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Estimating Volume by Counting on Frank

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In this lesson, students read the book Counting on Frank. They use information in the book to make estimates involving volume. In particular, students explore the size of humpback whales.

Story Summary  

The narrator likes to collect facts with the help of his dog, Frank.  Each two-page spread of this book includes a different fact involving such mathematical topics as counting, size comparison, and ratio along with delightful illustrations.

Structuring the Investigation 

Ask students to brainstorm what they already know about humpback whales. If they do not know a great deal about this topic, have a few website handy for students to read.

Next, ask students to predict the size of a humpback whale. Students should explain their estimates.

837 whale

Read the book, Counting on Frank, by Rod Clement aloud. Divide the class into small, cooperative working groups. Provide each student with a copy of the Getting The Facts activity sheet.

pdficonGetting The Facts Activity Sheet 

Ask students to calculate how large a box is needed to hold the average humpback whale. Because the data they have collected will most likely include weight and length, students will have to make inferences about several dimensions of the box. These decisions should be justified in questions 1 and 2 of the Getting The Facts activity sheet.

After estimating the size of the boy's house, encourage each group to describe the process they used to find the answer. Discuss the information needed to determine the number of whales that would fit inside the house.

Direct each group to complete the remainder of the activity sheet.


Assessment Options

You may wish to judge the reasonableness of the students' estimates. For example, based upon several internet resources, most humpbacks have an average length of 40-45 feet. The image shown suggests that the whale is about 4-5 times as long as it is tall, so 8-10 feet seems a fairly accurate estimate for the whale's width. And if the diver shown in that picture is assumed to be 6 feet tall, about 1.5 to 2 of him would fit in the height of the whale, so the height of a whale could be 9-12 feet. Once again, these are estimates only.


  1. Ask students to determine how many humpback whales would fit in their own house, including justifications for their estimations. This could be given as a family task.
  2. Move on to the next lesson, How Big is a Foot?

Questions for Students
1. Can you tell me how you are finding the size of the box?  What is the important information that you need? 

[Answers will vary.  Look for evidence that students know that they need to have information on the length, width, and height of the whale to create the box. Listen for mathematical arguments that justify the size and dimensions of the box.]

2. Do you think a humpback whale is larger or smaller than your car?  Your school?  Your house?

[Answers will vary.  Look for evidence that students are using mathematics to compare the two objects: the humpback whale's estimated dimensions and the estimated dimensions of common objects.]

Teacher Reflection 

  • As students research humpback whales, do they make casual comparisons or estimations such as "Wow! That's bigger than my house!" or "That's smaller than a brontosaurus!"
  • How do students resolve the different shapes of a humpback whale and a house?  The humpback whale is long, while students may live in a square or tall house.  Do they recognize that the volume can be the same and the shapes very different?
  • How can you provide an experience for students that makes the large size of the humpback whale real to them?

Shapes and Poetry

Students read the poem "Shapes" from A Light in the Attic, by Shel Silverstein, and create their own illustration of the poem. In this lesson, students explore geometric figures and positional words.

How Big Is a Foot?

In this lesson, students read the book How Big Is a Foot?, by Rolf Myller. They then create non-standard units (using their own footprints) and use them to make "beds." As a result, students explore the need for a standard unit of measure.

Fill 'Er Up

In this lesson, students use an interactive applet to investigate the formula for the volume of a rectangular prism. Students will construct two origami boxes and use centimeter cubes to measure and compare the volume of the boxes. Students will also analyze how changing the dimensions of the prism affects its volume.

Learning Objectives

Students will:

  • Predict and find the average size of humpback whales.
  • Solve problems involving estimation of volume.

NCTM Standards and Expectations

  • Understand that measurements are approximations and how differences in units affect precision.
  • Develop strategies for estimating the perimeters, areas, and volumes of irregular shapes.

Common Core State Standards – Mathematics

Grade 3, Measurement & Data

  • CCSS.Math.Content.3.MD.A.2
    Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem.

Common Core State Standards – Practice

  • CCSS.Math.Practice.MP1
    Make sense of problems and persevere in solving them.
  • CCSS.Math.Practice.MP4
    Model with mathematics.
  • CCSS.Math.Practice.MP5
    Use appropriate tools strategically.
  • CCSS.Math.Practice.MP7
    Look for and make use of structure.