Students struggling with ACT or SAT math often lack a solid foundation in basic math and pre-algebra skills. Without strengthening this foundation, it’s difficult to get students to improve. Why work on the quadratic formula or matrix multiplication when a student struggles with PEMDAS or dividing fractions? Below is a list of some ESSENTIAL basic math and pre-algebra skills that you need to master before moving on to intermediate algebra, coordinate geometry, plane geometry, and trigonometry. Let’s get started!
PEMDAS - Order of Operations
If you don’t know what “PEMDAS” means, learn it well! It’s the acronym that tells you the proper order in which to perform mathematical operations, and it stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. Failure to perform operations in this order will often result in an incorrect answer, and you can BET the SAT and ACT have listed that incorrect answer as an answer choice!
Even and Odd Numbers
Even numbers are divisible by 2. Odd numbers aren’t. -4, -2, 0, 2, and 4 are examples of even numbers. -1, -3, -5, 1, 3, and 5 are examples of odd numbers.
Odd + Odd = Even
Odd + Even = Odd
Even + Even = Even
Odd * Odd = Odd
Odd * Even = Even
Even * Even = Even
Positive and Negative Numbers
Anything greater than 0 is positive. Anything less than zero is negative.
Negative * Negative = Positive
Negative * Positive = Negative
Positive * Posite = Positive
Adding a negative is subtraction (E.g. 5 + (-3) = 2)
Subtracting a negative becomes addition! (E.g. 4 - (-3) = 7)
Multiples are the products of a number when multiplied by an integer. For example, the first 5 multiples of 3 are 3*1 = 3, 3*2 = 6, 3*3 = 9, 3*4 = 12, and 3*5 = 15.
Least Common Multiple
The least common multiple is the smallest multiple shared by two numbers. For example, the least common multiple of 3 and 4 is 12.
A factor of a number is an integer that divides evenly into the number. For example, factors of 12 include 1, 2, 3, 4, 6, and 12.
Greatest Common Factor
The greatest common factor is the largest factor that numbers share. For example, the greatest common factor of 12 and 6 is 6, while the greatest common factor of 12 and 16 is 4.
A prime number is a number whose only factors are itself and 1. The prime numbers under 100 are as follows:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
To find the prime factorization of a number, divide the number into factors until all you have are primes. For example, 45 can be factored into 9 and 5. 9 can be factored into 3 and 3. So the prime factorization of 45 is 3, 3, and 5.
Divisibility and Remainders
A number can be said to be divisible by another number if the result of the division is a whole number. For example, 14 is divisible by 7 because 14/7 is 2, and 2 is a whole number. A remainder is what is left over when a number is divided by a number by which it is not divisible. For example, 12 divided by 5 is 2 with a remainder of 2.
Percentages, Fractions, and Decimals
A percentage is part/whole relationship, where the whole is 100%, and the part is the percentage amount. A fraction is also a part out of a whole, and so is a decimal. For example, 30% is 30/100 and can also be represented as .3.
Working with Fractions
To add fractions, you must find a common denominator. To do so, find the least common multiple of the denominators, and multiply the numerator of the fraction you’re converting by how many times the old denominator goes into the new denominator. For example, to add 1/3 and 1/6, you use the common denominator of 6, and since 3 goes into 6 two times, you multiply the 1 in 1/3 by 2 to find that 1/3 is 2/6. 2/6 plus 1/6 is 3/6. To subtract fractions, follow the same process for adding fractions.
To multiply fractions, multiply the numerators to form the new numerator, then multiply the denominators to form the new denominator. For example, 2/5 times 1/3 is 2/15.
To divide fractions, you must multiply by the reciprocal. The reciprocal is what happens when you switch the numerator and denominator of a fraction. For example, to divide 1/3 by 1/2, first flip 1/2 to make it 2/1 and then multiply 1/3 times 2/1, giving you an answer of 2/3.
To reduce fractions, you must find the greatest common fraction of the numerator and denominator and divide the numerator and denominator by the GCF. For example, 3/15 has a greatest common factor of 3. 3/3 is 1, and 15/3 is 5, so 3/15 reduces to 1/5.
Ratios and Proportions
Ratios and proportions provide you a way to compare the number of one type of thing to the number of another type of thing. If there are 12 girls and 8 boys in a class, the ratio of girls to boys is 12:8, which reduces down to 3:2. To express this as a proportion, you would write 3/2. This will allow you solve for an unknown via cross multiplication. For example, if there is a 3:2 ratio of girls to boys in a class, and there are 9 girls, you can find out the number of boys by setting up the equation 3/2 = 9/x. Notice that the number of girls is in the numerator and the number of boys is in the denominator. To solve for x via cross multiplication, solve the equation 9*2 = 3x, giving you x = 6.
Mean, Median, Mode, and Range
The mean of a set of numbers is their average. To find an average, add up the terms and divide by the number of terms. For example, if you got an 80, a 90, and a 100 on three tests, to find your average, you would add 80+90+100, which equals 270, and then divide 270 by 3, because you took 3 tests, giving you an average of 90.
Median is the middle number when numbers are put in order from least to greatest.
Mode is the most frequently occurring number in a set.
Range is the high number minus the low number.
Probability is the likelihood of an event taking place. To calculate probability, you put the number of desirable outcomes over the number of possible outcomes. For example, the probability of drawing a green marble from a bag of 10 marbles that has 3 green marbles in it is 3/10. Probability can also be expressed as a decimal, fraction, or percentage.
When you raise a number to an exponent, you multiply that number by itself that many times. For example 2^3 equals 2*2*2=8.
To multiply terms with exponents that have a common base, such as 2^3 * 2^4, you add the exponents. 2^3 * 2^4 = 2^7.
To divide terms with exponents that have a common base, you subtract the exponents. x^5 / x^2 = x^3.
To raise an exponent to an exponent, you multiply the exponents. (x^2)^3=x^6.
An exponent of negative 1 is the same as one over the base. x^-1 = 1/x.
The square root of a number is the number that when squared results in the original number. For example, the square root of 64 is 8, because 8*8 is 64.
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