Download MathAct - Northside Middle School

Introduction to ACT Math
A quick review of concepts
Introduction to the ACT mathematics
Test
What to expect
• 33 Algebra Questions:14 pre algebra
10 elementary algebra
9 intermediate algebra
• 23 Geometry Questions:14 plane geometry
9 coordinate geometry
• 4 Trigonometry Questions: based on sine,
cosine, and tangent
Do NOT expect
• A formula page before the math section
The writers care more about what you know
then the SAT
PRACTICE PRACTICE
PRACTICE!!!
EASY
• 1) Cynthia, Peter, Nancy, and Kevin are all
carpenters. Last week, each built the following
number of chairs:
36-Cynthia 45-Peter 74-Nancy 13-Kevin
What was the average for the week?
A) 39
B)42
C)55
D)59
E)63
Answer
• Sum of everything/number of things=average
• 36+45+74+13/4+average
• ANSWER IS 42 (B)
Medium Problem
• Four carpenters built an average of 42 chairs each
last week. If Cynthia built 36, Nancy built 74, and
Kevin built 13 chairs, how many chairs did Peter
build?
F)24
G)37
H)45
J)53
K)67
ANSWER
•
•
•
•
36+74+13+Peter/4=42
36+74+13+Peter=168
168-123=45
Answer is 45(H)
Hard
• Four Carpenters each built an average of 42
chairs last week. If no chairs were left
uncompleted, and if Peter, who built 50 chairs,
built the greatest number of chairs, what is the
LEAST number of chairs one of the carpenters
could of built, if no carpenter built a fractional
number of chairs?
A) 18
D)39.33
B)19
E)51
C)20
ANSWER
•
•
•
•
•
50+x+y+z/4=42
50+x+y+z=168
50+49+49+z=168
Z=20
Answer is C
Ballpark
• Narrowing down your choices by guessing.
• Practice:
There are 600 schools children in the Lakeville
district. If 54 of them are high school seniors,
what is the percentage of high school seniors in
the Lakeville district?
A) .9%
D)11%
B) 2.32%
E)90%
C) 9%
Answer
• 10% of 600 is 60 so it is less then 10% so
choices D and E don’t work
• A and B don’t work because we need an
answer slightly less then 10%
• Answer is C
Partial Answers
• Students sometimes think they have
completed a problem before it is actually
complete.
• Watch out for these traps and read the
questions carefully
Practice
• A bus line charges $5 each way to ferry a
passenger between the hotel and an
archaeological dig. On a given day, the bus line
has a capacity to carry 255 passengers from the
hotel to the dig and back. If the bus line runs at
90% of capacity, how much money did the bus
line take in that day?
• F) $1,147.50
J) $2,550
• G) $1,275
K) $ 2,625
• H) $2,295
Answer
• Well, 255 passengers pay $5= $2,550
• If you were in a hurry you would probably stop
there but we need to find 90% of $2,550
• Both F and J are partial
• The answer is H) $2,295
Take Bite-Size Pieces
• Difficulty is determined by the number of
steps involved
• You have to break these questions into
manageable steps in order to avoid partial
answers
Sample
• Each member in a club had to choose an activity
for the day of volunteer work. 1/3 of the
members chose to pick up trash. ¼ of the
remaining members chose to paint fences. 5/6 of
the members still without tasks chose to clean
school busses. The rest of the members chose to
plant trees. If the club has 36 members, how
many of the members chose to plant trees?
• F) 3
G) 6
H) 9
J)12
K)15
work
1.
2.
3.
4.
5.
Write down 36 in your work area
Find 1/3 of 36=12 so 12 picked up trash
Find ¼ of 24=6 so 6 people painted fences
5/6 of 18 =15 so 15 people cleaned busses
Now it sais all of the remaining people
planted trees so 18-15=3 so 3 people planted
trees.
6. F is the answer
????????Calculators????????
• TI-89 and TI-92 are not allowed
• Plan to bring a TI-83 or other calculator on the
approved list make sure it can:
 Handle positive, negative, and fractional
exponents
 Use parenthesis
 Graph simple function
 Convert fractions to decimals and vice versa
 Change a linear equation into y=mx+b form
basics
Words to know
• Real numbers- are any number you can think
of.
• Rational numbers- any number that can be
written as a whole number, fraction, an
integer over another integer.
• Irrational numbers-cannot be written as an
integer over another integer
Negatives& positives
• Positive numbers are to the right of the 0 on
the number line. Negative numbers are to the
left of the 0 on the number line.
• Positive x positive= positive
• Positive x negative- negative
• Negative x negative= positive
• ex: 5+ (-3)=2
Prime numbers
• A prime number can be divided evenly by two
and only two distant factors.
• Thus, 2, 3, 5, 7, 11, 13 are all prime.
• There are no negative prime numbers
Absolute value
• The absolute value of a number is the distance
between that number and 0 on the number
line.
• Ex. |6|=6
|-6|=6
Variables and coefficients
• In the expression 3x+4y, x and y are the
variables because we don’t know what they
are.
• 3 and 4 are the coefficients because you
multiply the variables by them.
Basic opperations
• Divisibility Rules1. A number is divisible by 2 if its units digit
can be divided evenly by 2 ( in other words, if
it is even.) 46 is divisible by 2. So is 3,574
2. A number is divisible by 3 if it sums of its
digits can be divided evenly by 3.
3. A number is divisible by 4 if the number
formed by its last two digits is also divisible by
4. 316 is divisible by 4.
Factors & multiples
• a number is a factor of another number if it
can be divided evenly into that number.
• Ex: 3 is a factor of 15 because 3 can be divided
evenly into 15.
• A number is a multiple of another number if
it can be divided evenly by that number. Ex:
multiples of 15 include 15, 30, 45, and 60.
Standard symbols
•
•
•
•
•
•
Is not equal too ≠
Is equal too =
< is less then
> is greater then
≤ is less then or equal too
≥ is greater then or equal too
Exponents
• Base is called the lower and larger number
• Exponent is the upper number.
• 6^2 x 6^3 = 6^( 2+3) = 6^5
• (y^) (y^3) = y^(2+3)= y^5
Dividing Numbers with the same base
• When you divide numbers that
have the same base, you
simply subtract the bottom
exponent from the top
exponent.
Negative Powers
• A negative power is simply the reciprocal of a
positive power.
Fractional Powers
• Numerator: the number above the line in a
fraction, functions like a real exponent.
• Denominator: the number below the line in a
fraction, tells you what power radical to make
the number.
When wanting to raise a power to a power, you
Simply multiply the exponents.
Powers
• The Zero power anything to the zero power is
1
• The first power anything to the first power is
itself.
• Distributing exponents when several numbers
are inside parentheses, the exponent outside
the parentheses must be distributed to all of
the numbers within.
• Square root of a positive number x is the
number that when squared equals x.
• radical is the symbol for a positive square
root is √.
• Cube root of a positive number x is the
number that, when cubed, equals x.
Tips for act math
• Order of operations is: parentheses,
exponents, times, addition, and subtraction.
• Fractions ,decimals, ratios, percentages,
average charts and graphs combinations
• Calculators – students are permitted to use
calculators on act.
• The associative law: when adding a
string of numbers, you can add them
in any order you like. The same thing is
true when multiplying a string of
numbers.
• (-5)+4)2 {8/2 plus 4-8=0
•
• The Distributive Law: the distribute states that
if a problem gives you information in factor which is a(b+c) - you should distribute it
immededitately.
• If the information is given in distribute form
which is Ab + Ac you should factor it.
Fractions
• A fraction is just another way of expressing
division.
• A fraction is made up of a numerator and
denominator.
• The numerator is on the top and the
denominator is on the bottom.
• To reduce a fraction, see if the numerator and
the denominator have a common factor .
• Whatever factor they share can now be canceled.
Lets take the fraction 6/8. Is there a common
factor ? YES -2
• Sometimes a problem will involve deciding
which two fractions is larger.
• Which is larger 2/5 or 4/5 ? Think of these
parts of a whole. Which is bigger , two parts of
five or four parts out of five?
• 4/5 is clearly larger , they both had the same
whole, or the same denominator.
• Which ,is larger, 2/3 or 3/7 ? To decide, we
need to find a common whole , denominator
or denominator. You change the denominator
of a fraction by multiplying it by another
number. To keep an entire fraction the same,
however, you must multiply the numerator by
the same number.
• Change the denominator by 2/3 into 21.
• Which is the largest? 2/3, 4/7, 3/5?
• To compare these fraction directly, you need a
common denominator. Compare these
fractions two at a time, start with 2/3 and 4/7
an easy common denominator is 21.
Using the bowtie
• We get the common denominator by
multiplying the 2 denominators together.
• 2/3 --- 3/5
• 2/3 is larger
Adding and subtracting fractions
• Adding and subtracting fractions is simple.
Use the bowtie to add 2/5 and ¼
• 2/5 +1/4 = 8+5/20 = 13/20
• Use the bowtie to subtract 2/3 and 5/6
• 5/6 – 2/3 = 15-12/18 or 1/6
RATIOS
Ratio is always part over part. Not like
fractions which are part over whole. An
example of a ratio is like 4 cats over 3 dogs,
but the total would be 7 animals.
Percentages
A percentage is a fraction in which
the denominator equals 100. In
literal terms, percent means
“ divided by 100,”. You can always
express a percentage as a fraction.
Percentage Shortcuts
You can save time by remembering some
fractions and what their percentage is. For
example: 1/5 is 20 percent so whatever
number over 5, then you multiply that
number by 20. Another fast way is by using
decimal points. To find 10 percent you
move the point of the number over one
place to the left and to find 1 percent you
move the decimal point two places to the
left.
Averages
• Multiply # of things times avg then divided
total by what it equals
• 9 times 8 equals 72
• 72-42 or 30 divided by 2 equals 15
Weighted average
• Arithmetic mean the average
• Median =middle
• Mode=element appears most
Charts and graphs
• Decipher information presented in a graph
• If you can read a simple graph, then you can
solve the problem.
Combinations
The number of combinations is the
product of the number of things of
each type from which you have to
choose. The rule for combination
problems on the ACT is straightforward.
Info about ACT algebra
• Algebra is all about solving for an unknown
quantity.
• There are two general kinds of algebra
questions on the ACT.
• The first asked you to solve for a particular x.
• The second kind asks you to solve for a more
cosmic x.
The golden rule of algebra
• Whatever you do to one side, you have to do
to the other side of the equation.
Steps to working backwards
1.Start with the middle-(C or (H)
2.If its big, go to the next smaller choice
3.If its too small, go to the next larger choice
•
When you see numbers in the answer choices and when the question asked in the
last of the problem is relatively straight forward.
•
You don’t want to work backward . In the case, the answers wont give us a value
to try for either x or y .
Math terms
• Is = (any form of the
verb ”be” is the same as
• Of product times
• What a certain number
Percent
• 30 percent
• What percent
• More than
• Less than
• =
• X(multiplication)
• S , y , a z (your favorite
variable.
• 100(alternatively we
could use over 100)
• 300/100
• x/100
• +(addiction)
• -(subtraction)
The other method is plugging in
• The advantage of using a specific number is
that our minds do not think naturally in terms
of variables
• 1. Pick numbers for the variables in the
problem (and write them down).
• 2.using your numbers , find an answer to the
problem .
• 3.Plug your numbers into the answer choice to
see which choice equals the answer you found
How you spot a problem
• Any problem with variables in the answer
choices is a cosmic problem. You may not
choose to plug in every one of these, but you
could plug in on all of them.
FOIL
• First-multiply the first two terms in each
polynomial
• outer/inner – multiply the outer terms from
each polynomial and add the two terms then
the middle
• Last-multiply the last terms in each
polynomial.
The acts favorite factors
• Train yourself to recognize these quadratic
expressions instantly in both factored and unfactored form :
• X^2-y^2= (x+y)(x-y)
• X^2 +2xy+y^2= (x+y)^2
• X^2-2xy+y^2=(x-y)^2
Geometry
• There are 23 geometry questions on the math
ACT.
To Scale or Not To Scale?
• Every diagram is drawn exactly to scale.
• ACT diagrams were never intended to be
misleading.
P.O.E.
• Since the problems are always drawn to scale,
it will be possible to get very close
approximations of the correct answers before
you even do the problems.
Important Approximations
• You may want to eliminate problems that
contain answer choices with radicals or pie.
What Should I Do If There Is No
Diagram?
• Draw One!
• It’s always easier to understand a problem
when you can see it in front of you.
Triangles
• A triangle is a three-sided figure whose inside
angles always add up to 180 degrees.
• The largest angle of a triangle is always
opposite to the largest side.
Types of Triangles
•
•
•
•
There are 3 types of triangles.
Isosceles, Equilateral, and right.
Isosceles triangle has 2 equal sides.
Equilateral triangle has 3 equal sides and 3
equal angles.
• Right triangle has 1 inside angle that is equal
to 90 degrees.
Four-Sided Figures
• Rectangle: Four sided figure whose four
interior angles are equal to 90 degrees.
• Square: Rectangle whose four sides are all
equal in length.
• Parallelogram: Four sided figure made up of
two sets of parallel lines.
• Trapezoid: Four sided figure in which two sides
are parallel.
Circles
• Radius: Distance from the center of a circle to
any point on the circle.
• Diameter: Distance from one point on a circle
through the center of the circle to another
point.
Formulas
• The formula for the area of a circle is pie r
squared.
• The formula for the circumference is two pie
squared.
Slope Formula
•
•
•
•
All you need is two points
Helps you find the slope of a line
S= y1-y2/x1-x2 ex| (-2,5),(6,4)
5-4/-2-6 = -1/8
Midpoint Formula
• [x1+x2/2 + y1+y2/2]
Circles, ellipses, and parabolas
•
•
•
•
(x-h)^2 + (y-k)^2 = r^2
(h ,k) enter of circle
R= radius
*the Standard equ. For an ellipses just a squatlooking circle
• (x-h)^2/a^2 + (y-k)^2/b^2 = 1
• (h ,k) Center of ellipses
• 2a= horizontal axis(width) 2b= Vertical axis(Height)
• *Parabolas is a U-Shaped line
The distance Formulas
• You are able to do Pythagorean Theorem.
• A^2+B^2=C^2
• Example! What is the distance between
points A(2,2) & B(5,6)
•
•
•
•
•
A) 3
B) 4
C) 5
D) 6
E) 7
The distance Formulas
• You are able to do Pythagorean Theorem.
• A^2+B^2=C^2
• Example! What is the distance between
points A(2,2) & B(5,6)
• A) 3
• B) 4
• C) 5
• D) 6
• E) 7
Graphing Inequalities
3x+5>11
-5 -5
3x>6
X>2
-3
-2
-1
0
1
2
3 4
5
6
1. Which of the following represents the range of
solutions for inequality -5x-<x+5
-5x-7<x+5
-x
-x
-6x-7<5
+7 +7
-6x<12
X>-2
-4 -3 -2 -1 0 1 2 3 4
POE Poitets
-5 (-4) -7 <(-4) +5
20 -7 < 1
13 < 1
You plug in the answer to the exponent to see if it works
Graph in two dimension
X Is Negative
Y is Positive
X Is Negative
Y is Positive
Quadrant II
Quadrant I
X Is Negative
Y is Negative
X Is positive
Y is negative
Quadrant III
Quadrant IV
Trigonometry
SohCahToa
• Sine= Opposite over hypotenuse
• Cosine= Adjacent over hypotenuse
• Tangent= Opposite over adjacent
Hypotenuse
opposite
X
Adjacent
3 more relationships
• They evolve the reciprocals of the previous
three
• Cosecant= hypotenuse over opposite
• Secant= adjacent over hypotenuse
• Cotangent= adjacent over opposite
Example
• What is the sinǿ if the tanǿ=4/3
1. Draw a triangle with the opposite 4 and adjacent
3
2. Use A²+B²=C²
3. 3²+4²=C²
4. 9+16=25
5.Square root of 25=5 which is the hypotenuse
6. Sine is opposite over hypotenuse
7. The answer is 4/5
Harder Trigonometry
• Amplitude- Is the height of the curve. If
y=AsinØ the amplitude would be A. If A is
negative then the graph reflects over the xaxis
• Period- How long it take to get through a
complete cycle
• If there is no amplitude then A=1
Example
• y=1sin2x
2
1
1
2
π/2