6th Grade Unit – Expressions and Equations



Time for Unit: 2-3 weeks Pre Assessment:In order to assess where students currently are in their understanding of expressions, have students’ complete page 68 in their textbook. This will help you plan and adjust the lessons according to your students’ needs. Common Core Standards that will be addressed: 6EE1. Write and evaluate numerical expressions involving whole-number exponents6EE2 Write, read, and evaluate expressions in which letters stand for numbers. (broken down into a, b, and c)6EE2a. 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.”6EE2b. 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.6EE2c. 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 =?.6EE3. Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3(2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.6.EE.4. Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.6EE6. Use variables to represent numbers and write expressions when solving a real-world or mathematical problems; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.Strategies for Teaching the Math CCSSCreate “worthwhile” problems as a foundation for daily instruction.Use real data and current events to make mathematics more engaging and relevant.Ask quality questions and promote discourseConcepts – What students need to knowExpressionsMathematical terms sum, term, product, factor, quotient, coefficientOrder of OperationsProperties of operationsEquivalent expressionsProblemsEquationsVariablesSkills – What students need to be able to do Write (expressions and equations)Read (expressions) Evaluate (expressions)Identify (mathematical terms) Perform (order of operations)Apply (properties of operations)Generate (equivalent expressions)Solve (equations)Solve (real-world and mathematical problems) Big Ideas for UnitHow can expressions and equations be used in real-world contexts? What are expressions made of? Students must first have an understanding of:Order of OperationsSolving whole number exponents Lesson 1: Translating Between Words and Algebraic ExpressionsCommon Core standard addressed: 6EE2a. 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.”6EE6. Use variables to represent numbers and write expressions when solving a real-world or mathematical problems; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified mon Core Standard in plain language: Write expressions using variables. Understand a variable represents an unknown number and use variables to write expressions in real world contexts.Mathematical Practices: 6.MP.1. Make sense of problems and persevere in solving them.6.MP.2. Reason abstractly and quantitatively.6.MP.3. Construct viable arguments and critique the reasoning of others.6.MP.4. Model with mathematics.6.MP.6. Attend to precision.6.MP.7. Look for and make use of structure.Key Vocabulary: expression, variable, constant, sum, plus, subtract, minus, multiply, product, quotient, divided byIt is important for students to read algebraic expressions in a manner that reinforces that the variable represents a number. ? r + 21 as “some number plus 21 as well as “r plus 21” ? n ? 6 as “some number times 6 as well as “n times 6” ? and s ÷ 6 as “as some number divided by 6” as well as “s divided by 6” - Students should complete a close reading of the text in their textbook on page 54. Use the attached graphic organizer to help students become familiar with the vocabulary in the text. - It is important for students to become familiar with the terms algebraic expressions, variable, and constant. Use the graphic organizer provided to help students become familiar with the key vocabulary. Students should be allowed to work either with a partner or a group for this activity. Allow time for activity and bring the class back together and have a discussion to make sure that students fully understand the key vocabulary. Before the authentic discussion can take place, the class must have a set of group norms in place. Students should be trained on what an authentic discussion looks like. All students should have a copy of the discussion rubric available to them. A copy of the rubric is attached. - Students should do another close reading of the text on page 58 and 59 in their textbook. Students should pay close attention to the chart on page 59. Students should take notes in their Journal of the chart on page 59. Once students have completed the chart in their journal students should have a discussion about the contents of the chart. Students should first discuss the chart in small groups and then groups should share whole class what they found the chart to mean. - Once students have a full understanding of the key vocabulary, the text, and the chart on page 59, they should begin exploring algebraic expressions. Allow students to explore expressions using the handout titled, “Exploring Algebraic Expressions”. - Connecting writing expressions with story problems and/or drawing pictures will give students a context for this work. It is important for students to read algebraic expressions in a manner that reinforces that the variable represents a number. Use the following real world contexts and have students discuss in their group what algebraic expression that they could use to represent each situation. Have each group present what they did to the class and have a class discussion about each of the following problems. ? Maria has three more than twice as many crayons as Elizabeth. Write an algebraic expression to represent the number of crayons that Maria has. (Solution: 2c + 3 where c represents the number of crayons that Elizabeth has.) ? An amusement park charges $28 to enter and $0.35 per ticket. Write an algebraic expression to represent the total amount spent. (Solution: 28 + 0.35t where t represents the number of tickets purchased) ? Teri baked 48 cookies and divided them evenly into bags. Let n represent the number of cookies per bag. Write an algebraic expression for the number of bags filled. (Solution: 48 ÷ n)For more practice on writing expressions students should complete page 60 in their math textbook. Pay close attention to make sure that students are able to change a real world problem into an algebraic expression. More examples of this type of problem can be found in the problem solving book that goes along with the math textbook (lesson 2-2). Other resources that can be used during this lesson:Finish Line: Page 88 and 89Formative Assessment: Students should complete the lesson quiz on page 61 in the teacher’s edition Lesson 2: Identifying parts of an algebraic expression Common Core standard addressed: 6EE2b. 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 mon Core Standard in plain language: Identify parts of an expression.Mathematical Practices: 6.MP.1. Make sense of problems and persevere in solving them.6.MP.2. Reason abstractly and quantitatively.6.MP.3. Construct viable arguments and critique the reasoning of others.6.MP.4. Model with mathematics.6.MP.6. Attend to precision.Key Vocabulary: sum, term, product, factor, quotient, coefficient, variables, coefficients, constantsStudents should identify the parts of an algebraic expression including variables, coefficients, constants, and the names of operations (sum, difference, product, and quotient). Development of this common language helps students to understand the structure of expressions and explain their process for simplifying expressions. Terms are the parts of a sum. When the term is an explicit number, it is called a constant. When the term is a product of a number and a variable, the number is called the coefficient of the variable. Variables are letters that represent numbers. There are various possibilities for the number they can represent; Students can substitute these possible numbers for the letters in the expression for various different purposes. Consider the following expression: x2 + 5y + 3x + 6 The variables are x and y. There are 4 terms, x2, 5y, 3x, and 6. There are 3 variable terms, x2, 5y, 3x. They have coefficients of 1, 5, and 3 respectively. The coefficient of x2 is 1, since x2 = 1 x2. The term 5y represent 5 y’s or 5 ● y. There is one constant term, 6. The expression shows a sum of all four terms. - Go over the problem above with the class using whole group instruction. Have a discussion with the class about the different parts of an algebraic expression. Students should write down all important information about each of the parts in their math journals. - Once students have a good understanding of the parts of an algebraic expression students should complete the graphic organizer attached (Identifying Parts of an Algebraic Expression) to practice identifying the different parts of an algebraic expression. Place an algebraic expression on the board, one at a time. Students should copy the algebraic expression on their paper. (room for 6) Student’s should practice dissecting the expression and place each part into the correct box of the graphic organizer. Students should work in groups to complete activity and have discussions about where each piece of the expression should be placed. - In order to check further for student understanding, have students answer the following writing prompt: Cindy was absent from school the day her class learned about parts of algebraic expressions. Using the following expression x2 + 7y + 2x + 9, explain to Cindy what the parts of an algebraic expression are and how to identify them. Once students have answered the prompt, have students exchange papers with each other. Have students read each other’s answers and critique and add to their partner’s paper. Once students have had time to complete this, choose a few examples (cover the name of the student) to show using a document camera. Have a class discussion about what the student wrote. Have students participate in a notice and wonder protocol (“I notice…” and “I wonder what would happen if …”). Once you have completed a few examples, give students back their papers and have them edit their own writing. Collect the papers and look at them to see any misconceptions that you notice students may be having. Other resources that can be used during this lesson:Finish Line: Page 88 and 89Formative Assessment: Have students complete another chart like they did before on their own. (Identifying Parts of an Algebraic Expression)Lesson 3: Solving algebraic expressionsCommon Core standard addressed: 6EE2c. 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 = 6 s2 to find the volume and surface area of a cube with sides of length s = 1/2. (This standard will also be revisited during the Geometry unit) 6EE1. Write and evaluate numerical expressions involving whole-number exponentsCommon Core Standard in plain language: Write expressions using variables. Write and evaluate numerical expressions involving whole number exponents.Mathematical Practices: 6.MP.1. Make sense of problems and persevere in solving them.6.MP.2. Reason abstractly and quantitatively.6.MP.3. Construct viable arguments and critique the reasoning of others.6.MP.4. Model with mathematics.6.MP.6. Attend to precision.Key Vocabulary: evaluate, order of operations, exponents- Students should complete a close reading of the text in Finish Line on page 92 and pages 54 – 55 in their textbook. As students do a close reading, they should write down information that they find important in their math journal. After students have completed a close reading of the text, have a class discussion about important information that they found. Have students add any other important information that they left out of their math journal. It will be important to model how to substitute a value for a variable and solve. - Whole class instruction: complete the Guided Practice on page 56 in the math textbook. In order to check for further understanding after completing the Guided Practice, use the white boards and have the students’ complete page 57 (#’s 11 – 22). As you begin seeing students’ answers you will be able to better judge where the students are on understanding the concept of substituting values for variables. - Complete the Hands on Lab (Exploring Area and Perimeter of Rectangles) on page 66 and 67 in the math textbook.- To check students understanding of the vocabulary word evaluate, have students’ answer the following prompt. Write about situations, people, or things that are evaluated. Have students exchange papers and evaluate each others’ responses. - For extra practice, students can answer the questions on pages 93 – 95 of Finish Line. Other resources that can be used during this lesson:- my. (Problem solving 2-1, Challenge 2-1, Practice 2-1)Formative Assessment: Students should complete the lesson quiz on page 57 in the teacher’s edition. Lesson 4: Equivalent Expressions and PropertiesCommon Core standard addressed: 6EE.3. Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3(2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6(4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.6.EE.4. Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands mon Core Standard in plain language: Use the distributive, commutative, and associative properties to show equivalent expressions. Identify when 2 expressions are equivalent regardless of the value given to the variable.Mathematical Practices: 6.MP.2. Reason abstractly and quantitatively.6.MP.3. Construct viable arguments and critique the reasoning of others.6.MP.4. Model with mathematics.6.MP.6. Attend to precision.6.MP.7. Look for and make use of structureKey Vocabulary: commutative property, distributive property, associate property, factoring expressions, factor, equivalent expressions, equivalent - Students should complete a close reading of the text in their math textbook on page 26 and 27 as well as a close reading of the text in Finish Line on page 96. Students should write important information that they find in their math journal. Have students discuss with a partner what was important and tell why. Have students complete the graphic organizer titled Properties of Operations based on what they found in their close reading of the text. Once students have completed the graphic organizer go over it with the class using whole group instruction. At this point, it is important that students begin seeing and understanding the similarities and differences in the three properties. Students should also begin noticing that both sides of the equal signs are equivalent. - Have students complete the “Property Practice” sheet. Have the students share and discuss with a partner their answers. Go over the sheet whole group. - Let students explore making equivalent expressions. Give students a number. Ask students to come up with as many expressions that they can in order to equal the number that was chosen. For example ask students to make the number 15. Some students may come up with 10 + 5. Other student may come up with 10 x 2 – 5 etc. Give students about 3 minutes to come up with as many as they can. Have students compare their answers with a partner. Find some good examples to show and discuss the difference in difficulty and complexity in the different examples as a whole group. Remind students that they must follow order of operations when solving their expressions. Give students another number to make equivalent expressions for. Give them 3 additional minutes to come up with more equivalent expressions for the given number. Find the best expression (level of difficulty and complexity) and crown that person either King or Queen Expression writer. - Students should have a prior understanding of factors. Have students write what they think a factor is in their math journals. Ask students to compare their answers and then have a class discussion about what a factor is. Place an algebraic expression on the board making sure that each of the coefficients has a common factor. An example is: 12x2 + 9x – 6. Ask students to examine each of the coefficients. Ask them to find a common factor. Once everyone agrees on the common factor explain to the students that you are going to remove the factor from each of the coefficients by dividing by the common factor. When you remove the common factor of 3 you get an equivalent expression of 3(4x2 + 3x – 2). Have students complete the sheet “Factoring Expressions” with a partner. Ask students “why” it would be possible to have more than one correct expression while students are discussing. Students will need to discover that different numbers can be factored out if there are more than one factor the numbers have in common. Go over the sheet with the whole group. Other resources that can be used during this lesson:The following are also examples that can be used during this lesson: Students connect their experiences with finding and identifying equivalent forms of whole numbers and can write expressions in various forms. Students generate equivalent expressions using the associative, commutative, and distributive properties. They can prove that the expressions are equivalent by simplifying each expression into the same form. Example: ? Are the expressions equivalent? How do you know? 4m + 8 4(m+2) and 3m + 8 + m 2 + 2m + m + 6 + mSolution:ExpressionSimplifying the ExpressionExplanation4m + 84m + 8Already in simplest form4(m +2)4(m +2)4m + 8Distributive Property3m + 8 + m3m + 8 + m3m + m + 8 4m + 8Combined like terms2 + 2m + m + 6 + m2 + 2m + m + 6 + m2 + 6 + 2m + m + m(2 + 6) + (2m + m + m)8 + 4m4m + 8Combined like termsFormative Assessment: Using white boards, ask students questions about the different properties, equivalent expressions, and factoring expressions. Problem Solving: Conjectures About Properties (Implement the teaching strategies from Navigating the Mathematics Common Core State Standards)Learning Task – Exploration: With a partner, students will study sets of number sentences and determine if what they observe would be true for all numbers. Students will create statements with words about what they observe in each set of number sentences and then write the number sentences using variables to represent numbers. Activity is attached and can also be found on the Georgia DOE mon Core Discussion Rubric Total Points: /15?54321Quality of Comments*Builds on others’ ideas *Expresses their own ideas clearly *Contributes comments that are timely, appropriate, thoughtful and reflective *Responds respectfully to other student's remarks *Provokes questions and comments from the group.* Uses the language of the disciplineVolunteers comments, most are appropriate and reflect some thoughtfulness,leads to other questions or remarks from student and/or othersVolunteers comments but lacks depth, may or may not lead to other questions from studentsStruggles but participates, occasionally offers a comment when directly questioned, may simply restate questions or points previously raised, may add nothing new to the discussion or provoke no responses or questionsDoes not participate and/or only makes negative or disruptive remarks, comments are inappropriate or off topicResource/Document Reference*Comes to discussions prepared, having read or studied required material *Explicitly draws on that preparation and other information known about the topic to explore ideas under discussion*Clear reference to text being discussed and connects it to other texts or reference points from previous readings and discussions*Reviews the key ideas expressed and draws conclusions in light of information and knowledge gained from the discussionsHas done the reading with some thoroughness, may lack some detail or critical insightHas done the reading; lacks thoroughness of understanding or insightHas not read the entire text and cannot sustain any reference to it in the course of discussionUnable to refer to text for evidence or support remarksActive Listening*Posture, demeanor, eye contact, and behavior clearly demonstrates respect? and attentiveness to others*Follows agreed-upon rules or group norms for discussions and carries out assigned rolesListens to others most of the time, does not stay focused on other's comments (too busy formulating own) or loses continuity of discussion. Shows consistency in responding to the comments of othersListens to others some of the time, does not stay focused on other's comments (too busy formulating own) or loses continuity of discussion. Shows some consistency in responding to the comments of othersDrifts in and out of discussion, listening to some remarks while clearly missing or ignoring othersDisrespectful of others when they are speaking; behavior indicates total non-involvement with group or discussionBuilding Math Vocabulary Graphic OrganizerWord(s): Definition: Real World Examples of Expressions: Math definition of Expressions: Math examples of Expressions:Exploring Algebraic ExpressionsDirections: The table below has different forms of mathematical expressions. Some of the expressions are written in word form and others are written in mathematical terms. Cut each box out and then separate them into groups of expressions that say or mean the same thing. Once you have successfully separated the expressions answer the questions located at the bottom of this sheet. 4ySeventeen added to yFourteen subtracted from b y plus seventeen b minus fourteen Sixteen divided by my multiplied by 4Four groups of ySeventeen more than yFourteen less than by + 17The sum of fifteen and n16 ÷ mThe quotient of 16 and mb – 14 Fifteen times nThe sum of y and seventeenThe product of four and yn – 15 Take away fourteen from bFour times yThe difference of b and fourteenFifteen subtracted from n15nFifteen plus n15 + nThe difference of n and fifteenTake away fifteen from nn minus fifteenn added to fifteen n ÷ 15n multiplied by fifteenThe quotient of n and fifteenn more than fifteenn divided by fifteenThe product of fifteen and nFifteen groups of n Fifteen less than nUsing complete sentences, answer the following questions: In Language Arts, the word expression means a phrase or a sentence. How is an algebraic expression like a phrase? Write a situation that could be modeled by the expression r + 10. Factoring ExpressionsHow many different factors can be factored out of the expression 16y – 4? What are the numbers that can be factored out? Factor out each of the numbers out of the expression and write the equivalent expressions that are made. The following expression has been factored 5(6m – 2). What was the original expression before it was factored? The following expression has 5 factors that can be factored out. 24y2 + 36y + 12 What are the five factors? Using the five factors, write five equivalent expressions for the expression above. Factoring ExpressionsHow many different factors can be factored out of the expression 16y – 4? What are the numbers that can be factored out? Factor out each of the numbers out of the expression and write the equivalent expression that is made. The following expression has been factored 5(6m – 2). What was the original expression before it was factored? The following expression has 5 factors that can be factored out. 24y2 + 36y + 12 What are the five factors? Using the five factors, write five equivalent expressions for the expression above.Identifying Parts of an Algebraic ExpressionExpression 1: _____________________________ Expression 2: __________________________Expression 3: _____________________________ Expression 4: __________________________Expression 5: _____________________________ Expression 6: __________________________VariablesTermsVariable TermsCoefficientsConstant TermsThe expression shows: Properties of OperationsSimilarities inPropertiesDistributive PropertyAssociative PropertyCommutative PropertyThe differences in the properties that I noticed are: __________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________Directions: Identify the property as commutative, associative, or distributive and fill in the missing number. 1) 3 x 5 = m x 5 2) 6 x (7 x 5) = (6 x m) x 5 m = ____ m = ____ property ________________ property _________________3) 5 + y = 3 + 54) 4 x 8 x r = 6 x 8 x 4 y = ____ r = ____ property _________________ property __________________5) 5(4 + 5) = 5 x n6) 8 + (9 + 3) = (b + 9) + 3 n = ____ b = ____ property __________________ property __________________7) 4 x 8 = 4(5 + m) 8) b x 9 = 9 x 2 m = ____ b= ____ property __________________ property __________________9) 5(6 + 4) = 5 x p10) (5 x 2) x 3 = 3 x (2 x h) p = _____ h = _____ property __________________ property __________________Directions: Identify the property as commutative, associative, or distributive and fill in the missing number. 1) 3 x 5 = m x 5 2) 6 x (7 x 5) = (6 x m) x 5 m = ____ m = ____ property ________________ property _________________3) 5 + y = 3 + 54) 4 x 8 x r = 6 x 8 x 4 y = ____ r = ____ property _________________ property __________________5) 5(4 + 5) = 5 x n6) 8 + (9 + 3) = (b + 9) + 3 n = ____ b = ____ property __________________ property __________________7) 4 x 8 = 4(5 + m) 8) b x 9 = 9 x 2 m = ____ b= ____ property __________________ property __________________9) 5(6 + 4) = 5 x p10) (5 x 2) x 3 = 3 x (2 x h) p = _____ h = _____ property __________________ property __________________Task: Conjectures About PropertiesExploration With a partner look at the following sets of number sentences and determine if what you observe would be true for all numbers. Create statements with words about what you observe in each set of number sentences then write the number sentences using variables to represent numbers. 12 + 0 = 12 12 – 0 = 12 12?1=12 37 + 0 = 37 37 – 0 = 37 37?1=37 64 + 0 = 64 64 – 0 = 64 64?1=64 12÷1=12 12?0=0 12 ÷ 0 = 12 37÷1=37 37?0=0 45 ÷ 0 = 45 64÷1=64 64?0=0 64 ÷ 0 = 64 12(4 + 3) = 48 + 36 12(4?3)=48?36 6(7 + 2) = 42 6(7?2)=42?12 4(10 + 3) = 40 + 12 4(10?3)=40?12 4 x 8 = (2 x 8) + (2 x 8) 4 + 8 = (2 +8) + (2 + 8) 8 x 16 = (4 x 16) + (4 x 16) 8 + 16 = (4 +16) + (4 + 16) 5 x 14 = (2.5 x 14) + (2.5 x 14) 5 + 14 = (2.5 + 14) + (2.5 + 14) (32 + 24) + 16 = 32 + (24 + 16) 6?(4?3)=(6?4)?3 (450 + 125) + 75 = 450 + (125 + 75) 10?(5?2)=(10?5)?2 (33 + 17) + 3 = 33 + (17 + 3) (11?2)?3=11?(2?3) ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download