TEACHER GUIDE INCLUDES. Tier 1 Tier 2 Tier 3 Correlations. Diagnostic Interviews for Every Common Core Cluster

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1 TEACHER GUIDE FOR THE COMMON CORE STATE STANDARDS FOR MATHEMATICS INCLUDES Tier Tier Tier Correlations Diagnostic Interviews for Every Common Core Cluster Tier Lessons, Tier Prerequisite Skills, and Tier Scaffolded Examples with Answers GRADE 6

2 TEACHER GUIDE Response to FOR THE COMMON CORE STATE STANDARDS FOR MATHEMATICS GRADE 6 Includes Tier Tier Tier Correlations Diagnostic Interviews for Every Common Core Cluster Tier Lessons, Tier Prerequisite Skills, and Tier Scaffolded Examples with Answers

3 Expressions and Equations Apply and extend previous understandings of arithmetic to algebraic expressions. Lesson 49 CC.6.EE. Exponents Lesson 0 CC.6.EE. Evaluate Expressions Involving Exponents Lesson CC.6.EE.a Write Algebraic Expressions Lesson CC.6.EE.b Identify Parts of Expressions Lesson CC.6.EE.c Evaluate Algebraic Expressions and Formulas....0 Lesson 4 CC.6.EE. Problem Solving Combine Like Terms Lesson CC.6.EE. Generate Equivalent Expressions Lesson 6 CC.6.EE.4 Identify Equivalent Expressions Reason about and solve one-variable equations and inequalities. Lesson 7 CC.6.EE. Solutions of Equations Lesson 8 CC.6.EE. Solutions of Inequalities Lesson 9 CC.6.EE.6 Use Algebraic Expressions Lesson 60 CC.6.EE.7 Write Equations Lesson 6 CC.6.EE.7 Model and Solve Addition Equations Lesson 6 CC.6.EE.7 Solve Addition and Subtraction Equations Lesson 6 CC.6.EE.7 Model and Solve Multiplication Equations Lesson 64 CC.6.EE.7 Solve Multiplication and Division Equations Lesson 6 CC.6.EE.7 Problem Solving Equations with Fractions....9 Lesson 66 CC.6.EE.8 Write Inequalities Lesson 67 CC.6.EE.8 Graph Inequalities Represent and analyze quantitative relationships between dependent and independent variables. Lesson 68 CC.6.EE.9 Independent and Dependent Variables Lesson 69 CC.6.EE.9 Equations and Tables Lesson 70 CC.6.EE.9 Problem Solving Analyze Relationships Lesson 7 CC.6.EE.9 Graph Relationships Lesson 7 CC.6.EE.9 Equations and Graphs v

4 Exponents LESSON 49 CC.6.EE. OBJECTIVE Write and evaluate expressions involving exponents. An exponent tells how many times a number is used as a factor. The base is the number being multiplied repeatedly. For example, in, is the exponent and is the base. = = Write the expression 4 using equal factors. Then find the value. Step Identify the base. The base is 4. Step Identify the exponent. The exponent is. Step Write the base as many times as the exponent tells you. Place You should have one less multiplication a multiplication symbol between symbol than the value of the exponent. the bases =,04 Step 4 Multiply. So, 4 =,04. Write as an expression using equal factors. Then find the value ; 8 ; ; 64 ; ; 0, ;, ; 4,64 ; ; 90,6 0;0,000,000 Expressions and Equations 97

5 Exponents CC.6.EE. Use one or more exponents to write the expression Find the value , , Write 44 with an exponent by using as the base.. Write 4 with an exponent by using 7 as the base. 7 Problem Solving. Each day Sheila doubles the number of push-ups she did the day before. On the fifth day, she does push-ups. Use an exponent to write the number of push-ups Shelia does on the fifth day.. The city of Beijing has a population of more than 0 7 people. Write 0 7 without using an exponent. 0,000, Lesson 49

6 Evaluate Expressions Involving Exponents OBJECTIVE Use the order of operations to evaluate expressions involving exponents. LESSON 0 CC.6.EE. A numerical expression is a mathematical phrase that includes only numbers and operation symbols. You evaluate the expression when you perform all the computations. To evaluate an expression, use the order of operations. Order of Operations. Parentheses. Exponents. Multiply and Divide 4. Add and Subtract Evaluate the expression (0 + 6 ) Step Start with the parentheses. Use the order of operations for the computations inside the parentheses Find the value of the number with an exponent. Rewrite as multiplication: = Multiply and divide from left to right: = Add and subtract from left to right: = 46 Step Rewrite the original expression, using the value from Step for the part in parentheses. Step Now that the parentheses are cleared, look for exponents. Step 4 Multiply and divide from left to right. Step Add and subtract from left to right. So, (0 + 6 ) = 6. Evaluate the expression. (0 + 6 ) = There are no more exponents, so go on to the next step in the order of operations = (7 + ) (6-4 ) = (8-0 ). (0 - ) 6. (6 - ) Expressions and Equations 99

7 Evaluate Expressions Involving Exponents CC.6.EE. Evaluate the expression (7 - ) (8 + 6) (4 ) ( - 8) Place parentheses in the expression so that it equals the given value ; value: ; value: ( + ) (7 + - ) Problem Solving 9. Hugo is saving for a new baseball glove. He saves $0 the first week, and $6 each week for the next 6 weeks. The expression represents the total amount in dollars he has saved. What is the total amount Hugo has saved? 0. A scientist placed fish eggs in a tank. Each day, twice the number of eggs from the previous day hatch. The expression 6 represents the number of eggs that hatch on the sixth day. How many eggs hatch on the sixth day? $46 0 eggs 00 Lesson 0

8 Write Algebraic Expressions LESSON CC.6.EE.a OBJECTIVE Write algebraic expressions. Word problems use expressions that you can write with symbols. An algebraic expression has at least one variable. A variable is a letter or symbol that represents one or more numbers. Writing algebraic expressions for words helps you solve word problems. These are a few common words that are used for operations. add (+) sum increased by plus more than subtract (-) difference minus decreased by less less than multiply ( ) product times 7 more than x More than means add. x more than x means add 7 to x. four times the Times means multiply. sum of 7 and n Sum means add. 4 (7 + n) The words mean multiply 4 by (7 + n). A number next to a variable always shows multiplication. For example, n means the same as n. divide ( ) quotient divided by Write an algebraic expression for the word expression.. b divided by 9. c more than b 9 or b 9. d decreased by times g d c 8 g or 8g. p increased by 6. the quotient of k and 4 p + k 4 or k less than the product of and m 8. less than the quotient of d and 6 m - 7 or m - 7 d 6 - or d 6 - Expressions and Equations 0

9 Write Algebraic Expressions CC.6.EE.a Write an algebraic expression for the word expression.. less than p. the sum of x and 9. 6 more than the difference of b and p - x (b - ) 4. the sum of and the product of and v. the difference of and the product of and k 6. divided by the sum of h and + v - k (h + ) 7. the quotient of m and more than multiplied by f 9. 6 minus the difference of x and m 7 f (x - ) 0. 0 less than the quotient of g and. the sum of 4 multiplied. 4 more than the by a and multiplied by b difference of r and s (g ) - 0 4a + b (r - s) + 4 Problem Solving. Let h represent Mark s height in inches. Suzanne is 7 inches shorter than Mark. Write an algebraic expression that represents Suzanne s height in inches. 4. A company rents bicycles for a fee of $0 plus $4 per hour of use. Write an algebraic expression for the total cost in dollars for renting a bicycle for h hours. h h 0 Lesson

10 Identify Parts of Expressions LESSON CC.6.EE.b OBJECTIVE Identify and describe parts of expressions. Each part of an expression between the operation signs + or - is a term. A coefficient is a number multiplied by a variable, or letter. Describe the parts of the expression 6b - 7. Then write a word expression. Step Identify the terms. There are two terms: 6b and 7. Step Describe the terms. Step Identify the operation separating the terms. The first term shows multiplication: 6b = 6 b 6b is the product of 6 (the coefficient) and b (the variable). The second term is the number 7. Subtraction gives the difference of the two terms in the expression. Step 4 Write a word expression. the difference of 6 times b and 7 or 7 less than the product of 6 and b Identify the parts of the expression. Then write a word expression for the numerical or algebraic expression. -. Possible answers given.. (m - ) Identify the parts. Describe the parts. ; m - the number ; the difference of the variable m and the number Identify the operations. multiplication and subtraction Write a word expression. times the difference of m and y + ( ) The division is the quotient of The multiplications are the product of and. The addition is the sum 8 and y and the product of and. of the quotient and 7. Word expression: the quotient of and, plus 7 The addition is the sum of the products. Word expression: the product of 8 and y plus the product of and Expressions and Equations 0

11 Identify Parts of Expressions Identify the parts of the expression. Then write a word expression for the numerical or algebraic expression. CC.6.EE.b 4. Possible answers given.. (6-7) The subtraction is the difference of 6 and 7. The division is the quotient of the difference and. Word expression: the quotient of the difference of 6 and 7 and The multiplication is the product of and 9. The addition is the sum of 0 and the product. Word expression: the sum of 0 and the product of and 9. e - f q + q The expression is the difference of two terms. The first term is the product of and e, and the second term is f. Word expression: the difference of times e and f The expression is a sum of three terms. The first term is 8, the second term is the product of 6 and q, and the third term is q. Word expression: the sum of 8, the product of 6 and q, and q Identify the terms of the expression. Then give the coefficient of each term.. r + 7s 6. 6g - h Terms: r and 7s; coefficient of r : ; coefficient of 7s: 7 Problem Solving 7. Adam bought granola bars at the store. The expression 6p + n gives the number of bars in p boxes of plain granola bars and n boxes of granola bars with nuts. What are the terms of the expression? Terms: 6g and h; coefficient of 6g: 6; coefficient of h: 8. In the sixth grade, each student will get 4 new books. There is one class of students and one class of 0 students. The expression 4 ( + 0) gives the total number of new books. Write a word expression for the numerical expression. 6p and n The product of 4 and the sum of and Lesson

12 Evaluate Algebraic Expressions and Formulas OBJECTIVE Evaluate algebraic expressions and formulas. LESSON CC.6.EE.c To evaluate an algebraic expression or formula, substitute the value for the variable. Then follow the order of operations. Evaluate x + x for x =,,, and 0. x + x for x = x + x for x = x + x for x = x + x for x = To evaluate an expression with more than one variable, substitute each variable s value. Then follow the order of operations. Evaluate 4c d for c = and d = So, 4c d = for c = and d =. Evaluate the expression for x =,,, and x. x +. x + + x 4. x + x,, 9, Evaluate the expression for the given values of the variables.. 7x + y + 6 for x =, y = 7,, 7, 6. 8a + - b for a = 4, b = 8,, 6,, 8,, 0 7. b - c + for b =, c = Expressions and Equations 0

13 Evaluate Algebraic Expressions and Formulas CC.6.EE.c Evaluate the expression for the given values of the variables.. w + 6 for w =. r - 9 for r = c for c = b - 4 for b =. (h - ) for h = 6. x + x for x = m + m + for m = 8. 9a - a for a = ( - h) for h = m - 9n for m = 7 and n = 8. d - 9k + for d = 0 and k = 9 4. x + 4y for x = 7 and y = Problem Solving. The formula P = l +w gives the perimeter P of a rectangular room with length l and width w. A rectangular living room is 6 feet long and feet wide. What is the perimeter of the room? 4. The formula c = (f - ) 9 gives the Celsius temperature in c degrees for a Fahrenheit temperature of f degrees. What is the Celsius temperature for a Fahrenheit temperature of degrees? 94 feet 0 degrees Celsius 06 Lesson

14 Problem Solving Combine Like Terms LESSON 4 CC.6.EE. OBJECTIVE Combine like terms by applying the strategy use a model. Use a bar model to solve the problem. Each hour a company assembles 0 bikes. It sends 6 of those bikes to stores and keeps the rest of the bikes to sell itself. The expression 0h - 6h represents the number of bikes the store keeps to sell itself for h hours of work. Simplify the expression by combining like terms. Read the Problem What do I need to find? I need to simplify the expression 0h - 6h. What information do I need to use? I need to use the like terms 0h and 6h. Solve the Problem How will I use the information? I can use a bar model to find the difference of the like terms. Draw a bar model to subtract 6h from 0h. Each square represents h, or h. 0 h h h h h h h h h h h h h h h h h 6 h 4 h The model shows that 0h - 6h = 4h. So, a simplified expression for the number of bikes the store keeps is 4h.. Bradley sells produce in boxes at a farmer s market. He put 6 ears of corn and 9 tomatoes in each box. The expression 6b + 9b represents the total pieces of produce in b boxes. Simplify the expression by combining like terms.. Andre bought pencils in packs of 8. He gave pencils to his sister and pencils from each pack to his friends. The expression 8p - p - represents the number of pencils Andre has left from p packs. Simplify the expression by combining like terms. b p - Expressions and Equations 07

15 Problem Solving Combine Like Terms CC.6.EE. Read each problem and solve.. A box of pens costs $ and a box of markers costs $. The expression p + p represents the cost in dollars to make p packages that includes box of pens and box of markers. Simplify the expression by combining like terms. p + p = 8p. Riley s parents got a cell phone plan that has a $40 monthly fee for the first phone. For each extra phone, there is a $ phone service charge and a $0 text service charge. The expression 40 + e + 0e represents the total phone bill in dollars, where e is the number of extra phones. Simplify the expression by combining like terms. e A radio show lasts for h hours. For every 60 minutes of air time during the show, there are 8 minutes of commercials. The expression 60h - 8h represents the air time in minutes available for talk and music. Simplify the expression by combining like terms. h 4. A publisher sends 00 books to each bookstore where its books are sold. At each store, about books are sold at a discount to employees and about 40 books are sold during store weekend sales. The expression 00s - s - 40s represents the approximate number of the publisher s books sold at full price in s stores. Simplify the expression by combining like terms.. A sub shop sells a meal that includes an Italian sub for $6 and chips for $. If a customer purchases more than meals, he or she receives a $ discount. The expression 6m + m - shows the cost in dollars of the customer s order for m meals, where m is greater than. Simplify the expression by combining like terms. 7s 8m - 08 Lesson 4

16 LESSON Generate Equivalent Expressions OBJECTIVE Use the properties of operations to generate equivalent algebraic expressions. CC.6.EE. Equivalent expressions are two or more expressions that are equal for any value of the variable in the expressions. You can use the properties of operations to write equivalent expressions. Write an equivalent expression for 4c + + c. Step Identify like terms. 4c and c Step Use properties of operations to combine like terms. Commutative Property of Addition: switch and c Associative Property of Addition: group 4c and c Add 4c and c. 4c + + c = 4c + c + = (4c + c) + = c + Use properties of operations to write an equivalent expression by combining like terms.. 7x + x + x. 8a + - a. b - 8b + 4x 6a + 4b c c. 4p + - p 6. 8y - y + y 0c - 6 p + 7y Use the Distributive Property to write an equivalent expression. 7. (m + 7) 8. 4(t + ) m + 9. (9 + 6r) 0. 8(4n - n) 8t r n - 6n Expressions and Equations 09

17 Generate Equivalent Expressions CC.6.EE. Use properties of operations to write an equivalent expression by combining like terms.. 7h - h. x x. 6 + p - 9p 4h 7x p 4. y + y - 8y. (h + ) + h n + 7-4n y + y h n Use the Distributive Property to write an equivalent expression. 7. (9 + k) 8. (m + ) 9. 6(g + h) 8 + 0k m + 0 6g + 6h 0. 4d + 8. p + q. 8x + 9y Possible answer: 4(d + ) 7(p + q) Possible answer: 9(x + y) Problem Solving. The expression n + n + 00 represents the total cost in dollars for skis, boots, and a lesson for n skiers. Simplify the expression n + n Then find the total cost for 8 skiers. 4. Casey has n nickels. Megan has 4 times as many nickels as Casey has. Write an expression for the total number of nickels Casey and Megan have. Then simplify the expression. 7n + 00; $6 n + 4n; n 0 Lesson

18 LESSON 6 Identify Equivalent Expressions OBJECTIVE Identify equivalent algebraic expressions. CC.6.EE.4 Use properties to determine whether a + 7( + a) and a + are equivalent. Step Rewrite the first expression using the Distributive Property. Multiply 7 and and multiply 7 and a. Step Use the Commutative Property of Addition. Switch and 7a. Step Use the Associative Property of Addition to group like terms. a and 7a are like terms. a + 7( + a) = a + + 7a = a + 7a + = (a + 7a) + Step 4 Combine like terms. = a + Compare the expressions: a + and a +. They are the same. So, the expressions a + 7( + a) and a + are equivalent. Use properties to determine whether the expressions are equivalent.. 6(p + q) and 6p + q. 7y - + y and 9y -. + (8r + 9) and ( + 8) + 8r not equivalent equivalent equivalent n and n. 6s s and s 6. d and d equivalent 7. 0(e + 0.g) and 0e + g not equivalent 8. 8m + (9m - ) and 8m - 8 equivalent 9. 7( h) and h equivalent not equivalent not equivalent Expressions and Equations

19 Identify Equivalent Expressions CC.6.EE.4 Use properties of operations to determine whether the expressions are equivalent.. s + + s and 7s +. 7h and h v - v and 8 - v equivalent equivalent not equivalent 4. (9w 0) - and 9w -. (p + q) and p + (7q + 4q) 6. 6(4b + d) and 4b + d not equivalent equivalent not equivalent 7. 4m + 9-6m and 8m (y ) + and y (6t + ) and 9 + 0t equivalent equivalent equivalent 0. 9x x and 9x +. c - c and (4c - ). 6a 4 and 4a not equivalent not equivalent equivalent Problem Solving. Rachel needs to write book reports with b pages and science reports with s pages during the school year. Write an algebraic expression for the total number of pages Rachel will need to write. 4. Rachel s friend Yassi has to write (b + s) pages for reports. Use properties of operations to determine whether this expression is equivalent to the expression for the number of pages Rachel has to write. b + s equivalent Lesson 6

20 LESSON 7 Solutions of Equations OBJECTIVE Determine whether a number is a solution of an equation. CC.6.EE. An equation is a statement that two mathematical expressions are equal. Some equations include only numbers, operation signs, and an equal sign. Example: + 7 = 9 Other equations also include variables, such as x. Example: 0 - x = 7 For an equation with a variable, a solution is a value of the variable that makes the equation true. Equation: m = Is m =. a solution? Is m = 4.4 a solution? Step Write the equation m = m = Step Substitute the given number for the variable m Step Add..9 = Decide whether the equation is true. ( means does not equal) The equation is not true. So, m =. is not a solution. The equation is true. So, m = 4.4 is a solution. Determine whether the given value of the variable is a solution of the equation.. p - 4 = 6; p = y = ; y n + = 6; n = = 6 solution solution not a solution k = ; k =.4. + t = ; t = 6. 4x = 6; x = 8 not a solution solution not a solution Expressions and Equations

21 Solutions of Equations CC.6.EE. Determine whether the given value of the variable is a solution of the equation.. x - 7 = ; x = 8. c + = 0; c = 9. 7n = 7; n = not a solution solution not a solution 4. h = 6; h =. a - = 70; a = j = ; j = 8 8 not a solution solution solution d = ; d = = e; e = y = 7.9; y = not a solution solution not a solution Problem Solving 0. Terrance needs to score points to win a game. He has already scored 8 points. The equation 8 + p = gives the number of points p that Terrance still needs to score. Determine whether p = 7 or p = is a solution of the equation, and tell what the solution means. p = 7 is a solution; p = is not a solution; Terrance needs to score 7. Madeline has used 0 sheets of a roll of paper towels, which is _ 8 of the entire roll. The equation _ 8 s = 0 can be used to find the number of sheets s in a full roll. Determine whether s = or s = 80 is a solution of the equation, and tell what the solution means. s = is not a solution; s = 80 is a solution; There are 80 sheets in a points to win. full roll. 4 Lesson 7

22 Solutions of Inequalities LESSON 8 CC.6.EE. OBJECTIVE Determine whether a number is a solution of an inequality. An inequality is a mathematical sentence that compares expressions. A solution of an inequality is a value for a variable that makes the inequality true. For the inequality a < (a is less than ), a = is a solution because is less than. a = is not a solution because is not less than. Inequalities use these symbols: < (less than), > (greater than), (less than or equal to), and (greater than or equal to). Step Understand the inequality. Step Decide whether the value is a solution. For the inequality x, is x = a solution? x means x is less than or equal to. Any value that is equal to or less than is a solution. is less than, so x = is a solution. For the inequality y > 8, is y = a solution? y > 8 means y is greater than 8. Any value that is greater than 8 is a solution. is not greater than 8, so y = is not a solution. Determine whether the given value of the variable is a solution of the inequality.. m 4; m = m 4 means m is greater than or. k < 7; k =. z ; z = equal to m = is 4. not a solution solution solution 4. y ; y = 6. n > ; n = 8 6. t < 7; t = not a solution not a solution solution Give two solutions of the inequality. Possible answers are given. 7. x > 4 8. p 9. v 9 x = ; x = 6 p = ; p = v = 9 ; v = 0 Expressions and Equations

23 Solutions of Inequalities CC.6.EE. Determine whether the given value of the variable is a solution of the inequality.. s - ; s =? -. p < 0; p = 4. y - ; y = - solution not a solution not a solution 4. u > - ; u = 0. q 0.6; q = b < 4 ; b = solution not a solution solution 7. j -.7; j = a > - 8; a = w 4.; w = 4.4 solution solution not a solution Give two solutions of the inequality. Possible answers are given. 0. k <,. z -. f - k = 0; k = z = - ; z = f = - ; f = -. Problem Solving Possible answers are given.. The inequality s 9 represents the score s that Jared must earn on his next test to get an A on his report card. Give two possible scores that Jared could earn to get the A. 4. The inequality m $0 represents the amount of money that Sheila is allowed to spend on a new hat. Give two possible money amounts that Sheila could spend on the hat. 9; 9 $0; $9 6 Lesson 8

24 LESSON 9 Use Algebraic Expressions OBJECTIVE Use algebraic expressions to solve problems. CC.6.EE.6 You can use an algebraic expression to help solve a word problem. Use a variable to represent the unknown number. Ina wants to serve salad at her party. She will need one head of lettuce for every 6 guests who attend. Write an expression she could use for deciding how much lettuce she needs. Step Decide what operation the problem uses. Step Identify the unknown number. Step Write a word expression. Then use the word expression to write an algebraic expression. Each head of lettuce will serve 6 people. Divide the number of guests by 6. The problem does not state how many guests will attend. Use the variable g for the number of guests. the number of guests divided by 6 g 6 or g 6 Ina finds out that 8 guests will attend. Evaluate the expression for this number of guests. Step Substitute 8 for g. 8 6 Step Divide. 8 6 = So, Ina will need heads of lettuce. At her last party, Ina decorated with window stickers. For this party, she wants to use 4 times as many stickers.. Write an expression for the number of stickers, s, Ina used at her last party.. Ina wants to put an equal number of stickers on each of the windows. Write an expression to show how many stickers will go on each window, w.. Use the expression to find the new number of stickers if she used 4 stickers for her last party. s 4 or 4 s 4 4 = 6 4. Use the expression to find the number of stickers for each window if there are 8 windows. 6 w or 6 w 6 8 = 7 Expressions and Equations 7

25 Use Algebraic Expressions CC.6.EE.6 Jeff sold the pumpkins he grew for $7 each at the farmer s market.. Write an expression to represent the amount of money Jeff made selling the pumpkins. Tell what the variable in your expression represents. 7p, where p is the number. If Jeff sold 0 pumpkins, how much money did he make? $0 of pumpkins An architect is designing a building. Each floor will be feet tall.. Write an expression for the number of floors the building can have for a given building height. Tell what the variable in your expression represents. h where h is the height of the building 4. If the architect is designing a building that is feet tall, how many floors can be built? floors Write an algebraic expression for each word expression. Then evaluate the expression for these values of the variable:, 6,.. The quotient of 00 and the sum of b and 4 00 (b + 4); 4; ;. 6 Problem Solving 7. In the town of Pleasant Hill, there is an average of 6 sunny days each month. Write an expression to represent the approximate number of sunny days for any number of months. Tell what the variable represents. 6. more than the product of m and m + ; 8; 4; How many sunny days can a resident of Pleasant Hill expect to have in 9 months? 6s; s is the number of months 44 days 8 Lesson 9

26 LESSON 60 Write Equations OBJECTIVE Write algebraic equations. CC.6.EE.7 To write an equation for a word sentence, write the words as mathematical expressions and write = for equals or is. Write an equation for the word sentence. Example 6 fewer than a number is. Step Choose a variable. 6 fewer than a number is. Let n represent a number. Step Identify the operation. Step Write an equation. 6 fewer than n is. 6 fewer than a number is. Fewer than means subtract. So, the equation is n - 6 =. Example The quotient of 0.7 gallons and a number is 9 gallons. n - 6 = 0.7 p = 9 So, the equation is 0.7 p = 9. Write an equation for the word sentence.. 8 more than a number is 9... times a number is n + 8 = 9.n = less than a number is Four fifths of a number equals. n - 8 = 4 4 n =. The product of a number and 6 is The number of miles decreased by 9.8 is 9. 6p = 8 m = 9 Expressions and Equations 9

27 Write Equations CC.6.EE.7 Write an equation for the word sentence.. 8 is 4. times a number.. Eight more than the number of children is 4. 8 = 4.n 8 + c = 4. The difference of a number and 4. m minutes less than 80 minutes is _ is _ 8. minutes. n - = m =. A number divided by 0. is The product of the number of songs and $0.99 is $7.9. n 0. = 9 s 0.99 = 7.9 Write a word sentence for the equation. Possible answers are given. 7. x - 4 = 8..m = 0.46 The difference of x and 4 is. The product of. and m is = k q = 6 is the quotient of k and. q more than 4 is 6. Problem Solving. An ostrich egg weighs.9 pounds. The difference between the weight of this egg and the weight of an emu egg is.6 pounds. Write an equation that could be used to find the weight w, in pounds, of the emu egg.. In one week, the number of bowls a potter made was 6 times the number of plates. He made 90 bowls during the week. Write an equation that could be used to find the number of plates p that the potter made..9 - w =.6 6p = 90 0 Lesson 60

28 LESSON 6 Model and Solve Addition Equations OBJECTIVE Use models to solve addition equations. CC.6.EE.7 You can use algebra tiles to model and solve equations. Use a long rectangle to represent the variable, and a square to represent. Model and solve the equation x + 9 =. Step Model the equation using algebra tiles. = Step Get the variable by itself on one side of the equation. Remove the same number of tiles from each side. = Step Write the solution. = x = Solve the equation by using algebra tiles or by drawing a picture. Check students models.. x + 4 = 0. 8 = x + x = 6 x = 6 Expressions and Equations

29 Model and Solve Addition Equations Model and solve the equation by using algebra tiles. CC.6.EE.7. x + 6 = 9. x + = 6. 9 = x + x = x = x = x = 0. x + 7 = 6. 4 = + x x = x = 4 x = Solve the equation by drawing a model. 7. x + 4 = 7 8. x + 6 = 0 = = = = = = Problem Solving x = 9. The temperature at 0:00 was 0ºF. This is ºF warmer than the temperature at 8:00. Model and solve the equation x + = 0 to find the temperature x in degrees Fahrenheit at 8:00. x = 4 0. Jaspar has 7 more checkers left than Karen does. Jaspar has 9 checkers left. Write and solve an addition equation to find out how many checkers Karen has left. x = 7; 7ºF c + 7 = 9; c = ; checkers Lesson 6

30 LESSON 6 Solve Addition and Subtraction Equations OBJECTIVE Use algebra to solve addition and subtraction equations. CC.6.EE.7 To solve an equation, you must isolate the variable on one side of the equal sign. You can use inverse operations: undoing addition with subtraction or subtraction with addition. These actions are made possible by the Addition and Subtraction Properties of Equality. Solve and check. Example : y = 9.8 Example : 7 = x - 8 Step Look at the side with the variable. Subtract the number that is added to the variable, or add the number that is subtracted from the variable. Be sure to perform the same operation on both sides of the equation. y = = x - 8 y = Subtract 6.7 from = x Add 8 to both sides. both sides. Step Simplify both sides of the equation. y = = x - 8 y = = x y + 0 =. 6 = x + 0 y =. Step Check your answer in the original equation. 6 = x y = = x = = 7 So, y =. is the solution. Solve and check. So, x = 6 is the solution.. x + = 7. 8 = d -..4 = a w - = 4 x = 4 d = 60 a = 4. w = 7 Expressions and Equations

31 Solve Addition and Subtraction Equations Solve the equation, and check the solution.. y - 4 =. x + =. n + = 4 CC.6.EE.7 y = + 4 y = 7 x = n = 4. 6 = m - 4. w -.7 =.8 6. s + = m = 0 w = 6. s = 0 7. = x - 8. p - 4 = 4 9. m - 4 = 6 x = p = 8 m = t =.. = b = d + t = 0. b = 4 d = Problem Solving. A recipe calls for _ cups of flour. Lorenzo only has _ cups of flour. 4 Write and solve an equation to find the additional amount of flour Lorenzo needs to make the recipe. 4 + a = ; a = 4 ; 4 cups 4. Jan used. gallons of water in the shower. This amount is 7. gallons less than the amount she used for washing clothes. Write and solve an equation to find the amount of water Jan used to wash clothes. a - 7. =.; a = 0; 0 gallons 4 Lesson 6

32 LESSON 6 Model and Solve Multiplication Equations OBJECTIVE Use models to solve multiplication equations. CC.6.EE.7 You can use algebra tiles or a drawing to model and solve equations. Use a rectangle to represent the variable and a square to represent. Model and solve the equation x = 9. Step Model the equation using rectangles and squares. = x = 9 Step Divide the squares into equal groups. The number of groups should be the same as the number of rectangles. x = 9 Step Find the number of squares in each group. x = So, x = is the solution. Solve the equation by using algebra tiles or by drawing a picture.. 4x =. x = 6 x = x = 8 Check students models. Expressions and Equations

33 = Name Model and Solve Multiplication Equations CC.6.EE.7 Model and solve the equation by using algebra tiles.. x = 8. x = 0. = x x = 4 x = x = x = 0. 6x = = x x = x = x = Solve the equation by drawing a model = x 8. 4x = = Problem Solving x = x = 9. A chef used 0 eggs to make omelets. Model and solve the equation x = 0 to find the number of eggs x in each omelet. 0. Last month, Julio played times as many video games as Scott did. Julio played 8 video games. Write and solve an equation to find the number of video games Scott played. x = 4; 4 eggs v = 8; v = 6; 6 games 6 Lesson 6

34 LESSON 64 Solve Multiplication and Division Equations OBJECTIVE Use algebra to solve multiplication and division equations. CC.6.EE.7 A multiplication equation shows a variable multiplied by a number. A division equation shows a variable divided by a number. To solve a multiplication equation, you use the Division Property of Equality. To solve a division equation, you use the Multiplication Property of Equality. These properties state that both sides of an equation remain equal when you multiply or divide both sides by the same number. Solve and check. Example : a = 6 Example :.x = 0 Step Look at the side with the variable. Use the inverse operation to get the variable by itself. a = 6 a is divided by..x = 0 x is multiplied by.. a = 6 Multiply both sides by..x. = 0 Divide both sides by... Step Simplify both sides of the equation. a = 6.x = 0 a = 6.x. = 0. a = 0 x = 4 Step Check your answer in the original equation. a = 6.x = = 6 0 = 0 So, a = 0 is the solution. Solve and check. So, x = 4 is the solution.. x = 4. 4c = =.d 4. =.w x = 4. z = 9 6. d 6 4 c = = 7. = n.4 d = 4 w = 8 8. = 4 k z = 4 d = 0 n = 6.4 k = Expressions and Equations 7

35 Solve Multiplication and Division Equations CC.6.EE.7 Solve the equation, and check the solution.. 8p = 96. z = 8..x = p 8 = 96 8 p = z = 8 x = =.c. w = a 4 = 6.8 c = 0 w = 00 a = x = =.b 9. = t x = b = 6.8 t = x 7 = 0. 4n = 9. 4 g = 8 x = 0 Problem Solving. Anne runs 6 laps on a track. She runs a total of mile, or,80 feet. Write and solve an equation to find the distance, in feet, that she runs in each lap. 6d =,80; d = 880; 880 feet n = 9 4 or 4 g = 6 4. DeShawn uses _ of a box of rice to 4 cook dinner. The portion he uses weighs ounces. Write and solve an equation to find the weight of the full box of rice. w = ; w = 6; 6 ounces 4 8 Lesson 64

36 LESSON 6 Problem Solving Equations with Fractions OBJECTIVE Solve equations involving fractions by using the strategy solve a simpler problem. CC.6.EE.7 After driving miles, Kevin has traveled _ of the distance from his house to his friend s house. Use the equation = _ d to find the total distance d in miles to his friend s house. Read the Problem What do I need to find? I need to find the distance in miles from Kevin s house to his friend s house. What information do I need to use? I need to use the equation = d. Solve the Problem How will I use the information? I can use multiplication to change the equation to an equation with only whole numbers not fractions. Then I can solve the new equation., Step Write the original equation. Step Write a simpler equation without fractions. Multiply both sides by the denominator of the fraction. Step Solve the simpler equation. Use the Division Property of Equality. So, the total distance is 7. miles. Solve.. Alyssa s cat weighs pounds, which is _ of the weight of her dog. Use the 8 equation _ d = to find the weight 8 of Alyssa s dog. = d = ( )d 7 = 6 d 7 = d 7 = d 7. = d. Randall bought 6 baseball cards from Max, which is _ of Max s collection. Use the equation 6 = _ c to find the number of cards that were in Max s collection. pounds 40 cards Expressions and Equations 9

37 Problem Solving Equations with Fractions CC.6.EE.7 Read each problem and solve.. Stu is 4 feet tall. This height represents 6_ of his 7 brother s height. The equation 6_ h = 4 can be used 7 to find the height h, in feet, of Stu s brother. How tall is Stu s brother? h = 7 4 6h = 8 6h 6 = 8 6 h = 4 4 feet. Bryce bought a bag of cashews. He served 7_ 8 pound of cashews at a party. This amount represents _ of the entire bag. The equation _ n = 7_ can be used to find the 8 number of pounds n in a full bag. How many pounds of cashews were in the bag that Bryce bought? 6 pounds. In Jaime s math class, 9 students chose soccer as their favorite sport. This amount represents _ of the entire 8 class. The equation _ s = 9 can be used to find the 8 total number of students s in Jaime s class. How many students are in Jaime s math class? 4. There are blueberry muffins in a large basket. This represents _ of all the muffins that are in the basket. 9 The equation _ m = can be used to find the total 9 number of muffins m in the basket. How many muffins are in the basket? 4 students 7 muffins 0 Lesson 6

38 LESSON 66 Write Inequalities OBJECTIVE Write algebraic inequalities. CC.6.EE.8 Here are some ways to express each inequality symbol in words: < less than under not as much as less than or equal to at most no more than > greater than over more than greater than or equal to at least no less than Passengers at least years old pay full price for train tickets. Write an inequality to represent the situation. Step Choose a variable. Use a to represent age. Step Choose an inequality symbol. at least years old means greater than or equal to. a Step Write the inequality. a Write two word sentences to represent y < 9. Step Identify the inequality symbol. < means less than. Step Write a word sentence that uses the variable y is less than 9. and integer. Step Write another word sentence with the y is under 9. same meaning. Write an inequality for the word sentence.. The distance d Mr. Chin drove was no more than 6 miles.. The age a of Mia s sister is less than 8 years. Write two word sentences to represent the inequality.. The amount of juice c in the punch is more than cups. d 6 c > a < 8 4. The temperature t was at least 0 F. t 0. n 6. p > 6 n is greater than or equal to ; n is at least. p is greater than 6; p is more than 6. Expressions and Equations

39 Write Inequalities CC.6.EE.8 Write an inequality for the word sentence. Tell what type of numbers the variable in the inequality can represent.. The width w is greater than 4 centimeters. The inequality symbol for is greater than is >. w > 4, where w is the width in centimeters. w is a positive number.. The score s in a basketball game is greater than or equal to 0 points. s 0, where s is any whole number. The mass m is less than kilograms. m <, where m is any positive number 4. The height h is greater than. meters.. The temperature t is less than or equal to -. h >., where h is any positive number t -, where t is any negative number Write a word sentence for the inequality. Possible answers are given. 6. k < z 4 k is less than - 7. z is greater than or equal to 4. Problem Solving Possible answers are given. 8. Tabby s mom says that she must read for at least 0 minutes each night. If m represents the number of minutes reading, what inequality can represent this situation? 9. Phillip has a $ gift card to his favorite restaurant. He wants to use the gift card to buy lunch. If c represents the cost of his lunch, what inequality can describe all of the possible amounts of money, in dollars, that Phillip can spend on lunch? m 0 c Lesson 66

40 LESSON 67 Graph Inequalities OBJECTIVE Represent solutions of algebraic inequalities on number line diagrams. CC.6.EE.8 You can graph the solutions of an inequality on a number line. Graph the inequality n 9. Step Determine the meaning of the inequality. n 9 means n is greater than or equal to 9. Step Draw a number line and circle the number given in the inequality Step Decide whether to fill in the circle. For or, fill in the circle to show or equal to. For < or >, do not fill in the circle. Since the inequality uses, 9 is a possible solution. So, fill in the circle Step 4 Shade from the circle in the Since the inequality symbol is, the direction of the remaining shading covers all numbers greater than 9. solutions Graph the inequality.. k < 8. r w 4. x > Write the inequality shown by the graph x 7 x > Expressions and Equations

41 Graph Inequalities CC.6.EE.8 Graph the inequality.. h Draw a filled-in circle at to show that is a solution. Shade to the right of to show that values greater than are solutions x < -4. y > b < 8. m n 7. c Write the inequality represented by the graph Problem Solving x > x > - 0. The inequality x represents the elevation x of a certain object found at a dig site. Graph the solutions of the inequality on the number line The inequality x 44 represents the possible scores x needed to pass a certain test. Graph the solutions of the inequality on the number line Lesson 67

42 LESSON 68 Independent and Dependent Variables OBJECTIVE Write an equation to represent the relationship between an independent variable and a dependent variable. CC.6.EE.9 An equation with two variables shows a relationship between two quantities. The value of the dependent variable changes according to the value of the independent variable. Sam rides the bus almost every day. He pays $.0 for each bus ride. Identify the dependent and independent variables in this situation. Then write an equation to represent the relationship between the total cost and the number of bus rides. Step Understand the relationship and identify variables. Each bus ride costs $.0. The total cost c for Sam s bus rides depends on the number of rides r he takes. The value of c will change when the value of r changes. So, c is the dependent variable and r is the independent variable. Step Write an equation. The total cost will be $.0 multiplied by the number of rides. c =.0 r (or c =.0r) Use your equation to find out how much it would cost for Sam to take 4 bus rides. Step Think: 4 bus rides means r = 4. Step Substitute 4 for r in the equation. So, Sam s total cost will be $0.00 for 4 rides. c =.0 r c =.0 4 c = 0.00 Identify the dependent and independent variables. Write an equation to show the relationship between them. Then solve for the given value.. Janna is buying a netbook with a flash drive. The total cost c will include the price p of the netbook, plus $.0 for the flash drive. Find the total cost if the price of the netbook is $7.00. The total cost netbook price depends on the dependent variable: c independent variable: p equation: c = p +.0 Total cost: c = c = Expressions and Equations

43 Independent and Dependent Variables CC.6.EE.9 Identify the independent and dependent variables. Then write an equation to represent the relationship between them.. Sandra has a coupon to save $ off her next purchase at a restaurant. The cost of her meal c will be the price of the food p that she orders, minus $. The cost of her meal the price of her food. dependent variable: c independent variable: p equation: c = p - depends on. An online clothing store charges $6 for shipping, no matter the price of the items. The total cost c in dollars is the price of the items ordered p plus $6 for shipping. dependent variable: independent variable: equation: c = c p p + 6. Melinda is making necklaces. She uses beads for each necklace. The total number of beads b depends on the number of necklaces n. dependent variable: independent variable: equation: b = b n n 4. Tanner is years younger than his brother. Tanner s age t in years is less than his brother s age b. dependent variable: independent variable: equation: t = Problem Solving 6. Maria earns $4 for every lawn that she mows. Her earnings e in dollars depend on the number of lawns n that she mows. Write an equation that represents this situation. t b b -. Byron is playing a game. He earns 0 points for each question he answers correctly. His total score s equals the number of correct answers a times 0. dependent variable: independent variable: equation: s = 7. Martin sells cars. He earns $00 per day, plus any commission on his sales. His daily salary s in dollars depends on the amount of commission c. Write an equation to represent his daily salary. s a 0a e = 4n s = 00 + c 6 Lesson 68

44 LESSON 69 Equations and Tables OBJECTIVE Translate between equations and tables. CC.6.EE.9 You can use tables and equations to represent the relationship between two quantities. Use the equation to complete the table. y = x 4 Step Look at the equation to find the rule. The rule for finding y is x 4. Step Apply the rule and fill in the unknown values. Divide each x-value by = 6 4 = = = x y Write an equation for the relationship. Input, x Output, y Find a pattern. Think: What can I do to each x-value to find its corresponding y-value? The y-values are less than the x-values, so try dividing or subtracting. x y x y x y x y x y 0 = 6 = 7 40 = 8 4 = 9 0 = 0 The pattern is to divide x by to get y. The equation is y = x. Write an equation for the relationship shown in the table. Then find the unknown value in the table... x y 4 8 x 4 6 y y = x + y = 6x Use the equation to complete the table.. y = 7x 4. y = x - 6 Input, x 4 Input, x 8 4 Output, y Output, y 6 9 Expressions and Equations 7

45 Equations and Tables CC.6.EE.9 Use the equation to complete the table.. y = 6x. y = x - 7. y = x + 4 Input Output Input Output Input Output x y x y x y Write an equation for the relationship shown in the table. Then find the unknown value in the table. 4.. x 4 y 6? 40 y = 8x; 4 x y 9 0? y = x ; x x y 7? y? 4 Problem Solving y = x + ; 9 8. Tickets to a play cost $ each. There is also a service charge of $4 per order. Write an equation for the relationship that gives the total cost y in dollars for an order of x tickets. 9. Write an equation for the relationship shown in the table. Then use the equation to find the estimated number of shrimp in a -pound bag. Weight of bag (pounds), x y = x - 9; 8 4 Estimated number of shrimp, y y = x + 4 y = 4x; 0 shrimp 8 Lesson 69

46 LESSON 70 Problem Solving Analyze Relationships OBJECTIVE Solve problems involving relationships between quantities by using the strategy find a pattern. CC.6.EE.9 The table shows the number of miles an overnight train travels. If the pattern in the table continues, how far will the train travel in 0 hours? Overnight Train Travel Rate Time (hours) 4 Distance (miles) Read the Problem What do I need to find? I need to find the number of miles the train will travel in 0 hours. What information do I need to use? I need to find the relationship between time and distance shown in the table. Solve the Problem How will I use the information? I will find a pattern in the table and use the pattern to write an equation. Look for a pattern Overnight Train Travel Rate between the number of hours Time in hours, h 4 and the number of miles. Distance in miles, m Then write an equation to show the pattern. Equation: m = 60 h To find the miles the train will travel m = 60 0 in 0 hours, substitute 0 for h. m = 600. The table shows how much a restaurant pays for coffee. How much will the restaurant pay for 00 pounds of coffee? $400 Coffee Purchasing Pounds, p Cost, c $0 $40 $0 $40 Expressions and Equations 9

47 Problem Solving Analyze Relationships CC.6.EE.9 The table shows the number of cups of yogurt needed to make different amounts of a fruit smoothie. Use the table for. Batches, b 4 6 Cups of Yogurt, c 9 8. Write an equation to represent the relationship. The number of cups needed is multiplied by the number of batches, so c = b.. How much yogurt is needed for 9 batches of smoothie? 7 cups. Jerry used cups of yogurt to make smoothies. How many batches did he make? batches The table shows the relationship between Winn s age and his sister s age. Use the table for 4 6. Winn s age, w Winn s sister s age, s 4 4. Write an equation to represent the relationship. s =. When Winn is 4 years old, how old will his sister be? w When Winn s sister is years old, how old will Winn be? 9 40 Lesson 70

48 LESSON 7 Graph Relationships OBJECTIVE Graph the relationship between two quantities. CC.6.EE.9 You can use a graph to represent a relationship. Graph the relationship represented by the table to find the unknown value of y. Step Write ordered pairs that you know. (, 4), (, 6), (7, 0) x 7 y Step Plot the points. y x Step Find the unknown y-value. Use a ruler to draw a line through the points. Find the y-value that corresponds to an x-value of. So, when the x-value is, the y-value is 8. Graph the relationship represented by the table to find the unknown value of y... x 4 x y y x y y x y = y = 6 Expressions and Equations 4

49 Graph Relationships CC.6.EE.9 Graph the relationship represented by the table... x 4 x y y y x y x Problem Solving. Graph the relationship represented by the table. DVDs Purchased 4 Cost ($) Use the graph to find the cost of purchasing DVDs. $7 Cost ($) Cost of DVDs y DVDs Purchased x 4 Lesson 7