8th Grade OAT Math Study Guide
Measurement
Length : Shows distance (in units) from one point to another.
ex. You want to build a pond 6 feet long.
6ft.
•
6ft.
length
•
Area : The # of square units needed to cover a surface. In our
example we need to use the length & width of the pond.
ex. You want to make your pond 4 feet wide.
4ft.
6ft. x 4ft.= 24ft2
l x w = area
6ft.
•
•
Your pond now has an area of 24ft2 (24 square feet).
Perimeter: Add up the length of all the sides of your shape, in this case
your pond. The perimeter is necessary if you were going to figure out
how many bricks we need to go around the entire pond or if we wanted
to put a fence around it.
6ft. + 6ft. + 4ft. + 4ft. = 20ft.
1
Volume: The amount of space inside an object. Shows length and width
(area) plus depth (height).
ex. You decide to make your pond 3 feet deep.
6ft. x 4ft. x 3ft.=72ft3
l x w x h =volume
┬
3ft.
6ft.
┴
4ft.
Surface area : The area of all sides of your object added up.
ex. To find the amount of paper needed to wrap a present, we do not need
to know the volume such as in the previous problem. We are just covering
the box’s surface area.
10 in.
6 in.
5in.
There are six sides to this box.
Two sides each have an area of 6in. x 5in. = 30in. 2
x2
= 60in.2
Two sides each have an area of 10in. x 5in. = 50in. 2
Two sides each have an area of 10in. x 6in. = 60in2
60in.2 + 100in.2+120in.2= 280in.2
x2
= 100in.2
x2
=120in. 2
2
Customary and Metric units
In the U.S, we use customary units to measure distances, weight & volume.
These include miles, pounds and gallons. Most of the world uses the metric
system & Americans are slowly learning them. These include kilometers,
kilograms and liters.
!!The OAT test will not require you to have memorized the conversion
ratios. If they ask you to convert, the ratios will be given!!
What if your friend told you she lives 17 kilometers away and you wanted to
know how many miles this is?
We must convert kilometers to miles:
We need the conversion #’s which is 1km=0.62mi.
We are given: 17km
?miles
First we set up an equation: 17km x 0.62miles
1km
Put the equation in such a way that km is over km. This way they cancel out
leaving us with the miles unit.
17 x 0.62miles= 10.54miles in 17km
If a swimming pool holds 300gallons of water, how many liters does it hold?
Here’s the ratio: 3.7854liters=1 gallon
300gallons x 3.7854liters = 300 x 3.7854liters=1135.62 liters in 300gal.
1 gallon
3
You take your dog to the vet and he weighs in at a hefty 128pounds. How
much does he weigh in kilograms?
128
conversion ratio: 1 pound =0.4535924 kilograms
128pounds x 0.4535924kg = 58.1kg in 128lbs.
1 pound
This number is
rounded to the
nearest tenth.
Arrange the equation so pounds are over pounds so they cancel out leaving
kg units.
Sometimes the exact number may be necessary, especially concerning doses
of medicine or precise cuts of wood to design something of a particular size.
There are times when the exact conversion is not needed. If you are
traveling in the car and you convert with a calculator that your family has
13.89 miles to go, you will probably say “about 14 miles left”.
4
For measurement, recall how to convert units.
ex.→ If you have 128 ounces of copper, how many pounds of copper do you
have?
There will be a table on conversions if they ask you to convert or they will
give you the needed information such as 1 pound=16 ounces.
128 ounces x
1pound = 128 ounces x 1pound =128 pounds
16 ounces
16 ounces 16
= 8 pounds
Converting Fahrenheit to Celsius
If today’s temperature is 72° F, what is it in ° C?
Use formula→ °C = 5 (°F-32)
9
= 5 (72-32)
9
= 5 (40)
9
= 200 = 22.22°
9
22.22 °C = 72 ° F
5
Converting Celsius to Fahrenheit
Use formula →
°F = 9 (°C + 32)
5
If today’s temperature is 24° C, What is it in °F?
°F= 9 (24 °C + 32)
5
= 9 (56) = 504 = 100.8°
5
5
100.8 °F = 24 °C
To find area and volumes of certain shapes such as circles, triangles,
cylinders, etc, you will need to use specific formulas like the ones below.
6
Area of a circle
Formula →
r2 = (
Area =
x square of radius.)
(pi)=3.14
diameter
radius
5ft.
Area of this circle =
=
(5ft) 2
x 25ft.2
= 3.14 x 25ft.2
= 78.5 ft.2
To find volume of a cylinder or prism you multiply:
area of base x the height of the object.
ex. Find the volume of this cylinder.
5ft.
┬
15 ft.
┴
Area of this circle is 78.5ft2
(from previous problem)
x height of 15ft.= 1177.5ft3
7
Regular Polygons
Regular polygons-is a polygon with all sides the same length and all interior
angles equal.
To find the sum of interior angles we use the formula-
180degrees x (n-2)=
where n= # of sides of your polygon.
ex. Find the sum of interior angles of a polygon with
6 sides.
6 sides so we put 6 in the equation.
180 x (6-2)= 180 x 4= 720
How to find exterior angles of a regular polygon
The exterior angles of a regular polygon add up to 360˚
Use the formula:
360˚ = ? ˚ to find out how many degrees
n
each angle is
where n= # of sides
A 6 sided polygon
→
360˚ = 60˚
6
Note: when we measure the exterior angle we measure from one side to the
line that would extend outward from the next side as shown-
each exterior angle is 60˚ for this polygon.
Plug in the # of sides for n in the formula above to find the degree of
each angle.
8
Number, Number Sense
& Operations
Scientific Notation:
Is used to write very large or small numbers easily. To convert
scientific notation to standard notation, you move the decimal over
the number of places equal to the exponents value. When the exponent
is positive your decimal will move to the right.
moves 7 places to the right
moves 3 places to the right
When your exponent is negative, you will add that many places to the
left.
moves 3 places to the left.
standard notation to scientific notation :
If your number is very small such as 0.0024, move the decimal place
over until it is placed right after the single digit 2.
Then count how many places to get there: this is your exponent value.
→
2.4 x 10-3
move 3 places to the right
A # in standard notation with a zero(s) to the right of the decimal will
always have a negative exponent when using scientific notation to
express it.
ex. .0 1 & .0 0 0 0 0 0 1
9
If a number in standard notation has zeros to the left of the decimal
such as→
2.4 x 105
move 5 places to the left
to get the decimal
behind first digit (2).
To add & subtract #s in scientific notation
If we have to add numbers: 4.0 x 105 + 6.0 x 104
We first make the exponents the same- as we move the decimal over
to the left one place, the exponent gets bigger by one:
6.0 x 104 → 0.6 x 105
Now we can add: 4.0 x 105 + 0.6 x 105= 4.6 x 105
Subtraction is the same way: 3.0 x 107 – 5.3 x 105
We make their exponents the same.
Change the 5.3 x 105 into 0.053 x 107
3.0 x 107 –0.053 x 107= 2.947 x 107
To multiply #s in scientific notation
ex. ( 3.1 x 10 4)x(2.0 x 102)=
we multiply 3.1 x 2=6.2 & we add the exponents: 4+2=6
so the answer is 6.2 x 106
10
Dividing #s in scientific notation
ex.
6.0 x 107 = first we divide 6 by 2 to get 3.
2.0 x 104
Then we bring the exponent 4 in the denominator up to the
numerator so it will become negative→ -4
3.0 x 107-4 → = 3.0 x 103
Real # System
These include all the different types of #s. Get familiar with how they
are categorized. It’s pretty simple↓
Natural #s: These are also known as “counting #s” because these
are what you use to count such as how many slices of pizza you
have1, 2, 3 ,4…
Whole #s: These are the natural #’s plus zero.
0, 1, 2, 3, 4…
We have zero pieces left. Time to get some more!!
Integers: These are the set of #s that include the counting #s
along with negative versions of those same #s. Zero must also be
included.
…-3, -2, -1, 0, 1, 2, 3,…
11
Ex. You have a bank account of $100.00 and you buy a bike that
costs $150.00, charging it to this account. You will now be minus $50.00 not to mention charged for not having enough funds in
your account.
100- 150=-50
Rational #s: (“Ratio”)-These are fractions. If they are in the
form of:
a
b
Where a & b are integers and b cannot be 0.
Rational #s include all the previous #s described above (natural,
whole, integers), because all these #s can be put in a ratio or
fraction formex. 7 is also a rational # because it can be expressed in fraction
form-
Rational #s either stop such as the ones above or they
repeat like these→
Some #’s however cannot be expressed in fractions. These
numbers are called-Irrational #’s
12
Irrational #’s cannot repeat or end like rational #s do.
Does not repeat or end
ex.
↓
↓
This diagram will help you see their relation. Looking at it you
can see that all whole #s are rational #s, and natural #s are also whole
#s as well as integers etc.
Real
Rational
Irrational
Integers
Whole
Natural
http://www.jamesbrennan.org
Square roots:
Recall when we square a number: 32 = 9 it means 3 x 3= 9
Taking the square root of a number means you are finding a number
that, when it’s squared, it = your number being square rooted.
because 32 = 9
13
Perfect squares: are #’s such as 1,4, 9, 16, 25, etc. They result from
multiplying a whole # by itself.
All other #’s, especially large ones, can be solved using a calculator.
Question: Given the number line below, choose the point that
represents -
Type in 175 on calculator, then hit
key.
Sample question:
Which integer is a perfect square?
A. 120
B. 300
C. 400
D. 500
First, punch in your # in your calculator & then hit the square root key
.
→
120→ 10.95 not a perfect square.
300→ 17.32 not a perfect square.
400→ 20
a perfect square.
500→ 22.36 not a perfect square.
14
Percentages % - A ratio that compares a number to 100.
To find the % of a number(such as a cost) you can just convert that %
to a decimal (by dividing your % by 100) and multiply this decimal
number by your number(cost).
ex. what is 37% of $112.00 dollars
37 % off
37 % → 0.37 (because 37/100=0.37)
0.37 x $112.00 = $41.44
Final cost?
cost of stereo = $112.00
minus % off - $41.44
final cost
=
$70.56
Sam plays basketball for Maple Heights High School. For
the first game he attempted 20 shots and made 12 of them.
If Sam continues to shoot at this rate, how many baskets will he
make if he shoots 45 times?
We know his ratio is: 12
20
Make this
ratio= to x
45
12 = x
20 45
15
We cross multiply to solve for x.
20x = 12• 45
20x = 540
x = 540
20
27 = 0.60
45
x = 27
12 = 0.60
20
Same rate
Know how to convert a particular unit such as miles per second to
miles per hour to miles per year.
A comet is traveling at1.76 x 104 miles per second.
1.76x 10 4 mi.
sec.
How many miles per hour is this comet traveling?
1. convert seconds to hours: we do this by multiplying our miles per
second by a ratio that has seconds in the numerator and minutes in
16
the denominator. This numerator and denominator have to equal
each other. We then cancel like terms and what we’re left with is
our new unit.
2. 1.76 x 104 miles x 60 sec. x 60 min. =
sec.
1 min.
1 hr.
cross out like units to be left with miles per hour:
This number is rounded
to the nearest
hundredth
1.76 x 104 miles x 60 sec. x 60 min. = 6.34 x 107 miles
sec.
1 min.
1 hr.
hr.
3. convert miles/sec. to miles/year:
1.76 x 104 miles x 60 sec. x 60 min. x 24 hr. x 365 days =
sec.
1 min.
1 hr. 1 day 1 year
11
= 5.55 x 10 miles
year
This number is rounded
to the nearest
hundredth
Recall that negative numbers, when multiplied by a positive number
results in a negative number.
ex.
-4 • 4 = -16
A negative number multiplied by another negative number will yield
a positive number.
ex.
-4 • -4 = 16
17
This also applies when we raise a number to a certain power such as
squaring, cubing, etc).
If x is a negative number
Then this x2 = a positive number, because
A
-x · -x = +x2
negative · negative = +
Lets let x = -3
X2 = (-3) 2 = -3 · -3 = 9
If we let x= - 3 again, will x be + or - ?
X3 =(-3) 3 = -3 · -3 · -3 = - 27
9 · -3 = - 27
18
Patterns, Functions &
Algebra
Be able to predict the next logical sequence of numbers in a number
array.
Ex. The first four rows of a number array are shown.
Row 1
Row 2
Row 3
Row 4
45
30
50
20
35
55
15
25
40
60
Predict the number that will be in the far right end of row 7.
* As we can see, each row has one more number (or # of columns)
than the previous.
This means row 5 has 5 columns since row 4 had 4, etc.
* Also, as the numbers are moving down the rows, they are
increasing by an interval of 5. 15,20,25…etc.
Lets finish row 5 through 7.
Row 1
Row 2
Row 3
Row 4
Row 5
65
Row 6
90 95
Row 7 120 125 130
15
20 25
30 35 40
45 50 55 60
70 75 80 85
100 105 110 115
135 140 145 150
19
Patterns allow us to figure out how numbers grow exponentially.
ex. Darcy wants to breed hamsters and sell them as pets. She starts
with 2 hamsters and figures out they quadruple in number every 4
months. How many hamsters will she have in one year?
• There is more than on way to figure out most math problems.
Choose the technique that you feel most comfortable with☺
We will figure this problem out writing a rough table.
First, what does quadruple mean?
2 hamsters doubled is 4:
+
2 hamsters tripled is 6 :
+
+
2 hamsters quadrupled is 8:
+
+
1st 4 months
2+2+2+2=8
After 1st 4 months
she will have 8.
2nd 4 months
8+8+8+8=32
This 8 will again
quadruple after
4 more months
giving her 32.
+
last 4 months
32+32+32+32=128
Finally these 32
hamsters will
quadruple in last
4 months giving
her a total of 128.
20
We’ve just seen how numbers can increase in time. They can also
lose value in time such as automobiles (unless you’ve got an old
classic!)
ex. Daryll just bought a truck for $30,000 dollars. The value of his
truck will decrease linearly so after 10 years his truck will be worth
an average of $5,000 dollars.
How much does his truck decrease in value every year?
First lets figure out how much loss of value occurred in the 10
years.
$30,000
- $5,000
$25,000
In ten years, $25,000 dollars in value was lost. If we divide
$25,000 by ten years we get: $25,000/10years= $2,500 per year was
lost in value.
Using equations to solve patterns
Mrs. Jones is trying to sell items at her store within 60 days. She
has 2 options.
Option 1: She can sell items at $220.00 and give $1.00 off the
price for every day that it doesn’t sell.
Or
Option 2: She can sell items at $245.00 and give $2.00 off every
day the item doesn’t sell.
21
2pt. Question- Write equations for each option that expresses the
price of the item and number of days the item doesn’t sell. Also, use
these equations to find out which day the two options yield the same
price.
1st part: equations
for option 1: $220.00- $1(x)=
initial
cost
x= #of days
for option 2: $245.00- $2(x)=
initial
cost
x = # of days
If you answered just this part of the question correctly, you would
receive 1pt. out of 2. To get the final point we must figure out on
which day both equations yield the same price.
2nd part: finding the day both equations come out to be the same
price.
We could keep plugging in numbers for each equation until we
found a day that both options equal each other (takes a little more
time) or we can just ↓
22
Make the equations equal each other from the
start and solve for x.
$220- x= $245-2x
$220- x= $245-2x
+2x
Lets get positive x by itself because
were solving for x.
+2x
$220- x= $245-2x
+2x
+2x
$220+x =245
220+ x=245
- 220
-220
Subtract 220 from left side to get x by
itself. We must also subtract 220 from
right side.
220 + x = 25
-220
X = 25
on the 25th day is when
both options will yield
the same cost.
23
Using the nth term to find patterns.
In a sequence of numbers complete the pattern by finding a
formula (nth term) that satisfies the sequence.
ex.
Day
1
2
3
4
5
8
Number of
3
9
27
81
?
?
Mushrooms
• What formula can I put 1(day) into to get 3(# of mushrooms); 2
into to get 9, etc…?
Clue- the number of mushrooms are all a multiple of 3.
nth term = 3
n
→
3
1
=3
We say
“3 to the 1st power”
3 2 =9 (3x3)
“3 to the 2nd power”
3 3 =27 (3x3x3)
“3 to the 3rd power”
what is the nth term when n=5? 3 5 = ?
use your calculator to do powers 3 8 = 3x3x3x3x3=243
on day 5 there will be 243 mushrooms.
for day 8 → 3 8 = 3x3x3x3x3x3x3x3=6561
make sure you let the calculator do the work for you. Especially
for really large numbers! Day 20?! 320th power=
24
Solving linear equations
ex. 5x – 5 = 2x + 10
the goal is to get x terms on one side.
5x - 5 = 2x + 10
+5
+5
←
We do this by adding
5 to each side to get
5x by itself.
5x - 5 = 2x + 10
+5
+5
5x = 2x + 15
5x = 2x + 15
-2x
←
5x = 2x + 15
-2x -2x
3x = 15
←
Now we have to
bring 2x to the left
side so our x
variables are together
Finally, we have to get x by
itself. Since 3 is multiplied
by x (3x) we need to divide
both sides by 3 to get rid of
it on the left side.
3x = 15
3
3
3x = 15
3
3
x = 15
3
x= 5
put our x value which is 5 into original equation to check it.
5(5) – 5 = 2 (5) + 10 → 25-5=10+10 → 20=20
25
Solving and graphing inequalities
ex.
-6 - 5x < -2x + 3
-6 - 5x < -2x + 3
+6
+6
-5x < -2x + 9
Get x on one side.
To get rid of -6 on
left side we add +6
to each side
canceling out the -6
- 5x < -2x + 9
+2x +2x
We divide by -3 to
get x by itself.
- 5x < -2x + 9
+2x +2x
-3x < 9
When we divide by a negative
number in these inequality
problems, the inequality sign
becomes reversed!!
-3x < 9
-3 -3
x>9
-3
x > -3
-3 would have a filled in circle only if it was included by using a
greater than and equal sign: ≥
It would look like this x≥3
26
Graphing equations in slope-intercept form
ex.
Graph this equation below: -4x + 2y = 12
Slope-intercept form
1st: put your
equation in
slope-intercept
form
y = mx + b
m=slope of line
y-intercept (0,b)
-4x + 2y = 12
-4x + 2y = 12
+4x
+4x
4x + 2y = 12
+4x
+4x
2y = 4x + 12
2y = 4x + 12
2 2
2
y = 2x + 6
m = slope = 2
slope = rise = 2 = 2
run 1
27
2nd : Make a quick table of x and y values
with this new equation. We do this by plugging in some
numbers for x to get y values.
When x = 0, y is 6. these are your x and y points.
x y
0 6
-1 4
-2 2
-3 0
-4 -2
3rd: Plot these points on the graph.
28
Data Analysis and Probability
Scatterplots- these graphs look to see if there is any linear
relationship between two variables by plotting points.
ex. If we record the temperature of air as we go up a mountain
along with the elevation (height) we can see if there is a
relationship between these two factors.Here is the data we
collected:
1000ft.
3000ft.
5000ft.
7000ft.
9000ft.
11000ft.
80° F
70° F
60° F
50° F
40° F
30° F
E le v a t io n & T e m p e r a t u r e o f M o u n t
M a p le
12000
E le v a t io n in
fe e t
10000
8000
6000
4000
2000
0
0
10 20 30 40 50 60 70 80 90
T e m p e r a t u r e in d e g r e e s F
We can see that there is a relationship between the elevation and
the temperature because our plots create a straight line.
29
This particular example is a negative correlation meaning as one
factor goes up (elevation up mountain) the other (temperature)
goes down. From left to right the line goes down.
Also, the temperature and elevation are considered continuous
data because you can have temperatures between say 40 & 50 such
as 41.6 and the graph above could express this data by being in
between the 40 & 50 marks. Continuous data allows us to measure
in between values.
Scatterplots can also have a positive correlation. This means they
go up from left to right and as one factor goes up so does the other.
ex. As the # of hours you train for a race goes up so does your
speed.
20 hrs.
40 hrs.
60 hrs.
80 hrs.
100 hrs.
120 hrs.
15 mph
18 mph 20 mph
22 mph
24 mph
25 mph
T r a in in g t im e a n d S p e e d
S p e e d in
M PH
30
20
10
0
0
20
40
60
80
100
120
140
H o u r s s p e n t t r a in in g
This data is also considered to be continuous because someone
could train for time in between these values- say 60.50 hours.
An example of discrete data would be the number of shoes sold at
a store in one day. They cannot sell 17 and ¾ pairs, either 17 or 18
but nothing in between. Discrete data is exact and nothing in
between.
30
Graphs help us by giving us a visual comparison of groups and/or allows
us to see how a rate changes over time.
ex.
James cuts grass in his neighborhood and wants to compare the
amount he made each month for a whole year.
A bar graph would best illustrate this for him.
Dec
Nov
Oct
Sept
Aug
Jul
June
May
Apr
Mar
Feb
400
300
200
100
0
Jan
Amount in dollars $
James' monthly income
Month
The bar graph best compares the different months against each other.
A line graph would better illustrate how James’ income fluctuates from
month to month. ↓
400
300
200
100
Dec
Nov
Oct
Sept
Aug
Jul
June
May
Apr
Mar
Feb
0
Jan
Amount in dollars $
James' monthly income
Month
31
Pie graphs are used to compare parts of a whole.
If James wanted to illustrate what he uses his monthly income for, he
would use a pie graph.
How James spent his June income of $300.00
Savings
17%
Clothes &
shoes
33%
Gas & supplies
Going out
17%
Cell phone
13%
Clothes & shoes
Music
Cell phone
Music
10%
Gas &
supplies
10%
Going out
Savings
Bonus question: can you figure out how much he spent on each item
given the total income and the %’s for each.
ProbabilityWhat is the probability that the spinner will land on number 1?
1
1
0
2
2 out of the 4 sections are 1 so 2 out of 4 or 1 out of 2 spins will
land on 1.
What is the probability the spinner will land on the zero?
- zero takes up one out of the four sections so 1 out of 4 (1/4) spins
will land on zero.
32
Color
Purple
Orange
Black
Green
Red
Total
# of marbles
22
10
18
20
30
100
Maria and Julio empty all these marbles in a bag. If Maria reaches in
and grabs one marble, what is the probability that it will be green?
20 greens = 20 = 20 x 1 or 20 %
100 total
100 20 5
New situation- Maria pulls out a red marble. She keeps this marble
and Julio reaches in to get one marble. What is the probability he will
choose orange?
10 oranges
99 total (Maria has 1 red marble)
= 10 about 10% or 1
99
10
When looking at a multitude of #’s, such as test scores, we can find
the average score: mean, Most frequent scores: mode and middle
score: median.
Student sample
scores→
75% 75% 80%
85% 90% 92%
mean = add all scores = 75+ 75+80 + 85 + 90 + 92 = 497 = 82.8%
score
# of scores
6 scores
6
This answer has
been rounded to the
nearest tenth.
33
mode = 75 % was in the sample scores twice, the most frequent
score.
median = (middle score) you cut the # of scores equally in half from
least score to greatest. There are 6 scores. The middle score is between
80% and 85%.
75% 75% 80%
│
85% 90% 92%
To find the middle number you find the mean of these two numbers.
80 + 85 = 82.5
2
Note: If there is an extremely low or high number compared to the
others, then that extreme # will not be truly represented by the
average or mean score as belowex.
Class scores
40 43 50 53 55
58
Mean= 40+43+50+53+54+58+100=
7
your score
100
57
There are different ways to take samples in order to conduct a survey.
What if a girl wanted to figure out what most people in the U.S. prefer:
Pepsi or Coke. Of course she cannot ask every single person in the
country what they like best. She must take a sample of the population.
34
Here are some ways:
Convenience sample- She would just ask anyone who is available like
classmates.
Random sampling- everyone in her class has an equal chance of being
chosen to be surveyed. This could be done by drawing names from a hat.
Representative sample- This is your group of people being surveyed
that closely reflects the larger population. How people being surveyed
answer or choose is the survey response.
Stem and Leaf plot
Organizes data to show its shape and distribution.
In the plot, we put the first digit(s) in the stem column and the last
digit in the leaf column.
Stem
3
Leaf
6
Here are the math test scores (out of 50) placed from least to greatest. The
teacher can use a stem & leaf plot to see the score distribution.
36, 37, 38, 40, 42, 43, 44, 45, 45, 47, 48, 48, 50, 50, 50
Math Test Scores
(out of 50 pts)
Stem
Leaf
3
678
4
023455 788
5
000
35
Box & whisker plot
Is another way to show the distribution of data.
Recall from a few pages before that the median cuts your scores in equal
halfs.
If we have a set of test scores again:
70 70 75 80 85 │ 85 90 90 95 100
Median of all data (2nd quartile)
This median divides group into a lower and upper part.
We then find the median of the lower and upper parts too.
These are called 1st & 3rd quartiles.
70 70 75 80 85 │ 85 90 90 95 100
│
│
st
nd
2
3rd
1
quartile
quartile
quartile
= 75, 85, 90
Important Numbers
median = 85
first quartile = 75
third quartile = 90
smallest value = 70
largest value = 100
Place these values under a # line using dots:
70 75 80 85 90 95 100
● ●
● ●
●
36
Next, draw a box with ends through 1st & 3rd quartile and a
vertical line through the median point. Last, draw whiskers
(lines) from each end of the box to the lowest and highest #’s.
70 75 80 85 90 95 100
● ●
● ●
●
Geometry & Spatial Sense
A transversal is a line that intersects 2 other lines on the same plane.
If lines a and b are parallel, the angles formed when the transversal
intersecting them have special relationships.
The interior angles- are located between the parallel lines. They are
3,
4, 5,
6.
The exterior angles- are located outside the parallel lines. They are
1,
2,
7,
8.
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Alternate interior angles are always congruent.
They are-
4=
5,
3=
6.
Alternate exterior angles are always congruent.
They are-
2=
7,
1=
8.
Corresponding angles- have same position in relation to the lines
and the transversal.
2=
6,
4=
8.
1=
5,
3=
7.
Similar figures
50 ft.
4ft.
A tree casts a shadow 50 feet long. At the same time, a 6 feet tall
person standing perpendicular to the ground casts a shadow 4 feet
long. How tall is the tree?
Since the sun is casting these shadows at the same time, the shadows
will be equally proportional to the height of these figures. (this means
that the shadows height will be determined by the figures height).
38
?
6 ft.
50 ft.
4 ft.
We know the height of person and the length of their shadow. Lets
create a ratio:
=
tree’s height
Persons height
persons shadow
tree’s shadow
length
length
↓
6 feet
4 feet
↓
=
? tree’s height
50 feet
Lets cross multiply letting t be the variable that = tree’s height which
is not given.
4ft.(t) = 6ft. x 50ft.
↓
4ft.(t) = 300ft.
4t = 300
4t = 300
4
4
4t = 300
4
4
t = 300
4
t = 75 feet
39
Another Ratio Problem:
If a pan of water on the stove increases in temperature from
◦
◦
23 C to 38 C in 21 seconds, how long will it take the water to
◦
◦
increase from 23 C to 100 C?
First, create a ratio between change in temperature verse time:
◦
= 15
∆T = 38-23 = 15 C = 15
Time
21 sec. 21 sec. 21sec.
21
◦
◦
Now create a new ratio with new values (23 C to 100 C)
that equals the 15/21 ratio.
◦
◦
15 =100 C - 23 C =
21
time ?
cross multiply
Check it:
◦
15 = 77 C = 15t = 77 x 21
21
t
15t = 1617
t= 1617 = t= 107.8 sec.
15
15 = 77
21 107.8
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Angles of intersecting lines
What is the sum of angle B & D?
◦
1. We know that a straight line spans 180 . If part of line 1 is
◦
◦
95 as shown above, then the other angle (B) must be 85
◦
◦
◦
(95 +85 = 180 ).
◦
◦
Angle D must also be 85 because it plus 95 makes up
◦
◦
line 2. Angle C has to be 95 –when it plus 85
◦
makes straight line 1= 180 .
◦
Therefore, B= 85
◦
So B + D=170
&
◦
D=85
Finding the missing coordinates of shapes:
www.ode.state.oh.us
41
Find the missing coordinate that would make this a trapezoid.
(Remember a trapezoid has one pair of parallel sides.
top
base
The top and base are parallel to each other.
So a coordinate (an x & y point) that will create a pair of parallels
would be (-1,3)
Translations, reflections, rotations and dilations of shapes on a
coordinate plane.
Translations- these occur when a shape such as a polygon moves up
down, left or right. The object does not change its size or shape nor
does it flip or turn. All the object does is slide.
This trapezoid translates or slides 5
units to the right.
42
Reflections occur when the shape gets flipped over the x or y line
(called line of reflection).
In the example below, the rhombus gets reflected or flipped
over the x-axis. It’s like the x-axis is a mirror. Unlike translation, in
reflection the top becomes bottom and left becomes right.
A rotation occurs when an object rotates or spins on a fixed point
(center of rotation). The object keeps its same shape but rotates
clockwise
or counterclockwise
43
In the graph below the trapezoid rotates clockwise 180º. The center of
rotation is where the two shapes are touching (bottom left of top
trapezoid).
A Dilation occurs when an object changes its size but keeps its same
shape. It can either increase or decrease in size.
There may be a problem that wants you to figure out the length of a
side of a shape as it dilates.
ex. A trapezoid has a base of 4 inches and a top of 3. Can you figure
out its new top as it is enlarged?
3 inches
?
→
4 inches
6 inches
We know the ratio of the original to be ¾. We can make the unknown
top length our x and solve for it. Make these two ratios = to each
other because when an object gets dilated these ratios never change.
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Cross multiply
3 = x
4 x = (3x 6)
4
4 x =18
6
Divide each side
By 4 to get x by
itself.
4 x =18
4
4
x = 18/4
x= 4.5
Nets for cones
Nets are flat shapes that when you fold them correctly they will create
3-D shapes such as cones, pyramids and prisms.
ex. rectangular prism
http://jwilson.coe.uga.edu
45
ex. triangular prism
http://jwilson.coe.uga.edu
ex. cone
http://jwilson.coe.uga.edu
ex. cylinder
http://jwilson.coe.uga.edu
46
With right triangles there are special relationships between the three
sides. The Pythagorean Theorem explains this:
a2 + b2 = c2
a and b are the length of the legs and c is the length of the
hypotenuse ( the side opposite the right angle).
Simply, if you know the length of any two sides of a right triangle,
you can figure out the length of the third by using
a2 + b2 = c 2 .
a2 + b2 = c 2
ex.
?
6 feet
(6) 2 + (8) 2 = c2
36 + 64 = c2
8 feet
100 = c2
10 = c
c=10
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Have fun! Keep up the Great work!!
Hanna-Barbera
48
8TH GRADE MATH OAT VOCABULARY
NUMBER SENSE
SCIENTIFIC NOTATION- A way of expressing numbers as a product of a number that is at
least 1 but less than 10 and a power of 10. For example, 5500=5.5x103
RATIONAL NUMBERS- Numbers of the form a/b where a and b are integers and b does not
equal zero.
IRRATIONAL NUMBERS- A number that cannot be expressed as a/b where a and b are
integers and b is not equal to zero. The decimal form of the number never terminates and
never repeats.
REAL NUMBERS- The set of rational numbers together with the set of irrational numbers.
NATURAL NUMBERS- Also known as counting numbers. They are: 1,2,3,4,…
WHOLE NUMBERS- The numbers 0,1,2,3,4,…
INTEGERS- The whole numbers and their opposites. They are: …,-3,-2,-1,0,1,2,3,…
QUOTIENT- The answer to a division problem.
DIVIDEND- The number being divided.
RADICAL-The symbol used to indicate a nonnegative square root.
ORDER OF OPERATIONS- The rules to follow when more than one operation is used.
INVERSE PROPERTIES- The result of two real numbers that combine will give the identity
elements of zero or one. For addition 8+-8=0 For multiplication 2/3x3/2=1.
INVERSE OPERATION- An operation that will undo another operation.
PERFECT SQUARE- A rational number whose square root is a whole number.
SQUARE ROOT- One of the two equal factors of a number.
MEASUREMENT
RATE- In a percent proportion, the ratio of a number to 100.
QUADRILATERAL- A polygon having four sides.
PRECISION- Depends on the unit of measure. The smaller the unit the more accurate.
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CUSTOMARY UNITS- Units such as:miles,gallons,pounds.
METRIC UNITS- Units such as : kilometer, liter, kilogram.
SURFACE AREA OF A PRISM- The sum of the area of the base and the areas of the
rectangular faces.
VOLUME OF A PRISM- Found by multiplying the area of the base and the height of the
figure.
VOLUME OF PYRAMID- 1/3 times area of base times the height.
SURFACE AREA OF CYLINDER- 2 times pi times radius times height
VOLUME OF CYLINDER- pi times radius squared times height.
VOLUME OF SPHERE- 4/3 times pi times radius cubed.
VOLUME OF CONE- 1/3 times pi times radius squared times height
PERIMETER- The distance around a geometric figure.
CICUMFERENCE- The distance around a circle. Found by the formula : 2 times pi times
radius or pi times diameter.
AREA- the number of square units needed to cover a surface enclosed by a geometric figure.
AREA OF TRIANGLE- ½ times base times height.
AREA OF PARALLELOGRAM- base times height.
AREA OF CIRCLE- pi times radius squared.
GEOMETRY
TRANSVERSAL- A line that intersects two parallel lines to form eight angles.
COORDINATE PLANE- A plane determined by the intersection of two perpendicular
number lines in which a point can be located.
PARALLEL- Lines in the same plane that do not cross. The distance between the lines is
constant.
SIMILAR FIGURES- They have the same shape but may not have the same size.
VERTEX- The point where all the faces of a prism intersect.
TRANSLATION- A transformation where a figure is slid horizontally , vertically , or both.
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REFLECTION- A transformation that results in a mirror image of the original shape. In other
words , a figure is flipped over a line of symmetry.
ROTATION- A transformation when a figure is turned around a central point.
DILATION- A transformation that reduces or enlarges an image.
NET- Every solid with at least one flat surface can be formed from this two-dimensional
pattern.
ALGEBRA
COVARIANTS- Varying with another variable quantity in a manner that leaves a specified
relationship unchanged.
MONOMIAL- A number , a variable , or a product of a number and one or more variables.
POLYNOMIAL- The sum or difference of two or more monomials.
SLOPE- The steepness of a line. It is the ratio of the rise , or vertical change , to the run , or
horizontal change, as you move from one point on the line to another.
Y-INTERCEPT- The point(ordered pair) where a line crosses the y-axis.
QUADRATIC EQUATION- An equation in which the greatest power on the variable is 2.
DIRECT VARIATION- When the values of two variables maintain a constant ratio. This
relationship can be expressed as an equation of the form y=kx.
INVERSE VARIATION- The variables x and y vary inversely. If, for a constant k, yx=k or
y=k/x.
ORDERED PAIR- A pair of numbers used to locate a point in the coordinate system. It is
written in the form: (x-coordinate,y-coordinate).
COEFFICIENT- The numeric factor in a term. For example, the number 3 in the term 3xy.
LINEAR FUNCTION- A function whose graph on a coordinate grid is a straight line.
SIMPLIFY- A process used to make an expression have no like terms in it.
EVALUATE- To find the value of an expression by replacing the variables with numerals.
DATA ANALYSIS
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SCATTERPLOTS- A graph that shows the general relationship between two sets of data. It
is a graph with one point for each item measured. The coordinates of a point represent the
measures of two attributes of each item.
CIRCLE GRAPH- A type of statistical graph used to compare parts of a whole.
RANGE- The difference between the greatest number and the least number in a set of data.
QUARTILE- Values that divide data into four equal parts.
MEDIAN- The middle number or item in a set of numbers or objects arranged from least to
greatest , or the mean of the two middle numbers when the data has an even number of items.
RANDOM SAMPLE- A sample in which every event has an equal chance of selection and
each event is chosen by a random process.
DISJOINT EVENTS- Two events that have no outcomes in common.
INDEPENDENT EVENT- Two events in which the outcome of the first event does not
affect the outcome of the second event.
DISCRETE DATA- Data that can be counted. For example, the people in a town.
MEAN- the sum of a set of numbers divided by the number of elements in the set.
MODE- The number or object that appears most frequently in a set of numbers or objects.
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