Putting It All Together Extension Packet
Grade 4 • Unit 9
PUTTING IT ALL TOGETHER
Creative Math Inventor & Problem Solving Extensions
Inventor:
Date:
Welcome to your Math Inventor's Log! These extension quests are open-ended challenges designed to push your mathematical creativity. You will design, estimate, and build models using fractions and whole numbers. No special tools required—just your brilliant brain!
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Lesson 1: The Master Weaver's Ribbon Challenge
You have ribbon pieces with lengths of \( \frac{1}{4} \) yard, \( \frac{1}{2} \) yard, \( \frac{3}{4} \) yard, and \( \frac{5}{8} \) yard. Design two different ways to combine these pieces to make a total of exactly \( 2\frac{1}{2} \) yards. You may use each starting length at most twice in each design.
Design A (List pieces & write equation)
Design B (List pieces & write equation)
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Lesson 2: The Roller Coaster Support Blueprint
You are building a mini roller coaster model using exactly one 3-foot paper straw with no waste. You must cut it into three support pillars (Pillars A, B, and C):
• Pillar A must be between \( 1\frac{1}{4} \) feet and \( 1\frac{1}{2} \) feet tall.
• Pillar B must be exactly \( \frac{5}{12} \) of a foot shorter than Pillar A.
• Pillar C is the remaining length of the straw.
Find valid lengths for all three pillars. Prove your math works.
Your Calculations & Final Pillar Lengths (A, B, and C)
Imagine IM Grade 4 • Unit 9 Extensions Page 1
Unit 9 Extension Packet Section A & Section B
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Lesson 3: The Ultimate Relay Design Challenge
Diego wants to design a three-stage playground relay course. The total target completion time for the course is exactly \( 45\frac{5}{10} \) seconds.
• Stage 1 (Hurdles) takes \( 12\frac{75}{100} \) seconds.
• Stage 2 (Balance Beam) takes \( 18\frac{4}{10} \) seconds.
Write and solve an equation to find how many seconds are left for Stage 3 (Sack Race). Then, explain how the final time changes if the team cuts their Stage 2 time exactly in half!
Write your equation, solve for Stage 3, and write your explanation:
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Lesson 4: The Zero-Defeater Algorithm Trick
Priya claims: "To subtract any multi-digit number from \( 100,000 \), I can just subtract each of its digits from 9, and subtract the very last non-zero digit from 10! It completely avoids all borrowing and regrouping!"
Your Task: Test Priya's claim with the expression \( 100,000 - 34,728 \). Show both the standard subtraction algorithm and Priya's shortcut. Explain mathematically *why* her trick works.
Method A: Standard Algorithm
100,000
- 34,728
_________
Method B: Priya's Digits Shortcut
(9 - 3), (9 - 4), (9 - 7), (9 - 2), (10 - 8)
Why does this shortcut work?
Imagine IM Grade 4 • Unit 9 Extensions Page 2
Unit 9 Extension Packet Section B: Whole-Number Operations
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Lesson 5: The Digit Shuffle Challenge
Using the digits 2, 4, 6, and 8 exactly once each:
1. Create a 3-digit by 1-digit multiplication problem that gives the largest possible product. Solve it.
2. Create a 2-digit by 2-digit multiplication problem using those same digits to find the smallest possible product. Solve it.
1. Largest 3-digit x 1-digit Product Layout & Calculation:
2. Smallest 2-digit x 2-digit Product Layout & Calculation:
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Lesson 6: The Grid Quotient Designer
An engineer has exactly 4,352 solar cells to arrange into a massive rectangular grid power plant.
• If the layout must have exactly 8 rows, use the partial quotients method to find how many columns wide it will be.
• If they instead want to design a grid with a remainder of exactly 4 spare solar cells left over for backup, what could the number of rows be? Find two possibilities.
Case 1: Exactly 8 Rows (Show Quotient Work)
Case 2: Remainder of 4 (Find 2 row dimensions)
Imagine IM Grade 4 • Unit 9 Extensions Page 3
Unit 9 Extension Packet Section C: Multiplication & Division Problems
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Lesson 7: The Tech Hub Travel Budget
An engineer is comparing costs in three cities: Cyber-Town, Techno-City, and Silicon-Bay.
• A monthly train pass in Cyber-Town costs $45.
• A pass in Techno-City is 4 times as expensive as in Cyber-Town.
• A pass in Silicon-Bay is 3 times as expensive as in Techno-City.
Determine the cost in Silicon-Bay. If an engineer has a travel allowance of $1,000, could they buy 2 monthly passes in each city? Explain.
Your Equations & Allocation Reasoning:
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Lesson 8: The Fleet Rover Optimization
A science mission must transport exactly 360 research scientists to base camp. They have two sizes of transport rovers:
• Rover Alpha: holds exactly 15 passengers and costs $120 per trip.
• Rover Beta: holds exactly 40 passengers and costs $300 per trip.
Find two different combinations of rovers that transport exactly 360 scientists with zero empty seats. Which combination costs less? Prove it!
Combination A (Show passenger & cost math)
Combination B (Show passenger & cost math)
Imagine IM Grade 4 • Unit 9 Extensions Page 4
Unit 9 Extension Packet Sections C & D
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Lesson 9: Reverse Engineering the Algorithm
An automated math app generated the following sequence of operations to solve a multi-step word problem:
1. \( 125 \times 12 = 1,500 \)
2. \( 1,500 - 340 = 1,160 \)
3. \( 1,160 \div 8 = 145 \)
Write a creative, real-world story problem about an amusement park, a concert, or a school event that matches these steps.
Write your story problem and explain what the final answer "145" represents:
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Lesson 10: The Dripping Faucet Waste Estimate
A leaky school faucet drips exactly 15 milliliters of water every minute.
• Estimate about how many liters are wasted in one day. (Note: \( 1\text{ liter} = 1,000\text{ mL} \)).
• If the school has 8 identical dripping faucets, estimate the total water wasted during a school year of 180 days. Provide a value that is *too low*, *about right*, and *too high*.
1. Daily Leak Estimate (Show math steps)
2. 8 Faucets for 180 Days (Too Low / About Right / Too High)
Imagine IM Grade 4 • Unit 9 Extensions Page 5
Unit 9 Extension Packet Section D: Creation & Design
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Lesson 11: The Expression Alchemist
Analyze these four expressions. Determine the exact value of each:
A: \( (8 + 4) \times (10 - 6) \) | B: \( 12 \times 4 \) | C: \( 24 + 24 \) | D: \( 96 \div 2 \)
Write two different "Which Three Go Together" arguments. For each, tell which letter is left out and explain the mathematical reason why the other three belong together.
Argument 1: (Left Out: ___) Explain why the other three belong together:
Argument 2: (Left Out: ___) Explain why the other three belong together:
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Lesson 12: Design a Decimal Number Talk
Create a sequence of 4 expressions starting with \( 4 \times 0.25 \) that acts as "stepping stones" to help classmates solve \( 16 \times 2.75 \) mentally. Write a brief note explaining the strategy connection.
Stepping Stone Expressions:
1. 4 × 0.25 = ______
2. _________________
3. _________________
4. 16 × 2.75 = ______
Your strategy guide for classmates:
Imagine IM Grade 4 • Unit 9 Extensions Page 6
Teacher Resource
EXTENSION ANSWER KEY
Lessons 1 to 6
Grading & Pedagogy Guidance: These extension challenges are designed to evaluate deep mathematical reasoning. Student work should prioritize logic, labeled diagrams/equations, and precision over identical formatting.
Lesson 1: Ribbon Challenge
Target length is exactly \( 2\frac{1}{2} \) yards. Sample solutions:
• Design A: \( \frac{3}{4} + \frac{3}{4} + \frac{1}{2} + \frac{1}{2} = 2\frac{1}{2} \) yards.
• Design B: \( \frac{5}{8} + \frac{5}{8} + \frac{3}{4} + \frac{1}{2} = \frac{10}{8} + \frac{6}{8} + \frac{4}{8} = 2\frac{1}{2} \).
Lesson 2: Support Blueprint
Pillars must total 3 ft. If Pillar A = \( 1\frac{1}{3} \) ft (\( \frac{16}{12} \)):
• Pillar B: \( 1\frac{4}{12} - \frac{5}{12} = \frac{11}{12} \) ft.
• Pillar C: \( 3 - (\frac{16}{12} + \frac{11}{12}) = \frac{9}{12} \) ft (\( \frac{3}{4} \) ft).
Other combinations are correct if Pillar A is between \(1\frac{1}{4}\) and \(1\frac{1}{2}\) and total sum is 3.
Lesson 3: Relay Design
Equation: \( 12\frac{75}{100} + 18\frac{40}{100} + x = 45\frac{50}{100} \)
• Stage 3 Time: \( x = 14\frac{35}{100} \) seconds.
• Cutting Stage 2 in half reduces it to \( 9.2 \) seconds, reducing total target time to \( 36.3 \) seconds.
Lesson 4: Zero-Defeater Algorithm
Correct Difference: 65,272.
• Priya's shortcut works because \( 100,000 - n \) is mathematically identical to \( (99,999 - n) + 1 \). Subtraction from \( 99,999 \) requires zero borrowing.
Lesson 5: Digit Shuffle
Using exactly 2, 4, 6, and 8:
• Largest Product: \( 642 \times 8 = 5,136 \).
• Smallest Product: \( 26 \times 48 = 1,248 \).
Lesson 6: Grid Quotient Designer
• Case 1 (8 Rows): \( 4,352 \div 8 = 544 \) columns.
• Case 2 (Remainder of 4): Must find a divisor of \( 4,348 \) larger than 4. Possibilities include 6 rows (gives 724 columns) or 12 rows (gives 362 columns).
Imagine IM Grade 4 • Unit 9 Answer Key Page 7
Teacher Resource
EXTENSION ANSWER KEY
Lessons 7 to 12
Grading & Pedagogy Guidance: These creative tasks emphasize standard algorithms, real-world context application, modeling, and strategic computational estimation.
Lesson 7: Tech Hub Travel Budget
• Cyber-Town: \( \$45 \)
• Techno-City: \( 4 \times 45 = \$180 \)
• Silicon-Bay: \( 3 \times 180 = \$540 \)
• Total cost for 2 passes in each city: \( 2 \times (45 + 180 + 540) = \$1,530 \).
This is not affordable since \( \$1,530 > \$1,000 \) budget limit.
Lesson 8: Fleet Rover Optimization
Target is exactly 360 scientists:
• Combo 1: 24 Rover Alphas. Cost: \( 24 \times 120 = \$2,880 \).
• Combo 2: 9 Rover Betas. Cost: \( 9 \times 300 = \$2,700 \).
• Combo 3: 8 Alphas & 6 Betas. Cost: \( 960 + 1800 = \$2,760 \).
Rover Beta Only (Combo 2) is the most economical.
Lesson 9: Reverse Engineering
Stories must follow the given steps:
1. Total seats: \( 125 \times 12 = 1,500 \)
2. Remaining seats: \( 1,500 - 340 = 1,160 \)
3. Divided equally: \( 1,160 \div 8 = 145 \).
Students should clearly explain that the final quotient 145 represents equal portions of the leftover items.
Lesson 10: Dripping Faucet
• Daily Leak: \( 15\text{ mL} \times 1,440\text{ minutes} = 21,600\text{ mL} = 21.6\text{ liters} \).
• Yearly Waste: \( 21.6 \times 8 \times 180 = 31,104\text{ liters} \).
• Estimates: Too low is ~15,000 L, too high is ~50,000 L, about right is ~31,000 L.
Lesson 11: Expression Alchemist
All expressions evaluate to 48.
• Argument 1 (Exclude A): A is the only multi-step parenthetical expression.
• Argument 2 (Exclude C): C is purely additive, whereas the others are multiplicative/divisive in nature.
Lesson 12: Number Talk
Stepping stones should build logically:
1. \( 4 \times 0.25 = 1 \)
2. \( 16 \times 0.25 = 4 \)
3. \( 16 \times 2 = 32 \)
4. \( 16 \times 2.75 = 32 + 12 = 44 \).
Decomposing decimal multipliers is a powerful mental strategy.
Imagine IM Grade 4 • Unit 9 Answer Key Page 8
