7 th Grade Mathematics

Transcription

1 7 th Grade Mathematics Unit #3: Applying Algebraic Reasoning to Write and Evaluate Equations, Expressions, and Inequalities Pacing: 42 Days Unit Overview This unit consolidates and expands upon students understanding of equivalent expressions as they apply the properties of operations to write expressions in both standard form and in factored form. They use linear equations to solve unknown angle problems and other problems presented within context to understand that solving algebraic equations is all about the numbers. Students use the number line to understand the properties of inequality and recognize when to preserve the inequality and when to reverse the inequality when solving problems leading to inequalities. They interpret solutions within the context of problems. Students extend their sixth-grade study of geometric figures and the relationships between them as they apply their work with expressions and equations to solve problems involving area of a circle and composite area in the plane, as well as volume and surface area of right prisms Source: Prerequisite Skills Vocabulary Mathematical Practices 1) Order of Operations 2) Commutative, Associative, and Distributive Properties 3) Find equivalent fractions by multiplying the numerator and denominator by the same number 4) Compute efficiently using all four operations Associative Property Commutative Property Distributive Property Identity Property Expression Coefficient Like Terms Expand Simplify Expand Linear Equivalent Variable Equation Inequality Solution Set Angle Complimentary Supplemental MP.1: Make sense of problems and persevere in solving them MP.2: Reason abstractly and quantitatively MP.3: Construct viable arguments and critique the reasoning of others MP.4: Model with mathematics MP.5: Use appropriate tools strategically MP.6: Attend to precision MP.7: Look for and make use of structure MP.8: Look for and express regularity in repeated reasoning

2 Common Core State Standards Progression of Skills Additional Standards (10%) Major Standards (70%) 7.G.5: Solve Multi- Step Angle Problems 7.EE.1: Add, subtract, factor, and expand linear expressions 7.EE.2: Understand contextual equivalent expressions 7.EE.3: Solve multi-step problems with rational numbers 7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. 7.EE.4a: Solve Problems with 2-Step Linear Equations 7.EE.4b: Solve Problems with 2-Step Linear Equations According to the PARCC Model Content Framework, Standard 7.EE.4 is an end of year fluency expectation in 7 th grade: In solving word problems leading to one-variable equations of the form px + q = r and p(x + q) = r, students solve the equations fluently. This will require fluency with rational number arithmetic (7.NS.1 3), as well as fluency to some extent with applying properties operations to rewrite linear expressions with rational coefficients (7.EE.1). According to the PARCC Model Content Framework, Solving an equation such as 4 = 8(x 1/2) requires students to see and make use of structure (MP.7), temporarily viewing x 1/2 as a single entity. 6 th Grade 7 th Grade 8 th Grade 6.EE.6: 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. 6.EE.5: 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 set makes an equation or inequality true. 6.EE.9: 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. 6.EE.9: 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. 7.EE.1: Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. 7.EE.2: Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. 7.EE.3: Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. 7.EE.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. 8.EE.7: Solve linear equations in one variable. N/A 8.EE.8C: Solve real-world and mathematical problems leading to two linear equations in two variables. 8.EE.C.8: Solve real-world and mathematical problems leading to two linear equations in two variables. 2 Page

3 Big Ideas How can I formulate and use different strategies to solve one and two step equations? How can I use the properties of equality to express an equation in a different but equivalent way? What is the difference between expressions, equations and inequalities? What is the difference between an algebraic solution and an arithmetic solution? How do you graph the solution set of inequalites in the form px + q > r or px + q < r? Know/Understand What it means to expand or simplify a linear expression. Linear expressions can be expanded and simplified by using the properties of operations with addition, subtraction, combining like terms, and factoring. Different expressions are equivalent if they can be simplified or rewritten to have the same value. Expressions can be decomposed and recomposed in different ways to generate equivalent expressions. That rewriting an expression in different forms can better explain or highlight different parts of the relationship between quantities in an expression. That estimation and mental fluency can be used to determine the reasonability of answers. Variables are letters or symbols are used to represent an unknown quantity in equations and inequalities. Real world and mathematical problems may be modeled using equations or inequalities. Equations are used to model real world or mathematical problems when trying to find the values of two expressions that are equal and may take the form of px + q = r and/or p(x+q)=r, where p, q, and rare rational numbers. When solving an equation it is possible to use inverse operations (algebraic) or substitution (arithmetic) by following the sequence of operations. Inequalities are used to model real world or mathematical problems when trying to find the values of two expressions that are not equal and may take the form of px + q > r and/or p(x+q)<r, where p, q, and r are rational numbers. Students Will Be Skilled At Writing a linear expression following a given sequence of operations. Adding, subtracting, factoring and expanding linear expressions with rational coefficients using the properties of operations, including but not limited to the distributive, commutative, and associative properties. Using properties of operations in a linear equation to identify equivalent expressions or to simplify an expression. Justify the reasonableness of answers by using fluency and estimation Writing equations (px + q = r and p(x + q) = r) and inequalities (px + q > r and p(x + q) < r), using variables to represent unknown quantities. Using inverse operations fluently to solve equations and inequalities. Finding the value of the variable in an algebraic equation quickly and accurately using inverse operations and substitution Comparing the solutions found to an equation when using inverse operations (algebraic) versus substitution (arithmetic). Writing equations (px + q = r and p(x + q) = r) and inequalities (px + q > r and p(x + q) < r) based off of real world problems. Solving word problems that are in the form of equations (px + q = r and p(x + q) = r) and inequalities (px + q > r and p(x + q) < r). 3 Page

4 Student Friendly Objective SWBAT 1 Evaluate and write algebraic expressions 2 Write algebraic expressions to describe and extend sequences Key Points/ Teaching Tips Unit Sequence Note: since is largely an opportunity to review key prerequisite skills, take some time to explicitly practice writing expressions to match a sequence given in words (i.e. see exit ticket and provided links) Exit Ticket Write an expression to describe each sequence of operations. (a) Add 3 to x, subtract the result from 1, then double what you have. (b) Add 3 to x, double what you have, then subtract 1 from the result. Instructional Resources Chapter 5 Lesson 1 th/grade-7/writevariable-expressions s.com/lessons/vol7/exp ressions.html Chapter 5 Lesson Create equivalent forms of expression to represent realworld scenarios Apply properties of operations to simplify expressions and generate equivalent expressions Pacing: 2 days Module 2 Lessons Pacing: 2 days The lesson listed here is intended to be used either for additional practice or a remediation resource; the Engage NY lessons should be at the heart of your planning for these two days For parts (a)-(e), select Yes or No to indicate whether each of these expressions is equivalent to 2(2x + 1). a) 4x + 2 Yes No b) 2(1 = 2x) Yes No c) 2(2x) + 1 Yes No d) 2x x + 1 Yes No e) x + x + x + x Yes No Module 3 Lessons 1 2 Remediation /Extra Practice Resource: Chapter 5 Lesson 3 om/prealgebra/06- properties/index.html 4 Page

5 7-8 Use area models and the distributive property to write products as sums and sums as products Pacing: 2 days Note: use the Learning Task resource as the heart of your lesson for day 1, then use Engage NY (you may choose only 1 or select problems from each) and lesson as additional practice after students have conceptually connected area models to the distributive property Learning Task: Distributing and Factoring Area Module 3 Lessons 3-4 Additional Practice: Chapter 5 Lesson 4 Module 3 Lesson 5 9 Apply the identity properties of 0 and 1 and the existence of inverses (opposites and reciprocals) to write equivalent expressions 10 Simplify algebraic expressions by combining like terms. Attend to precision to describe each component of an expression (terms, like terms, coefficients, constants, etc) It is recommended to use the real world link from page 387 of My Math as the hook for the lesson, so that students can conceptualize like terms by modeling a real world scenario; the remaining My Math resources in this lesson should be used for additional practice and/or reteach/remediation, as the Engage NY practice problems are more appropriate for 7 th grade. 1) Simplify the following expressions: 1. p + q + p + q + p 2. 5c + 2d 3c 4d 3. xy + yx 4. 2xy 4ac + 5yx + 4ac 2 Module 3 Lesson 6 Chapter 5 Lesson 5 essons/810 m/lessons/812- simplify-anexpression-with-afraction-by-addingor-subtracting-termswith-fractions 5 Page

6 11 Use substitution and the properties of operations to determine whether two expressions are equivalent Students must understand that two different expressions are equivalent if they can be simplified or rewritten to have the same value. x 1) If we multiply by 4, we get 2x Is 2x + 3 and equivalent expression to x? Why or why not? Public/PreviewResourc e/preview/ lessons/816 2) Look at each expression. Is it x + 3y equivalent to? 2 Select Yes or No for each expression. 3) The students in Mr. Sanchez's class are converting distances measured in miles to kilometers. To estimate the number of kilometers, Abby takes the number of miles, doubles it, then subtracts 20% of the result. Renato first divides the number of miles by 5, then multiplies the result by 8. (a) Write an algebraic expression that shows the process each student used. 6 Page

7 12 Flex Day (Instruction Based on Data) Recommended Resources: Chapter 5 Mid-Chapter Check (Page 386) Problem Solving Investigation (Pages ) Learning Task: Area and Algebra Guess My Number Algebra Magic 13 Apply properties of operations to add linear expressions 14 Apply properties of operations to subtract linear expressions Consider beginning the lesson with a problem students will need to come back to after the lesson so they are able to use the lesson in order to create their linear expression Encourage students to think about the importance of order of operations/sequence by asking them to explain how changing the sequence of operations for an expression affects the final result. Consider beginning the lesson with a problem students will need to come back to after the lesson so they are able to use the lesson in order to create their linear expression Encourage students to think about the importance of order of operations/sequence by asking them to explain how changing the sequence of operations for an expression affects the final result. 1) Which of the following is equivalent to the expression below? (-3m + 5) + (m-11) A. 4m 16 B. -4m 6 C. 2m 16 D. -2m 6 2) Which expression represents the sum of (2x 5y) and (x + y)? A. 3x 4y B. 3x 6y C. x 4y D. x 6y When 5 8 x +11 is subtracted from x 5 1, the result is 6 A. 5 8 x B. 5 8 x C. 5 8 x D. 5 8 x Chapter 5 Lesson 7 om/prealgebra/06- properties/index.html Chapter 5 Lesson 7 om/prealgebra/06- properties/index.html 7 Page

8 15 Factor and expand linear expressions 16 Write expressions to represent and solve real world problems involving taxes, discounts and mark-ups Review the root words of the properties to help students understand them and how to identify them Briefly review order of operations and different answers that can be produced when operations are moved and grouped differently Application Problem: You are the manager of a store and business is a little slow. The best way to attract more customers is to have a sale. Here are the sales you are considering: a. Buy one, get one of equal value half off b. Buy three of equal value for the price of two c. Apply a 40% discount on the item As a business owner, which sale should you run? Write an expression to represent each sale, then use substitution to plug in different values to help you make your decision: 1) Why is the following true? 2(x + y) = 2x + 2y 2) Rearrange, using the Associative Property: 2(3x) 3) Why is it true that 3(4x) = (4x)(3)? 4) Simplify 3a 5b + 7a. Justify your steps. 1) Leo bought a used car for x dollars. One year later the value of the car was 0.88x. Which expression is another way to describe the change in the value of the car? A. 0.12% decrease B. 0.88% increase C. 12% decrease D. 88% increase Chapter 5 Inquiry Lab Chapter 5 Lesson 8 m/lessons/1128-factorlinear-expressions m/lessons/815- rewrite-anexpression-byexpanding-it lessons/3441-write-anexpression-to-find-thecost-of-an-item-withtax lessons/3442-write-apercent-markupexpression lessons/813-write-apercent-increaseproblem-as-a-productof-the-original-amount lessons/814-write-apercent-decreaseproblem-as-a-productof-the-original-amount 8 Page

9 17 Flex Day (Instruction Based on Data) Recommended Resources: Chapter 5 21 st Century Career (Pages ) Chapter 5 Review (Pages ) 18 Deduce that if a number sentence is true, then adding, subtracting, multiplying or dividing each value by the same number will result in the same outcome Students understand that if a number sentence is true and we make any of the following changes to the number sentence, the resulting number sentence will be true: Module 2 Lesson 21 If a = b, then a + c = b + c If a = b, then a - c = b c If a = b, then a( c )= b( c ) If a = b and c 0, then a c = b c 19 Apply if-then reasoning to solve one-step addition and subtraction equations We will use this language/thought process in lesson 22 (see it s accompanying resource). To lay the foundation have students think through problems using the same if/then reasoning we discussed yesterday. Example: X + 6 = 4 1) The difference between a number b and 7.4 is Write as an algebraic equation to represent this scenario, then solve for b 2) Solve for x: x = ) Solve for p: p = Chapter 6 Lesson 1 Algebra- Notes/Identify-Parts- Of-An-Equation.html If x + 6 = 4, then x = 4 6 Therefore, x = -2 9 Page

10 20 Apply if-then reasoning to solve one-step multiplication and division equations Example: 7x = 63 1) Solve for h: 2h = -57 Chapter 6 Lesson 2 If 7x = 63, then 7x divided by 7 = 63 divided by 7. Therefore, x = 9 21 Apply if-then reasoning to solve one-step equations with rational coefficients 1) Solve for the unknown: - b 7 = 4 Chapter 6 Lesson Use algebra and if/then reasoning to solve two-step equations (of the form px+_q=_r _and p(_x+_q)_=_r) Pacing: 2 days True or False: The following equations are true: 1) = ) * 2 / 6 = ) 4b = b * 4 4) 2(-3 + d) = -6 2d Chapter 6 Lessons 4-5 Module 2 Lesson 22 om/itext/math/sample_ chapter/ch02/02-02/ph_alg1_ch02-02_obj1.html 10 Page