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In this lab, you will have the opportunity to become familiar with

hands-on measurements, some error analysis, and build upon the concepts

covered in this week’s lectures: ratios, volume, and surface area.

**Equipment required for this week’s lab
**

- At least three different sizes of cups, containers, or beakers with circular bases (per working group)
- 27 cubes (dice would work, wooden blocks, etc.) (per working group)
- At least three different rectangular containers
- Ruler
- String
- Graduated cylinders
- Water source
- Scale

**Week 1: General
Measurements and Ratios**

**Invitation
to Inquiry**

If you have popped a batch

of popcorn, you know that a given batch of kernels might pop into big and

fluffy popcorn. But another batch might not be big and fluffy and some of the

kernels might not pop. Popcorn pops because each kernel contains moisture that

vaporizes into steam, expanding rapidly and causing the kernel to explode, or

pop. Here are some questions you might want to consider investigating to find

out more about popcorn: Does the ratio of water to kernel mass influence the

final fluffy size of popped corn? (Hint: Measure mass of kernel before and

after popping). Is there an optimum ratio of water to kernel mass for making

bigger popped kernels? Is the size of the popped kernels influenced by how

rapidly or how slowly you heat the kernels? Can you influence the size of

popped kernels by drying or adding moisture to the unpopped kernels? Is a

different ratio of moisture to kernel mass better for use in a microwave than

in a conventional corn popper? Perhaps you can think of more questions about popcorn.

Summarize your

findings here:

Figure 1.1

**Background**

The purpose of this introductory laboratory

exercise is to investigate how measurement data are simplified in order to

generalize and identify trends in the data. Data concerning two quantities will

be compared as a **ratio**, which is generally defined as a relationship

between numbers or quantities. A ratio is usually simplified by dividing one

number by another.

**Procedure**

**Part A: Circles and
Proportionality Constants**

1. Obtain

three different sizes of cups, containers, or beakers with circular bases.

Trace around the bottoms to make three large but different-sized circles on a

blank sheet of paper.

Figure 1.2

2. Mark

the diameter on each circle by drawing a straight line across the center.

Measure each diameter in mm and record the measurements in Data Table 1.1.

Repeat this procedure for each circle for a total of three trials.

3. Measure

the circumference of each object by carefully positioning a length of string

around the object’s base, then grasping the place where the string ends meet.

Measure the length in mm and record the measurements for each circle in Data

Table 1.1. Repeat the procedure for each circle for a total of three trials.

Find the ratio of the circumference of each circle to its diameter. Record the

ratio for each trial in Data Table 1.1.

4. The ratio of

the circumference of a circle to its diameter is known as **pi** (symbol?), which has a value of 3.14… (the periods mean

many decimal places). Average all the values of ? in Data Table 1.1 and

calculate the experimental error.

**Part B: Area and
Volume Ratios**

1. Obtain

one cube from the supply of same-sized cubes in the laboratory. Note that a

cube has six sides, or six units of surface area. The side of a cube is also

called a *face*, so each cube has six identical faces with the same area.

The overall surface area of a cube can be found by measuring the length and width

of one face (which should have the same value) and then multiplying

(length)(width)(number of faces). Use a metric ruler to measure the cube, then

calculate the overall surface area and record your finding for this small cube

in Data Table 1.2.

2. The

volume of a cube can be found by multiplying the (length)(width)(height).

Measure and calculate the volume of the cube and record your finding for this

small cube in Data Table 1.2.

3. Calculate the

ratio of surface area to volume and record it in Data Table 1.2.

4. Build

a medium-sized cube from eight of the small cubes stacked into one solid cube.

Find and record (a) the overall surface area, (b) the volume, and (c) the

overall surface area to volume ratio, and record them in Data Table 1.2.

5.

Build a large cube from 27 of the

small cubes stacked into one solid cube. Again, find and record the overall

surface area, volume, and overall surface area to volume ratio and record your

findings in Data Table 1.2.

6. Describe

a pattern, or generalization, concerning the volume of a cube and its surface

area to volume ratio. For example, as the volume of a cube increases, what

happens to the surface area to volume ratio? How do these two quantities change

together for larger and larger cubes?

**Part C: Mass and
Volume Ratios**

1. Obtain

at least three straight-sided, rectangular containers. Measure the length,

width, and height *inside*the container (you do not want the container

material included in the volume). Record thesemeasurements in Data

Table 1.3, in rows 1, 2, and 3. Calculate and record the volume of each

container in row 4 of the data table.

Width

Length

Height

Figure 1.3

2. Measure

and record the mass of each container in row 5 of the data table. Measure and

record the mass of each container when “level full” of tap water. Record each

mass in row 6 of the data table. Calculate and record the mass of the water in

each container (mass of container plus water minus mass of empty container, or

row 6 minus row 5 for each container). Record the mass of the water in row 7 of

the data table.

Measure

the volume here

Figure 1.4

3. Use

a graduated cylinder to measure the volume of water in each of the three

containers. Be sure to get *all* the water into the graduated cylinder.

Record the water volume of each container in milliliters (mL) in row 8 of the

data table.

4. Calculate

the ratio of cubic centimeters (cm^{3})

to mL for each container by dividing the volume in cubic centimeters (row 4

data) by the volume in milliliters (row 8 data). Record your findings in the

data table.

5.

Calculate the ratio of mass per

unit volume for each container by dividing the mass in grams (row 7 data) by

the volume in milliliters (row 8 data). Record your results in the data table.

6. Make

a graph of the mass in grams (row 7 data) and the volume in milliliters (row 8

data) to picture the mass per unit volume ratio found in step 5. Put the volume

on the *x*-axis (horizontal axis) and the mass on the *y*-axis (the

vertical axis). The mass and volume data from each container will be a data

point, so there will be a total of three data points.

7. Draw

a straight line on your graph that is as close as possible to the three data

points and the origin (0, 0) as a fourth point. If you wonder why (0, 0) is

also a data point, ask yourself about the mass of a zero volume of water!

8.

Calculate the slope of your graph. The slope of a line is the change in rise

divided by the change in run.

Mathematically slope=

9. Calculate your

experimental error. Use 1.0 g/mL (grams per milliliter) as the accepted value.

10.

Density is defined as mass per

unit volume, or mass/volume. The slope of a straight line is also a ratio,

defined as the ratio of the change in the *y*-value per the change in the *x*-value.

Discuss why the volume data was placed on the *x*-axis and mass on the *y*-axis

and not vice versa.

11.

Was the purpose of this lab

accomplished? Why or why not? (Your answer to this question should show

thoughtful analysis and careful, thorough thinking.)

**Results**

1. What is a ratio?

Give several examples of ratios in everyday use.

2. How is the value of? obtained? Why does?not have units?

3. Describe

what happens to the surface area to volume ratio for larger and larger cubes.

Predict if this pattern would also be observed for other geometric shapes such

as a sphere. Explain the reasoning behind your prediction.

4. Why does

crushed ice melt faster than the same amount of ice in a single block?

5. Which

contains more potato skins: 10 pounds of small potatoes or 10 pounds of large

potatoes? Explain the reasoning behind your answer in terms of this laboratory

investigation.

6. Using your own words, explain the meaning of

the slope of a straight-line graph. What does it tell you about the two graphed

quantities?

7. Explain

why a slope of mass/volume of a particular substance also identifies the

density of that substance.

**Problems**

An aluminum block that is 1 m× 2 m× 3 m has a mass of 1.62× 10^{4}

kilograms (kg). The following problems concern this aluminum block:

2 m

One

face

1 m

3 m

Figure 1.5

l. What is the

volume of the block in cubic meters (m^{3})?

2. What are the

dimensions of the block in centimeters (cm)?

3. Make a sketch

of the aluminum block and show the area of each face in square centimeters (cm^{2}).

4. What is the

volume of the block expressed in cubic centimeters (cm^{3})?

5. What is the

mass of the block expressed in grams (g)?

6.

What is the ratio of mass (g) to

volume (cm^{3}) for

aluminum?

7. Under

what topic would you look in the index of a reference book to check your answer

to question 6? Explain.