Pre Algebra, Unit 1: Variables, Expression, and Integers

Syllabus Objectives (1.1) Students will evaluate variable and numerical expressions using the order of operations. (1.2) Students will compare integers. (1.3) Students will order integers (1.4) Students will add integers. (1.5) Students will subtract integers. (1.6) Students will multiply integers. (1.7) Students will divide integers. (1.8) Students will identify point in a coordinate plane. (1.9) Students will plot points in a coordinate plane Evaluating Algebraic Expressions Variable a letter or symbol that represents a number. Examples a, b, c, _,,,... Variable (algebraic) expression an expression that consists of numbers, variables, and operations. Examples - a+ b,2x+ 7 y, 5+ 3( m 2)2 1 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) 1. 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. Pre-Algebra Unit 1: Variables, Expressions, and Integers Page 1 of 15

Example1 Evauate a+ bwhen a= 5and b= 4. a+ b = 5+ 4 = 9 Example 2 Evauate 2x+ 7y when x= 3and y = 6 2x+ 7 y = 2(3) + 7(6) = 6+ 42 = 48 Example 3 Evauate 5+ 3( m 2)2 1when m= 6 5+ 3( m 2)2 1= 5+ 3(6 2)2 1 = 5+ 3(4)2 1 = 5 + (12)2 1 = 5+ 24 1 = 29 1 = 28 Exponentials Exponential base number times itself exponent number of times. 2 3 is read 2 to the third power or 2 cubed. 2 is the base and 3 is the exponent. 4 4 4 2 2222 16 3 3333 81 4 4444 256 = i i i = = i i i = = i i i = 3 3 3 2 2 2 2 8 3 3 3 3 27 4 4 4 4 64 = i i = = i i = = i i = 2 2 2 2 2 2 4 3 3 3 9 4 4 4 16 = i = = i = = i = Do you see a pattern that would allow you to determine the values of exponential expressions with exponents of 1 and/or 0? 2 1 =? 3 1 =? 4 1 =? and 2 0 =? 3 0 =? 4 0 =? 2 1 = 2 3 1 = 3 4 1 = 4 and 2 0 = 1 3 0 = 1 4 0 = 1 Any number to the power of 1 = the number. n 1 = n Any number to the power of 0 = 1. n 0 = 1 Also notice that if there is no exponent, the exponent is always 1. Pre-Algebra Unit 1: Variables, Expressions, and Integers Page 2 of 15

Examples - Pre Algebra, Unit 1: Variables, Expression, and Integers To write81 in exponential form: 81= 9i9 = 3333 i i i = 3 4 To write125 in exponential form: 125= 25i5 = 5i5i5 = 5 3 Integers Integers are defined as the negative and positive whole numbers and 0. Integers = {... 3, 2, 1,0,1, 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. Compare the following. Use > or <. Ex. 1) 24 5 Ex. 2) 13 2 Ex. 3) 8 12 24 is to the right of 5 2 is to the right of -13 8 is to the right of 12 24 > 5 13 < 2 8 > 12 Ex. 4) List the following integers in order from least to greatest. 7, 12, 5, 0, 3, 15, 20, 2, 2 20, 12, 5, 2, 0, 2, 3, 7, 15 The Absolute Value of a number is the distance from zero (0) on a number line without regard to the sign. The symbol for Absolute Value is and is read as the absolute value of. ex.1) 7 = 7 ex.2) 5 = 5 ex.3) 0 = 0 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. Pre-Algebra Unit 1: Variables, Expressions, and Integers Page 3 of 15

Example - ( + 3) + ( + 2) or ( 3) ( 2) Sign of value Sign of value Sign of value Sign of value Sign of operation Sign of operation Notice that the signs of value and the sign of operation are identical. Please note that numbers written without a sign are assumed to be positive; +3 = 3 and +2 = 2. 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 tells 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 problem above, (+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 1: 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 start with (-3) + (-2). We start at 0 on the number line and move in a negative direction 3 units. From the point -3, we move another 2 units in a negative direction. Our ending point is the sum of (-3) + (-2) = -5. Pre-Algebra Unit 1: Variables, Expressions, and Integers Page 4 of 15

Rule 2: A negative number plus a negative number is equal to a negative number. Try a few sample problems with the students. You might want to 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). 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. We will answer the why a little later. Our next example is ( 2) + (+5); We start at 0 on the number line and move in a 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. Pre-Algebra Unit 1: Variables, Expressions, and Integers Page 5 of 15

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 absolute values and use the sign of the number with the greatest absolute value. additional examples - ( 12) + (+12) = 0 ( 12) + (+16) = +4 (+7) + ( 5) = +2 ( 7) + (+5) = 2 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; Ex. 1 ( 2) (+8) = We will use zero as our starting point just like we did in addition and move to the left, 2 units to a 2. We want to subtract from 2, +8. 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. Ex. 2 ( 1) ( 5) = ( 2) (+8) = 10 We start at zero and move 1 unit in a negative direction to 1. The second number is 5, but because we are subtracting, we move in a positive direction 5 units. ( 1) ( 5) = +4 Pre-Algebra Unit 1: Variables, Expressions, and Integers Page 6 of 15

Ex. 3 6 (+13) = We start at zero and move 6 units in a positive direction to +6. The second number is 13, but because we are subtracting, we move in a negative direction 13 units. 6 (+13) = 7 Let s go back and look at the above examples on the number line. Ex. 1, ( 2) (+8) = 10 looks like the addition problem ( 2) + ( 8) = 10 Ex. 2, ( 1) ( 5) = +4 looks like the addition problem ( 1) + (+5) = +4 Ex. 3, 6 (+13) = 7 looks like the addition problem 6 + ( 13) = 7 A close examination of our examples and additional practice problems leads us to a rule for subtraction of integers. Rule 4: When subtracting integers, change the sign of the problem from subtraction to addition, and change the sign of the subtrahend (second number) and use rules for Addition of Integers. Pre-Algebra Unit 1: Variables, Expressions, and Integers Page 7 of 15

Multiplying Integers Definition - A mathematical operation that is an abbreviated process of adding an integer to itself a specified number of times. For example; 5 x 3 = 5 + 5 + 5 = 15 or 5 x 3 = 3 + 3 + 3 + 3 + 3 = 15 A couple of points to be made here; 1. Review of the commutative property of Multiplication 5 x 3 = 3 x 5 2. The above is the product of positive numbers Positive x Positive = Positive [P x P = P] So, let s take a look at the product of a negative number times a positive number. By our definition of multiplication, 5 x 3 = 5 + 5 + -5 = 15 or 5 x 3 = 3 + 3 + 3 + 3 + 3 = 15 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. 4 x 3 = 12 5 x 3 = 15 8 x 3 = 24 4 x 2 = 8 5 x 2 = 10 8 x 2 = 16 4 x 1 = 4 5 x 1 = 5 8 x 1 = 8 4 x 0 = 0 5 x 0 = 0 8 x 0 = 0 4 x 1 = 4 5 x 1 = 5 8 x 1 = 8 4 x 2 = 8 5 x 2 = 10 8 x 2 = 16 4 x 3 = 12 5 x 3 = 15 8 x 3 = 24 4 x 4 = 16 5 x 4 = 20 8 x 4 = 32 By now, as you work with your students, you will have done enough problems to establish the following rules of multiplication; Positive x Positive = Positive (P x P = P) Positive x Negative = Negative (P x N = N) Let s continue the pattern development to show Negative x Positive = Negative 4 x 3 = 12 5 x 3 = 15 8 x 3 = 24 4 x 2 = 8 5 x 2 = 10 8 x 2 = 16 4 x 1 = 4 5 x 1 = 5 8 x 1 = 8 4 x 0 = 0 5 x 0 = 0 8 x 0 = 0 Pre-Algebra Unit 1: Variables, Expressions, and Integers Page 8 of 15

Be prepared to discuss the identity property and multiplication by zero. You are now ready to discuss; Negative number x Negative number = Positive number 4 x 3 = 12 5 x 3 = 15 8 x 3 = 24 4 x 2 = 8 5 x 2 = 10 8 x 2 = 16 4 x 1 = 4 5 x 1 = 5 8 x 1 = 8 4 x 0 = 0 5 x 0 = 0 8 x 0 = 0 4 x 1 = 4 5 x 1 = 5 8 x 1 = 8 4 x 2 = 8 5 x 2 = 10 8 x 2 = 16 4 x 3 = 12 5 x 3 = 15 8 x 3 = 24 4 x 4 = 16 5 x 4 = 20 8 x 4 = 32 Rules for Multiplication Positive x Positive = Positive (P x P = P) Positive x Negative = Negative (P x N = N) Negative x Positive = Negative (N x P = N) Negative x Negative = Positive (N x N = P) 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 doable. 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. 12 4 = 3 15 5 = 3 24 8 = 3 8 4= 2 10 5= 2 16 8= 2 4 4= 1 5 5= 1 8 8= 1 0 4= 0 0 5= 0 0 8= 0 4 4= 1 5 5= 1 8 8= 1 8 4= 2 10 5= 2 16 8= 2 12 4 = 3 15 5 = 3 24 8 = 3 16 4 = 4 20 5 = 4 32 8 = 4 The patterns developed in the above examples give the students two rules for division; Positive Positive = Positive and Negative Positive = Negative Pre-Algebra Unit 1: Variables, Expressions, and Integers Page 9 of 15

Making a few slight modifications, we can develop patterns to show the last two rules of division 8 4= 2 10 5= 2 16 8= 2 4 4= 1 5 5= 1 8 8= 1 Negative Negative = Positive (N N = P) Pi P= P P P= P Pi N= N P N= N Ni P= N N P= N Ni N= P N N= P 12 4 = 3 15 5 = 3 24 8 = 3 0 4= 0 0 5= 0 0 8= 0 4 4= 1 5 5= 1 8 8= 1 8 4= 2 10 5= 2 16 8= 2 12 4 = 3 15 5 = 3 24 8 = 3 16 4 = 4 20 5 = 4 32 8 = 4 Positive Negative = Negative(P N = N) and Now is the time to compare the rules for multiplication and division; Further investigations and discussion will lead us to two rules which can be used with both multiplication and division; Rule 5: Like signs equal a positive. Rule 6: Unlike signs equal a negative. To review our rules; Addition Rule 1: 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. Pre-Algebra Unit 1: Variables, Expressions, and Integers Page 10 of 15

Subtraction Rule 4: When subtracting integers, change the sign of the problem from subtraction to addition, and change the sign of the subtrahend (second number) and use rules 1, 2, or 3 for Addition of Integers. Multiplication and Division Rule 5: Like signs equal a positive. Rule 6: Unlike signs equal a negative. When working with these rules, we must understand the rules work for only two numbers at a time. In other words, if I asked you to simplify ( 3) ( 4) ( 5), the answer would be 60. The reason is ( 3) 4 ( )=+12, then a ( +12) ( 5)= 60 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 x sign because in algebra it might be confused with the variable x. 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, it still equals + 13 or 8 + 5 = 13. Example: Example: ( 8)+ ( 5) can be written without the sign of operation 8 5, it still equals 13 or 8 5 = 13 ( 8)+ ( +5), can be written without the sign of operation 8 + 5, it still equals 3 or 8+ 5= 3. For ease, we have eliminated the x 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. Pre-Algebra Unit 1: Variables, Expressions, and Integers Page 11 of 15

A subtraction problem will always have an additional sign, the sign of operation. For ( ) example, 12 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, 12 + 5, and 9 12 are all samples of addition problems. Naturally, you would have to use the rule that applies. The Coordinate Plane A coordinate plane is formed by the intersection of a horizontal number line called the x-axis and a vertical number line called the y-axis. The x-axis and y-axis meet or intersect at a point called the origin. The coordinate plane is divided into six parts; the x-axis, the y-axis, Quadrant I, Quadrant II, Quadrant III, and Quadrant IV. Quadrant II Qu adrant I y 6 5 4 3 2 1-6 -5-4 -3-2 -1 1 2 3 4 5 6-1 -2-3 -4-5 -6 x Q uadrant III Quadrant IV Pre-Algebra Unit 1: Variables, Expressions, and Integers Page 12 of 15

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 x-axis and the second number of the ordered pair corresponds to a value on the y-axis. Our movements in the coordinate plane are similar to movements on the number line. As you move from left to right on the x-axis, the numbers are increasing in value. The numbers are increasing in value on the y-axis as you go up. y 6 5 4 3 2 1-6 -5-4 -3-2 -1 1 2 3 4 5 6-1 -2-3 -4-5 -6 x To find the coordinates of the red point in Quadrant I, start from the origin and move 2 units to the right, and up 3 units. The red point in Quadrant I has coordinates (2, 3). To find the coordinates of the black point in Quadrant II, start from the origin and move 3 units to the left, and up 4 units. The black point in Quadrant II has coordinates ( 3, 4). To find the coordinates of the blue point in Quadrant III, start from the origin and move 4 units to the left, and down 2 units. The blue point in Quadrant III has coordinates ( 4, 2). To find the coordinates of the pink point in Quadrant IV, start from the origin and move 2 units to the right, and down 5 units. The pink point in Quadrant IV has coordinates (2, 5). Notice that there are two other points on the above graph, one point on the x-axis and the other on the y-axis. For the point on the x-axis, 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 y-axis, you do not move left or right, but you do move up 2 units on the y-axis. This point has coordinates of (0, 2). Points on the x-axis will have coordinates of (x, 0) and points on the y-axis will have coordinates of (0, y). Pre-Algebra Unit 1: Variables, Expressions, and Integers Page 13 of 15

The first number in an ordered pair tells you to move left or right along the x-axis. The second number in the ordered pair tells you to move up or down along the y-axis. Pre-Algebra Unit 1: Variables, Expressions, and Integers Page 14 of 15

Plot the following points on the coordinate plane below; 1. A (2,5) 2. B ( 3,8) 3. C (4, 5) 4. D ( 3, 2) 5. E (0, 4) 6. F (3, 0) y 6 5 4 3 2 1-6 -5-4 -3-2 -1 1 2 3 4 5 6-1 -2-3 -4-5 -6 x Pre-Algebra Unit 1: Variables, Expressions, and Integers Page 15 of 15