ParamountUnifiedSchoolDistrict EducationalServices Mathematics7–Unit2 StageOne–DesiredResults Unit 2: Ratios and Proportions In this unit, students extend their understanding of ratios, rates, and unit rates to formally define proportional relationships and the constant of proportionality. They distinguish proportional relationships from other relationships and explore the multiple representations of proportional relationships, including tables, graphs, and equations. Students extend their reasoning about ratios and proportional relationships to compute unit rates for ratios specified by rational numbers. Students learn that the unit rate of a collection of equivalent ratios is called the constant of proportionality and can be used to represent proportional relationships with equations of the form y = kx, where k is the constant of proportionality. They graph proportional relationships and understand what the points (0, 0), (1, r) and other points (x, y) on the line mean in the context of a real‐world situation. Students apply their understandings of proportional relationships to the context of scale drawings. The identify scale factor as the constant of proportionality, calculate actual lengths and areas of objects in the drawing, and create their own scale drawings of a two‐dimensional view. Students solve problems about scale drawings and extend their understanding of proportional relationships to similar figures. Common Misconceptions: Students may have difficulty with ratio and proportion vocabulary. Students may not understand the difference between additive reasoning versus multiplicative reasoning. Students may not recognize different strategies for solving proportions are equivalent. Students may have difficulty calculating unit rate, recognizing unit rate when it is graphed on a coordinate plane, and realizing that unit rate is also the slope of a line passing through the origin. Students may fail to realize that conversions of scale drawing area to actual length area are not achieved by simply multiplying the area of the drawing by the scale. M7, U2 pg. 1 Unit 2 Overview: Ratios and Proportions Transfer Goals 1) Demonstrate perseverance by making sense of a never‐before‐seen problem, developing a plan, and evaluating a strategy and solution. 2) Effectively communicate orally, in writing, and using models (e.g., concrete, representational, abstract) for a given purpose and audience. 3) Construct viable arguments and critique the reasoning of others using precise mathematical language. Standards Meaning‐Making Analyze proportional relationships and use them to solve real‐world and mathematical problems. 7.RP.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction ½/¼ miles per hour, equivalently 2 miles per hour. 7.RP.2 Recognize and represent proportional relationships between quantities. a. 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. b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. c. Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn. d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. Draw, construct, and describe geometrical figures and describe the relationships between them. 7.G.1 Solve problems using scale drawings of geometric Understandings Essential Questions Students will understand that… Students will keep considering… What is a proportion? Why are multiplicative relationships proportional? What is the difference between a unit rate and a ratio? How are equivalent ratios, values in a table, and ordered pairs connected? What characteristics define the graphs of all proportional relationships? How can you apply ratios and proportional reasoning to real‐world situations? How can scale factor be applied to scale drawings? Reasoning with ratios involves attending to and coordinating two quantities. A ratio is a multiplicative comparison of two quantities, or it is a joining of two quantities in a composed unit. Ratios can be meaningfully reinterpreted as quotients. A proportion is a relationship of equality between two ratios. In a proportion, the ratio of two quantities remains constant as the corresponding values of the quantities change. Superficial cues presented in the context of a problem do not provide sufficient evidence of proportional relationships between quantities. Acquisition Knowledge Students will know… Skills Students will … Proportional relationships are relationships between two equivalent ratios. To determine if two quantities are in a proportional relationship, test for equivalent ratios in a table. When graphed as coordinates, pairs of quantities that sit on the same straight line passing through the origin are proportional to one another. A proportion is an equation that states that two 5 10 . equations are equal, such as = 2 4 Determine whether two quantities, shown in various forms, are in a proportional relationship. Compute unit rates and find the constant of proportionality of proportional relationships in various forms. Represent proportional relationships between quantities using equations. Interpret specific values from a proportional relationship in the context of a problem situation. Create scale drawings. Solve problems involving scale drawings using proportional M7, U2 pg. 2 figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different rate. Unit rate is a useful means for comparing ratios and their rates when measured in different units. The constant of proportionality, or unit rate, is the constant ratio found in a proportional relationship. The points (0, 0) and (1, r), where r is the unit rate, will always appear on the line representing two quantities that are proportional to each other. A scale drawing is a proportional image of an actual figure. The constant of proportionality used in a scale drawing, called the scale factor, can be calculated from the ratio of any length in the scale image to its corresponding length in the actual picture. Lengths and areas are affected in different ways when scaled. Key Vocabulary Terms: proportional relationship, ratio, unit rate, constant of proportionality, origin, scale drawing, scale, scale factor reasoning. M7, U2 pg. 3 ParamountUnifiedSchoolDistrict EducationalServices Mathematics7–Unit2 StageTwo–EvidenceofLearning Unit 2: Ratios and Proportions Transfer is a student’s ability to independently apply understanding in a novel or unfamiliar situation. In mathematics, this requires that students use reasoning and strategy, not merely plug in numbers in a familiar‐looking exercise, via a memorized algorithm. Transfer goals highlight the effective uses of understanding, knowledge, and skills we seek in the long run – that is, what we want students to be able to do when they confront new challenges, both in and outside school, beyond the current lessons and unit. These goals were developed so all students can apply their learning to mathematical or real‐world problems while simultaneously engaging in the Standards for Mathematical Practices. In the mathematics classroom, assessment opportunities should reflect student progress towards meeting the transfer goals. With this in mind, the revised PUSD transfer goals are: 1) Demonstrate perseverance by making sense of a never‐before‐seen problem, developing a plan, and evaluating a strategy and solution. 2) Effectively communicate orally, in writing, and using models (e.g., concrete, representational, abstract) for a given purpose and audience. 3) Construct viable arguments and critique the reasoning of others using precise mathematical language. Multiple measures will be used to evaluate student acquisition, meaning‐making, and transfer. Formative and summative assessments play an important role in determining the extent to which students achieve the desired results in stage one. Formative Assessment Summative Assessment Aligning Assessment to Stage One What constitutes evidence of understanding for this lesson? What evidence must be collected and assessed, given the desired results defined in stage one? Through what other evidence during the lesson (e.g. response to questions, observations, journals, etc.) will students demonstrate achievement of the What is evidence of understanding (as opposed to recall)? desired results? Through what task(s) will students demonstrate the desired understandings? How will students reflect upon, self‐assess, and set goals for their future learning? Opportunities Inquiry lesson discussions Unit assessments Checking for understanding (response boards) Teacher‐created chapter tests or mid‐unit assessments Ticket out the door, Cornell note summary, and error analysis Illustrative Mathematics tasks (https://www.illustrativemathematics.org/) Illustrative Mathematics tasks (https://www.illustrativemathematics.org/) Performance tasks Quizzes M7, U2 pg. 4 The following pages address how a given skill may be assessed. Assessment guidelines, examples and possible question types have been provided to assist teachers in developing formative and summative assessments that reflect the rigor of the standards. These exact examples cannot be used for instruction or assessment, but can be modified by teachers . Skill Standard Assessment Guidelines Determine whether two quantities, shown in various forms, are in a proportional relationship. 7.RP.2a Students identify tables of values that represent proportional relationships. Tables should be labeled and have four to five sets of data. All tables within an item should follow the same format. Difficulty can be adjusted: Table values are whole numbers or fractions. Fractions may be mixed numbers. Distractors should be tables that do not show a proportional relationship, which may include a relationship following an equation in the form of y = mx + b (where b ≠ 0) or y = x2. Example Select all tables that represent a proportional relationship between x and y. Possible Response Type(s) Multiple Choice, Multiple Correct Response Matching Tables Use of Calculator Yes (A and C) Select all tables that represent a proportional relationship between x and y. (A and D) M7, U2 pg. 5 Skill Standard Assessment Guidelines (cont.) (cont.) Students identify which graphs represent proportional relationships. Difficulty can be adjusted: Unit rate is a whole number or fraction. Distractors should be graphs that do not show a proportional relationship, which may show a nonlinear relationship or a relationship following an equation in the form of y = mx + b (where b ≠ 0) or y = x2. Example Select all the graphs that show a proportional relationship. Compute unit rates and find the constant of proportionality of proportional relationships in various forms. 7.RP.1 7.RP.2b Students give the constant of proportionality (unit rate) of two proportional quantities. The described ratio should involve ratios of fractions. Difficulty can be adjusted: Mixed number fractions increase the difficulty compared to fractions which are not mixed numbers. Unit rates can be whole numbers or fractions. 1 1 4 2 to make a fruit drink. Find the number of cups of apple juice David uses for 1 cup of carrot juice. 1 Equation/Numeric Multiple Choice, Single Correct Response Short Text Yes David uses cup of apple juice for every cup of carrot juice ( Use of Calculator (cont.) (B and C) Possible Response Type(s) Multiple Choice, Multiple Correct Response Multiple Choice, Single Correct Response Equation/Numeric ) 2 M7, U2 pg. 6 Skill Standard (cont.) (cont.) Assessment Guidelines Students give the constant of proportionality (unit rate) of two proportional quantities. Tables should be labeled and contain two to four sets of data. Difficulty can be adjusted: Values can be whole numbers and/or fractions with or without mixed numbers. Unit rate can be a whole number or fraction. Example This table shows a proportional relationship between the number of cups of sugar and flour used for a recipe. Find the number of cups of sugar used for 1 cup of flour. Possible Response Type(s) Equation/Numeric Multiple Choice, Single Correct Response Short Text Use of Calculator Yes ( 2 ) 5 This table shows a proportional relationship between the number of cups of sugar and flour used for a recipe. Find the number of cups of sugar used for 1 cup of flour. ( 1 ) 3 Students give the constant of proportionality (unit rate) of two proportional quantities. Tables should be labeled and contain two to four sets of data. Difficulty can be adjusted: Values can be whole numbers and/or fractions with or without mixed numbers. Unit rate can be a whole number or fraction. This diagram shows how much apple juice is mixed with carrot juice for a recipe. Apple Juice: Carrot Juice: Find the number of cups of apple juice used for 1 cup of carrot juice. (2) M7, U2 pg. 7 Skill Standard (cont.) (cont.) Assessment Guidelines Example Students give the constant of proportionality (unit rate) of two proportional quantities. Tables should be labeled and contain two to four sets of data. Difficulty can be adjusted: Values can be whole numbers and/or fractions with or without mixed numbers. Unit rate can be a whole number or fraction. This diagram shows how much apple juice is mixed with carrot juice for a recipe. Apple Juice: Carrot Juice: Find the number of cups of apple juice used for 1 cup of carrot juice. Students give the constant of proportionality (unit rate) of two proportional quantities. The equation should come in the following forms: For a drink recipe, the amount of papaya juice is proportional to the amount of carrot juice. This equation represents the proportional relationship between the number of quarts of papaya juice (p) and carrot juice (c) in a recipe. y = rx, where r is the unit rate [coefficient][variable]=[coefficient][variable] Difficulty can be adjusted: Unit rates include whole numbers, positive fractions, and mixed numbers. Coefficients include whole numbers, fractions, and exclude the number one. Possible Response Type(s) Equation/Numeric Multiple Choice, Single Correct Response Short Text Use of Calculator Yes ( 5 ) 3 2p = 8c Find the number of quarts of papaya juice used for 1 quart of carrot juice. (1/4) For a drink recipe, the amount of papaya juice is proportional to the amount of carrot juice. This equation represents the proportional relationship between the number of quarts of papaya juice (p) and carrot juice (c) in a recipe. 1 1 3 3 (1 )p = (3 )c Find the number of quarts of papaya juice used for 1 quart of carrot juice. ( 5 ) 2 M7, U2 pg. 8 Skill Represent proportional relationships between quantities using equations. Standard Assessment Guidelines 7.RP.2c Students give an equation that represents the proportional relationship between two given quantities. Graph is linear and begins at (0, 0) or a set of plotted points which includes (0, 0). Graphs axes are labeled and include whole numbers and/or fractions. Tables should be labeled, represent the relationship between two variables, and have 3‐5 sets of data. The unit rate, r, is a whole number or fraction. Difficulty can be adjusted: The graph of the line that intersects at point (1, r) where r is the unit rate in rational number form. Scaling of the graph may be fractional or in units other than multiples of 2 or 10. Table values are whole numbers or fractions. Fractions are not mixed numbers. Example This graph shows a proportional relationship between the number of hours (h) a business operates and the total cost of electricity (c). Possible Use of Response Type(s) Calculator Yes Equation/Numeric Multiple Choice, Single Correct Response Find the constant of proportionality (r). Using the value for r, enter an equation in the form of c = rh that represents the relationship between the number of hours (h) and the total cost (c). (c = 10h) This graph shows a proportional relationship between x and y. Find the constant of proportionality (r). Using the value for r, enter an equation in the form of y = rx. (y = 2x) This table shows a proportional relationship between x and y. Find the constant of proportionality (r). Using the value for r, enter an equation in the form of y = rx. (y = 12x) M7, U2 pg. 9 Skill Standard Assessment Guidelines Interpret specific values from a proportional relationship in the context of a problem situation. 7.RP.2d Students select specific values from a proportional relationship in the context of a problem situation. Graph is linear and begins at (0, 0) or a set of plotted points which includes (0, 0). Graphs axes are labeled and include whole numbers and/or fractions. The unit rate, r, is a whole number or fraction. Difficulty can be adjusted: One answer choice which assesses the interpretation of a single point on the graph that is not the unit rate is easier than an answer choice that compares the interpretation of two different points. Example This graph shows a proportional relationship between the number of hours (h) a business operates and the total cost (c) of electricity. Select True or False for each statement about the graph. (T, F, T) Create scale drawings. 7.G.1 Student creates a scale drawing of a polygon on a grid. Difficulty can be adjusted: Figures may consist of polygons such as quadrilaterals, trapezoids, or parallelograms. Lengths and angles may be positive integers or rational numbers. Scale factor may be a positive rational number. Inclusion of extraneous information. Possible Use of Response Type(s) Calculator Yes Matching Tables Multiple Choice, Multiple Correct Response Multiple Choice, Single Correct Response Equation/Numeric Short Text A scale factor of 2 is applied to this figure. Draw the resulting figure. (example) Graphing Multiple Choice, Single Correct Response Yes M7, U2 pg. 10 Skill Standard Assessment Guidelines Solve problems involving scale drawings using proportional reasoning. 7.G.1 Student gives the area of an actual figure based on a scale drawing and scale factor. Difficulty can be adjusted: Types of polygons (square, rectangle, parallelogram, or right triangle) Linear dimensions can be a combination of rational numbers. Area can be a combination of rational numbers. Student gives the length of one or more sides of a polygon or the scale factor being applied based on a scale drawing. Difficulty can be adjusted: Scale factor and side lengths may be positive rational numbers. Types of polygons (square, rectangle, parallelogram, or right triangle) Lengths of corresponding sides of similar polygons are not all labeled. Inclusion of extraneous information. Example The scale drawing of the right triangle shown was drawn using a scale factor of 1 . 20 Each square on the grid is 3 units in length. What is the area of the actual figure, in square units, on which this scale drawing is based? (2700) Possible Use of Response Type(s) Calculator Yes Equation/Numeric Multiple Choice, Single Correct Response Short Text Figure A is a scale image of Figure B, as shown. The scale that maps Figure A onto 1 Figure B is 1:3 . Find the value 2 of x. (17.5) Figure B is a scale image of Figure A, as shown. Find the scale factor applied to Figure A to produce Figure B. (3) M7, U2 pg. 11 Skill Standard (cont.) (cont.) Assessment Guidelines Student gives the length of one side of an actual figure based on a scale drawing. Difficulty can be adjusted: Scale factor may be given in a key. Numbers can be rational. Combinations of area and length are provided. Extra information is provided. Unit conversion is used. Scale factor may be a positive rational number. Example The front side of a playhouse is shown in this scale drawing. The height of the door in the drawing is 1.8 inches. The scale that maps the drawing to the actual playhouse is 1 inch to 2.5 feet. Possible Use of Response Type(s) Calculator (cont.) Equation/Numeric Multiple Choice, Single Correct Response Short Text Using the scale given, enter the actual height of the playhouse door, in feet. (4.5) Student gives the area of a polygon based on a scale drawing. Difficulty can be adjusted: Dimensions can be a combination of positive and rational numbers. Scale factor may be a positive rational number. This scale drawing of a rectangular rug has dimensions 8 inches by 5 inches. The length of the longer side of the actual rug is 32 feet. Find the area, in square feet, of the actual rug. (640) M7, U2 pg. 12 ParamountUnifiedSchoolDistrict EducationalServices Mathematics7–Unit2 StageThree–LearningExperiences&Instruction Unit 2: Ratios and Proportions Prior to planning for instruction, it is important for teachers to understand the progression of learning and how the current unit of instruction connects to previous and future courses. Teachers should consider: What prior learning do the standards and skills build upon? How does this unit connect to essential understandings of later content? How can assessing prior knowledge help in planning effective instruction? What is the role of activating prior knowledge in inquiry? Looking Back Looking Ahead In Mathematics 6, students: In Mathematics 8, students will: Understood the concept of a ratio and used ratio language to describe a relationship between two quantities. Graph proportional relationships, interpreting the unit rate, or constant of proportionality, as the slope of the graph. Compare two different proportional relationships represented in different ways. Understood the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and used rate language in the context of a ratio relationship. Explained how equivalent ratios were represented in tables, tape diagrams, double number line diagrams, and equations. Describe the effect translations, reflections, dilatations, and rotations have on two‐ dimensional figures. Made tables of equivalent ratios relating quantities with whole‐number measurements (scaling up and scaling back), found missing values in the tables, and plotted the pairs of values on the coordinate plane. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. Solved unit rate problems including those involving unit pricing and constant speed. Graphed points on the coordinate plane. Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real‐world and mathematical problems. M7, U2 pg. 13 Mathematics7 Timeframe: Oct. 10 – Nov. 18 Unit 2: Ratios and Proportions Course Textbook: McGraw Hill, California Math Transfer Goals: 1) Demonstrate perseverance by making sense of a never‐before‐seen problem, developing a plan, and evaluating a strategy and solution. 2) Effectively communicate orally, in writing, and using models (e.g., concrete, representational, abstract) for a given purpose and audience. 3) Construct viable arguments and critique the reasoning of others using precise mathematical language. Understandings: Reasoning with ratios involves attending to and coordinating two quantities. A ratio is a multiplicative comparison of two quantities, or it is a joining of two quantities in a composed unit. Ratios can be meaningfully reinterpreted as quotients. A proportion is a relationship of equality between two ratios. In a proportion, the ratio of two quantities remains constant as the corresponding values of the quantities change. Superficial cues presented in the context of a problem do not provide sufficient evidence of proportional relationships between quantities. Essential Questions: What is a proportion? Why are multiplicative relationships proportional? What is the difference between a unit rate and a ratio? How are equivalent ratios, values in a table, and ordered pairs connected? What characteristics define the graphs of all proportional relationships? How can you apply ratios and proportional reasoning to real‐world situations? How can scale factor be applied to scale drawings? Time & Skills Learning Goals Possible Lessons & Knowledge Focus Questions Teacher Notes Content Activities 4 days Activate Prior Knowledge – Possible activity: In pairs, students create a graphic organizer to record what they know about ratios. During a whole‐class discussion, Proport. students add or remove statements. Relation. Review ratio Review ratio relationships, Activity – Review Math 6 Proportional relationships are How are ratios represented relationships and ratio tables, double in different models? Understandings relationships between two representational number line diagrams, equivalent ratios. models. How can you tell ratios are tape diagrams, and graphs. equivalent? proportional relationship M7, U2 pg. 14 Time & Content 4 days (cont.) Skills Determine whether two quantities, shown in various forms, are in a proportional relationship. Learning Goals Use a variety of strategies to determine whether two quantities are proportional. in a table of equivalent ratios (calculator allowed) Possible Lessons & Activities Inquiry Task A: Proportional Relationships – Tables (revised) Knowledge Focus Questions Teacher Notes To determine if two quantities are in a proportional relationship, test for equivalent ratios in a table. Two quantities are proportional if one constant number is multiplied by each measure in the first quantity to give the corresponding measure in the second quantity. How can we tell if two quantities are proportional to each other? Possible teacher‐ developed practice needed. nonproportional relationship Use a variety of strategies to determine whether two quantities are proportional. examine a graph to determine whether two quantities are proportional Describe how to use a graph to determine if two or more quantities are proportional. Inquiry Task B: Proportional Relationships – Graphs (revised from Proportional and Nonproportional Relationships) When graphed as coordinates, pairs of quantities that sit on the same straight line passing through the origin are proportional to one another. How can we tell if two quantities are proportional to each other? origin M7, U2 pg. 15 Time & Content 4 days (cont.) Skills (cont.) Learning Goals Use a variety of strategies to determine whether two quantities are proportional. use algebraic strategies to determine whether two quantities are proportional. Possible Lessons & Activities Possible task: Students use previously explored equivalent ratios (e.g. 5 2 = 10 4 ) to discover a new way to prove the ratios are equivalent. 20 20 5 2 = 10 4 Knowledge Focus Questions A proportion is an equation How can we tell if two that states that two equations quantities are proportional to each other? 5 10 . are equal, such as = 2 4 proportion (This same strategy may have been used by students to determine if two fractions are equivalent.) Teacher Notes This learning goal provides conceptual understanding for why cross products can be used to solve a proportion. Teacher‐ developed practice needed or selected problems from pp. 58 – 62. Independent practice with transfer goals Generate a real‐world scenario that could be represented proportionally (open‐ended) Explain why a context does or does not represent a proportional relationship Represent real‐world situations with ratios and determine if those ratios are proportional Illustrative Mathematics tasks (https://www.illustrativemathematics.org/7.RP) Depend. Variables 2 days Compute unit rates and find the constant of proportionality of proportional relationships in various forms. (calculator allowed) Recognize the dependent variable when determining unit rate. Inquiry Task C: Dependent Variables & Unit Rate When determining unit rate, the unknown quantity is the dependent variable and is the dividend when calculating. dependent variable independent variable rate unit rate What is the difference between a dependent and independent variable? What role do these variables play when determining unit rate? Teacher‐ developed practice needed M7, U2 pg. 16 Time & Content 2 days (cont.) Skills (cont.) Unit Rate & Constant of Prop. Table & Tape Diagram 2 days Compute unit rates and find the constant of proportionality of proportional relationships in various forms. (calculator allowed) Learning Goals Recognize how dependent and independent variables are represented on a graph. Determine the constant of proportionality given: a table of equivalent ratios a tape diagram Possible Lessons & Activities Inquiry Task D: Dependent Variables & Graphs Extend the Candy Barrels task (inquiry task B) for students to look at the Cooper’s Candies table in a different way (division/multiplication can be used to find the unit rate, which is constant of proportionality) OR Engage NY Lesson 7 (Examples 2 and 3) ** Tape diagram problem needs to be created. Knowledge Focus Questions Teacher Notes When graphing a ratio on a coordinate plane, the dependent variable must be identified and graphed as the x‐axis. The independent variable is graphed on the y‐ axis. How are dependent and independent variables represented on a graph? Teacher‐ developed practice needed Unit rate is a useful means for comparing ratios and their rates when measured in different units. Division and/or multiplication can be used to determine unit rate. The constant of proportionality, or unit rate, is the constant ratio found in a proportional relationship. How is unit rate helpful when making comparisons about quantities? From Math 6: Students know unit rate is also called the constant because it is a fixed number. Possible teacher‐ developed practice needed. constant of proportionality 2 days Opportunities for Review: * Make number line models * Review procedural skills * Fluency practice * Open‐ended questions * Illustrative Mathematics tasks (https://www.illustrativemathematics.org/7.RP) * Independent practice with transfer goals Review & Assess Formative Assessment M7, U2 pg. 17 Time & Content Unit Rate & Constant of Prop. Equation & Graph 5 days Skills Represent proportional relationships between quantities using equations. Learning Goals Determine the constant of proportionality given: the equation of a proportional relationship Use unit rate of (calculator allowed) proportional relationship to generate an equation. Write an equation to represent a proportional relationship. Possible Lessons & Activities Engage NY Lesson 8 Example 1 Engage NY Lesson 9 Examples 1‐2 Knowledge Focus Questions How can unit rate be used to write an equation x relating two variables that constant of proportionality, k. are proportional? y What type of relationship If k , then y kx . can be modeled using an x equation in the form y kx , and what do you The constant of proportionality can be need to know to write an multiplied by the independent equation in this form? variable to find the dependent variable. The dependent variable can be divided by the constant of proportionality to find the independent variables. To find the constant of proportionality, divide to find y the unit rate: k . x To write an equation for a proportional relationship, use y kx and substitute the value of the constant of proportionality in place of k. The unit rate of y is the Teacher Notes M7, U2 pg. 18 Time & Content 5 days (cont.) Skills Compute unit rates and find the constant of proportionality of proportional relationships in various forms. (calculator allowed) Interpret specific values from a proportional relationship in the context of a problem situation. (calculator allowed) Learning Goals Discover unit rate on a graph is the constant of proportionality. Determine the constant of proportionality given: a graph Identify key points on a graph of a proportional relationship including (0, 0) and (1, r). Explain what the points (0, 0), (1, r) and other points (x, y) on the line mean in the context of a real‐world situation. Generalize that the graph of any proportional relationship may be modeled by a line passing through the origin. Possible Lessons & Activities Engage NY Lesson 8 Example 2 (connects graphs to equations) Engage NY Lesson 10 Examples 1‐2 Knowledge Focus Questions The points (0, 0) and (1, r), where r is the unit rate, will always appear on the line representing two quantities that are proportional to each other. * The unit rate, r, in the point (1, r) represents the amount of vertical increase for every horizontal increase of 1 unit on the graph. * The point (0, 0) indicates that when there is zero amount of one quantity, there will also be zero amount of the second quantity. These two points may not always be given as part of the set of data, but they will always appear on the line that passes through the given data points. Where r is the unit rate, how are points (0, 0) and (1, r) related? Why can there be infinitely many points (x, y) on the graph of a proportional relationship? How can you use the unit y rate of to create a table, x equation, or graph of a relationship of two quantities that are proportional to each other? Teacher Notes Independent practice with transfer goals Examine a never‐before‐seen situation to identify the constant of proportionality Explain relationships between the table/equation/graph of a proportional relationship Open‐ended questions Illustrative Mathematics tasks (https://www.illustrativemathematics.org/7.RP) 2 days Opportunities for Review: * Return to essential questions Make number line models * Review procedural skills * Fluency practice * Open‐ended questions * Illustrative Mathematics tasks (https://www.illustrativemathematics.org/7.RP) * Independent practice with transfer goals Review & Assess Mid‐Unit Summative Assessment (teacher‐created) M7, U2 pg. 19 Time & Content Ratios of Fractions 1 day Scale Drawing 6 days Skills Compute unit rates and find the constant of proportionality of proportional relationships in various forms. Learning Goals Knowledge Focus Questions How is unit rate calculated when both values in the ratio are fractions? Additional real‐ world problems are needed. (calculator allowed) A fraction whose numerator or denominator is itself a fraction is called a complex fraction. Simplify complex fractions to compute unit rate. complex fractions How is constant of proportionality represented in scale drawings? How are ratios used in scale drawings? Teacher‐ developed practice needed. Calculate unit rates involving ratios of fractions. Identify corresponding parts of scale drawings and similar figures. Apply proportional (calculator allowed) reasoning to identify the constant of proportionality, or scale factor, in scale drawings. Create scale Use a different scale to drawings. produce a similar figure. (calculator allowed) Solve problems involving scale drawings using proportional reasoning. Possible Lessons & Activities 1‐2 Complex Fractions and Unit Rates TE pp. 17‐24 Inqiry Task E: Relating Ratios to Scale Drawings (revised) Inquiry Task F: Creating Scale Drawings A scale drawing is a proportional image of an actual figure. Each length of the figure in the image is reduced or enlarged proportionally to the original image. The constant of proportionality used in a scale drawing is called the scale factor. Scale factor can be calculated from the ratio of any length in the scale image to its corresponding length in the actual picture. Teacher Notes How is the scale used to create a scale image? By looking at a scale, how can you if the scale image is going to be a reduction or an enlargement? scale drawing scale image scale factor Solve problems involving scale drawings using proportional reasoning. Determine an unknown length of a figure or image using the scale provided for the scale drawing. (calculator allowed) Use a scale on a map, blueprint, or other scale drawing to find an actual measure or distance. Engage NY Lesson 18 Lengths and areas are Examples 2‐3 affected in different ways when scaled. * Students should continue to use the methods from Inquiry Task E. In what ways are lengths affected when scaled? Use a variety of scale factor representations (1 to 2.5, 1:2.5, 1 ). 2.5 M7, U2 pg. 20 Time & Content 6 days (cont.) Skills (cont.) Learning Goals Determine an unknown area of a figure or image using the scale provided for the scale drawing. Possible Lessons & Activities Engage NY Lesson 19 Examples 1‐3 Knowledge Lengths and areas are affected in different ways when scaled. Focus Questions Teacher Notes In what ways are area affected when scaled? Opportunities for Review: * Make number line models * Review procedural skills * Fluency practice * Open‐ended questions * Illustrative Mathematics tasks (https://www.illustrativemathematics.org/7.RP) * Independent practice with transfer goals Formative Assessment Independent practice with transfer goals Reproduce a scale drawing with a different scale Open‐ended questions Illustrative Mathematics tasks (https://www.illustrativemathematics.org/7.RP) 5 days Opportunities for Review: * Return to essential questions * Make number line models * Review procedural skills * Fluency practice Review * Open‐ended questions * Illustrative Mathematics tasks (https://www.illustrativemathematics.org/) * Independent practice with transfer goals & Assess Nov. End‐of‐Unit Summative Assessment 14‐18 M7, U2 pg. 21

© Copyright 2021 Paperzz