T 12 California Topics
Topic 5: Dimensional Analysis
for use after
Comparing and Scaling
Investigation 3
To convert a measurement from one unit to another, you can use a
conversion factor. A
is a rate equal to 1. For example,
12 in. = 1 ft, so you can use the rate
to convert feet to inches.
A. 1. Use the conversion factor
to convert 100 feet to inches.
2. Use a conversion factor to convert 100 inches to feet.
B. 1. What conversion factor can you use to change seconds to minutes?
2. What conversion factor can you use to change minutes to
seconds?
C. Which unit belongs in the denominator of the conversion factor, the
given measurement or the resulting measurement?
is a method of checking the units that result from
using conversion factors. You can use dimensional analysis to check
whether your methods and answers are reasonable.
A. 1. Use conversion factors for hours to minutes and minutes to seconds
to write a rate for converting hours to seconds.
2. Write a conversion factor for changing seconds to hours.
B. 1. You want to convert 1,000 seconds to hours. Which method below is
correct?
1,000 s 3
3
1,000 s 3
3
= 3,600,000 h
= 0.28 h
2. You want to convert 240 miles per second to miles per hour. Which
method below is correct?
240
3
3
240
3
3
= 864,000 mi/h
= 0.67 mi/h
1 h
60 min
1 min
60 s
mi
s
60 min
1 h
60 s
1 min
mi
s
1 h
60 min
1 min
60 s
60 min
1 h
60 s
1 min
Dimensional analysis
12 in.
1 ft
12 in.
1 ft
conversion factor
Exercises
1. The table shows equivalent measurements.
a. Write a conversion factor for changing meters
to feet.
b. Write a conversion factor for changing feet
to meters.
c. How many feet equal 100 meters?
d. How would you find a conversion factor for
changing square meters to square feet?
For Exercises 2–3 below, do parts (a) and (b).
a. Use a conversion factor to solve the problem.
b. Use dimensional analysis to check your answer.
2. Change 432 square inches to square feet.
3. Change 2,232 minutes to days.
4. You bike for 45 minutes at a rate of 10 mi/h. You turn around and
return by the same route. Your return trip takes 30 minutes. What was
your average speed over the entire trip?
5.
Density
is a unit rate. It is the mass of a
substance per unit volume. The table gives
data for the masses and volumes of four
metal samples.
a. Which metal has the greatest density?
b. Convert the density of copper from
8,930 kilograms per cubic meter to
grams per cubic centimeter. Use
dimensional analysis to check that
your answer is reasonable.
c. Which sample below shows the correct first
step for converting the density of titanium
to grams per cubic centimeter?
1. 4,500
3
333
2. 4,500
3
333
d. Write the density of titanium in grams per cubic centimeter.
100 cm
1 m
100 cm
1 m
100 cm
1 m
1 kg
1,000 g
kg
m
3
1 m
100 cm
1 m
100 cm
1 m
100 cm
1,000 g
1 kg
kg
m
3
Copper
Gold
Silver
Titanium
8,930
9,660
20,980
4,500
Metal
Mass
(kilograms)
1
0.5
2
1
Volume (cubic
centimeters)
Valuable Metals
1
2
3
4
3.28
6.56
9.84
13.12
Length
in Meters
Length
in Feet
Measurements
Topic 5
for use with
Comparing and Scaling T 13
Topic 5
for use with
Dimensional Analysis
33
Topic 5: Dimensional Analysis
Teaching Guide
Mathematical Goals
• Use conversion factors to convert units
• Use dimensional analysis to check units for reasonableness
Vocabulary
•
conversion factor
•
dimensional analysis
At a Glance
Students may have trouble deciding whether to multiply or divide when
converting between units. In Topic 5, students will learn to use dimensional
analysis to check the units of a converted quantity. Students can avoid
writing unreasonable answers by writing out all conversion factors and
canceling units correctly.
After Problem 5.1, ask:
•
When you convert from a smaller unit to a larger unit, will the numerical
result be greater than or less than the original measure?
•
When you convert from a larger unit to a smaller unit, will the numerical
result be greater than or less than the original measure?
•
Is it easier to think of a length as 5 feet or 60 inches?
Summarize Problem 5.2A by asking:
•
How do you know that the rate you found for converting hours to
seconds is a conversion factor?
•
How can you use the rate you found for converting hours to seconds to
write a conversion factor for changing seconds to hours without writing
the conversion factors for hours to minutes and minutes to seconds?
After Problem 5.2B, ask:
•
How does dimensional analysis help you decide which conversion
factors to use?
•
How can you keep track of which units remain after you multiply by a
conversion factor?
Homework Check
When reviewing Exercise 1, ask:
•
How can you use the table to write conversion factors for changing
meters to inches and inches to meters?
After reviewing Exercises 2–4, ask:
•
Can you write a conversion factor to change square inches to feet?
Why or why not?
•
Can you write a conversion factor to change from minutes to feet?
Why or why not?
PACING
1 day
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California Implementation Guide
Assignment Guide for Topic 5
Core
1–5
Answers to Topic 5
Problem 5.1
A. 1.
1,200 inches
2.
ft
B. 1.
2.
C.
the given measurement
Problem 5.2
A. 1.
2.
B. 1.
the second method, 0.28 h
2.
the first method, 864,000 mi/h
Exercises
1. a.
b.
c.
328 ft
d.
Square the conversion factor for changing
meters to feet.
2. a.
3 ft
2
b.
Check students’ work. Sample:
432 in.
2
3
3
3. a.
1.55 days
b.
Check students’ work. Sample:
2,232 min 3
3
4.
12 mi/h
5. a.
gold
b.
8.93 g/cm
3
c.
1
d.
4.5 g/cm
3
1 day
24 h
1 h
60 min
1 ft
12 in.
1 ft
12 in.
1 m
3.28 ft
3.28 ft
1 m
1 h
3,600 s
3,600 s
1 h
60 s
1 min
1 min
60 s
8
1
3
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