PreAlgebra Notes Unit One: Variables, Expressions, and Integers


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1 PreAlgebra Notes Unit One: Variables, Expressions, and Integers Evaluating Algebraic Expressions Syllabus Objective: (.) The student will evaluate variable and numerical expressions using the order of operations. A variable is defined as a letter or symbol that represents a number. Examples: a, b, c, _,,,... A variable (algebraic) expression is an expression that consists of numbers, variables, and operations. Examples: ab,2x7 y, 53( m2)2 General Strategy To evaluate algebraic expressions, substitute the assigned number (or value) to each variable and simplify the resulting arithmetic expression. When expressions have more than one operation, mathematicians have agreed on a set of rules called the Order of Operations. Order of Operations (PEMDAS or Please excuse my dear Aunt Sally s loud radio)*. Do all work inside the Parentheses and/or grouping symbols. 2. Evaluate Exponents. 3. Multiply/Divide from left to right.* 4. Add/Subtract from left to right.* *Emphasize that it is NOT always multiplythen divide, but rather which ever operation occurs first (going from left to right). Likewise, it is NOT always addthen subtract, but which of the two operations occurs first when looking from left to right. Example : Evaluate a b when a 5 and b4. ab Example 2: Evaluate 2x 7y when x 3 and y 6. 2x7 y McDougal Littell, Chapter HS PreAlgebra, Unit 0: Variables, Expressions, Integers Page of 3
2 Example 3: Evaluate 5 3( m2) when m 6. 53( m 2) 53(6 2) 53(4) 5 (2) 7 6 Exponential Notation An exponent is the superscript which tells how many times the base is used as a factor. 3 2 base exponent = { factors An exponent tells you how many times to write the base as a factor. The base 2 is used as a factor 3 times. 2 3 is read 2 to the third power or 2 cubed. 2 is the base and 3 is the exponent. Students must be able to convert numbers into exponential form and vice versa. Examples: (a) Write 8 in exponential form (b) Evaluate Let us look at a pattern that will allow you to determine the values of exponential expressions with exponents of or ? 0 2? ? 0 3? ? 0 4? Any number to the power of is equal to the number. That is, n = n. Any number to the power of 0 is equal to one. That is, n 0 =. Also point out to students that if there is no exponent, the exponent is always. McDougal Littell, Chapter HS PreAlgebra, Unit 0: Variables, Expressions, Integers Page 2 of 3
3 Integers Syllabus Objective: (.2) The student will compare and order integers. Integers are defined as whole numbers and their opposites. Integers =... 3, 2,, 0,, 2, 3,... The negative numbers lie to the left of 0 on the number line and the positive numbers lie to the right of 0. As you move from left to right on the number line, numbers are always increasing in value. Likewise, as you move from right to left, the numbers are decreasing in value. Examples: Compare the following. Use > or < (a) 24 5 (b) 3 2 (c) is to the right of 5 2 is to the right of 3 8 is to the right of 2 24 > 5 3 < 2 8 > 2 (d) List the following integers in order from least to greatest. 7, 2, 5, 0, 3, 5, 20, 2, 2 Answer: 20, 2, 5, 2, 0, 2, 3, 7, 5 The absolute value of a number is the distance from zero on a number line without regard to the sign. The symbol for absolute value is and is read as the absolute value of. Examples: (a) 7 7 (b) 5 5 (c) 0 0 Let s take a look at the graph of x. What numbers are one unit from zero without regard to sign? The points and. When we work with signed numbers, we are often working with two different signs that look exactly alike. They are signs of value and signs of operations. A sign of value tells you if the number you are working with is greater than zero (positive) or less than zero (negative). Signs of operations tell you to add, subtract, multiply, or divide. ( 3) ( 2) or ( 3) ( 2) sign of value sign of operation sign of value sign of value sign of operation sign of value McDougal Littell, Chapter HS PreAlgebra, Unit 0: Variables, Expressions, Integers Page 3 of 3
4 Notice that the sign of value and the sign of operations are identical. Please note that numbers written without a sign are assumed to be positive; for example, 3 3, Operations with Integers Syllabus Objective: (.3) The student will perform operations with integers. Adding Integers Sometimes we refer to integers as directed numbers, because the sign of the number tells us to move a certain number of units in a specific direction along the number line. Positive numbers tell us to move to the right on the number line or in a positive direction; negative numbers tell us to move to the left on the number line or in a negative direction. Our starting point will always be zero or the origin. Now, let s look at the previous problem mentioned, (+3) + (+2). We start at 0 on the number line and move in a positive direction 3 units. From the point +3, we move another 2 units in a positive direction. Our ending point is the sum of (+3) + (+2) = +5 or 5. We now have our first rule for adding integers. Rule : A positive number plus a positive number is equal to a positive number. Try a few sample problems with the students. Let s look at what happens when we add a negative number to a negative number. We will begin with3 2. We start at 0 on the number line and move in a negative direction (left) 3 units. From the point 3, we move another 2 units in the negative direction. Our ending point is the sum of Rule 2: A negative number plus a negative number is equal to a negative number. Try a few sample problems with the students. You should devote some time reviewing the commutative property of addition. In the previous two examples, it should be clear to the students that (+3) + (+2) = (+2) + (+3) and that ( 3) + ( 2) = ( 2) + ( 3). McDougal Littell, Chapter HS PreAlgebra, Unit 0: Variables, Expressions, Integers Page 4 of 3
5 It is now time to look at what happens when we add a positive number to a negative number or a negative number to a positive number. Let s start with the problem ( 3) + (+3) We start at 0 on the number line and move in a negative direction 3 units. From the point 3, we now move 3 units in a positive direction. As you can see from the above picture, you end at zero, which was our starting point: ( 3) + (+3) = 0. Our next example is ( 2) + (+5): We start at 0 on the number line and move in the negative direction 2 units. From the point 2, we now move 5 units in a positive direction. ( 2) + (+5) = +3. We have one more addition to look at, (+2) + ( 5): We start at 0 on the number line and move in a positive direction 2 units. From the point +2, we now move 5 units in a negative direction, (+2) + ( 5) = 3. If we play with the above examples and enough similar type problems, we ll come up a rule that will allow us to add positive and negative numbers without the use of the number line. Rule 3: When adding one negative number and one positive number, find the difference between their absolute values and use the sign of the number with the greatest absolute value. Examples: ( 2) + (+2) = 0 ( 2) + (+6) = +4 (+7) + ( 5) = +2 ( 7) + (+5) = 2 McDougal Littell, Chapter HS PreAlgebra, Unit 0: Variables, Expressions, Integers Page 5 of 3
6 Subtracting Integers Remember that subtraction is the opposite operation for addition. So if we subtract on the number line, we want to do the opposite of what the number tells us to do. So let s look at our first example. Example : 28? We will use zero as our starting point just like we did in addition and move to the left, 2 units to 2. We want to subtract +8 from 2. But remember, we are subtracting. Moving 8 units in the positive direction is what we would do if we were adding. Since we are subtracting and subtraction is the opposite of addition, we will do the opposite and move 8 units in the negative direction. Example 2: 5? We start at zero and move unit in a negative direction to. The second number is 5, but because we are subtracting, we move in a positive direction 5 units Example 3: 63? 5 4 We start at zero and move 6 units in a positive direction to +6. The second number is 3, but because we are subtracting, we move in a negative direction 3 units McDougal Littell, Chapter HS PreAlgebra, Unit 0: Variables, Expressions, Integers Page 6 of 3
7 Let s go back and look at the above examples on the number line. Example : 28 0 looks like the addition problem Example 2: 5 4 looks like the addition problem Example 3: 63 7 looks like the addition problem A close examination of our examples and additional practice problems leads us to a rule for the subtraction of integers. Rule 4: When subtracting integers, change the sign of the problem from subtraction to addition, change the sign of the subtrahend (second number), and use the rules for addition of integers. Another way to state this is add the opposite. Multiplying Integers Multiplying integers is a mathematical operation that is an abbreviated process of adding an integer to itself a specified number of times. For example, (five groups of 3) or (three groups of 5) A couple of points to be made here:. Review of the commutative property of multiplication The above is the product of positive numbers Positive Positive = Positive P P P So, let s take a look at the product of a negative number times a positive number. By our definition of multiplication, or Notice that in each of the above examples, we are looking at a negative number added a positive number of times. Since we cannot use our definition of multiplication to show a number added to itself a negative number of times, we will use pattern development. McDougal Littell, Chapter HS PreAlgebra, Unit 0: Variables, Expressions, Integers Page 7 of 3
8 ( ) 4 5 ( ) 5 8 ( ) 8 4 ( 2) 8 5 ( 2) 0 8 ( 2) 6 4 ( 3) 2 5 ( 3) 5 8 ( 3) 24 By now, as you work with your students, you will have done enough problems to establish the following rules of multiplication: Positive Positive = Positive Positive Negative = Negative P P P P N N Let s continue the pattern development to show what the sign is when you multiply a negative number times a negative number. We know the following to be true: Be prepared to discuss the identity property and multiplication by zero. You are now ready to discuss that a negative number times a negative number equals a positive number. Continuing the pattern development: 4 ( ) 4 5 ( ) 5 8 ( ) 8 4 ( 2) 8 5 ( 2) 0 8 ( 2) 6 4 ( 3) 2 5 ( 3) 5 8 ( 3) 24 Therefore, the rules for the multiplication of integers are: Positive Positive = Positive Positive Negative = Negative Negative Positive = Negative Negative Negative = Positive P P P P N N N P N N N P McDougal Littell, Chapter HS PreAlgebra, Unit 0: Variables, Expressions, Integers Page 8 of 3
9 Division of Integers Division is the opposite operation to multiplication, but we can still use pattern development to explain the rules of division for integers. The pattern development is a little trickier, but we can do it. Notice that the examples below are tied to the problems used in the pattern development for multiplication. If necessary, each problem can be rewritten as a multiplication problem which has been used with the students The patterns developed in the above examples give the students two rules for division: Positive Positive Positive PP P Negative Positive Negative N P N Making a few slight modifications, we can develop patterns to show the last two rules of division: 2 ( 4) 3 5 ( 5) ( 4) 2 5 ( 5) ( 4) ( 4) ( 4) ( 4) ( 4) Positive Negative Negative PN N Negative Negative Positive N N P Now is the time to compare the rules for multiplication and division: PPP PPP PN N PN N N PN N PN N N P N N P McDougal Littell, Chapter HS PreAlgebra, Unit 0: Variables, Expressions, Integers Page 9 of 3
10 Further investigations and discussion will lead us to two rules which can be used with both multiplication and division: Rule 5: When multiplying or dividing, like signs equal a positive. Rule 6: When multiplying or dividing, unlike signs equal a negative. To review our rules: Addition Rule : A positive number plus a positive number is equal to a positive number. Rule 2: A negative number plus a negative number is equal to a negative number. Rule 3: When adding one negative number and one positive number, find the difference between absolute values and use the sign of the number with the greatest absolute value. Subtraction Rule 4: When subtracting integers, change the sign of the problem from subtraction to addition, change the sign of the subtrahend (second number), and use rules, 2, or 3 for addition of integers. Another way to state this is add the opposite. Multiplication and Division Rule 5: When multiplying or dividing, like signs equal a positive. Rule 6: When multiplying or dividing, unlike signs equal a negative. When working with these rules, we must understand the rules work for only two numbers at a 3 4 5, the answer would be 60. time. In other words, if I asked you to simplify The reason is 34 2, and then Have students investigate what happens to the sign if you have more than 3 numbers by experimenting with several possibilities. Students should discover that if you have an even number of signs, the product is positive. If you have an odd number of signs, the product is negative. In math, when we have two parentheses coming together without a sign of operation, it is understood to be a multiplication problem. We leave out the sign because in algebra it might be confused with the variable x. McDougal Littell, Chapter HS PreAlgebra, Unit 0: Variables, Expressions, Integers Page 0 of 3
11 Stay with me on this! Often times, for the sake of convenience, we also leave out the + sign when adding integers. Example: 8 5 can be written without the sign of operation 8 5, still equals 3 or Example: 8 5 can be written without the sign of operation 8 5, still equals 3 or Example: 8 5 can be written without the sign of operation 8 5, still equals 3 or85 3. For ease, we have eliminated the sign for multiplication and the + sign for addition. That can be confusing. Now the question is: How do I know what operation to use if we eliminate the signs of operation? The answer: If you have two parentheses coming together as we do here, 5 3, you need to recognize that as a multiplication problem. A subtraction problem will always have an additional sign, the sign of operation. For example, 2 5, you need to recognize the negative sign inside the parentheses is a sign of value, the extra sign outside the parentheses is a sign of operation. It tells you to subtract. Now, if a problem does not have two parentheses coming together and it does not have an extra sign of operation, then it s an addition problem. For example, 8 4, 25, and 9 2 are all samples of addition problems. Naturally, you would have to use the rule that applies. The Coordinate Plane Syllabus Objectives: (4.) the student will identify points in a coordinate plane. (4.2) The student will plot points in a coordinate plane. A coordinate plane is formed by the intersection of a horizontal number line called the xaxis and a vertical number line called the yaxis. The xaxis and yaxis meet or intersect at a point called the origin. The coordinate plane is divided into six parts: the xaxis, the yaxis, Quadrant I, Quadrant II, Quadrant III, and Quadrant IV. (Refer to the diagram on the next page.) Hint: a way to remember the order of the quadrants is to think of writing a C (for coordinate plane) around the origin. To create the C you start in quadrant I and move counterclockwise (and so does the numbering of the quadrants). McDougal Littell, Chapter HS PreAlgebra, Unit 0: Variables, Expressions, Integers Page of 3
12 Quadrant II Quadrant III yaxis y x Quadrant I xaxis Quadrant IV origin (0, 0) The coordinate plane consists of infinitely many points called ordered pairs. Each ordered pair is written in the form of (x, y). The first coordinate of the ordered pair corresponds to a value on the xaxis and the second number of the ordered pair corresponds to a value on the yaxis. Our movements in the coordinate plane are similar to movements on the number line. As you move from left to right on the xaxis, the numbers are increasing in value. The numbers are increasing in value on the yaxis as you go up. To find the coordinates of point A in Quadrant I, start from the origin and move 2 units to the right, and up 3 units. Point A in Quadrant I has coordinates (2, 3). B y x 2 C 34 D 56 A To find the coordinates of point B in Quadrant II, start from the origin and move 3 units to the left, and up 4 units. Point B in Quadrant II has coordinates ( 3, 4). To find the coordinates of the point C in Quadrant III, start from the origin and move 4 units to the left, and down 2 units. Point C in Quadrant III has coordinates ( 4, 2). To find the coordinates of the point D in Quadrant IV, start from the origin and move 2 units to the right, and down 5 units. Point D in Quadrant IV has coordinates (2, 5). McDougal Littell, Chapter HS PreAlgebra, Unit 0: Variables, Expressions, Integers Page 2 of 3
13 Here are a few phrases that teachers have used with students to help them remember how to determine the coordinates of a point. Taxi before you take off implies moving right or left before you move up or down. Same with Run before you jump. Notice that there are two other points on the above graph, one point on the xaxis and the other on the yaxis. For the point on the xaxis, you move 3 units to the right and do not move up or down. This point has coordinates of (3, 0). For the point on the yaxis, you do not move left or right, but you do move up 2 units on the yaxis. This point has coordinates of (0, 2). Points on the xaxis will have coordinates of (x, 0) and points on the yaxis will have coordinates of (0, y). The first number in an ordered pair tells you to move left or right along the xaxis. The second number in the ordered pair tells you to move up or down along the yaxis. Plot the following points on the coordinate plane given below.. A (2, 5) 2. B ( 3, 8) 3. C (4, 5) 4. D ( 3, 2) 5. E (0, 4) 6. F (3, 0) y x McDougal Littell, Chapter HS PreAlgebra, Unit 0: Variables, Expressions, Integers Page 3 of 3
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