Table of Contents Teacher Pages 6-CORE2 General Information 0 6-CORE2.1 6-CORE 2.2 Algebraic Expressions and Equations Solve problems involving money and decimals. Use variables in expressions and equations. Use the distributive property and the conventions for order of operations to evaluate expressions. Find values that make equations true. Write verbal statements symbolically. Beginning Inequalities Solve problems involving money and decimals. Graph solutions to inequalities and recognize inequalities that have infinitely many solutions. Determine whether inequalities are true or false. Find a value from a specified set that makes an inequality true. Find and write solutions to inequalities. 6- CORE 2.3 Saving for a Purchase Analyze the relationship between dependent and independent variables. Use tables, graphs, equations, and words to solve problems. 6-CORE2 TEACHER PAGES GRADE 6: MATHEMATICS COMMON CORE SUPPLEMENT EXPRESSIONS, EQUATIONS, AND INEQUALITIES Student Pages 12 1 18 10 22 16 6-CORE 2.4 Vocabulary, Skill Builders, and Review 26 6-CORE2 TP
General Information GENERAL INFORMATION PACING PLAN SUGGESTIONS TRADITIONAL MATH PERIOD Day Lesson Review/Practice Notes 1 (6-CORE2.1) SP0, 1-4 SP27 2 (6-CORE2.1) SP0, 5-9 SP28-29 3 (6-CORE2.2) SP0, 10-12 SP30 4 (6-CORE2.2) SP0, 13-15 SP31 5 (6-CORE2.3) SP0, 16-21 SP32 6 (6-CORE2.3) SP0, 22-25 SP33 7 Catch up, Task 3 SP26, 34, 36-37 8 (6-CORE2) Quiz 5-6 BLOCK SCHEDULE Day Lesson Review/Practice Notes 1 (6-CORE2.1) SP0, 1-7 SP27-29 2 3 (6-CORE2.1) SP0, 8-9 (6-CORE2.2) SP0, 10-14 (6-CORE2.2) SP0, 15 (6-CORE2.3) SP0, 16-21 SP30-31 SP32-33 4 (6-CORE2.3) SP0, 22-25 SP26, 34, 36-37 5 Catch up, Task 3 6 (6-CORE2) Quiz 5-6 6-CORE2 TP0
General Information COMMON CORE STATE STANDARDS MATHEMATICS 6.EE.2a 6.EE.2b 6.EE.3 6.EE.4 6.EE.5 6.EE.6 6.EE.8 6.EE.9 MP1 MP2 MP3 MP4 MP5 MP6 STANDARDS FOR MATHEMATICAL CONTENT Write, read, and evaluate expressions in which letters stand for numbers: Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation Subtract y from 5 as 5 y. Write, read, and evaluate expressions in which letters stand for numbers: Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms. Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) t o produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y. Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for. Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams. Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time. STANDARDS FOR MATHEMATICAL PRACTICE Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. MP7 MP8 Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. 6-CORE2 TP1
General Information Assessments A5: 6-CORE2 Quiz Form A A6: 6-CORE2 Quiz Form B Reproducibles and Tasks R4: Pizza Shop Menu T4: My Menu Vocabulary, Skill Builders, and Review SP26: Focus on Vocabulary SP27-33: Skill Builder 1-7 SP34: Test Preparation SP36: Knowledge Check SP37: Home-School Connection Options for a Substitute Teacher Any time: SP27 After 6-CORE2.1: SP28-29 After 6-CORE2.2: SP30-31 After 6-CORE2.4: SP26, 32-35 PACKET PLANNING INFORMATION Materials Calculators (optional) Menus (optional) Index cards (optional) Colored pencils or highlighters (optional) Rulers Prepare Ahead For 6-CORE2.1 Collect some take-out menus for students to use for the homework, or ask students to bring in their favorite take-out menu. General Tip: Stay in contact with the parents and support staff for students with special needs. Giving them instructional materials ahead of time may allow them to help students be more successful with these lessons, which require interpretation of stories and language. 6-CORE2 TP2
General Information boundary point of a solution set coefficient constant term dependent variable distributive property equality TEACHER CONTENT INFORMATION WORD BANK A boundary point of a solution set is a point for which any segment surrounding it on the number line contains both solutions and non-solutions. If the solution set is an interval, the boundary points of the solution set are the endpoints of the interval. Example 1: Example 2: The boundary point for 2 + x > 3 (or x > 1) is x = 1. In this case the boundary is NOT part of the solution set. 1 The boundary point for 3x 6 (or x 2) is 2. In this case the boundary IS part of the solution set. A coefficient is a number or constant factor in a term of an algebraic expression. 2 Example: In the expression 3x + 5, 3 is the coefficient of the linear term 3x, and 5 is the constant term. A constant term in an algebraic expression is a term that has a fixed numerical value. Example: In the expression 5 + 12x 2 + 7, 5 and 7 are constant terms. A dependent variable is a variable that represents output values of a function. See function. Example: For the function y = x 2 + 3x +1, y is the dependent variable and x is the independent variable. When a value is assigned to x, the value of y is completely. The distributive property states that a(b + c) = ab + ac and (b + c)a = ba + ca for any three numbers a, b, and c. This property relates two operations (multiplication and addition). It is called the distributive property because it distributes the factor outside the parentheses over the two terms within the parentheses. Example: 3(4 + 5) = 3(4) + 3(5) and (4 + 5)8 = 4(8) + 5(8) An equality is a mathematical statement that asserts that two numbers or expressions are equal. Examples: 5 + 3 = 4 + 4 and x + 1 = 7 are equality statements. 6-CORE2 TP3
General Information equation evaluate an expression expression function independent variable inequality WORD BANK (Continued) An equation is a mathematical statement that asserts the equality of two expressions. When the equation involves variables, a solution to the equation consists of values for the variables which, when substituted, make the equation true. Example 1: Example 2: 5 + 6 = 14 3 is an equation that involves only numbers. 10 + x = 18 is an equation that involves numbers and a variable; the value for x must be 8 to make this equation true. Evaluate refers to finding a numerical value. To evaluate an expression, replace each variable in the expression with a value and then calculate the value of the expression. Examples: To evaluate the expression 2x + 5 when x = 10, we calculate 2x + 5 = 2(10) + 5 = 20 + 5 = 25. To evaluate the expression u 2 v when u = 5 and v = 6, we calculate u 2 v= (5) 2 6 =25 6 = 19. A mathematical expression is a combination of numbers, variables, and operation symbols. When values are assigned to all the variables, an expression represents a number. Examples: Some mathematical expressions are 7x, a + b, 4v w, and 19. A function is a rule that assigns to each input value a unique output value. Example: Example: The function y = 2x + 3 assigns to each input value x the output value y = 2x + 3. The input variable x is the independent variable, and the output variable y is the dependent variable. In this example, if the input value is 7, then the output value is 2(7) + 3 = 17. The function y = x 2 + 1 assigns to the input value x = 2 the output value y = 2 2 + 1 = 5. An independent variable is a variable that represents input values of a function. See function. An inequality is a mathematical statement that asserts the relative size or order of two objects. When the expressions involve variables, a solution to the inequality consists of values for the variables which, when substituted, make the inequality true. Example 1: Example 2: 5 > 3 is an inequality. x + 3 > 4 is an inequality. Its solution (which is also an inequality) is x > 1. 6-CORE2 TP4
General Information linear function order of operations simplify an expression slope of a line WORD BANK (Continued) A linear function (in variables x and y) is a function that can be expressed in the form y = mx + b. The graph of y = mx + b is a straight line with slope m and y-intercept b. Example: The graph of the linear function y = 2x + 1 is a straight line with slope m = 2 and y-intercept b = 1. An order of operations is a convention, or set of rules, that specifies in what order to perform the operations in an expression. The standard order of operations is as follows: 1. Do the operations in parentheses first. (Use rules 2-4 inside the parentheses.) 2. Calculate all the expressions with exponents. 3. Multiply and divide in order from left to right. 4. Add and subtract in order from left to right. In particular, multiplications and divisions are carried out before additions and subtractions. Example: 2 2 2 3 + ( 6 2 1) 3 + ( 12 1) 3 + ( 11) 9 + ( 11) 20 5 5 5 5 5 = = = = = 4 Simplify refers to converting a numerical or variable expression to a simpler form. A variable expression might be simplified by combining like terms. A fraction might be simplified by dividing numerator and denominator by a common divisor. Example 1: 2x + 6 + 5x + 3 = 7x + 9 Example 2: 8 2 = 12 3 The slope of a line is the vertical change (change in the y-value) per unit of horizontal change (change in the x-value). The slope is sometimes described as the ratio of the rise to the run. y Example: The slope of the line through (1, 2) and (4, 10) is 8 3 : rise (difference in y ) (10 2) 8 run (difference in x) (4 1) 3 slope = = = = 4 2 0-2 -4 y = 2x + 1 2 4 6 8 10 8 6 4 2 0 2 4 6 8 x y x 6-CORE2 TP5
General Information slope-intercept form solve an equation terms variable y-intercept WORD BANK (Continued) The slope-intercept form of the equation of a line is the equation y = mx + b, where m is the slope of the line and b is the y-intercept of the line. Example: The equation y = 2x + 3 determines a line with slope is 2 and y-intercept 3. Solve an equation refers to finding all values for the variables in the equation that, when substituted, make the equation true. Values that make an equation true are called solutions to the equation. Example : Solve the equation 2x = 6 x = 3, so 3 is the solution to the equation. The terms in a mathematical expression involving addition (or subtraction) are the quantities being added (or subtracted). Terms that are constant multiples of the same quantity are referred to as like terms. Example: Expression: 2x + 6 + 3x + 6 Terms: 2x, 6, 3x, and 5 Like terms: 2x and 3x are like terms because they have the same variable part (x). 6 and 5 are like terms because they are both constants (or their variable part can be considered as x 0 ). A variable is a quantity whose value has not been specified. Variables are used in many different ways. They may refer to functions, to quantities that vary in a relationship, or to unknown quantities in equations and inequalities. Examples: In the equation d = rt, the quantities d, r, and t are variables. In the equation 2x + 6 = 4x, the variable x may be referred to as the unknown. The y-intercept of a line is the y-coordinate of the point at which the line crosses the y-axis. It is the value of y that corresponds to x = 0. y Example: The y-intercept of the line y = 3x + 6 is 6. (0, 6) 6 If x = 0, then y = 6. 4-2 2 0 x 6-CORE2 TP6
General Information MATH NOTES MN1: Evaluating an Expression (6-CORE2.1, 2.2) To evaluate an expression, substitute each variable in the expression with a number, and then calculate the value of the expression. Example: To evaluate 6 + 3x at x = 2, substitute 2 for x and calculate: 6 + 3(2) = 12. MN2: Variables in Algebra (6-CORE2.1, 2.2) Loosely speaking, variables are quantities that can vary. They have many different uses in mathematics. Variables are represented by letters or symbols. The introduction of literal notation together with the rules of arithmetic makes algebra a powerful tool. The three most important ways that variables appear in algebra are the following. An unknown quantity in an equation: In this case, the equation is valid only for specific value(s) of the unknown, and we are challenged to find those value(s). Quantities that vary in a relationship: In this case, there is always more than one variable in the equation. The equation might be a formula, it might be a rule describing a function, or it might be a more complicated relation. Quantities that generalize rules of arithmetic, quantities in identities, or inequalities that flow from rules of arithmetic: In this case, there may be one or more variables. The identity or inequality is true, except possibly for certain specific values as noted. x + 4 = 9 5x = 20 x 2 + 2x < 1 (in this case, unknown quantities in an inequality) Formula: P = 2l + 2w, A = s 2 Function: y = 5x, y = x + 3 More complicated relation: y 2 = x 3 + 4x + 1 Commutative property of addition: x + y = y + x Distributive property: x(y + z) = xy + xz Multiplicative identity: x( 1 x ) = 1 for x 0 Identities: 5(x + 2) = 5x + 10 Inequalities: x < x + 1 6-CORE2 TP7
General Information MATH NOTES (Continued) MN3: Order of Operations (6-CORE2.1, 2.2) There are many necessary mathematical conventions that enable us to communicate about common situations. For example, when using the coordinate plane, an agreed-upon convention is that we will call the horizontal axis the x-axis, and the vertical axis the y-axis. This allows for easier, common communication. It may be beneficial to have a discussion with students about what it means for something to be a convention in mathematics. The rules we follow for evaluating arithmetic expressions are called the order of operations. These rules are also a matter of convention, though some of the rules are quite natural. For instance, the rule that we perform multiplications before we perform additions is quite natural. However, the left-to-right rule is more arbitrary. To see that this rule affects the result, consider the following example: You purchase 2 bottles of water for $1.00 each and 3 bags of peanuts for $3.00 each. Write an expression for the total cost, and simplify the expression. Expression: 2 1 + 3 3 Simplification: 2 + 9 = 11 (total cost is $11.00) To the uninitiated, performing the operations in order from left to right might make sense, especially considering that we read English from left to right. However, if we evaluate the operations in the expression from right to left, as is natural in some languages, we obtain a different result: Expression: 2 (1) + 3 (3) Evaluation left to right: ( ( 2 1) + 3 ) 3 = ( 2 + 3 ) 3 = 5 3 = 15 (total cost is $15.00) In this case, if we operate from left to right instead of following the order-of-operations convention, we would pay $15 instead of $11 for the water and peanuts. MN4: Equality and Inequality (6-CORE2.1, 2.2) Here are some symbols for equality and inequality. In some sense, they behave as mathematical verbs because they connect two phrases (expressions) to make a sentence. Symbol Word phrase Examples = is equal to 5 + 3 = 2 + 6 is less than or equal to is greater than or equal to 3 1 10 + 5 3 1 2 + 0-5 6 14-5 6 11 < is less than 0.33 < > is greater than.55505 >.55 1 3 6-CORE2 TP8
General Information MATH NOTES (Continued) MN5: Independent and Dependent Variables (6-CORE2.3) Common Core State Standard 6.EE.9 states, Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable We view this standard as an informal introduction to functions, where each input (independent variable) is assigned to exactly one output (dependent variable). For linear functions (the type of relationships studied here), the selection of the independent variable is arbitrary and generally depends on context. Consider this situation: Suppose a text message costs $0.05 Let n = the number of text messages sent. Let C = the cost of a text message bill If we know the number of text messages sent, then we can determine the cost of the text message bill as C = 0.05n. In this case n is the natural independent variable and C is the dependent variable because the total cost depends on the number of text messages sent. If we are operating within a budget and we have a limit C on what we can spend for text C messages, then we can determine the number of text messages we can send as n =. In this 005. case C is the natural independent variable and n is the dependent variable because the number of text messages depends on the total cost. MN6: Steeper Lines Have Greater Slope (6-CORE2.3) Though slope is not formally presented in this lesson, important ideas relating to slope are addressed in context to lay the foundation for slope. rise vertical change The slope of a line is the ratio of the rise to the run ( run horizontal change ) =. The slope of a horizontal line is 0. A vertical line has no slope. In other words, the slope of a vertical line is undefined. The variable m (think: mountain) is typically used to represent slope. If (a, b) and (c, d) are two points on a line, then the slope between these two points can be calculated as: difference between y-coordinates b d. difference between x-coordinates a c m = = Example: The slope of the line that passes through the points (3, 5) and (1, 9) is 5 9-4 3 1 2 m = = = -2. Geometrically, slopes with greater absolute values are represented by a steeper lines. In this lesson, the amount of money each girl saves each month gives a contextual representation of slope. 6-CORE2 TP9
General Information MP1 MP3 MP4 MP7 MP8 TEACHING NOTES TN1: Standards for Mathematical Practice (6-CORE2.1, 2.2, 2.3) Make sense of problems and persevere in solving them. Students have the opportunity to make sense of information given when saving for a purchase and solve problems using multiple representations. Construct viable arguments and critique the reasoning of others. Students pose potential solutions to equations and inequalities and discuss the reasonableness and completeness of these answers. Model with mathematics. Students are informally introduced to the linear function in 6-CORE2.3 as a reasonable mathematical formula to model that amount of money saved after a given number of months. Look for and make use of structure. Students record money saved in organized tables and create an equation based on patterns in the table. Look for and express regularity in repeated reasoning. Students observe patterns in tables and write explicit rules of the form y = mx + b. TN2: Caution! Please Be Careful with Aunt Sally (6-CORE2.1, 2.2) The statement Please Excuse My Dear Aunt Sally is commonly used to help students remember the correct order of operations because the first letter of each word roughly corresponds to the conventions (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction). However, caution should be used in presenting this mnemonic because the operations are not always performed in that order. According to the conventions, multiplication and division have equal status in the hierarchy, as do addition and subtraction. For instance, when there are only multiplications and divisions, we perform them in order from left to right. Example: Right: 8 4 2 = 2 2 = 4 Wrong! 8 4 2 = 8 8 = 1 In this case, we throw My Dear Aunt Sally overboard and perform the division before the multiplication. In practice, one should avoid using an ambiguous expression such as 8 4 2. Instead, use parentheses to make the order of operations clear, in this case (8 4) 2 or 8 4 2. 6-CORE2 TP10
General Information TEACHING NOTES (Continued) TN3: Learning the Language of Algebra (6-CORE2.1, 2.2) It takes time for students to learn the specialized system of notation and rules used in algebra because students may be confused by apparent inconsistencies in notation. The following examples illustrate some different meanings when symbols or numbers are adjacent to one another. 54 means the sum of five tens and four ones, or 5(10) + 4 5 1 2 means the sum of five and one-half, or 5 + 1 2 5x means the product of five and x or 5 groups of x, or x + x + x + x + x = 5 x. Even when students are comfortable with arithmetic, the language of algebra can be confusing to them. Teachers may want to include extra parentheses within expressions to make meaning clear. Practice, combined with explicit explanations, will help students learn the language of algebra. TN4: Similar Phrases with Different Meanings (6-CORE2.1, 2.2) When students see a word problem, they may find it useful to translate words into symbols. While this can be a useful tool, encourage students to use caution when applying this technique. Example 1: Example 2: Is less than vs. less than 4 is less than 10 translates to the statement 4 < 10 4 less than 10 translates to the expression 10 4 Is greater than vs. greater than 7 is greater than 2 + 3 translates to the statement 7 > 2 + 3 7 greater than 2 + 3 translates to the expression (2 + 3) + 7 One way students may be able to distinguish between these ideas is to think about the parts of speech in English. Is greater than behaves like a mathematical verb. Greater than is a phrase. In English, we connect phrases with verbs to make sentences. The same is true in mathematics. TN5: Avoiding Learned Helplessness (6-CORE2.1, 2.2, 2.3) When problem solving, students may want immediate reassurance from an expert that their work is correct. We encourage instructors to refrain from providing answers too quickly. For students to become independent, confident thinkers, they must be given adequate time to work through a problem and sufficient opportunities to resolve misconceptions and unclear concepts through discussion with peers. Teachers unwittingly contribute to learned helplessness when they provide too much feedback without allowing, and demanding, adequate effort from students to work through problems themselves. 6-CORE2 TP11
2.1 Algebraic Expressions and Equations Summary ALGEBRAIC EXPRESSIONS AND EQUATIONS Students write and evaluate algebraic expressions. Students represent algebraically the costs of items on a menu. Students find values that make equations true. Strategies for English Learners For students who cannot read and comprehend written text: Allow them to give an example of the vocabulary words, and later relate examples to written definitions. Preview vocabulary: exactly one means only one. Preview the meaning of instructions with a sample exercise. Reproducibles and Tasks R4: Pizza Shop Menu T3: My Menu Math Notes and Teaching Notes MN1: Evaluating an Expression MN2: Variables in Algebra MN3: Order of Operations MN4: Equality and Inequality Goals Solve problems involving money and decimals. Use variables in expressions and equations. Use the distributive property and the conventions for order of operations to evaluate expressions. Find values that make equations true. Write verbal statements symbolically. Strategies for Special Learners Use letters that represent the size of drinks (s, m, L, x) as variables to connect the contexts to abstract ideas. Materials TN1: Standards for Mathematical Practice TN2: Caution! Please Be Careful with Aunt Sally TN3: Learning the Language of Algebra TN4: Similar Phrases with Different Meanings TN5: Avoiding Learned Helplessness Calculators (optional) Index Cards (optional) Menus (optional) Student Pages Technology Connections SP1: Algebraic Expressions and Equations SP2: Pizza Shop Variables and Expressions SP3-4: More Expressions SP5-6: Pizza Shop Equations SP7: From Words to Symbols SP8: Boom Burgers Expressions SP9: Boom Burgers Equations Strategies for Advanced Learners Use Task 3 to challenge students to create and share restaurant orders using algebra. Management Ideas Cutting a corner off the cards prior to distributing them makes it easy to determine the front from the back. Permission is granted to project the student pages from the 6-CORE CCSS Mathematics Supplement CD. front back 6-CORE2 TP12
2.1 Algebraic Expressions and Equations Ø SP0 Word Bank Ø SP1 Algebraic Expressions and Equations (Ready, Set, Go) Ø SP2 Pizza Shop Variables and Expressions Ø R4 Pizza Shop Menu PREVIEW / WARMUP Introduce the goals and standards of the lesson. Discuss important vocabulary as relevant. Students find the cost of 6 granola bars in multiple ways. Review decimal operations as needed. Some strategies are provided below. Method 1 Standard addition algorithm 1.15 1.15 1.15 1.15 1.15 +1.15 6.90 1.15 1.15 1.15 1.15 1.15 1.15 Method 2 Derived addition strategy 2.30 2.30 2.30 6.90 INTRODUCE 1 Method 3 Standard multiplication algorithm 1.15 6 6.90 Method 4 Proportional reasoning 2 granola bars: 2(1.15) 2.30 Then triple that: 3(2.30) 6.90 Method 5 Breaking into parts (distributive property) 6(1.00) 6.00 6(0.15) 0.90 OR 6(1 + 0.15) = 6 + 0.90 = 6.90 Display the menu only. For simplicity, in this series of lessons, assume that tax is included. 6.90 What do you think c + p means? The cost of a cheese slice plus the cost of a pepperoni slice. What is the value of c + p (problem 2)? $2.25. Remind students that variables must stand for numbers or values, so c and p must stand for the cost of these slices of pizza, not the items themselves. Have students define all variables from the menu before the proceeding to evaluate expressions. Choosing Sense-Making Variables Identify the cost of each menu item with a variable as it arises from context. To help students transition to abstract notation, the first letter of the menu item is typically selected as the variable. The letter x is used here for an extralarge drink because x is a commonly used variable, and students will likely be familiar with the abbreviation XL for extra large. Let c = the cost of a cheese slice Let p = the cost of a pepperoni slice Let d = the cost of a deluxe slice Some variables used in this lesson: Let s = the cost of a small drink Let m = the cost of a medium drink Let L = the cost of a large drink Let x = the cost of an extra-large drink 6-CORE2 TP13
2.1 Algebraic Expressions and Equations Ø SP2 Pizza Shop Variables and Expressions Ø R4 Pizza Shop Menu Materials: Calculators (optional) Ø SP3-4 More Expressions INTRODUCE 1 (Continued) Using variables, read orders below aloud and display them visually. Ask students to interpret the meaning of the variables and to evaluate the expressions. Draw attention to various algebraic properties and conventions associated with variables. Problem 3: What is the value of 4p? $5.00. Illustrate variable notation. Students usually accept implied multiplication in context like this, even if they have never seen it before. Problem 4: What is the value of 3p + 2d + 3s + 2m? $12.50.Illustrate correct order of operations. Students are usually aware of the correct order of operations (multiply before adding) when a problem is in context like this, even without formal exposure to the order of operations. Discuss and review the order of operations conventions as needed. For problem 5: Find the value of 2c + d + L + x. $7.25. Illustrate good choices of letters for variables. Using certain letters as variables can be confusing. The lower case L, for example, may look like the number one, especially with sloppy writing. Try having the class generate a list of other potentially confusing variables potentially that may be confusing letters mistaken for b 6 g 9 i, l 1 o 0 s 5 t + x multiplication z 2 Problem 6: Evaluate 3(p + d + s) $11.85. Is there another way to do this computation? Illustrate the distributive property. This problem gives a meaningful context for the distributive property. Using the terminology the cost of three orders of a pepperoni slice, a daily special, and a small drink is the same as 3(p + d + s) may be a helpful transition. This presentation can be effective as well: ( p+ d + s) ( p+ d + s) + ( p+ d + s) 3 p+3 d+3s Problem 7: Find the value of 2(3c + s). $7.90. Illustrate distributive property again. Review the vocabulary associated with the parts of a multiplication equation. Then discuss the vocabulary terms, constants, variables, and coefficients in the contexts of the provided examples and be sure that students record them in their word banks. For the variable term y in example 2, why is 1 its coefficient? The term y is equivalent to one y, or 1 y. 6-CORE2 TP14
2.1 Algebraic Expressions and Equations Pairs/Individuals Ø SP3-4 More Expressions Ø SP3-4 More Expressions /Small Groups Materials: Index cards EXPLORE 1 Students do the remaining problems on these two pages. Share and discuss. SUMMARIZE 1 Ask questions that utilize vocabulary and increase understanding of variable expressions. How are problems 3 and 4 the same? They both have only one term. How are they different? 11r is a variable term, while 12 is a constant term. The coefficient of r in 11r is 11, while 12 is the constant term. For problem 6, what is the variable term and its coefficient? 3x; 3. What is the constant term? 6. For problem 10, what are two ways to find the total cost of the orders? (1) Evaluate the variable expression for total cost. (2) Add all of the expressions evaluated for each friend. For problem 11, what happens if the owner changes the price of his pizza? Does the formula still hold? Of course. This is the point of using a variable. PRACTICE 1 Activity 1: Students individually create orders on index cards in their group. On the back of the card they evaluate the expression. Students pass around their cards, one at a time, and other members of the group evaluate each expression and then check the back to see that their answer agrees with the author. Encourage students to selfcorrect. Then pass and do the next one. Activity 2: Each group writes their collective group order (sum of their individual orders) as a variable expression. Then, one spokesperson from each group reads the group expression aloud while the other students in the class record them. Each group writes the collective whole class order, and spokespersons read out that collective order to see if groups agree with the result. This result should be the same as the sum of the individual orders. 6-CORE2 TP15
2.1 Algebraic Expressions and Equations Ø SP5-6 Pizza Shop Equations Pairs/Individuals Ø SP5-6 Pizza Shop Equations Ø SP5-6 Pizza Shop Equations INTRODUCE 2 Refer to the Pizza Shop menu. Re-emphasize that the letters stand for prices, not the item itself. Note that the same letters are used as previously defined. What is the difference between an expression and an equation? An expression is a combination of numbers, variables, and operation symbols. An equation is a statement that two expressions are equal to each other. An equation has an equal sign, which represents the verb in the statement. An expression does not have an equal sign an expression can be thought of as a phrase without a verb. Discuss the meanings of the equations in questions 1 and 2. What does the equation p+ = 3cmean? The cost of a slice of pepperoni pizza plus the cost of something else on the menu is the same as the cost of three cheese pizza slices. Find the cost of the unknown item. $1.75. How did you get this? Answers will vary. Students are likely to substitute known values and then use guess-and-check. Find the unknown item. A slice of the daily special. How did you get this? Look at menu for an item that costs $1.75. EXPLORE 2 Students complete the pages while referring to the menu. Be sure they understand that with the same problem, the refers to the same item. In different problems, the need not represent the same menu item. Variables were defined on a previous page. Share answers and discuss. A variety of approaches should be shared. Guess and check may be the preferred solution method. Encourage students to share methods that involve using inverse operations or performing operations to both sides of the equation. Discuss problems. SUMMARIZE 2 What makes problem 6 different than the others? It has two squares, which represent the unknown in the problem. Must both squares represent the same item in problem 6? Yes. What is the value for the square? $1.25. The menu item? A slice of pepperoni pizza. 6-CORE2 TP16
2.1 Algebraic Expressions and Equations Small Groups/ Paris/Individuals Ø SP7 From Words to Symbols Individuals Ø SP8 Boom Burgers Expressions Ø SP9 Boom Burgers Equations Individuals Ø T3 My Menu Materials: Menus Ø SP0 Word Bank Ø SP1 Algebraic Expressions and Equations (Ready, Set) PRACTICE 2A Students write variable expressions for each situation, then write the corresponding equation. For problem 4b, what mathematical operation suggests putting things into equal m groups? Students will probably connect this to division, 5 =3. Consider making the connection to multiplication, namely 1 5 PRACTICE 2B m = 3. These pages are appropriate for classwork or homework. EXTEND Students create a menu or use a take-out menu from a local restaurant. They write expressions and equations to represent costs. CLOSURE Review the vocabulary, goals, and standards for the lesson. 6-CORE2 TP17
2.2 Beginning Inequalities Summary Students write linear inequalities in one variable and graph the solutions. Students find values that make inequalities true. Strategies for English Learners Emphasize the importance of the verb in an inequality sentence. 6 is greater than 4 (6 > 4) 6 greater than 4 (4 + 6) Reproducibles and Tasks Math Notes and Teaching Notes MN1: Evaluating an Expression MN2: Variables in Algebra MN3: Order of Operations MN4: Equality and Inequality BEGINNING INEQUALITIES Goals Solve problems involving money and decimals. Graph solutions to inequalities and recognize inequalities that have infinitely many solutions. Determine whether inequalities are true or false. Find a value from a specified set that makes an inequality true. Find and write solutions to inequalities. Strategies for Special Learners One way to remember the correct meaning of the directional inequality sign is to think of the Less than symbol as a tilted L. Materials TN1: Standards for Mathematical Practice TN2: Caution! Please Be Careful with Aunt Sally TN3: Learning the Language of Algebra TN4: Similar Phrases with Different Meanings TN5: Avoiding Learned Helplessness Colored pencils or highlighters (optional) Student Pages Technology Connections SP10: Beginning Inequalities SP11: Number Sentences and their Graphs SP12: Words, Symbols, and Graphs SP13-14: Pizza Shop Revisited SP15: Boom Burgers Revisited Strategies for Advanced Learners On SP14, problem 13, challenge students to write the inequality in a simpler form. (c + d < c + x d < x) Management Ideas When discussing problems with multiple solutions, record all solutions on the board and then have students defend their answers. Permission is granted to project the student pages from the 6-CORE CCSS Mathematics Supplement CD. 6-CORE2 TP18
2.2 Beginning Inequalities Ø SP0 Word Bank Ø SP10 Beginning Inequalities (Ready, Set, Go) Ø SP11 Number Sentences and their Graphs Materials: Colored pencils or highlighters (optional) PREVIEW / WARMUP Introduce the goals and standards of the lesson. Discuss important vocabulary as relevant. Students select two symbols that make each statement correct, though there may be more in some cases. Then they write number sentences as verbal sentences. Address misconceptions about inequality symbols as they arise. For problems 9 and 10, be sure students write 0.75 as seventy-five hundredths and 1.5 as one and five tenths. INTRODUCE 1 Review some basic symbol notation and graphing conventions in a realistic context. For example: In some states, one must be at least 16 years of age to drive. How can we write this symbolically? x 16 if x represents the age in years of a person eligible for a driver s license. If you could plot a point on a number line for all the possible ages of legal drivers, what would that look like? A closed circle at 16 with an arrow pointing to the right. What are some values that make this inequality true? Students will likely identify integer values (16, 17, 18 ). Could x = 16.75? Yes, because this would indicate an age of 16 years and nine months. Make sure that students recognize that nonintegers greater than 16 are valid solutions. What would the graph of x < 16 represent in this context? Non-legal drivers. What would it look like? An open circle at 16 with an arrow pointed to the left. Students match the number sentences to their graphs. Reinforce the meaning of the closed circle (includes the value) and the open circle (excludes the value). What is the difference between x 3 and x > 3? The former includes 3 and all numbers that are greater, while the latter includes only the numbers that are greater, not including 3. Students often consider only the integer values greater than 3, namely, 4, 5, 6, Make sure students understand for x > 3 that also included is every number between 3 and 4, 4 and 5, 5 and 6 as well. Which number sentence (and graph) does not represent an inequality? x = 3. This is the only number sentence above that does NOT represent an infinite number of values. Discuss the meaning of a boundary point and reinforce the difference between the open and closed circles. What is the boundary point for all of the inequalities above? x = 3. 6-CORE2 TP19
2.2 Beginning Inequalities Pairs/Individuals Ø SP12 Words, Symbols, and Graphs Ø SP12 Words, Symbols, and Graphs Ø SP13-14 Pizza Shop Revisited EXPLORE 1 Students write each verbal statement using correct symbols, graph with the appropriate type of dot/circle at the boundary point, and list possible integer and noninteger solutions, if any. Share and discuss as desired. SUMMARIZE 1 How are problems 1 and 2 the same? They both have the same boundary point, 4, and the graphs both have arrows pointing right. Different? The first has a closed circle at 4, the second an open circle at 4. Do any problems NOT have any integer solutions? No, they all do. Share some as desired. Do any problems have only integer solutions? The solution to problem 5 is x = 4. This is a whole number (subset of integers). INTRODUCE 2 Revisit the now familiar Pizza Shop menu, reminding students that we had previously defined variables to represent the cost of each item. What does x stand for? The cost of an extra large drink (not the drink itself). Work through the first problem, focusing student attention on the directions that ask for more than one inequality for each situation. Chris has $1.00. What is the most that he can spend? $1.00. Is there an item that is $1.00? Yes, a slice of cheese. Is the inequality 1.00 1.00 true? Yes. c 1.00 1.00 1.00 Can Chris spend less than $1.00? Yes. How do we know that? Because the sign means that the item can be less than one dollar OR equal to one dollar. We already addressed the equal to 1.00 part. Are there any items that are less than $1.00? Yes, a small drink costs $0.95. How can we use the less than symbol to show that the small drink costs less than $1.00? s < 1.00 0.95 < 1.00. Read the second problem with the students and ask questions similar to those above, making sure that they understand the words less than $1.20 prohibits them from choosing the cost of a medium drink as a solution. 6-CORE2 TP20
2.2 Beginning Inequalities Pairs/Individuals Ø SP13-14 Pizza Shop Revisited Ø SP13-14 Pizza Shop Revisited Individuals Ø SP15 Boom Burgers Revisited Ø SP0 Word Bank Ø SP10 Beginning Inequalities (Ready, Set) EXPLORE 2 Students finish the problems on the first page, and then continue to the next. Point out that problems 10-13 ask for one solution for each problem, though some may have more. Share and discuss answers as desired. SUMMARIZE 2 Can Ariella buy a slice of pepperoni? Yes. Why? She can spend the entire $1.25 (less than or equal to symbol). Can Maricella buy a slice of the daily special? No. Why not? She cannot spend the entire $1.75 (strictly less than symbol). What is different about problems 8 and 9? What is similar? Other than the obvious answers about having different variables, expressions, inequality symbols, etc., these questions are designed to address the difference in the fact that both number sentences have equivalent expressions on either side of the inequality symbol, but problem 8 is a false statement (3.00 > 3.00), while problem 9 is true (3.75 3.75). Problem # Possible solutions 10 all 11 c, p s, m, L 12 none 13 x Are there any problems for which every menu item represents a solution? Problem 10. No solutions? Problem 12. Exactly one solution? Problem 13. Challenge students to rewrite problem 13 in a simpler form. Though not required at this point, students may recognize that c + d < c + x is equivalent to d < x or 1.75 < 1.90. PRACTICE This page is appropriate for class work or homework. CLOSURE Review the vocabulary, goals, and standards for the lesson. 6-CORE2 TP21
2.3 Saving for a Purchase Summary Students identify independent and dependent variables in contexts. Students use real-world problems to understand the relationship between independent and dependent variables. Students use input-output equations, tables, and graphs to determine how much time is needed to save for purchases. Strategies for English Learners Students not yet speaking English can indicate answers to questions by pointing to evidence on the graph. Reproducibles and Tasks Math Notes and Teaching Notes SAVING FOR A PURCHASE Goals Analyze the relationship between dependent and independent variables. Use tables, graphs, equations, and words to solve problems. Strategies for Special Learners Link the formula for a linear function to a familiar context: y = mx + b m = (money invested monthly) b = (start amount in the bank) Materials MN5: Independent and Dependent Variables MN6: Steeper Lines Have Greater Slope TN1: Standards for Mathematical Practice TN5: Avoiding Learned Helplessness Rulers Colored Pencils Student Pages Technology Connections SP16: Saving for a Purchase SP17: Independent and Dependent Variables SP18: Camera: Instructions SP19: Camera: Table SP20: Camera: Graph SP21: Camera: Questions SP22: Printer: Instructions and Tables SP23: Printer: Graph and Questions SP24: Brian s Problem: Instructions and Table SP25: Brian s Problem: Graph and Questions Strategies for Advanced Learners Challenge students to use repeated reasoning to find explicit rules for input-output tables based on entries in the tables. Management Ideas Allow adequate time for summarizing. Students may need a lot of help making important connections between the different representations (numbers in the table; symbols in the equations; graphs; using vocabulary and expressing solutions properly). Permission is granted to project the student pages from the 6-CORE CCSS Mathematics Supplement CD. 6-CORE2 TP22
2.3 Saving for a Purchase Ø SP0 Word Bank Ø SP16 Saving for a Purchase (Ready, Set, Go) Ø SP17 Independent and Dependent Variables PREVIEW / WARMUP Introduce the goals and standards for the lesson. Discuss important vocabulary as relevant. Students fill in the t-table and discuss why the input/output equation should be y = 3x + 5. Note: we informally introduce the slope-intercept form of a line through the context of this lesson without using formal vocabulary. INTRODUCE 1A Introduce the concept of independent and dependent variables using the four situations posed. These terms are introduced in familiar settings, linking common vocabulary to mathematical language. One author s interpretation of each situation is stated here: 1. The cost of a cell phone bill (c) depends on the number of text messages sent (t). 2. The cost of an order (c) depends on the number of pizzas ordered (n). 3. The perimeter of the square (p) depends on the length of the side of a square (s). 4. The amount of Victor s paycheck (a) depends on the number of hours Victor works (h). It may be argued that either variable may depend on the other. Be open to different student interpretations, and accept all answers that are backed by sound reasoning. Does the cost of a cell phone bill depend on the number of text messages sent or does the number of text messages sent depend upon the cost of a cell phone bill? Students will most likely reason that it is the former. When a function (such as this one) is invertible, either variable might be regarded as the independent variable. If students choose to take cost as the independent variable, be sure they make consistent statements about the independent and dependent variable. Give students time to do the remaining problems on the page. 6-CORE2 TP23
2.3 Saving for a Purchase Ø SP18 Camera: Instructions Ø SP19 Camera: Table Ø SP20 Camera: Graph Materials: Rulers Colored pencils Ø SP21 Camera: Questions INTRODUCE 1B Preview the directions, and link the information to the equation. To encourage persistence in problem solving, allow sufficient time for them to talk about the context and make sense of it. How much money must Julie save to purchase a camera? $240. If Julie deposited $100 in the bank, and 0 months have passed, what is her balance? Be sure students understand the meaning of account balance. After 0 months she will not have saved any extra, so she will still have $100. Using the equation and substituting 0 for x, model for students that 10(0) + 100 = 100. Do the same for another month or two. Do we have to compute the balance for months 1, 2, 3, 4, etc. to find the balance after month 3? No. If desired, we can use the formula to check any month. Students may find going one month at a time sequentially to be inefficient. However, allowing them to be inefficient now will help them recognize and express regularity in repeated reasoning. It will also help them recognize the efficiency of a formula. How much will Julie have after five months? 10(5) + 100 = 150. For how many months must Julie save until she has the $240? Verify using the equation. 14 months: 10(14) + 100 = 240. Students make a graph that shows how much Julie is saving each month. What are the dependent and independent variables? Total amount saved (y) depends on the number of months of saving (x). What are the coordinates on the graph that show how much Julie has at the start? Graph the coordinate (0, 100). This is a good time to discuss appropriate labeling and scaling for the graph. Convention dictates that the number of months, the independent variable, will go on the horizontal (x) axis. The total amount saved in dollars, the dependent variable, will go on the vertical (y) axis. Do the points form any pattern? They appear to be in a line and they are. However, in this situation, the actual points are not connected because we are recording monthly data and fractions of months do not apply in this context. We call these discrete points. However, it is permissible to draw a trend line, solid or dashed, to show the linear relationship. Students answer comprehension questions to summarize the important points when Julie saves for a camera. 6-CORE2 TP24
2.3 Saving for a Purchase Pairs/Individuals Ø SP22 Printer: Instructions and Table Ø SP23 Printer: Graph and Questions Materials: Rulers Colored pencils Ø SP22 Printer: Instructions and Table Ø SP23 Printer: Graph and Questions Materials: Rulers Colored pencils EXPLORE Students represent another context (saving for a printer) using tables and graphs, and they answer questions about the problem using their mathematical representations. Remind students to consider the following prior to graphing: (1) the independent and dependent variables, and on which axes they go; (2) scaling the axes; (3) whether it is appropriate or not to draw a solid line. Encourage students to use the formula rather than checking each month sequentially. The formula is an efficient way to find the desired solution, rather than a slower, iterative process. Although we are working informally at this point, take advantage of teachable moments to identify and discuss the meaning of slope and intercept in the context and equation. Share and discuss solutions. SUMMARIZE How many months does it take for Theresa to save for the printer? 7 months. What is the equation for the amount that Theresa saved at the end of any month? y = 20x + 10. How can this equation be helpful in these problems? It is more efficient and quicker to find the amount saved at any month. Consider Theresa s information, and ask questions to help students explicitly link the graph, the table, and the equation to the linear function y = mx + b. What is her monthly increase? $20. Where does the monthly increase appear in her table? Each increase, from one month to the next, is 20 (if recorded month-to-month). In her graph? Each successive month, from left to right, requires a vertical increase of 20 on the graph. In her equation? The multiplier of x is 20. This is a teachable moment for students who are not familiar with the term coefficient. What was Theresa s starting amount in the bank? $10. Where does that appear in her graph? The point (0, 10) is on the y-axis. In her table? It is the output value that corresponds to the input value of x = 0. In her equation? It is added to the 20x. We call this the constant term. Why is a solid line NOT appropriate for this graph? We plot discrete points because we are only considering whole months. However, it is permissible to draw a solid or dashed trend line and note it as such. 6-CORE2 TP25