OGT Math Study Guide - Hillsboro City Schools

OGT 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.= 24ft.2
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
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.
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:
3
area of base x the height of the object.
ex. Find the volume of this cylinder.
5ft.
┬
15 ft.
┴
Area of this circle is 78.5ft 2
(from previous problem)
x height of 15ft.= 1177.5ft3
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
4
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
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)
5
= 504 = 100.8
5
100.8 °F = 24 ° C
5
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
6
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.
7
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.
8
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 ↓
Make the equations equal each other from the
start and solve for x.
9
$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.
Using the nth term to find patterns.
10
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
→
1
3
=3
We say
“3 to the 1st power”
32 =9 (3x3)
“3 to the 2nd power”
33 =27 (3x3x3)
“3 to the 3rd power”
what is the nth term when n=5? 35 = ?
use your calculator to do powers 35 = 3x3x3x3x3=243
on day 5 there will be 243 mushrooms.
for day 8 → 38 = 3x3x3x3x3x3x3x3=6561
make sure you let the calculator do the work for you. Especially
for really large numbers! Day 20?! 320th power=
Solving linear equations
11
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
Solving and graphing inequalities
12
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
Graphing equations in slope-intercept form
13
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
2nd : Make a quick table of x and y values
14
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.
Data Analysis and Probability
Probability15
What 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
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
= 10 about 10% or 1
16
99 total (Maria has 1 red marble)
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
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
17
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
Pie graphs are used to compare parts of a whole.
18
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.
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
19
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
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
1
2
3rd
quartile
quartile
quartile
= 75, 85, 90
20
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
● ●
● ●
●
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
● ●
● ●
●
21
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.
22
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).
23
?
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
24
With right triangles there are special relationships between the three
sides. The Pythagorean Theorem explains this:
a2 + b 2 =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 = c2
a2 + b2 = c2
ex.
?
6 feet
(6)2 + (8)2 = c2
36 + 64 = c2
8 feet
100 = c2
√100 = c
10 = c
c=10
25
Number, Number Sense and Operations
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.
26
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.
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
27
If a number in standard notation has zeros to the left of the decimal
such asmove 5 places to the left
to get the decimal
behind first digit (2).
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
28
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
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
29
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 104 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
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
30
11
This number is rounded
to the nearest
hundredth
= 5.55 x 10 miles
year
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
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 x 2 = a positive number, because
A
-x · -x = +x2
negative · negative = +
Lets let x = -3
x 2 = (-3)2 = -3 · -3 = 9
If we let x= - 3 again, will x be + or - ?
x 3 =(-3) 3 = -3 · -3 · -3 = - 27
9 · -3 = - 27
31