I. Mathematics Test Strategies

Review the test preparation and test taking strategies, go over the special strategies for answering the mathematics multiple-choice items to make a study plan and get yourself ready for the LAST.

The mathematics tested is the kind you probably had in high school. It is the kind of mathematics you will use as you teach and go about your everyday life. Computational ability alone is expected but is held to a minimum. Remember to use the general test strategies while taking the mathematics portion of the test.

Strategies for answering mathematics multiple-choice items:

1. Write in the Test Booklet. It is particularly important to write in the test booklet while taking the mathematics portion of the test. Use these hints for writing in the test booklet:

• Do Your Calculations in the Test Booklet. Do all your calculations in the test booklet to the right of the question. This makes it refer to the calculations as you choose the correct answer.;

• Draw Diagrams and Figures in the Test Booklet. When you come across a geometry problem or related problem, draw a diagram in the booklet to help;

• Circle important Information and Key Words and Cross Out Information You Don’t Need. This approach will draw your attention to the information needed to answer the question. A common mistake is to use from the question information that has nothing to do with the solution.

2. Estimate to Be Sure Your Answer Is Reasonable. You can use estimation and common sense to be sure that the answer is reasonable. You may make a multiplication error or misalign decimal points. You may be so engrossed in a prob­lem that you miss the big picture because of the details. These difficulties can be headed off by making sure your answer is reasonable. Stand back for a second after each question and ask, “Is this reasonable? Is this at least approximately correct? Does this make sense?” Check answers to computation, particularly division and subtraction. When you have completed a division or subtraction example, do a quick, approximate check. Your check should confirm your answer. If not, your answer is probably is not reasonable.

3. Work from the Answers. If you don’t know how to solve a formula or relation, try out each answer choice until you get the correct answer.

4. Try Out Numbers.

5. Eliminate and Guess. Use this approach when all else has failed. Begin by eliminating the answers you know are wrong. Sometimes you know with certainty that an answer is incorrect. Other times, an answer looks so unreasonable that you can be fairly sure that it is not correct. Once you have eliminated in correct answers, a few will probably be left. Just guess among these choices. There is no method that will increase your chances of guessing correctly.

II. Mathematics Review

Study a comprehensive subject review to prepare for the mathematics items on the LAST. It also includes completed examples and reading comprehension questions that accompany the graphs. There are about 12 mathematics items on the LAST. This review section targets skills and concepts you need to know to pass mathematics part of the LAST.

Number Sense And Numeration

Understanding and Ordering Whole Numbers

Whole numbers are the numbers you use to tell how many. They include 0, 1, 2, 3, 4, 5, 6 ... The dots tell us that these numbers keep going on forever. There are an infinite number of whole numbers, which means you will never reach the last one. Cardinal numbers such as 1, 9, and 18 tell how many. There are 9 players on the field in a baseball game. Ordinal numbers such as 1st, 2nd, 9th, and 8th tell about order. You can use the number line to compare numbers. Numbers get smaller as we go to the left and larger as we go to the right. We use the terms equal to (=), less than (<), greater than (>), and between to compare numbers.

12 equals 10 +2 12 = 10 + 2

2 is less than 5 2 < 5

9 is greater than 4 9 > 4

6 is between 5 and 7 5 < 6 < 7

Place Value

We use ten digits, 0-9 to write out numerals. We also use a place value system of numeration. The value of a digit depends on the place it occupies. Look at the following place value chart: millions — hundred thousands — ten thousands — thousands — hundreds — tens — ones.

Positive Exponents

You can show repeated multiplication as an exponent. The exponent shows how many times the factor appears. Sometimes we use scientific notation to represent large numbers.

Understanding and Ordering Decimals

Decimals are used to represent numbers between 0 and 1. We also use ten digits 0-9 and a place value system of numeration to write decimals. Look at the following place value chart: ones — tenths — hundredths — thousandths — ten thousandths — hundred thousandths — millionths — ten millionths — hundred millionths — billionths.

Comparing Whole Numbers and Decimals

To compare two numbers line up the place values. Start at the left and keep going until the digits in the same place are different.

Understanding and Ordering Fractions

A fraction names a part of a whole or of a group. A fraction has two parts, a numerator and a denominator. The denominator tells how many parts in all. The numerator tells how many parts you identified. Two fractions that stand for the same number are called equivalent fractions. Multiply or divide the numerator and denominator by the same number to find an equivalent fraction.

Mixed Numbers and Improper Fractions

Change an improper fraction to a mixed number. Change a mixed number to an improper fraction.

How and When to Add, Subtract, Multiply, and Divide

Use this phrase to remember the order in which we do the operations: Please Excuse My Dear Aunt Sally

(1) Parentheses;

(2) Exponents;

(3) Multiplication or Division;

(4) Addition or Subtraction.

Number Theory

Number theory explores the natural numbers {1, 2, 3, 4, …}. We’ll review just a few important number theory concepts: 1) The factors of a number evenly divide the number with no remainder; 2) A prime number has exactly two factors, itself; 3) A composite number has more than two factors.

Real Number Systems And Subsystems

Add, Subtract, Multiply, and Divide Decimals

1) Line up the decimal points and add or subtract.

2) Multiply as with whole numbers. Count the total number of decimal places in the factors. Put that many decimal places in the product. You may have to write the leading zeros.

3) Make the divisor a whole number. Match the movement in the dividend and then divide.

Add, Subtract, Multiply, and Divide Fractions and Mixed Numbers

1) Write any mixed number as an improper fraction. Multiply numerator and denominator. Write the product in simplest form.

2) Write any mixed numbers as improper fractions. Invert the divisor and multiply. Write the product. Write in simplest form.

3) Write fractions with common denominators. Add and then write in simplest form.

4) Write fractions with common denominators. Subtract and then write in simplest form.

Square Roots

The square root of a given number, when multiplied by itself, equals the given number. This symbol means the square root of 25 Ö25. The numbers with whole-number square roots are called perfect squares.

Ratio and Proportion

A ratio is a way of comparing two numbers with division. It conveys the same meaning as a fraction. A proportion shows two ratios that have the same value; that is, the fractions representing the ratios are equivalent. Use cross multiplication. If the cross products are equal, then two ratios form a proportion. You may have to write a proportion to solve a problem: 1. write a proportion; 2. cross multiply to solve.

Percent

Percent comes from per centum, which means per hundred. Whenever you see a number followed by a percent sign it means that number out of 100. To write a decimal as a percent, move the decimal point two places to the right and write the percent sign. To write a fraction as percent, divide the numerator by the denominator. Write the answer as a percent or write an equivalent fraction with 100 in the denominator. Write the numerator followed by a percent sign.

To write a percent as a decimal, move the decimal point two places to the left and delete the percent signTo write a percent as a fraction, write a fraction with 100 in the denominator and the percent in the numerator. To find a percent of a number, write a number sentence with a decimal for the percent and solve. To find what percent one number is of another, write a number sentence and solve to find the percent. To find a number when a percent of it is known, write a number sentence with a decimal or a fraction for the percent and solve to find the number.

Probability And Simple Statistics

Probability

The probability of an occurrence is the likehood that it will happen. Most often, we write the probability as a fraction. Flip a fair coin and the probability that it will come up heads (H) is Ѕ. The same is true for tails (T). Write the probability this way: P(H) = 1/2 P(T) = 1/2 If something will never occur the probability is 0. If something will always occur, the probability is 1. Therefore, if you flip a coin, P(7) = 0 P(H or T) = 1

The events are independent when the outcome of one event does not affect the probability of the other event. Each coin flip is an independent event. No matter the outcome of one flip, the probability of the next flip remains the same. Coin flips are independent events.

Events are dependent where the outcome of one event does affect the probability of the other event. For example you have a full deck of cards. The probability of picking the Queen of Hearts is 1/52. You pick one card and it’s not the Queen of Hearts. You don’t put the card back. The probability of picking the Queen of Hearts is now 1/51. Cards picked without replacement are dependent events.

Statistics

Descriptive statistics are used to explain or describe a set of numbers. Most often we use the mean, median, or mode to describe these numbers. The mean (average) is a position midway between two extremes. To find the mean: 1) add the items or scores; 2) divide by the number of items. The mean or average of 24, 17, 42, 51, 36 = 170 ¸ 5 = 34.

The median is the middle number. To find the median: 1. arrange the numbers from least to greatest. 2. if there are an odd number of scores, then find the middle score. 3. if there is an even number of scores, average the two middle scores. Find the median of these numbers: 6, 9, 11, 17, 21, 33, 45, 72. There are an even number of scores (17+21) ¸2 = 19. The median is 19. Don’t forget to arrange the scores in order before finding the middle score!

The mode is the number that occurs most often. Not all sets of numbers have a mode. Some sets of numbers may have more than one mode.

Algebra

The number line can also show negative numbers. There is a negative whole number for every positive whole number. Zero is neither positive nor negative. The negative whole numbers and the positive whole numbers are called integers.