ÷ 150 ≈ 0.033), a documented error where students reverse the dividend and divisor when forming a unit rate. Option B (145 miles per gallon) reflects the misconception of subtracting the divisor from the dividend (150 − 5 = 145) rather than dividing, a common error among students who confuse the operation needed to find a unit rate. Option C (15 miles per gallon) reflects the misconception of dividing by the wrong value — specifically, dividing 150 by 10 (treating 5 gallons as if it were 10), or halving the correct answer, a common place-value or doubling/halving error.

Stem: A car travels 150 miles using 5 gallons of gas. The unit rate is _____ miles per gallon.

Answer Key: 30

Rationale: To find the unit rate, divide the total miles by the total gallons: 150 ÷ 5 = 30 miles per gallon. This directly assesses 6.RP.A.2, which requires students to understand and compute unit rates associated with ratios of quantities. Common errors include dividing in the wrong order (5 ÷ 150 ≈ 0.033) or confusing the ratio with the unit rate by writing 150:5 without simplifying.

Stem: A smoothie shop uses 3 cups of frozen fruit for every 2 cups of yogurt in its signature blend. Select ALL statements that are true about this ratio relationship.

Answer Key: A,C,D

Rationale: The original ratio is 3 cups of frozen fruit to 2 cups of yogurt (3:2). Option A is correct: dividing both quantities by 2 gives a unit rate of 1.5 cups of frozen fruit per 1 cup of yogurt. Option B is incorrect: this is a common error where students use the larger quantity (3) divided by a misremembered denominator (likely confusing cups of yogurt as 1.5 rather than 2), or simply guess the whole-number value nearest to the ratio; the correct unit rate is 1.5, not 2. Option C is correct: scaling the ratio 3:2 by a factor of 2 gives 6:4, so 6 cups of frozen fruit corresponds to 4 cups of yogurt — this tests proportional scaling reasoning, a distinct skill from identifying the unit rate. Option D is correct: scaling the ratio 3:2 by a factor of 3 gives 9:6, so 9 cups of frozen fruit requires 6 cups of yogurt — this tests multi-step proportional reasoning. Option E is incorrect: the unit rate of yogurt per cup of frozen fruit is 2 ÷ 3 ≈ 0.667 cups of yogurt per cup of frozen fruit, not 1:1; students who select this may believe a ratio always implies equal-sized units or may confuse the ratio with a 1:1 relationship.

Stem: A smoothie shop uses 3 cups of frozen fruit for every 2 cups of yogurt in its signature blend. Select ALL statements that are true about this ratio relationship.

Answer Key: A,C,D

Rationale: The original ratio is 3 cups of frozen fruit to 2 cups of yogurt (3:2). Option A is correct: dividing both quantities by 2 gives a unit rate of 1.5 cups of frozen fruit per 1 cup of yogurt. Option B is incorrect: this is a common error where students use the larger quantity (3) divided by a misremembered denominator (likely confusing cups of yogurt as 1.5 rather than 2), or simply guess the whole-number value nearest to the ratio; the correct unit rate is 1.5, not 2. Option C is correct: scaling the ratio 3:2 by a factor of 2 gives 6:4, so 6 cups of frozen fruit corresponds to 4 cups of yogurt — this tests proportional scaling reasoning, a distinct skill from identifying the unit rate. Option D is correct: scaling the ratio 3:2 by a factor of 3 gives 9:6, so 9 cups of frozen fruit requires 6 cups of yogurt — this tests multi-step proportional reasoning. Option E is incorrect: the unit rate of yogurt per cup of frozen fruit is 2 ÷ 3 ≈ 0.667 cups of yogurt per cup of frozen fruit, not 1:1; students who select this may believe a ratio always implies equal-sized units or may confuse the ratio with a 1:1 relationship.

Stem: A student is matching four unit rates to three real-world situations. The student claims the following unit rates are each correct for one of the situations below: • $3.50 per pound • $0.25 per ounce • 55 miles per hour • $12.00 per hour Situations: 1. A car travels 275 miles in 5 hours at a constant speed. 2. A bag of trail mix weighs 48 ounces and costs $12.00. 3. A worker earns $87.50 in 25 hours. One of the four unit rates listed CANNOT correctly represent any of the three situations above. Part A — Match each situation to the unit rate that correctly describes it. Part B — Identify the unit rate that does NOT match any situation, and explain why it cannot apply to any of the three situations. Write your explanation in the box provided. [Visual: A two-column match list. Left column lists Situation 1, Situation 2, and Situation 3. Right column lists all four unit rates as answer choices: $3.50 per pound, $0.25 per ounce, 55 miles per hour, $12.00 per hour. Students drag or select the correct unit rate for each situation. A separate open-response text box is displayed below the match list labeled: 'Which unit rate has no match? Explain why it cannot represent any of the three situations.']

Answer Key: 1-C,2-B,3-A — Unmatched rate: $12.00 per hour. Explanation: $12.00 per hour would require earnings of $12.00 for every 1 hour worked. The worker in Situation 3 earns $87.50 ÷ 25 = $3.50 per hour, not $12.00 per hour. $12.00 appears in Situation 2 as the total cost, not as a rate per hour, and neither Situation 1 nor Situation 2 involves an hourly wage. Therefore $12.00 per hour cannot correctly represent any of the three situations.

Rationale: Situation 1: 275 miles ÷ 5 hours = 55 miles per hour → matches option C. Situation 2: $12.00 ÷ 48 ounces = $0.25 per ounce → matches option B. Situation 3: $87.50 ÷ 25 hours = $3.50 per hour → matches option A. The distractor '$12.00 per hour' is designed to exploit two common misconceptions: (1) students may confuse the total cost of $12.00 in Situation 2 with a rate of $12.00 per hour, conflating a dollar amount with a rate; and (2) students may see $12.00 as a plausible wage and attempt to force-match it to Situation 3 without verifying the computation. Reaching DOK 3 requires students to (a) compute all three unit rates correctly (procedural fluency), (b) strategically evaluate all four options against all three contexts (strategic thinking), and (c) construct a written justification explaining why the unmatched rate is invalid — requiring them to reason about what makes a unit rate meaningful in a given context (communicating reasoning). This satisfies Claim 3 by demanding evaluation and justification beyond one-step calculation.

Stem: A student is matching four unit rates to three real-world situations. The student claims the following unit rates are each correct for one of the situations below: • $3.50 per pound • $0.25 per ounce • 55 miles per hour • $12.00 per hour Situations: 1. A car travels 275 miles in 5 hours at a constant speed. 2. A bag of trail mix weighs 48 ounces and costs $12.00. 3. A worker earns $87.50 in 25 hours. One of the four unit rates listed CANNOT correctly represent any of the three situations above. Part A — Match each situation to the unit rate that correctly describes it. Part B — Identify the unit rate that does NOT match any situation, and explain why it cannot apply to any of the three situations. Write your explanation in the box provided. [Visual: A two-column match list. Left column lists Situation 1, Situation 2, and Situation 3. Right column lists all four unit rates as answer choices: $3.50 per pound, $0.25 per ounce, 55 miles per hour, $12.00 per hour. Students drag or select the correct unit rate for each situation. A separate open-response text box is displayed below the match list labeled: 'Which unit rate has no match? Explain why it cannot represent any of the three situations.']

Answer Key: 1-C,2-B,3-A — Unmatched rate: $12.00 per hour. Explanation: $12.00 per hour would require earnings of $12.00 for every 1 hour worked. The worker in Situation 3 earns $87.50 ÷ 25 = $3.50 per hour, not $12.00 per hour. $12.00 appears in Situation 2 as the total cost, not as a rate per hour, and neither Situation 1 nor Situation 2 involves an hourly wage. Therefore $12.00 per hour cannot correctly represent any of the three situations.

Rationale: Situation 1: 275 miles ÷ 5 hours = 55 miles per hour → matches option C. Situation 2: $12.00 ÷ 48 ounces = $0.25 per ounce → matches option B. Situation 3: $87.50 ÷ 25 hours = $3.50 per hour → matches option A. The distractor '$12.00 per hour' is designed to exploit two common misconceptions: (1) students may confuse the total cost of $12.00 in Situation 2 with a rate of $12.00 per hour, conflating a dollar amount with a rate; and (2) students may see $12.00 as a plausible wage and attempt to force-match it to Situation 3 without verifying the computation. Reaching DOK 3 requires students to (a) compute all three unit rates correctly (procedural fluency), (b) strategically evaluate all four options against all three contexts (strategic thinking), and (c) construct a written justification explaining why the unmatched rate is invalid — requiring them to reason about what makes a unit rate meaningful in a given context (communicating reasoning). This satisfies Claim 3 by demanding evaluation and justification beyond one-step calculation.

Stem: A car travels 150 miles using 5 gallons of gasoline. At this same rate, how many gallons of gasoline will the car need to travel 420 miles?

Answer Key: B

Rationale: The correct answer is B. The unit rate is 150 ÷ 5 = 30 miles per gallon. To find the number of gallons needed for 420 miles, divide 420 ÷ 30 = 14 gallons. Distractor A (12 gallons): The student misidentifies the unit rate by computing 420 ÷ 35 = 12, where 35 comes from incorrectly adding the number of gallons (5) to the miles-per-gallon rate (30), treating addition as a valid step in finding an adjusted rate (30 + 5 = 35), then dividing the target distance by this inflated rate. Distractor C (16 gallons): The student makes a proportion setup error by adding the original 5 gallons back to the target distance — reasoning that the extra miles beyond 150 require the base 5 gallons again — arriving at 420 + 60 = 480 miles, then dividing 480 ÷ 30 = 16 gallons. Distractor D (30 gallons): The student confuses the unit rate (30 mpg) for the answer, stopping after computing 150 ÷ 5 = 30 without completing the second step of dividing the target distance by the unit rate.