ACT 1-on-1 CurriculumMath: Pre-algebra & Algebra

ACT Math

• 60 questions• 60 minutes

• Three sections of questions:

– Pre-Algebra/Elementary Algebra• 24 questions

– Intermediate Algebra/Coordinate Geometry• 18 questions

– Plane Geometry/Trigonometry• 18 questions

Math Section at a Glance

DIRECTIONS: Answer each question and then plug in the matching bubble on your answer sheet. Do not stay on problems that you are struggling with. Answer the questions you know first; then go back complete the other problems as time permits. Calculators are allowed on this section of the test. Note that some of the problems can be answered without a calculator. Note: unless the questions explain something differently, assume the following. 1. Figures, graphs and charts may not be drawn to actual measurements.2. Linear data, figures and shape are represented in a plane.3. A straight, 180° line will be introduced as a line.4. Arithmetic mean will be introduced as an average.

Math Directions:

ACT Math

• Concepts range from pre-algebra up to algebra II and trigonometry

• There are only 4 trigonometry questions

• Questions are generally in order of difficulty

• To answer all questions, you need to work generally at a pace of one minute per question

The Basics – Math

ACT Math

• Write down your work!– Do not just go to your calculator to find the answer. Write it down.– If you work it out, you can review the steps to get the right answer if you

make a mistake.• Pace yourself

– It is easy to lose track of time on one problem.– Do not spend too much time on any one problem.

• Read the questions carefully!– This is a way to avoid easy mistakes.

• Pick & Plug– Use the answer choices to help you solve the problem.

• Oddball Answers are OK– In math, many times the odd answer out is the correct one.

• Assign Values to Variables– If a question has numerous variables, assign a consistent numerical value

to help you solve it.

General Strategies

ACT Math

• Translate words into an equation or statement that you can solve.

Word Problems – Pre-Algebra/Elementary Algebra

1. 50 is what percent of 450?

2. 21 is 6 more than what number?

3. 8 is 5 less than half of what number?

4. 21 more than what number is five times the number?

ACT MathWord Problems – Pre-Algebra/Elementary Algebra

ACT Math

• Mean:– Mean is the average of a list of numbers.

• Median– The middle number between a list of numbers is the median.– If the list has an even set of numbers, the median is the average of the

two middle values.

• Mode– The number that occurs the most in a list of numbers is the mode.

Mean, Median, and Mode – Pre-Algebra/Elementary Algebra

ACT Math

• Primes:– A prime number is only divisible by one and itself.– Special rules:

• 1 is not a prime number.• 2 is the only even prime number.

• Factors:– Factors are numbers that you multiply together to get a product.

• Multiples:– Multiples are formed by multiplying a given number by the counting numbers.

• Greatest Common Factor (GCF) and Least Common Multiple (LCM)– Greatest common factor is the largest factor of a set of numbers.– Least Common Multiple is the smallest common multiple of a set of numbers.

Primes, Multiples, and Factors– Pre-Algebra/Elementary Algebra

ACT Math

• For distance problems, remember that:

Distance= Rate x Time

• Use the given units to help on this.– Many will use miles and hours.

Distance – Pre-Algebra/Elementary Algebra

ACT Math

• For measurement questions, identify the units that are used.

• You need to remember some basic conversions:

Measurements – Pre-Algebra/Elementary Algebra

Length Conversions

12 inches (in.) = 1 foot (ft.)

3 ft. = 36 in. = 1 yard (yd.)

5280 ft. = 1760 yd. = 1 mile (mi.)

Surface Area Conversions

ACT Math

• Absolute value is the number of integers a number is away from zero.

|-5| = 5

• For difficult absolute value questions, account for the positive and negative values of a variable when the variable is between the brackets.

Absolute Value – Pre-Algebra/Elementary Algebra

ACT Math

• Probability of x:

• Combinations: For these problems, multiply the numbers provided by each other.

• Permutations: For these problems, order matters. To find the answer, multiply the number of items by one less for each placement.

Probability– Pre-Algebra/Elementary Algebra

ACT Math

• Remember rules for simplifying expressions with exponents:

– * = – = – = – = – =

Exponents and Roots– Pre-Algebra/Elementary Algebra

ACT Math

• Ratios are comparisons between two variables.

• Solving Ratio Problems:– Write down both ratios as fractions, setting them equal to one another.– Verify that your units match on both sides of the equation.– Use cross-multiplication to solve the problem.

Ratios and Proportions – Pre-Algebra/Elementary Algebra

ACT Math

• Arithmetic sequences are separated by a common number, called a common term or difference.– Example: 28, 24, 20, 16 is a arithmetic sequence with a difference of -

4.

• Geometric sequences are separated by a common ratio that is either multiplied or divided by the previous number.– Example: 2, 6, 18, 54 is a geometric sequence with a ratio of 3.

Series – Pre-Algebra/Elementary Algebra

ACT Math

• These test if you can simplify or manipulate algebraic expressions.

• Remember the order of operations: PEMDAS.– Parentheses, Exponents, Multiplication and Division, Addition and

Subtraction

• Remember the distributive property.• A(b+c) = ab + ac

Simplifying Algebraic Expressions – Pre-Algebra/Elementary Algebra

ACT Math

• Work on the following Practice Problems from 61D:

Pre-Algebra/Elementary Algebra

– # 1– # 3– # 4– # 5– # 6– # 14– # 15– # 16

– # 24– # 27– # 32– # 35– # 39– # 43

ACT Math

• System of equations provide a pair of equations with two variables for which you have to solve.

• One way to solve for x is to add or subtract the equations to eliminate one of the variables.

• Another way to solve for x is by substitution, where you solve for one of the variables and plug it into the other equation.

System of Equations – Intermediate Algebra

ACT Math

• Multiplying binomials:– Use FOIL to multiply binomials: First, Outer, Inner, Last.

• Example: (x+5)(x-3)

• Solving quadratics by factoring:– Factor x2+5x+6.

Solving & Factoring Quadratic Equations – Intermediate Algebra

ACT Math

• Functions are a way of restating y=ax+b as f(x) = ax+b.

• To solve functions, you substitute x for some number or variable.

• Example: f(x) = x2+2x-4. What is f(3)?

Functions – Intermediate Algebra

ACT Math

• In an inequality, one of the sides of the equation is greater than the other.

• Solve inequalities similar to the way you solve linear equations.

• One note on solving inequalities: if you multiply or divide by a negative number, you must flip the sign.

Inequalities – Intermediate Algebra

ACT Math

• Matrix questions occasionally pop up on the ACT.

• Adding and subtracting matrices:– To add or subtract matrices, add or subtract corresponding numbers.

• Multiplying matrices:– Take the first row of the first matrix and the first column of the second matrix

and multiply the corresponding numbers.

• Determinant:– The determinant is found by subtracting the product of the diagonals.

Matrices – Intermediate Algebra

ACT Math

• The basic form of a logarithm:

Logb y = x means bx = y

• Logarithmic identities:– Logb (xy) = Logb x + Logb y – Logb (x/y) = Logb x - Logb y – Logb (xd) = d Logb x

Logarithms – Intermediate Algebra

ACT Math

• Work on the following Practice Problems from 61D:

Intermediate Algebra

– # 10– # 11– # 12– # 21– # 34– # 57