Introduction to ACT Math

1
Introduction to ACT Math

• A quick review of concepts

2

• Introduction to the ACT mathematics Test

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What to expect

• 33 Algebra Questions14 pre algebra 10
  elementary algebra 9 intermediate algebra
• 23 Geometry Questions14 plane geometry
  9 coordinate geometry
• 4 Trigonometry Questions based on sine, cosine,
  and tangent

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Do NOT expect

• A formula page before the math section
• The writers care more about what you know then
  the SAT

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• PRACTICE PRACTICE PRACTICE!!!

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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

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Answer

• Sum of everything/number of thingsaverage
• 36457413/4average
• ANSWER IS 42 (B)

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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

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ANSWER

• 367413Peter/442
• 367413Peter168
• 168-12345
• Answer is 45(H)

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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

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ANSWER

• 50xyz/442
• 50xyz168
• 504949z168
• Z20
• Answer is C

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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?
• .9 D)11
• 2.32 E)90
• 9

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Answer

• 10 of 600 is 60 so it is less then 10 so
  choices D and E dont work
• A and B dont work because we need an answer
  slightly less then 10
• Answer is C

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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

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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

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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

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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

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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

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work

1. Write down 36 in your work area
2. Find 1/3 of 3612 so 12 picked up trash
3. Find ¼ of 246 so 6 people painted fences
4. 5/6 of 18 15 so 15 people cleaned busses
5. Now it sais all of the remaining people planted
   trees so 18-153 so 3 people planted trees.
6. F is the answer

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????????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 ymxb form

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basics
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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

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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

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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

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Absolute value

• The absolute value of a number is the distance
  between that number and 0 on the number line.
• Ex. 66
• -66

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Variables and coefficients

• In the expression 3x4y, x and y are the
  variables because we dont know what they are.
• 3 and 4 are the coefficients because you multiply
  the variables by them.

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Basic opperations

• Divisibility Rules-
• 1. 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.

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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.

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Standard symbols

• Is not equal too ?
• Is equal too
• lt is less then
• gt is greater then
• is less then or equal too
• is greater then or equal too

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Exponents

• Base is called the lower and larger number
• Exponent is the upper number.
• 62 x 63 6( 23) 65
• (y) (y3) y(23) y5

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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.

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Negative Powers

• A negative power is simply the reciprocal of a
  positive power.

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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.

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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.

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• 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 v.
• Cube root of a positive number x is the number
  that, when cubed, equals x.

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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.

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• 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 4-80

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• The Distributive Law the distribute states that
  if a problem gives you information in factor -
  which is a(bc) - you should distribute it
  immededitately.
• If the information is given in distribute form
  which is Ab Ac you should factor it.

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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

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• 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.

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• 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.

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• 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.

43
Using the bowtie

• We get the common denominator by multiplying the
  2 denominators together.
• 2/3 ---? 3/5
• 2/3 is larger

44
Adding and subtracting fractions

• Adding and subtracting fractions is simple. Use
  the bowtie to add 2/5 and ¼
• 2/5 1/4 85/20 13/20
• Use the bowtie to subtract 2/3 and 5/6
• 5/6 2/3 15-12/18 or 1/6

45
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.

46
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.

47
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.

48
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

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Weighted average

• Arithmetic mean the average
• Median middle
• Modeelement appears most

50
Charts and graphs

• Decipher information presented in a graph
• If you can read a simple graph, then you can
  solve the problem.

51
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 straight-forward.

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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.

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The golden rule of algebra

• Whatever you do to one side, you have to do to
  the other side of the equation.

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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 dont want to work backward . In the case,
  the answers wont give us a value to try for
  either x or y .

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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)

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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 in step
  2.

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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.

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F O I L

• 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.
• When you add your first , outer , inner, and last
  terms together, you get back to where you
  started.

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The acts favorite factors

• Train yourself to recognize these quadratic
  expressions instantly in both factored and
  un-factored form
• X2-y2 (xy)(x-y)
• X2 2xyy2 (xy)2
• X2-2xyy2(x-y)2

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Geometry
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• There are 23 geometry questions on the math ACT.

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To Scale or Not To Scale?

• Every diagram is drawn exactly to scale.
• ACT diagrams were never intended to be misleading.

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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.

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Important Approximations

• You may want to eliminate problems that contain
  answer choices with radicals or pie.

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What Should I Do If There Is No Diagram?

• Draw One!
• Its always easier to understand a problem when
  you can see it in front of you.

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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.

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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.

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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.

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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.

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Formulas

• The formula for the area of a circle is pie r
  squared.
• The formula for the circumference is two pie
  squared.

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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

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Midpoint Formula

• x1x2/2 y1y2/2

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Circles, ellipses, and parabolas

• (x-h)2 (y-k)2 r2
• (h ,k)? enter of circle
• R radius
• the Standard equ. For an ellipses just a
  squat-looking circle
• (x-h)2/a2 (y-k)2/b2 1
• (h ,k)? Center of ellipses
• 2a horizontal axis(width) 2b Vertical
  axis(Height)
• Parabolas is a U-Shaped line

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The distance Formulas

• You are able to do Pythagorean Theorem.
• A2B2C2
• Example! What is the distance between points
  A(2,2) B(5,6)
• A) 3
• B) 4
• C) 5
• D) 6
• E) 7

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The distance Formulas

• You are able to do Pythagorean Theorem.
• A2B2C2
• Example! What is the distance between points
  A(2,2) B(5,6)
• A) 3
• B) 4
• C) 5
• D) 6
• E) 7

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Graphing Inequalities
3x5gt11 -5 -5 3xgt6 Xgt2
-3 -2 -1 0 1 2 3 4
5 6
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1. Which of the following represents the range of
solutions for inequality -5x-ltx5
-5x-7ltx5 -x -x -6x-7lt5 7
7 -6xlt12 Xgt-2
-4 -3 -2 -1 0 1 2 3 4
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POE Poitets
-5 (-4) -7 lt(-4) 5 20 -7 lt 1 13 lt 1 You plug in
the answer to the exponent to see if it works
79
Graph in two dimension
X Is Negative Y is Positive Quadrant I
X Is Negative Y is Positive Quadrant II
X Is positive Y is negative Quadrant IV
X Is Negative Y is Negative Quadrant III
80
Trigonometry
81
SohCahToa

• Sine Opposite over hypotenuse
• Cosine Adjacent over hypotenuse
• Tangent Opposite over adjacent

Hypotenuse
opposite
X
Adjacent
82
3 more relationships

• They evolve the reciprocals of the previous three
• Cosecant hypotenuse over opposite
• Secant adjacent over hypotenuse
• Cotangent adjacent over opposite

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Example

• What is the sin? if the tan?4/3
• Draw a triangle with the opposite 4 and adjacent
  3
• Use A²B²C²
• 3. 3²4²C²
• 4. 91625
• 5.Square root of 255 which is the hypotenuse
• 6. Sine is opposite over hypotenuse
• 7. The answer is 4/5

84
Harder Trigonometry

• Amplitude- Is the height of the curve. If yAsinØ
  the amplitude would be A. If A is negative then
  the graph reflects over the x-axis
• Period- How long it take to get through a
  complete cycle
• If there is no amplitude then A1

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Example

• y1sin2x

2
1
-1
p/2
-2