CCGPS Frameworks Teacher Edition Mathematics

Transcription

1 CCGPS Frameworks Teacher Edition Mathematics 6 th Grade Unit4: One Step Equations and Inequalities

2 Sixth Grade Mathematics Unit 4 Unit 4 One Step Equations and Inequalities TABLE OF CONTENTS Overview...3 Key Standards...4 Enduring Understandings...7 Essential Questions...8 Concepts & Skills to Maintain...8 Selected Terms and Symbols...8 Classroom Routines...10 Strategies for Teaching and Learning...10 Instructional Resources/Tools...11 Evidence of Learning...11 Tasks...12 Set It Up...13 Building with Toothpicks...20 Fruit Punch...26 Picturing Proportions...30 Making Sense of Graphs...36 Analyzing Tables...42 When is it Not Equal...46 Culminating Task: Want Ads...52 References...57 April 2012 Page 2 of 57

3 Sixth Grade Mathematics Unit 4 OVERVIEW In this unit students will: Determine if an equation or inequality is appropriate for a given situation Represent and solve mathematical and real world problems with equations and inequalities Interpret the solutions to equations and inequalities Represent the solutions to inequalities on a number line Analyze the relationship between dependent and independent variables through the use of tables, equations and graphs Beginning experiences in solving equations will require students to understand the meaning of the equation as well as the question being asked. The use of illustrations, drawings, and balance models to represent and solve equations and inequalities will help students to develop this understanding. Solving equations and inequalities will also require students to develop effective strategies such as fact families, and inverse operations. As effective strategies are developed students will revisit rate and proportional reasoning problems and solve them using strategies developed in solving similar one-step equations. Students will represent, model and solve equations and inequalities that are based on mathematical and real world problems. Presented with these situations, students must determine if a single value is required as a solution or if the situation allows for multiple solutions will be included. This creates the need for both equations (single solution for the situation) and inequalities (multiple solutions for the situation). When working with inequalities, students will work with situations in which the solution is not limited to the set of positive whole numbers but includes positive rational numbers. As an extension to this concept, certain situations may require a solution between two numbers. Therefore, the exploration with students as to what this would look like both on a number line and symbolically will be explored. The process of translating between mathematical phrases and symbolic notation is essential in the writing of equations/inequalities for a situation. This is a two-way process and students will be able to write a mathematical phrase for an equation. The goal is to help students connect the pieces. This is done by having students use multiple representations for mathematical relationships. Students will translate freely among the story, words (mathematical phrases), models, tables, graphs and equations/inequalities. Given any one of these representations students should be able to develop the others. Although the units in this instructional framework emphasize key standards and big ideas at specific times of the year, routine topics such as estimation, mental computation, and basic computation facts should be addressed on an ongoing basis. Ideas related to the eight process standards should be addressed constantly as well. To assure that this unit is taught with the appropriate emphasis, depth, and rigor, it is important that the tasks listed under Evidence of April 2012 Page 3 of 57

4 Sixth Grade Mathematics Unit 4 Learning be reviewed early in the planning process. A variety of resources should be utilized to supplement this unit. This unit provides much needed content information, but excellent learning activities as well. The tasks in this unit illustrate the types of learning activities that should be utilized from a variety of sources. STANDARDS ADDRESSED IN THIS UNIT Mathematical standards are interwoven and should be addressed throughout the year in as many different units and activities as possible in order to emphasize the natural connections that exist among mathematical topics. KEY STANDARDS Reason about and solve one-variable equations and inequalities. MCC6.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 specified set makes an equation or inequality true. MCC6.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. MCC.6.EE.7 Solve real-world and mathematical problems by writing and solving equations of the form x p q and px q for cases in which p, q and x are all nonnegative rational numbers. MCC.6.EE.8 Write an inequality of the form x c or x c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x c or x c have infinitely many solutions; represent solutions of such inequalities on number line diagrams. Represent and analyze quantitative relationships between dependent and independent variables. MCC6.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. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. April 2012 Page 4 of 57

5 Sixth Grade Mathematics Unit 4 Understand ratio concepts and use ratio reasoning to solve problems. MCC.6.RP.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. MCC.6.RP.3a Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios. MCC.6.RP.3b Solve unit rate problems including those involving unit pricing and constant speed. MCC.6.RP.3c Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole given a part and the percent. MCC.6.RP.3d Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. Standards for Mathematical Practice: The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important processes and proficiencies with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately) and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one s own efficacy). 1. Make sense of problems and persevere in solving them. In grade 6, students solve problems involving ratios and rates and discuss how they solved them. Students solve real world problems through the application of algebraic and geometric concepts. Students seek the meaning of a problem and look for efficient ways to represent and solve it. They may check their thinking by asking themselves, What is the most efficient way to solve the problem?, Does this make sense?, and Can I solve the problem in a different way? 2. Reason abstractly and quantitatively. In grade 6, students represent a wide variety of real world contexts through the use of real numbers and variables in mathematical April 2012 Page 5 of 57

6 Sixth Grade Mathematics Unit 4 expressions, equations, and inequalities. Students contextualize to understand the meaning of the number or variable as related to the problem and decontextualize to manipulate symbolic representations by applying properties of operations. 3. Construct viable arguments and critique the reasoning of others. In grade 6, students construct arguments using verbal or written explanations accompanied by expressions, equations, inequalities, models, and graphs, tables, and other data displays (i.e. box plots, dot plots, histograms, etc.). They further refine their mathematical communication skills through mathematical discussions in which they critically evaluate their own thinking and the thinking of other students. They pose questions like How did you get that?, Why is that true? Does that always work? They explain their thinking to others and respond to others thinking. 4. Model with mathematic contextually s. In grade 6, students model problem situations symbolically, graphically, tabularly, and contextually. Students form expressions, equations, or inequalities from real world contexts and connect symbolic and graphical representations. Students begin to explore covariance and represent two quantities simultaneously. Students use number lines to compare numbers and represent inequalities. They use measures of center and variability and data displays (i.e. box plots and histograms) to draw inferences about and make comparisons between data sets. Students need many opportunities to connect and explain the connections between the different representations. They should be able to use all of these representations as appropriate to a problem context. 5. Use appropriate tools strategically. Students consider available tools (including estimation and technology) when solving a mathematical problem and decide when certain tools might be helpful. For instance, students in grade 6 may decide to represent similar data sets using dot plots with the same scale to visually compare the center and variability of the data. Additionally, students might use physical objects or applets to construct nets and calculate the surface area of three dimensional figures. 6. Attend to precision. In grade 6, students continue to refine their mathematical communication skills by using clear and precise language in their discussions with others and in their own reasoning. Students use appropriate terminology when referring to rates, ratios, geometric figures, data displays, and components of expressions, equations or inequalities. 7. Look for and make use of structure. Students routinely seek patterns or structures to model and solve problems. For instance, students recognize patterns that exist in ratio tables recognizing both the additive and multiplicative properties. Students apply properties to generate equivalent expressions (i.e x = 3 (2 + x) by distributive property) and solve equations (i.e. 2c + 3 = 15, 2c = 12 by subtraction property of equality), c=6 by division property of equality). Students April 2012 Page 6 of 57

7 Sixth Grade Mathematics Unit 4 compose and decompose two and three dimensional figures to solve real world problems involving area and volume. 8. Look for and express regularity in repeated reasoning. In grade 6, students use repeated reasoning to understand algorithms and make generalizations about patterns. During multiple opportunities to solve and model problems, they may notice that a/b c/d = ad/bc and construct other examples and models that confirm their generalization. Students connect place value and their prior work with operations to understand algorithms to fluently divide multi digit numbers and perform all operations with multi digit decimals. Students informally begin to make connections between covariance, rates, and representations showing the relationships between quantities. Related Standards Apply and extend previous understandings of multiplication and division to divide fractions by fractions. MCC6.NS.1 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. Compute fluently with multi-digit numbers and find common factors and multiples. MCC6.NS.2 Fluently divide multi-digit numbers using the standard algorithm. MCC6.NS.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. MCC6.NS.4 Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers with a common factor as a multiple of a sum of two whole numbers with no common factor. ENDURING UNDERSTANDINGS Represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules; Relate and compare different forms of representation for a relationship; Use values from specified sets to make an equation or inequality true. Develop an initial conceptual understanding of different uses of variables; Graphs can be used to represent all of the possible solutions to a given situation. April 2012 Page 7 of 57

8 Sixth Grade Mathematics Unit 4 Many problems encountered in everyday life can be solved using proportions, equations or inequalities. ESSENTIAL QUESTIONS How is an equation like a balance? How can the idea of balance help me solve an equation? What strategies can I use to help me understand and represent real situations using proportions, equations and inequalities? How can I write, interpret and manipulate proportions, equations, and inequalities? How can I solve a proportion, equation or inequality? How can I tell the difference between an expression, equation and an inequality? How can proportions be used to solve problems? How can proportional relationships be described using the equation y = kx? How can proportional relationships be represented using rules, tables, and graphs? How can the graph of y = kx be interpreted for different contexts? How does a change in one variable affect the other variable in a given situation? Which tells me more about the relationship I am investigating, a table, a graph or a formula? CONCEPTS/SKILLS TO MAINTAIN It is expected that students will have prior knowledge/experience related to the concepts and skills identified below. It may be necessary to pre-assess in order to determine if time needs to be spent on conceptual activities that help students develop a deeper understanding of these ideas. Using parentheses, brackets, or braces in numerical expressions and evaluate expressions with these symbols. Write and interpret numerical expressions. Generating two numerical patterns using two given rules. Interpret a fraction as division Operations with whole numbers, fractions, and decimals SELECTED TERMS AND SYMBOLS The following terms and symbols are often misunderstood. These concepts are not an inclusive list and should not be taught in isolation. However, due to evidence of frequent difficulty and misunderstanding associated with these concepts, instructors should pay particular attention to them and how their students are able to explain and apply them. April 2012 Page 8 of 57

9 Sixth Grade Mathematics Unit 4 The definitions below are for teacher reference only and are not to be memorized by the students. Students should explore these concepts using models and real life examples. Students should understand the concepts involved and be able to recognize and/or demonstrate them with words, models, pictures, or numbers. The websites below are interactive and include a math glossary suitable for middle school children. Note At the middle school level, different sources use different definitions. Please preview any website for alignment to the definitions given in the frameworks. This web site has activities to help students more fully understand and retain new vocabulary Definitions and activities for these and other terms can be found on the Intermath website. Intermath is geared towards middle and high school students. Addition Property of Equality: Adding the same number to each side of an equation produces an equivalent expression. Constant of proportionality: The constant value of the ratio of two proportional quantities x and y; usually written y = kx, where k is the constant of proportionality. In a proportional relationship, y = kx, k is the constant of proportionality, which is the value of the ratio between y and x. Direct Proportion (Direct Variation): The relation between two quantities whose ratio remains constant. When one variable increases the other increases proportionally: When one variable doubles the other doubles, when one variable triples the other triples, and so on. When A changes by some factor, then B changes by the same factor: A=kB, where k is the constant of proportionality. Division Property of Equality: States that when both sides of an equation are divided by the same number, the remaining expressions are still equal Equation: A mathematical sentence that contains an equal sign Inequality: A mathematical sentence that contains the symbols >, <,, or. Inverse Operation: A mathematical process that combines two or more numbers such that its product or sum equals the identity. Multiplication Property of Equality: States that when both sides of an equation are multiplied by the same number, the remaining expressions are still equal. April 2012 Page 9 of 57

10 Sixth Grade Mathematics Unit 4 Proportion: An equation which states that two ratios are equal. Subtraction Property of Equality: States that when both sides of an equation have the same number subtracted from them, the remaining expressions are still equal. Term: A number, a variable, or a product of numbers and variables. Variable: A letter or symbol used to represent a number or quantities that vary CLASSROOM ROUTINES The importance of continuing the established classroom routines cannot be overstated. Daily routines must include such obvious activities as estimating, analyzing data, describing patterns, and answering daily questions. They should also include less obvious routines, such as how to select materials, how to use materials in a productive manner, how to put materials away, how to access classroom technology such as computers and calculators. An additional routine is to allow plenty of time for children to explore new materials before attempting any directed activity with these new materials. The regular use of routines is important to the development of students' number sense, flexibility, fluency, collaborative skills and communication. These routines contribute to a rich, hands-on standards based classroom and will support students performances on the tasks in this unit and throughout the school year. STRATEGIES FOR TEACHING AND LEARNING The skill of solving proportions, equations, inequalities must be developed conceptually before it is developed procedurally. This means that students should be thinking about what numbers could possibly be a solution to the proportion, equation, or inequality prior solving them. For example, in the equation x + 21 = 32 students know that = 30 therefore the solution must be 2 more than 9 or 11, so x = 11. Provide multiple situations in which students must determine if a single value is required as a solution, or if the situation allows for multiple solutions. This creates the need for both types of equations (single solution for the situation) and inequalities (multiple solutions for the situation). Solutions to equations should not require using the rules for operations with negative numbers since the conceptual understanding of negatives and positives is being introduced in Grade 6. When working with inequalities, provide situations in which the solution is not limited to the set of positive whole numbers but includes rational numbers. This is a good way to practice fractional numbers and introduce negative numbers. Students need to be aware that numbers less than zero could be part of a solution set for a situation. As an extension to this concept, certain situations may require a solution between two numbers. For example, a problem situation may have a solution that requires more than 10 but not greater than 25. Therefore, the exploration with students as to what this would look like both on a number line and symbolically April 2012 Page 10 of 57

11 Sixth Grade Mathematics Unit 4 is a reasonable extension. Provide multiple situations for the student to analyze and determine what unknown is dependent on the other components. For example, how far I travel is dependent on the time and rate that I am traveling. INSTRUCTIONAL RESOURCES/TOOLS Use graphic organizers as tools for connecting various representations. Hands On Equations Pedal Power NCTM illuminations lesson on translating a graph to a story. Interactive grapher from the National Library of Virtual Manipulatives Algebra Balance Scales from the National Library of Virtual Manipulatives Manipulatives in order to solve equations EVIDENCE OF LEARNING By the conclusion of this unit, students should be able to demonstrate the following competencies: graph data that occurs as a result of relationships between varying quantities in the coordinate plane; analyze graphs and tables to determine the relationship between varying quantities; describe how change in one variable affects the other; use written descriptions, tables, graphs and equations to represent relationships between varying quantities use the addition and multiplication properties of equality to solve one-step linear equations and inequalties solve problems by defining a variable, writing and solving a proportion, equation, or inequality and interpreting the solution in the context of the original problem. Draw pictures and use manipulatives to demonstrate a conceptual understanding of proportion; Solve problems using proportional reasoning; April 2012 Page 11 of 57

12 Sixth Grade Mathematics Unit 4 TASKS: The following tasks represent the level of depth, rigor, and complexity expected of all sixth grade students. These tasks or tasks of similar depth and rigor should be used to demonstrate evidence of learning. It is important that all elements of a task be addressed throughout the learning process so that students understand what is expected of them. While some tasks are identified as a performance task, they may also be used for teaching and learning (learning / scaffolding task) Task Name Set It Up Building Toothpicks Fruit Punch Picturing Proportions Making Sense of Graphs Analyzing Tables When is it Not Equal Want Ads Task Type Grouping Strategy Performance Task/ Individual/Partner Task Performance Task/ Individual/Partner Task Performance Task/ Partner/Group Task Performance Task/ Individual/Partner Task Performance Task/ Individual/Partner Task Performance Task/ Individual/Partner Task Performance Task/ Individual/Partner Task Culminating Task/ Individual Content Addressed One Step Equations Algebraic Expressions Proportional Relationships Proportional Relationships Proportions, Rules, and Algebra Equations Inequalities Culminating April 2012 Page 12 of 57

13 Task: Set It Up Sixth Grade Mathematics Unit 4 ESSENTIAL QUESTIONS Why do we need conventions in mathematics? How do I solve a one step equation? STANDARDS ADDRESSED MCC6.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 specified set makes an equation or inequality true. MCC6.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. MCC.6.EE.7 Solve real-world and mathematical problems by writing and solving equations of the form x p q and px q for cases in which p, q and x are all nonnegative rational numbers. TASK COMMENTS In this task, students will use the balance method to write and solve one-step equations and is designed as a learning task. Students will also learn why conventions are an integral part of mathematics. Set It Up Adapted from Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School. Part I. Marcus has 6 pet rabbits. He keeps them in two cages that are connected so they can go back and forth between the cages. One cage is orange and the other cage is blue. 1. Show all the ways that 6 rabbits can be in two cages. Solution Orange Blue April 2012 Page 13 of 57

14 0 6 Sixth Grade Mathematics Unit 4 Comment Students may choose to illustrate the above situation with a pictorial representation. 2. Write an equation that represents the rabbits. Solution r + b = 6 or b = 6 r or r = 6 b Comment Students may need to add an additional column to their table to assist them in writing an equation. Orange Blue Total Students may also use their understanding of fact families to write multiple equations that represent the rabbits. 3. Write a different equation that represents the rabbits. Solution r + b = 6 or b = 6 r or r = 6 b 4. Write a different equation that represents the rabbits. Solution r + b = 6 or b = 6 r or r = 6 b Note: Students should analyze the different equations and reflect on the addition and subtraction properties of equalities. Teachers can also begin a discussion on the relationship between the properties of equality and inverse operations. April 2012 Page 14 of 57

15 Sixth Grade Mathematics Unit 4 Part II. Find the weight of the pair of shoes and pair of socks. = 13.9 ounces 1. Write an equation that represents the above balance scale. Solution + = 13.9 ounces 2. What does 13.9 represent in the equation? Solution Thirteen and 9 tenths (13.9) represents the combined weight of the shoes and socks. 3. What do you notice about the shoes if the pair of socks weighs 0.8 ounces? How can you find the weight of the pair of shoes if the pair of socks weighs 0.8 ounces? Solution The shoes weigh more than the socks and less than 13.9 ounces if the total weight is 13.9 ounces = 13.1 Comment If students use trial and error to determine the weight of the socks, guide them to use a different method to also find the weight. Guiding questions may be necessary to help students discover and use the inverse operation to undo the given operation to find the weight. Students need to determine that they can use the inverse operation to solve problems. This will help students understand why conventions are put in place to solve equations and will assist them when solving more complicated equations. 4. How can you find the weight of the pair of socks if the pair of shoes weighs 13.1 ounces? Solution The socks weigh less than the shoes and less than 13.9 ounces if the total weight is 13.9 ounces = 0.8 April 2012 Page 15 of 57

16 Sixth Grade Mathematics Unit 4 Comment If students use trial and error to determine the weight of the socks, guide them to use a different method to also find the weight. Guiding questions may be necessary to help students discover and use the inverse operation to undo the given operation to find the weight. Students need to determine that they can use the inverse operation to solve problems. This will help students understand why conventions are put in place to solve equations and will assist them when solving more complicated equations = 13.9 ounces a. Select a variable to represent the athletic shoes (tennis shoes). Comment Students may select any letter to represent the shoes. To stay consistent we will select a to represent the athletic shoes. b. Select a variable to represent the socks. Comment Students may select any letter to represent the shoes. To stay consistent we will select s to represent the athletic shoes. c. Write an equation that represents the above equations using variables instead of pictures. Solution a + s = 13.9 Comment Students may need to first write the equation in words then write the equation using variables. d. Write an equation in terms of athletic shoes. Solution a = 13.9 s Comment Students may think about fact families to help them develop this equation. e. Write an equation in terms of socks. Solution s = 13.9 a April 2012 Page 16 of 57

17 Sixth Grade Mathematics Unit 4 Set It Up Adapted from Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School. Part I. Marcus has 6 pet rabbits. He keeps them in two cages that are connected so they can go back and forth between the cages. One cage is orange and the other cage is blue. 1. Show all the ways that 6 rabbits can be in two cages. 2. Write an equation that represents the rabbits. 3. Write a different equation that represents the rabbits. 4. Write a different equation that represents the rabbits. Note: Students should analyze the different equations and reflect on the addition and subtraction properties of equalities. Teachers can also begin a discussion on the relationship between the properties of equality and inverse operations. April 2012 Page 17 of 57

18 Sixth Grade Mathematics Unit 4 Part II. Find the weight of the pair of shoes and pair of socks. = 13.9 ounces 6. Write an equation that represents the above balance scale. 7. What does 13.9 represent in the equation? 8. What do you notice about the shoes if the pair of socks weighs 0.8 ounces? How can you find the weight of the pair of shoes if the pair of socks weighs 0.8 ounces? 9. How can you find the weight of the pair of socks if the pair of shoes weighs 13.1 ounces? April 2012 Page 18 of 57

19 Sixth Grade Mathematics Unit = 13.9 ounces a. Select a variable to represent the athletic shoes (tennis shoes). b. Select a variable to represent the socks. c. Write an equation that represents the above equations using variables instead of pictures. d. Write an equation in terms of athletic shoes. e. Write an equation in terms of socks. April 2012 Page 19 of 57

20 Sixth Grade Mathematics Unit 4 Building with Toothpicks ESSENTIAL QUESTIONS Why do we use letters to represent numbers in mathematics? Why do we need conventions in mathematics? How do I evaluate an algebraic expression? How can variables be used to describe patterns? How do I solve a one step equation? STANDARDS ADDRESSED MCC6.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 specified set makes an equation or inequality true. MCC6.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. MCC.6.EE.7 Solve real-world and mathematical problems by writing and solving equations of the form x p q and px q for cases in which p, q and x are all nonnegative rational numbers. MATERIALS toothpicks TASK COMMENTS In this task, students will describe and generalize patterns using algebraic expressions. Students will also write and solve one-step equations. Building with Toothpicks The shapes shown below are made with toothpicks. Look for patterns in the number of toothpicks in the perimeter of each shape. April 2012 Page 20 of 57

21 Sixth Grade Mathematics Unit 4 Shape 1 Shape 2 Shape 3 Shape 4 1. Use a pattern from the shapes above to determine the perimeter of the fifth shape in the sequence. Show or explain how you arrived at your answer. Solution Shape Perimeter 1 4 = 4(1) = 4(2) = 4(3) = 4(4) = 4(5) 20 n 4(n) 4n April 2012 Page 21 of 57

22 Sixth Grade Mathematics Unit 4 2. Graph the relationship between the shape number and the perimeter. Based on your graph identify the dependent and independent variable. 3. What is the perimeter for shape n? Solution The perimeter for shape n is 4n. 4. Write a formula that you could use to find the perimeter of any shape n. Explain how you found your formula. Solution The formula that can be used to find the perimeter of any shape n is 4n. The product of the shape number and the perimeter is 4 times the shape number. Students should recognize that the perimeter of each shape is 4 more than the perimeter of the previous shape. April 2012 Page 22 of 57

23 Sixth Grade Mathematics Unit 4 5. What is the shape number if it had a perimeter of 128? Solution Shape Perimeter n = What is the perimeter for shape 10? Solution The perimeter for shape 10 is 40. 4n = perimeter 4(10) = Is there a figure with a perimeter of 62? Explain your reasoning. Solution There is not a figure with a perimeter of 62. Each figure has a perimeter is that is a multiple of 4 and 62 is not a multiple of 4.. April 2012 Page 23 of 57

24 Sixth Grade Mathematics Unit 4 Building with Toothpicks The shapes shown below are made with toothpicks. Look for patterns in the number of toothpicks in the perimeter of each shape. Shape 1 Shape 2 Shape 3 Shape 4 1. Use a pattern from the shapes above to determine the perimeter of the fifth shape in the sequence. Show or explain how you arrived at your answer. 2. Graph the relationship between the shape number and the perimeter. Based on your graph identify the dependent and independent variable. April 2012 Page 24 of 57

25 Sixth Grade Mathematics Unit 4 3. What is the perimeter for shape n? 4. Write a formula that you could use to find the perimeter of any shape n. Explain how you found your formula. 5. What is the shape number if it had a perimeter of 128? 6. What is the perimeter for shape 10? 7. Is there a figure with a perimeter of 62? Explain your reasoning. April 2012 Page 25 of 57

26 Sixth Grade Mathematics Unit 4 Task: Fruit Punch This task was taken from ACHIEVE ESSENTIAL QUESTIONS What is a proportion? How can proportional relationships be represented using rules, tables, and graphs? STANDARDS ADDRESSED MCC6.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 specified set makes an equation or inequality true. MCC6.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. MCC.6.EE.7 Solve real-world and mathematical problems by writing and solving equations of the form x p q and px q for cases in which p, q and x are all nonnegative rational numbers. MCC.6.RP.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. MATERIALS Tiles (optional chips TASK COMMENTS In this task, students will demonstrate a deeper understanding of proportional reasoning. Students will connect their understanding of similarity to direct proportions. Teachers should support good student dialogue and take advantage of comments and questions to help guide students into correct mathematical thinking. Fruit Punch This task was taken from ACHIEVE John was making fruit punch for a party using crystals that you mix with water. He mixed four scoops of crystals with nine cups of water and it tasted just right. His sister, Sarah, who likes April 2012 Page 26 of 57

27 Sixth Grade Mathematics Unit 4 sweet drinks, walked by and dumped another scoop of crystals into the pitcher. How much water does John need to add so that the fruit punch will taste exactly the same as it did before? Comment: In unit 2 students learned about ratios. In unit 4 the use of ratios is in the context of equations and solving for unknown parts of proportions. Solution (part 1) Let s find out how much water John s recipe needs per scoop. Let w be the number of cups of water and s be the number of scoops of crystals. For the drink to taste right, we know from John s original mixture that. So we solve for w: This means he needs cups of water total to make the punch taste just right. Since John originally used nine cups of water, he will need to add cups of water to restore the original flavor. Differentiation Students may benefit from using manipulatives to determine both ratios and a proportion. Students may draw the manipulatives to determine both ratios and a proportion. Students should use the drawing to write a proportion using variables. Students should write a proportion and solve. April 2012 Page 27 of 57

28 Sixth Grade Mathematics Unit 4 Solution (part 2) John s punch has 9 scoops of water for every 4 scoops of crystals. That is the same as 2 ¼ scoops of water for every scoop of crystals. Because Sarah added one scoop of crystals, John should add 2 ¼ cups of water. Ask students to define a variable for the number of cups of water and the number of scoops of crystals. w = 1 cup of water s = 1 scoop of crystals April 2012 Page 28 of 57

29 Sixth Grade Mathematics Unit 4 Fruit Punch This task was taken from ACHIEVE John was making fruit punch for a party using crystals that you mix with water. He mixed four scoops of crystals with nine cups of water and it tasted just right. His sister, Sarah, who likes sweet drinks, walked by and dumped another scoop of crystals into the pitcher. How much water does John need to add so that the fruit punch will taste exactly the same as it did before? April 2012 Page 29 of 57

30 Sixth Grade Mathematics Unit 4 Task: Picturing Proportions ESSENTIAL QUESTIONS What is a proportion? How can proportional relationships be represented using rules, tables, and graphs? STANDARDS ADDRESSED MCC6.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 specified set makes an equation or inequality true. MCC6.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. MCC.6.EE.7 Solve real-world and mathematical problems by writing and solving equations of the form x p q and px q for cases in which p, q and x are all nonnegative rational numbers. MCC.6.RP.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. MATERIALS graph paper colored pencils TASK COMMENTS In this task, students will create and analyze graphs to demonstrate their understanding of direct proportions. It is important for students to compare multiple graphs and describe the variation that occurs between the graphs and their corresponding equations. Teachers should support good student dialogue and take advantage of comments and questions to help guide students into correct mathematical thinking. Picturing Proportions Draw a picture and write an explanation to convince someone that Solution April 2012 Page 30 of 57

31 Rectangle A Sixth Grade Mathematics Unit 4 Rectangle B Rectangle A and rectangle B are the same size. Rectangle A is split into 4 equal pieces and rectangle B is split into 12 equal pieces. When you shade 2 out of the 4 pieces and 6 out of the 12 pieces, the shaded regions are the same size. 2. Triangle ABC has side lengths 3 units, 4 units, and 5 units. Draw two triangles, one smaller than Triangle ABC, and one larger than Triangle ABC so that the sides of your triangles vary proportionally with the sides of Triangle ABC. Possible Solution Let x equal a dimension of Triangle ABC. Let y equal the dimension of a smaller triangle whose side lengths vary proportionally with the side lengths of Triangle ABC. Let. Then the sides of the smaller triangle can be expressed as y = kx: A E 2.5 B 4 C F 2 G April 2012 Page 31 of 57

32 Sixth Grade Mathematics Unit April 2012 Page 32 of 57

33 Sixth Grade Mathematics Unit 4 A X 3 5 B 4 C 6 10 Y 8 Z Similarly, a larger triangle can be generated by letting k = 2. April 2012 Page 33 of 57

34 Sixth Grade Mathematics Unit 4 3. Isabella takes 4 2 of a candy bar and gives her little brother Lucas 2 1 of a different candy bar. Look at the picture representing the two candy bars. Isabella explains to Lucas that 1 2, so she is giving him the same amount of candy that she takes for herself. Is 2 4 Isabella s explanation correct or is she trying to trick her little brother? Explain your reasoning. Isabella s portion Lucas s portion Solution Isabella is trying to trick her brother. of something is equal to of the same thing, but in this case, Isabella s candy bar was larger than Lucas s to begin with, so of her candy bar will be more than of Lucas s smaller candy bar. April 2012 Page 34 of 57

35 Sixth Grade Mathematics Unit 4 Picturing Proportions Draw a picture and write an explanation to convince someone that Triangle ABC has side lengths 3 units, 4 units, and 5 units. Draw two triangles, one smaller than Triangle ABC, and one larger than Triangle ABC so that the sides of your triangles vary proportionally with the sides of Triangle ABC. 3. Isabella takes 4 2 of a candy bar and gives her little brother Lucas 2 1 of a different candy bar. Look at the picture representing the two candy bars. Isabella explains to Lucas that 1 2, so she is giving him the same amount of candy that she takes for herself. Is 2 4 Isabella s explanation correct or is she trying to trick her little brother? Explain your reasoning. Isabella s portion Lucas s portion April 2012 Page 35 of 57

36 Sixth Grade Mathematics Unit 4 Task: Making Sense of Graphs ESSENTIAL QUESTIONS How can proportional relationships be described using the equation y = kx? How can proportional relationships be represented using rules, tables, and graphs? How can the graph of y = kx be interpreted for different contexts? How can algebraic expressions be used to model real-world situations? How can we solve simple algebraic equations, and how do we interpret the meaning of the solutions? STANDARDS ADDRESSED MCC6.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 specified set makes an equation or inequality true. MCC6.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. MCC.6.EE.7 Solve real-world and mathematical problems by writing and solving equations of the form x p q and px q for cases in which p, q and x are all nonnegative rational numbers. MCC6.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. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. MCC.6.RP.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. MATERIALS graph paper colored pencils April 2012 Page 36 of 57

37 Sixth Grade Mathematics Unit 4 TASK COMMENTS In this task, students will create and analyze graphs to demonstrate their understanding of direct proportions. It is important for students to compare multiple graphs and describe the variation that occurs between the graphs and their corresponding equations. Teachers should support good student dialogue and take advantage of comments and questions to help guide students into correct mathematical thinking. Making Sense of Graphs The graph below shows the amount of money required to buy gasoline if the cost per gallon is $2.00. April 2012 Page 37 of 57

38 Sixth Grade Mathematics Unit 4 a. What two quantities vary proportionally in this situation? Solution The price of gas varies proportionally with the number of gallons of gas purchased. b. What is the value of the constant of proportionality? What does this value represent in the context of the problem? How is the constant of proportionality represented on the graph? Solution The constant of proportionality is 2. This value represents the price per gallon of gas. c. Write an equation to represent this situation. Solution The constant of proportionality is 2 which means. Therefore, y = 2x. d. Suppose gas prices rose to $3.00 per gallon. How would the graph change? Explain your reasoning. Solution The graph would get steeper, because for every one increase in the number of gallons purchased, there is an increase of $3.00 in the cost instead of $2.00. Over the same amount of horizontal increase, there is a larger vertical increase, which would make the graph steeper. April 2012 Page 38 of 57

39 Sixth Grade Mathematics Unit 4 e. Write an equation to represent the situation in part d. Solution The constant of proportionality is 3 which means. Therefore, y = 3x. April 2012 Page 39 of 57

40 Sixth Grade Mathematics Unit 4 Making Sense of Graphs The graph below shows the amount of money required to buy gasoline if the cost per gallon is $2.00. a. What two quantities vary proportionally in this situation? April 2012 Page 40 of 57

41 Sixth Grade Mathematics Unit 4 b. What is the value of the constant of proportionality? What does this value represent in the context of the problem? How is the constant of proportionality represented on the graph? c. Write an equation to represent this situation. d. Suppose gas prices rose to $3.00 per gallon. How would the graph change? Explain your reasoning. e. Write an equation to represent the situation in part d. April 2012 Page 41 of 57

42 Sixth Grade Mathematics Unit 4 Task: Analyzing Tables ESSENTIAL QUESTIONS How can proportional relationships be described using the equation y = kx? How can proportional relationships be represented using rules, tables, and graphs? How can the graph of y = kx be interpreted for different contexts? How can algebraic expressions be used to model real-world situations? How can we solve simple algebraic equations, and how do we interpret the meaning of the solutions? STANDARDS ADDRESSED MCC6.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 specified set makes an equation or inequality true. MCC6.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. MCC.6.EE.7 Solve real-world and mathematical problems by writing and solving equations of the form x p q and px q for cases in which p, q and x are all nonnegative rational numbers. MCC6.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. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. MCC.6.RP.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. MATERIALS graph paper (optional) TASK COMMENTS In this task, students will demonstrate a deeper understanding of direct proportions through tables, graphs, and rules (equations). It is important for students to communicate their April 2012 Page 42 of 57

43 Sixth Grade Mathematics Unit 4 understanding of direct proportions using multiple representations. Students should also explain how each representation demonstrates proportional reasoning. Teachers should support good student dialogue and take advantage of comments and questions to help guide students into correct mathematical thinking. Extension Ask students to graph each data set. Compare graphs to proportional data sets and sets that are not proportional. Analyzing Tables Consider the tables below where the x- and y-values represent two quantities. For each table, do the following: a. Do the quantities vary proportionally? Explain how you know. b. Write a rule for each table in words. c. Write the rule as an equation. Table 1 x y Solution a. The quantities x and y vary proportionally because for each pair of values. b. In other words, the ratio between the two quantities y and x is constant (2). c. For each entry in the table, y is two times x. Rule: y = 2x Table 2 x y Solution a. The quantities x and y vary proportionally because for each pair of values. b. In other words, the ratio between the two quantities y and x is constant. April 2012 Page 43 of 57