Gary School Community Corporation Mathematics Department Unit Document Unit Number: 5 Grade: 7 Unit Name: Equations and Inequalities Duration of Unit: 20-25 Days UNIT FOCUS Standards for Mathematical Content 7.AF.2: Solve equations of the form px + q= r and p(x + q) = r fluently, where p, q, and r are specific rational numbers. Represent real-world problems using equations of these forms and solve such problems. 7.AF.1: Apply the properties of operations (e.g., identity, inverse, commutative, associative, distributive properties) to create equivalent linear expressions, including situations that involve factoring (e.g., given 2x - 10, create an equivalent expression 2(x - 5)). Justify each step in the process. 7.AF.3: Solve inequalities of the form px +q (> or ) r or px + q (< or ) r, where p, q, and r are specific rational numbers. Represent real-world problems using inequalities of these forms and solve such problems. Graph the solution set of the inequality and interpret it in the context of the problem. 7.AF.4: Define slope as vertical change for each unit of horizontal change and recognize that a constant rate of change or constant slope describes a linear function. Identify and describe situations with constant or varying rates of change. 7.AF.5: Graph a line given its slope and a point on the line. Find the slope of a line given its graph. 7.AF.6: Decide whether two quantities are in a proportional relationship (e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin). 7.AF.8: Explain what the coordinates of a point on the graph of a proportional relationship mean in terms of the situation, with special attention to the points (0, 0) and (1,r), where r is the unit rate. Standard Emphasis Critical Important Additional Vertical Articulation documents for K 2, 3 5, and 6 8 can be found at: http://www.doe.in.gov/standards/mathematics (scroll to bottom)
7.AF.9: Identify real-world and other mathematical situations that involve proportional relationships. Write equations and draw graphs to represent proportional relationships and recognize that these situations are described by a linear function in the form y = mx, where the unit rate, m, is the slope of the line. Mathematical Process Standards: PS.1: Make sense of problems and persevere in solving them. PS.2: Reason abstractly and quantitatively PS.3: Construct viable arguments and critique the reasoning of others PS.4: Model with mathematics PS.5: Use appropriate tools strategically PS.6: Attend to Precision PS.7: Look for and make use of structure PS.8: Look for and express regularity in repeated reasoning Big Ideas/Goals Students will solve real-life and mathematical problems using numerical and algebraic expressions and equations. Essential Questions/ Learning Targets How are equations and inequalities used for solving realworld or mathematical problems? I Can Statements I can use a variable to represent an unknown quantity. I can write a simple algebraic equation of the form px + q=r where (p, q, and r) are specific numbers to represent a real -world problem. I can use inverse operations and the properties of equality to solve word problems leading to equations of the form px + q = r where p, q, and r are specific numbers. I can use inverse operations and the properties of equality to solve word problems leading to equations of the form p(x Revised July 2012 9 + q) = r where p, q, and r are specific numbers. I can compare the algebraic solution to an arithmetic solution. 2
Students will use properties of How can the properties of operations to generate equivalent operations be used to transform expressions. linear expressions? I can apply the properties of operations as strategies to add linear expressions with rational coefficients. I can apply the properties of operations as strategies to subtract linear expressions with rational coefficients. I can apply the properties of operations as strategies to factor linear expressions with rational coefficients. I can apply the properties of operations as strategies to expand linear expressions with rational coefficients. Students will solve real-life and mathematical problems using numerical and algebraic expressions and equations. How are equations and inequalities used for solving realworld or mathematical problems? I can write a simple algebraic inequality in the form px+q=r where (p, q, and r) are specific numbers to represent a real-world problem. I can use inverse operations and the properties of inequality to solve word problems leading to inequalities of the form px + q > r where p, q, and r are specific numbers I can useinverse operations and the properties of inequalities of the form px + q < r where p, q, and r are specific numbers. I can use inverse operations and the properties of I can interpret the solution set in relation to the problem. Students will understand the connections between proportional relationships, lines, and linear equations. What are multiple ways to compare two different proportional relationships? I can define the slope as a vertical change per unit of its horizontal change. I can recognize that a straight line has constant slope or rate of change. I can demonstrate a rate as a measure of one quantity with respect to another quantity. I can find the slope of line from its graph. I can graph a line given the slope and one point on the line. I can graph a line given two points on the line. Students will analyze proportional relationships and How can ratios of fractions and quantities measured in like or I can determine if two quantities are proportional using a variety of 3
use them to solve a variety of real-world and mathematical problems. different units be expressed as unit rates? How can proportional relationships be represented? methods (table, graphs, diagrams, equations, or verbal description). I can represent proportional relationship in an equation. Students will graph proportional relationships and understand the unit rate informally as a measure of the steepness of the related line, called the slope. Students will analyze proportional relationships and use them to solve a variety of real-world and mathematical problems. Students will understand the connections between proportional relationships, lines, and linear equations. How can a proportional relationship be represented by an equation? I can explain what a point (x, y) on the graph of a proportional relationship means. I can identify the unit rate by using the point (1, r). I can explain what the point (0, 0) on the graph of a proportional relationship means. How can ratios of fractions and I can recognize and represent quantities measured in like or proportional relationships between different units be expressed as quantities unit rates? I can represent proportional How can a proportional relationships in an equation. relationship be represented by an I can graph proportional relationships. equation? I can interpret the unit rate as a slope of the graph of a proportional What are multiple ways to relationship. compare two different I can compare two different proportional relationships? proportional relationships represented in different ways. UNIT ASSESSMENT TIME LINE Beginning of Unit Pre-Assessment Assessment Name: Grade 7 Pre-Assessment Equations and Inequalities Assessment Type: Pre-assessment Equations and Inequalities Assessment Standards: AF.1, AF.2, AF.3, AF.4, AF.5, AF.6, AF.8, & AF.9 Assessment Description: This Unit helps the student understand linear equations (slopes, y- intercepts, relationships, patterns, graphing, etc.), solving equations and inequalities in real world situations. Throughout the Unit Formative Assessment Assessment Name: Evaluating, Factoring and Simplifying Linear Expressions Assessment Type: Performance Assessment 4
Assessing Standards: AF.1 Assessment Description: Students will review order of operations and evaluating, factoring and simplifying linear expressions, as well as identifying the coefficients for linear expressions. Assessment Name: Solving Equations and Inequalities Assessment Type: Performance Assessment or Exit Slip Assessing Standards: AF.2 & AF.3 Assessment Description: Students will solve equations and inequalities for mathematical practice and connection to real-world problems. Assessment Name: Understanding Slope Assessment Type: Performance Assessment Assessing Standards: AF.4 & AF.5 Assessment Description: Students will identify slope and y-intercepts from equations, graphs and tables, and students will graph linear equations. Assessment Name: Graphing and Writing Linear Equations Assessment Type: Performance Assessment Assessing Standards: AF.6, AF.8, & AF.9 Assessment Description: Students will graph, compare and locate various proportional relationships. End of Unit Summative Assessments Assessment Name: Grade 7 Summative Assessment Equations and Inequalities Assessment Type: Benchmark Tests or Sample Standardized Assessment Questions Assessing Standards: AF.1, AF.2, AF.3, AF.4, AF.5, AF.6, AF.8, & AF.9 Assessment Description: Summative assessment of equations and inequalities, as well as linear equations using Mathematical Processes and real life situations. 5
PLAN FOR INSTRUCTION Unit Vocabulary Key terms are those that are newly introduced and explicitly taught with expectation of student mastery by end of unit. Prerequisite terms are those with which students have previous experience and are foundational terms to use for differentiation. Key Terms for Unit Prerequisite Math Terms properties of operations linear expressions expanding expressions coefficient rational number expression inequality properties of inequality algebraic solution arithmetic solution inverse operation properties of equality solution set proportion proportional relationship quantities equivalent ratio cross product ordered pair quadrants graph unit rate slope (rate of change) factor variable evaluate coordinate plane origin x-axis y-axis x-coordinate y-coordinate coordinates axis (axes) ordered pair quadrants 6
Unit Resources/Notes Include district and supplemental resources for use in weekly planning Pre-Assessment: Algebraic Functions and Expressions and Equations istep Strands Formative Assessment 1 AF.1 Use EE.A.1 resources when you click the link below. https://grade7commoncoremath.wikispaces.hcpss.org/unit+3+expressions+and+equations Formative Assessment 2 AF.2 - Use EE.4A when you click the link below. https://grade7commoncoremath.wikispaces.hcpss.org/unit+3+expressions+and+equations http://www.rda.aps.edu/mathtaskbank/fi_html/pfactask.htm The Triangle Train problem can be found on this website. The site includes the teacher instructions for possible student answers and misconceptions. http://learner.org/workshops/algebra/workshop1/index2.html This website provides explanations for using manipulatives to solve equations as well as a real-world problem involving two-step equations with and without the distributive property. http://www.math-play.com/two-step-equations-game.html This is a basketball game where students determine the solutions of two-step equations. http://betterlesson.com/common_core/browse/420/ccss-math-content-7-ee-b-4-use-variables-to-represent-quantitiesin-a-real-world-or-mathematical-problem-and-construct-simple-eq Text Resources/Tasks: Driscoll, M. (1999). Fostering Algebraic Thinking: A guide for Teachers Grades 6-10. Heinemann. p. 22 "Golden Apples" Math Task p. 22: This math task allows students to develop a strategy to solve a story problem. This would be a nice introduction for connecting their strategy to constructing equations. AF.3 - Use EE.4B when you click the link below. https://grade7commoncoremath.wikispaces.hcpss.org/unit+3+expressions+and+equations http://www.cut-the-knot.org/simplegames/frogsandtoadsc.shtml This website includes a puzzle where students are asked to use slides and jumps to switch the places of frogs and toads. This activity could be a lead into an activity for writing an equation for the minimum number of moves necessary for any number of frogs/toads. The current puzzle has three of each. Formative Assessment 3 - AF.4 Use EE.B.5 when you click the link below. https://grade8commoncoremath.wikispaces.hcpss.org/unit+3+analyzing+functions+and+equations http://education.ti.com/calculators/timathnspired/us/activities/detail?sa=1008&t=9447&id=16866 Lesson: This lesson involves students using the emulation of a spring scale stretched by a weight to record and graph the direct proportional relationship between the weight and the stretch. (Note: This resource will also be referenced with 7.RP.1.) http://www.learner.org/workshops/algebra/workshop7/index.html Direct Variation Teacher Workshop with activities. (Note: This resource will also be referenced with 7.RP.2c.) http://education.ti.com/calculators/timathnspired/us/activities/detail?sa=1008&t=9447&id=16896 TI Lesson: Students compute unit rates, find linear equations using unit rates and examine ordered pairs to confirm that linear equations represent proportional relationships. (Note: This resource will also be referenced with 7.RP.2c.) AF.5 Use EE.B.5 when you click the link below. https://grade8commoncoremath.wikispaces.hcpss.org/unit+3+analyzing+functions+and+equations 7
http://education.ti.com/calculators/timathnspired/us/activities/detail?sa=1008&t=9447&id=16866 Lesson: This lesson involves students using the emulation of a spring scale stretched by a weight to record and graph the direct proportional relationship between the weight and the stretch. (Note: This resource will also be referenced with 7.RP.1.) http://www.learner.org/workshops/algebra/workshop7/index.html Direct Variation Teacher Workshop with activities. (Note: This resource will also be referenced with 7.RP.2c.) http://education.ti.com/calculators/timathnspired/us/activities/detail?sa=1008&t=9447&id=16896 TI Lesson: Students compute unit rates, find linear equations using unit rates and examine ordered pairs to confirm that linear equations represent proportional relationships. (Note: This resource will also be referenced with 7.RP.2c.) Formative Assessment 4 AF.6 Use RP.2 & RP.2A when you click the link below. https://grade7commoncoremath.wikispaces.hcpss.org/unit+2+ratios+and+proportional+relationshi ps PARCC/UTA Dana Center Prototype Assessment Item: "Proportional Relationships" (7.RP.A.2a, MP.3, MP.6) http://www.parcconline.org/sites/parcc/files/parcc_sampleitems_mathematics_g7proportionalrelationships_081913 _Final.pdf http://education.ti.com/calculators/timathnspired/us/activities/detail?sa=1008&t=9447&id=16897 TI Lesson: Students analyze a scatter plot of data, generalize a rule to describe the data, and graph the function that describes the rule to verify that it matches the scatter plot. http://illustrativemathematics.org/illustrations/100 Task: Art Class, Variation 1. Student analyze the mixtures of different shades of green paint, including graphing the mixtures on a coordinate plane. Text Resources for Lessons/Task: Lobato, J.E., Ellis, A.B, Charles, R.I., & Zbiek, R.M. (2010). Developing Essential Understanding of Ratios, Proportions, & Proportional Reasoning for Teaching Mathematics in Grades 6-8. Reston, VA: National Council of Teachers of Mathematics. p. 66. Task p. 66: Students compare two heart rates to determine whether they are beating at the same pace. AF.8 Use RP.2D when you click the link below. https://grade7commoncoremath.wikispaces.hcpss.org/unit+2+ratios+and+proportional+relationships PARCC Prototype Assessment Item: "Speed" (7.RP.2b and 7.RP.2d) http://www.parcconline.org/samples/mathematics/grade-7-speed http://education.ti.com/calculators/timathnspired/us/activities/detail?sa=1008&t=9447&id=16892 TI Lesson: Students use a ratio to plot points and determine a mathematical relationship for plotted points. They also compute the unit rate and predict ordered pairs. http://illustrativemathematics.org/illustrations/181 Task: Robot Races. Student analyze graphs of the travel of three robots. AF.9 Use RP.2, RP.2C, & EE.5 when you click the link below. https://grade7commoncoremath.wikispaces.hcpss.org/unit+2+ratios+and+proportional+relationships http://www.learner.org/workshops/algebra/workshop7/index.html Direct Variation Teacher Workshop with activities. (Note: This resource will also be referenced with 8.EE.5.) http://education.ti.com/calculators/timathnspired/us/activities/detail?sa=1008&t=9447&id=16896 TI Lesson: Students compute unit rates, find linear equations using unit rates and examine ordered pairs to confirm that linear equations represent proportional relationships. (Note: This resource will also be referenced with 8.EE.5.) 8
http://illustrativemathematics.org/illustrations/101 Task: Art Class, Variation 2. Student analyze the mixtures of different shades of green paint, including writing equations to represent each of the shades of paint. http://illustrativemathematics.org/illustrations/180 Task: Sore Throats, Variation 1. Student analyze salt water mixtures and determine equations for them. Summative Assessment Algebraic Functions and Expressions and Equations istep Strands or Acuity Targeted Process Standards for this Unit PS.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. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, Does this make sense? and "Is my answer reasonable?" They understand the approaches of others to solving complex problems and identify correspondences between different approaches. Mathematically proficient students understand how mathematical ideas interconnect and build on one another to produce a coherent whole. PS.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. PS.3: Construct viable arguments and critique the reasoning of others Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They analyze situations by breaking them into cases and recognize and use counterexamples. They organize their mathematical thinking, justify their conclusions and communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument explain what it is. They justify whether a given statement is true always, sometimes, or never. Mathematically proficient students participate and collaborate in a mathematics community. They listen to or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. 9
PS.4: Model with mathematics Mathematically proficient students apply the mathematics they know to solve problems arising in everyday life, society, and the workplace using a variety of appropriate strategies. They create and use a variety of representations to solve problems and to organize and communicate mathematical ideas. Mathematically proficient students apply what they know and 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, two-way tables, graphs, flowcharts and formulas. They 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. PS.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. PS.6: Attend to precision Mathematically proficient students communicate precisely to others. They use clear definitions, including correct mathematical language, 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 express solutions clearly and logically by using the appropriate mathematical terms and notation. They specify units of measure and label axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently and check the validity of their results in the context of the problem. They express numerical answers with a degree of precision appropriate for the problem context. PS.7: Look for and make use of structure Mathematically proficient students look closely to discern a pattern or structure. They step back for an overview and shift perspective. They recognize and use properties of operations and equality. They organize and classify geometric shapes based on their attributes. They see expressions, equations, and geometric figures as single objects or as being composed of several objects. PS.8: Look for and express regularity in repeated reasoning Mathematically proficient students notice if calculations are repeated and look for general methods and shortcuts. They notice regularity in mathematical problems and their work to create a rule or formula. Mathematically proficient students maintain oversight of the process, while attending to the details as they solve a problem. They continually evaluate the reasonableness of their intermediate results. 10