Gary School Community Corporation Mathematics Department Unit Document Unit Number: 3 Grade: 7 Unit Name: Real Numbers, Ratios, and Proportional Relationships Duration of Unit: 30 Days Standards for Mathematical Content UNIT FOCUS 7.C.6: Use proportional relationships to solve ratio and percent problems with multiple operations, such as the following: simple interest, tax, markups, markdowns, gratuities, commissions, fees, conversions within and across measurement systems, percent increase and decrease, and percent error Standard Emphasis Critical Important Additional 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.7: Identify the unit rate or constant of proportionality in tables, graphs, equations, and verbal descriptions of proportional relationships. 7.C.5: Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. 7.GM.3: Solve real-world and other mathematical problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing. Create a scale drawing by using proportional reasoning. 7.NS.2: Understand the inverse relationship between squaring and finding the square root of a perfect square integer. Find square roots of perfect square integers 7.NS.3: Know there are rational and irrational numbers. Identify, compare, and order rational and common irrational numbers ( 2, 3, 5, ) and plot them on a number line. 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)
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 understand the concept of ratio and unit rate and apply it to multiple situations. Students solve a wide variety of problems involving ratios and rates. Essential Questions/ Learning Targets How can proportional relationships be used to solve percent and ratio problems? How are rates and percent problems similar? How do rates and percent problems differ? I Can Statements I can use proportional relationships to solve multi-step ratios. I can use proportional relationships to solve multi-step percent problems. Students will understand the connections between proportional relationships, lines, and linear equations. Students will analyze proportional relationships and use them to solve a variety of real-world and mathematical problems. What are multiple ways to compare two different proportional relationships? What are the properties of a proportional relationship and how can they be identified? How can these properties be identified when the relationship is modeled in various ways? I can define the rate of change in relation to the situation. I can interpret the unit rate of a proportional relationship as the slope of the graph. I can solve real-world problems and mathematical problems dealing with systems of linear equations and interpret the solution in the context of the problem. proportionality in a table. proportionality in a diagram. proportionality in a graph. proportionality in an equation. proportionality in a verbal description. 2
Students will analyze proportional relationships and use them to solve a variety of real-world and mathematical problems. How can ratios of fractions and quantities measured in like or different units be expressed as unit rates? I can compute unit rates with ratios of fractions. I can compute unit rates with ratios of lengths, areas, and other quantities. I can compute unit rates with ratios measured in like units. I can compute unit rates with ratios measured in different units. Students will draw, construct and describe geometrical figures and describe the relationships between them. How do scale drawings assist in problem solving? I can solve problems involving scale drawings of geometric figures. I can compute the actual length of a geometric figure from a scale drawing. I can compute the actual area of a geometric figure from a scale drawing. I can reproduce a scale drawing at a different scale. Students will apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. Students will work with radicals and integer exponents How can a number line be used to demonstrate the properties and processes of addition and subtraction of rational numbers? How can the previous learned properties of multiplication and division be extended to multiplication and division of rational numbers? How are the properties of integer exponents used to generate equivalent numerical expressions? I can use the properties of integer exponents to simplify expressions. I can recognize taking a square root as the inverse of squaring a number. I can use reasoning to determine between which two consecutive whole numbers a square root will fall. I can recognize taking a cube root of a perfect square. I can evaluate the cube root of a perfect cube. Students will know that there are In what ways can rational I can classify a number as rational or numbers that are not rational, numbers be represented and how irrational based on its decimal and approximate them by rational can they be used? expansion. numbers. I can convert a repeating decimal into a rational number. I can plot the estimated value of an irrational number on a number line. I can estimate the value of an irrational number by rounding to a specific place value. I can use estimated values to compare two or more irrational numbers. I can justify that the square root of a non-perfect square will be irrational. UNIT ASSESSMENT TIME LINE 3
Beginning of Unit Pre-Assessment Assessment Name: Grade 7 Pre- Assessment Ration and Proportions Acuity or istep Assessment Type: Pre-assessment Ratio and Proportions Assessment Standards: 7.C.6, 7.AF.4, 7.AF.7, 7.C.5, 7.GM.3, 7.NS.1, 7.NS.2, 7.NS.3 Assessment Description: Understand ratios and proportions using real numbers and in real-world situations. Throughout the Unit Formative Assessment Assessment Name: Computing Unit rates and Using Proportional Relationships to solve ration and percent problems Assessment Type: Performance Assessment Assessing Standards: 7.C.5 & 7.5.6 Assessment Description: Use proportional relationships to solve ratio and percent problems with multiple operations. Assessment Name: Prime and Composite Numbers Rational and Irrational numbers Assessment Type: Performance Assessment - Task Assessing Standards: 7.NS.1, 7.NS.2, & 7.NS.3 Assessment Description: Apply and extend previous understandings of operations with prime and composite numbers to prime factorization and squares and square roots Assessment Name: Show me the slope Assessment Type: Exit Slip and Performance Assessment - Task Assessing Standards: 7.GM.3, 7.AF.4 & 7.AF.7 Assessment Description: Apply and extend understanding of proportions to compute the slope/rate of change of a line through points, graphs, lines, etc. 4
End of Unit Summative Assessments Assessment Name: Grade 7 Summative Assessment Ratio and Proportions Assessment Type: Benchmark Tests or Sample Standardized Assessment Questions Assessing Standards: 7.C.6, 7.AF.4, 7.AF.7, 7.C.5, 7.GM.3, 7.NS.1, 7.NS.2, 7.NS.3 Assessment Description: Summative Assessment of ratios and proportions using real numbers and in real-world situations. 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 unit rate ratio fraction percent markups markdowns simple interest tax gratuity commissions discounts percent increase percent decrease percent error reciprocal unit rate rate tape diagram double number line diagrams equivalent ratios equations proportions Prerequisite Math Terms Operations with whole numbers Fractions Decimals Rational numbers Rounding Estimating Exponents 5
coordinate plane table point constant constant speed part whole convert Customary system constant of proportionality Metric system scale scale factor scale drawings area polygon triangle quadrilateral Unit Resources/Notes Include district and supplemental resources for use in weekly planning Pre-Assessment Grade 7 Acuity or istep pre-assessment Ratio and Proportions Formative Assessment 1 The student will complete a performance assessment that connects to the real-world situation for percent problems and unit rates. http://www.insidemathematics.org/assets/common-core-math-tasks/mixing%20paints.pdf Use any of the assessments in the section for 7.RP.1 & 7.RP.3 Also, use any of the assessments in the section for 6.RP.3 http://www.insidemathematics.org/common-core-resources/mathematical-contentstandards/standards-by-grade/6th-grade Formative Assessment 2 The student will complete a performance assessment that manipulates factors using prime and composite numbers to prime factorization and squares and square roots. Use any of the assessments in the section for 4.OA.4 (http://www.insidemathematics.org/common-coreresources/mathematical-content-standards/standards-by-grade/4th-grade), 8.EE.2, 8.NS.1 & 8.NS.2 http://www.insidemathematics.org/common-core-resources/mathematical-contentstandards/standards-by-grade/8th-grade Formative Assessment 3 - Apply and extend understanding of proportions to compute the slope/rate of change of a line through points, graphs, lines, etc. Use any of the assessments from 7.RP.2 (http://www.insidemathematics.org/common-core-resources/mathematical-contentstandards/standards-by-grade/7th-grade) 7.GM.3 (use the link for any of the assessments for 7.G.3 - http://www.insidemathematics.org/common-core-resources/mathematical-contentstandards/standards-by-grade/7th-grade) 6
Summative Assessment Ratio and Proportion istep Stands Other resources Textbook: Glencoe McGraw-Hill Math Connects Course 2 (2012 ed.), www.connected.mcgrawhill.com, ISTEP Coach, www.myskillstutor.com, ISTEP Reference Sheet, www.mathnook.com, Hands on Equations, www.edhelper.com, Glencoe Mathematics Study Guide & Practice Workbook (Past textbook adopt.), Brain Pop, www.acuityathome.com, Scholastic Math, Daily Math Practice 6+ 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. 7
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. 8