1.
166
Cycle 2
6. When trying to find the solution to the problem 6 , 3, we can think “? * 3 = 6?”
Let’s apply this idea to the following problems. Rewrite each division problem as a
multiplication problem and then find the solution. Verify your answer by using your
calculator. For the answer that results in an error on the calculator, write “undefined.”
a. 0 , 6 6 * ? = 0 ? = 0
b. 6 , 0 0 * ? = 6 undefined
c. Try a few more similar problems in which 0 is divided by a number or a number is
divided by zero. What conclusions can be drawn?
We cannot divide by zero. Zero divided by any number except zero is zero.
7. For each problem, state positive, negative, or it depends. If you state it depends, give
two examples to support your answer.
a. Positive + Positive Positive
b. Positive + Negative depends Possible examples: 3 + (-7) = -4; 7 + (-3) = 4
It
c. Negative + Negative Negative
d. Positive - Negative Positive
e. Positive * Positive Positive
f. Positive * Negative Negative
g. Negative - Positive Negative
8. Suppose your friend reads the following instructions to you from a book of math tricks:
Pick a number.
Subtract 1.
Multiply by 3.
Add 6.
Divide by 3.
Subtract the original number.
She then correctly guesses your final number is 1. You repeat this and arrive at the same
result.
Try this once with a negative number and once with a fraction. Do you get 1 each
time as an answer? Write out your calculation steps.
Yes. The result is always 1, even if you start with a negative number or a fraction.
45–50 min
Explore
5–10 min
Instructor note: Instruct
students to work in groups on the
Explore. Discuss responses.
Instructor note: If you are using them, distribute the counters or
chips that students can use in the
following problem while groups are
working on the Explore.
ALMY8454_01_AIE_C02.indd 166
2.5
An Ounce of Prevention
1. A student is taking a math class that has five exams, each worth 100 points. After four
exams, her average in the class is 80%. She tells her friend, “If I get a 100 on the last exam,
my grade will be an A. 80 and 100 average to be a 90, which is an A.” Her friend replies,
“No, it’s not even possible for you to get an A in the class.” Who is right and why?
The friend is correct. Currently, the student has 80% of 400 points, or 320 points. To earn
an A in the class, she needs 90% of 500 points, or 450 points. She would need to earn
130 points on the last exam, which is not possible.
5/31/13 5:12 AM
2.
LESSON 2.5 An Ounce of Prevention
Discover
5 min
Instructor
note: Discuss the
Look It Up.
look
it up
167
Most grades are found by calculating an average or mean. Let’s learn more about this measure of a data set.
Mean
The mean or average of a set of numbers is one measure of the data’s center or middle.
To find the mean of a data set, add the data values and divide by the number of values.
For example, John buys dinner at the same restaurant three times in one month with
bills of $24, $38, and $40. His average meal price is:
$24 + $38 + $40
3
= $34>meal.
This means if he had spent the same total on three meals, $102, but paid the same
amount each time, he would have spent $34 on each meal.
45
38
40
35
30
40
34
34
24
25
24
20
38 40
15
10
5
0
Instructor note: Mention that the amount in the bars above the 34 line would fill the gap above
the first bar.
In the second picture, we can think of the mean as the balancing point (or fulcrum) of a teeter-totter.
20 min
Instructor note: Students will
complete #2–6 in groups. Walk
around to keep groups progressing.
2. Two months into taking a biology course, you have taken three quizzes with these
scores (out of 10 points each): 8, 8, 5. Your friend has the same quiz average as you,
but she earned the same score on each of her quizzes. What was her score each time?
If your friend scored the same on each quiz with the same total, 21, her score was 7 each
time.
3. Let’s visualize the average of these quizzes. Draw squares, stacked vertically, to represent counters. Each stack should represent the points on one of the quizzes: 8, 8, 5.
Now, earrange the stacks into three even stacks. How many counters are in each
r
stack? How could you have predicted that number?
7; This is the average, the number of counters in each stack so the stacks have the same
height.
Start:
End:
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3.
168
Cycle 2
4. Suppose on the fourth quiz, you score 3 points out of 10. Compute your new average.
How does this low score affect your grade? Why is that?
6; It lowers the average. Since you scored below your old average of 7 on the last quiz, it
brought down your average. Since the total did not increase by 7 and the total has to be
d
istributed into four columns, the average will be lower than before.
Start:
End:
Remember?
5. How many points lower is your average after quiz #4 compared to after quiz #3? What
is the percent change in your average as a result of the last score?
To compute the percent
change, find the amount of
change and divide it by the
original value.
An average of 7 dropping to 6 is one point;
1
≈ 14%
7
6. Unhappy with your current average, you decide that you want to raise it to a C (7 out
of 10). If the fifth quiz is also worth 10 points, what will you need to score on it to
bring your average back to the C range? To find the answer, use the counters to model
the first four quiz scores. Explain how to use the counters to answer the question.
If you make stacks for 8, 8, 5, and 3, how high would a new stack need to be so you could
make them all at least 7 counters high? You would take one off the first stack, one off the second stack, and add them both to the third stack. But you would need four more to add onto
the fourth stack and still need 7 for the last stack. So you would need 11 counters.
Start:
Middle:
End:
Check your answer by averaging the five scores.
The scores 8, 8, 5, 3, and 11 average to 7.
Is this possible? If the instructor offered a bonus point on the quiz, would it be
p
ossible? Is it likely?
Not possible on a 10-point quiz. If there was a bonus point, then it could happen. It is
u
nlikely since the student never scored that high on the earlier material in the course.
ALMY8454_01_AIE_C02.indd 168
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4.
LESSON 2.5 An Ounce of Prevention
Instructor note: Set
up the balance for demonstration. After all the quiz
scores are plotted physically,
students should notice that
the average is the balancing
point.
5 min
If you do have not have access to a
balancing model (options described
in the accompanying instructor
page), complete #7 by drawing a
teeter-totter similar to the one in the
Look It Up. 7 will be the fulcrum.
169
7. Use a number line to represent a balance. Let 7 represent the balancing point. Plot
your original three quiz scores: 8, 8, 5. What is the average in this context?
The balancing point
Now plot the fourth quiz score, 3. What will you need to score on the fifth quiz to
bring the average back to 7?
11
Instructor note: Ask students whether the balance model eminds them of anything a teeter-totter or seesaw). Ask what they did to
r
(
balance it if there was a lot of weight on one side. Go out very far on the other side.
Another example involves shutting a door. Try to close a door by pushing next to the hinge with one finger—it’s difficult, if not impossible. Now
push with one finger near the handle—it’s very easy to close the door. The reason has to do with the distance compounding with the force to
create the desired leverage.
Connect
5 min
8. Suppose you have 60% of 100 possible points in a course so far. If you get 90 points
out of 100 points on the next test, what is your average in the course?
75%
Instructor note: Discuss these
problems as a class.
9. Suppose you have 60% of 900 possible points in a course so far. If you get 90 points
out of 100 points on the next test, what is your average in the course?
63%
5 min
Instructor note: Explain
the contents of the Wrap-Up.
Have tudents write an answer to
s
the ycle question, “Why does it
c
m
atter?” A prompt is provided to
help students. Discuss homework.
Wrap-Up
lesson
Reflect
What’s the point?
Means are one measure of a data set’s center. They are often used when calculating
grades. Understanding the math behind means can help you succeed in your classes.
What did you learn?
How to find the mean of a data set
How to use means in applied problems
Cycle 2 Question: Why does it matter?
What does this expression mean? “An ounce of prevention is worth a pound of cure.”
It is often easier and simpler to prevent a problem than to deal with the consequences
later.
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5.
170
Cycle 2
2.5 Homework
Skills
MyMathLab
• Find the mean of a data set.
• Use means in applied problems.
1. Find the mean of this data set: -2, 7, 8, 4, -1, -10, 1
1
2. A student has a class with four tests, each worth 100 points, and has earned these
scores on the first three: 75, 74, 71. What is her average now? What must she earn on
the fourth test to get a B (80) in the class?
Her current average is 73.3. She must earn a 100 on the last test to have a B average.
Concepts and Applications
• Use means in applied problems.
3. Suppose you have 3 quizzes and want an average of 7 points. List at least five ways this
average can be accomplished. Keep in mind that each score can be no more than
10 points.
Any 3 scores, between 0 and 10, that add to 21 will work.
4. One of the take-away points of the lesson is that some things are difficult to undo.
Several things fall into the category of being easy to do but hard to undo. List two
examples from real life and two examples from math.
Easy to gain weight, hard to lose it. Easy to get in debt, hard to get out of it.
Addition is usually easier than subtraction. Multiplication is easier than division.
5. Suppose you have dinner with four friends, and the total check (with tax and tip) is
$155. How much should you each pay if you split the check evenly? What does this
amount represent?
You would each pay $31. This amount represents the average of your individual bills.
6. Consider the following two sets of incomes. Each income is in thousands of dollars
per year.
Group 1: 32, 36, 38, 39, 42, 43, 44, 47, 49, 50
Group 2: 32, 36, 38, 39, 42, 43, 44, 47, 49, 150
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6.
LESSON 2.5 An Ounce of Prevention
171
a. Find the average or mean salary for each list.
Group 1: $42,000/year; Group 2: $52,000/year
b. Make a conjecture about what happens to the mean of a data set when the data
set includes an extreme value.
The mean is affected by an extreme value.
c. Do you think the mean of the second set is a good measure of the center of data
for the salaries? Explain.
No. When there is an extreme value, the mean is not always a good measure of the
center of the data.
7. . Create a data set with 5 different values that have a mean of 50.
a
Answers will vary.
b. Add 10 to each of your 5 data values. List the new values.
Answers will vary.
c. Find the new mean.
Answers will vary.
d. What do you notice?
By adding 10 to each score, the mean also increases by 10.
e. If a teacher wanted to increase a class test average of 62 to 70, what could he or she
do to each student’s test score to achieve that?
Add 8 points to each student’s test score.
8. . A student scores a 50 and 100 on two tests in a class. What is his average?
a
75
b. Another student scores a 75 and 75 on two tests in a class. What is her average?
75
c. The mean gives us information about a data set’s center. What does it not describe?
It does not tell whether the data is close to the mean or spread out.
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7.
172
Cycle 2
9. . Find the mean of this data set: 7, 13, 6, 14, 5, 15
a
10
b. If you were to draw a number line and label all the points in the data set as well as
the mean, where should the mean fall? Do this to confirm your conjecture.
10 is the balancing point if we think of the number line as a teeter-totter.
10. Looking forward, looking back For each statement, incorrect units have been used.
Correct the statement and explain why it was wrong.
a. The average height of the males in our class is 69 square inches.
It should be 69 inches. Height is a linear measurement, not an area.
b. The average square footage of homes in this town is 1,500 feet.
It should be 1,500 square feet. Square footage is a measurement of area, which uses
square units.
65–70 min
Explore
5–10 min
2.6
Measure Up
Here are some commonly used geometric formulas.
A = lw
Parallelogram
A = bh
Triangle
A =
1
2 bh
Trapezoid
A =
1
2 h (B
Circle
d = 2r, A = πr 2, C = 2πr = πd
Cylinder
V = πr 2h, S = 2πr 2 + 2πrh
Cone
V =
1
2
3 πr h
Sphere
V =
4
3
3 πr ,
Rectangular Prism
Instructor note: Instruct
students to work in groups on the
Explore.
Rectangle
V = lwh
+ b)
S = 4πr 2
1. There are many variables in these formulas. List what each variable represents.
A area
V volume
S surface area
B longer base of a trapezoid
l length
ALMY8454_01_AIE_C02.indd 172
w width
b base
r radius
d diameter
h height
5/31/13 5:12 AM
8.
LESSON 2.6 Measure Up
173
2. Ana is making brownies and debating between an 8-inch round cake pan and
an 8-inch square pan.
a. If she likes thick brownies, which pan should she use? Explain why and use an
a
ppropriate calculation to support your answer.
She should use the round pan because the bottom has a smaller area, forcing the
brownies to be thicker. The area for the round pan is approximately 50.3 square inches,
whereas the area for the square pan is 64 square inches.
Instructor note: Bring the
class together and go over answers.
Discuss the Discover introduction.
Discover
30 min
Instructor note: To help
students see a calculation with
units, write this volume calculation:
V = π(4 cm)2(8 cm).
Instructor note: Instruct
students to work in groups to
complete the third column of #3
without a calculator. Give students
5 minutes.
1
b. If she wants as much crispy edge as possible, which pan should she use? Explain
why and use an appropriate calculation to support your answer.
She should use the square pan because that pan has a larger perimeter, producing
more crust on the brownies. The perimeter for the square pan is 32 inches, whereas the
p
erimeter for the round pan is approximately 25 inches.
When using geometric formulas to find perimeter, area, volume, or surface area, it is common to work only with numbers in the formulas and then list the units with our answer.
This method of leaving the units until last can sometimes be problematic since it is easy to
forget to include the units at the end. And even if we do remember to list the units, we may
forget which type of unit goes with the problem being solved. Including units in the calculation solves both of these problems, but it presents new ones. How do we work with units?
3. Recall that exponents are used to express repeated multiplication. Instead of writing
2 # 2 # 2 # 2, we can write 24. The factor being repeated is 2, and the number of repetitions is 4. In this example, 2 is the base, and 4 is the exponent. Let’s work with this
notation to learn more about it.
2
3
4
5
Expression
Rewrite expression
without exponents
Rewrite result from
the third column
using exponents
Conjecture
a.
53 # 54
5#5#5#5#5#5#5
57
When multiplying numbers with
like bases, keep the base and add
the exponents.
b.
(3 # 5)2
(3 # 5) # (3 # 5)
32 # 52
When taking a product to a power,
apply the exponent to each factor.
c.
2 2
a b
3
2
2
a b#a b
3
3
d.
(73)2
(7 # 7 # 7)2 = (7 # 7 # 7) # (7 # 7 # 7)
76
When taking a power to a power,
multiply the exponents.
e.
115
11 # 11 # 11 # 11 # 11
11 # 11 # 11 # 11 # 11
=
11 # 11
11 # 11
113
When dividing numbers with like
bases, keep the base and subtract
the exponents (top – bottom).
112
22
32
When taking a quotient to a
power, apply the exponent to the
numerator and denominator.
Instructor note: As a class, go through each row, checking the 3rd column result and completing
the 4th column together.
For each row, instruct students to look only at the 2nd and 4th columns and make a conjecture
allowing them to skip directly from the 2nd to the 4th column. List conjectures in the last column.
Point out that examples do not prove a rule but are helpful in this case.
ALMY8454_01_AIE_C02.indd 173
5/31/13 5:12 AM
9.
174
Cycle 2
Instructor note: Instruct
students to work in groups on
#4 and #5. Give them 5 minutes.
4. Simplify the four expressions using exponents. Do not use your calculator.
44
43
42
41
=
=
=
=
256
64
16
4
Look at the problems and results to find a pattern. Write the next problem and result.
40 = 1
Make a conjecture about an exponent of zero.
Answers will vary. A number to the zero power is one.
5. Simplify the four expressions using exponents. Do not use your calculator.
04 = 0
03 = 0
02 = 0
Instructor note: Go over #4–5
as a class. Explain that since 00
doesn’t have a unique answer, we
say this is an indeterminate form.
01 = 0
Based on the pattern, what do you think the result should be for 00?
0
Based on #4, what do you think the result should be?
1
Use your calculator to find 00. Make a conjecture for the result you see on your
calculator.
Calculators should give an error message. Zero raised to the zero power is not defined since
it seems it could be equal to two different numbers.
We have investigated some important properties of whole-number exponents. Let’s
s
ummarize them. Since we are generalizing these number properties, we will use variables
to represent numbers.
ALMY8454_01_AIE_C02.indd 174
5/31/13 5:12 AM
10.
LESSON 2.6 Measure Up
175
To simplify expressions with exponents: a and b are whole numbers; x and y
are real numbers.
1. x ax b = x a + b
W
hen multiplying two numbers with the same base, keep the base and
add the exponents.
Instructor note: Talk about
the How It Works, focusing on the
version of the rules in words.
2.
note
Sticky
nt
If you forget an expone
the probrule, write out
.
lem without exponents
example, should you add
For
nts for
or multiply the expone
4 2? Writing it out, you get
x x
is
x · x · x · x · x · x which
6 We can see the rule is to
x .
the
keep the base and add
exponents.
xa
xb
= xa-b
W
hen dividing two numbers with the same base, keep the base and subtract
the exponents, top exponent – bottom exponent.
3. x 0 = 1
A
number to the zero power is 1. This is true for all numbers x
except zero.
4. (x a)b = x a b
When taking a power to a power, multiply the exponents.
5. (x y) a = x a y a
When taking a product to a power, apply the exponent to each factor.
x a
xa
6. a b = a
y
y
W
hen taking a quotient to a power, apply the exponent to the numerator and
denominator.
For example:
Work with the coefficients and
1. ( - 3x 2x 5) ( - 6x 2) = ( -3x 7) ( -6x 2) = 18x 9
the exponents separately.
Be
2. ( - 2x 4)4 = ( -2)4 (x 4)4 = 16x 16 careful to apply the exponent
of 4 to -2, not just to 2.
x 15
x 15
Simplify the numerator
3. 2 3 = 6 = x 9
and denominator first.
(x )
x
Instructor note: Work #6 as a
class. Talk through how you choose
to start and which rules you are
using.
Instructor note: Ask students
for the base in 6d and 6e. Identifying
the base will help when simplifying
the expression.
Instructor note: Point out that
6f can be simplified in more than
one way: Apply the exponent to the
numerator and denominator and then
simplify or simplify first and then apply the exponent. You might want to
simplify the expression both ways and
discuss the pros and cons of each.
ALMY8454_01_AIE_C02.indd 175
Applying exponent properties can be a challenge. Practice using them to improve your
understanding.
6. Simplify each expression.
a. 17x2 # 2x5 34x7
b. (4x4)2 16x 8
c. ( -5x 3)2 25x 6
d. ( -8)0 1
e. -80 -1
f. a
3x4 2 x 2
b
4
6x3
5/31/13 5:12 AM
11.
176
Cycle 2
We now have the skills to work with geometric formulas and units simultaneously.
10-15 min
Instructor note: Complete
7a with the class to model the
correct work. Instruct students to
work in groups on #7b–8. 7b allows
s
tudents to see where cc’s occur.
Go over answers.
tech Tip
Finding half a number is
the same as dividing by 2.
So if you have to calculate
1
2 (5)(20), you can type
(1 , 2) # 5 # 20 or 5 # 20 , 2.
7. . ind the area of a triangle with base 5 inches and height 20 inches.
a F
A =
1
1
bh = (5 in.)(20 in.) = 50 in.2
2
2
b. Find the volume of a cylinder whose radius is 2 centimeters and height is
1 centimeter. Find an approximate answer by using 3.14 for π and round to the
nearest tenth.
V = πr 2h = π (2 cm)2 (1 cm) = π (4 cm2)(1 cm) ≈ 12.6 cm3 or 12.6 cc
c. For a sphere, find the ratio of its volume to its surface area.
4 3
4
r
πr
4
1
V
3
3
4 1
r
= r , 4 = r# =
or r
=
=
S
4
3
3 4
3
3
4πr 2
8. . Find the area of a square whose sides have lengths of 3 centimeters.
a
Why do you think 32 is sometimes read as “three squared”?
9 cm2; The quantity 32 gives the area of a square with sides of length 3.
b. Find the volume of a cube whose sides have lengths of 5 feet.
Why do you think 53 is sometimes read as “five cubed”?
125 feet3; The quantity 53 gives the volume of a cube with sides of length 5.
Connect
5–10 min
Instructor note: Explain the
formula in #9 and emphasize the
difference between exponents and
subscripts. The idea of magnets may
help illustrate the formula. Have
students complete #9 and go over
answers.
Instructor note: Part a
p
reviews the concepts of direct
and inverse variation, developed in
Cycle 3.
Instructor note: Point out
that understanding formulas and
the units involved will be useful if
students will be taking more science classes.
ALMY8454_01_AIE_C02.indd 176
9. The following formula calculates the gravitational force between two objects based on
their masses (m1 and m2) and the distance between them (r).
m1m2
F = G 2
r
a. If the masses increase, what happens to the force between them? It increases as well.
If the distance between the objects increases, what happens to the force between
them? It decreases.
b. m1 and m2 have units of kilograms, and r has units of meters. What units must the
constant G have in order for the units on F to be newtons (N)?
Nm 2
kg 2
5/31/13 5:12 AM
12.
LESSON 2.6 Measure Up
5 min
Instructor note: Explain
the contents of the Wrap-Up.
Have tudents write an answer to
s
the ycle question, “Why does it
c
m
atter?” A prompt is provided to
help students. Discuss homework.
Wrap-Up
lesson
Reflect
177
What’s the point?
Including units when working with formulas illuminates the units for the answer.
To do calculations this way, we must understand the properties and rules
of exponents.
What did you learn?
How to apply whole-number exponent rules
How to use basic geometric formulas
Cycle 2 Question: Why does it matter?
Working with calculations and units at the same time can be more work. Why
does it matter that we do this?
It makes clear the type of units to be included in the answer. Area and surface area
will be in square units. Volume will be in cubic units. If we do not use units in calculations, the units of the results must be memorized and might have little meaning. Also,
it is easy to forget to put units in the answer if you have not been using them in the
calculations.
2.6 Homework
Skills
MyMathLab
• Apply whole-number exponent rules.
• Use basic geometric formulas.
1. Simplify.
a. (14ab)3(ab) 2,744a 4b 4
b. 150(3x2)3 27x 6
c.
ALMY8454_01_AIE_C02.indd 177
4m9 m 6
8m3n 2n
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13.
178
Cycle 2
2. a. n interior designer is designing a round table whose radius is 24 inches. She
A
would like to use a special kind of trim, which is quite expensive, for the edge of
the table. Instead of building the table and measuring for the length of trim, she
is working on paper in case the trim is too expensive and she needs to make the
table smaller. How much trim will she need? Round up to the nearest whole inch
to ensure that she has enough.
151 inches
b. If she wants to cover the top of the table with a special crackle paint, how many
square feet of surface will the paint need to cover? Round up to the nearest whole
square foot to ensure she has enough.
13 ft2
Concepts and Applications
• Apply whole-number exponent rules.
• Use basic geometric formulas.
3. The volume of a substance is related to its mass and density by this formula: V = m. If
d
the mass is in grams and the density is in grams per cubic centimeter, what will be the
units for volume? cubic centimeters
4. The distance a projectile will travel when it is fired at an initial angle of 30 degrees
2
can be found using the formula D = vg # 0.87.
If velocity is in meters/second and g is in meters/second2, what will be the units for
distance? meters
5. The formula T = 2π
L
gives the time T for a full cycle of a pendulum’s swing where
Ag
the pendulum’s length in feet is L, and g is the acceleration due to gravity, 32.2 ft/sec2.
When T is calculated from this formula, what will the units be? seconds
FP
50–55 min
Explore
5–10 min
Instructor note: Instruct
students to work in groups on the
Explore. Go over answers.
2.7
Count Up
1. Find the following sums. List the items being counted in each problem.
a. 2 cats + 3 cats
5 cats
Items: cats
2
1
b.
+
5
5
3
5
Items: fifths
3
2
c.
+
7
7
5
7
Items: sevenths
ALMY8454_01_AIE_C02.indd 178
5/31/13 5:12 AM
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