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Variables and Expressions Lesson 1-1 Pre-Algebra Objectives: 1. To identify variables, numerical expressions, and variable expressions 2. To write expressions for word phrases Variables and Expressions Lesson 1-1 Pre-Algebra Terms: 1. Variable – is a letter that stands for a number 2. Variable Expression – a mathematical phrase that uses variables, numerals, and operation symbols Variables and Expressions Lesson 1-1 Additional Examples Pre-Algebra Identify each expression as a numerical expression or a variable expression. For a variable expression, name the variable. a. 7 - 3 numerical expression b. 4t variable expression; t is the variable. Variables and Expressions Lesson 1-1 Pre-Algebra Additional Examples Write a variable expression for the cost of p pens priced at 29¢ each. Words 29¢ Let p Expression 29 times number of pens = number of pens. • p The variable expression 29 • p, or 29p, describes the cost of p pens. The Order of Operations Lesson 1-2 Pre-Algebra Objectives: 1. To use the order of operations 2. To use grouping symbols The Order of Operations Lesson 1-2 Pre-Algebra Terms: 1.Order of Operations – the order in which you perform operations 2. Simplify – perform the order of operations until you get to the simplest value Tips: make sure to enter an expression into the calculator correctly, otherwise, you will get an incorrect solution because the calculator uses the order of operations. The Order of Operations Lesson 1-2 Additional Examples Simplify 8 – 2 • 2. 8–2•2 8–4 First multiply. 4 Then subtract. Pre-Algebra The Order of Operations Lesson 1-2 Pre-Algebra Additional Examples Simplify 12 ÷ 3 – 1 • 2 + 1. 12 ÷ 3 – 1 • 2 + 1 4 – 2 + 1 2+1 3 Multiply and divide from left to right. Add and subtract from left to right. Add. The Order of Operations Lesson 1-2 Pre-Algebra Additional Examples Simplify 20 – 3[(5 + 2) – 1]. 20 – 3[(5 + 2) – 1] 20 – 3[ 7 – 1] Add within parentheses. 20 – 3 [6] Subtract within brackets. 20 – 18 2 Multiply from left to right. Subtract. Writing and Evaluating Expressions Lesson 1-3 Pre-Algebra Objectives: 1. To evaluate variable expressions 2. To solve problems by evaluating expressions Writing and Evaluating Expressions Lesson 1-3 Pre-Algebra Terms: 1. Evaluate – first replace each variable with its number, the use the order of operations to simplify Writing and Evaluating Expressions Lesson 1-3 Pre-Algebra Additional Examples Evaluate 18 + 2g for g = 3. 18 + 2g = 18 + 2(3) Replace g with 3. = 18 + 6 Multiply. = 24 Add. Writing and Evaluating Expressions Lesson 1-3 Pre-Algebra Additional Examples Evaluate 2ab – c for a = 3, b = 4, and c = 9. 3 2ab – c = 2 • 3 • 4 – 9 3 3 Replace the variables. =2•3•4–3 Work within grouping symbols. =6•4–3 Multiply from left to right. = 24 – 3 Multiply. = 21 Subtract. Writing and Evaluating Expressions Lesson 1-3 Pre-Algebra Additional Examples The Omelet Café buys cartons of 36 eggs. a. Write a variable expression for the number of cartons the Café should buy for x eggs. b. Evaluate the expression for 180 eggs. a. x eggs x 36 b. 180 eggs x 180 = 36 36 = 5 Evaluate for x = 180. Divide. The Omelet Café should buy 5 cartons to get 180 eggs. Writing and Evaluating Expressions Lesson 1-3 Pre-Algebra Additional Examples The One Pizza restaurant makes only one kind of pizza, which costs $16. The delivery charge is $2. Write a variable expression for the cost of having pizzas delivered. Evaluate the expression to find the cost of having two pizzas delivered. Words $16 Let p Expression 16 for each pizza plus $2 delivery charge + 2 = number of pizzas. • p Evaluate the expression for p = 2. 16 • p + 2 = 16 • 2 + 2 = 32 + 2 = 34 It costs $34 to have two pizzas delivered. Integers and Absolute Value Lesson 1-4 Pre-Algebra Objectives: 1. To represent, graph, and order integers 2. To find opposite and absolute values Integers and Absolute Value Lesson 1-4 Pre-Algebra Terms: 1. Opposites – numbers that are the same distance from zero on a number line but in opposite directions 2.Integers – the whole numbers and their opposites 3. Absolute Value – a number’s distance from zero on a number line Integers and Absolute Value Lesson 1-4 Additional Examples Pre-Algebra Write a number to represent the temperature shown by the thermometer. The thermometer shows 2 degrees Celsius below zero, or –2°C. Integers and Absolute Value Lesson 1-4 Additional Examples Graph 2, –2, and –3 on a number line. Order the numbers from least to greatest. The numbers from least to greatest are –3, –2, and 2. Pre-Algebra Integers and Absolute Value Lesson 1-4 Pre-Algebra Additional Examples Use a number line to find |–5| and |5|. |–5| = 5 |5| = 5 Adding Integers Lesson 1-5 Pre-Algebra Objectives: 1. To look at models to add integers 2. To use rules to add integers Adding Integers Lesson 1-5 Pre-Algebra Terms: 1. When you add opposites the sum is zero. So opposites are also called Additive Inverses Adding Integers Lesson 1-5 Pre-Algebra Additional Examples Use tiles to find (–7) + 3. (–7) + 3 Model the sum. –4 Group and remove zero pairs. There are four negative tiles left. (–7) + 3 = – 4 Adding Integers Lesson 1-5 Additional Examples Pre-Algebra From the surface, a diver goes down 20 feet and then comes back up 4 feet. Find –20 + 4 to find where the diver is. Start at 0. To represent –20, move left 20 units. To add positive 4, move right 4 units to –16. –20 + 4 = –16 The diver is 16 feet below the surface. Adding Integers Lesson 1-5 Pre-Algebra Additional Examples Find each sum. a. –20 + (–15) –20 + (–15) = –35 Since both integers are negative, the sum is negative. b. 13 + (–17) |–17| – |13| = 17 – 13 =4 13 + (–17) = – 4 Find the difference of the absolute values. Simplify. Since –17 has the greater absolute value, the sum is negative. Adding Integers Lesson 1-5 Pre-Algebra Additional Examples A player scores 22 points. He then gets a penalty of 30 points. What is the player’s score after the penalty? 22 + (–30) Write an expression. |–30| – |22| = 30 – 22 Find the difference of the absolute values. =8 22 + (–30) = – 8 The player’s score is – 8. Simplify. Since –30 has the greater absolute value, the sum is negative. Adding Integers Lesson 1-5 Pre-Algebra Additional Examples Find –7 + (– 4) + 13 + (–5). –7 + (– 4) + 13 + (–5) –11 + 13 + (–5) 2 + (–5) –3 –7 + (– 4) + 13 + (–5) = –3 Add from left to right. The sum of the two negative integers is negative. |13| – |11| = 2. Since 13 has the greater absolute value, the sum is positive. |5| – |2| = 3. Since –5 has the greater absolute value, the sum is negative. Subtracting Integers Lesson 1-6 Pre-Algebra Objectives: 1. To look at models to subtract integers 2. To use a rule to subtract integers Subtracting Integers Lesson 1-6 Pre-Algebra Additional Examples Find –7 – (–5). Start with 7 negative tiles. Take away 5 negative tiles. There are 2 negative tiles left. –7 – (–5) = –2 Subtracting Integers Lesson 1-6 Pre-Algebra Additional Examples Find 2 – 8. Start with 2 positive tiles. There are not enough positive tiles to take away 8. Add 6 zero pairs. Take away 8 positive tiles. There are 6 negative tiles left. 2 – 8 = –6 Subtracting Integers Lesson 1-6 Pre-Algebra Additional Examples An airplane left Houston, Texas, where the temperature was 42°F. When the airplane landed in Anchorage, Alaska, the temperature was 50°F lower. What was the temperature in Anchorage? 42 – 50 Write an expression. 42 – 50 = 42 + (–50) To subtract 50, add its opposite. = –8 Simplify. The temperature in Anchorage was –8°F. Inductive Reasoning Lesson 1-7 Pre-Algebra Objectives: 1. To write rules for patterns 2. To make predictions and test conjectures Inductive Reasoning Lesson 1-7 Pre-Algebra Terms: 1.Inductive Reasoning – making conclusions based on patterns you observe 2. Conjecture – a conclusion you reach by inductive reasoning 3. Counterexample – an example that proves a statement false Tips: all you need is one counter example to prove a conjecture is not true Inductive Reasoning Lesson 1-7 Additional Examples Pre-Algebra Use inductive reasoning. Make a conjecture about the next figure in the pattern. Then draw the figure. Observation: The circles are rotating counterclockwise within the square. Conjecture: The next figure will have a shaded circle at the top right. Inductive Reasoning Lesson 1-7 Pre-Algebra Additional Examples Write a rule for each number pattern. a. 0, – 4, – 8, –12, . . . Start with 0 and subtract 4 repeatedly. b. 4, – 4, 4, – 4, . . . Alternate 4 and its opposite. c. 1, 2, 4, 8, 10, . . . Start with 1. Alternate multiplying by 2 and adding 2. Inductive Reasoning Lesson 1-7 Pre-Algebra Additional Examples Write a rule for the number pattern 110, 100, 90, 80, . . . Find the next two numbers in the pattern. 110, 100, 90, – 10 – 10 – 10 80 The first number is 110. The next numbers are found by subtracting 10. The rule is Start with 110 and subtract 10 repeatedly. The next two numbers in the pattern are 80 – 10 = 70 and 70 – 10 = 60. Inductive Reasoning Lesson 1-7 Additional Examples A child grows an inch a year for three years in a row. Is it a reasonable conjecture that this child will grow an inch in the year 2015? No; children grow at an uneven rate, and eventually they stop growing. Pre-Algebra Inductive Reasoning Lesson 1-7 Additional Examples Is each conjecture correct or incorrect? If it is incorrect, give a counterexample. a. Every triangle has three sides of equal length. The conjecture is incorrect. The figure below is a triangle, but it does not have three equal sides. b. The opposite of a number is negative. The conjecture is incorrect. The opposite of –2 is 2. Pre-Algebra Inductive Reasoning Lesson 1-7 Additional Examples (continued) c. The next figure in the pattern below has 16 dots. The conjecture is correct. The diagram below shows the next figure in the pattern. Pre-Algebra Problem Solving Strategy: Look for a Pattern Lesson 1-8 Pre-Algebra Objectives: 1. To find number patterns Problem Solving Strategy: Look for a Pattern Lesson 1-8 Pre-Algebra Tips: there are many ways to find a solution to a problem that involves a pattern, such as tables or a tree diagram Problem Solving Strategy: Look for a Pattern Lesson 1-8 Pre-Algebra Additional Examples Each student on a committee of five students shakes hands with every other committee member. How many handshakes will there be in all? The pattern is to add the number of new handshakes to the number of handshakes already made. 4 the number of handshakes by 1 student 4+3=7 the number of handshakes by 2 students Problem Solving Strategy: Look for a Pattern Lesson 1-8 Pre-Algebra Additional Examples (continued) Make a table to extend the pattern to 5 students. Student 1 2 3 4 5 Number of original handshakes 4 3 2 1 0 Total number of handshakes 4 4+3 =7 There will be 10 handshakes in all. 7+2 =9 9+1 = 10 10 + 0 = 10 Multiplying and Dividing Integers Lesson 1-9 Pre-Algebra Objectives: 1. To multiply integers using repeated addition, patterns, and rules 2. To divide integers using rules Multiplying and Dividing Integers Lesson 1-9 Pre-Algebra Tips: when only multiplying or dividing integers, you can count the number of negative signs to see if your answer will be negative or positive. Even number of negative signs the answer is positive and an odd number of negative signs the answer will be negative. Multiplying and Dividing Integers Lesson 1-9 Additional Examples A diver is descending from the surface of the water at a rate of 5 ft/s. Write an expression with repeated addition to show how far the diver is from the surface of the water after four seconds. Use a number line to show repeated addition. 4 (–5) = (–5) + (–5) + (–5) + (–5) = –20 The diver is 20 feet below the surface of the water. Pre-Algebra Multiplying and Dividing Integers Lesson 1-9 Pre-Algebra Additional Examples Use a pattern to find each product. a. –2(7) 2(7) = 14 Start with products you know. 1(7) = 7 0(7) = 0 –1(7) = –7 –2(7) = –14 Continue the pattern. Multiplying and Dividing Integers Lesson 1-9 Pre-Algebra Additional Examples (continued) b. –2(–7) 2(–7) = –14 Start with products you know. 1(–7) = –7 0(–7) = 0 –1(–7) = 7 –2(–7) = 14 Continue the pattern. Multiplying and Dividing Integers Lesson 1-9 Pre-Algebra Additional Examples Multiply 6(–2)(–3). 6(–2)(–3) = (–12)(–3) = 36 Multiply from left to right. The product of a positive integer and a negative integer is negative. Multiply. The product of two negative integers is positive. Multiplying and Dividing Integers Lesson 1-9 Pre-Algebra Additional Examples Use the table to find the average of the differences in the values of a Canadian dollar and a U.S. dollar for 1999–2002. –33 + (–33) + (–35) + (–36) 4 Write an expression for the average. Multiplying and Dividing Integers Lesson 1-9 Pre-Algebra Additional Examples (continued) = –137 4 = –34.25 Use the order of operations. The fraction bar acts as a grouping symbol. The quotient of a negative integer and a positive integer is negative. For 1999–2002, the average difference was –34¢. The Coordinate Plane Lesson 1-10 Pre-Algebra Objectives: 1. To name coordinates and quadrants in the coordinate plane 2. To graph points in the coordinate plane The Coordinate Plane Lesson 1-10 Pre-Algebra Terms: 1.Coordinate Plane – formed by the intersection of two number lines 2. x-axis – the horizontal number line 3.y-axis – the vertical number line 4. Quadrants – the x and y axes divide the coordinate plane into 4 sections 5.Orgin – the point where the x and y axes intersect 6. Ordered Pair – gives the coordinates (x , y) and location of a point 7.x-coordinate – shows the position left or right of the y-axis 8. y-coordinate – shows the position above or below the x-axis The Coordinate Plane Lesson 1-10 Pre-Algebra Additional Examples Write the coordinates of point G. In which quadrant is point G located? Point G is located 2 units to the left of the y-axis. So the x-coordinate is –2. The point is 3 units below the x-axis. So the y-coordinate is –3. The coordinates of point G are (–2, –3). Point G is located in Quadrant III. The Coordinate Plane Lesson 1-10 Additional Examples Graph point M(–3, 3). Pre-Algebra