Lesson 4

Measuring Volume

Est. Class Sessions: 3–4

Developing the Lesson

Part 1: Measure Volume by Displacement

Introduce Measurement by Displacement with Story. Read the book Mr Archimedes' Bath by Pamela Allen. When Mr. Archimedes takes a bath with three of his friends, Kangaroo, Wombat, and Goat, the water always overflows and makes a mess. Mr. Archimedes is determined to find the culprit. Using a measure and taking turns getting out, they finally discover who it is. This book is an effective springboard for discussion about measurement of volume by displacement. "The Crow and the Pitcher," one of Aesop's fables, can also be used to illustrate the concepts of volume and displacement.

X

The Crow and the Pitcher. "The Crow and the Pitcher," one of Aesop's fables can be used to illustrate the concepts of volume and displacement. This is a very old story of a very thirsty crow. The crow, ready to die of thirst, flew with joy to a pitcher which he saw some distance away. When he came to the pitcher, he found water in it, but so near the bottom that he was not able to drink. He tried to knock over the pitcher so he might at least get a little of the water, but he did not have enough strength for this. At last, seeing some pebbles nearby, he dropped them one by one into the pitcher, so little by little, he raised the water to the very brim and satisfied his thirst.

image

You can ask a student to act out the part of the crow as you tell the story. He or she can drop marbles in the container of water to illustrate how the objects displace the water. There are also many animated versions of the story available. You may choose to show the animated version to better illustrate the action of displacement described in this story.

Show students the items you collected before the lesson and tell them that in this lab they will determine the items' volumes. Remind students that volume is the measure of the space occupied by an object. Students should recall that in the lab Marshmallows and Containers in Unit 5, they found the volume of three containers by filling them. In Unit 10, Addition Properties Using Volume, they found volume by counting cubes. Hold up a train of 8 centimeter connecting cubes as a model.

Explain that the unit of volume in the lab is a cube that is one centimeter on each side. This unit is given the special name 1 cubic centimeter or 1 cc.

Ask:

X
  • What is the volume of this train of connecting cubes? How do you know? (I count the cubes. There are 8 cubes, so it is 8 cubic centimeters.)

Remind students that a cubic centimeter is equivalent to a milliliter, particularly if your graduated cylinders are marked in milliliters. See the discussion in Lesson 2 of this unit.

Hold up the set of objects you have collected and ask:

X
  • How can you find the volume of an object that cannot be filled or that doesn't have cubes to count?
  • How can you find out how much total space this object takes up? Think about the story.

Connect the discussion to the books or stories you used to open the lesson. Explain that students will measure the volume of an object by placing it in a graduated cylinder that contains 80 cc of water.

Show a graduated cylinder filled with 80 cc of water and ask questions similar to the following:

X
  • What happens to the water level in a container when you add something to it? (It will rise.)
  • What will happen to the water level when I put something in this cylinder? (The water level will rise.)
  • How much will it go up? (By how big the object is.)
  • If the water goes up 10 cc, what does that say about the size of the object? (It is 10 cc in size.)
  • How much space does the object take up in the water? (10 cc)
  • Is that its volume? (yes)

Show the train of centimeter connecting cubes again.

X
  • What is the volume of this train? How did you find it? (8 cc; by counting the cubes in the train)

Show students the graduated cylinder you have filled with 80 cc of water and the train of centimeter connecting cubes.

Ask:

X
  • There are 80 cc of water in this cylinder. What will happen to the water if I add this train of 8 connecting cubes? (Possible response: The volume will rise.)
  • What volume will show on the graduated cylinder? (Possible response: 88 cc)
  • How do you know? (Possible response: You started with 80 cc and added 8 cc. The new volume is 88 cc.)
X

Mass vs. Volume. Students often confuse the amount of matter (mass) with the space occupied (volume). Students may assume that a heavy object will occupy more space than a light object. But if students consider a balloon filled with air and a small steel sphere, they will see that sometimes the lesser mass has the greater volume.

X

Students may suggest measuring the object to find volume. Do not dismiss this suggestion since students will learn to determine volume of some objects by measurement in later grades. Rather, focus the discussion on the difficulty of measuring the small, irregular objects you have collected, especially since all the faces would have to be measured. In addition, the measurement of any one face would not give the volume.

Demonstrate Measurement by Displacement. Demonstrate finding the volume of the train of centimeter connecting cubes using a graduated cylinder filled with 80 cc water. Explain that objects must be put into the graduated cylinder very carefully so that no water will splash out. Drop the centimeter connecting cube train gently into the cylinder or tilt the graduated cylinder and let the cube train slide in. The water level rises but the volume of water does not change when the object is placed in the graduated cylinder.

Some objects, such as the links and centimeter connecting cubes, will float to the top. Since the object must be submerged to get an accurate reading of volume, students will have to push the objects under the water to read the volume. Tell them not to push the object under with a finger since a finger has a large volume that will distort the reading. Instead, they should use something with a small volume, such as a pencil tip. Show students how to push the object until it is just under the water.

After inserting the cube train, ask:

X
  • Is there more water in the graduated cylinder now? Why or why not? (No, you didn't put any more water into it.)
  • How much water is still in the cylinder? (80 cc)
  • What happened to the level of the water? (It went higher in the cylinder.)
X

Some students may have a tendency to read the volume of an object as the placement of the top of the object in the graduated cylinder after it sinks to the bottom, rather than reading the meniscus of the water level above it. Point out during demonstration that it is the water level that should be read, not looking at where the top of the object "stops" in the cylinder.

Direct students' attention to the Reading a Graduated Cylinder Tips chart that is on display from Lesson 2.

Then ask:

X
  • Can someone come read what the level says now? (It is 88 cc now.)
  • What was the volume of the water before the object was put into the graduated cylinder? (80 cc)
  • What is the total volume of the water and the train of centimeter connecting cubes? (88 cc)
  • Can you think of a way to find the volume of the connecting cube train alone? (Possible response: We can count up from 80 to 88, or count back from 88 to 80.)
  • What number sentence shows how to find the difference in the water level before and after I put the train of connecting cubes into the graduated cylinder? (Possible responses: 88 − 80 = 8 cc; 80 + = 88 cc)
  • What number sentence shows the sum of the water in the graduated cylinder and the volume of the train of centimeter connecting cubes together or the total volume? (80 + 8 = 88 cc)
  • Was your prediction correct? How do you know? (Yes, we were correct because we started with 80 cc of water and now the cylinder says 88 cc, so the water went up 8 cc to show the volume of the cube train.)
  • The volume of this graduated cylinder is 100 cc. How many trains of 8 connecting cubes are needed to make a total volume greater than 100 cc? Explain your thinking. (Possible response: One train makes a total volume of 88 and two trains, 96 cc. Three trains add too much volume. So, the volume of two trains and 80 cc of water is a little less than 100 cc.)

Demonstrate how to remove the train of cubes by emptying the water from the graduated cylinder into a small cup. Remove the cube train and pour the water back into the cylinder. Explain that you need to check the water level again as it will be a little less than 80 cc; add water with the eyedropper until it reaches 80 cc again.

Guide students in finding the volume of an object that is not easily predicted. Hold up the object you selected for demonstration and place it into the graduated cylinder. Have a student read the meniscus. Have students talk with a partner to determine the volume of the object, then encourage them to explain their strategies and share a number sentence that represents their thinking. Some students may think of the problem as an addition situation and others may think of a subtraction sentence for finding the volume of the object (V). Here are two possible solution strategies for an object with a volume of 4 cc:

  • 80 + = 84 cc; volume is 4 cc because 80 + 4 = 84 cc
  • 84 − 80 = 4 cc

Display the Volume Math Master. Discuss Armando's problem and his solution path. Ask students to share their strategies for solving the problem. Have them write number sentences to show their thinking.

Include prompts similar to the following:

X
  • What is the difference in the water level before and after Armando put the marbles into in the graduated cylinder? What number sentence shows how to find the difference? (3 cc; 68 − 65 = 3 cc)
  • What number sentence shows the sum of the water in the graduated cylinder and the volume of the marbles, or the total volume? (65 + 3 = 68 cc)
Adventure Book:
Student Activity Book:
X
SAB_Mini
+
Master: VM
X
SG_Mini
+
Two strategies for finding the volume of Building A on the City Buildings page
X
+