How Much Space Do Objects Occupy?

7th Grade Mathematics

QAR

SQRQCQ
During Reading
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After Reading

Text: Middle School Math. (2002) Upper Saddle River, NJ: Prentice Hall. p. 476-479.

Before Reading

Objectives

1. The students will be able to define volume and state the formula for finding the volume of a rectangular prism.

2. The students will be able to successfully solve word problems regarding volume using the SQRQCQ strategy.

3. The students will be able to name at least one real-world application where they would need to solve for volume of an object.

I. Motivation

Teacher: "In the first part of the chapter we discussed the area of various sorts of polygons. Can somebody tell me the basic formula for the area of a rectangle? Right, the formula is length times width. So, we all know that area is two-dimensional because it does not have any height. But we also know that we do not live in a two-dimensional world. Every object around us is in three dimensions with some sort of space that it occupies. In the last few sections of the chapter we need to figure out information about objects in three dimensions and how much space they occupy. Does anybody know what new concept is that includes all three dimensions? That concept is volume. What is volume exactly, well in the next couple of days we are going to be looking at some of the characteristics of volume.

II. Prior Knowledge

Teacher: "In the first part of the chapter, we learned how to find the area of something like a farm field. So if I know that a field is 300 yards long and 400 yards wide, what is the area of that field? Right, the answer is 120,000 square yards. So we know the formula to find the area of something that is a rectangle is A= length * width. We also learned some other formulas to figure the area of other objects. Can anybody tell me some of these formulas and we will make a list on the board."

Area of a Parallelogram: length*width

Area of a Triangle: 1/2 * base * height

Area of a Circle: Pi * radius^2

Teacher: "So now we need to take what we know about the area of objects and use what we know to figure out how to find the volume of certain objects."

III. Purpose

Teacher: "Volume is what occupies the world around us. Can you imagine a world in only two dimensions? We need a third dimension to define everything that takes space around us. The purpose of this lesson is so you can understand how to determine the special characteristics of volume. By reading the information in the chapter and working through some word problems regarding volume, you should have a better understanding of the world in three dimensions."

During Reading

IV. Text Structure

Teacher: "This text structure is expository, with both enumeration/simple listing and problem and solution. The first part of the section tells some specific characteristics of volume. The later part of the section has word problems that have a solution that can be found by finding the volume. A good strategy to use to solve word problems and critically understand them is the SQRQCQ strategy that we have used previously in the year."

Teacher Note: At the bottom of the page there is a link to information about SQRQCQ.

V. Vocabulary and Formulas

Teacher: "Before we start to solve problems using volume, lets find a definition for volume. The book lays down a concrete definition for volume.

Volume is the number of cubic units in a solid.

"Now let's break down that definition. Like we said earlier, volume is the amount of space a certain object takes up. Can anybody tell me why that would be in cubic units? Right, because volume is found in three dimensions, the units we use must be cubed. Remember when we did area, the units we used were squared units. Now that we have added another dimension, we must use cubed units. In this first section we are dealing with rectangular solids that we want to find the volume. Can anybody find the formula for the volume of a rectangular solid?"

Volume of a Rectangular Solid = B (area of the base) * h (height of the solid)

"So now we know what the definition is for volume and a basic formula we can use to solve for the volume of a rectangular solid object. We can now go on to the word problems to see how much space some objects take up."

VI. Strategy Instruction

Teacher: "The strategy we will be using today to solve the word problems in the examples in the chapter is the SQRQCQ method that we have used before to solve word problems. For those of us who do not quite remember what to do, here are the basic steps to the SQRQCQ strategy.

1. Survey -- Read the problem rapidly, skimming to determine its nature.

2. Question - Decide what is being asked of you, or what the problem really wants you to do.

3. Read - Read the problem for details and interrelationships.

4. Question - Decide which processes and strategies should be used to address the problem.

5. Compute - Carry out the necessary computations.

6. Question - Ask whether the answer seems correct. Check computations against the facts presented in the problem.

Modeling

Teacher: "Now that we have refreshed ourselves on the steps of SQRQCQ, I will model how I want you to use the method to help solve word problems regarding volume. If you still have questions once I have finished the problem using SQRQCQ, please make sure to raise your hands and ask. We will be looking at Example 1 on page 477 of your textbooks.

Problem: A next-day delivery service delivers a package with dimensions 12 in. by 7 in. by 30 in. What is the volume of the package?

Survey: The box has certain dimensions, 12 x 7 x 30 inches and we are told to find the volume of this package.

Question: We need to find out what the volume is of a package with those certain dimensions. So we must find the area of the base of the package, as well as the height.

Read: The relevant information is the dimensions. 12 x 7 x 30. Also that the units of the box are in inches.

Question: We need to find the area of the base of the box as well as the height of the box. Since the problem did not specifically give that information, we can use any of the dimensions for the area of the base and the height. Let's say that the area of the box is 12 x 30 inches with a height of 7 inches.

Compute: First, we must find the area of the box. So we take 12 x 30 inches and get 360 square inches for the area of the base. Then we multiply that area by the height of the box because the formula for V = B * h. So 360 square inches x 7 inches tall. We get that the volume of the box is 2520 inches cubed.

Question: Does that answer make sense? Well we could think if we could fit 2520 cubic-inch cubes into a box that size. That is a lot of cubes, but because the box is so long at 30 inches, it would make sense that the volume of the box is 2520 inches cubed.

Guided Practice

Teacher: "Now to make sure we all know how to correctly use this strategy, we are going to complete one more example as a class. Let's see if we can use SQRQCQ to solve Example 2 on page 477. Would someone like to read the problem."

Find the height of a rectangular solid with volume 72 cubed centimeters and base dimensions 6 cm by 3 cm.

Teacher Note: Have a different student walk through each step and find the necessary information. Write the steps and process on the chalkboard. Below is some concepts that should fall under each step.

Survey: We know the volume of the box and the dimensions of the base.

Question: We need to find out the height of the rectangular object given the volume and base dimensions.

Read: Total volume is 72 cubic cm with a base that has dimensions 6 cm by 3 cm.

Question: We need to find the area of the base first and then find the height by dividing the area from the total volume.

Compute: 6 cm by 3 cm is an are of 18 cm squared. We then divide 72 cm cubed/ 18 cm squared to get a height of 4 inches

Question: This makes sense because 18 can be rounded to 20 and 20 * 4 is 80, a little more than 72.

VII. Content

Teacher: "Complete Example 3 on page 478 using the SQRQCQ problem solving method. Also, solve the following problem that could apply to your lives at some point in time."

Problem: "You are planning a trip to Florida over your spring break this year. You are planning on flying down to Florida so you do not have to make the long drive. Your airline says that the maximum dimensions of a package anybody can take on the plane is 30 inches long by 14 inches wide and 7 inches deep. If your family has 5 members that can each take one suitcase that meets the airline requirements for luggage dimensions, how many cubic inches do you have available that your family can use to pack in?"

After Reading

VIII. Questioning

Teacher Note: The questioning method being used is Question-Answer-Relationships (QAR). The students must answer the question and then also explain where it came from.

Teacher: "Okay class, now that we have completed the lesson on volume of rectangular solid objects, let's make sure we fully understand the concepts discussed today.

Question

Answer

What is the equation we can use to find the volume of a rectangular solid?

Right There on page 476

 Volume = Area of Base * Height

How can SQRQCQ help to solve word problems regarding volume?

On your own

SQRQCQ helps students organize the information in the problem and find a solution that makes sense given the problem.

What kind of life situation would require finding the volume of a rectangular object?

On your Own

 When shipping a package when cost is based on the volume of the package.

IX. Writing to Learn Activity

Teacher Note: The students should be using examples done as models in class as well as the guided practice and content problems as references in their notes. Students should write these examples as notes of how to complete the volume problems using SQRQCQ. The students will then have a compiled list of problems as well as any definitions or formulas used in this section.

X. Closure

Teacher: "Your assignment for tomorrow will be to complete 5-12, and 14. I would like you complete problems 11, 12, and 14 using SQRQCQ. So today we learned about some characteristics of volume in rectangular solids. In the next few days, we are going to be looking at volume in shapes that are non-rectangular, as well as how to use scaling in regards to volume. We need to take our understanding of the principle of volume and relate it to non-rectangular objects in the next few sections. The formulas that we are using may change, but the principle of volume will not change throughout. In this chapter you will gain a deeper understanding of a world in three dimensions."

 
SQRQCQ Reading Strategy
 
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