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Math Skill Builders
Physics A First Course
Math Skill Builders
Credits
CPO Science Curriculum Development Team
Author: Thomas Hsu, Ph.D.
Vice Presidents: Thomas Narro and Lynda Pennell
Writers: Patricia Tremblay and Mary Beth Hughes
Graphic Artists: Polly Crisman, Bruce Holloway, and Jim Travers
Curriculum Contributor
Scott Eddleman
Technical Consultants
Tracy Morrow
Physics A First Course
Teacher Resource CD-ROM
Copyright
2005 CPO Science
ISBN 1-58892-144-1
1 2 3 4 5 6 7 8 9 - QWE - 09 08 07 06 05
All rights reserved. No part of this work may be reproduced or transmitted in any form or by an
means, electronic or mechanical, including photocopying and recording, or by any information
store or retrieval system, without permission in writing. For permission and other rights under
this copyright, please contact:
CPO Science
26 Howley Street,
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http://www.cposcience.com
Printed and Bound in the United States of America
Math Skill Builders
The Math Skill Builders are listed in the order in which you may use them in your classroom. The chapter or chapters for which these skill builders best apply is written in parentheses next to each title. These skill builders were
created in order to help you work with students who are concurrently taking algebra and Physics A First Course.
Decimals and Place Value (1)
Slope (2)
Slope from a Graph (2, 11)
Order of Operations, Part 1 (3, 10, 14)
Order of Operations, Part 2 (3, 10, 14)
Evaluating Algebraic Expressions (3, 10, 13, 14)
Inverses (3, 10, 14)
Inverse Operations (3, 10, 14)
Negative Exponents and Exponents of Zero (4, 8, 17, 18)
Scientific Notation: Standard to Scientific (4)
Scientific Notation: Scientific to Standard (4)
Scientific Notation: Products and Quotients (4)
Two-Dimensional Vectors (5)
Pythagorean Theorem (5)
Pythagorean Triples (5)
Special Right Triangles (5)
Measuring Angles with a Protractor (5)
Constructing Angles with a Protractor (5)
Functions: Conversion between Celsius and Fahrenheit (7)
Using a Graphing Calculator: Conversion between Celsius and Fahrenheit (7)
Ratios (8, 17)
Ratio and Proportions in a Recipe (8, 17)
Probability (9)
Reciprocals and Negative One as an Exponent (19)
Problem Solving Boxes (template for solving problems)
Physics A First Course
Name:
Date:
Decimals and Place Value
The place-value chart below is to help you write and read decimals and to understand their values.
The decimal 1364.2895 is shown in the chart at the right. The chart may be extended in either direction. Notice
that the columns on either side of the ones columns appear to have similar names. The columns to the right of the
decimal point always end with -ths.
Notice that the decimal point separates the ones and tenths places. It is read as and.
The decimal 1,364.2895 is read as one thousand, three hundred sixty-four and two thousand eight hundred
ninety-five ten-thousandths.
1.
Write thirty and five hundred seventy-four thousandths as a number.
One way to think about this is to see the decimal point as splitting the number in two parts. Now read just the
part of the number that comes before the word and. Write that part of the number. Then put a decimal point
for the word and. Now read the right side of the number. The number must end in the last column that is
named. To help do this correctly, you could draw small segments to fill in the places for the decimal part of
the number. If there are not enough digits to end in the last column on the right, place as many zeros as it
necessary.
30. ___ ___ ___ 574 fits into the lines so no zeros are necessary.
30.574
2.
Write 345.029 in words.
First you write the words for the part of the number that is to the left of the decimal point. Then write and for
the decimal point. Now write the number to the right of the decimal point as if it is a whole number. Finally,
write the name of the column that the total number ends in. (Remember the ths at the end of the column
name.)
Three hundred forty-five and twenty nine thousandths
Page 2 of 2
1.
2.
3.
Write each of the following as a decimal:
a.
fourteen hundredths
b.
four thousand, six hundred and twenty-five thousandths
c.
one thousand and one thousandths
d.
nine hundred eighty-five and sixty-three hundredths
e.
eight thousand thirty-five and four tenths
Write each number in words:
a.
105.064
b.
23.0049
c.
36.7
d.
45.003
e.
74.998
In the 1996 Olympics, Michael Johnson won both the men’s 200-meter and 400-meter track competitions.
These records have held through the 2000 Olympics.
a.
His time for the 200-meter competition was 19.32 seconds. Write this decimal in words.
a.
His time for the 400-meter competition was forty-three and forty-nine hundredths seconds. Write this as
a decimal.
Name:
Date:
Slope
Slope is a word used to describe the steepness of a line or the rate of change of a linear relationship. It may have
a positive or negative value.
The formula for the slope of the line passing through point 1 with the coordinates (x1, y1) and point 2 with
coordinates (x2, y2) is:
vertical change
change in y- = y--------------2 – y1
- = rise
-------- = -----------------------------------------slope = --------------------------x2 – x1
run
horizontal change
change in x
A line goes through the points (1, 2) and (5, 4).
To find the slope we do the following:
4----------– 2- = 2--- = 1--5–1
4
2
OR
2----------– 4- = –-----2- = –-----1- = 1--1–5
–4
–2
2
Although both of these divisions give the same final, positive answer, they can be interpreted in two different
ways. One interpretation is that as x changes to the right 2 units then y changes up 1 unit. Another interpretation
is that as x changes to the left 2 units y changes down 1 unit.
When the slope is a negative number, such as - 1/2, then the two interpretations could be: (1) that as x changes to
the right 2 units then y changes down 1 unit, or (2) that as x changes to the left 2 units y changes up 1 unit.
Find the slope for the following points and then give an interpretation of the slope. The letter m is the usual
variable used for slope. It probably comes from the French word monter meaning to climb.
Points
Example:
(0, 5), (3, 4)
Slope (m)
1m = ----–3
Interpretation
As x changes ____ unit(s) left or right, y changes _____ unit(s)
up or down.
As x changes 3 units to the left, y changes 1 unit up.
1. (-2, 3), (2, 5)
m=
2. (0, 0), (-3, 7)
m=
3. (-10, -8), (-3, -4)
m=
4. (7, 0), (0,7)
m=
5. (3.5, 1), (0.5, 0)
m=
Name:
Date:
Slope from a Graph
To determine the slope of a line choose two points on the line. Then count how many steps up or down you must
move to be on the same horizontal line as your second point. Multiply this number by the scale factor.
Put the result along with the positive or negative sign in the numerator of your slope ratio if the scale is one. Then
count how many steps you must move right or left to land on your second point. Multiply the number of steps by
the scale factor. Place the results in the denominator of your slope ratio.
A
The chosen points for the example are (0, 0) and (3, 9). (There are
many choices for this graph, but only one slope. If you have the point
(0, 0), you should choose it as one of your points.)
It takes 9 vertical steps to move from (0, 0) to (0, 9). Put a 9 in the
numerator of your slope ratio (or put 9 – 0). Then count the number of
steps to move from (0, 9) to (3, 9). This is your denominator of your
slope ratio. Again, you can do this by subtraction (3 - 0).
9
3
m =
=
3
1
B
The two points that have been chosen for this example are (0, 24) and
(6, 15). Be careful of the scales on each of the axes.
It takes 3 vertical steps to go from (0, 24) to (0, 15). But each of these
steps has a scale of 3. So you put a -9 into the numerator of the slope
ratio. It is negative because you are moving down from one point to the
other. Then count the steps over to (6, 15). There are 3 steps but each
counts for 2 so you put a 6 into the denominator of the slope ratio.
−9
−3
m =
=
6
2
Find the slope of the line in each of the following graphs:
Graph #1:
Graph #2:
Page 2 of 2
Graph #3:
Graph #4:
Graph #5:
Graph #6:
Graph #7:
Graph #8:
Graph #9:
Graph #10:
Name:
Date:
Order of Operations, Part 1
In arithmetic there are four operations: multiplication, division, addition, and subtraction. If you have an
expression that has only these symbols, then the rule to evaluate them is to do multiplication or division moving
from left to right. When this is completed, go back to the left and do the addition or subtraction moving left to
right.
Example 1:
Step 1
3+2×4–5
= 3+8–5
Multiply 2 and 4.
Step 2
3+8–5
= 11 - 5
Step 3
11 - 5
= 6
Do the addition and subtraction moving left to
right—add 3 and 8.
Then, subtract 5 from 11.
Step 1
1--× 48 – 6 × 2
3
=
Step 2
16 – 12
Example 2:
16 – 12
= 4
Evaluate the following expressions:
1.
25 – 5 × 3
2.
15 + 5 × 7
3.
5 + 0.5 × 7
4.
18 – 3 × 2 – 5
5.
18 × 3 – 2 × 5
6.
18 – 5 × 2 – 3
7.
3×2+5×2 +6
8.
7×2+3×2+6×2
9.
4×2+5×2+6
10. 7 + 3 × 2 – 1
11. 8 – 4 × 2 + 7
12. 25 – 6 + 7 – 2
1
Multiply --- and 48. Multiply 6 and 2.
3
Subtract 12 from 16.
Page 2 of 2
13. 1/2 × 20 + 4
14. 36 ÷ 3 × 2 – 5
15. 1/4 × 8 + 9 – 1
16. A large computer store has certain software on sale at 4 for $25.00 with a limit of 4 at the sale price.
Additional software is available at the regular price of $8.95 each.
a.
Write an expression you could use to find the cost of 6 software packages.
b.
How much would 6 software packages cost?
17. Valerie is signing up with a new internet provider. The service costs $5.99 a month, which includes
100 hours of access. If she is online for more than 100 hours she must pay an additional $0.95 per hour.
a.
Suppose Valerie is online for 120 hours the first month. Write an expression that represents what
Valerie must pay for the month.
b.
Now evaluate the cost.
18. Most bacteria reproduce by dividing into identical cells. This process is called binary fission. A certain type
of bacteria can double its numbers every 20 minutes. Suppose 150 of these cells are in one culture dish and
250 of the cells are in another culture dish.
a.
Write an expression that shows the total number of bacteria cells in both dishes after 20 minutes.
b.
Now, evaluate the expression to find out how many bacteria are in both dishes after 20 minutes.
19. Jamal and Alexandria are selling tickets for their school talent show. Floor seats cost $7.50 and balcony
seats cost $5.00. Alexandria sells 60 floor seats and 70 balcony seats. Jamal sells 50 floor seats and 90
balcony seats.
a.
Write an expression to show how much money Alexandria and Jamal have collected for tickets.
b.
Evaluate the expression to determine how much they collected.
Name:
Date:
Order of Operations, Part 2
In evaluating expressions sometimes grouping symbols such as parentheses, square parentheses, set notation, and
fraction bars are used. If these are present then the operations within them are done first. Expressions may also be
raised to a power. General steps for evaluating expressions with grouping symbols and values raised to a power
are as follows:
Step 1
Step 2
Step 3
Step 4
A
B
Evaluate expressions inside grouping symbols. If grouping symbols enclose grouping symbols start
at the innermost part of the expression and work out.
Evaluate all power expressions.
Do all multiplication or division from left to right. Note: Both “·” and “×” can be used to represent
“multiplied by.”
Do all additions or subtractions from left to right.
Step 1
2(8) + 5(4 + 3)
= 2(8) + 5(7)
Add within grouping symbol.
Step 2
2(8) + 5(7)
=
There are no power expressions to evaluate.
Step 3
2(8) + 5(7)
=
Step 4
16 + 35
= 41
Step 1
5 + 42
22 ⋅ 3
Step 2
5 + 42
22 ⋅ 3
16 + 35
This expression means: 2 × 8 + 5 × 7.
Multiply moving left to right.
Add 16 and 35.
This means (5 + 42) ÷ (22 × 3).
=
5 + 16
4⋅ 3
Evaluate the power number in the numerator and denominator.
=
21
12
Simplify the numerator and denominator.
=
7
4
Reduce the fraction.
Step 3
Step 4
Evaluate these expressions using the steps for working with grouping symbols and values raised to a power.
1.
(6 – 4) · 3 =
2.
(8 + 5) · 2 =
3.
10 + 5 × 6 =
Page 2 of 2
4.
10(5 + 6) =
5.
60 – 12 ÷ 4 =
6.
250 ÷ (5[(3 · 7) + 4]) =
7.
28 ÷ 4 · 2 – 32 =
8.
64 ÷ (2 · 4) + 2 =
9.
(6 + 5)(4 + 3) =
=
10.
11.
15 + 60
=
30 − 5
10(3) + 2(3)
32 − 3
=
12.
36 ÷9 ÷4 =
25⋅ 3+ 6 + 9
13.
82 − 2 2 =
(2 ⋅ 8) + 4
14. 390 ÷ [5(7 + 6)] =
15. 15 ÷ 3 · 5 – 42 =
16. Use grouping symbols so that the equation is true. If one grouping symbol needs to enclose another, use
brackets to enclose parentheses.
a.
10 · 5 + 4 = 90
b.
6 + 5 – 2 · 8 = 30
c.
3 + 4 · 5 – 1 = 28
d.
20 ÷ 2 · 5 + 8 = 10
e.
20 ÷ 2 · 5 + 8 = 58
f.
20 ÷ 2 · 5 + 8 = 130
17. Jared is signing up with a new cable company. The service costs $50.00 a month which includes 100 hours
of access. If he is online for more than 100 hours he must pay an additional $3.95 per hour. Suppose Jared is
online for 110 hours the first month.
a.
What is the expression that represents what Jared must pay for the first month?
b.
How much will the bill be for the first month?
Name:
Date:
Evaluating Algebraic Expressions
Algebraic expressions often contain more than one operation. Algebraic expressions can be evaluated when the
values of the variables are known.
Step 1
Step 2
Replace the variables with their values.
Find the value of the numerical expression using the rules for the order of operations.
Evaluate a2 – (b2 + 4c) when a = 8, b = 4, and c = 5.
Step 1
a2 – (b2 + 4c)
=
Step 2
82 – (42 + 4 · 5)
= 82 – (16 + 4 · 5)
Evaluate 42.
82 – (16 + 4 · 5)
= 82 – (16 + 20)
Multiply 4 and 5.
82 – (16 + 20)
= 82 – (36)
= 64 – (36)
Add 16 and 20.
82 – (36)
64 – (36)
1.
2.
82 – (42 + 4 · 5)
= 28
πr2 when π = 3.14 and r = 5 cm
4.
2(L + W) when L = 8 m and W = 12 m
6.
Evaluate 82.
Subtract 36 from 64.
m
---- when m = 0.30 grams and V = 0.040 cm3
V
v2 – v1
--------------- when v2 = 4 m/sec, v1 = 1 m/sec and t = 1 sec
t
3.
5.
Replace variables.
y2 – y1
---------------- when (x1, y1) = (4, 4) and (x2, y2) = (0, 0)
x2 – x1
The Pyramid of the Sun in Mexico is the third largest pyramid in the world. It
stands 60 meters high with a base of 200 meters2. The volume of any pyramid is
one-third the product of the area of the base (B) and its height (h).
a.
Write an expression that represents the volume (V) of a pyramid.
b.
Evaluate your expression using 200 m2 for B and 60 m for h to find the
volume of this pyramid.
Name:
Date:
Inverses
Every positive rational number can be paired with a negative rational number. These pairs are called opposites.
A number and its opposite are additive inverses of each other. When you add two opposites, the sum is always
zero.
Two numbers whose product is 1 are called multiplicative inverses or reciprocals.
Zero has no reciprocal because any number times zero equals zero.
Additive inverse examples:
+5 + (-5) = 0
–4 +(+4) = 0
Multiplicative inverse examples:
2/
3
× 3/2 = 1
8 × 1/8 = 1
2 3/5 × 5/13 = 1
1/
Give the additive inverse for the following:
1.
29
5.
–0.75
2.
–101
6.
24.7
3.
5/
9
7.
0
4.
– 4/3
8.
– 4.32
Give the multiplicative inverse for the following:
1.
5/
9
5.
-10
2.
2 2/3
6.
0
3.
-3/4
7.
–1 1/2
4.
0.06
8.
1/
0.5
4
×4=1
Name:
Date:
Inverse Operations
An inverse operation “undoes” the original operations. Addition and subtraction are the inverse operations of
each other. Division and multiplication are the inverse operations of each other.
Example 1:
Step 1
Step 2
Step 3
x + 4 = 10
(x + 4) – 4 = 10 – 4
x=6
Subtraction is the inverse operation of addition.
Subtract 4 from both sides of the equation.
Example 2:
Step 1
Step 2
Step 3
y – 7 = 12
(y – 7) + 7 = 12 + 7
y = 19
Addition is the inverse operation of subtraction.
Add 7 to both sides of the equation.
Example 3:
Step 1
Step 2
Step 3
3z = 4
3z
----- = 12
-----3
3
z=4
Division is the inverse operation of multiplication.
Divide both sides of the equation by 3.
Example 4:
Step 1
--t- = 25
5
Multiplication is the inverse operation of division.
Step 2
--t- × 5 = 25 × 5
5
Multiply both sides of the equation by 5.
Step 3
t=5
Find the value of the variable using inverse operations.
1.
a – 12 = 12
2.
3a – 12 = 12
3.
3a + 8 = 32
4.
b + 7 = 21
5.
2b + 7 = 21
6.
2b – 7 = 17
Page 2 of 2
7.
3c = 24
8.
3c – 5 = 28
9.
3c + 5 = 23
d
10. --- = 16
4
11.
d--- + 8 = 16
4
12.
d--- – 4 = 16
4
5
13. --- f = 45
9
5
14. --- f – 36 = 9
9
5
15. --- f + 36 = 117
9
Name:
Date:
Negative Exponents and Exponents of Zero
For any nonzero value of b and any value of n,
b-n =
1
bn
1
= bn
-n
b
b0 = 1
[00 is undefined.]
Negative exponents are not restricted to -1. To find a number or expression to a negative power rewrite the
number as a fraction and then invert the numerator and the denominator. Then apply the power.
A.
⎛2⎞
⎜ ⎟
⎝3⎠
B.
C.
−3
3
2-3
1
1
=
3 =
3
24
3× 2
1,0000 = 1
D.
0
⎛ -1 ⎞
⎜ ⎟ =1
⎝ 24 ⎠
1.
5-2 =
2.
109-1 =
3.
(-1,000,000)0 =
4.
(-30)-2 =
5.
⎛ -2 ⎞
⎜ ⎟
⎝9⎠
6.
⎛ 2 ⎞
⎜
⎟
⎝ -3 ⎠
7.
5–3 =
8.
10
(2)5 =
0
=
-2
=
3
3
27
3
= ⎛⎜ ⎞⎟ = 3 =
2
8
⎝2⎠
We invert the two in the numerator with the three in the denominator
and then raise each of these to the third power.
In this example the exponent only applies to the numerator. The 2–3
is moved into the denominator and becomes 23. A one must be
placed into the numerator.
We say that this is true because of the definition of b0.
We say that this is true because of the definition of b0.
Page 2 of 2
9.
10
=
(-2)5
10.
25
=
25
11.
3x5
where x ≠ 0
3x5
12.
a 2b −3
where a and b ≠ 0
ab −3
13.
7 −3
=
7 −3
14.
8c−2 d 4
where c and d ≠ 0
2c−2 d
15.
0.5m−2n−3
m−3n−2
16.
10-2
=
10−4
17.
1000
=
10−4
18.
50
=
52
where m and n ≠ 0
19. 36(2–3)(3–2) =
-1
⎛ 1 ⎞⎛ 5 ⎞
⎜
⎜
⎟
3
−
20. ⎝ 2 ⎠ 8 ⎟ =
⎝
⎠
Name:
Date:
Scientific Notation: Standard to Scientific
Do you know what 300,000,000 m/sec is the measure of? It’s the speed of light.
Do you recognize what 0.000 000 000 753 kilograms is the measure of? This is the mass of a dust particle.
Scientists have developed a shorthand method for writing very large numbers. This method is called scientific
notation. A number is expressed in scientific notation when it is written as a product of a factor and a power of
10. The factor must be greater than or equal to 1 and less than 10.
The form for scientific notation is written as a × 10n, where 1 ≤ a < 10 and n is an integer.
Example 1:
Express 28,500,000 in scientific notation.
28,500,000 = 2.85 × 107
The decimal point moved 7 places to the left, so the point is between the 2 and the 8. Since 28,500,000 > 1, the
exponent is positive. When the decimal point moves left the exponent is positive.
Example 2:
Express 0.0000432 in scientific notation.
0.0000432 = 4.32 × 10-5
The decimal point moved 5 places to the right, so the point is between the 4 and the 3. Since 0.0000432 < 1, the
exponent is negative. When the decimal point moves right the exponent is negative.
Example 3:
Express each of the numbers in scientific notation.
0.000781 = 0007.81 × 10n
0.000781 = 7.81 × 10-4
Move the decimal point 4 places to the right.
21,845,000 = 2.1845 × 10n
21,845,000 = 2.1845 × 107
Move the decimal point 7 places to the left.
Remember if the number you start with is larger than 1 than you will get a positive exponent in your
answer.
Page 2 of 2
1.
Convert the number from standard notation to scientific notation:
Standard Notation
a.
0.0453
b.
18,700,000.0
c.
0.257000
d.
999.0
e.
264,000
f.
761,000,000
g.
1,030
h.
0.00120
i.
0.03040
j.
0.000 000 000 000 000 052
k.
42,000,000,000,000
l.
7,650,000
Scientific Notation
m. 0.000999
2.
Write the numbers in the following sentences in scientific notation:
a.
The national debt in 2000 was about $5,670,000,000,000.
b.
In 2000, the U.S. population was 281,000,000.
c.
Earth’s crust contains approximately 120 trillion metric tons of gold (120 trillion =
120,000,000,000,000).
d.
The mass of an electron is 0.000 000 000 000 000 000 000 000 000 000 91 kilograms.
e.
The usual growth rate of human hair is 0.00033 meters per day.
f.
The sun burns about 4,400,000 tons of hydrogen per second.
g.
In 1995 the population of Iran was about 65,100,000.
h.
In the middle layer of the sun's atmosphere, called the chromosphere, the temperature averages
27,800°C.
i.
There are approximately 200,000,000,000 stars in the Androeda Galaxy.
j.
Alex Rodriguez signed a contract with the Texas Rangers in 2000 that guarantees him $25,200,000 a
year for l0 seasons.
k.
The Great Pyramid of Giza stands about 137 meters high.
l.
A normal, healthy body temperature for a human being is 98.6°F.
Name:
Date:
Scientific Notation: Scientific to Standard
In this skill sheet, you will practice converting numbers from scientific notation to standard (decimal) numbers.
Example 1:
Express 3.75 × 104 in standard notation.
3.75 × 104
=
3.75 × 10,000
=
37,500
A positive exponent means the
power of ten is greater than one.
Move the decimal 4 places to the
right. Add zeros as place holders.
Example 2:
Express 1.05 × 10-5 in standard notation.
1.05 × 10-5
=
1
1.05 × -------510
=
=
1.05 × 0.00001
0.000 010 5
A negative exponent means the
power of ten is less than one.
Move decimal 5 places to the left.
Add zeros as place holders.
Express the number in each statement in standard notation.
1.
An electron has a negative charge of 1.6 × 10-19 coulombs.
2.
There are approximately 5.58 × 1021 atoms in a gram of silver.
3.
Americans make almost 2 × 1010 phone calls each day.
4.
The moon’s average distance from Earth is 2.39 × 105 miles.
5.
The mass of a proton is about 1.67 × 10-27 kilograms.
6.
1.5 × 108 km is the approximate distance from Earth to the sun.
7.
9.29 × 107 miles is the approximate distance from Earth to the Sun.
8.
5 × 10-4 inches is the thickness of a piece of paper.
9.
In 1995 the population of the United States was about 2.63 × 108.
10. In 1995 the population of China was about 1.22 × 109.
11. One millimeter equals 1 × 10-3 meters.
12. The speed of sound in air is about 3.4 × 102 m/sec.
Name:
Date:
Scientific Notation: Products and Quotients
You can use properties of powers to compute numbers written in scientific notation.
Example 1:
Evaluate (5.9 × 103)(3 × 10-5). Express the result in scientific and standard notation.
(5.9 × 103)(3 × 10-5)
= (5.9 × 3)(103 × 10-5)
= 17.7 × 10-2
Associative and communicative properties
= (1.77 × 101) × 10-2
= 1.77 × (101 × 10-2)
17.7 = 1.77 × 101
Associative property
= 1.77 × 10-1
Product of powers
Compute the product of factors and the
product of powers.
The solution is 1.77 × 10-1 or 0.177.
Example 2:
8
2.45 × 10 Evaluate -----------------------. Express the results in scientific and standard notation.
5
7.1 × 10
8
2.4495 × 10 -----------------------------5
7.1 × 10
=
⎛ 10 8 ⎞
⎛ 2.45
---------- ⎞ ⎜ -------- ⎟
⎝ 7.1 ⎠ ⎝ 5 ⎠
10
=
0.345 × 103
Quotient of powers (not in scientific notation)
=
(3.45 × 10-1) × 103
=
3.45 × (10-1 × 103)
0.345 = 3.45 × 10-1
Associative property
=
3.45 × 102
The solution is 3.45 × 102 or 345.
Evaluate. Express each result in scientific and standard notation.
4
2.5
×
10
1. --------------------2
2 × 10
– 12
2.
× 10 7--------------------– 15
2 × 10
3.
(2.4 × 10-2)(3.0 × 102)
Associative property
Page 2 of 2
–8
4.
2.35 × 10 -------------------------– 12
2.5 × 10
5.
(6.60 × 105)(3.10 × 102)
6.
1.35 × 10
------------------------10
2.5 × 10
7.
(9.1 × 105)(3.1 × 10-4)
8.
2.2 × 10
-------------------------– 14
1.1 × 10
9.
6 × 10 --------------------2
1.5 × 10
8
– 12
–4
10. (4.5 × 10-6)(2.25 × 10-2)
4
5 × 10 11. ----------------------–4
2.5 × 10
12. Human red blood cells carry oxygen from one place to another in your body. A cubic millimeter of human
blood contains about 5 × 106 red blood cells. An adult human body may contain about 5 × 106 cubic
millimeters of blood. About how many red blood cells does an adult human body contain?
13. A space probe that is 6.4 × 1012 meters away from Earth sends signals to NASA. If the radio signals travel
at the speed of light (3 × 108 m/sec) how long will it take the signals to reach NASA?
14. The minimum distance from Earth to the moon is approximately 2.26 × 105 miles. There are approximately
6.34 × 104 inches in one mile. What is the minimum distance from Earth to the moon in inches?
15. The population of Arizona is about 4.78 × 106. The land area of Arizona is about 1.14 × 105 square miles.
What is the population density per square mile?
16. During the year 2000, 1.65 billion credit cards were in use in the United States. During that same year,
$1.54 trillion was charged to these cards. ( Hint: 1 trillion = 1 × 1012).
a.
Express each of these values in standard and then scientific notation.
b.
Find the average amount charged per credit card.
Name:
Date:
Two-Dimensional Vectors
Definitions
A vector is a directed line segment that represents a vector quantity.
A scalar is a quantity such as time, speed, or volume that has only
magnitude but no direction.
A vector quantity is a quantity such as force, velocity, or displacement
that has both magnitude (length) and direction.
The tail of a vector is the point where it begins. The head of a vector is
the point where it ends. An arrowhead is drawn at the head of a vector.
(Note: A vector is not a ray. It has two endpoints.)
Two vectors are equal if they have the same magnitude and the same
direction. So you may translate a vector without changing it, but you
cannot rotate or dilate it.
In the diagram at the left, vectors a and b are equal even though they have
different starting and ending points.
The sum of vectors can be determined by representing the vectors as
arrows joined in sequence with the initial point of each vector coinciding
with the terminal point of the preceding one. The sum is then the vector
with the initial point of the first vector and the terminal point at the
terminal point of the last vector.
Geometrically, the sum of the vectors a and b is the vector c along the
diagonal of the parallelogram determined by a and b.
This is known as the parallelogram law of forces.
Name:
Date:
Pythagorean Theorem
The Pythagorean theorem states that the sum of the squares of the lengths of the
legs of a right triangle is equal to the square of the hypotenuse. The following
expression represents the Pythagorean theorem:
a2 + b2 = c2 where c is the hypotenuse of a right triangle and a and b are the
measures of the legs.
Geometrically, this theorem is that the area of ABGF in the figure at right is
equal to the sum of the areas of ACDE and BCKH.
Use the Pythagorean theorem expression (a2 + b2 = c2) to solve the following problems.
Example 1: What is the length of c if
a = 6 and b = 8?
2
2
6 +8 = c
2
36 + 64 = c
100 = c
2
100 = c
10 = c
Example 2: What is the length of a if
b = 5 3 and c = 10?
2
2
2
2
a + ( 5 3 ) = 10
2
2
a + 75 = 100
2
a = 25
2
a = 25
a = 5
All of the following values apply to right triangles. Find the measure of the missing side of the triangle using the
Pythagorean theorem. If the measure has a square root (like 3 ) leave it in the answer.
1.
a=5
b = 12
c=
2.
a=
b = 15
c = 17
3.
a=7
b=
c = 25
4.
a=
b=4
c= 4 2
5.
a= 8 3
b=8
c=
6.
a = 15
b = 20
c=
Name:
Date:
Pythagorean Triples
A Pythagorean triple is three whole numbers that satisfy the equation a2 + b2 = c2 where c is the greatest number.
If the measures of the sides of any right triangle are whole numbers, the measures form a Pythagorean triple.
Which of these sets of measures form the sides of a right triangle? Which of these sets form a Pythagorean triple?
1.
3, 4, and 5
a2 + b2 = c2
32 + 42 = 52
9 + 16 = 25
25 = 25
2.
8, 15, and 16
a2 + b2 = c2
82 + 152 = 162
64 + 225 = 256
289 ≠ 256
3.
------3- , ------6- , and 3--5
5 5
2
2
2
⎛ ------3-⎞ + ⎛ ------6-⎞ = ⎛ 3
---⎞
⎝ 5⎠
⎝ 5⎠
⎝ 5⎠
3- + ----6- = ----9----25
25 25
•
In example 1, the segments form the sides of a right triangle since they satisfy the Pythagorean theorem. The
measures are whole numbers and form a Pythagorean triple.
•
In example 2, the segments with these measures cannot form a right triangle. Therefore, they do not form a
Pythagorean triple.
•
In example 3, the segments with these measures form a right triangle. However, the three numbers are not
whole numbers. Therefore, they do not form a Pythagorean triple.
Determine if the following segments form a right triangle and determine if they are Pythagorean triples:
1.
5, 12, and 13
2.
8, 15, and 17
3.
1, 2, and 3
4.
12, 35, and 37
5.
4, 16, and 3 48
6.
7, 24, and 25
7.
When we take multiples of the Pythagorean triples above, they form new Pythagorean triples. These new
Pythagorean triples are considered families of the original triple. For example, you found that 3, 4, and 5
formed a triple. If you multiple each of these by a whole number (other than zero) you will find new triples.
An example is 18, 24,and 30.
a.
How were these formed?
b.
Use the triples in #1, to find at two additional triples in the same family.
c.
Use the triples in #2, to find two additional triples in the same family.
Name:
Date:
Special Right Triangles
Two special kinds of right triangles are the 45° – 45° – 90° right triangle and the 30° – 60° – 90° right triangle.
They are used so frequently that you should learn the pattern for finding the measures of the sides.
In a 45° – 45° – 90° triangle the hypotenuse is 2
times as long as a leg. The legs of these triangles are
equal in length. This is indicated by the small tick
marks made on each leg.
In a 30° – 60° – 90° triangle, the hypotenuse is twice
as long as the shorter leg and the longer leg is 3
times as long as the shorter leg.
Find the value of x.
Example 1: Since the triangle is a 45° – 45° – 90°
triangle, the hypotenuse is 2 times as long as the
leg. So, x = 5 2 .
Example 2: Since the triangle is a 30° – 60° – 90°
triangle, the hypotenuse is 3 times as long as the
leg. So, x = 2(8)=16.
Page 2 of 2
Find the value of x for each triangle below.
1.
2.
3.
4.
5.
6.
Name:
Date:
Measuring Angles with a Protractor
Measure each of these angles (A - Q) with a protractor. Record the angle measurements in the table below.
Letter
Angle
Letter
A
J
B
K
C
L
D
M
E
N
F
O
G
P
H
Q
I
Angle
Name:
Date:
Constructing Angles with a Protractor
Use a protractor to construct the angles listed below. Use the arrow beneath each letter to make the angle.
A. 50°
B. 90°
C. 130°
D. 35°
E. 100°
F. 146°
G. 78°
H. 15°
I. 64°
J. 45°
K. 112°
L. 160°
Name:
Date:
Functions: Conversion between Celsius and Fahrenheit
A function is a relationship between input and output values. Each input has exactly one output. Sometimes
people like to describe a function as a “machine.” The idea is that you put a thing in, something happens to it, and
some (related) thing comes out.
The point of this picture is not that a function is magic, but that it is an automatic routine. Once we know the
routine, we can substitute any number into the place of the input symbol, and we get the corresponding output
without any hard work.
If your machine has a rule to square the input number and then add 2 to the results, what will the values for the
following output be?
Rule: x2 + 2
Input
Output
10
-3
.5
4
-2
A first degree equation of the form y = ax + b can be used as a function. This means that some value will be input
for x. The machine will multiply by the determined a; add a value that has been determined for b; and finally give
the output value y.
y = 3x + 10 is the rule for your machine. You may chose any real number for x, and your machine will take your
number as the input; multiply it by 3; add 10; and give the final value as output.
Step 1
Step 2
y = ax + b
y = 3(x) + 10
y = 3(6) + 10
y = 18 + 10
y = 28
Replace variables.
Multiply 3 and 6.
Add 18 and 10.
Page 2 of 3
Fill in the following table using the rule for the machine (the function).
Rule for the machine (the function)
y = 3(x) + 10
Input
Output
6
28
0
7
– 9/5
–4
– 5/9
In science we often want to convert between Fahrenheit and Celsius. There are two formulas, or rules, that we
use for these conversions.
9
T Fahrenheit = --- T Celsius + 32
5
5
T Celsius = --- ( T Fahrenheit – 32 )
9
We could use these rules for our function machines. One function machine converts Celsius to Fahrenheit. The
second machine converts Fahrenheit to Celsius.
If you have trouble remembering which fraction you should be multiplying by think about this:
•
When you multiply by a fraction larger than 1 like ( 9/5), the product will be greater than the number
you started with. If you are converting from Celsius to Fahrenheit, then you are trying to get a much larger
number than the Celsius measure. Therefore, you multiply by 9/5 and then add 32.
•
When you multiply by a fraction less than 1 like ( 5/9), then the product will be less than what you
started with. The Fahrenheit scale uses larger numbers. Therefore, if you are converting from Fahrenheit to
Celsius, you are looking for a smaller number, so you must be multiplying by 5/9 after you reduce the
number by 32.
Page 3 of 3
Use the temperature conversion formula on the previous page to convert these Fahrenheit values to Celsius.
Compare your answers to the thermometer below.
°Fahrenheit
°Celsius
45
81
0
–4
–10
100
– 2.5
0
– 18.3
37.8
Name:
Date:
Using a Graphing Calculator: Conversion between Celsius
and Fahrenheit
Imagine that you are a weather reporter for a television network in your town. During the nightly broadcast you
must speak about the temperature in various places around the world. In order to have up-to-date information,
you must have your report ready 15 minutes before air time. Since most countries report temperature in degrees
Celsius, you need to have a way to do quick conversions to Fahrenheit for your American audience.
The conversion formula for Celsius to Fahrenheit is:
9
T Fahrenheit = --- T Celsius + 32
5
The calculation can be done using a simple calculator. However, a graphing calculator provides other options.
Two are being presented here: (1) using the LIST option of your calculator, and (2) using the calculator create a
graph and table of the values.
Using the LIST option with a graphing calculator
To use the LIST option (this is found under the STAT button) you should input the values that you have for each
location in L1. Then bring your cursor to the top of L2. This will highlight L2. You should type in your function
using L1 for your Celsius measure. At the bottom of the second list you would see:
9
L2 = ⎛ --- ⎞ L1 + 32
⎝ 5⎠
Press enter, and you will have your output values for each of the places in L2. (If you did not remember to put a
location’s temperature in your L1, you will need to go through the process again.) Add the missing
temperature(s). Then go to the top of L2, and reenter the formula. All the answers will be generated.)
Find the Fahrenheit degree measure for the following places:
City (Country)
Athens (Greece)
Berlin (Germany)
Buenos Aires (Argentina)
Calgary (Canada)
Cairo (Egypt)
Istanbul (Turkey)
Melbourne (Australia)
Mexico City (Mexico)
Nairobi (Kenya)
Rio de Janeiro (Brazil)
Tokyo (Japan)
°Celsius (L1)
22
4
29
7
18
10
9
20
33
24
15
°Fahrenheit (L2)
Page 2 of 2
Creating a graph with a graphing calculator
A graph comparing Celsius and Fahrenheit temperatures is a quick way to make conversions. Use x for the
Celsius degree measure and y for the Fahrenheit degree measure. Put the conversion equation into y = and set the
Window.
Try the following settings for your graph: Xmin = -1, Xmax = 105, Xscale = 5, Ymin = -1, Y max = 215, Yscale = 10.
With these dimensions you can see the axis, and you are able to get some values. You could then look at the
TABLE of the values for the graph. You should think about the settings for this. You could trace for some values
but this is not always as accurate as you would like. You could adjust the increments (∆Tbl) in your table set to
see if you get better answers.
Finally, you have the CALC button. You may now select this. Under this menu, you would select the “value”
choice. It would give you a prompt to input a Celsius value. Then, it would give you the output value for
Fahrenheit.
It is important for you to know how the output value is calculated. Once you understand this, it is not necessary
for you to do the calculations by hand every time.
Use the graphing method to fill in the values for temperature in Fahrenheit in the table below.
°Fahrenheit
°Celsius
°Fahrenheit
°Celsius
32
38
-4
100
0
-28
-273
31.8
25.8
50
99
-3
-24
15.5
Name:
Date:
Ratios
A ratio is a comparison of two quantities that tells the scale between them. Ratios may be expressed as quotients,
fractions, decimals, percents, or given in the form a:b. The word per in a ratio means “for every.”
1.
The ratio of females to males on a swim team is 3 to 4, or 3/4.
2.
The airplane is flying at a constant speed of 150 meters per second or 150 meters for every second flown or
150
meters------------------------.
1 second
A ratio can be a comparison of a part to a whole or a part to a part. In the example above about the swim team
members, the team is made up of two parts, females and males. The whole is the total number of people on the
-------------- , Females
-------------------- , or Females
-------------------- .
team. There are three ratios that can be constructed given what we need. They are: Males
Team
Males
Team
Because ratios are not always fractions, we can have numbers in either position without regard to size. This also
allows us to use two different notations when comparing males and females. We need to be concerned with the
order in which the ratio is stated.
•
In the above example, we have
•
However, we also know what
Females
-------------------- .
Males
Males ------------------Females
We know from the information that this is 3/4.
would be from the same statement. It would be 4/3.
The first quantity mentioned in the comparison is the numerator—the number on the upper position of the
fraction.
Reduce all fractions as you solve the following problems.
1.
2.
A research study shows that 3 out of every twenty pet owners got their pets from a breeder, 14 out of the
twenty got their pets from an animal rescue agency, 1 got their pet from the street, and 2 got their pets from
a friend or relative.
a.
What is the ratio of pets received from a breeder compared to the total number?
b.
What is the ratio of pets received from a friend compared to an animal rescue agency?
c.
What is the ratio of pets from an animal rescue agency compared to the total number of pets?
d.
Do you know how many pets were in this research study? Explain your answer.
The ratio of two measurements having different units of measure is called a rate. Examples of this are a price
of $1.99 per dozen, a speed of 55 miles per hour, and a salary of $30,000 per year. A ratio or rate called a
scale is used when making a model or drawing of something that is too large or too small to be drawn
conveniently at actual size. Maps and blueprints are two commonly used scale drawings.
Give a ratio (or rate, or scale) for the following situations:
a.
The scale of a map for Blue Hills Regional Park is 2 inches = 9 miles.
Page 2 of 2
3.
b.
Alicia goes on a 30-mile bike ride every Saturday. She rides the distance in 4 hours.
c.
The Tremblays’ minivan requires 5 gallons of gasoline to travel 112 miles.
d.
A blueprint for a house states that 2.5 inches equals 10 feet.
e.
A collector's model racecar is scaled so that 1 inch on the model equals 6 3/8 feet on the actual car.
f.
On average, the basketball player scores 19 points per game.
g.
These cookies have 35 grams of fat in one serving of 5 cookies.
h.
Wynonna’s prepaid calling card cost $5 and allows her to make 80 minutes worth of calls.
i.
Bill's 85-pound Husky eats 40 pounds of dog food every two weeks.
Use the following chart to answer the questions:
Mr. Little’s Class
Right-handed
Left-handed
Ambidextrous
Boys
11
5
1
Girls
8
4
2
a.
Write the ratio of right-handed boys to right-handed girls.
b.
Write the ratio of left-handed boys to the total number of students in the class.
c.
Write the ratio of ambidextrous girls to the total number of girls in the class.
d.
Write the ratio of ambidextrous girls to ambidextrous boys.
e.
Write the ratio of right-handed boys to ambidextrous boys.
f.
Write the ratio of boys in the class to girls in the class.
g.
Write the ratio of right-handed girls to the entire class.
Name:
Date:
Ratio and Proportions in a Recipe
Double Fudge Brownies
Ingredients:
3/
4
c. sugar
2 eggs
6 tablespoons unsalted butter
1 teaspoon vanilla extract
2 tablespoons milk
2 cups semi-sweet chocolate chips
3/
4
1/
3
1/
4
2 tablespoons confectioner’s sugar
teaspoon salt
cup all-purpose flour
teaspoon baking soda
Makes 16 brownies.
2-------------------------------tablespoons
2 cups
1.
What is the ratio of milk to chocolate chips?
2.
When we know the ratios, we can make proportions by setting two ratios equal to one another. This will help
us to find missing answers.
Suppose Patricia only needs 8 brownies. She does not want to ruin her diet by having leftovers. Find out the
ingredients she needs. The original recipe will make 16 brownies. You will use the ratio of 8/16 = 1/2 to find
the amount for each of the ingredients. Use cross-multiplication to solve the proportions.
For flour:
Step 1
8- = --------x----16
3⁄4
Step 2
3
8 × --- = 16x
4
Step 3
6 = 16x
Step 4
6- = -------16x----16
16
Step 5
3--- = x
8
1.
Patricia needs 3/8 cup of flour to
make 8 brownies.
What is the ratio of unsalted butter to eggs?
For every __________ tablespoons of butter, you will need __________ eggs.
2.
What is the ratio of flour to baking soda?
For every __________ cups of flour, you will need __________ teaspoons of baking soda.
Page 2 of 2
3.
What is the ratio of salt to flour?
For every __________ teaspoon(s) of salt, you will need __________ cups of flour
4.
Find the correct amount of each ingredient to make 8 brownies (1/2 of the recipe).
Ingredient
Flour
Amount
3/
8
cup
Sugar
Butter
Milk
Chocolate chips
Eggs
Vanilla extract
Baking soda
Salt
Confectioner’s sugar
5.
Why are the ingredients, eggs and confectioner’s sugar, easy to work with to make 8 brownies?
6.
Patricia has a surplus of all the ingredients. How many brownies can be made using 3 cups of chocolate
chips?
7.
How much vanilla will she need when she makes the batch of brownies using 3 cups of chocolate chips?
Name:
Date:
Probability
The probability of an event is the chance of getting a possible outcome of a system.
Probability is defined as the ratio of the number of ways an event can occur to the
number of possible outcomes.
of ways the event can occurprobability of an event = number
------------------------------------------------------------------------------------number of possible outcomes
What is the probability of rolling a number less than 5 when you roll a six-sided die?
Answer: There are four numbers less than 5 on the die out of 6 possible outcomes.
Therefore, the probability of the number rolled being less than 5 is 4/6 or 2/3.
1.
2.
3.
A box of colored pencils contains 2 red ones, 5 blues, 3 greens, and 2 yellows. The total number of pencils in
the box is ______. If you choose at random, what is the probability that you will pick each of the following?
a.
a yellow pencil
b.
a blue pencil
c.
a red or green pencil
d.
a pencil that is not blue
e.
a pencil that is not red
A card is chosen randomly from a deck of 52 cards. A standard deck of cards has four suits: spades, clubs,
diamonds, and hearts. The spades and clubs are black; the hearts and diamonds are red. Each suit has 13
cards. Each type of card has one from each suit, so there are 4 cards of each type. For example, there are 4
nines. Find the probability of choosing each of the following.
a.
a red card
b.
the queen of spades
c.
an ace
d.
a black 9
e.
not a diamond
f.
a heart
A spinner is divided into 5 equal pie shapes. Each one is of the same size. The shapes are colored red, purple,
red, yellow, and red. Find the probability of spinner landing on each of the following.
a.
a red section
b.
a yellow section
c.
a blue section
d.
a purple or red section
e.
not a red section
Name:
Date:
Reciprocals and Negative One as an Exponent
All real numbers, except zero, have a reciprocal. To find the reciprocal, you write the number as a fraction and
then invert the numerator and the denominator. The reciprocal is another name for the multiplicative inverse.
In algebra, we use a negative one as an exponent for a number or expression to indicate the reciprocal. If the
numerator or denominator has more than a single number then parentheses must be used to indicate what the base
of the exponent is.
Example 1:
⎛3
--- ⎞
⎝ 7⎠
–1
7
= ⎛ --- ⎞ Notice that the number did not become negative.
⎝ 3⎠
Example 2:
( 5a )
–1
1= ----5a
Example 3:
⎛1
--- ⎞
⎝ 2⎠
---------3
–1
=
1 ----------1
3 ⎛⎝ ---⎞⎠
2
=
1-------3
--2
=
⎛3
---⎞
⎝ 2⎠
=
2--3
Convert the fraction to a number
without a negative exponent.
Notice that because of the
parentheses, the negative exponent
only applies to the numerator of
this fraction.
This fraction can easily be
expressed as a reciprocal with a
negative exponent.
–1
The reciprocal is found by
inverting the numerator and
denominator.
Page 2 of 2
Find the reciprocal for the following numbers:
1.
3/
4
2.
1
3.
7
4.
8 1/4
5.
– 1/4
6.
–4 1/8
7.
– 6/13
8.
–100
/5
Convert the following to numbers that do not have a negative exponent.
9.
20-1
2 –1
10. ⎛ --- ⎞
⎝ 3⎠
11. (-3b)-1
–1
12. ⎛ 3--- ⎞
⎝ 4⎠
---------5
2 –1
13. ⎛ 5 --- ⎞
⎝ 3⎠
14. (6d) -1
15. 6d (-1)
( 2 × 8 ) –1
16. ⎛ ----------------- ⎞
⎝ (3 × 6) ⎠
17. (-8)-1
18. 8(3a)-1
3
19. ⎛ --- ⎞
⎝ 5⎠
–1
6 –1
× ⎛ ------ ⎞
⎝ 15 ⎠
Name:
Date:
Problem Solving Boxes
Looking for
Solution
Given
Relationships
Looking for
Solution
Given
Relationships
Looking for
Given
Relationships
Solution
Math Skill Builder Answer Keys
Decimals and Place Value
1. Answers are:
a. 0.14
b. 4,600.025
c. 1,000.001
d. 985.63
e. 8,035.4
2. Answers are:
a. one hundred five and sixty-four thousandths
b. twenty-three and forty nine ten-thousandths
c. thirty-six and seven tenths
d. forty-five and three thousandths
e. seventy-four and nine hundred ninety-eight thousandths
3. Answers are:
a. nineteen and thirty-two hundredths
b. 43.49
Slope
1. m = 2/4 = 1/2
As x changes 4 units to the left, y changes 2 units down. OR
As x changes 4 units to the right, y changes 2 units up.
4. m = 7/-7 = m = -1
As x changes 1 units to the left, y changes 1 units up. OR
As x changes 1 units to the right, y changes 1 units down.
2. m = 7/-3
As x changes 3 units to the left, y changes 7 units up. OR
As x changes 3 units to the right, y changes 7 units down.
5. m = 1/3
As x changes 3 units to the left, y changes 1 units down. OR
As x changes 3 units to the right, y changes 1 units up.
3. m = 4/7
As x changes 7 units to the left, y changes 4 units down. OR
As x changes 7 units to the right, y changes 4 units up.
Slope from a Graph
Numbers correlate to graph numbers:
6. m = –-----2- = – 1
3 × 3- = –-----31. m = –-------------6×2
4
7. m = 0
2
2 × 3- = –-----32. m = –-------------4×2
8. m = 3---
4
4
3. m = 5--9
9. m = 2
4. m = 4--- = 2
10. m = –-----1-
2
2
5. m = 3--- = 1
3
Order of Operations, Part 1
1. 10
14. 19
2. 50
15. 10
3. 8.5
16. Answers are:
a. $25.00 + 2 × $8.95
b. $42.90
4. 7
5. 44
17. Answers are:
a. $5.99 + 20 × $0.95
b. $24.99
6. 5
7. 22
18. Answers are:
a. 2 × 150 + 2 × 250
b. 800 cells of bacteria
8. 32
9. 24
10. 12
19. Answers are:
a. 60 × $7.50 + 70 × $5.00 + 50 × $7.50 + 90 × $5.00
b. $1,625.00
11. 7
12. 24
13. 14
1
Order of Operations, Part 2
1. 6
13. 3
2. 26
14. 6
3. 40
15. -17
4. 110
16. Answers are:
a. 10 · (5 + 4) = 90
b. 6 + [(5 – 2) · 8] = 30
c. (3 + 4) · (5 – 1) = 28
d. 20 ÷ (2 · 5) + 8 = 10
e. (20 ÷ 2) · 5 + 8 = 58
f. (20 ÷ 2) · (5 + 8) = 130
5. 57
6. 2
7. -18
8. 10
9. 77
17. Answers are:
a. $50.00 + (10 × $3.95)
b. $89.50
10. 3
11. 6
12. 1/90 = 0.111
Evaluating Algebraic Expressions
1. 7.5 g/cm3
2. 3
5. 1
6. Answers are:
a. 1/3(B × h)
m/sec2
3. 78.5
cm2
b. 4000 m3
4. 40 m
Inverses
Additive inverse:
Multiplicative inverses:
1. -29
5. 0.75
2. 101
6. – 24.7
4
4
5. -0.1
6. Zero has no reciprocal.
5
7. Zero has no reciprocal.
3. – /3
7. – 2/3
/3
8. 4.32
4. 100/6 = 50/3
8. 0.5
3. – /9
4.
1. 9/5
2. 3/8
Inverse Operations
1. a = 24
9. c = 6
2. a = 8
10. d = 64
3. a = 8
11. d = 32
4. b = 14
12. d = 80
5. b = 7
13. f = 81
6. b = 12
14. f = 81
7. c = 8
15. f = 145.8
8. c = 11
Negative Exponents and Exponents of Zero
1. 1/25
8.
2. 1/109
5----16
5
9. ---------
3. 1
– 16
10. 1
4. 1/900
11. 1
5. 1
12. a
6. 9/4
13. 1
7. 1/125
2
14. 4d3
17. 10,000,000
0.5m
15. -----------n
18. 2
19. 1/2
20. 1/5
16. 100
Scientific Notation: Standard to Scientific
1. Answers are:
a. 4.53 × 10-2
b. 1.87 × 107
c. 2.57 × 10-1
d. 9.99 × 102
e. 2.64 × 105
f. 7.61 × 108
g. 1.03 × 103
h. 1.20 × 10-3
i. 3.04 × 10-2
j. 5.2 × 10-17
k. 4.2 × 1013
l. 7.658 × 106
m. 9.99 × 10-4
2. Answers are:
a. $5.67 × 1012
b. 2.81 × 108 people
c. 1.2 × 1014 metric tons
d. 9.1 × 10-31 kg
e. 3.3 × 10-4 m
f. 4.4 × 106 tons
g. 6.51 × 107 people
h. 2.78 × 104 °C
i. 2.0 × 1011 stars
j. $2.52 × 107
k. 1.37 × 102 m
l. 9.86 × 101 °F
Scientific Notation: Scientific to Standard
1. 0.000 000 000 000 000 000 16 coulombs
7. 92,900,000 miles
2. 5,580,000,000,000,000,000,000 atoms
8. 0.0005 inches
3. 20,000,000,000 phone calls
9. 263,000,000 people
4. 239,000 miles
10. 1,220,000,000 people
5. 0.000 000 000 000 000 000 000 000 001 67 kg
11. 0.001 meters
6. 150,000,000 km
12. 340 m/sec
Scientific Notation: Products and Quotients
1. 1.25 × 102 = 125
12. (5 × 106 red blood cells/mm3) × (5 × 106 mm3)
= 2.5 × 1013 = 25,000,000,000,000 red blood cells
2. 3.5 × 103 = 3,500
13. (6.4 × 1012 meters) ÷ (3 × 108 m/sec)
= 2.1 × 104 sec =
21,000 sec
3. 7.2 × 100 = 7.2
4. 9.4 × 103 = 9,400
5. 2.05 × 108 = 205,000,000
14. (2.26 × 105 miles) × (6.34 × 104 in/mile) =
1.43 × 1010 inches = 14,300,000,000 inches
6. 5.4 × 10-3 = 0.0054
7. 2.8 × 102 = 280
15. (4.78 × 106 people) ÷ (1.14 × 105 mi2) =
4.19 × 101 people/mi2 = 41.9 people/mi2
2
8. 2.0 × 10 = 200
9. 4 × 10-6 = 0.000004
16. Answers are:
a. 1.65 billion credit cards =
1,650,000,000 credit cards = 1.65 × 109 credit cards
$1.54 trillion = $1,540,000,000,000 = $1.54 × 1012
b. ($1.54 × 1012) ÷ (1.65 × 109 credit cards) =
$9.33 × 102 = $933
-7
10. 1.0 × 10 = 0.000 000 1
11. 2 × 108 = 200,000,000
Two-Dimensional Vectors
There are no questions to answers for this skill builder.
3
Pythagorean Theorem
1. c = 13
4. a = 4
2. a = 8
5. c = 16
3. b = 24
6. c = 25
Pythagorean Triples
1. 52 + 122 = 132 = 169; yes, this is a Pythagorean triple.
6. 72 + 242 = 252 = 625; yes, this is a Pythagorean triple.
2. 82 + 152 = 172 = 289; yes, this is a Pythagorean triple.
7. Answers are:
a. The triple (18, 24, 30) was formed by multiplying each
number in the triple (3, 4, 5) by the whole number 6.
b. Example answer: (10,24,26) by multiplying by 2; and (15,
36, 39) by multiplying by 3.
c. Example answer: (16, 30, 34) by multiplying by 2; and
(24, 45, 51) by multiplying by 3.
2
2
2
3. 1 + 2 = 5; 3 = 9; no, this is not a Pythagorean triple.
4. 122 + 352 = 372 = 1,369; yes, this is a Pythagorean triple.
5. 42 + 162 = 272; ( 3 48 )2 = 432; no, this is not a Pythagorean
triple and one of the values is not a whole number.
Special Right Triangles
1. x = 7 2 mm
4. x = 40 ft
2. x = 8 cm
5. x = 16 2 inches
3. x = 10 m
6. x = 14 yards
Measuring Angles with a Protractor
Answers are:
Letter
Angle
Letter
Angle
A
56°
J
153°
B
110°
K
131°
C
10°
L
148°
D
96°
M
81°
E
167°
N
90°
F
122°
O
73°
G
34°
P
27°
H
45°
Q
139°
I
19°
Constructing Angles with a Protractor
There are no questions to answers for this skill builder; students
need only to construct the angles listed.
4
Functions: Conversions between Celsius and Fahrenheit
Example:
Practice set 2:
Input
10
-3
.5
4
-2
Output
102
11
2.25
18
6
Practice set 1:
Rule for the machine: y = 3(x) + 10
°Fahrenheit
°Celsius
45
7.2
81
27.2
0
-17.8
–4
-20
–10
-23.3
212
100
27.5
-2.5
Input
Output
32
0
6
28
–1
-18.3
0
10
100
37.8
7
31
– 9/5
43/5
–4
-2
– 5/9
81/3
Using a Graphing Calculator: Conversions between Celsius and Fahrenheit
Practice set 1:
City (Country)
Athens (Greece)
Berlin (Germany)
Buenos Aires
(Argentina)
Calgary (Canada)
Cairo (Egypt)
Istanbul (Turkey)
Melbourne (Australia)
Mexico City (Mexico)
Nairobi (Kenya)
Rio de Janeiro (Brazil)
Tokyo (Japan)
Practice set 2:
°Celsius (L1)
22
4
29
°Fahrenheit (L2)
71.6
39.2
84.2
7
18
10
9
20
33
24
15
44.6
64.4
50
48.2
68
91.4
75.2
59
°Fahrenheit
°Celsius
89.6
32
24.8
-4
32
0
-459.4
-273
78.4
25.8
210.2
99
-11.2
-24
100.4
38
212
100
-18.4
-28
89.24
31.8
122
50
26.6
-3
59.9
15.5
Ratios
1. Answers are:
a. 3/20
b. 2/14 =
c. 14/20 =
d. No, you only know the ratios and not the actual numbers.
The total number of pets could be 20 or 20,000—you
can’t tell from a ratio.
1
7
/7
/10
2. Answers are:
5
a. scale: 2 inches/9 miles = 0.22 inches per mile
h. $5/80 minutes = $0.0625 per minute or 6.25 cents per minute
b. 30 miles/4 hours = 7.5 mph
i. 40 pounds/2 weeks = 20 pounds of dog food per week
3. Answers are:
a. 11/8
b. 5/31
c. 2/14 = 1/7
d. 2/1
e. 11/1
f. 17/14
c. 112 miles/5 gallons = 22.4 mpg
d. scale: 2.5 inches/10 feet = 0.25 inches per foot
e. scale = 1 inch/6 3/8 feet = 0.157 inches per foot
f. 19 points per game
g. 35 grams per 5 cookies = 7 grams fat/cookie
Ratios and Proportions in a Recipe
1. Answers for the blanks in order: 6 tablespoons; 2 eggs
2. Answers for the blanks in order: 3/4 cup; 1/3 teaspoon
3. Answers for the blanks in order: 1/4 teaspoon; 3/4 cup
4. Table answers:
Butter
1 tablespoon
Chocolate chips
1 cup
Eggs
1 egg
Vanilla extract
1
Baking soda
1/
6
1
Confectioner’s sugar
1 tablespoon
/8 teaspoon
5. To make 16 brownies, you need two eggs and 2 table spoons
of sugar. Therefore, to make 8 brownies, you only need 1 of
each unit for each ingredient: 1 egg and 1 tablespoon.
3 tablespoons
Milk
Salt
6. Since 8 brownies requires 1 cup of chocolate chips, 3 cups of
chocolate chips will make 24 brownies.
7. 1.5 teaspoons vanilla is needed to make 24 brownies.
/2 teaspoon
teaspoon
Probability
d. 2/52 = 1/26
e. 39/52 = 3/4
f. 13/52 = 1/4
3. Answers are:
a. 3/5
b. 1/5
c. 0
d. 4/5
e. 2/5
1. 12 total
a. 2/12 = 1/6
b. 5/12
c. 5/12
d. 7/12
e. 10/12 = 5/6
2. Answers are:
a. 26/52 = 1/2
b. 1/52
c. 4/52 = 1/13
Reciprocals and Negative One as an Exponent
1. 4/3
11. -1/3b
12. 4/15
2. 5
3. 1/7
13. 3/17
5. -4
15. 6/d
6. -8/33
16. 9/8
8. -1/100
9. 1/20
18. 8/3a
14. 1/6d
4. 4/33
17. -1/8
7. -13/6
19. 25/6
10. 3/2
6