Algebra 1 Unit 3 Systems of Equations：1单元3的代数方程组.docx

6th Grade Math Unit 1: Algebra (Common Core Investigations)Enduring understanding (Big Idea): Students will understand how to translate word phrases into algebraic expressions and simplify them using the properties of operations.Essential Questions: How can I use algebraic expressions to represent a relationship? How can I use algebraic expressions to solve problems?BY THE END OF THIS UNIT:Students will be able to Write, read, and evaluate expressions in which letters stand for numbers Write and evaluate numerical expressions involving whole number expressions Identify parts of an expression Evaluate expressions which variables are specific values Apply the properties of operations Identify and apply the properties of operations Identify when two expressions are equivalent Use variables to write expressions to solve problems Write equations using variables to represent two related quantities in a real-world problem Analyze relationships between dependent and independent variables using graphs, tables, and equations Us the distributive property to express the sum of two whole numbers with a common factor as multiple of a sum of two whole numbers with no common factorStudents will knowDistributive Property a(b + c) = ab + acab + ac = a(b + c)Vocabulary:VariableExpressionDependent VariableIndependent VariableCommutative PropertyAssociative PropertyDistributive PropertyInverse OperationMathematical Practices Focus:1. Make sense of problems and persevere in solving them 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoningUnit ResourcesLearning Task: Mathematical ReflectionsPerformance Task: Check-UpProject: NoneUnit Review: Additional PracticeUnit PlansInvestigationSuggested ACE Questions Standard 6.EE.2; 6.EE.2a; 6.EE.2b; 6.EE.2c; 6.EE.3; 6.EE.4; 6.EE.6; 6.EE.7; 6.EE.9 CC Investigation 2Number Properties and Algebraic Equations2.1 Variables and Expressions2.2 Independent/Dependent variables2.3 Calculating profits2.4 Algebraic Expressions2.5 Algebraic expressions2.6 Writing Equations/Properties2.7 Combining Like Terms/Distributive Property2.1: ACE1-4, 9-13, 22-25, 39-44, 53, 66, 792.2: ACE 5-6, 14-21, 26, 51-52, 892.3: ACE 35-372.4: ACE 7-8, 45-502.5: ACE 54-64, 702.6: ACE 27-34, 38, 65, 67-69, 71, 90-942.7: ACE 72-78, 80-88. 95Standard 6.EE.8CC Investigation 3Integers and the Coordinate Plane3.6 InequalitiesACE 56-69CORE CONTENTCluster Title: Expressions and Equations: Apply and extend previous understandings of arithmetic to algebraic expressions. Standard 6.EE.2a: Write, read, and evaluate expressions in which letters stand for numbers. a. 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. Concepts and Skills to Master Understand that letters, called variables, represent unknown numbers. Know that the same rules apply in operations with numbers also apply in operations with variables. Translate an expression from its word form and vice versa. Represent variables with letters (e.g., a, x,. except e and i).SUPPORTS FOR TEACHERSCritical Background Knowledge Recognize that expressions use one or more mathematical symbols to represent a number or quantity. Know that expressions do not include equal, greater than, or less than signs. Perform the four operations of addition, subtraction, multiplication, and division. Represent a problem such as “seven plus a number is twelve” with symbolic notation, such as 7+=12 Represent the four operations with manipulatives, diagrams, and number lines.Academic Vocabulary Equation, expression, notation for multiplication (e.g., 3x, 3(4), variable Suggested Instructional Strategies Within your classroom, have the students find situations where they can role play to compare known and unknown quantities (e.g., Student A (Dory) and Student B (Colleen); Dory says, “I have two sisters.” Colleen says, “I have Dory 1 sisters.” Dory says, “You have d 1 sister. You have one sister.”) Use all operations. Give each pair of students an expression such as x + 957. Challenge them to find a way to evaluate each expression for x = 35, 825, and 373. Then have the students write a real-life context for each expression. Resources Textbook Correlationo Common Core Investigations (CMP2) Investigation 2: Number Properties and Algebraic EquationsSample Formative Assessment TasksSkill-Based TaskWrite each word phrase as an algebraic expression. 6 less than 3t The product of w and 8 r divided by 15Problem TaskHannah is 3 years younger than Katie. Joey is twice as old as Hannah. Let k stand for Katies age. Write an expression to represent Hannahs age. Using k, write an expression for Joeys age.CORE CONTENTCluster Title: Expressions and Equations: Apply and extend previous understandings of arithmetic to algebraic expressions. Standard 6.EE.2b: Write, read, and evaluate expressions in which letters stand for numbers. b. 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.Concepts and Skills to Master Understand that terms are values in an expression separated by addition and subtraction (e.g., x + 3 contains two terms; 2x 5 contains two terms). Understand that multiplication and division in an expression represent a single term (e.g., 3y is a term, x is a term). Know that a coefficient is a number that multiplies a variable (e.g., 3 is the coefficient of 3y). Read an expression correctly using appropriate mathematical terms (e.g., 2 (8 + 7) is read as the product of 2 times the quantity or sum of 8 + 7). Identify parts of an expression (e.g., in the expression 2x 5, 2x is the first term where 2 is the coefficient of the variable x and 5 is the second term, a constant). Represent the parts of an expression with manipulatives, diagrams, or coordinate plane. SUPPORTS FOR TEACHERSCritical Background Knowledge Recognize that variables represent unknown quantities. Know the meaning of the terms sum, product, factor, quantity, and quotient. Read expressions of single operations (e.g., 3y, x + 7, 4 + 3). Represent the four operations (addition, subtraction, multiplication and division) with manipulatives, diagrams, number lines, and story context. Academic VocabularyCoefficient, Dividend, Divisor, Equation, Expression, Factor, Multiplier, Product, Quotient, Sum, Term (optional term Constant), Parentheses Suggested Instructional Strategies Give an example of an expression, such as 4x + 7 + x, and identify the parts. The terms are 4x, 7 and x. The coefficients are 4 and 1 (since there is one x). The values 4 and x are both factors of the product 4x. Give additional examples to students to identify parts of expressions in small groups. Give the students word problems such as the following and have them find the solutions: The sum of twice a number plus 13 is 75. Write the equation and find the number. Use the algebra tiles to represent terms in an expression (e.g., 2x requires two x pieces) Game 1: Give students two sets of cards. The first set contains equations and the second set contains the academic vocabulary. Students turn over an equation card. Then students match academic vocabulary to the parts of the equation. Game 2: Start with a small expression, such as 2x - 4. Break the expression into distinct parts. Write on separate cards 2, x, -, and 4. Resources Textbook Correlation:Common Core Investigations (CMP2) - Investigation 2: Number Properties and Algebraic Equations Math Goodies - Writing Algebraic Expressions - Tutorial and Practice Exercises -/lessons/vol7/expressions.html Study Guides and Tutorials - Teacher Tutorial - /mathproblems.htm Mathwire - I Have, Who Has - Game - /whohas/whalgA.pdf Math Play - Who Wants to Be a Millionaire - Game /Algebraic-Expressions-Millionaire/algebraic-expressions-millionaire.html Sample Formative Assessment TasksSkill-Based TaskProvide an expression (e.g., 6x + 4) for students to identify the parts. Write an expression that has two terms and a product. Write an expression that has a coefficient and a sum. Problem TaskThis is a skill-based standard. Therefore, there is no problem task.CORE CONTENTCluster Title: Expressions and Equations-Apply and extend previous understandings of arithmetic to algebraic expressions. Standard 6.EE.2c.:Write, read, and evaluate expressions in which letters stand for numbers. c. Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s3 and A = 6s2 to find the volume and surface area of a cube with sides of length s = 1/2.Concepts and Skills to Master Know that when using an expression to solve a problem, numbers replace variables in the expression and answers will vary depending on which numbers are substituted for variables. Extend understanding that we use standard formulas because of their potential to efficiently express relationships to include fractional units Extend understanding of standard formulas through composition or decomposition to include fractional units. Find and use correct units of measure in solutions to real world problems. Substitute values for variables in expressions (e.g., If I buy 2 bags of lemon drops, the expression can be written 2x. If the cost of the bag is $1.59 I can substitute that value in for x. If the next week the price changes to $1.69 I can still use the same expression, assigning the new value to the variable). Use order of operations including, exponents. Represent expressions, including standard formulas, with manipulatives, drawings, and diagramsSUPPORTS FOR TEACHERSCritical Background KnowledgeRecognize that variables represent unknown quantities. Standards: 6.ee.2a and 6.ee.2b Understand that we use standard formulas because of their potential to efficiently express relationships with whole numbers. Make sense of standard formulas through composition or decomposition using whole numbers. Use order of operations. Create and use area models. Represent volume graphically and with manipulativesAcademic Vocabulary, expression, equations, formula, order of operations, superscripted numbers, variable Suggested Instructional Strategies1. Write an expression such as x + 6 on the board. Leave space between the symbols and numbers. Give each student an index card with a different number from 0 20. Have a volunteer use his or her card to cover the x. Have another volunteer give the value for x and evaluate the expression. Summarize by saying, for example, When x = 4, then x + 6 = 10. Repeat with other students cards and different expressions involving any of the four operations. Vary the degree of difficulty. 2. In small groups, give students the expression 3 + 4 x 5. This represents the number of people coming to a party. There are three individuals and four families each with five members. List the steps you would use to solve this problem and justify each step. Next, have students write this expression for a variable number of families. Allow students to reason which number becomes the variable. Resources Textbook Correlationo Common Core Investigations (CMP2) Investigation 2: Number Properties and Algebraic Equations /interactivate/activities/SurfaceAreaAndVolume/Sample Formative Assessment TasksSkill-Based TaskComplete the table by evaluating the algebraic expression. x037125x + 8Problem TaskYou know that you can find the area of a triangle using the formula A = bh. If a triangle has an area of 48 cm2, what can its base and height be? Draw diagrams to justify your thinking.CORE CONTENTCluster Title: Apply and extend previousunderstandingsof arithmeticto algebraicexpressions.Standard 6.EE.3 Applythepropertiesof operations togenerateequivalent expressions.For example, apply thedistributive propertytotheexpression3(2+ x) toproducetheequivalent expression6+3x;apply thedistributiveproperty tothe expression24x+ 18y to producetheequivalent expression6(4x + 3y); applyproperties ofoperations toy + y + yto producetheequivalentexpression 3y.Concepts and Skills to Master Understand that the properties used with numbers also apply to expressions with variables. Apply the properties of operations with expressions involvingvariables togenerate equivalentexpressions. Use manipulatives or diagrams to represent the distributive property. Show properties of operations to be equivalent with manipulatives, diagrams or story contexts.SUPPORTS FOR TEACHERSCritical Background Knowledge Variables are letters that stand for numbers. Understandingproperties of operationsandapplyingeachofthemin numeric representations (e.g.,3+ 2= 2+ 3). Multiplication is repeated addition. Understandtheassociative propertyofadditionandmultiplication,thecommutative propertyof additionand multiplication,theidentitypropertyof additionandmultiplicationandthe distributive propertyofmultiplicationover addition or subtraction. Recognize howto usecommoncalculations (such asfactfamilies,basic mathfacts,number bonds,composing anddecomposingnumbers) togeneratesolutions to problems. Find common factors of two whole numbers. Model operations of addition, subtraction, multiplication and divisionAcademic VocabularyAssociative property, Commutative property, distributive property, equivalent, identity property, identity element, variableSuggested Instructional Strategies Students must be able to make connections between what they already know about the properties of operations when used with numbers and how those properties apply when used with algebraic notation (using variables). Therefore, teachers should first present the topic of equivalent expressions using numbers and then shift to variables. Have students open a package of M&Ms and use the distributive property to write expressions showing how many of each color would be in 5 bags, 10 bags etc. (e.g. There are 10 red, 15 green, and 12 yellow in one bag. That could be represented by the expression 10r + 15g +12y. To represent the number in 5 bags you would obtain the expression 5(10r + 15g + 12y). Using the distributive property, the expression is 50r + 75g + 60y. So there are 50 red, 75 green, and 60 yellow M&Ms in 5 bags.)Resources Textbook Correlation - Common Core Investigations (CMP2) - Investigation 2: Number Properties and Algebraic Equations Math League - Teacher Tutorial - /help/wholenumbers/wholenumbers.htm Purplemath - Teacher Tutorial - /modules/numbprop.htm /Common-Core/Common%20Core%20Frameworks/CCGPS_Math_6_6thGrade_Unit3SE.pdfSample Formative Assessment TasksSkill-based taskGenerate an equivalent expression for each of the following:4 (x - 2)15x - 24yx + x + y + y5x + 2y5r + (2s + 2t)Problem TaskIn one packet of nuts, there are two different types of nuts. There are 5 peanuts (p) and 7 cashews (c) in each container. I have 6 packets of nuts; write two expressions that show how many nuts I have all together.Possible answers: 6 (5 + 7) or (6 x 5) + (6 x 7)CORE CONTENTCluster Title: Apply and extend previous understandings of arithmetic to algebraic expressions. Standard 6.EE.4Identify 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. Concepts and Skills to Master: Patterns of Reasoning Recognize equivalent expressions. Substitute values into expressions to prove equivalency. Model equivalent expressions with manipulatives, diagrams or story contexts.SUPPORTS FOR TEACHERSCritical Background Knowledge Variables are letters that stand for numbers. Substitute values into expressions to solve an equation, Model algebraic expressions with manipulatives, diagrams or story contexts.Academic Vocabulary =, equation, equivalent, expression, variableSuggested Instructional StrategiesTeach this standard concurrently with 6.EE.3 1. Use substitution to verify that both expressions are equivalent (e.g., 3(2 + x) = 6 + x) is not true. Substitute any number for x; the expressions have different values, therefore they are not equivalent. (Make sure students try more than one number to make sure it works, because 0 will make this st