Standards of Learning Content Review Notes Grade 7 Mathematics 3 rd Nine Weeks, 2016-2017 1
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Content Review: Standards of Learning in Detail Grade 7 Mathematics: Third Nine Weeks 2016-2017 This resource is intended to be a guide for parents and students to improve content knowledge and understanding. The information below is detailed information about the Standards of Learning taught during the 3 rd grading period and comes from the Mathematics Standards of Learning Curriculum Framework, Grade 7 issued by the Virginia Department of Education. The Curriculum Framework in its entirety can be found at the following website: http://www.doe.virginia.gov/testing/sol/frameworks/mathematics_framewks/2009/fra mewk_math7.pdf SOL 7.11 The student, given data in a practical situation, will a) construct and analyze histograms; and b) compare and contrast histograms with other types of graphs presenting information from the same data set. All graphs tell a story and include a title and labels that describe the data. A histogram is a form of bar graph in which the categories are consecutive and equal intervals. The length or height of each bar is determined by the number of data elements falling into a particular interval. A frequency distribution shows how often an item, a number, or range of numbers occurs. It can be used to construct a histogram. Comparisons, predictions and inferences are made by examining characteristics of a data set displayed in a variety of graphical representations to draw conclusions. The information displayed in different graphs may be examined to determine how data are or are not related, ascertaining differences between characteristics (comparisons), trends that suggest what new data might be like (predictions), and/or what could happen if (inference). 3
SOL Practice Items provided by the VDOE, http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/index.shtml Answers are located on the last page of the booklet. SOL 7.11 (Histograms) 1 2 4
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SOL 7.2 The student will describe and represent arithmetic and geometric sequences using variable expressions. In the numeric pattern of an arithmetic sequence, students must determine the difference, called the common difference, between each succeeding number in order to determine what is added to each previous number to obtain the next number. Example 1: 3, 7, 11, 15, 19, The common difference is 4 (add 4 to the previous number). Example 2: 30, 25, 20, 15, 10, The common difference is -5 (add -5 to the previous number). In geometric sequences, students must determine what each number is multiplied by in order to obtain the next number in the geometric sequence. This multiplier is called the common ratio. Sample geometric sequences include: Example 1: 2, 4, 8, 16, 32, The common ratio is 2 (multiply times 2). Example 2: 80, 20, 5, 1.25, The common ratio is 4 1 (multiply times 4 1 or 0.25). Example 3: Below is a geometric sequence. What is the 8 th term in the sequence? 3, 9, 27, 81, 243, 729, The common ratio is 3 (each number is multiplied by 3 to get the next number). Multiply 729 3 to get the 7 th term. The 7 th term is 2,187. Next, multiply the 7 th term by 3. 2,187 3 = 6,561 The 8 th term in the geometric sequence is 6,561. 6
A variable expression can be written to express the relationship between two consecutive terms of a sequence. Examples: If n represents a number in the sequence 3, 6, 9, 12, the next term in the sequence can be determined using the variable expression n + 3. If n represents a number in the sequence 1, 5, 25, 125, the next term in the sequence can be determined by using the variable expression 5n. SOL Practice Items provided by the VDOE, http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/index.shtml Answers are located on the last page of the booklet. SOL 7.2 (Patterns & Sequences) 1 2 What is the common difference of the arithmetic sequence shown below? -5, -1, 3, 7 A 2 B 4 C 6 D 8 3 7
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SOL 7.13 The student will a) write verbal expressions as algebraic expressions and sentences as equations and vice versa; and b) evaluate algebraic expressions for given replacement values of the variables. An expression is a name for a number. An expression that contains a variable is a variable expression. An expression that contains only numbers is a numerical expression. A verbal expression is a word phrase. (e.g., the sum of two consecutive integers ) A verbal sentence is a complete word statement. (e.g., The sum of two consecutive integers is five. ) An algebraic expression is a variable expression that contains at least one variable. (e.g., 2x 5) Examples of Algebraic Expressions and Equivalent Verbal Expressions: Algebraic Expression x + (x + 1) Verbal Expression The sum of two consecutive integers 2x 4 3x + 8 Four less than twice a number Three times a number increased by eight Key words in translating verbal expressions/sentences to algebraic expressions/equations may include words and their translations such as: is to =, of to multiplication, more than to +, less than to, increased by to +, and decreased by to. An algebraic equation is a mathematical statement that says that two expressions are equal. (e.g., 2x + 1 = 5) Examples of Algebraic Equations and Equivalent Verbal Sentences: Algebraic Equation Verbal Sentence 30 40 = x Forty less than thirty is a number. x + 5 = 8 The sum of a number and five is eight. 3 + 2x = 15 Three more than twice a number is fifteen. 9
To evaluate an algebraic expression, substitute a given replacement value for a variable and apply the order of operations. For example, if a = 3 and b = -2 then 5a + b can be evaluated as: 5(3) + (-2) = 15 + (-2) = 13 The replacement values are the numbers that replace the variables in an algebraic expression. Example: If x = (-5), what is the value of this expression? x + 4 10 Step 1: x + 4 10 Step 2: (-5) + 4 10 Step 3: -5 + 40 Step 4: The answer is 35. SOL Practice Items provided by the VDOE, http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/index.shtml Answers are located on the last page of the booklet. SOL 7.13 (Verbal Expressions and Algebraic Expressions & Sentences) 1 2 10
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SOL 7.14 The student will a) solve one- and two-step linear equations in one variable; and b) solve practical problems requiring the solution of one- and two-step linear equations. An equation is a mathematical sentence that states that two expressions are equal. A one-step equation is defined as an equation that requires the use of one operation to solve. 1 Example: x + 3 = 4 x + 3 3 = 4 3 x = 7 The inverse operation for addition is subtraction, and the inverse operation for multiplication is division. A two-step equation is defined as an equation that requires the use of two operations to solve. Examples: 2x + 1 = -5; 2x + 1 1 = -5 1 2x = -6 2x 6 2 2 x = -3 x 7 4 3 x 7 3 4 3 3 x 7 12 x 7 7 12 7 x 19 SOL Practice Items provided by the VDOE, http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/index.shtml Answers are located on the last page of the booklet. SOL 7.14 (One- And Two-Step Linear Equations) 1 15
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SOL 7.15 The student will a) solve one-step inequalities in one variable; and b) graph solutions to inequalities on the number line. A one-step inequality is defined as an inequality that requires the use of one operation to solve. Examples: x 4 > 9 2n -14 The inverse operation for addition is subtraction, and the inverse operation for multiplication is division. When both expressions of an inequality are multiplied or divided by a negative number, the inequality symbol reverses. Example: 3x < 15 is equivalent to x > 5. Solutions to inequalities can be represented using a number line. Example 1: x < 2½ Example 2: s 2 2 s 2 + 2 2 + 2 s 4 0 1 2 3 4 5 6 7 Note: When the solution to an inequality is > or <, it is represented on a graph using an open circle (Example 1 above). When the solution to an inequality is or, it is represented on a graph using a closed circle (Example 2 above). 19
SOL Practice Items provided by the VDOE, http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/index.shtml Answers are located on the last page of the booklet. SOL 7.15 (One-Step Inequalities in One Variable) 1 2 3 4 20
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9 10 11 SOL 7.12 The student will represent relationships with tables, graphs, rules, and words. Rules that relate elements in two sets can be represented by word sentences, equations, tables of values, graphs, or illustrated pictorially. A relation is any set of ordered pairs. For each first member, there may be many second members. A function is a relation in which there is one and only one second member for each first member. Example: Which number replaces the? in the table? x -2-1 0 1 2 x + (-2) -2 + (-2)? 0 + (-2) 1 + (-2) 2 + (-2) y -4? -2-1 0 The value of x is given in the top row. The work for the function is given in the second row. Replace the x in x + (-2) so that it reads -1 + (-2) to determine the missing value. The correct answer is -3. 22
As a table of values, a function has a unique value assigned to the second variable for each value of the first variable. As a graph, a function is any curve (including straight lines) such that any vertical line would pass through the curve only once. Examples: These are examples of graphed functions. y = x + 3 y = x + 2 x y -3 0-2 1-1 2 0 3 1 4 Some relations are functions; all functions are relations. 23
SOL Practice Items provided by the VDOE, http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/index.shtml Answers are located on the last page of the booklet. SOL 7.12 (Relations & Functions) 1. Which table contains only values that satisfy the following? y = 2x 2. Which table contains only values that satisfy the following? y = x + 5 3. The graph displays the relationship between time and profit. Which equation represents the relationship between time (t) and profit (P)? 24
4. Which is true for all values in the table? 5. Which contains only values that make the following true? y = 2x + 7 6. Which table contains only values that satisfy the following? y = x 1 7. Which graph appears to contain all the ordered pairs in the table? 25
8. Graph three points that are on the line represented by y = 2x -1. The coordinates of the points must be integers. 9. Which table of values represents the same relationship as the rule? 10. Which rule is best represented by this graph? 26
11. Larry charges a customer a one-time fee of $15 plus $40 each week. Which table has values that represent this situation? 12. Plot three points on the coordinate plane that lie on the relation represented by The coordinates of the points must be integers. 13. Which number sentence represents the relation shown in this table? 27
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SOL Link QR Code 7.11 Creating Histograms http://www.youtube.com/watch?v=g1wuk-jv7ju 7.2 Finding the common difference of an arithmetic sequence http://tinyurl.com/ob5e97a 7.2 Compare arithmetic and geometric sequences http://tinyurl.com/pzlegf3 7.13 Writing and evaluating verbal expressions & sentences as algebraic expressions & equations https://www.youtube.com/watch?v=g6jesuhyhyy 7.13 Writing and evaluating verbal expressions & sentences as algebraic expressions & equations https://www.youtube.com/watch?v=miyybr9bto0&list=pl557a7d37c 1462AC3 29
7.14 Solving one-step equations https://www.youtube.com/watch?v=9dxrf6ttws4 7.14 Solving two-step equations https://www.youtube.com/watch?v=tcxfe8eh-du 7.14 Solving two-step equations https://www.youtube.com/watch?v=mapb3v-vlwi 7.14 Solving two-step equations using manipulatives http://www.youtube.com/watch?v=r0ex1bsu0jw 7.15 Solving and graphing one-step inequalities https://www.youtube.com/watch?v=oim2rvmgwcq 7.15 Solving and graphing one-step inequalities https://www.youtube.com/watch?v=jmr9jhla2-y 30
cumulative frequency frequency distribution histogram arithmetic sequence common difference common ratio geometric sequence Vocabulary SOL 7.11 Includes a running total of the frequencies of all the previous groups Shows how often an item, a number, or a range of numbers occurs A special kind of bar graph in which the bars are used to represent the frequency of numerical data that have been organized in intervals SOL 7.2 A sequence in which each term is found by adding the same number to the previous term In an arithmetic sequence, the number that is added to each previous number to obtain the next number In a geometric sequence, the number that is multiplied by each previous number to obtain the next number A sequence in which each term can be found by multiplying the previous term by the same number algebraic equation algebraic expression SOL 7.13 A mathematical statement that states that two expressions are equal An expression with at least one variable 31
verbal expression verbal sentence expression equation A word phrase A complete word statement A name for a number A mathematical sentence that states that two expressions are equal inverse operation one-step equation two-step equation one-step inequality function relation SOL 7.14 Operations that "undo" each other; addition and subtraction are inverse operations. Multiplication and division are inverse operations. An equation that requires only one operation to solve An equation that requires two operations to solve SOL 7.15 An inequality that requires only one operation to solve SOL 7.12 A rule that pairs exactly one element of a set with one and only one element of another set A set of ordered pairs, in which for each first number there may be many second numbers 32
SOL 7.13 Practice Activity Directions: Cut out each card and match the verbal expressions (on the left) with the algebraic expressions (on the right). The product of four and a number, decreased by seven 4x - 7 Triple the number of students in Erin s class divided by four is seven. 3x = 7 4 A number subtracted from seven is four. 7 x = 4 Seven less than a number is four. x 7 = 4 Four less than seven is a number. 7 4 = x The sum of a number and four is seven. x + 4 = 7 Four more than twice a number is seven. 4 + 2x = 7 The sum of two consecutive integers x + (x + 1) Four less than twice a number 2x - 4 Three times a number increased by seven 3x + 7 33
Released Test Answers (3 rd Nine Weeks) SOL 7.11 (Histograms) 1. G 2. 3. C 4. 5. B SOL 7.2 (Patterns & Sequences) 1. A 2. B 3. C 4. D 5. A 6. a) common difference b) 4 c) 136 7. 5, 1, 3, -3 5 SOL 7.13 (Verbal Expressions and Algebraic Expressions & Sentences) 1. G 2. J 3. F 4. A 5. B 6. A 7. C 8. G 9. F 10. F 11. F 12. G 13. A 14. G SOL 7.13 Continued 15. J 16. J 17. A 18. F 19. A 20. C 21. A SOL 7.14 (One- And Two-Step Linear Equations) 1. A 2. B 3. G 4. G 5. J 6. D 7. A 8. A 9. B SOL 7.15 (One-Step Inequalities in One Variable) 1. A 2. J 3. A 4. G 5. 6. C 7. A 8. B 9. 10. B 11. C 10. C 11. C 12. D 13. C 14. J 15. A 16. B 17. D 18. D 19. 13.5 hours 34
SOL 7.12 (Relations and Functions) 35