Const. C:
[ 50 ]
Const. B:
[ 8 blocks of 50 = 400 ]
Const. A:
[ 4 blocks of 400 = 1,600 ]
Lesson 5: The Toy Store Venture Answer: $84 left over
• Game = \$24. Rocket = \(2 \times \$24 = \$48\). Combined = \(24 + 48 = \$72\).
• Elena spent 3 times that combined amount last year: \(3 \times \$72 = \$216\).
• Savings remaining from \$300: \(300 - 216 = \$84\).
300 - 3 × (24 + (24 × 2)) = 84
Lesson 6: The Multi-Digit Grid Lock Constraint: Place Value Shifts
• Value of '5' in X is \(10 \times\) value of '5' in Y. ('5' in X is one place value left of '5' in Y).
• Value of '3' in Y is \(100 \times\) value of '3' in X. ('3' in Y is two place values left of '3' in X).
• Number X: 8,530 → Value of 5 = 500, value of 3 = 30.
• Number Y: 3,850 → Value of 5 = 50, value of 3 = 3,000.
Verify: 500 is 10 times 50; 3,000 is 100 times 30. Correct!
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Lesson 7: The Ribbon Maze Equation: 2R + G = 350 cm
Mathematical Proof: Total length = \(4\text{ m} = 400\text{ cm}\). Blue (\(B\)) is Red (\(R\)) + 50. Total formula: \(R + (R + 50) + G = 400 \implies 2R + G = 350\text{ cm}\).
Three Valid Combinations (Red, Blue, Green in cm):
Combo 1
Red: 100 cm
Blue: 150 cm
Green: 150 cm
Combo 2
Red: 50 cm
Blue: 100 cm
Green: 250 cm
Combo 3
Red: 120 cm
Blue: 170 cm
Green: 110 cm
Lesson 8: The Forest Hike Answer: Jada = 4,550 meters
Step-by-Step Calculations:
• Lin's distance: \(2.5\text{ km} = 2,500\text{ m}\).
• Diego's distance: \(2,500\text{ m} + 450\text{ m} = 2,950\text{ m}\).
• Andre's distance: \(2 \times 2,500\text{ m} = 5,000\text{ m}\).
• Combined three friends: \(2,500 + 2,950 + 5,000 = 10,450\text{ m}\).
• Total Target: \(15\text{ km} = 15,000\text{ m}\).
• Jada's distance: \(15,000\text{ m} - 10,450\text{ m} = 4,550\text{ m}\) (or 4.55 km).
Lesson 9: The Science Lab Puzzler Equation: 2T + 5J = 40
Mathematical Proof: Total capacity = \(2\text{ L} = 2,000\text{ mL}\). Test tubes = 100 mL (\(T\)), Jars = 250 mL (\(J\)). Equation: \(100T + 250J = 2,000 \implies 2T + 5J = 40\). \(J\) must be an even number to ensure a whole-number solution for \(T\).
Three Valid Options:
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Lesson 10: The Animal Relay Answer: 530 m (53/100 of race)
Step-by-Step Math: Total race = \(1\text{ km} = 1,000\text{ m}\).
• Snail distance = \(12,000\text{ cm} = 120\text{ m}\).
• Sloth distance = \(350\text{ m}\). Snail + Sloth = \(120 + 350 = 470\text{ m}\).
• Tortoise remaining = \(1,000 - 470 = 530\text{ m}\).
• Fraction of total race = \(\frac{530}{1000} = \frac{53}{100}\) of the race.
Lesson 11: The Baker's Balance Answer: 8 loaves (0 oz left)
Step-by-Step Math:
• 1 loaf = \(1\text{ lb } 4\text{ oz} = 16\text{ oz} + 4\text{ oz} = 20\text{ oz}\).
• Flour bag = \(10\text{ lbs} = 10 \times 16\text{ oz} = 160\text{ oz}\).
• Whole loaves = \(160 \div 20 = 8\) loaves exactly. Leftover = 0 ounces.
Lesson 12: Space Launch Pacing Answer: 175 minutes (LESS)
Step-by-Step Math:
• Phase 1: \(1\frac{1}{2}\text{ hours} = 90\text{ minutes}\).
• Phase 2: \(45\text{ minutes}\).
• Phase 3: \(2,400\text{ seconds} = 2,400 \div 60 = 40\text{ minutes}\).
• Total = \(90 + 45 + 40 = 175\text{ minutes}\) (or 2 hr 55 min).
• Seconds verify: \(175 \times 60 = 10,500\text{ seconds}\). Since \(10,500 < 15,000\), it is LESS.
Lesson 13: The Secret Trail Mixture Answer: 48 oz added
Step-by-Step Math:
• Peanuts: \(\frac{3}{4}\text{ lb} = 12\text{ oz}\). Raisins = \(12\text{ oz}\). Chocolate Chips: \(\frac{1}{2}\text{ lb} = 8\text{ oz}\).
• Combined current weight = \(12 + 12 + 8 = 32\text{ oz} = 2\text{ lbs}\).
• Target = \(5\text{ lbs} = 80\text{ oz}\). Extra ounces needed: \(80 - 32 = 48\text{ ounces}\).
Sample mix: 24 oz almonds and 24 oz dried cranberries.
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L14: Picnic Feast
• Lemonade: \(3\text{ gal} = 48\text{ cups}\). \(48 \div 4 = 12\) jugs.
• Clay: \(5 \times 24\text{ oz} = 120\text{ oz}\). \(120 \div 16 = 7.5\text{ lbs}\) (or \(7\frac{1}{2}\text{ lbs}\)).
L15: Giant Disc Throw
• Tyler: \(15\text{ yd} = 45\text{ ft}\).
• Elena: \(4 \times 45 = 180\text{ ft}\).
• Andre: \(180 + 45 = 225\text{ ft}\).
• Ratio: \(225 \div 45 = 5\) times as far.
Lesson 16: The Fence Designer
• Rectangle A: \(24 = 2(8 + W) \implies W = 4\text{ in}\).
• Rectangle B (scaled \(\times 3\)): Length = 24 in, Width = 12 in. Perimeter = \(2(24 + 12) = 72\text{ in}\).
• Ratio: \(72 \div 24 = 3\) times greater. This is always true because scaling a shape scales its perimeter proportionally by the exact same factor: \(P_B = 2(3L + 3W) = 3 \times P_A\).
Lesson 17: The Sandbox Masterpiece
• Two equal sides: \(8\frac{1}{2} + 8\frac{1}{2} = 17\text{ feet}\).
• Remaining four sides: \(40 - 17 = 23\text{ feet}\).
• Since they are equal: \(23 \div 4 = 5.75\text{ feet}\) or \(5\frac{3}{4}\text{ feet}\) (or 5 feet 9 inches) per side.
Lesson 18: The Ultimate Mythbuster
• Statement 3 is the LIE. Kangaroo jump is \(9\text{ m} = 900\text{ cm}\). Tortoise crawl is \(300\text{ cm}\). The true ratio is \(900 \div 300 = 3\) times farther, NOT 30 times.
• Statement 1 is TRUE (\(250\text{ lbs} = 4,000\text{ oz}\)). Statement 2 is TRUE (\(2,700\text{ kg} = 9 \times 300\text{ kg}\)).
Open-Ended Response Grading Rubric
Mastery (4 pts) All conversions are accurate; equations are mathematically correct; student reasoning is thorough and complete.
Developing (2-3 pts) Minor math error or conversion mistake exists, but the logical structure and comparative reasoning are sound.
Beginning (0-1 pt) Work is incomplete, fails to model comparisons, or shows major unit-conversion misconceptions.
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