Rigorous Curriculum Design



Rigorous Curriculum Design

Unit Planning Organizer

|Subject(s) |Middle Grades Mathematics |

|Grade/Course |8th |

|Unit of Study |Unit 5: Linear Functions |

|Unit Type(s) |❑Topical X Skills-based ❑ Thematic |

|Pacing |18 days |

|Unit Abstract |

|In this unit, students will graph proportional relationships; interpret unit rate as the slope; compare two different proportional |

|relationships represented in different ways and use similar triangles to explain why the slope is the same between any two points on a |

|non-vertical line. Students will use the equation y = mx for a line through the origin; the equation y = mx + b for a line intercepting the |

|vertical axis at b; and interpret equations in y = mx + b form as linear functions. |

|Common Core Essential State Standards |

|Domain: Expressions and Equations (8.EE), Functions (8.F) |

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|Clusters: Understand the connection between proportional relationships, lines, and |

|linear equations. |

|Define, evaluate and compare functions. |

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|Standards: |

|8.EE.5 GRAPH proportional relationships, INTEPRETING the unit rate as the slope of the graph. COMPARE two different proportional relationships|

|represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects|

|has greater speed. |

|8.EE.6 USE similar triangles to EXPLAIN why the slope m is the same between any two distinct points on a non-vertical 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. |

|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 = s2 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. |

|Standards for Mathematical Practice |

|1. Make sense of problems and persevere in solving them. |

|2. Reason abstractly and quantitatively. |

|3. Construct viable arguments and critique the reasoning of others. |

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|4. Model with mathematics. |

|5. Use appropriate tools strategically. |

|6. Attend to precision. |

|7. Look for and make use of structure. |

|8. Look for and express regularity in repeated reasoning. |

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| “UNPACKED STANDARDS” |

|8.EE.5 Students build on their work with unit rates from 6th grade and proportional relationships in 7th grade to compare graphs, tables and |

|equations of proportional relationships. Students identify the unit rate (or slope) in graphs, tables and equations to compare two |

|proportional relationships represented in different ways. |

|Example 1: |

|Compare the scenarios to determine which represents a greater speed. Explain your choice including a written description of each scenario. Be |

|sure to include the unit rates in your explanation. |

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|Scenario 1: Scenario 2: |

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|y = 55x |

|x is time in hours |

|y is distance in miles |

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|Solution: Scenario 1 has the greater speed since the unit rate is 60 miles per hour. The graph shows this rate since 60 is the distance |

|traveled in one hour. Scenario 2 has a unit rate of 55 miles per hour shown as the coefficient in the equation. |

|Given an equation of a proportional relationship, students draw a graph of the relationship. Students recognize that the unit rate is the |

|coefficient of x and that this value is also the slope of the line. |

|8.EE.6 Triangles are similar when there is a constant rate of proportionality between them. Using a graph, students construct triangles |

|between two points on a line and compare the sides to understand that the slope (ratio of rise to run) is the same between any two points on a|

|line. |

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|Example 1: |

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|The triangle between A and B has a vertical height of 2 and a horizontal length of 3. |

|The triangle between B and C has a vertical height of 4 and a horizontal length of 6. |

|The simplified ratio of the vertical height to the horizontal length of both triangles is 2 |

|to 3, which also represents a slope of [pic] for the line, indicating that the triangles are similar. Given an |

|equation in slope-intercept form, students graph the line represented. |

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|Students write equations in the form y = mx for lines going through the origin, recognizing that m represents the slope of the line. |

|Example 2: |

|Write an equation to represent the graph to the right. |

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|Solution: y = [pic] x |

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|Students write equations in the form y = mx + b for lines not passing through the origin, recognizing that m represents the slope and b |

|represents the y-intercept. |

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|Solution: y = [pic]x - 2 |

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|8.F.3 Students understand that linear functions have a constant rate of change between any two points. Students use equations, graphs and |

|tables to categorize functions as linear or non-linear. |

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|Example 1: |

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|Determine if the functions listed below are linear or non-linear. Explain your reasoning. |

|1. y = -2x2 + 3 |

|2. y = 0.25 + 0.5(x–2) |

|3. A = ( r2 |

|4. 5. |

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|Solution: |

|Non-linear |

|Linear |

|Non-linear |

|Non-linear; there is not a constant rate of change |

|Non-linear; the graph curves indicating the rate of change is not constant. |

|“Unpacked” Concepts |“Unwrapped” Skills |COGNITION |

|(students need to know) |(students need to be able to do) |DOK |

|8.EE.5 | | |

|Proportional relationships |I can identify unit rate (slope) in a graph, table or |2 |

| |equation. |2 |

| |I can compare proportional relationships represented in two | |

| |different ways. | |

|8.EE.6 | | |

|Points in a linear function |I can demonstrate that points that lie on the same line have|2 |

| |the same slope | |

| |I can develop the equation of a line that passes through the|2 |

| |origin. ( y=mx ) | |

| |I can develop the equation of a line not passing through the|2 |

| |origin. ( y=mx + b) | |

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|8.F.3 | | |

|Linear and nonlinear functions |I can determine if a table, graph or function represents a |2 |

| |linear or non-linear function and explain my reasoning | |

|Essential Questions |Corresponding Big Ideas |

|8.EE.5 | |

|How can I determine the unit rate, rate of change, slope or constant |Students will work with data sets, tables, graphs, and functions to |

|of proportionality from a data set, table, graph or function? |determine the unit rate, rate of change, slope, or constant of |

| |proportionality. |

|How can I compare proportional relationships represented in two |Students will compare proportional relationships represented in two |

|different ways? |different ways. |

|8.EE.6 | |

|How can I demonstrate that points on the same line have the same |Students will demonstrate that points on the same line have the same |

|slope? |slope. |

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|How can I write the equation of line that passes through the origin on|Students will write the equation of line that passes through the origin|

|the coordinate plane? |on the coordinate plane. |

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|How can I write the equation of line that does not go through the |Students will write the equation of line that does not go through the |

|origin on the coordinate plane? |origin on the coordinate plane. |

|8.F.3 | |

|How do I know when a data set, data table, function or graph is linear|Students will be able to recognize that a constant rate of change |

|or nonlinear? |represents a linear function, the equation can be written as y = mx + b|

| |and the graph is a straight line. |

|Vocabulary |

|unit rate, proportional relationships, slope, rate of change, vertical, horizontal, similar triangles, y-intercept, linear, non-linear |

|Language Objectives |

|Key Vocabulary |

|8.EE.5 – 8.EE.6 |Students will be able to define, give an example of and use the key vocabulary when working |

|8.F.3 |with linear functions: unit rate, proportional relationships, slope, rate of change, vertical, |

| |horizontal, similar triangles, y-intercept, linear, non-linear |

|Language Function |

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|8.EE.5 |SWBAT recognize that unit rate, rate of change, slope, and constant of proportionality are all |

| |equivalent. Students will also be able to calculate each measure for a set of data. |

|Language Skill |

|8.EE.6 |SWBAT demonstrate through graphic models that similar triangles used to represent vertical |

| |change compared to horizontal change between two points will produce points on the same line. |

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| |SWBAT construct oral or written arguments to show that a data set is linear if there is a |

|8.F.3 |constant of proportionality between points. |

|Language Structures |

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| |SWBAT demonstrate through graphic models, to their partner or whole class that similar |

|8.EE.6 |triangles used to represent vertical change compared to horizontal change between two points |

| |will produce points on the same line. |

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|Language Tasks |

|8.EE.5 |SWBAT explain to a partner or whole class that unit rate, rate of change, slope, and constant |

| |of proportionality are the same in terms of their meaning, by examining data sets, data table, |

| |functions and graphs and demonstrating that the calculations produce the same number and |

| |meaning in the context of the given situation. |

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| |SWBAT construct oral or written arguments to show that a data set is linear if there is a |

| |constant of proportionality between points. |

|8.F.3 | |

|Language Learning Strategies |

|8.EE.6 |SWBAT write an equation of a line that passes through the origin on the coordinate plane. |

| |Students will justify their equation using the correct vocabulary. |

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| |SWBAT determine if a data set, data table, function or graph is linear or nonlinear by looking |

|8.F.3 |for proportionality to determine a constant rate of change and justifying their answers using |

| |correct vocabulary. |

|Information and Technology Standards |

|8.TT.1.1 Use appropriate technology tools and other resources to access information (search engines, electronic databases, digital magazine |

|articles). |

|8.TT.1.2 Use appropriate technology tools and other resources to organize information (e.g. graphic organizers, databases, spreadsheets, and |

|desktop publishing). |

|8.RP.1.1 Implement a project-based activity collaboratively. |

|8.RP.1.2 Implement a project-based activity independently. |

|Instructional Resources and Materials |

|Physical |Technology-Based |

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|Connected Math 2 Series |WSFCS Math Wiki |

|Common Core Investigation 2 | |

|Thinking With Mathematical Models, Inv. 2, ACE |NCDPI.Wikispaces Eighth Grade |

|Say It With Symbols, Inv.4 | |

| |Illuminations.NCTM Walk the Plank |

|Partners in Math Materials | |

|Sticks and No Stones I & II |Georgia Unit |

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|Lessons for Learning (DPI) |Education.calculators/Activities=5875 |

|Perplexing Puzzle | |

|Non-Linear Functions |LinearFunctMachine/ |

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|Mathematics Assessment Project (MARS) |cgraph/cslope/ |

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| |ENLVM.usu.edu/eqns_lines |

|Book | |

|A Visual Approach to Functions by Frances Van Dyke |2.mathpartners |

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| |linear |

| |commcore/G.8 |

| |Math.fullerton.edu/Linear_Equations |

| |alg/ |

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| |Lessonplan/Grade=8 |

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| |KATM.Flip Book8l |

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| |Pre-AlgebraLessons |

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