Multiplicative Comparison Student Packet
5th Grade Math Week 10 • CCSS.Math.Content.4.OA.A.2
“Great mathematicians are made, not born.”
Multiplicative Comparison Word Problems
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Day 1: Model | Understanding Multiplicative Comparison with Tape Diagrams
“Times as Many”
Core Math Definitions
Additive Comparison:
Asks how much more or less one quantity is than another (\(+\) or \(-\)).
Multiplicative Comparison:
Asks how many times as many or times as much one is than another (\(\times\) or \(\div\)).
Factor / Multiplier:
The number showing how many times a base quantity is multiplied.
Analyze the Model: Maya & Leo's Stickers
Maya has \(6\) stickers. Leo has \(3\) times as many stickers as Maya. How many stickers does Leo have?
Maya:
6
1 unit (\(6\) stickers)
Leo:
6
6
6
3 equal units (\(3 \times 6\))
Equation:
\(L = 3 \times 6 = 18\)
Answer: 18 stickers
Unknown Factor Scenario: Ribbon Lengths
A red ribbon is \(28\) inches long. A blue ribbon is \(7\) inches long. How many times as long is the red ribbon as the blue ribbon?
1. Write equation with unknown multiplier (\(m\)): \(7 \times m = 28 \quad \text{or} \quad 28 \div 7 = m\)
2. Solve for \(m\): \(m = 4\)
Answer: \(4\) times as long
How can drawing a tape diagram help you decide whether to multiply or divide when solving a comparison word problem?
Key Takeaway:
Day 1
Independent Practice
Multiplicative Comparison • Finding Unknowns
Problem 1 Pages Read
Sam read \(8\) pages of a book on Monday. On Tuesday, he read \(4\) times as many pages as he did on Monday. How many pages did Sam read on Tuesday?
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Problem 2 Tree & Sapling
A large tree is \(36\) feet tall. A small sapling is \(6\) feet tall. How many times as tall is the tree as the sapling?
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Problem 3 Savings
Sophia saved \(\$9\). Her older brother saved \(5\) times as much money as Sophia. How much money did her brother save?
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Problem 4 Dog & Puppy
A dog weighs \(45\) pounds. A puppy weighs \(9\) pounds. How many times heavier is the dog than the puppy?
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Problem 5 Apple Trees
An apple orchard has \(7\) green apple trees. The orchard has \(6\) times as many red apple trees as green apple trees. How many red apple trees are in the orchard?
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Problem 6 Milk Cartons
A cafeteria served \(64\) cartons of chocolate milk and \(8\) cartons of strawberry milk. How many times as many cartons of chocolate milk were served compared to strawberry milk?
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Problem 7 High Jump
Ben jumped \(12\) inches off the ground. A gymnast jumped \(3\) times as high as Ben. How high did the gymnast jump?
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Problem 8 Electronics Cost
A laptop costs \(\$400\), which is \(8\) times as much as a tablet keyboard. How much does the tablet keyboard cost?
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Detach or Keep with Packet
Day 1 Exit Ticket
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1. Bakery Muffins:
A bakery sold \(9\) blueberry muffins on Saturday. They sold \(6\) times as many corn muffins as blueberry muffins. How many corn muffins did the bakery sell?
Tape Diagram / Work:
Answer:
2. Pool Lengths:
A swimming pool is \(50\) meters long. A paddling pool is \(10\) meters long. How many times as long is the swimming pool as the paddling pool?
Equation / Work:
Answer:
Day 2: Model 5th Grade Math • 4.OA.A.2
Finding the Unknown Smaller Quantity
Solving for Unknown Smaller Quantities
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When to Divide: When a word problem gives us the larger total quantity and the comparison factor, but asks for the smaller unknown quantity, we divide or solve an equation with a missing factor (\(\text{total} \div \text{factor} = \text{base unit}\)).
Model Problem: The Bicycle Sales
Division Scenario
A store sold \(35\) bicycles in May. That is \(5\) times as many bicycles as they sold in January. How many bicycles did the store sell in January?
Step 1: Identify the Comparison Statement
\(\text{May Bicycles} = 5 \times \text{January Bicycles}\) (May is the large known amount; January is the smaller unknown).
Step 2: Set Up the Tape Diagram
May (\(35\) total):
?
?
?
?
?
= 35
January (?):
?
1 unit (smaller base amount)
Step 3: Write & Solve the Equation
\(5 \times J = 35 \quad \implies \quad J = 35 \div 5 = 7\)
Answer: \(7\) bicycles in January
Key Takeaways: Multiplication vs. Division Comparisons
Day 2
Independent Practice
Solving for Unknown Smaller Quantities
Problem 1 Movie Tickets
A movie theater sold \(48\) adult tickets. That was \(6\) times as many tickets as child tickets sold. How many child tickets were sold?
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Problem 2 Basketball Points
Ethan scored \(32\) points in a basketball game. He scored \(4\) times as many points as Noah. How many points did Noah score?
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Problem 3 Harvest Weight
A farmer harvested \(72\) kilograms of potatoes. This was \(8\) times the weight of the carrots harvested. How many kilograms of carrots were harvested?
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Problem 4 Fundraiser
A school club raised \(\$54\) selling bake sale items. That is \(9\) times as much money as they raised selling lemonade. How much money did they raise selling lemonade?
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Problem 5 Sports Balls
A sports store has \(60\) soccer balls on display. This is \(5\) times as many as the number of basketballs on display. How many basketballs are on display?
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Problem 6 Table Prices
An oak table costs \(\$210\). That is \(3\) times as much as a plastic foldable table. How much does the plastic table cost?
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Problem 7 Juice Pitcher
A pitcher holds \(42\) ounces of juice, which is \(7\) times as much juice as a glass holds. How many ounces of juice does a glass hold?
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Problem 8 Favorite Food
In a school survey, \(81\) students selected pizza as their favorite food. That is \(9\) times as many students as those who selected salad. How many students selected salad?
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Detach or Keep with Packet
Day 2 Exit Ticket
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1. Coat & Hat Cost:
A coat costs \(\$72\), which is \(8\) times as much as a hat. How much does the hat cost?
Equation / Work:
Answer:
2. Library Books:
A library has \(40\) non-fiction books on a display shelf. This is \(4\) times as many non-fiction books as fiction books on the shelf. How many fiction books are on the shelf?
Equation / Work:
Answer:
Day 3: Model Critical Analysis & Precision
Operation Traps: Multiply or Divide?
Error Analysis: Choosing the Correct Operation
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Case File: Jackson's Seashell Problem
Analyze the Mistake
The Word Problem:
“Maya collected \(24\) seashells. This is \(4\) times as many seashells as her brother Caleb collected. How many seashells did Caleb collect?”
Jackson's Student Work (SW):
24 × 4 = 96
“Caleb collected 96 seashells.”
Reality Check:
Does it make sense for Caleb to have \(96\) if Maya's \(24\) is \(4\) times as many as Caleb's? No! Caleb must have fewer than Maya!
What was the error and how do we fix it?
1. Explain Jackson's Error:
2. Provide the Correct Work & Solution:
Word Problem Operation Guide
✔ Multiply when: You know the small base quantity and want to find the larger quantity (\(\text{small} \times \text{factor} = \text{large}\)).
✔ Divide when: You know the larger quantity and need to find the smaller base quantity (\(\text{large} \div \text{factor} = \text{small}\)).
Key Takeaway for Day 3:
Day 3
Independent Practice
Mixed Word Problems & Error Analysis
Problem 1 Bakery Flour
A recipe requires \(3\) cups of flour. A bakery recipe requires \(7\) times as much flour. How many cups of flour does the bakery recipe require?
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Problem 2 Water Container
A container holds \(30\) liters of water, which is \(6\) times as much water as a bucket holds. How many liters of water does the bucket hold?
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Problem 3: Error Analysis Lila's Work
Marcus ran \(18\) miles this week. That is \(3\) times as far as he ran last week. How many miles did Marcus run last week?
SW: 18 × 3 = 54 miles
What is the error in Lila's work?
How should she fix it?
Correct:
Problem 4 Book & Video Game
A book costs \(\$15\). A video game costs \(4\) times as much as the book. How much does the video game cost?
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Problem 5 Smartphones
An electronics store sold \(56\) smartphones today. That is \(7\) times as many smartphones as smartwatches sold today. How many smartwatches were sold?
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Problem 6 Grid Lines
A red line on a grid is \(27\) units long. A blue line is \(3\) units long. How many times as long is the red line as the blue line?
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Problem 7 Bus & Van
A school bus can carry \(48\) passengers. A van can carry \(8\) passengers. How many times as many passengers can the school bus carry compared to the van?
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Problem 8 Harvest Baskets
An orchard harvested \(63\) baskets of peaches. That is \(9\) times as many baskets as the plums harvested. How many baskets of plums were harvested?
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Detach or Keep with Packet
Day 3 Exit Ticket
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1. Car & Airplane Speed:
A car travels \(45\) miles per hour. An airplane travels \(9\) times as fast as the car. How fast does the airplane travel?
Equation / Work:
Answer:
2. Daniel's Jacket Error:
Jacket costs \(\$63\), which is \(7\) times as much as a scarf.
SW: 63 × 7 = 441. The scarf costs $441.
What is Daniel's error & how to fix?
Correct:
