# Math 8 Assessment Review Diagnostic

Document Sample

```					Math 8 Assessment
Review

Absolutely
Positively
Everything You Need To Know
New York State says there are
certain math skills and concepts
they expect you to know. These

 Algebra
 Geometry
Measurement
Number Sense and Operations
How To Use This Program
This program is designed to give you a good
idea of what you already know and what
you need to learn a little better.

Each slide has a topic (written in yellow) you
are expected to know and some sample
questions. If you get all of the questions on
the slide right, click to the next slide.

If you miss any questions, click on the link
and you will go to a tutorial that will help
you learn what you need to know.
Number Sense
Let’s see if you know your integer rules.

1) 5 + (-7)    -2       7) -2 x 9        -18
2) -3 + (-1)   -4       8) -2 x -9          18
3) -4 + 9      5        9) 5 x -4        -20
4) 5 – (-2)    7        10) -10 x (-5)      50
5) 2 – 5       -3       11) 15 / (-5)       -3
6) -7 – 3      -10      12) -15 / (-5)      3
Use the order of operations to simplify
expressions. Do you know it?

Simplify each:
1)18 – 5 + 3 16

2)50 - 4·9   14

1) If a = 5 and b = 3, evaluate ab2   45

2) If a = 4, b = -3 and c = 5, evaluate
a(b + c) – (a – b) 1
Be able to express large and small numbers in
scientific notation. Do you know it?

Express each in scientific notation:

1)872,000,000       8.72 x 108

2)0.00000104        1.04 x 10-6
Apply the Laws of Exponents for Multiplication
and Division. Do you know it?

1) x2·x3= x5

2) (3x4) (5x5) = 15x9

3) v8/v2 = v6

4) 12a6/3a2 = 4a4
Evaluate Expressions with Integral Exponents.
Do you know it?

1) 34    81

2) 50    1

3) 102       100

4) 3-2   1/9

5) 199   1
Read, write and identify percents less than 1 and
greater than 100. Do you know it?

1) Express 125% as a decimal in simplest form.
1.25
2) Convert 0.0003 to a percent.
0.03%
3) Describe a situation where a percent can be
greater than 100%.
Invest money and get back more than you had.

4) Convert 3/8 to a percent. 37.5%
Apply percents to tax. Do you know it?

Suffolk County has an 8.5% sales tax. If Jenna buys
some cleats for \$50 ………………….
a)How much tax does she have to pay?

\$4.25

b) How much does she give the cashier?

\$50 + \$4.25 = \$54.25
Find the percent of increase or decrease.
Do you know it?

Todd buys a painting for \$25 and then sells it on
eBay for \$30. What is the percent of increase on
the value of he painting?

20% increase
Use percents to find the sale price of an item.
Do you know it?
Overstock is having a Christmas sale. Originally
a pair of earrings cost \$78 and are on sale for
45% off. What is the new price?

\$42.90
Calculate markup on an item and determine
the price the store sells an item for (retail.)
Do you know it?

Fred’s Flyfishing Emporium marks up every
item he sells by 120%. If the store buys a
DVD for \$8, how much does he sell it for?

\$17.60
Calculate simple interest. Do you know it?

In order to buy a jeep, Jimmy is thinking of taking
out a loan. He would need to borrow \$5,000 for
2 years at an interest rate of 9%.
a)How much interest would Jimmy have to pay?

\$900

b)What is the total Jimmy would have to pay back?

\$5,000 + \$900 = \$5,900
Estimate the percent of a quantity in an
application. Do you know it?

1)Two out of nine kittens are black. Estimate
what percent are black.

20%

2) Students in a school in Tennessee voted on
their favorite country singer. 52% of them
voted for Tim McGraw. If 250 students were
surveyed, how many voted for Tim McGraw?

125
Use estimation to analyze how reasonable an
answer is. Do you know it?

Tara buys 12 different magic markers when she
shops at Staples. Each marker costs \$1.49. She
thinks the total cost of the markers will be \$15. Is
she right?

No. The cost is closer to \$18.
(1.5 x 12) = 18
Add and subtract monomials. Do you know it?

1)5x + 2x = 7x        3) 3c – 7c =   -4c

2) 3x2 + 7x2 = 10x2   4) ab – 4ab= -3ab
Solve multistep equations by combining like
terms, distributing or moving variables. Do you
know it?

1)4w + 3 – 2w = 17       3) 5(2x – 3) = 25

w=7                     x=4

2) 7y – 3 = 4y - 15

y = -4
Graph a table of values or pattern from an equation.
Do you know it?

Using a table of values, graph y = 3x – 4.
x   Y                y

0   -4

1   -1
(0,0)              x
2   2

3   5
Create algebraic tables using charts/tables,
graphs, equations and expressions. Do you
know it?
Tara reads 3 pages per minute. One Saturday
minutes. Create a table of values to show what
page (p) she is on after any minute (m).
m p              m   p
0   17
1   20
2   23
3   26
4   29
5   32
Build a pattern to develop the sum of the interior
angles of a polygon. Do you know it?

How many degrees in the interior angles of a …..
Triangle        180
Pentagon        540
Hexagon         720

In general?     (Number of sides – 2) x 180
Write an equation to represent a function from
a table of values. Do you know it?
Write a function rule that explains the
relationship between x and y in each table.

x     y                   x    Y
0     3                   -1    3
1     5                   1     5
2     7                   3     7
3     9                   5     9

Y=2x+3
Rule: ________                   Y=x+4
Rule: ________
Translate verbal sentences into algebraic
expressions. Do you know it?

Write each as an algebraic expression:
1) A number, n, increased by five.   n+5
2) Seven less than a number, n.      n-7
3) The product of three and a number, n. 3n
4) Two more than four times a number, n. 4n + 2
5) The sum of a number squared and one. n2+ 1
Create a graph given a description or an
expression for a situation involving a linear
or nonlinear relationship. Do you know it?
Jamal bought a plant that was 3 cm tall. Each week
it grew 2 cm. Graph the height (h) on any week
(w).                      h
w h         w    h

0    3

1    5

2    7

3    9
2

1

0                         w
Add and subtract polynomials. Do you know
it?

1)Find the sum of 3x – 2 and x – 5. 4x - 7

2) Add: (5a + b) + (2a – 2b)        7a - b

2              2
3) Add: 3x – 2x + 5 – 2x – 7x – 5            x2 – 9x

4) Subtract: (5d – 3f) – (2d – f)     3d – 2f
Multiply a binomial by a monomial or
binomial. Do you know it?

Simplify each expression:
1) 3(x – 7)     3x - 21

2) -2(2d – 9)    -4d + 18

3) x(x + 2)     x2 + 2x

4) (x + 3) (x + 5)        x2 + 8x + 15

5) (x – 5) (x + 2)    x2 – 3x - 10
Factor algebraic expressions using the Greatest
Common Factor (GCF). Do you know it?

Factor each:
1) 3x + 12        3(x + 4)

2) 5h – 20        5(h – 4)

3) x2 + 5x        x(x + 5)

4) x2 – 8x + 12      (x – 6) (x – 2)
Solve multistep inequalities and graph the answer on
a number line. Do you know it?

Solve the following in equalities and graph each on
a number line:
1) 3x – 7 < 5

x<4
4

2) 4 – 2x < 10

x>3                     3
Solve multistep linear inequalities by combining
like terms, distributing or moving variables
from one side to the other. Do you know it?

Solve each inequality for the variable:
1)3d – 5d + 4 > 12       2) 4x + 6 < x – 3

d < -4               x < -3
Geometry
Identify the hypotenuse, right angle and legs of
a right triangle. Do you know it?

In the picture of the right triangle below,
identify the legs and the hypotenuse.

hypotenuse
leg

leg
Be able to find a missing leg or hypotenuse
using the Pythagorean Theorem. Do you know
it?

Find n in each diagram:

13
4        n               5

n
3                       n = 12
n=5
Identify pairs of vertical angles as congruent.
Do you know it?

5n - 1      3n + 27

Find n
n = 14
Identify pairs of supplementary and complementary
angles. Do you know it?

Label each diagram as complementary or
supplementary.

supplementary

complementary
Calculate the missing angle in a complementary
or supplementary pair. Do you know it?

Find the value of n in each diagram:

67                              n     71
n

n = 109
n = 23
Determine the angle relationships when two
parallel lines are cut by a transversal. Do you
know it?

List all of the
angles that are                       1 2
equal to each                        3 4
other.

<2, <3, <5, <8                 6 5
8 7
equal eachother.
<1, <4, <6, <7
equal eachother.
Calculate the missing angle measurements
when two line are cut by a transversal. Do you
know it?
m<a = 115
a b
c d                m<b = 65
m<c = 65
m<d = 115
115 e                      m<e = 65
g f
m<f = 115
m<g = 65
Identify different transformations in the plane,
using proper function notation. Do you know
it?

1)                        2)       rotation

reflection

3)

translation
Identify horizontal and vertical line symmetry
as well as point symmetry. Do you know it?

Which of the following letters have horizontal,
vertical and/or point symmetry?

C     horizontal

Y     vertical

S     point

J     no symmetry
Be able to draw a line reflection. Do you
know it?
Reflect the line segment over the x-axis.
y

(0,0)        x
Be able to draw a rotation of 90, 180 and 270
degrees. Do you know it?
Rotate the rectangles (one for each) below 90
degrees, 180 degrees and 270 degrees.

90

180

270
Be able to draw and compute the result of a
translation. Do you know it?

Translate the line segment two units to the
right and down four.
Measurement
Calculate distance using a map scale. Do you
know it?
The scale of a map is 1 inch = 80 miles. On the
map, the distance from Patchogue to Callicoon is
2.5 inches. How far is it from Patchogue to
Callicoon?

200 miles
Calculate unit price using proportions. Do you
know it?

A box of Corn Flakes cereal sells for \$3.78. The
volume of the box is 18 ounces. What is the
unit price for an ounce of Corn Flakes?

21 cents or \$0.21 per ounce
Compare unit prices for least and most
expensive. Do you know it?

Costco sells two different size packages of chicken.
One package sells 3.5 pounds for \$8.19 and the
other sells 5 pounds for \$10.79. Which one is a
better buy and how much do you save per pound?

The 5 lb bag is cheaper by
\$0.18 per pound.
Set up and use proportions to solve word
problems. Do you know it?

Emily babysits in order to make some money. On
Saturday, she gets paid \$12 for 2.5 hours of
babysitting. On Monday, she gets paid \$30. How
long did Emily babysit?

6.25 hours
Determine if two figures are congruent or similar. Do
you know it?

1)What does congruent mean?       same size & shape
2)What does similar mean?
same shape sides in proportion
Is the pair of triangles below similar or congruent
(assume all corresponding angles are equal.)

similar
Use a proportion to solve for a missing side in
similar figures. Do you know it?

Find x.
12
9
6
x

x=8
Absolutely, positively
everything you need to
know!

There is one question you need to ask
yourself when adding integers: Are the
signs the same?

Are the signs the same?

Yes                               No

Subtract. Keep
Add. Keep sign.                  Sign of bigger #.
Adding Integers With The Same Sign
Remember: Add and keep the same sign!
1) 3 + (+5) 8
2) -3 + (-5) -8
3) 7 + 4     11
4) -7 + -4 -11
5) -3 + (-100) -103
6) -2 + -2     -4
7) -12 + (-3) -15
If the signs are different, subtract and keep the sign of the
bigger number.

1)   5 + (-3)   2
2)   5 + (-4)   1
3)   5 + -5     0
4)   5 + (-6)   -1
5)   -3 + 4      1
6)   -9 + 7     -2
7)   8 + -5      3
Mix & Match – How Many Can You Get?
1)   -3 + 5 2            6) -2 + -3 -5

2)   7+8   15            7) 9 + -9 0

3)   1 + -5 -4           8) -7 + (-7) -14

4)   -10 + -5 -15        9) -4 + 1 + 8   5

5)   -8 + -2 -10         10) -2 + (-1) + (-4)   -7

You understand when you can get 9 or more
right  !
Subtracting Integers
If you let it, subtracting integers can get very confusing. If
you are having trouble, let’s try a new way and hope we
don’t get confused. DON’T SUBTRACT!!!! That’s right –
The hardest part of integers is when all of the different rules
start confusing us. We already know the rules for
number we are adding is positive or negative.
Here’s what I mean: 4 – 2 can also mean (+4) plus (-2).
So just read it that way. 4 + (-2) = 2. Let’s try another:
5 – 7 should be read as +5 + -7. And we already know
that is 2.
Practice. Practice. Practice.

3–7         3 plus -7                 -4

-5 – 2      -5 plus -2                -7

-8 – 2      -8 plus -2                -10

-8
-4 – 4      -4 plus -4

9-5         9 plus -5 OR just 9 - 5   4
The Last Detail of Subtraction
In case you had not noticed, the last set of problems lacked
the type of question that looked like 5 – (-3). Let’s just
write this as 5 “-” (-3). In order to do this type, we need
to remember that when you negate or minus ( “-” ) a
negative, you make a positive. So 5 – (-3) is 5 “+” 3.

4 – (-5)      4+5           9
2 – (-3)      2+3           5
-5 – (-2)     -5 + 2        -3
-4 - 10       -4 plus -10   -14
Subtraction Practice

1)   7 – (-3)       7+3            10

2)   5–8           5 plus -8       -3

3)   -7–2          -7 plus - 2     -9

4)   10 – (-5)     10 + 5          15

5)   3-7           3 plus - 7      -4
What Do We Know So Far?
1)   When you add and the signs are the same, you
______ and keep the ____________.

2)   When you add and the signs are different, you
subtract               sign of the bigger number
_________ and keep the _____________________.

DON’T Subtract!!!! Treat the problem like _________
positive
and the sign in front tells you if it is ___________ or
negative
___________. If you see two negatives in a row (like
+
5 – (-2) ), change the “-” “-” to a ___.

1)   8 + -4   4        6) -3 – 9 -12

2)   12 – (-5) 17      7) -10 – (-2) -8

3)   -2 – 1 -3         8) 5 + -4 1

4)   -3 + -4 -7        9) 7 + (-7) 0

5)   -4 + -1 + -3 -8   10) -4 – (-1) + -7 -10
Multiplication & Division
There are several ways to explain multiplying and dividing
with signed numbers, but let’s try a less common
method. Here’s the deal: just multiply (or divide) the
numbers and only when you are doing so BY a negative
do you change the sign. It’s easy!

-3 x 5       3 times 5 is 15 and keep the neg. -15
-3 x -5      -3 x 5 is -15, BUT -5 means change sign +15
8 x -4       8 x 4 = 32,BUT -4 means change sign -32
-5 x -9      -5 x 9 = 45, BUT -9 means change sign -45
Multiplication and Division Practice
1)     -5 x – 2 10            5) (-10) / 2   -5

2)     -18 / 9 -2             6) -1 x -3 3

3)     12 / -4 -3             7) -10 x -8 80

4)     5 x -5 -25             8) -20 / 5 -4

In #s 1, 3, 4, 6 and 7, you multiplied or divided BY a
negative, so the sign changed. In all of the others,
you multiplied or divided BY a positive, so it stayed the
same  !
Completely Mixed Practice
1)   7 + (-5) 2        6) -3 x -8 24

2)   -3 – (-4) 1       7) 8 – 10   -2

3)   -22 / 11 -2       8) -9 + (-2) -11

4)   -6 x -5 30        9) 10 – (-4) 14

5)   -4 + 3 -1         10) -25 / -5 5
Challenging Questions
1)   Explain why when you add two negatives you get a
negative but when you multiply them, you get a
positive.
When adding, you keep the same sign but when
multiplying by a negative you change the sign.
2)   When you see a problem that says 5 – (-3) with those
two minus signs in a row, what do you have to
remember to do?
Change the “-” “-” to a “+”.

There is no subtraction. It is really ___________ and
the sign in the front tells you if it is positive or negative.
Absolutely, positively
everything you need to
know!
The Order of Operations
The order or operations tells us what to do and when to do
it when trying to evaluate an expression. Some people
remember PEMDAS and others PLEASE EXCUSE MY
DEAR AUNT SALLY. Either way, it all means the order
is:
Parenthesis (aka Packages)
Exponents
Multiplication / Division (left to right, they are equal)
Add / Subtract (left to right, they are equal)
What Operation Would You Do First?
1)   9–2+9      subtract

2)   17 + 4 x 5 multiply

3)   10 (5 – 3) subtract

4)   10 – 14 ÷ 2 divide

5)   83 + 5 x 100 exponent
Step by Step Order of Operations
Go slow and answer this question step by step. As you
click on the mouse, each step will appear.

Evaluate: 25 – 3 x 22
OK. First you have to do 22 = 4
So it becomes 25 – 3 x 4
Next you have to do 3 x 4 = 12
Finally, it is 25 - 12
13
Let’s Do Another Step BY Step
Evaluate 5(7 – 3) – 8 + 1
First things first …… do the package 7 – 3 = 4
So we have 5(4) – 8 + 1
I sure hope you realize multiplication comes next. 5 x 4 = 20
OK. Now 20 – 8 + 1
Don’t get fooled!!! Addition and subtraction are equal. 20 – 8= 12
Finally, 12 + 1

13
Perfect Practice Makes Perfect
1)   5(7 – 9)          3) 2(4 + 3)2

5 (-2)            2(7)2

-10               2 · 49
98
2) 22 – 7 + 10 ÷2      4) 102 – 43

22 – 7 + 5         100 – 64
15 + 5             36
20
Algebra & The Order of Operations
Use the values a = 10, b = -5, c = 4, d = 2 and e = -2:

1)    ab2            2) e3 + ab           3) ac + be
10 · (-5)2      (-2)3 + 10(-5)       10 · 4 + (-5)(-2)
10 · 25         -8 + 10(-5)          40 + 10
250             -8 + (-50)           50
-58
Summary
The Order of Operations is:
1) Parenthesis (aka Packages)
2) Exponents
3) Multiplication / Division

*** DANGER!!! Be very careful with the last two steps.
Multiplication and Division are equal to each other. If
they are the only steps remaining, just go left to right.

The same holds true for addition and subtraction.
Absolutely, positively
everything you need to
know!
Equation Basics
There are 4 concepts we must remember at all times with
basic equations:
1) The strategy is to get letters on one side and numbers
on the other.

2) First identify what is happening to the variable.

3) Do the opposite, undo or do the inverse operation.

4) Be sure to do the same thing to both sides!!!!!
Basic Two Step Equation
Solve for x: 3x – 1 = 14
+1 things are happening: subtracting and
Two +1
multiplying. First attack the subtraction.
3x = 15
3 3        Now attack the multiplication.
x = 5

More Practice:
n/2 + 5 = 3        5h + 4 = 9                  12 = g/3 + 9
-5 -5            - 4 -4                   -9       -9

n/2    = -2        5h     =5                   3 = g/3

n   = -4             h =1                   9 = g
More Two Step & A Multi Choice Trick
Before we can possibly understand the harder equations,
we have to master these. Solve for x:
4x + 7 = 3           10 = x/9 – 1           12 = 3c - 12
x = -1                99 = x                  8=c
In a multiple choice question, if you have a hard time
solving the equation, you can substitute and check the
Solve for x: 7x – 5 = -33
a) -5        b) 5           7(-4) – 5 =
c) 8         d) -4          -28 – 5 = -33 check!
Two Step Inequalities
The biggest difference between an equation and an
inequality is that an equation has only one answer and
an in equality has an infinite number of answers. For the
most part, they are exactly the same. Here’s a
comparison:
2x + 1 = 21                      2x + 1 < 21
-1    -1                         -1    -1
2x        = 20                   2x        < 20
x        = 10                    x        < 10

Step by step is identical. The only difference is the
symbol – for now. Let’s practice some more  .
More Inequalities
5f + 7 ≤ 22           f/3 – 2 > 8      -3d – 1 < 11
f≤3                   f > 30            d > -4
+ right!!!!
That’s 1 +1
Let’s try a multiple choice:             -3d     < 12
d > -4 NOT less
Which is a solution of 2x + 7 < -13      -3
than! Why?  -3
a) -12                                 Whenever you x
SEE IT !!!!! You
If we solve, we get x<-10.      ÷ an inequality
ordivided BY a neg.
b) -11    Only (d) is less than -10.   BY a negative, sign!
SWITCH the
c) -10                                 you have to
You can also check each          d      <
change the -4
the steps! Click
now.
Graphing Inequalities
The < and > symbols are nothing more than the “less” and
“greater” ends of the number line. As long as the
variable is on the left, the graph on a number line goes
the way the symbol points. Let’s graph each:
x>2                            x≥2

2                        2

What is the difference between the two graphs? The ≥
has the circle filled in. Graph these:
x<2                          x≤2

2                                2
More Graphing
What happens if you solve an in equality and the answer
looks like this: 7 < x
The letter is on the right……. that’s not the way w want it.
All you need to do is put it in the left and MAKE SURE
the inequality points to the same thing.
7 < x is the same exact thing as x > 7
Solve and graph each of the following:
-4x + 11 < -1                -10 > 3x - 1
x>3                        -3 > x
x < -3
3
-3
Solving Multistep Equations
There are two primary types of equations we will have to
deal with here. One has the variables on the same side
and the other has them on opposite sides.
SAME SIDE                 OPPOSITE SIDE
3n + 7 + 4n = 21               5b – 3 = 3b + 17
-3b     -3b
7n + 7 = 21
2b – 3 = 17
-7     -7
+3 +3
n = 2
b = 10

Remember this KEY SAYING: Same Side, Same
Operation. Opposite Side Opposite Operation.
What Would Your First Step be?
Would you do the same operation and combine like terms
(same side) or use the opposite operation (opposite side).

3x + 7x – 3 = 11               5d + 7 = d - 9
same operation
opposite operation
combine terms

2n – 9 = n + 11                 2x + 7x – x = 42

opposite operation              same operation
combine terms
Perfect Practice Makes Perfect
Let’s try these four. Don’t forget about it! Same side
same operation                        opposite operation
_________________. Opposite side ________________.

3w + 2 + w – 5 = 5              6f + 1 = 28 – 3f
4w – 3 = 5                     9f = 27
w     =2                        f=3
2x + 3x + 7x – 10 = 134         5x + 2x – 3 = 4x - 15

12x – 10 = 134                 7x – 3 = 4x – 15

x      = 12                      3x = -12
x = -4
Equations Using Distribution
If you see parenthesis (or a package) in an equation, you will
probably have to distribute.

3(x – 5) = 9         2(x + 4) = 7x – 12     3(x – 2) + x – 3 = 11
3x – 15 = 9           2x + 8 = 7x – 12       3x – 6 + x – 3 = 11
+ 15 +15         -2x      -2x           4x – 9 = 11
3x       = 24               8 = 5x – 12         x =5
x   = 8             +12          +12
20 = 5x
4=x
Perfect Practice Makes Perfect
1) 3n + 8 – n + 2 = 40   3) 3(h – 7) – 2 = 7

n = 15                  h = 10

2) 8q – 3 = 5q + 18      4) x + x + 10 + 2x + 20 = 110

q=7                     x = 20
The Get It - Got Its
scale
An equation is like a __________. As long as you do the
same thing
_____________ to both _______ it stays balanced.
sides

multiply     divide
When solving an inequality, if you _________ or ________
negative
by a _________, you have to switch the sign.

With multiple step equations, if you see parenthesis, you
distribute
can expect to _____________. If the variables or
same
numbers are on the same side, use the _____
operation
___________, but if they are on opposite sides, use the
____________ ____________.
opposite        operation
Absolutely, positively
everything you need to
know!
To be good at adding and subtracting monomials, you need
to know two things:
#1: How to add and subtract integers
#2: That whenever you add or subtract anything on the
face of the earth, the type of thing stays the same.

2 “puppy dogs” + 3 “puppy dogs” = 5 “puppy dogs”
2 “kitty cats”  + 3 “kitty cats” = 5 “kitty cats”
But you can’t add 2 puppies + 3 kittens!!!!!
\$2 + \$3 = \$5         2x + 3x = 5x         2ab + 3ab = 5ab

you end with!!!
Identifying Polynomial Types
What is a polynomial? The word is large enough to look
intimidating, but it is actually made up of two simple
parts. Poly means many (polygon, polytheism) and
nomial means term (x, 3y, 4ab and x2 are all terms.) So
a polynomial has many terms.
3w – 5y               2x2 – 8x + 11y            2p – q + 1 – w
are all polynomials.

We can be a bit more specific. 2w – 5y is a binomial
because it has two terms and 3a – 2b + c is a trinomial
because it has three terms. 5x is a monomial.
Perfect Practice Makes Perfect
Simplify each expression:

1) \$4 + \$3          3) 3w – 2w          5) 4ab + 3ab
\$7                     w                 7ab

2) 4d + 3d          4) 8h – h           6) 4xy - xy

7d                  7h                   3xy

There are two critical things you absolutely MUST remember:

1) Only like things can be added.
More Perfect Practice
Simplify each expression:

1) 3c – 7c         3) -4w + 3w       5) ab + ab

-4c                     -w           2ab

2) 3x2 – x2        4) -q + 8q – 2q   6) 3a2b + 2a2b

2x   2
5q           5a2b

Remember: Things must be the same. Things stay the same.
This is pretty much the same – but we have to really
emphasize that only like things can be put together! We
can use Poppa’s Bucket Principle to clean up the mess.
Everything has a bucket and only the same things go in
a bucket!
3a + 5b – 2c + a – 2b – 3c

3a          5b          -2c
Find the            a       -2b           -3c
sum in
each            +4a         +3b         -5c
bucket.
Final Answer: 4a + 3b – 5c (like the buckets say)
Perfect Practice Makes Perfect
Simplify each by combining like terms:

1) 2a – 3b + a – 2b       4) 2q2 – 3q + 5 – q2 + q – 3

3a – 5b                      q2 – 2q + 2

2) 3x + 5 + 5x – 7        5) a – 3b – 5c – a + 2b – c
8x - 2                       -b – 6c

3) n2 – 3n + n2 + 2n      6) 3x – 2y + 3y - x

2n2 - n                      2x + y
Perfect Practice Makes Perfect
Simplify each by combining like terms:

1) 2a – 3b + 5a – b       4) 4q2 – q + 4 – q2 + q – 2

3a – 4b                      3q2 + 2

2) 4x + 5 + 5x – 3        5) a – 7b – 3c – 2a + 4b – c
9x + 2                     -a -3b – 4c

3) n2 – 3n + 3n2 + n      6) 5x – 3y + 3y - x

4n2 - 2n                      4x
The Distributive Property
The distributive property distributes or gives out a term to
the others. Examples:
5(3x – 7) = 15x – 35        2(x2 + 5x – 1) = 2x2 + 10x – 2
Notice we multiplied the 5 by 3x to get 15x and the 5 by -7
to get -35. The “2” is similarly given to the trinomial in
the second example. Now try these:
1) 3(7x + 2)                3) 5(3x2 – 2x + 9)
21x + 6                    15x2 – 10x + 45
2) 2(5a – 3b + c)           4) -7(2x – 5)

10a – 6b + 2c               -14x + 35
Double Distributing
Frequently in algebra we are required to multiply a binomial
by a binomial. For instance (x + 3) (x + 5). The key is to
distribute one on t he binomials at a time.
(x + 3) (x + 5) = x2 + 5x + 3x + 15
Distributing the blue “x” gives the blue part of the answer
and the yellow “3” gives the yellow part. The only thing
left to do is put the two like terms together. Click now!
You can see that the 5x and 3x are like terms. Combine
them to get 8x. So the final answer is x2 + 8x + 15
Now try this one:
(x + 10)(x + 4) (first the “x”) x2 + 4x (then the 10) +10x+40

x2 + 4x + 10x + 40 = x2 + 14x + 40
Double Distribute Practice
Find the product of each pair of binomials.

1) (x – 3)(x + 5)          4) (x + 10)(x – 1)
x2 + 5x – 3x – 15          x2 – x + 10x – 10
x2 + 2x - 15               x2 + 9x - 10
2) (a + 8)(a – 4)          5) (c – 2)(c – 7)
a2 -4a + 8a – 32
c2 – 9c + 14
a2 + 4a - 32
3) (m + 1)(m + 1)          6) (2x – 3)(x + 4)
m2 + m + m + 1              2x2 + 8x - 12
m2 + 2m + 1
Factoring Using the GCF
GCF is short for Greatest Common Factor. While
distributing multiplies, factoring divides or pulls out the
GCF. For instance:
5x + 10 = 5(x + 2)
5 is the GCF so we divide it out and write the quotient in
the package. The GREAT thing about factoring is that
all we have to do to know if we are right is distribute our
answer and see if we get the question! Try these:

10x + 70             4x – 12               x2 + 5x
10(x + 7)            4(x – 3)              x(x + 5)
Perfect Practice Makes Perfect
Factor each of the following:
1) 3c + 21                  4) 5a + 5b – 15c

3(c + 7)                   5(a + b – 3c)

2) x2 – 3x                 5) c2 + 9c
x(x – 3)                    c(c + 9)

3) 8b – 80                 6) 2q - 38

8(b – 10)                  2(q – 19)
If you are ever worried that you are wrong, just
distribute back and see if you get what you started with.
Factoring Trinomials
The last thing we need to talk about is factoring trinomials.
The state says that this topic can be on the test, so we
should know how to do it. The thing is that the topic has
not yet been on the test, and it takes quite a while to
learn. So I am going to suggest that if you want to learn
this so that you can factor any trinomial, go to extra help.
But it is quick enough and easy enough to answer a
question in multiple choice format – just double distribute
the choices! Example:
Factor x2 – 7x + 12                The correct answer is c,
just double distribute
a) (x+3)(x+4) b) (x+6)(x+2)
each of the choices to
c) (x-3)(x-4) d) (x-6)(x-2)        see why.
Perfect Practice Makes Perfect
1)   What are the factors of x2 + 9x + 20?
a)   (x + 10) (x + 2)
b)   (x – 10) (x – 2)
c
c)   (x + 5) (x + 4)
d)   (x – 5) (x – 4)

2)   What are the factors of x2 – x – 30?
a)   (x – 6) (x – 5)
b)   (x – 6) (x + 5)         b
c)   (x + 6) (x – 5)
d)   (x – 10) (x – 3)
Key Points to Ponder
Remember when combining like terms, everything is
positive or negative.

Distributing is multiplication. Factoring is division.

When factoring, all you need to do is distribute back and
see if you get what you started with.

When double distributing, always look for the two terms in
the middle to combine.
Absolutely, positively
everything you need to
know!
What You Need to Know
There are a handful of very important exponent concepts
you need to know. Tops among the concepts are:

1) Scientific Notation
2) Evaluating Integral Exponents
3) The Laws of Exponents
Scientific Notation
Scientific notation is most commonly used as a shortcut for
expressing and working with large and small numbers.
In general, the format for a number expressed in
scientific notation is
(Number between 1 & 10) X 10# places decimal moves

What is wrong with each of these?

1)   235,000 = 235 x 103 Numbers is not between 1 & 10

2)   5,000,000 = 5 x 56     The base is not 10, it is 5
Express Each in Scientific Notation
1)   42,000,000,000

4.2 x 1010; Some common mistakes are ……….
42 x 109 (42 is not between 1 & 10)
.42 x 109 (.42 is not between 1 & 10)

2)   705,000
7.05 x 105; Some common mistakes are ………..
705 x 103 (stopped moving the decimal as 0’s stopped)
70.5 x 104 (70.5 is not between 1 & 10)
Scientific Notation Done Right
1)   Convert 52,000,000,000 to scientific notation.
10 places to move

We need to get the number into proper form. So ….
In order to get 52,000,000,000 to proper form I need
to move the decimal (and count) until I get 5.2.
The answer is 5.2 x 1010
2) Convert 0.00000073 to scientific notation.
7 places to move

Now we need to move the decimal to the right
until we get a number between 1 & 10. Count the
places! The exponent is negative because the
number is small (0.0000……)
The answer is 7.3 x 10-7
Perfect Practice Makes Perfect
1)   507,000              3) 11,000,000
5.07 x 105                 1.1 x 107

2)   0.0000004            4) 0.00000003108
4 x 10-7                   3.108 x 10-8

5) The sun is 93,000,000 miles from the earth. Express
this number in scientific notation.
9.3 x 107
Evaluating Integral Exponents
We already know that an exponent means how many times
you use the base as a factor. For instance:
23 = 2 x 2 x 2 = 8     and 15 = 1 x 1 x 1 x 1 x 1 =1

It gets a little tricky when the exponent is negative or zero.
First point to remember: Anything (except 0) to the zero = 1.
Evaluate each:

50 = 1              x0 = 1               ab0 if a = 3 and b=9.
3 · 90
3x1=3
Negative Exponents
The hardest exponent to compute is a negative exponent!
Just looking at 5-2 makes you think the answer ought to
have a negative sign, like say ……. -25. It doesn’t.
A negative exponent has absolutely, positively nothing to
do with a negative sign!!!!!!
A negative exponent means to use the reciprocal or put the
exponent in the denominator. For example:
3-2 = 1 / 32 = 1/9. Notice the neg. exp. means “one over”
the positive exponent.

7-2 = 1 / 72 = 1/49.
Perfect Practice Makes Perfect
Evaluate each:

1)   5-3 1/125             4) 10-3 1/1000

2)   3-2 1/9               5) 7-1 1/7

3)   2-5 1/32              6) 9-2   1/81
The Laws of Exponents
This is another topic students can find confusing. If you
stay focused, I think you will find that the rules make
sense.

xaxb = xa+b What we are seeing here is that the base stays
the same when multiplying and you add the exponents.

xa ÷ xb = xa – b What we are seeing here is that when divide,
the base stays the same and you subtract the
exponents. For instance:

8
x5x3 = x                    x8 ÷ x2 = x6 the exponent is 6 not 4
because you subtract.
Some Practice & Clarification
1)   x3x7= x10      2) 3233 35          3) c6÷c3 c3

I hope they seemed easy for you  . Just remember that
the base ALWAYS stays the same and you add or
subtract the exponent depending on the operation.
Sometimes students get confused if you put numbers in
front (called coefficients.) The thing you have to
remember is that numbers are numbers and that they
behave like numbers – not like exponents!!!!! Examples:
(3w2)(5w) = 15w3 The 5 and 3 are numbers not exp’s! 5x3=15
10x8÷5x2 = 2x6 The 10 and 5 are numbers not exp’s 10/5 = 2
Perfect Practice Makes Perfect
1)   12c5÷4c 3c4        4) (7v3)(-3v6) -21v9

2)   (3x)(4x2) 12x3     5) 20a9b4÷(5a3b) 4a6b3

3)   100x2÷10x 10x      6) (-4xy2)(-5xy) 20x2y3
Remember to keep the bases the same,
treat exponents like exponents and
coefficients (numbers) like numbers!
What You Need to Know
In scientific notation, you must have a number between 1 &
10 times 10# places you move the decimal.

Anything to the zero power is 1

A negative exponent means 1 over the positive exponent.

With respect to Exponent Laws, treat exponents like
exponents             bases
__________, keep the _______ the same and treat the
coefficients
numbers (also called _____________) like numbers.
Absolutely, positively
everything you need to
know!
The Perpetually Problematic Percents
Percents drive some kids NUTS! One big reason for that is
that you don’t solve all of the problems the same way.
First the bad news: Any student who just wants to be
told what to do instead of having to think with percents
is going to have a hard time. Now the good news:
Any student willing to remember and work with a
couple of basic ideas can master percents. Key ideas:

1)   Percent means “out of 100.”
2)   The percent is the “part” out of 100.
3)   Fractions, decimals and percents are all the same
thing (like hello, hola and bon jour).
Percent Means “out of 100”
Simply put, percent means out of 100. So if 55 out of 100
dentists prefer Trident gum, 55% prefer it.

If 99 out of 100 students think Math is the best subject,
99% think Math is the best subject.

But what percent think Math is the best subject if only 2 out
of 5 think it is the best? Do you know what a lot of
students write? 2%!!!!! Can you believe that? The
question said 2 out of 5 – not 2 out of 100! To find the
percent we must find out how many out of 100.
Converting Between Decimals and Percents
Converting between decimals and percents is one of he
easiest math skills. Remember, percent means out of a
hundred. Also remember that two decimal places is the
equivalent of “100”.

% to Decimal         Decimal to %
Move decimal         2 places left        2 places right

Percents are bigger so we move to the right
Examples:
50% = .50 (or just .5)              .721 = 72.1%
115% = 1.15 (2 places only)         .005 = 0.5% (2 places)
Converting to a Percent
There are two main ways to convert fractions like 2 out of 5
(2/5) to a percent.
since 5x20 =   2/5 means 2 ÷ 5 which
1) 2 = ?
100, 2x20=40   is .4000………. To
5 100 …                  convert to a percent,
40/100 = 40%   move the decimal two
places right or 40%.

Convert each to a percent:
1) 7/10             2) 3 out of 25       3) 3/8
70%                  12%                 37.5%
Convert Percent to a Fraction
To convert a percent to a fraction, just write it “out of 100”
and reduce if necessary.

25% = 25 out of 100 = 25/100 reduced = ¼. Now you try.
45% = 45 out of 100 = 45/100 reduce to 9/20
90% = 90 out of 100 = 90/100 reduce to 9/10
37% = 37 out of 100 = 37/100 this is reduced.
What if we get one like 37.5%? Then we need to do
37.5/100 but this looks funky. Convert to 375/1000 to get
rid of the decimal (move numerator & denominator one
place each. You try 31.7%. 31.7 out of 100 = 317/1000
Perfect Practice Makes Perfect
Convert each to a percent:
1) 0.57 57%         2) 12/100 12%   3) 12/50 24%

4) 1.39 139%       5) 5/8 62.5%     6) 0.001 0.1%

Convert each to a decimal:
7) 77%   0.77      8) 29% 0.29      9) 0.2% 0.002

10) 145% 1.45      11) 100% 1       12) 25.5% 0.255
Converting to Fractions
We already discussed how, in order to go from a percent to
a fraction, we should “put it “over” or “out of” 100 and
reduce. To go from a decimal to a fraction is just as
easy. Just read it properly and reduce …….
Remember, one decimal place is tenths, two are
hundredths, three thousandths and so on. For
example:

1)   0.5 read properly is 5 “tenths” = 5/10 = ½.
2)   0.34 read properly is 34 “hundredths” = 34/100= 17/50
3)   1.233 read properly is 1 and 233 “thousandths” =
1 233/1000
Perfect Practice Makes Perfect
Convert to a fraction in simplest form:

1) 0.1                      2) 0.79       3) 0.433
1/10                  79/100         433/1000

4)    0.38                  5) 75%        6) 0.009
19/50                  3/4           9/1000

7) 110%                     8) 0.747      9) 0.5%
1 1/10                 747/1000       1/200
Fractions, Decimals, Percents Summary
Besides remembering that fractions, decimals and percents
are just different ways of saying the same thing, we need
to remember how to convert amongst them.

two
Decimals and percents move the decimal ________ places
right                      left
to the ________ for a percent and _________ for a
decimal.

Decimals to fractions you just read it properly and
______
reduce                                    divide
_______. Fractions to decimals you just ________.

over
Percents to fractions, just put it ______ 100.
The Percent Proportion
Most students use the percent proportion in percent
problems, so we will might as well, too. The percent
proportion is:
Percent = part (is)
100        Whole (of)

Your job as a student is to determine what number goes in
which place. Now, the 100 always stays where it is. The
word is usually tells the part and of usually tells the
whole. A lot of people think underlining can be a big
help in these word problems. DON’T try to solve the
question right away – try to find the parts.
Problem One
Jenny takes a science test that has 25 questions. She gets
21 of them right. What percent did Jenny get right?
First things first ……. Let’s underline the important stuff!
Two numbers are underlined, 21 and 25. Which one
represents the whole number of questions and which
one the part?
% = part         Now the part = 21 and whole = 25, so
100    whole
N     = 21      cross multiply 25n = 2100
100     25                       25       25
n = 84
Problem Two
Remember, find the important stuff and underline!!!!!

In order to pay her way through college, Tammy works and
earns \$800 a month. If 65% of this money goes to
tuition, how much money does Tammy spend each
month on tuition? The whole amount Tammy earns is \$800

%      = part (is)
100       whole (of)
65   = n                   100 n = 52,000
100      800                    n = 520
The Hardest Part of Some Percent Problems
Why do some percent problems give students a hard time?
I think one of the biggest reasons is that many percent
problems, and for that matter many math questions in
general, require only one process to get the answer.

Typically, all of he following percent related questions
require TWO STEPS!!!!!!

1)   Sales tax
2)   Sale price
3)   Tips (gratuities)
4)   Mark up
Problem Three: Sales Tax
Probably the easiest way to understand a sales tax
problem is first find the sales tax amount and add it to
the cost of the item; after all, isn’t that how it works in a
store?
Rob buys a fishing rod for \$80. The tax rate is 8.5%.
a) How much tax does Rob pay?
8.5 = n (tax is part)          100 n = 680
100      80 (whole cost)            n = \$6.80

b)   How much does Rob give the cashier?
Rob gives the cashier \$80 + \$6.80 = \$86.80
Another Sales Tax Question
John buys two bats for \$150 each. If the sales tax rate is
8%, how much does he have to pay the cashier?
Maybe it will help to keep underlining…….
First off, John spends \$300 (two bats)
8     = n
100       300
100 n = 2400
n = 24

John has to give the cashier \$300 + \$24 = \$324.
Problem Four: Sale Price
This should be easy to remember. Everybody shops
during a sale because items cost less and less means to
subtract! Find the amount you save and subtract it.

Darla wanted to buy Alfalfa some flowers. They usually
cost \$20, but this week only are on sale for 40% off.
How much does Darla have to spend on the flowers?
%     = part          100n = 800
100      whole            n=8
40 = n
100   20              Darla pays \$20 - \$8 = \$12.
Another Sale Price Question
Sue is looking to buy a new rug for the dining room. A rug
she likes ordinarily costs \$1800 but is on sale for 25%
off. Sue has \$1500 to spend. Can she afford the rug?

%    = part            100 n = 45,000
100      whole               n = 450

25    = n                The new price is \$1800 - \$450

100      1800            which is \$1350.
Yes, Sue can afford the rug!
Tips and Gratuities
A normal tip for a meal at a restaurant is 15% of the bill. If
Bob and Betty went to dinner and the meal cost \$45.00,
how much should the leave the waitress as a tip?
much should they give the waitress?

15    = part (tip is part)       100 n = 675
100     whole bill                    n = \$6.75

15    = n                    Rounded to the nearest whole
dollar, the tip is \$7. They should
100     45
give the waitress \$45 + \$7 = \$52.
Mark Up Problems: More Two Step Questions
Mark up is how stores make money. Stores buy the items
they sell for much cheaper than a customer pays for
them. The amount the increase the price is called “mark
up,” and because it increases the price you have to add.
Remember, many of these questions are tricky because
you often need to do two steps.
Rob sells fishing supplies. He marks up all of the items he
sells 120%. If he pays \$40 for a fishing reel, how much
does he charge?
% = part                 100 n = 4800     Now for Step
#2: So the cost
100 whole price               n = 48.00   Rob sells the
120 = n               So the mark      rod for is \$40 +
up is \$48.00.    \$48 = \$88
100      40
Another Mark Up Question
Bob owns a pet supply store. With fish tanks, he marks the
price up 40%. If Bob pays \$150 for a fish tank, how
much does he sell it for?

%     = part (mark up)     100 n = 6000
100        whole cost            n = 60
40    =    n               This is the mark up!!!
100       150               The tank sells for
\$150 + \$60 = \$210.
A Commission Question
Mrs. Lindquist is a real estate agent. Recently, she sold a
piece of land for \$510,000. The commission (amount of
money the real estate agent makes) she gets is 1.55%.
How much money does Mrs. Lindquist make?
%     =    Part             100 n = 790,500
100        Whole cost             n = \$7,905

1.55 =      n                Mrs. Lindquist makes
100        510,000           \$7,905 on the sale of
the land.
Percent of Increase and Decrease
Percent of increase and decrease problems still depend on
your being able to determine the part and the whole.
The thing is that, with respect to increase and decrease
problems, the amount of change is always the part and
the whole is always the starting amount.
% = part                    % = amount of change
100     whole                 100      original amount
Last year, the SOA wrestling team won 8 matches. This year
they won 10 matches. What is the percent of increase?
%     = amnt change      n    = 2 (10 – 8)     8n = 200
100     orig. amount    100     8                n = 25

The percent of increase is 25%
Another Percent of Increase / Decrease
Mr. Smith figures it is about time he retires and gets to go
fishing whenever he wants. Ahhhhh, what a life! He is
looking over his financial information and determines that
he makes \$95,000 a year now, but when he retires, he
will only make \$75,000. To the nearest whole percent,
what is Mr. Smith’s percent of decrease?
%       = amount of change   75,000n = 2,000,000
100       starting amount            n = 26.6666667%
n    = 20,000            To the nearest whole
percent, Mr. Smith has a
100       75,000
27% decrease in income.
Simple Interest
Simple interest is really easy and if you don’t know it, by
the time you are done with these questions you’ll agree it
really is easy. The formula for simple interest is:
I = prt
I is interest.
p is principle (amount of money).
r is interest rate as a DECIMAL.
t is time in YEARS.
How much interest does Johairy get if she puts \$5,000 in
an account that earns 7% interest for 5 years?
I = prt
I = (5000)(.07)(5)
I = \$1,750
Two More Simple Interest Questions
Cathy invests \$200,000 in a money market fund at 4.5%
interest. If she leaves it in the account for 4 years……
A) How much interest does she earn?
I = prt
I = (200,000)(.045)(4)
I = \$36,000
B) How much is in the account?
The amount in the account is \$200,000 +
\$36,000 = \$236,000.
Percent Summary
Percents are not easy! Some of the hardest percent
questions are those that involve mark up, discount,
sales, sales tax and tips. The thing that makes them
two steps
tricky is that they often have __________________.

One strategy that often helps is being sure to ___________
underline
the important parts of each question.

Finally, be sure to write and then substitute into the percent
proportion. The percent proportion says
%     = part (is)
100     whole (of)
Absolutely, positively
everything you need to
know!
Measurement on the State Test
On the 8th Grade Assessment Test, New York State is not
limited to measuring with a ruler. The state wants to see
much more. Can you use some pieces of measured
information to compute others. These situations include
but are not limited to:

1)    distances on a map.
2)    similar figures.
3)    unit price.
4)    convert units like inches and feet as well as Celsius and
Fahrenheit temperatures.
Measurement
The scale of a map is 1 inch = 80 miles. On the
map, the distance from Patchogue to Callicoon is
2.5 inches. How far is it from Patchogue to
Callicoon?

First things first –     1 inch = 2.5 inches
write a proportion.     80 miles n miles
inches = inches         n = 80 x 2.5
miles     miles         n = 200 miles.
The scale of a map is 1 inch = 60 miles. On the
map, the distance from Patchogue to Monticello is
2.5 inches. How far is it from Patchogue to
Callicoon?

First things first –     1 inch = 2.5 inches
write a proportion.     60 miles n miles
inches = inches         n = 60 x 2.5
miles     miles         n = 150 miles.
Perfect Practice Makes Perfect
The scale of a map is 1
inch = 15 miles.

If the map shows that
it is 1.8 inches across
Brookhaven, how
many miles wide is
Brookhaven?
1 = 1.8
inches = inches              15   n
miles    miles               n = 27 miles
A box of Corn Flakes cereal sells for \$3.78. The
volume of the box is 18 ounces. What is the
unit price for an ounce of Corn Flakes?

unit means “one” so the question is s aying
what is the price for one ounce.
\$3.78 ÷ 18 ounces = \$0.21 per ounce
More Practice
John Haag was a locally famous fly tier on Long Island. He
used to buy enormous quantities of feathers. He would
buy 5 pounds of feathers for \$240.00. What was the unit
price (per ounce) that John paid for feathers?
dollars = dollars
ounces ounces
80 n = 240
(16 oz)(5lbs) = 80
n   = 3
\$240 = n
80 oz   1 oz
Costco sells two different size packages of chicken.
One package sells 3.5 pounds for \$8.19 and the
other sells 5 pounds for \$10.79. Which one is a
better buy and how much do you save per pound?

\$8.19 ÷ 3.5 = \$2.34

\$10.79 ÷ 5 = \$2.158 = \$2.16
\$2.34 - \$2.16 = \$0.34

The 5 lb bag is cheaper by \$0.18 per
pound.
Emily babysits in order to make some money. On
Saturday, she gets paid \$12 for 2.5 hours of
babysitting. On Monday, she gets paid \$30. How
long did Emily babysit?

dollars = dollars        12n = 75
hours        hours         n = 6.25
\$12 = \$30              Emily works 6.25 hrs.
2.5    n
Perfect Proportion Practice
Mr. Lindquist goes fishing for rainbow trout. In his first 2.5
hours of fishing, he catches 10 beautiful rainbow trout.
At this pace, how long will it take Mr. Lindquist to catch
18 trout?

time = time           10 n = 45
trout   trout             n = 4.5

2.5 = n
It takes Mr. Lindquist
10      18             4.5 hours
Perfect Practice Makes Perfect
One dozen tomatoes cost \$3.15. How much do 30
tomatoes cost?

dollars = dollars
tomatoes tomatoes

\$3.15 = n
12 tom.   30 tom.

12 n = \$94.50
n = \$7.875
n = \$7.88
Determine if two figures are congruent or similar. Do
you know it?

1)What does congruent mean?       same size & shape
2)What does similar mean?
same shape sides in proportion
Is the pair of triangles below similar or congruent
(assume all corresponding angles are equal.)

similar
Use a proportion to solve for a missing side in
similar figures. Do you know it?

Find x.
12
9
6
x

li’l = li’l      9 x = 72
big       big     x=8
9 = 6
12    x
Perfect Practice Makes Perfect
1) Two triangles are similar. Find the value of x.

20               n
n = 10
4
10

2) A photograph is enlarged. The original picture had a
length of 5 and a width of 3. If the new picture has a
length of 7.5, what is the width?
length = length               5 = 7.5      5 n = 22.5

width         width           3   n         n = 4.5
What You Need To Know
There are two main types of questions that fall under the
heading of “measurement.” One involves unit price and
the other involves proportions.

divide
When finding unit price, always _________ by the quantity.

When using a proportion, first write the proportion in
words         substitute
______, then ___________ the values. Finally, _______ cross
__________ to get the answer  !
multiply
Absolutely, positively
everything you need to
know!
The Three Angles of a Triangle
180
The three angles of a triangle add up to _____. Find n in
each diagram:

57                                      n-6

90        n

n            n
90 + 57 = 147
180 – 147 = 33                   n + n + n – 6 = 180
3n – 6 = 180
n = 62
Word Problems & Triangles
1) The three angles of a triangle can be represented by x,
2x – 5 and 2x + 20. Find the value of x.
x + 2x – 5 + 2x + 20 = 180
5x + 15 = 180
x = 33
2) The three angles of a triangle are n, n and n + 90. Is
the triangle acute, right or obtuse.

n + n + n + 90 = 180       The triangle is obtuse
3n + 90 = 180              because one of the angles
is 30 + 90 = 120 degrees.
n = 30
The Pythagorean Theorem
The Pythagorean Theorem says a2 + b2 = c2. You can use
this formula as long as you always remember that c is
the hypotenuse. Another way to do the Pythagorean
theorem is leg2 + leg2 = hyp2.
Find n in each diagram:
n2 + 62 = 102
10       n2 + 36 = 100
n       12          6
- 36   - 36
n
5                                  n2 = 64
52 + 122 = n2
N    = 8
25 + 144 = n2
13 = n
Word Problem
A ladder is leaning against the side of a building. The
ladder is 15 ft long and the base of the ladder is at a
point that is 9 feet from the house. How high up the

n2 + 92 = 152
n2 + 81 = 225
n         15
n2     = 144
n      = 12
9
Parallel Lines Cut By a Transversal
One of the most famous
situations in geometry is
when 2 parallel lines
are cut by a transversal.
There are three things
you must know:
1) All acute angles are =.

2) All obtuse angles are =.

3) Any acute angle + any
obtuse angle = 180.
Parallel Lines & Angles

Name all of the acute angles in
the diagram. <1, <4, <5, <8
1       2
What do you know about these                   3       4
angles? They are all equal.
5       6
Name all of the obtuse angles in                           7       8
the diagram. <2, <3, <6, <7

What do you know about these angles?
They are all equal.

If a question pairs an acute angle with an
supplementary
obtuse angle, they are _________.
Parallel Lines & Angles

Name all of the angles in the
diagram that are congruent to            1       2
<1. Once you select them, click              3       4
and they will appear light blue.
5       6
7       8

Name all of the angles in the diagram
that are congruent to <7.
Once you select, click and they will    <2 and <8 are
appear red.
supplementary
______________.
Parallel Lines & Angles
Which angle is the alternate
interior angle to <4? <5
1       2
3       4
Which angle is the alternate
5       6
interior angle to <3? <6
7       8

Which angle is the consecutive interior angle to <5? <3

Which angle corresponds to <4? <8
Parallel Lines & Angles
Which angle is the alternate
interior angle to <5? <4
1       2
3       4
Which angle is the alternate
5       6
interior angle to <6? <3
7       8

Which angle corresponds to <1? <5

Which angle corresponds to <2? <6
Parallel Lines & Angles

If m<5 = 70, find the
measures of all the other         1       2
angles:                               3       4

m<1 = 70
5       6
m<2 = 110                                         7       8

m<3 = 110
m<4 = 70
m<6 = 110
m<7 = 110
m<8 = 70
Parallel Lines & Angles

If m<5 = 75, find the
measures of all the other         1       2
angles:                               3       4

m<1 = 75
5       6
m<2 = 105                                         7       8

m<3 = 105
m<4 = 75
m<6 = 105
m<7 = 105
m<8 = 75
Parallel Lines & Angles

If m<4 = 3x – 18 and
m<5 = x + 24, find the m<4.             1     2
3       4
Equal or Supplementary? Find x.
Substitute          5       6
<4 & <5 are both acute, so
7       8
they are equal.
3x – 18 = x + 24
-x       -x
2x – 18 = 24                    m<4 = 3(21) – 18
+ 18 +18                          = 63 – 18 = 45
2x = 42        x = 21
Parallel Lines & Angles

If m<4 = 2x + 20 and
1       2
m<6 = 3x + 10, find x.
3       4
<4 is acute and <6 is obtuse, so the
two are supplementary!                                 5       6
7       8
2x + 20 + 3x + 10 = 180
(since they are on the same side, like terms go together)
5x + 30 = 180
- 30    -30
5x          = 150
x     = 30
Identifying Shapes: recall the name & click

Number of Sides

3            Triangle

5            Pentagon

6            Hexagon

8            Octagon
Angle Types

There are four angle types you need to know.
Name them and their definitions.
Acute      Between 0 and 90

Obtuse     Between 90 and 180
Right      90 degrees

Straight   180 degrees
Label Each Angle Type

Acute
Right

Straight                            Obtuse
A Few More Questions

What is the largest acute angle that
you could double and still get an
acute angle?
44. If you tried 45 and doubled it you
get 90 – a rt <.

What is the smallest acute angle you
could triple and get an obtuse angle?
Well, we need to find the smallest number that tripled is
bigger than 90. Since 30 x 3 = 90, the angle we are
looking for is 31………. 31 x 3 = 93.
A Few Questions ……….
1) An angle that measures 54 degrees is _________.
a) Acute     b) Obtuse c) Straight      d) Right

2) A 180 degree angle is a __________ angle.
a) Acute     b) Obtuse     c) Straight   d) Right

3) Perpendicular lines make what kind of angles?
Right Angles!!!!!
Problems to Ponder

Two congruent acute angles add up to a right angle. How
many degrees in each angle?
45 degrees. 45 + 45 = 90

Is it possible for a quadrilateral to have no acute angles?

Sure  ! A rectangle has 4
right angles.
Triangle Types: Choose ALL possible choices from the words below:
Scalene Equilateral Acute Right Isosceles Obtuse

1) This triangle has all sides equal. Equilateral
2) This triangle has no sides equal and all
angles less than 90 degrees. Scalene. Acute.
3) This triangle has two equal sides. Isosceles
4) This triangle has an angle greater than 90
and less than 180 degrees.     Obtuse
5) The angles of this triangle are 90, 45 and 45.
Right. Isosceles.
How many degrees in the 3 <s of a triangle?
_______
180

How many degrees in the missing
angle below?                               58 + 52 = 110
180 – 110 = 70
58

52                       70
Find the value of n

This is isosceles, so it has two
congruent angles as well as two
48
congruent sides. What does the
base angle on the left equal?
n – like the one on the right 

Therefore ……………..
n
2n + 48 = 180. Solve this!
2n + 48 = 180
- 48     -48
2n          = 132      n = 66  !
The angles of a triangle measure n, 2n and 3n.
1) Find n.
2) Is the triangle best classified as acute, right or obtuse?
3 angles add up to 180 degrees!

n + 2n + 3n = 180
6n    =        180
30
n =       30
n = 30
2n = 60                               90      60
3n = 90
Find the value of n

The vertex angle of an isosceles
triangle measures 80. Find the
80
measure of each of the base
angles.
The three angles sum to 180, so
N + N + 80 = 180
n           n
2N + 80 = 180
- 80     -80
2N          = 100
N = 50
Tricky Triangle Questions

Johnny says an equilateral triangle is not isosceles. Is
Johnny right or wrong?
Johnny is wrong. An isosceles triangle has two sides
equal. Two of the sides of the equilateral triangle ARE
equal. It does not matter that the third sides is, too.

Is it possible for a triangle to have two right angles?
No. Numerical explanation: two of the angles would
already sum to 180 (90 + 90). There would be no
room for a third angle. Geometric explanation: Two
“sides” of the “triangle” would never meet.

90                90
Angle & Triangle Questions

1) An acute triangle has three acute angles. Does an
obtuse triangle have 3 obtuse angles? Why or why
not?
No! The three angles of a triangle sum to 180.
All obtuse angles are greater than 90, so three of
them would sum to 270 or more.
2) True or false and why: An obtuse triangle can
never be isosceles?
False! One example is a triangle with angles of
100, 40 and 40. Also, see the diagram below.

Papa Parallelogram
Opposite Sides Parallel & Equal
Opposite Angles Equal     Angles Sum to 360

Randy Rectangle                 Rebecca Rhombus
Parallelogram + 4 Rt <s         Parallelogram +

Congruent Diagonals             4 Equal Sides
Squiggy Square
The Tricky Trapezoid

What makes a trapezoid
tricky?
110   110
It is kind of like a
parallelogram – but it only
70               70
has one pair of parallel
sides! A parallelogram has
two pairs.
How is a trapezoid like a parallelogram?

The interior angles add up to 360.
Check out the angles………..
Always. Sometimes. Never.

A rectangle is a parallelogram.
Always – the opposite sides of a rectangle are
congruent.

A rhombus is a square.

Sometimes. But a rhombus does not have to
have a right angle and a square does.
A trapezoid is a parallelogram.
Never! A trapezoid only has one pair of parallel sides.
Always. Sometimes. Never.

A parallelogram has 4 congruent sides.
Sometimes. When it does, it is a special type of
rhombus
parallelogram called a rhombus.

The opposite angles of a square are each 100 degrees.
Never. They are each 90 degrees.

The sum of the angles of a parallelogram is 360.
Absolutely! Think of the 4 rt angles of a square.
90 x 4 = 360                                        90

Two consecutive sides of a rhombus are 3b – 5 and b + 11.
All Sides of a
Find b. Find the length of a side. rhombus are equal!
Strategies:                             3b – 5 = b + 11
3b - 5
1) Diagram                              -b      -b
b + 11
2) Equation                             2b – 5 = 11

3) Solve                                     +5    +5
2b        = 16

Side: 3x8 – 5 = 19        b = 8

Two opposite sides of a parallelogram measure 4(x – 5)
and 28. Find the value of x.
Opposite sides of a                       4(x – 5)
parallelogram are equal!
28
4(x – 5) = 28 DISTRIBUTE!
4x – 20 = 28
+ 20 +20
4x       = 48
x = 12

The measure of an angle of a rectangle can be
expressed as 2x – 10. Find x.
Strategies:
1) Diagram
2) Equation
2x + 10
3) Solve

The angles of a rectangle are right angles…… =90
2x + 10 = 90
- 10 -10
2x        = 80      x = 40
Match the Name to the Pic
1) Rectangle only

2) Parallelogram

3) Square

4) Rhombus
Key Terms & Key Pictures

1) Define Complementary.     Two <s add up to 90

2) Define Supplementary      Two <s add up to 180
What is the trick for remembering that complementary
means 90 and supplementary means 180?
C comes before S, 90 before 180
3) What is the complement of 40?
50
4) What is the supplement of 100?
80
Find the complement and supplement of an angle of 40
degrees.

Complement = 50; Supplement = 140.

What is the difference between the supplement and
complement of 40? 90

Will the difference between the complement and
supplement of any given angle always be 90? Why or
why not?
Complementary adds up to 90 and
supplementary to 180. The difference
between those two numbers is 90  .
Complementary or Supplementary?

Which picture is which and find the value of n.

Supplementary
2n + 68 = 180; n = 56
2n   68

Complementary
63                      8; 3n + 66 = 90
3n - 3
Pairs of Angles
Find the value of n in each diagram.

7n + 8      2n + 43

3n - 17   2n + 12

These angles are called               These angles are called
________ and they are
vertical                               supplementary
___________ and they
_______.
equal                                sum to ______.
180
7n + 8 = 2n + 43
3n – 17 + 2n + 12 = 180
-2n         -2n
5n – 5 = 180
5n + 8 = 43
+5      + 5
- 8   - 8
5n = 185          n = 37
5n =       35    n=7
Counterexamples in Geometry

Ellen says that every pair of supplementary angles must
have one acute and one obtuse angle.
Is Ellen right or wrong? If you think she is right, why? If
you think she is wrong, give a counterexample.

At first thought, we might think Ellen is right because
the sum of two acute angles is less than 180 and the
sum of two obtuse angles is greater than 180. So we
think that we need one of each (example: 80 + 100).
But Ellen is wrong! Two right angles are also
supplementary and are neither acute nor obtuse.
Perimeter and Area

Perimeter measures the
outside
_____________ of a figure.           10 in
32 in
Area measures the
inside
_____________ of a figure.
Which operation?    Perimeter ______. Area __________.
Find the perimeter and area of the
rectangle above.
Perimeter                  Area
Add all sides. 84 inches   Area = length x width
10 in+32 in+10 in+ 32in    10 in x 32 in = 320 sq. in.
Circles: Circumference & Area

Circumference is like
perimeter – but only applies
to a circle.
Area doesn’t change – it is
inside.

What is the trick to remember the formulas?

Cherry Pie is delicious.     Apple pies are, too.
C = πd                           A = πr2
Parts of a Circle

For our purposes, there are two main parts of a circle
we need to know ………….. diameter and radius. What
is the difference?

The diameter goes all the
diameter
goes halfway.

If the diameter if a circle is 18
The radius only goes halfway across so it
must be half of 18 inches …… 9 inches.
Find the Circumference and Area of the circle.

C=πd              (d is diameter)
10 m
C = 3.1416 x 20 m
C = 62.832 m

A = π r2          (r is radius)

A = 3.1416 x 10 m x 10 m
A = 314.16 m2
Surface Area of a Rectangular Prism

How many faces does
this prism have? 6

I’ll name one side.
You name its opposite
partner.
FRONT                 LEFT                    TOP
BACK                  RIGHT                   BOTTOM
Finding The Surface Area

Step 1: Find the front’s area.
TOP
5 in x 8 in = 40 in2 x 2 = 80 in2
Why double the 40? The Back!         5 in FRONT
2) Find the top area and double it.
8 in x 4 in = 32 in2 x 2 = 64 in2         8 inches      4 in

3) Find the area of the right and double it because   RIGHT
of the left “partner” or equal face.
4 in x 5 in = 20 in2 x 2 = 40 in2

80 in2 + 64 in2 + 40 in2 = 184 in2
Surface Area of a Cylinder

What shapes do you get when
you unfold a cylinder?
Two congruent circles on the
top and bottom and a rectangle.
If the width of the rectangle on
the side is the height, how do
you find the length?
Radius = 10; Height = 25
It is the circumference of the circles.
Computing the Surface Area

The reference sheet gives the following formula
for the surface area of a cylinder:
SA = 2πrh + 2πr2
Substitute and crank out the antza  .
SA = 2(3.1416)(10)(25) + 2(3.1416)(102)
= 1570.8 + 628.32
= 2199.12

Find the area of the shaded
region.
8 in   6 in              5 in
Area of green rectangle:                               8 in
8 in
A = l x w = 8in x 25in=200in2                  25 in
Area of the triangle: A = ½ b h = ½ (6 in) (8 in) = 24 in2

Area of the parallelogram = b x h = (5 in) (8 in) = 40 in2
Area of the shaded region is
200 in2 – (24 in2 + 40 in2)
136 in2
Find the Area of the Shaded Region

Area Square – Area Circle

Area of the square = side x side
5 inches
Since the radius is 5, side is 10 in
10 in x 10 in = 100 in2

Area of the circle = πr2
A = (3.1416) x (5 in) x (5 in)
A = 78.54 in2

100 in2 – 78.54 in2 = 21.46 in 2
Key Points to Remember
180
The three angles of a triangle add up to ______.

When two parallel lines are cut by a transversal,
=
1) all acute angles are ___.
=
2) all obtuse angles are ___.
3) any acute angle + any obtuse angle =180  ___.

=
Vertical angles are always ___.

Complementary means two angles that add up to ____ and
90
180
supplementary means adds up to ____.

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