Solving Equations with One Variable


 Gillian Richard
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1 Grade 8 Mathematics, Quarter 1, Unit 1.1 Solving Equations with One Variable Overview Number of Instructional Days: 15 (1 day = 45 minutes) Content to Be Learned Solve linear equations in one variable with one solution, infinitely many solutions, or no solution. Solve linear equations in one variable including use of the distributive property. Solve linear equations in one variable including rational number coefficients (whole numbers, decimals, fractions, and their opposites). Solve linear equations in one variable combining like terms (on one or both sides of the equal sign). Essential Questions What steps would you use to solve any linear equation in one variable? What are the possible types of solutions to any linear equation in one variable? How would you write an equation that demonstrates each of the following: one solution, no solution, or infinitely many solutions? Mathematical Practices to Be Integrated Make sense of problems and persevere in solving them. Understand steps and their order before beginning the problem. Check solutions and ask if they make sense. Understand different approaches to the same problem. Reason abstractly and quantitatively. Use the distributive property to combine like terms. Know and use properties to solve linear equations. Look for and express regularity in repeated reasoning. Justify the steps for solving an equation in one variable. Understand patterns in problems to gain efficiency in solving them. When would you use the distributive property to simplify a linear equation in one variable? How can you combine like terms to simplify a linear equation? 1
2 Grade 8 Mathematics, Quarter 1, Unit 1.1 Solving Equations with One Variable (15 days) Written Curriculum Common Core State Standards for Mathematical Content Expressions and Equations 8.EE Analyze and solve linear equations and pairs of simultaneous linear equations. 8.EE.7 Solve linear equations in one variable. a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Common Core Standards for Mathematical Practice 1 Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, Does this make sense? They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. 2 Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. 2
3 Grade 8 Mathematics, Quarter 1, Unit 1.1 Solving Equations with One Variable (15 days) 8 Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y 2)/(x 1) = 3. Noticing the regularity in the way terms cancel when expanding (x 1)(x + 1), (x 1)(x 2 + x + 1), and (x 1)(x 3 + x 2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. Clarifying the Standards Prior Learning In sixth grade, students were exposed to applying the distributive property and simplifying algebraic expressions by combining like terms. In seventh grade, students were fluent in working with rational numbers and multistep equations with one variable on one side of the equation. Students applied the distributive property and they have knowledge of combining like terms with rational number coefficients. Current Learning Students solve linear equations with one variable resulting in one solution, infinitely many solutions, or no solution. They also solve linear equations with rational number coefficients, requiring the distributive property and combining like terms. This is a major cluster (PARCC) and a critical area of CCSS. Later this year, students will use their knowledge from this unit for the following units: Linear Functions, Slope, and Systems of Equations. Future Learning In high school algebra, students will create, solve, and graph linear inequalities in one variable. Students will solve linear equations in one variable with variable coefficients. Students will solve quadratic equations in one variable. Additional Findings According to Principles and Standards for School Mathematics, in grades 6 8, all students should recognize and generate equivalent forms for simple algebraic expressions and solve linear equations (p. 222). According to Curriculum Focal Points, Students in eighth grade will use linear functions, linear equations, and systems of linear equations to represent, analyze, and solve a variety of problems (p. 20). According to the PARCC Model Content Frameworks, Onevariable linear equations culminate in grade 8 with the solution of general onevariable linear equations, including cases with infinitely many solutions or no solutions as well as cases requiring algebraic manipulation using properties of operations. Coefficients and constants in these equations may be any rational number. (p. 35) 3
4 Grade 8 Mathematics, Quarter 1, Unit 1.1 Solving Equations with One Variable (15 days) 4
5 Grade 8 Mathematics, Quarter 1, Unit 1.2 Introduction to Linear Functions Overview Number of Instructional Days: 15 (1 day = 45 minutes) Content to Be Learned Definition of a function (for each input there is exactly one output). Use a function rule to generate ordered pairs. Graph a function using ordered pairs. Determine rate of change of one function represented in multiple ways (table, graph, description, ordered pairs). Describe and compare the rate of change of two functions. Compare the properties of two functions in different representations (graphs, tables, and verbal descriptions). Construct a function to model a linear relationship between two quantities. Given a graph or a table, interpret what rate of change and initial value mean in the context of the linear function. Given a realworld situation, describe what the rate of change and initial value means. Essential Questions What is a function? What are the different ways to represent a function? How can you determine the rate of change of a function given different representations? Mathematical Practices to Be Integrated Make sense of problems and persevere in solving them. Explain correspondences between multiple representations. Continuously ask, Does this make sense? Check and evaluate progress. Reason abstractly and quantitatively. Represent a situation with a function. Identify how functions relate to realworld situations. Write a verbal description of a given function. Make sense of quantities and their relationships in problem situations. Model with mathematics. Identify important quantities in a function rule. Apply functions to everyday situations. Use mathematical tools to show relationships and draw conclusions. How could you use a function to model a realworld situation? Given two different functions, how can you tell which function has a greater rate of change? What is the relationship between an input/output table and a graph? 5
6 Grade 8 Mathematics, Quarter 1, Unit 1.2 Introduction to Linear Functions (15 days) Written Curriculum Common Core State Standards for Mathematical Content Functions 8.F Define, evaluate, and compare functions. 8.F.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. 1 1 Function notation is not required in Grade 8. 8.F.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Use functions to model relationships between quantities. 8.F.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Common Core Standards for Mathematical Practice 1 Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, Does this make sense? They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. 2 Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the 6
7 Grade 8 Mathematics, Quarter 1, Unit 1.2 Introduction to Linear Functions (15 days) representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. 4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, twoway tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. Clarifying the Standards Prior Learning In sixth grade, students learned about independent and dependent variables and analyzed the relationship between them using graphs, tables, and equations. In seventh grade, students identified the unit rate in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. Students explained what a point (x, y) means in terms of a problem situation. Students represented proportional relationships with equations. Current Learning These concepts are a critical area and major cluster of the grade 8 CCSS standards and should be taught at the developmental and reinforcement level. Students define a function (they do not need to use function notation). They make a function table to generate ordered pairs as a means of graphing a function. Students determine the rate of change of a function represented in multiple ways and are able to describe and compare two functions represented in multiple ways. Students construct a function to model a linear relationship between two quantities. They interpret what rate of change and initial value mean in the context of the function. Students will use their knowledge of functions later in the school year while studying slope. They need to be fluent with using and interpreting linear functions by the end of grade 8. Future Learning In high school algebra, students will be deepening their knowledge of functions to expand their studies of nonlinear functions. 7
8 Grade 8 Mathematics, Quarter 1, Unit 1.2 Introduction to Linear Functions (15 days) Additional Findings According to Principles and Standards for School Mathematics, in grades 6 8, All students should identify functions as linear or nonlinear and contrast their properties from tables, graphs, or equations. (p. 222). According to Curriculum Focal Points, Students use linear functions to represent, analyze, and solve a variety of problems (p. 39). According to the PARCC Model Content Framework, students also formalize their previous work with linear relationships by working with functions rules that assign to each input exactly one output (p. 35). 8
9 Grade 8 Mathematics, Quarter 1, Unit 1.3 Slope Overview Number of Instructional Days: 15 (1 day = 45 minutes) Content to Be Learned Graph proportional relationships on a coordinate plane. Understand that rate of change is slope. Compare proportional relationships represented in different ways. Derive the formula y = mx + b (understand m is slope and b is yintercept) by visual representations of similar triangles. See Progressions: Expressions and Equations p. 12; commoncoretools.wordpress.com. Understand that any two points on a line can be used to find the slope. Distinguish between linear and nonlinear functions. Mathematical Practices to Be Integrated Make sense of problems and persevere in solving them. Explain correspondences between multiple representations. Continuously ask, Does this make sense? Check and evaluate progress. Use appropriate tools strategically. Decide when tools (i.e., graphing calculators) are appropriate. While graphing, know to use graphing paper, pencil, and rulers. Attend to precision. Use appropriate scales. Use proper definitions and symbols. Look for and express regularity in repeated reasoning. Pay attention to the calculation of slope while repeatedly checking to see if points are on the line. Know that the slope represents a pattern of constant change. Essential Questions What is the relationship between rate of change and slope? How do you know what the yintercept is? What makes a function nonlinear? Using a graph and an equation in the form y = mx + b, how do you compare the slope? How can you find the slope? What does the equation y = mx + b look like on a graph? 9
10 Grade 8 Mathematics, Quarter 1, Unit 1.3 Slope (15 days) Written Curriculum Common Core State Standards for Mathematical Content Expressions and Equations 8.EE Understand the connections between proportional relationships, lines, and linear equations. 8.EE.5 8.EE.6 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distancetime graph to a distancetime equation to determine which of two moving objects has greater speed. Use similar triangles to explain why the slope m is the same between any two distinct points on a nonvertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. Functions 8.F Define, evaluate, and compare functions. 8.F.3 Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s 2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Common Core Standards for Mathematical Practice 1 Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, Does this make sense? They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. 10
11 Grade 8 Mathematics, Quarter 1, Unit 1.3 Slope (15 days) 5 Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. 6 Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. 8 Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y 2)/(x 1) = 3. Noticing the regularity in the way terms cancel when expanding (x 1)(x + 1), (x 1)(x 2 + x + 1), and (x 1)(x 3 + x 2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. Clarifying the Standards Prior Learning In sixth grade, students grow their ability to analyze proportional relationships. In seventh grade, students were able to analyze proportional relationships and they solved problems involving unit rates. They have also worked with solving problems involving scale drawings and geometric figures. Students have reproduced images at a different scale. 11
12 Grade 8 Mathematics, Quarter 1, Unit 1.3 Slope (15 days) Current Learning In eighth grade, all components of this unit are critical areas and major clusters. Students build on previous knowledge with unit rate, proportional relationships, and graphing to connect these ideas and understand that points (x, y) on a nonvertical line are the solutions of the equation y = mx + b, where m is the slope of the line as well as the unit rate. In eighth grade, students interpret unit rate as the slope of the graph. They interpret and compare different slopes. Students use similar triangles on a graph to explain why the slope of a line is the same between any two points (for example, 4/8 = 1/2). They then extend that graph to find the yintercept. Students understand that y = mx + b defines a linear function and what a nonlinear function looks like. Future Learning In high school algebra, students will use graphing to interpret multiple functions. They will construct linear functions. Additional Findings According to Principles and Standards for School Mathematics, Students will explore relationships between symbolic expressions and graphs of lines, paying particular attention to the meaning of intercept and slope (p. 222). According to the PARCC Model Content Framework, Students build on previous work with proportion relationships, unit rates, and graphing to connect these ideas and understand that points (x, y) on a nonvertical line are the solutions of the equation y = mx + b, where m is the slope of the line as well as the unit rate of a proportional relationship (in the case b = 0) (p. 35). According to Common Core Progressions 68, Expressions and Equations, the connection between the unit rate in a proportional relationship and the slope of its graph depends on a connection with the geometry of similar triangles (p. 12). 12
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