Download DIAGRAMMING A WORD PROBLEM

DIAGRAMMING A WORD PROBLEM
Often in a word problem, it may be easiest to draw a diagram to truly ‘see’ what is
happening in the problem. This concept can be applied to all kinds of word
problems.
One such example of a word problem that can be diagrammed is the following:
Paul is taking medications for his illness and needs to take an antibiotic every 6 hours, a pain
reliever every 4 hours, and every 3 hours he needs to drink an entire glass of water. Paul starts
this regime at 8am. At what time will he be taking both of the medicines and the water?
We can diagram this by setting up a schedule for Paul. I will use ‘A’ for antibiotic,
‘P’ for pain reliever, and ‘W’ for the glass of water. Since everything must be taken
in hours increments, the schedule can be set up by hours starting at 8am.
If Paul starts his regime at 8am, then they everything would be taken at 8am and that
is the hour we will count from for all of the other times the medication is taken. First
you could start by filling in all of the antibiotics.
8am 9am 10am 11am 12pm 1pm 2pm 3pm 4pm 5pm 6pm 7pm 8pm 9pm 10pm 11pm 12am
APW
A
A
Next, the pain reliever could be filled in.
8am 9am 10am 11am 12pm 1pm 2pm 3pm 4pm 5pm 6pm 7pm 8pm 9pm 10pm 11pm 12am
APW
P
A
P
AP
P
And then finally, the water can be added. Make note of where it is that all three fall
in the same hour!
8am 9am 10am 11am 12pm 1pm 2pm 3pm 4pm 5pm 6pm 7pm 8pm 9pm 10pm 11pm 12am
APW
W
P
AW
P
W
APW
You can tell by the clearly designed diagram, that all three, the antibiotic, the pain
reliever, and the glass of water, will be taken again at 8pm.
So we have our answer!
W
P
Making a chart is just one way of diagramming a problem. You may also want to
draw pictures. Drawing pictures and delegating quantities to areas of the picture can
help you organize the information.
Example:
A piece of wood 15ft long must be cut into 4 segments each 3.5ft in length. Once these pieces are
cut away, what is the remaining length of wood?
Start by drawing a picture of the wood and partition it into four segments: the three
cuts and the leftover piece.
3.5 feet
3.5 feet
3.5 feet
the leftover
15 feet total length
This shows us that in order to find the leftover portion, we must take our 15 feet and
subtract (cut away) 3 separate 3.5 feet lengths.
15 − 3.5 − 3.5 − 3.5
Or you could have demonstrated that you were cutting 3 segments of 3.5 feet with
multiplication.
15 − 3(3.5)
Either way you get 4.5 feet as your answer.
Sample Problems:
1. Angelica is starting a house-sitting service and needs to decide if she needs to
hire extra help. She will be visiting houses several times a week. She has four
clients already set up. Mrs. Meyers needs her house visited on Mondays,
Wednesdays, and Fridays. The Roberts need her to come by on Wednesday,
Thursday, and the weekend days. Mr. Haskins needs her to come every other
day starting Monday, and her neighbor Sam needs her to come by every 3
days starting Monday. If she has to visit more than 3 houses on any day
within the first three weeks, she will need to hire some help. Can Angelica do
this on her own?
2. Pearl is making a sandwich and she just discovered that it needs to serve 5
people. The sandwich is 17.5 inches long. How long will each person’s
individual sandwich be?
3. Someone cancelled on Pearl and now her sandwich (from the last problem)
only needs to serve 4 people. They each want an individual sandwich 4
inches long. Will there be any sandwich left over for Pearl? If yes, how long
will it be?
EXPONENTS
Exponents (or powers) are used to denote repeated multiplication. The exponent
(the little number in the upper right hand corner) tells you how many times to
multiply the base (the bigger number). For example, 32 = 3 ⋅ 3 = 9 and
53 = 5 ⋅ 5 ⋅ 5 = 125 .
When negative numbers are used with exponents, it is necessary to determine
whether the negative sign is part of the base (and therefore being included in the
repeated multiplication) or if it is not, and should be applied separately. This is
shown with the use of parentheses. If the negative is within the parentheses, then it
is part of the base. If there are no parentheses or the negative is displayed outside of
the parentheses, then the negative is not part of the base. For example, (−4) 2 means
that −4 is being multiplied twice. (−4) 2 = (−4)(−4) = 16 because a negative times a
negative is a positive. However, −42 which is the same as −(4) 2 is −(4 ⋅ 4) = −16 .
Sample problems:
Simplify.
1. 52
2. (−6)3
3. (−1) 4
4. −7 2
5. −(3)3
6. 43
7. −102
FRACTIONS
Fractions consist of two parts, a numerator (the top) and a denominator (the
bottom). To approximate the value of the fraction, take the top number (numerator)
and divide it by the bottom number (denominator) to find its decimal value
approximation. This way, you can compare it to other numbers to determine how
large or small it is.
An example:
3
= 3 ÷ 5 = 0.6 .
5
Sample Problems:
Find the decimal value of each fraction.
6
1.
5
2.
15
2
3.
3
4
4.
4
9
GREATEST COMMON FACTOR
A factor is something that is multiplied. When looking for the factors of a number,
you look for all of the numbers that can be multiplied to give you that number. For
example, factor of 12 would be 1, 2, 3, 4, 6, and 12, because 1× 12 = 12 , 2 × 6 = 12 ,
and 3 × 4 = 12 .
The greatest common factor (or GCF) is the largest factor that two or more numbers
have in common. So, if we are looking for the GCF of 12 and 18, we would locate
the factors of 12 and 18 and then pick out which is the largest one they have in
common.
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 8, 18
Although these two numbers have many factors in common, such as 1, 2, 3, and 6,
the number 6 is the largest of these and so it is the GCF.
GCF = 6.
Another way to think of it is that factors are numbers that evenly divide into your
number. For example, since 2 × 6 = 12 , then 12 can be divided by both 2 and 6. So,
you can look at two numbers and try to determine their GCF by finding the largest
number that divides into these numbers. For 12 and 18, although they are divisible
by many numbers that are the same, 6 is the largest number that divides into both.
Sample Problems:
Find the GCF of the following sets of numbers.
1. 14, 42
2. 16, 30
3. 120, 165
RADICALS
The most common radical, and the one you will be dealing with, is the square root.
Square roots ask you to determine what number squared (raised to a power of 2) will
give you the number placed under the radical sign. For example, 36 asks you to
determine what number squared will give you 36. Since, 62 = 36 , then the 36 = 6 .
Only certain numbers under the radical sign come out to be perfectly determined
numbers. These are the perfect squares. For example, 4, 16, and 64 can be
determined quite easily. But 5 or 12 are not as easy. An approximation of the
value of these can be found by using your calculator. Type in the number and then
hit the square root key to find its square root. This will give you a decimal value
approximation for this number so you can determine how large or small it is in
relationship to other numbers.
You may also be asked about a number multiplied by a radical, such as 3 2 . To
approximate this in your calculator, find the square root first and then multiply by
the number. So you would type 2, then the square root key, then times 3, and Enter
to get the decimal value approximation.
Sample Problems:
Approximate each radical.
1. 9
2.
15
3.
72
4.
121
5. 4 3
6. 8 4
ORDER OF OPERATIONS
You may have heard the phrase “Please Excuse My Dear Aunt Sally” or PEMDAS.
This is a common phrase intentioned to help you remember the order of operations.
There are six letters in this acronym but only FOUR steps in order of operations.
Step 1: Parentheses. This step requires you to perform any operations within
parentheses. If there are two sets of parentheses, then you start at the inner-most set
and work your way out. This step does not just apply to parentheses, but any other
grouping symbols as well. So, procedures inside of brackets, braces, or absolute
value bars would also need to be done first.
Step 2: Exponents. This step requires you to evaluate any number raised to an
exponent.
Step 3: Multiplication and Division done from LEFT to RIGHT. This step requires
you to perform any multiplication or division as you find it from left to right. That
means, if you find division before multiplication, you should perform it first.
Step 4: Addition and Subtraction done from LEFT to RIGHT. This step, like the
previous, requires you perform and addition or subtraction as you find it from left to
right. Again, this may require you to perform subtraction before addition.
Example:
2(3 + 8) 2 − 10 ÷ 2
First the parentheses: (3+8) = 11
2(11) 2 − 10 ÷ 2
Then, take care of the exponent: (11) 2 = 121
2(121) − 10 ÷ 2
Next, any multiplication/division from left to right. In this case, there is
multiplication first: 2(121) = 242 and then division: 10 ÷ 2 = 5 .
242 − 10 ÷ 2
242 − 5
There is only one step to perform and it should be either addition, subtraction, or
both. Take care of anything you find from left to right. For this problem, there is
only subtraction: 242 − 5 = 237 .
So, the answer is 237.
Example: 8 − 2 ÷ 2 + 3(4)
Although there are parentheses, there is nothing to do inside of them, so there is no
parentheses step. Also, there are no exponents. So, we start with step 3 and do any
multiplication or division from left to right.
There is division first: 2 ÷ 2 = 1 , and then multiplication: 3(4) = 12
8 − 1 + 3(4)
8 − 1 + 12
Then, finish up any addition/subtraction from left to right. For this problem, first we
perform subtraction: 8 − 1 = 7 , and then the addition: 7 + 12 = 19
8 − 1 + 12
7 + 12
= 19
So, the answer is 19.
You may be asked for the order of operations to a problem and not the actual
answer. The answer choices will be a series of operations that you would perform.
You just have to pick the right order.
Example: 15 + 8 ÷ 2 × 7
Since there are no parentheses or exponents, you would do the division first (because
it is step 3 left) and then the multiplication (because it is step 3 right), and lastly the
addition (because it is step 4).
So your answer would be: ÷× + in that order.
Sample Problems:
Simplify.
1. 5 + 3(2 − 8) 2
2. −3(9 + 2 − 1) + 23
3.
4 + 5(2) 2
7 −1
List the order of operations you would use to solve the problem.
4. 12 − 6 ÷ 3 − 4 ÷ 2
5.
16
+ (13 − 5)(−1)
4
PERCENTS
Percents are characterized by a percent sign at the end of the number. For example,
23% is twenty-three percent.
The value of a percent can be determined by converting into decimal form. This is
done by moving the decimal point in the percent 2 places to the LEFT. So, in 23%,
the decimal is at the end of the number 23.
Moving the decimal two places left would make 0.23
This is also found in the calculator by dividing by 100. For example, 15% can be
converted to a decimal by typing 15 ÷ 100 into your calculator. A result of 0.15
should be found.
Also, you may need to convert a decimal into a percent. This is simply the opposite
process as above. The decimal point should be located in the decimal value and
moved 2 places to the RIGHT. So, in the number 0.45, the decimal point is before
the 4.
After it is moved two places to the right, the percent would be 45%
This is also found in the calculator by multiplying by 100. For example, 0.981 can be
converted to a percent by typing 0.981× 100 into your calculator. A result of 98.1
should be found and once you add the percent sign, you will have 98.1%
T2 − T1
where T1
T1
represents the first value in the problem and T2 represents the second value in the
problem. The first and second values are according to time, not placement in the
problem. If you buy a house and then sell it, the value at which you bought it
occurred first and the value at which it is sold is the second value.
Percent increase or decrease is found by using the formula
You will have multiple choice answers to choose from, so if your first value is
smaller than your second value, then that is an obvious increase. Where if your first
value is larger than your second value, there is an obvious decrease. With a
decrease, you should also result in a negative value. Positive values resulting from
the formula are increases and negative values are decreases. This can help you rule
out some of your answer choices. If, for example, you are dealing with a problem
that says you bought a house for $150,000 and then sold it for $250,000, there is an
obvious increase and so any answer choices that say there was a decrease can be
ruled out.
Example:
A house is bought for $150,000 and then sold for $250,000. Find the percent
increase or decrease.
The first value ( T1 ) is $150,000 and the second ( T2 ) is $250,000. As stated above, this
is an obvious increase which may rule out some of your answer choices.
T2 − T1 250000 − 150000 100000
=
=
T1
150000
150000
So taking 100000 ÷ 150000 = 0.66666667 and then converting this into a percent, this
is approximately a 66.67% increase.
Example:
A house is bought for $600,000 and was sold later for $525,000. Find the percent
increase or decrease.
T1 is $600,000 and T2 is $525,000. This is an obvious decrease which would rule out
answer choices stating it was an increase.
T2 − T1 525000 − 600000 −75000
=
=
T1
600000
600000
So taking −75000 ÷ 600000 = −0.125 and then converting this into a percent and
taking the negative as a decrease, this is approximately a 12.5% decrease.
Sample Problems:
Convert these percents to decimals.
1. 33%
2. 19.4%
3. 2.7%
4. 125%
5. 76%
Convert these decimals to percents.
6. 0.17
7. 0.2355
8. 2.99
9. 0.002
10. 0.684
Find the percent increase or decrease.
11. In 1999, Jane bought a house for $180,000. In 2005 she sold the house for
$229,000.
12. Tim bought a stock at $14 a share and later sold it for $10 a share.
SCIENTIFIC NOTATION
The format for scientific notation is a decimal number between zero and ten,
multiplied by 10 raised to a positive or negative power.
Some example of scientific notation: 1.5 ×108 , 6.11×10−3 , 3.459 ×104
In order to find the value of these numbers written in scientific notation, you convert
them to what is called “standard form”. This is done by moving the decimal point in
the first number however many times the exponent on the 10 says. The sign on the
exponent tell you the direction to move it. So, if your exponent is − 5, you would
move the decimal 5 places to the left. If the exponent, is 7, then you move the
decimal 7 places to the right.
Examples:
1.5 ×108 tells you to move the decimal that is between the 1 and the 5, eight places to
the right. So, 1.5 ×108 = 150000000
6.11× 10−3 tells you to move the decimal that is between the 6 and 11, three places to
the left. So, 6.11×10−3 = .00611
3.459 ×104 tells us to move the decimal that is between the 3 and the 459, four places
to the right. So, 3.459 ×104 = 34590
By converting to standard form, you can see the true value of the number and
compare it to other numbers to determine if it is larger or smaller.
Sample Problems:
Convert to standard form.
1. 3.8 ×10−6
2. 9.12 × 10−2
3. 2.2 ×105
4. 1.07 × 103
SURFACE AREA
Surface area refers to area of all of the faces of a 3-dimensional figure, in other
words, the ‘area of the surface’. Surface area is another type of area and so its units
are similar to area units and will again be squared.
The surface area formulas are given on the formula sheet.
The surface area of a prism or pyramid is the sum of the areas of all the faces. So,
whatever shape the faces are, you will use that formula from the area section and
compute it until you have the area of every face in the figure. Add them all up and
you have your surface area.
Example:
12 ft
5 ft
26 ft
This is a rectangular prism and its faces are
all rectangles. So, the area of the front and
back are the same, the top and bottom are
the same, and the right and left side are the
same. The area formula for a rectangle is
A = lw so that is what we will use for each
of the faces.
First we will compute the area of the front
and back faces. The dimensions of this
rectangle are 26 by 12 ft. So,
A = lw = (26)(12) = 312 . Since there are two
of these faces (front AND back), we will
have (312)(2) = 624 ft 2 .
The dimensions of the top and bottom face are 26 by 5 ft. So, A = lw = (26)(5) = 130 .
There are two of these faces (the top AND bottom) so we will have
(130)(2) = 260 ft 2 .
The right and left face are both 12 by 5 ft rectangles. So, A = lw = (12)(5) = 60 .
Again there are two of these faces (right AND left) so we will have (60)(2) = 120 ft 2 .
Don’t worry, we’re almost done! All we have left to do is add up all of the individual
areas for the faces to get the total surface area (in other words, the area of the entire
surface). There are six faces but half of them were duplicates so we already created
our three individual areas .
S . A. = 624 + 260 + 120 = 1004 ft 2
Example:
The following figure has face triangles each with a height of 20 meters.
6.25 m
14.5 m
This figure is a rectangular pyramid. It has a
rectangle base but the other faces are all
triangles. The base area can be found using
A = lw where the length and width are 14.5
meters and 6.25 meters.
A = lw = (14.5)(6.25) = 90.625m 2 .
Although all four of the sides are triangles, they have different bases. The front and
back faces have a base of 14.5 meters while the side triangles have a base of 6.25
meters. So we will compute each separately with a height of 20 meters.
1
1
So we can compute the front and back triangle by: A = bh = (14.5)(20) = 145m 2 .
2
2
We need two of these because there is one in front and one in the back:
(312)(2) = 624 ft 2
1
1
The two side triangles can be computed by: A = bh = (6.25)(20) = 62.5m 2 . Again,
2
2
we need two of these because there are two side triangles: (62.5)(2) = 125m 2 .
Ok, now we just need to add everything up! To find the surface area of this figure we
need to add up the areas of all four triangles and the rectangular base.
S . A. = 90.625 + 624 + 125 = 839.625m 2
The surface area of a cylinder equals the sum of the bases and its rectangular wrap.
The formula is: S . A. = 2(π r 2 ) + 2(π r )h . A value of 3.14 will probably be given to use
for π . h represents the height of the cylinder and r the radius (or distance from the
center of the circle at the top or bottom to the edge of the cylinder).
Example:
r = 3 in
This cylinder has a radius of 3 cm and a
height of 8 cm.
S . A. = 2(π r 2 ) + 2(π r )h = 2(π 32 ) + 2(π 3)(8) = 2(9π ) + 2(3π )(8) = 18π
h = 8 in
Let π = 3.14 and then we have
S . A. = 66π = 66(3.14) = 207.24 .
So the surface area of this cylinder is 207.24
in 2 .
The surface area of a sphere is given by the formula S . A. = 4π r 2 again with π and r
for the radius.
Example:
Find the surface area of this sphere with radius 91 mm.
Using the surface area formula provided, and using π = 3.14 we find:
S . A. = 4π r 2 = 4(3.14)(91)2 = 4(3.14)(8281) = 104009.36mm 2 .
Sample Problems:
Find the surface area of the following figures.
1.
r = 24 ft
h = 25 ft
2.
7.5 ft
2 ft
4 ft
3. A sphere with radius 7 inches. Use π = 3.14 .
AREA
The area of a figure is determined by finding a value for the space within the figure. There
are several formulas for area depending on the figure. These can be found on your provided
formula sheet listed under AREA. There is a formula for a triangle, rectangle, trapezoid,
parallelogram, and a circle. The units of area are always squared. So, if your measurements
are all in feet , then your answer for the area will be in ft 2 .
1
The triangle formula is A = bh where b is the base, or the part the triangle is resting on,
2
and h is the height, or the vertical distance from the highest point of the triangle to the base.
The height and base are perpendicular, a word which means these two lines intersect at a
90° angle. The height is not necessarily one of the sides of the triangle but can be located
inside or even outside the triangle.
Example:
30 m
The height of this triangle is inside the triangle. The height is 30
meters and
21 mthe base is 21 meters.
1
1
A = bh = (21)(30) = 315 . So, the area of this triangle is
2
2
2
315 m .
Example:
12 ft
The height of this triangle is one of the sides
because it is a right
(which means it contains that desired angle of 90° ).
15 triangle
ft
1
1
A = bh = (15)(12) = 90 . So, the area of this
2
2
triangle is 90 ft 2 .
The rectangle formula is A = lw where l is the length, or the longer side of the rectangle, and
w is the width, or the shorter side of the rectangle. The sides across from each other are the
same length or width value.
Example:
4 km
The length of this rectangle is 12 km and
the width is 4 km.
A = lw = (12)(4) = 48 . So, the area of the
rectangle is 48 km .
12 km
2
1
h(b1 + b2 ) where h is the height just like in the triangle
2
formula, b2 is a base the trapezoid is resting on, and b1 is the other base the trapezoid is not
resting on. Bases b1 and b2 are parallel, a word which means that these lines will never
intersect, to each other. Make sure to use order of operations when using this formula. The
bases should be added first and then the multiplication should be done.
The trapezoid formula is A =
Example:
25 cm
10 cm
The trapezoid has a height of 10 cm, and the two bases
are 25cm and 12 cm.
12 cm
the area of the trapezoid is 185 cm 2 .
A=
1
1
1
h(b1 + b2 ) = (10)(12 + 25) = (10)(37) = 185 . So,
2
2
2
The parallelogram formula is A = bh where b is the base, or the side the parallelogram is
resting on, and h is the height which is again, just like in the triangle formula.
Example:
72 mm
The parallelogram has a height of 72 mm and the base is
182
182 mm.
A = bh = (182)(72) = 13104 . So, the area of the
parallelogram is 13,104 mm .
2
The circle formula is A = π r 2 where π is the symbol for pi, and r is the radius or distance
from the center to the outside of the circle. You should be instructed in the problem if you
are to use a value for pi. Usually you will be asked to use π = 3.14 when doing your
calculations. Again, be careful of order of operations. Make sure to square the radius value
first, and then multiply by pi.
Example:
Let π = 3.14 for the following problem.
8m
The circle has a radius of 8 meters. A = π (8) 2 = 64π .
Since π = 3.14 , A = 64π = 64(3.14) = 200.96 . So, the area of the circle is 200.96 m 2 .
You may be asked to find the area of something that is not one of these defined figures but is
actually a combination of figures. Assess which figures are being combined and separate
your drawing into these smaller figures. You can find the area of each of the smaller figures
and add those areas to find the total area of the combined figure.
Example:
Find the area of the building layout below.
48 ft.
20 ft.
41 ft.
20 ft.
Separate the figure into smaller defined figures. This will help you assign values to those
figures more carefully.
48 ft.
20 ft.
41 ft.
20 ft.
Since one of the figures is a square and one of the sides is 20ft., then all the sides of the
square are 20ft. The other figure is a rectangular and we know one of the sides is 41ft. The
other side, combined with the square, is 48ft. Since the square takes up 20ft of the 48ft, we
can subtract to find the other side of the rectangle: 48 – 20 = 28ft.
20 ft.
28 ft.
20 ft.
20 ft.
41 ft.
Now, we can calculate the area of each smaller figure. The square is the same formula as the
rectangle except its sides are the same so A = 20 ⋅ 20 = 400 . The rectangle area can be
computed by A = 28 ⋅ 41 = 1148 . So, the total area can be calculated by adding the area of the
two smaller figures: A = 400 + 1148 = 1548 .
So, the area of the building is 1548 ft 2 .
Sample Problems:
Find the area of the following figures.
1.
16 ft.
7.5 ft.
9 ft.
2.
1.25 m
0.6 m
3. A circle with a radius of 22 cm. Let π = 3.14 .
4. A parallelogram with base 103 m and height 217 m.
5.
13 ft
17 ft
17 ft
DISTANCE
The distance formula provided on your formula sheet is d = rt which is distance
equals rate × time . In whatever problem you do, you will need to determine which
two values are given and solve for the third one. Distance will be a measured in feet,
or meters, or another length measurement. Rate is the speed the vehicle or person is
going, and time will be measured in hours or minutes or some time increment.
Example:
Matt is traveling at a speed of 60mph toward his destination. After 5 hours of
driving, how many miles did he travel?
In this problem, we have been given the speed (60 mph) and the time (5 hours) and
we are left to find the distance.
d = rt
d = (60)(5)
d = 300
So, the answer is a distance of 300 miles.
Example:
Francine is driving to the Keys for Spring Break from her home in Orlando. The trip
will be a distance of 378 miles. Francine is going to leave at 10am but she will stop
for a 45 minute lunch and a 15 minute fuel stop. She arrives in Key West at 5:45pm.
What was her average rate of speed during her drive?
A) 54 mph
B) 49 mph
C) 63 mph
D) 56 mph
If we are going to use our distance formula, we need to identify the pieces we have
and what we are missing. We are given a distance of 378 miles. We also know what
time she leaves and arrives. She leaves at 10am and arrives at 5:45pm. That is a
total of 7.75 hours (since 45 minutes is 0.75 of an hour). However, she made stops
that we will subtract from the total time. She made a 45 minute stop for lunch and a
15 minute stop for fuel. This is a total of 1 hour of time she was not actually driving.
So we need to take this amount out of the total driving time.
7.75 – 1 = 6.75 hours.
So we have a distance and a time but no rate. That is what we need to solve for.
d = rt
378 = r (6.75)
In order to solve for the rate, or speed, we can plug in the various multiple choice
answers to see which one works.
Plugging in (A) we get 378 = r (6.75) = (54)(6.75) = 364.5 . But since 378 ≠ 364.5 , this
is not our answer.
Plugging in (B) we get 378 = r (6.75) = (49)(6.75) = 330.75 . But since 378 ≠ 330.75 ,
this is not our answer.
Plugging in (C) we get 378 = r (6.75) = (63)(6.75) = 425.25 . But since 378 ≠ 425.25 ,
this is not our answer.
Plugging in (D) we get 378 = r (6.75) = (56)(6.75) = 378 .
And since 378 = 378 , the rate of 56 mph is our answer!
Sample Problems:
1. Tonya goes for a 2 hour jog in the park. The path she jogs is 8 miles long. At
what rate is Tonya jogging?
2. Melvin starts his bike ride a 7am and finishes at 10am. He stopped for a bite
to eat along the way at a local street vendor and stayed for about 30 minutes.
He was riding his bike at a rate of 10 mph. How far did he ride?
A) 75 miles
miles
B) 25 miles
C) 30 miles
D) 3
3. How long did it take Sam to drive a distance of 9.5 miles to his friend’s house
if he was driving 25 mph?
A) 0.38 hours
D) 2.63 hours
B) 0.4 hours
C) 237.5 hours
MEASUREMENT CONVERSIONS
There are many conversions available on your formula sheet. You just need to know
how to use that information to convert from one measurement to another.
Here are the conversions you are given on your formula sheet and thus what you
have to work with:
1 yard = 3 feet = 36 inches
1 mile = 1,760 yards = 5,280 feet
1 acre = 43,560 square feet
1 hour = 60 minutes
1 minute = 60 seconds
1 cup = 8 fluid ounces
1 pint = 2 cups
1 quart = 2 pints
1 gallon = 4 quarts
1 pound = 16 ounces
1 ton = 2,000 pounds
1 liter = 1000 milliliters = 1000 cubic centimeters
1 meter = 100 centimeters = 1000 milliliters
1 kilometer = 1000 meters
1 gram = 1000 milliliters
1 kilogram = 1000 grams
Also, it is mentioned on your given sheet that metric numbers are customarily
presented without a comma when four digits long (Example: 9960 kilometers) and if
the number is greater than four digits will use a space instead of a comma (Example:
12 5000 liters).
So, here are some examples of problems where you will need to convert the numbers
you are working with in the problem to another measurement for your answer.
Example:
Anastasia is starting her run at 8:30am. She is training for a marathon and allows
herself only 150 minutes to complete her run and reach her desired destination. At
what time will she finish her run?
Since every hour is 60 minutes, then 150 minutes can be broken down into groups of
60 to determine the number of hours.
150 = 60 + 60 + 30 minutes
So there are two sets of 60 minutes, or two hours, and an additional 30 minutes, or
half an hour.
She started her run at 8:30am. 2 hours and 30 minutes later would be 11:00am.
Anastasia will finish her run at 11:00am.
Example:
A recipe for 4 servings of the non-alcoholic drink ‘Fuzzy Noel’ calls for 3 cups peach
1
sorbet, 1 cups orange juice, and 4 cups sparkling water. How many fluid ounces is
2
this?
1
1
This recipe calls for 3 + 1 + 4 cups of ingredients. This is a total of 8 or 8.5 cups.
2
2
One cup is 8 fluid ounces. We have 8.5 cups. So, (8)(8.5) = 68 fluid ounces.
There are 68 fluid ounces in this recipe.
Sample Problems:
1. Pete has to mow three lawns. The first is an acre, the second is 21,000 square
feet and the third is 3 acres. How many total square feet must he mow?
2. Jane needs a total 8 yards of fabric. She finds six feet in one bin and 184
inches in another. Does she have enough?
3. Talia is told she will have 205 minutes and 300 seconds to complete her
exam. How many hours is that?
PERIMETER & CIRCUMFERENCE
Perimeter is the distance around an object. It is all of the sides of your figure added
together. You don’t need a formula for perimeter because no matter what your
figure is, you can add the lengths of all the sides to find its perimeter.
Example: A rectangle has a length of 4 ft and width of 3 ft. Find the perimeter of
this rectangle.
4 ft
3 ft
3 ft
The perimeter is all the sides added
4 ft
up so that would be: P = 4 + 3 + 4 + 3 = 14 . The perimeter of the rectangle is 14 ft.
Example: Find the perimeter of the following figure.
4.5 in
1 in
1 in
2 in
This figure has 4 different lengths
that repeat as we go around the
figure. Starting at the top and
moving around the figure
clockwise matching it to the given
side lengths we can calculate the
perimeter by adding the top +
corner + right side + corner +
bottom + corner + left side +
corner.
P = 4.5 + 1 + 1 + 2 + 1 + 1 + 4.5 + 1 + 1 + 2 + 1 + 1 = 21
So, the perimeter of this figure is 21
i
n
c
h
e
s
.
Circumference is just like perimeter in that it is the distance around the object but it
refers specifically to a circle. Circles only really have one side so you can’t add up all
the sides of this figure. The formula for circumference is located on the formula
sheet and it is C = π d = 2π r . The value of 3.14 may be given for π (pi) in the actual
problem. In this case, you should replace π with 3.14 when computing the
circumference. There are two formulas provided here for computing the
circumference. Both use pi but one uses the radius r and the other uses the diameter
d . The radius is the distance from the center to the outside of the circle and the diameter is the
distance from one side of the circle to the other, passing through the center. So, it will just
depend on what you are given in the problem as to which version of the formula you
should use.
Example: Find the circumference of the given circle. Use π = 3.14 .
5 cm
This circle shows a diameter of 5
cm. So, the circumference is C = π d = π (5) . With π = 3.14 ,
C = π (5) = (3.14)(5) = 15.7 .
The circumference of this circle is
15.7 cm.
Example: Find the circumference of the given circle. Use π = 3.14 .
2.35 mm
This circle shows a radius of 2.35
mm. So, the circumference is
C = 2π r = 2π (2.35) .
With π = 3.14 ,
C = 2π (2.35) = 2(3.14)(2.35) = 14.758
.
The circumference of this circle is
14.758 mm.
Sample Problems:
Find the perimeter of the following figures.
1.
12 in
12 in
18 in
2.
9 cm
4 cm
10 cm
8 cm
Find the circumference of the following circles. Use π = 3.14 .
3.
17 ft
4.
4 in
RATIOS & PROPORTIONS
Ratios are fractions pairing two values into the numerator and the denominator.
The word “per” is often used for the position of the fraction bar. Some common
miles
ratios are ‘miles per gallon’ =
and ‘cost per unit’ such as a bag of potato chips
gallon
$1.50
which cost $1.50 for a 3 ounce bag which is the same as
.
3ounces
Proportions are equations that set two ratios equal. There will often be a variable
12 3
somewhere in the proportion. An example is:
= .
x 5
Since we need to find the solution to this equation, use the multiple choice answers
to plug in for the variable until you get a true statement.
Example:
12 3
=
x 5
A) 15
B) 20
C) 22
D) 26
12 3
12
3
= . But,
= 0.8 whereas = 0.6 .
(15) 5
15
5
Since 0.8 ≠ 0.6 , answer choice (A) is not correct.
So, plugging in answer choice (A), we get:
Trying answer choice (B), we get:
12
3
12
= . The first fraction is:
= 0.6 and the
(20) 5
20
3
= 0.6 . So these fractions (or ratios) are equal and answer
5
choice (B) is correct.
second fraction is :
You may also need to set up your own proportion. Remember the proportion
above? Proportions are commonly read in this way: 12 is to x as 3 is to 5. This can
help when trying to write your own proportion because you want to try to find that
relationship among the words in the problem. Look for two numbers that have a
relationship and another two numbers that have the same relationship. These will be
your two ratios you will set equal.
Here’s an example:
Linda was pumping gas in her car and stopped part way through to notice that she
had put in 4 gallons and it already cost her $9.40. She continued pumping and when
she was finished her bill was $20.68. How many total gallons did Linda put in her
car?
A) 8.8 gallons
B) 9.2 gallons
C) 48.6 gallons
D) 2.2
gallons
There is a relationship between the number of gallons and the cost of those gallons.
9.40
because that represents the ‘cost
So, a ratio that would make sense would be
4
per gallon’. We can set up another ratio that models our first one (cost per gallon)
dealing with the total and set the two equal. Since we do not know the total number
20.68
of gallons, we will let that be our variable x. So our second ratio would be
.
x
Our proportion will be
9.40 20.68
=
.
4
x
Solving this proportion, like in the first example, would simply consist of plugging in
our answer choices.
Plugging in our first answer choice (A) we get:
9.40 20.68
=
. Checking each ratio,
4
(8.8)
9.40
20.68
= 2.35 and
= 2.35 . So these ratios are equal and answer (A) is correct.
4
8.8
No need to check any further!
One application of proportions you may encounter is the idea of similar triangles.
This is where you will have two figures that are similar in shape but are different
sizes. Think of a dollhouse that is built as a replication of a real house. Everything
in the dollhouse is smaller but it is scaled the same as the larger house. These two
‘similar’ figures can be compared with a proportion and we call them ‘proportional’.
Example:
For the following similar triangles, find the length of the missing side.
x
14 ft
16 ft
A) 9.25 ft
10 ft
B) 8.75 ft
C) 11.43 ft
D) 12.35 ft
We will set up a ratio for each triangle keeping the sides we are comparing in the
same placements of the ratio. In other words, we can put the left side of the triangle
in the numerator and the bottom side of the triangle in the denominator. We will do
this for both ratios.
The bigger triangle ratio is
14
x
and the smaller triangle ratio is
.
16
10
So our proportion for these similar triangles is
14 x
= .
16 10
Next we check our answer choices until we find the one that works:
Plugging in our first answer choice (A) we get:
14 (9.25)
=
. Checking each ratio,
16
10
14
9.25
= 0.875 and
= 0.925 . Since 0.875 ≠ 0.925 , answer choice (A) is not correct.
16
10
Plugging in our first answer choice (B) we get:
14 (8.75)
=
. Checking each ratio,
16
10
14
8.75
= 0.875 and
= 0.875 . These ratios are equal and so answer choice (B) our
16
10
answer!
Sample Problems:
Solve the Proportion.
1.
10 a
=
16 8
A) 2
2.
B) 4
C) 5
D) 12
B) 5
C) 8.1
D) 4.8
1.35 2.70
=
3
y
A) 6
Set up and solve a proportion for the given information.
3. James is trying to duplicate a recipe he found in a magazine but he doesn’t
want to make as many servings as the recipe calls for. He needs 0.75 pounds
of asparagus for every 2 tbsp of almonds. But he only has 0.3 pounds of
asparagus. How many tablespoons of almonds should he use?
A) 1 tbsp
B) 1.2 tbsp
C) 5 tbsp
D) 0.8 tbsp
4. The distance between two cities is 14.5 cm on a map. 1 cm is equivalent to 18
miles. How far is it between these two cities?
A) 261 miles
B) 1.24 miles
C) 0.8 miles
D) 255 miles
5. A 5.5 ft woman standing in the park in the afternoon casts a shadow 10 ft
long. A nearby tree casts a shadow 18 ft long. How tall is the tree?
A) 10.5 ft
B) 9.9 ft
C) 8.9 ft
D) 32.7 ft
VOLUME
Volume is the word we use to describe the space within a 3-dimensional figure. For
example, what it would take to fill a box would be considered its volume.
The units for volume are always based on the 3-dimensions of that figure. So, the
units are cubed. For example, if a box has dimensions all measured in feet , then the
volume would be in ft 3 .
There are a few formulas provided for you in the formula sheet. You are given the
formula for the volume of a prism or cylinder, a pyramid or cone, and a sphere.
A prism and cylinder both use the same formula: V = Bh where B is the area of the
base and h is the height. So, you may have to find the area of whatever shape the
base is first, and then multiply by whatever the height is.
Example:
12 ft
5 ft
26 ft
This figure is a rectangular prism. We
need to find the area of the base first.
The base is a rectangle so
A = lw = (26)(5) = 130 . Then we
multiply that by the height to find our
volume: V = (130)(12) = 1560 ft 3 .
Example:
r = 3 in
h = 8 in
This figure is a cylinder which has a
circle base. First we will find the area of
the base: A = π r 2 = π (3) 2 = 9π . If we
are asked to use π = 3.14 , then
A = 9π = (9)(3.14) = 28.26 . Then we
multiply by the height to find our
volume: V = (28.26)(8) = 226.08in3 .
1
A pyramid and cone also use the same formula. That formula is V = Bh where
3
again B is the area of the base and h is the height. When computing the final answer
in your calculator, divide out the fraction first and then continue by multiplying by
the base area and the height.
Example:
h = 20 m
This is a pyramid with a rectangular
6.25 m
base. B14.5
is the
area
of
that
base
so
B
=
lw
=
(14.5)(6.25)
= 90.625.
m
Now implement the volume formula
rounding to two decimal places:
1
1
V = Bh = (90.625)(20) ≈ 604.17m3 .
3
3
4
The formula for a sphere is V = π r 3 where a value for π may be given in the problem.
3
Usually π = 3.14 is the most common value given to use. The radius r is the distance
from the center of the sphere to a point on its surface. Be careful when computing
this in your calculator. You may want to find the value of the fraction, write it down
and set it aside as you follow the order of operations, starting with the radius cubed.
Example:
Find the volume of this sphere with radius 91 mm using π = 3.14 . Approximate to
two decimal places.
This one is as easy as plugging in the
4
4
4
numbers: V = π r 3 = (3.14)(91)3 = (3.14)(753571) ≈ 3154950.59 . So, the volume
3
3
3
3
is 3,154,950.59 mm .
Sample Problems:
Find the volume of the following figures.
1.
2.
3. A sphere with radius 7 inches and π = 3.14 .
TRANSLATIONS & SYMMETRY
Geometric figures can be moved without altering the shape or size of the figure.
Such figure motions can be referred to as reflections, rotations, and translations.
A reflection of a figure is a mirror image of the original across a reflection line.
B
B’
A
A’
D
D’
C
C’
Reflection Line
A rotation of a figure is the movement of the original around a fixed point called the
point of rotation.
A
C’
A’
C
B
Rotation Point
A translation, or glide, moves a figure by sliding it across a straight line.
B
C
A
B’
A’
D
C’
E
D’
E’
If a reflection or rotation causes the image to be the same as the original, then the
figure has reflective or rotational symmetry, respectively.
A
D’
D
B
C’
A’
B’
C
Rotational Symmetry
through a 90 degree angle
Reflective Symmetry
with 2 Lines of Reflection
Sample Problems:
A
B
F
C
E
D
C’
B’
D’
A’
E’
F’
1. What type of transformation moved ABCDEF to A’B’C’D’E’F’ ?
A.
B.
C.
D.
translation
rotation
reflection
regular
2. If figure ABCDEF were reflected over the y-axis, state the coordinates of the
reflected image’s vertices.
3. If figure ABCDEF is translated by sliding vertex A to (-4, -1), what will be
the coordinates of the rest of the translated image’s vertices?
THREE-DIMENSIONAL SOLIDS
There are many types of three-dimensional solids. We classify them by the shape
and number of surfaces or faces.
A rectangular solid has six rectangular faces. It can also be referred to as a
rectangular prism, but we commonly think of rectangular solids as boxes. A cube is
a special box with 6 congruent faces.
A pyramid can have 4 or more faces, at least three of which must be triangles. We
call one of the faces the base, which can be any regular polygon.
A cone looks like a pyramid, but with a circular base.
A cylinder has two circular bases connected by a rectangular wrap. Think of a soup
can with a circular top and bottom, and the label is the rectangular wrap.
A sphere is a three-dimensional circle, or a ball. Every point on the surface of the
ball is the same distance from the center.
You may be asked to find the Surface Area or Volume of these shapes. There are
separate sections you can reference for information about how to find the volume or
surface area. The formulas for these are located on the provided formula sheet.
ANGLES
An angle is the union of two rays. The two rays meet at a common point called the
vertex of the angle. Angles are named by 3 letters and the vertex must be the middle
letter. For example, this is angle ABC with vertex B. We use m∠ ABC to represent
the ‘measure’ in degrees of angle ABC.
A
vertex B
C
Angles can be classified according to their angle measure. A straight angle measures
exactly 180 degrees and a right angle measures exactly 90 degrees. An angle with a
measure between 0 and 90 degrees is an acute angle. An angle with measure
between 90 and 180 degrees is obtuse.
E
A
B
C
G
F
Right Angle
m∠ EFG = 90 degrees
Straight Angle
m∠ ABC = 180 degrees
I
J
K
Acute Angle
m∠ IJK is less than 90 degrees
M
N
O
Obtuse Angle
m∠ MNO is greater than 180 degrees
Pairs or groups of angles can also be classified according to the sum of their angle
measures. Supplementary angles have a sum of 180 degrees while complimentary
angles have a sum of 90 degrees.
D
A
B
C
Supplementary Angles
m∠ ABC + m∠ DBC = 180 degrees
E
H
G
F
Complimentary Angles
m∠ EFG + m∠ HFG = 90 degrees
Adjacent angles are angles that share a vertex and a side, but do not overlap. For
example, the supplementary angles above are adjacent because they are next to each
other.
Certain types of angles have relationships that cause them to be equal in measure.
When two lines intersect, the non-adjacent angles formed are called vertical angles.
Vertical angles are equal in measure. There are two pairs of vertical angles in the
following figure.
1
4
2
3
Vertical Angles
m∠ 1 = m∠ 3 and m∠ 2 = m∠ 4
The following figure shows two parallel lines cut by a transversal. This figure
demonstrates alternate interior angles, alternate exterior angles, and corresponding
angles.
Alternate interior angles are those between the parallel lines, but on the opposite side
of the transversal. For example, there are two pairs of alternate interior angles in the
figure. Alternate interior angles are also equal in measure. So, m∠ 3 = m∠ 5 and
m∠ 4 = m∠ 6.
Alternate exterior angles are those NOT between the parallel lines, and on the
opposite side of the transversal. There are also two pairs of alternate exterior angles
in the figure. Alternate exterior angles are also equal in measure. So, m∠ 1 = m∠ 7
and m∠ 2 = m∠ 8.
Corresponding angles are on the same side of the transversal, but one in an interior
angle and the other is an exterior angle. There are four pairs of corresponding angles
in the figure. Corresponding angles are equal in measure. So, m∠ 1 = m∠ 5, m∠
2 = m∠ 6, m∠ 3 = m∠ 7, and m∠ 4 = m∠ 8.
1
4
5
8
6
7
Sample Problems:
3
2
Using the figure, answer the following questions.
1
4
5
8
2
3
6
7
1. Name all pairs of adjacent angles.
2. Name all pairs of vertical angles.
3. Name all pairs of corresponding angles.
4. Name all pairs of supplementary angles.
5. Name all pairs of complimentary angles.
6. Name all pairs of alternate interior angles.
7. Name all pairs of alternate exterior angles.
8. If m∠ 6 = 37 degrees, find the measures of every other angle.
9. Which type of angle is the complement of an angle with measure of 45
degrees?
A.
B.
C.
D.
right
acute
obtuse
straight
LINES & SEGMENTS
A point is a location that is sometimes represented by a capital letter.
A line is infinite in length, meaning it goes forever in both directions. A line is
determined by two points. The symbol for a line is the letters of the
suurtwo points with
a line above them. For example, the symbol for the line below is AB . Notice the
arrows in both directions on the line and the symbol.
A
B
A line segment is part of a line. It is represented by its endpoints’ letters with a
segment above them. For example, the symbol for the segment below is CD
C
You may be asked to find the midpoint, length, or slope of a given line segment.
These formulas are given on your formula sheet. You can find the midpoint, length,
or slope of a segment if you are given a graph or if you are given the endpoints of the
line segment.
SLOPE
MIDPOINT
m=
rise y2 − y1
=
run x2 − x1
⎛x +x y +y ⎞
M =⎜ 2 1, 2 1⎟
2 ⎠
⎝ 2
LENGTH or DISTANCE between two points
D=
( x2 − x1 ) + ( y2 − y1 )
2
2
For example, given the following graph, we will find the slope, length, and
midpoint.
D
We have two options for finding the slope.
We can ‘count’ the slope on the graph by using the relationship “rise over run”.
You must choose a point to start at. Let’s start at B. Count the rise (change in y or
vertical movement) between the two points. Moving up indicates a positive value
and down indicates a negative value. The rise value is the numerator of the slope.
Then count the run (change in x or horizontal movement) and that will be our
denominator of the slope. Moving right indicates a positive value and left indicates a
negative. So, put your two values together in a fraction for the slope.
For our problem, we rise up 3 places so the rise is positive 3, and we run right 10
places so our run is positive 10.
Combining these, we get the following fraction for our slope:
m=
3
10
Or, you can use the formula for slope. If we are going to use the formula, we must
first identify the coordinates of the given endpoints. They are B(-6, 0) = ( x1 , y1 ) and
A(4, 3) = ( x2 , y2 ) . Let’s plug these into the equation.
3−0
3
3
=
=
4 − (−6) 4 + 6 10
Now lets find the length of BA
D=
( x2 − x1 ) + ( y2 − y1 )
2
2
=
( 4 + 6) + (3 − 0)
2
2
= 109 = 10.44
And the midpoint of BA
⎛ x + x y + y ⎞ ⎛ 4−6 3+ 0 ⎞
M =⎜ 2 1, 2 1⎟=⎜
,
⎟ = ( −1,1.5 )
2 ⎠ ⎝ 2
2 ⎠
⎝ 2
Sample Problems:
Use the following graph for questions 1-3.
1. Find the slope of AB
2. Find the length of AB
3. Find the midpoint of AB
4. Find the slope of the line that passes through (-7, 3) and (2, 4).
5. Find the length of the segment connecting the points (2, 1) and (8, -14).
.
6. Find the midpoint of the segments with end points (-3, -6) and (9, 2)
7. Find the distance between (0, 6) and (7, 0).
QUADRILATERALS
A quadrilateral is a two-dimensional closed shape (polygon) with four sides. We
can classify quadrilaterals by some distinctive features. We call the ‘corners’ of the
figure, where two edges meet, the angles of the quadrilateral. If we draw a line from
one corner to an opposite corner, we call this a diagonal of the quadrilateral.
A trapezoid is a quadrilateral with only one pair of parallel sides.
A parallelogram is a quadrilateral with exactly two pairs of parallel sides.
Rectangles, squares, and rhombi are all quadrilaterals that fall under the
parallelogram category. The opposite sides of a parallelogram are both parallel and
equal in length.
A rhombus is a parallelogram in which all four sides are equal and whose
diagonals are perpendicular. A diamond is a common example of a rhombus.
A rectangle is a parallelogram with four right angles whose diagonals are
equal in length.
A square is a special rectangle in which all four sides are equal in
length. The diagonals of a square are perpendicular and bisect each other.
You will be asked to identify quadrilaterals by a description involving sides, angles,
and diagonals. Your formula sheet has pictures of a parallelogram, rectangle, and
trapezoid for reference.
Sample Problems:
1. If a quadrilateral has exactly one pair of sides parallel, it must be a:
2. If the diagonals of a quadrilateral are perpendicular and equal, it must be a:
RECTANGULAR COORDINATE SYSTEM
The rectangular coordinate system, also known as the xy-plane, is used for
graphing. The plane is divided into four parts, or quadrants, by two axes that
intersect at the origin. The horizontal axis is the x-axis and the vertical axis is the yaxis.
Quadrant II
Quadra
origin
Quadrant
III
Locations on the graph are represented by an order pair in the form (x, y), also
called coordinates. The origin has coordinates (0,0). The x-axis is positive to the
right while the y-axis’s positive direction is upwards. Identifying or plotting points is
done by finding the vertical line corresponding to the x value and then moving up or
down this vertical line to the correct y-value.
For example, to plot the point (2, -3), first find x = 2 on the x-axis. Then move down
to y = -3.
Quadra
IV
The place where a graphed line crosses an axis is called an intercept. Any point on
the x-axis (an x-intercept, for example) has the coordinates (x, 0). Similarly, any
point on the y axis, (a y-intercept) has the coordinates (0, y). Points that lie on the
axes cannot be classified as being in any of the quadrants.
The graph below has an x-intercept of ( -4, 0) and a y-intercept of (0, 4).
Points on a graph can be reflected across an axis. For example, the reflection of A
across the x-axis is B, while the reflection of A across the y-axis is C.
When you are asked to find the reflection of a point across a certain axis, imagine
creasing the paper that the graph is drawn on, and folding the graph in half. If our
original point was wet with ink, where would it leave a mark, when you fold the
paper? That is your reflection point.
Sample Problems:
Use the given graph to answer questions 1 and 2.
1. Which point has the coordinates ( -3, -6)?
2. Name the points in quadrant II.
3. In which quadrants are the following points located?
(3, -2)
(-4, 3)
(4, 0)
4. Which statement best describes the points in quadrant I?
A.
B.
C.
D.
The x coordinate is positive and the y coordinate is negative.
The x coordinate is negative and the y coordinate is negative.
Both the x and y coordinates are positive.
Both the x and y coordinates are negative.
5. What is the reflection of point (8, -3) across the y-axis?
6. If you reflect (-1, -5) across the x-axis, where is its reflection point?
REGULAR POLYGONS
A regular polygon is a two-dimensional figure whose sides are all equal in length
and whose angles are all equal in measure. We say it is both equilateral and
equiangular, meaning that all sides are equal in length and all the angles are equal in
measure.
Polygons are named according to their number of sides, which is also the number of
angles. Here are some common names of polygons.
3
4
5
6
7
8
9
10
12
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
Dodecagon
The sum of the angles in an n-sided polygon is (n − 2)180o .
For example, if asked to find the sum of the measures of the angles in a heptagon,
first, recognize that a heptagon has seven sides. Since we know n = 7, then
(7 − 2)180o = (5)180o = 900 o .
Sample Problems:
1. A gazebo in a local park is built in the shape of a regular octagon. What is the
measure of each interior angle?
2. Crunchy Hexagons is a breakfast cereal whose pieces are in the shape of regular
hexagons. Two pieces are stuck together and sharing a side of the hexagon.
Determine the measure of the angle between the two pieces of cereal.
TRIANGLES
A triangle is a three-sided polygon. A triangle also has three angles and three
vertices. If we were to measure the angles of a triangle, and add them up, the sum
would always be 180o . That is, the sum of the angles of any triangle is 180o . There
are many types of triangles that are classified according to the properties of their
sides and angles.
An equilateral triangle has all sides equal in length, and all angles equal in measure.
This means that each interior angle of an equilateral triangle must be
60o because 180o ÷ 3 angles = 60o per angle.
An isosceles triangle has exactly two sides equal in length and two angles equal in
measure.
An acute triangle has three acute angles.
An obtuse triangle has one obtuse angle.
A right triangle has one right angle ( 90o ). We name the sides of a right triangle
according to their relationship with the right angle. The legs are the two sides that
form the right angle (labeled a and b), while the hypotenuse is the longest side which
is across from the right angle (labeled c).
a 2 + b2 = c2
We label the sides of a right triangle this way so that we can use the Pythagorean
Theorem. The Pythagorean Theorem relates the lengths of the sides of a right
triangle.
Sample Problems:
1. Identify the type of triangle.
c
a
b
2. Use the Pythagorean Theorem to find the third side of the right triangle with
hypotenuse 5 in. and a leg of length 3 in.
3. A 40 foot electrical pole is supported by 3 cables, each of which is 8 feet from the
base of the pole. How much cable is being used to support the pole?
4. The measure of the largest interior angle of a triangle is 80 degrees. Which
statement about the triangle must be true?
A)
B)
C)
D)
The triangle is a right triangle
The other two angles are congruent
Each of the other angles measures 80
The measures of the two other angles have a sum of 100.
SOLVING EQUATIONS & INEQUALITIES
To solve an equation or inequality means to find the value(s) of the variable(s) that
make(s) the mathematical statement true. You will be asked to choose the correct
solution from the given multiple choice answers. Plug each of the multiple choice
answers into the equation to see which one produces a true statement. The one that
produces a true statement is your answer! It is good habit to put parentheses around
the value plugged in for the variable. This will help prevent you from making a sign
error. Also, remember to follow order of operations when simplifying.
To determine whether the mathematical statement is true, you must know the
meanings of the following symbols. The symbols definition is followed by an
example of a true statement using the symbol.
=
Is Equal To
Ex. 5 = 5
<
Is Strictly Less Than
Ex. 3 < 9
>
Is Strictly Greater Than
Ex. 7 > -2
≤
Is Less Than or Equal To
Ex. -18 ≤ -9
≥
Is Greater Than or Equal To
Ex. 37 ≥ 37
For example, let’s try an equation. An equation is a mathematical statement
containing an equal sign.
Solve 3 x + 7 = 28 .
A)
B)
C)
D)
x = 63
x=9
x = -4
x=7
Only one of the following multiple choice answers will work in this equation. Let’s
try them. Our goal is to have the expression on the left simplify to the number on the
right, 28.
A) 3 (63) + 7 = 189 + 7 = 196 ≠ 28
So this is not our answer.
B) 3 (9) + 7 = 36 + 7 = 43 ≠ 28
So this is not our answer.
C) 3 (-4) + 7 = -12 + 7 = -5 ≠ 28
So this is not our answer.
D) 3 (7) + 7 = 21 + 7 = 28 YES!
This is our answer!
Now, let’s try an inequality. We must realize that there are inclusive inequality
symbols (≤ and ≥), and exclusive inequality symbols (< and >). Our answer must
match the type of symbol (inclusive or exclusive) given in the problem.
Solve 3 x + 7 < 28 .
A)
B)
C)
D)
x > 63
x≤7
x > 12
x<7
Only one of the following multiple choice answers will work in this inequality. We
need to try a number from each choice that fits the description. Our goal is to have
the expression on the left simplify to a number less than, 28.
A) Try a number bigger than 63.
3 (70) + 7 = 210 + 7 = 217 > 28
so this is not our answer.
B) Because this symbol is less than or equal to, this cannot be our answer.
Even if it seems to work, our original inequality used a strictly less than
symbol (exclusive), which does not match the symbol in the original
problem.
3 (7) + 7 = 21 + 7 = 28 ≤ 28
C) Try a number greater than 12.
3 (12) + 7 = 36 + 7 = 43 > 28
so this is not our answer.
D) Try a number less than 7.
3 (0) + 7 = 0 + 7 = 7 < 28 YES!
This is our answer!
To solve a system of equations or inequalities means to find the ordered pair that
makes both mathematical statements in the system true. You will be asked to choose
the correct solution from the given multiple choice answers. Plug each of the
multiple choice answers into the system to see which ordered pair ‘works’ in the
system. Your solution, (x, y) must work in both equations or inequalities!
Now, let’s try a system of equations.
Solve the following system:
y = x2 + 4 x + 9
y = −x + 3
A)
B)
C)
D)
(1, 2)
(-1, 6)
(5, 1)
(-3, 6)
Remember, we want the ordered pair that works for both equations! Only one of the
following multiple choice answers will make both statements true.
A)
2 = (1) 2 + 4(1) + 9 = 1 + 4 + 9 = 14
2 = −(1) + 3 = 2
The ordered pair (1, 2) works in the first equation, but not in the
second. So this is not our answer.
B)
6 = (−1) 2 + 4(−1) + 9 = 1 − 4 + 9 = 6
6 = −(−1) + 3 = 4
Again, the ordered pair (-1, 6) works in the first equation, but not the
second, so this is not our answer.
C)
our answer.
1 = (5) 2 + 4(5) + 9 = 25 − 20 + 9 = 14
1 = −(5) + 3 = −2
The ordered pair (5, 1) doesn’t work in either equation, so this is not
D)
6 = (−3) 2 + 4(−3) + 9 = 9 − 12 + 9 = 6
6 = −(−3) + 3 = 6
YES!
This is our answer!
Sample Problems:
1. Solve for x. 9 x + 2 = 29
A)
B)
C)
D)
x = 22
x=3
x = 18
x=1
2. Solve for t. 4t − (3 − t ) = 7(t − 3) + 10
A)
B)
C)
D)
t=4
t = -3
t = -9
t=1
3. Solve for m. 4m − 8 ≤ 16
A)
B)
C)
D)
m ≤ -6
m<6
m≤6
m ≥ -6
4. Solve for z. 13 z − 5 > 12 z − 9
A)
B)
C)
D)
z > -14
z < -14
z > -4
z < -4
5. Solve the system of equations.
2x − y = 2
3x + 2 y = 10
A)
B)
C)
D)
(2, 2)
(2, -5)
(3, -2)
(5, 7)
SEQUENCES
A sequence is a list of numbers, or terms, that follows a pattern. When trying to find
the pattern, look for a common difference between terms, or a common ratio
between terms. Once the pattern is established use this information to find the next
term in the sequence.
For example, if asked to find the next term in the following sequence, start by
looking for a common difference
-3, 1, 5, 9,…..
Subtract the first term from the second, then subtract the second term from the third.
Do they have the same difference?
t2 − t1 = 1 – (-3) = 1 + 3 = 4
t3 − t 2 = 5 – 1 = 4
The differences match! You should make sure that this difference applies to the
entire sequence. Now, add the difference to the last term in order the get the next
one
t next = tlast + difference = 9 + 4 = 13
Let’s try another. What is the next term in the following sequence?
2, -2, 2, -2, 2,….
Try for a common difference first. Subtract the first term from the second, then
subtract the second term from the third. Do they have the same difference?
t2 − t1 = -2 – 2 = -4
t3 − t2 = 2-(-2) = 2 + 2 = 4
Their differences are NOT the same, so we will try to find a common ratio. Divide
the second term by the first, then divide the third term by the second. Are the ratios
the same?
t2 −2
=
= −1
t1
2
t3
2
=
= −1
t2 −2
Their ratios match! So, to find the next term in the sequence, multiply the last term
by the common ratio.
tnext = tlast ⋅ ratio = 2 ⋅ −1 = −2
Sample Problems:
What is the next term in the sequence of numbers?
1.
1 5 1 1
, ,1 ,1 , ….
2 6 6 2
2. -2, 3, 8, 13, …..
3. 100, 10, 1, 0.1, 0.01, …..
4. 5, 15, 45, 135, ……
5. 17, 10, 3, -4, -11, …..
TRANSLATING BETWEEN ENGLISH AND ALGEBRA
When given a mathematical word problem, it is imperative that you understand the
meanings of those words. Even though some words have multiple definitions, many
have very specific meanings when it comes to math. For example, the word ‘IS’ can
be replaced by an equal sign in math. Here are some other mathematical key words
grouped by the mathematical operation that they imply.
Addition: sum, total, plus, increased by, more than
Subtraction: difference, decreased by, less than, taken from
Multiplication: product, times, of, by
‘Twice’ and ‘double’ mean multiply by two
Division: quotient, ratio, split equally
Equality: is, equals, is equivalent to, same as
Every time you are presented with a word problem, you should follow a few simple
steps.
1. Read, reread, and understand what is being asked of you.
2. Write down pertinent information given in the problem.
3. Plan and strategize your method of finding a solution. This might entail
defining variables, writing an equation, and drawing a picture.
4. Execute your plan and find a solution.
5. Read the problem again. Have you answered the question? Is your
answer reasonable?
Some word problems may ask you about consecutive integers. Consecutive integers
are numbers that would follow each other when counting. For example, 1, 2, and 3
are consecutive integers. You may be asked to find two or more consecutive integers
with a certain sum or product. We represent the first integer with a variable, x. The
next integer is just the next counting number, or one larger than the last, x + 1. Let’s
try an example.
Find three consecutive integers whose sum is 156.
First write expressions for each of your integers.
x = first integer
x + 1 = second integer
x + 2 = third integer
The problem tells us that the sum of these three numbers is 157. Sum implies
addition, so we will add the expressions together.
x + (x + 1) + (x + 2) = 156
Combine like terms and solve the equation for x.
3x + 3 = 156
3x = 153
x = 51
Have we answered the question? It asked us to find three consecutive integers.
We’ve only found one! Go back to the expressions you wrote for the three integers,
and plug in your x value. Our three integers are 51, 52, and 53.
Sample Problems:
1. Which expression represents the height of a tree that is 7 feet taller the flag pole
represented by f ?
A)
B)
C)
D)
7f
7+ f
7÷ f
7− f
2. Which expression represents the cost of 24 pencils if the cost of one pencil is p?
A)
B)
C)
D)
24 − p
24 ÷ p
24 p
24 + p
3. Which expression represents having x less than a dozen eggs.
A)
B)
C)
D)
12 – x
x – 24
24 ÷ x
x – 12
4. Brett’s age is 7 more than twice Irma’s age, and Irma’s age is a. Which expression
represents Brett’s age?
A) 7 (2a)
B) 3a + 7
C) 14 + a
D) 2a + 7
5. Which expression represents the following statement? The product of 15 and w is
75.
A) 15 + w = 75
B) 15 ⋅ 75 = w
C) 15w = 75
15
D)
= 75
w
6. Which expression represents the following statement? Five times k decreased by 4
is 21.
A)
B)
C)
D)
5k – 4 = 21
4 – 5k = 21
4 ÷ 5k = 21
5k + 4 = 21
7. The sum of two consecutive integers is 37. Write the equation and find the two
integers.
A)
B)
C)
D)
x + x + 1 = 37
x + y = 37
2 x = 37
x( x + 1) = 37
8. Scooter rental costs $19.95 per day and $0.05 per mile. If m = the number of
miles and c = the total cost for one day, write an equation that represents the total bill
for one day of scooter rental.
A)
B)
C)
D)
c = 0.05m + 19.95
c = 19.95m + 0.05
c = 19.95(m − 0.05)
c = 19.95 − 0.05m
COUNTING PRINCIPAL, PERMUTATIONS, &
COMBINATIONS
The counting principal says that if a first experiment can be performed M distinct
ways, and a second experiment can be performed N distinct ways, then the two
experiments (in that order) can be performed M ⋅ N distinct ways.
For example, let’s say that a certain account must have a password that consists of
three uppercase letters followed by three digits. We want to determine how many
different passwords are possible, but must consider two cases; with or without
repetition. Each symbol in the password is considered an ‘experiment’ from the
definition of the counting principal above. There are six symbols, so six
experiments.
LLLDDD
Each experiment has a certain number of possible outcomes. There are 26 letters in
the alphabet, so there are 26 possibilities for each of the first three experiments. For
each of the last three spots in the password (experiments) there are 10 possibilities
because our number system has 10 digits.
Case 1: If repetition of letters and digits is permitted,
26 26 26 10 10 10
= 17,576, 000 possible passwords
L L L D D D
Case 2: If repetition of letters and digits is not permitted, then we must
consider a digit or letter ‘used up’ by the previous experiment.
26 25 24 10 9 8
= 11, 232, 000 possible passwords
L L L DDD
A combination is a distinct set of objects whose order does not matter. The number
of combination possible when r objects are selected from n objects is found by the
formula
n!
this can be read ‘n choose r’
n Cr =
(n − r )!r !
The exclamation point you see in this formula means factorial. When a number is
followed by the factorial symbol, we will multiply that number by every smaller
integer until you get to 1. For example, 6! means 6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅1 = 720
Let’s try a combination problem. An essay exam consists of seven questions, five of
which must be selected and answered. In how many ways can the test be completed?
There are 7 questions and we must choose 5. So, n = 7 and r = 5 and we can say ‘7
choose 5’ combinations are possible.
n!
7!
7 ⋅ 6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1 30
=7 C5 =
=
=
= 15 ways to
(n − r )!r !
(7 − 5)!5! (2 ⋅1) ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1 2
complete the test
n
Cr =
A permutation is any ordered arrangement of a given set of objects. The number of
permutations possible when r objects are chosen from n objects is found by the
formula
n!
n Pr =
(n − r )!
Let’s try a permutation problem. The student council is electing a president, vice
president, and secretary from its 18 members. How many different arrangements, or
permutations, of officers are possible?
We need to choose 3 officers from 18 people, so r = 3 and n = 18. This is a
permutation, NOT a combination, because the officers’ positions are ranked, or
ordered.
n
Pr =
n!
18!
18 ⋅17 ⋅16 ⋅ (15!)
=18 P3 =
=
= 18 ⋅17 ⋅16 = 4896
(n − r )!
(18 − 3)!
(15!)
Now, if there are r groups of identical objects, n1 identical objects, n2 identical
objects, …. nr identical objects, then the number of distinct permutations is found by
the formula
n!
n1 !n2 !⋅⋅⋅ nr !
For example, in how many ways can the letters in the word TENNESSEE be
arranged? Consider that there are 4 E’s, 2 N’s, and 2 S’s. We are not concerned
with the one T because 1! = 1.
n!
9!
=
= 3780
n1 !n2 !⋅⋅⋅ nr ! 4!2!2!
Sample Problems:
1. A certain key pad has a 3 digit code. If repetition of digits is permitted, how
many possible codes are there?
2. How many ways can a class of 12 children be arranged in a line?
3. In how many ways can the letters of the word SUCCESS be arranged?
4. A commercial airliner has 5 seats available on a flight. There are 9 passengers on
standby. How many ways can the airline choose its 5 lucky passenger?
CHARTS & GRAPHS
A bar graph is used to show relationships between groups. The bar graph will
compare two or more items measured along the side of the graph. One item is
represented by the bars and the other is classified at the bottom by descriptions. The
items are measured by the scale on the left side of the graph. The title of the graph
can also help you understand what two items are being compared.
Example:
Determine the amount of men in an intermediate socio-economic status that smoke.
When reading a graph, use the key to help you discern between items being
compared. The dark blue bar represents men while the light blue bar represents
women. Also, along the bottom of the graph we see that these bars are classified into
three socio-economic statuses: managerial and professional, intermediate, and
routine and manual.
We are looking for men in the intermediate socio-economic group. So, we should
look to the middle grouping of bars and at the dark blue bar. Now, we simple use
the height of the bar and the scale on the let to determine the answer. Notice the
scale on the left is marked with a percent sign showing that these values are percents.
The dark blue bar in the center measures up to the 29 mark along the left-hand scale.
So, to answer the question, we can say that 29% of men in the intermediate socioeconomic group are smokers.
Example:
Determine the total number of people who claim that there is more than one gangrelated homicide in 2000,
A) 25 people
B) 37 people
C) 50 people
D) 92 people
This question asks for a total number of people - specifically, those who claimed the
number of gang-related homicides is more than one. So, looking only in the section
marked ‘more than one’, we should total all of the bars which indicate different
groups of people.
The dark blue bar is approximately 8 people, the light blue bar is approximately 4
people, and the black bar is approximately 25 people. So, we will total these values:
8+4+25=37 people.
So this would be answer choice B.
A pie-chart is a circular chart divided into sectors representing different percentages.
The total percentage, being the whole pie, is 100%.
Example:
The University of Washington Facilities Department took a survey of facility usage.
If the following graph represents a total usage of 67 million gallons, how many
millions of gallons were used by Cafeteria/Food Service?
Note: Due to rounding of decimal values, this chart does not have percents totaling 100.
The entire pie-chart represents 67 million gallons of usage and Cafeteria/Food
Service represents 9% of that value. We convert 9 % to a decimal making it 0.09.
So, nine percent of 67 million is (0.09)(67) = 6.03 million gallons.
Sample Problems:
Use the following charts and graphs to answer the questions.
1. What are the top three sources of bacteria in the Four Mile Run watershed?
2. If this chart represents a total of 54 samples checked for bacteria, how many
of those samples were sourced back to human bacteria?
A) 18
B) 9.72
C) 36
D) 11.45
3. Approximately what percentage of females ages 65-74 died from CVD?
A) 22%
B) 42%
C) 12%
D) 32%
4. If this graph represented a total of 1500 people’s deaths, about what
percentage of men ages 85+ died due to CVD?
A) 675
B) 610
C) 450
D) 400
MEASURES OF CENTRAL TENDENCY
There are three measures of central tendency: mean, median, and mode.
Mean is another word for average. To find the average, you must first add up all the
values in your data. Then divide that number by however many values you had to
add up. This is your mean.
Example:
Find the mean of the following data: 12, 16, 21, 11, 6, 14, 18.
First we add up all of the data values: 12 + 16 + 21 + 11 + 6 + 14 + 18 = 98 .
Then we divide by the number of values we added. There are seven values so we
will divide by seven: 98 ÷ 7 = 14
So, the mean is 14.
The median is the ‘middle’ number – think of a median in the street which is located
in middle of the lanes of traffic. Sort your data from smallest to largest in value and
then count down to the number that is in the middle. If there are even amounts of
numbers, there will be two numbers that split the middle. Take the mean of these
two numbers and that will be your median.
Example:
Find the median of the following data: 12, 16, 21, 11, 6, 14, 18.
We need to sort the data from smallest to largest: 6, 11, 12, 14, 16, 18, 21.
Then we count down to the middle number:
6, 11, 12, 14, 16, 18, 21
6, 11, 12, 14, 16, 18, 21
6, 11, 12, 14, 16, 18, 21
So, our middle number, and thus our median, is 14.
Example:
Find the median of the following data: 15, 18, 29, 2, 66, 45.
First we will sort the values: 2, 15, 18, 29, 45, 66.
Then we will count down to the middle:
2, 15, 18, 29, 45, 66
2, 15, 18, 29, 45, 66
There are two numbers in the middle so we will find the mean (or average) of these
18 + 29 47
two numbers:
=
= 23.5 .
2
2
So, the median is 23.5.
The mode is the number that occurs most often. So when you are looking through
your data, pick out which number is repeated the most times. That is your mode. If
every number is displayed the same number of times then there is no mode. If two
numbers are displayed the same number of times and they are both displayed the
most, then we list them both as modes and say it is ‘bimodal’.
Example:
Find the mode of the following data: 15, 18, 29, 2, 66, 45.
In this data set, no number repeats so there is no mode.
Example:
Find the mode of the following data: 21, 16, 18, 11, 6, 4, 18, 4, 22, 18, 11.
In this data set, the number 18 appears three times while all other numbers only
appear once or twice.
So, 18 is the mode.
Sample Problems:
Find the mean, median, and mode.
1. 12, 12, 14, 18, 19, 20, 20, 20
2. 65, 50, 37, 77, 86, 90, 68, 85, 77
3. 145, 187, 100, 112, 92, 98, 156, 186, 99, 173, 118, 166
PROBABILITY
The probability of a specific event occurring is found either by
-dividing the number of times the event has occurred by the total number of times the
experiment has been performed or
-dividing the number of times the event is likely to occur by the total number of
possible outcomes.
The probability of an event occurring is always between 0 and 1 and can be
expressed as a decimal or fraction.
Example:
A fair coin is tossed 100 times, and lands heads up 66 times. What is the probability
of the coin landing heads up?
P=
66
= 0.66
100
Sometimes, we are asked the find the probability of events happening in succession.
For this situation, find the probability of each event and multiply them together.
For example:
What is the probability of rolling a 3 twice in a row on a fair die?
1
The probability of rolling one 3 is P = , so the probability of rolling two 3’s
6
1 1 1
is ⋅ =
6 6 36
The words AND and OR are key words in problems requiring you to compute
probability.
When calculating the probability of event A and B occurring, if the problem does not
specify whether replacement occurs, we will assume that event A has already occurred
when we calculate P(B).
P ( AandB) = P( A) ⋅ P( B)
P ( AorB) = P( A) + P( B) − P( AandB)
Example:
Two cards are to be selected with replacement from a deck of cards. Find the
probability that two sevens will be selected.
P (two7 s ) = P(1st seven) ⋅ P(2nd seven) =
4 4
1
⋅ =
52 52 169
Now lets try the same problem, without replacement.
P (two7 s ) = P(1st seven) ⋅ P(2nd seven) =
4 3
1
⋅ =
52 51 221
What is the probability of pulling a face card or an ace?
P ( FaceOrAce) = P( Face) + P( Ace) − P( FaceAndAce) =
12 4
16 4
+ −0 =
=
52 52
52 13
Sample problems:
1. A child’s piggy bank contains 15 pennies, 2 nickels, 11 dimes, and 9 quarters. If
the child reaches in and pulls out a coin without looking, what is the probability that
the coin will be
A)
B)
C)
D)
a dime?
a penny?
a quarter or a nickel?
NOT a quarter?
2. A standard deck of playing cards contains 26 red cards and 26 black cards. What
is the probability that you will draw two black cards in a row, if the first card is
placed back in the deck before you draw again?
3. A fair coin is tossed 3 times, what is the probability that it will show tails all three
times?
4. There are 5 red marbles, 3 green marbles, 2 blue marbles, and 1 white marble in a
bag. What is the probability that the first two marbles drawn are red if the first
marble is not put back in the bag?
ANSWER KEY TO SAMPLE PROBLEMS
STRAND 1
Fractions:
1. 1.2
2. 7.5
3. 0.75
4. 0.444444….
Radicals:
1. 3
2. ≈ 3.87298
3. ≈ 8.48528
4. 11
5. ≈ 6.92820
6. 16
Percents:
1. 0.33
2. 0.194
3. 0.027
4. 1.25
5. 0.76
6. 17%
7. 23.55%
8. 299%
9. 0.2%
10. 68.4%
11. ≈ 27.22% increase
12. ≈ 28.57% decrease
Exponents:
1. 25
2. -216
3. 1
4. -49
5. -27
6. 64
7. -100
Scientific Notation:
1. 0.0000038
2. 0.0912
3. 220,000
4. 1,070
GCF:
1. 14
2. 2
3. 15
Order of Operations:
1. 113
2. -22
3. 4
4. ÷ ÷ − −
5. ÷ −× +
Diagramming:
1. Yes
2. 3.5 in
3. 1.5 in
STRAND 2
Perimeter & Circumference:
1. 42 in
2. 54 cm
3. 53.38 ft
4. 25.12 in
Area:
1.
2.
3.
4.
5.
93.75 ft2
0.375 m2
1519.76 cm2
22351 m2
399.5 ft2
Surface Area:
1. 7385.28 ft2
2. 106 ft2
3. 615.44 in2
Volume:
1. 45216 ft3
2. 60 ft3
3. ≈ 1436.0267 in3
Distance:
1. 4 mph
2. B
3. A
Measurement Conversions:
1. 195240 square feet
2. No
3. 3.5 hours
Ratios & Proportions:
1. C
2. A
3. D
4. A
5. B
STRAND 3
Rectangular Coordinate System:
1. F
2. A, B
3. (3, -2) Quadrant IV; (-4, 3) Quadrant II; (4, 0) none
4. C
5. (-8, -3)
6. (-1, 5)
Lines & Segments:
2
1. −
5
2. ≈ 10.77
3. (0, 3)
1
4.
9
5. ≈ 16.16
6. (3, -2)
7. ≈ 9.22
Angles:
1. 1,2; 1,4; 2,3; 3,4; 5,6; 5,8; 6,7; 7,8
2. 1,3; 2,4; 5,7; 6,8
3. 1,5; 4,8; 2,6; 3,7
4. 1,2; 1,4; 2,3; 3,4; 5,6; 5,8; 6,7; 7,8
5. none
6. 4,6; 3,5
7. 2,8; 1,7
8. m∠2 = m∠4 = m∠6 = m∠8 = 37 degrees
m∠1 = m∠3 = m∠5 = m∠7 = 143 degrees
9. B
Regular Polygons:
1. 135 degrees
2. 120 degrees
Triangles:
1.
c
a
b
Right
Acute
Obtuse
2. 4 inches
3. ≈ 122.38 feet of cable
4. D
Quadrilaterals:
1. Trapezoid
2. Square
Symmetry:
1. B
2. A’ (4.5, 6)
B’ (1.5, 6)
C’ (0, 4)
D’ (1.5, 2)
E’ (4.5, 2)
F’ (6, 4)
3. A’ (-4, -1)
B’ (-1, -1)
C’ (-0.5, -3)
Isosceles
Equilateral
D’ (-1, -5)
E’ (-4, -5)
F’ (-5.5, -3)
3-Dimensional Solids:
No Sample Problems
STRAND 4
Translating Between English and Algebra:
1. B
2. C
3. A
4. D
5. C
6. A
7. A
8. A
Solving Equations & Inequalities:
1. B
2. A
3. C
4. C
5. A
Sequences:
5
1. 1
6
2. 18
3. 0.001
4. 405
5. -18
STRAND 5
Charts & Graphs:
1. Waterfowl, Raccoon, Human
2. B
3. D
4. A
Measures of Central Tendency:
1. Mean 16.875, Median 18.5, Mode 20
2. Mean ≈ 70.56, Median 77, Mode 77
3. Mean 136, Median 131.5, No Mode
Probability:
11
1. A)
or 0.30
37
15
B)
or 0.41
37
C) 0.28
28
D)
or 0.76
37
1
or 0.25
2.
4
1
3.
or 0.125
8
4. 0.18
Permutations & Combinations:
1. 729
2. 479,001,600
3. 420
4. 126
GLOSSARY
acute angle – an angle measuring less than 90 degrees
bisect – to cut in half
congruent – to have the same measure
coordinates – the value of x and y in an ordered pair that determine the location of a point on an xy-plane.
denominator – the bottom of a fraction
difference – the result found by subtracting
face – surface of a three-dimensional figure
infinite – indefinite in length
integers – the positive and negative whole number values: {…-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5…}
numerator – top of a fraction
obtuse angle – an angle measure between 90 and 180 degrees
parallel lines – lines in the same plane that never intersect
perpendicular lines – lines that intersect at a 90 degree angle
polygon – a two dimensional figure with many sides
ray – part of a line that has an endpoint and extends indefinitely in one direction
right angle – an angle measuring exactly 90 degrees
sum – the result found by adding
transversal – a line that cuts through two other lines
three-dimensional – exists in three dimensions like most objects in our world
two-dimensional – exists in two dimensions or a plane
vertex - the point where the rays or lines in an angle meet