4.5 Multiplicative Comparison and Measurement

Unit Goals

  • Students interpret, represent, and solve multiplicative comparison problems using an understanding of the relationship between multiplication and division. They use this thinking to convert units of measure within a given system from larger to smaller units.

Section A Goals

  • Analyze, describe, and represent multiplicative comparison situations.
  • Solve one-step and two-step problems involving multiplicative comparison.

Section B Goals

  • Convert from larger units to smaller units within a given system of measurement.
  • Solve multi-step problems involving multiplicative comparison and measurement.
  • Understand the relative sizes of kilometers, meters and centimeters, liters and milliliters, kilograms and grams, and pounds and ounces.

Section C Goals

  • Solve multi-step problems involving multiplicative comparison and measurement.
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Section A: Multiplicative Comparison

Problem 1

Pre-unit

Practicing Standards:  3.OA.A.1

Write a multiplication expression represented by each diagram.

Solution

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Problem 2

Pre-unit

Practicing Standards:  3.OA.A.3

Each bottle contains 2 liters of water. How many liters of water are there in 6 bottles?

Solution

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Problem 3

Pre-unit

Practicing Standards:  4.NF.B.4.b

Find the value of each product. Explain or show your reasoning.

  1. \(5 \times \frac{1}{8}\)

  2. \(5 \times \frac{3}{8}\)

  3. \(4 \times \frac{7}{12}\)

Solution

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Problem 4

Pre-unit

Practicing Standards:  4.NF.B.3.a, 4.NF.B.3.c

Find the value of each expression.

  1. \(\frac{2}{5} + \frac{4}{5}\)
  2. \(1\frac{5}{8} - \frac{4}{8}\)
  3. \(2\frac{1}{6} + 1\frac{5}{6}\)

Solution

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Problem 5

Here are some connecting cubes.

connecting cube tower, 2.

Select all pictures that show 4 times as many connecting cubes.

A:
Connecting Cube Tower. Blue, 2. Orange, 4.
B:
Connecting Cube Tower. 8, blue.
C:
Connecting Cube Tower. Blue, 2. Orange, 2. Yellow, 2. Green, 2
D:
Connecting Cube Tower, 4, blue.
E:
Connecting Cube Tower. Orange, 4. Blue, 2.

Solution

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Problem 6

Han read 4 books during the summer. Priya read 3 times as many books as Han.

  1. Draw a diagram that represents the situation.
  2. Write an equation that represents the situation.

Solution

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Problem 7

Mai exercised 3 times as many hours as Tyler last week. Mai exercised for 15 hours.

  1. Write an equation to represent the situation.
  2. How many hours did Tyler exercise last week?

Solution

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Problem 8

Diego has 8 songs on his playlist. Mai has 3 times as many songs as Diego.

  1. Write an equation that represents the situation.

  2. Draw a diagram that represents the situation.
  3. Explain how the diagram and equation are related.

  4. How many songs are on Mai's playlist?

Solution

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Problem 9

Lin scored 6 points in her basketball game on Saturday. She scored 3 times as many points in her basketball game on Sunday. How many points did Lin score altogether in the two games? Explain or show your reasoning.

Solution

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Problem 10

  1. If diagram A represents 15, what does diagram B represent? Explain your reasoning.

    diagram. two rectangles. Bottom rectangle, B. partitioned into 5 equal parts. Top rectangle, A. Same size as one of the 5 parts of bottom rectangle.
  2. If diagram B represents 100, what does diagram A represent? Explain your reasoning.

Solution

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Problem 11

Exploration

The diagram represents the books that Diego and Lin brought to the book fair.

diagram. two rectangles.

For each question, explain or show your reasoning.

  1. If Lin brought 10 books, how many books did Diego bring?
  2. If Diego brought 21 books to the fair, how many books did Lin bring?
  3. If Lin and Diego brought 60 books altogether, how many did they each bring?

Solution

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Problem 12

Exploration

Write a comparison story problem.

Trade problems with a partner.

  1. Draw a diagram and write an equation to represent the situation.
  2. Solve the story problem.

Solution

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Section B: Measurement Conversion

Problem 1

Select all measurements that are greater than 1 meter.

A:

the width of a paper clip

B:

the length of your math workbook

C:

the width of the classroom

D:

the height of the door in the classroom

E:

the length of a finger

Solution

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Problem 2

  1. The Burj Khalifa is a skyscraper in Dubai. It is 828 meters tall. Is that more or less than 1 kilometer? Explain your reasoning.
  2. An airplane flies 9,500 meters above the ground. Is that more or less than 1 kilometer? Explain your reasoning.

Solution

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Problem 3

  1. A bottle of water holds 250 milliliters.

    • How many of those bottles does it take to make 1 liter?

    • How many liters of water does the bottle hold?

  2. Jada's backpack weighs 3.5 kg. How many grams is Jada's backpack? Explain or show your reasoning.

Solution

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Problem 4

A tuna fish can swim 200 kilometers in a day.

  1. The camel travels 40 kilometers in a day. How many times farther does the tuna fish travel than the camel?

  2. The giant tortoise travels 500 meters in a day. How many kilometers is that?

  3. How many times as far as the giant tortoise does the tuna fish travel?

Solution

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Problem 5

The average weight of a newborn baby is 7 pounds and 8 ounces. How many ounces is that?

Solution

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Problem 6

Diego said it took him 1,800 seconds to get home from school yesterday. Lin thinks he is exaggerating.

  1. It took Diego \(\frac{1}{2}\) hour to get home from school yesterday. How many minutes is that? Explain your reasoning.
  2. How many seconds is that? Explain or show your reasoning.
  3. Was Diego's statement correct or is he exaggerating?

Solution

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Problem 7

Han makes sparkling juice by mixing 1.5 liters of juice and 500 milliliters of sparkling water.

  1. How many milliliters of sparkling juice did Han make?

  2. How many times more juice did Han use than sparkling water?

Solution

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Problem 8

Exploration

Jada is trying to decide which option will take less time to make a trip. Here are the details.

Option 1 (Fly)

  • 30 minutes to get to the airport
  • \(1\frac{1}{2}\) hours waiting at the airport
  • 45 minute flight
  • \(1 \frac{1}{4}\) hours to get to destination

Option 2 (Drive)

  • \(3\frac{3}{4}\) hour drive
  • \(\frac{1}{2}\) hour stop for lunch

Which option takes less total time? Explain your reasoning.

Solution

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Problem 9

Exploration

There are different hikes leaving from the same location. Here are the distances:

Trail A: 800 m

Trail B: 2.5 km

Trail C: 1,300 m

Trail D: 3 km

  1. Andre hiked a total of more than 3 km but less than 4 km. Which trails might Andre have taken?
  2. Jada hiked more than 2 km but less than 3 km. Which trails might Jada have taken?

Solution

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Section C: Let’s Put it to Work

Problem 1

  1. Jada’s class drank 2 gallons of milk at lunch. How many cups is that?

  2. The third graders at Jada’s school drank 3 times as much milk as Jada’s class. How many quarts is that?

Solution

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Problem 2

Diego saw a snake that is 9 inches long at the pet store. The world's longest snake, a python, is 10 yards long. How many times longer is the python than the snake Diego saw? Explain or show your reasoning.

Solution

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Problem 3

  1. Andre drew a rectangle that is 5 centimeters long and 6 centimeters wide. What is the perimeter of Andre’s rectangle?
  2. Jada drew a rectangle whose side lengths are 4 times as long as those of Andre’s rectangle. What is the perimeter of Jada's rectangle?

Solution

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Problem 4

  1. A rectangle has perimeter 19 cm. One side is 4 cm. How long is the other side of the rectangle?

  2. A square has perimeter 19 cm. How long is each side of the square? Explain or show your reasoning.

Solution

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Problem 5

Exploration

A furlong is 220 yards. It is a unit usually used to measure the distance of horse races. A mile is 5,280 feet. Mai read that a furlong is \(\frac{1}{8}\) mile. Is this true? Explain or show your reasoning.

Solution

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Problem 6

Exploration

Many of the length measurements that we use today have interesting histories. Choose one of them, such as a foot, a yard, or a marathon and investigate its origin.

Solution

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