COMMON CORE MATHEMATICS I – Unit 2: Modeling with Linear Relationships
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Unit Overview:Quarter x One Two
Three Four / Instructional Time: approximately 25 days (based on 90-minute classes)
Grade Level: Common Core Mathematics 1
Unit Theme: Modeling with Linear Relationships / Depth of Knowledge: Level 3, Strategic Thinking
Unit Summary: This unit takes students through creating equations with one variable, solving literal equations and formulas, and creating equations with two variables given the context. Students use real world contexst to create linear equations and identify the graphs of linear functions by their shapes and specific characteristics. Students create lines of best fit taking into account residuals. Students examine correlation coefficients and the connection between correlation and causation. Lastly, students solve for one variable in terms of the other. Using this framework, students solve for y, in terms of x, in order to help students see the connection between the terms; as well as identify the slope and y-intercept of the equation (given contextual information).
Critical Area of Concentration: (1) Create equations that describe numbers or relationships, (2) Understand solving equations as a process of reasoning and explain the reasoning, (3) Solve equations and inequalities in one variable, (4) Interpret functions that arise in applications in terms of the context.
North Carolina Information and Technology Essential Standards:
HS.TT.1
Use technology and other resources for assigned tasks.
HS.TT.1.3
Use appropriate technology tools and other resources to design products to share information with others (e.g. multimedia presentations, Web 2.0 tools, graphics, podcasts, and audio files).
HS.SE.1
Analyze issues and practices of responsible behavior when using resources.
Common Core State Standards
Interpreting Functions: Understand the concept of a function and use function notation.
F-IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
F-IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
Functions: Linear, Quadratic, and Exponential Models Construct and compare linear and exponential models and solve problems.
F-LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.
(Focus on the linear relationships in this unit.)
a. Prove linear functions grow by equal differences over equal intervals and exponential functions grow by equal factors over equal intervals.
b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
F-LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
Algebra: Creating Equations Create equations that describe numbers or relationships
A-CED.1 Create equations and inequalities in one variable and use them to solve problems. Instructional focus should include equations arising from linear functions.
A-CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
Algebra: Reasoning with Equations and Inequalities Understand solving equations as a process of reasoning and explain the reasoning.
A-REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
Algebra: Reasoning with Equations and Inequalities Solve equations and inequalities in one variable.
A-REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
Functions: Building Functions Build a function that models a relationship between two quantities
F-BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. Formal recursive notation is not used.
Interpreting Categorical and Quantitative Data - Summarize, represent, and interpret data on two categorical and quantitative variables.
S.ID.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
S.ID.6 Represent data on two quantitative variables on a scatterplot, and describe how the variables are related.
a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear models.
b. Informally assess the fit of a function by plotting and analyzing residuals.
c. Fit a linear function for scatterplots that suggest a linear association.
Interpreting Categorical and Quantitative Data - Interpret linear models
S-ID.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
S-ID.8 Compute (using technology) and interpret the correlation coefficient of a linear fit.
S-ID.9 Distinguish between correlation and causation.
Italicized standards notate Gap Standards that are new for North Carolina.
Bold standards notate Power Standards that are heavily weighted on Standardized Tests.
Italicized and Bold indicates the standard is both a Gap and a Power Standard.
Essential Question(s):
· How would creating a table help you see the pattern of a set of data?
· How do equations help us in finding missing information?
· How do addition and multiplication properties help us solve for unknowns?
· Why is the idea of balance so important in solving equations?
· How do you write inequalities and what do they mean?
· How do you represent solutions for inequalities on a number line?
· How do we identify and describe patterns?
· How do we use patterns to make predictions and solve problems?
· How do you convert a recursive relationship into an explicit function?
· How can technology help us analyze sequences?
· How can you identify equal changes in intervals from a table, a graph, or a linear function?
· What is a function?
· What are the different ways that functions may be represented?
· How can functions be used to model real world situations, make predictions, and solve problems?
· How can technology help us analyze sequences and functions?
· How can an equation, table, and graph be used to analyze the rate of change and other applicable information, related to a real-world problem and the representative function?
· How do we use residuals to help create a line of best fit?
· How do we find and interpret a correlation coefficient for a linear equation?
· How do we distinguish between correlation and causation?
Enduring Understanding(s):
· Linear equations and inequalities represent real-world problems and situations which grow by equal differences over equal intervals.
· There is a clear relationship between the variables in a function and that relationship is evident in tables, equations and graphs.
· Each step in solving an equation corresponds with an algebraic property.
· There are specific characteristics of linear functions and equations.
· Functions are a mathematical way to describe relationships between two quantities that vary.
· Correlation does not necessarily imply causation.
I Can Statement(s):
· I can create one-variable linear equations and inequalities from contextual situations.
· I can solve and interpret the solution to multi-step linear equations and inequalities in context.
· I can justify the steps in solving equations by applying and explaining the properties of equality, inverse and identity.
· I can use the names of the properties and common sense explanations to explain the steps in solving an equation.
· I can justify the fact that linear functions grow by equal difference over equal intervals using tables and graphs.
· I can find the correlation coefficient—and interpret it—for a linear equation.
· I can distinguish between correlation and causation.
· I can find the line of best fit for a set of data which appears to be linear.
Vocabulary:
greater than
no less than / less than
no more than / at most
at least / =, <, >, / equations
inequalities / variables
function
linear functions / coordinate plane / constant / coefficient / properties of operations
properties of equalities / like terms / variable / evaluate / justify / viable
constant rate of change / interval / rate / literal equation / relationship / input
output
sequences
arithmetic sequence / arithmetic / recursive / explicit / slope / correlation coefficient
residual
scale / independent variable
dependent variable / domain
range
Transdisciplinary Connections
WHST.9-10.1 - Write arguments focused on discipline-specific content.
· Introduce precise claim(s), distinguish the claim(s) from alternate or opposing claims, and create an organization that establishes clear relationships among the claim(s), counterclaims, reasons, and evidence.
· Develop claim(s) and counterclaims fairly, supplying data and evidence for each while pointing out the strengths and limitations of both claim(s) and counterclaims in a discipline-appropriate form and in a manner that anticipates the audience’s knowledge level and concerns.
WHST.9-10.6 - Use technology, including the Internet, to produce, publish, and update individual or shared writing products, taking advantage of technology’s capacity to link to other information and to display information flexibly and dynamically.
RST.9-10.4 - Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in a specific scientific or technical context relevant to grades 9–10 texts and topics.
RST.9-10.5 - Analyze the structure of the relationships among concepts in a text, including relationships among key terms (e.g., force, friction, reaction force, energy).
Evidence of Learning (Formative Assessment):
· Daily Math Journal/Reflective Blog Entries
· Q & A
· Graphics Design
· HW responses
· Durham Public Schools Small Goal Assessments / Summative Assessment:
· End of Unit Assessment
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Unit Implementation:
The primary resource for Unit 2 is: Core-Plus (Glencoe; 2008; Course 1). The estimated days are based on 90-minute periods and include review time and a period for assessment in each block of time.
Days 1-10: Unit 1: Patterns of Change: Lessons 3, 4 (Looking Back)
Days 11-25: Unit 3: Linear Functions: Lessons 1, 2, 3, and 4 (Looking Back)
Optional Resources:
Robo-Pull - Although this was originally designed for middle school, this would be a great modified lesson for a class project. Students can design their own robots using a classroom robotics kit, and create the story context for solving equations and inequality based on the design of their robot. The teacher can either assign the problems or have the students create the problem based on the tested information known about the robots (e.g. speed it can travel in a given distance, amount it can haul in one trip, etc.) For example: Groups will determine the maximum load their robot can carry. As an alternative or extension, students can set up different ramps for the robot to travel. A robot may only be able to pull “x grams” up the first ramp, but may pull 5x on a flat surface, or (1/3) x up a steeper ramp. Have students record their observations using Google Docs, and have them present their findings in a Google Presentation. They may also choose to use animation to illustrate the movement of their robot, or record their trials using their iPads, Google Tablets, or digital cameras.
NCTM Interactive Introduction to Linear Functions Lesson - In this interactive lesson, students explore functions, function rules, and data tables. They explore terms of linear functions. Students use computers to analyze the functions. Students write short sentences to explain data tables and simple algebraic expressions.
· Video Lessons for Identifying Functions in a Table - These are instructional videos that explain functions shown in a table, given a “story” scenario.
· An Introduction to Arithmetic and Geometric Sequences - In this Interactive lesson, students explore the concept of sequences. In this sequences lesson, students create arithmetic and geometric sequences using an applet. Students pick the initial value, multiplier, and the add-on. Students discuss the differences and similarities between arithmetic and geometric sequences.
· Practice with Arithmetic and Geometric Sequences Word Problems
Solve the following problems dealing with Arithmetic and Geometric Sequences and Series. Look carefully at each question to determine with "which" sequence you are working. You may wish to have your graphing calculator handy.
· The Exploration of the Slope of a Line - Students graph lines and find their slope. In this graphing lesson, students used Geometer's Sketchpad software to plot several lines and compare the slope of parallel and perpendicular lines. In the second portion of the lesson, students gather data by measuring different rubber band lengths and graphing the results.
Supportive Unit Resources: (Please note that these are resources that can be used to supplement instruction before or during a lesson.)
Scaffolding Option 1:
Intervention / Scaffolding Option 2:
Maintenance / Scaffolding Option 3:
Extension
Instructional Activities: / (CCSS A.CED.4)
Algebra Lab Tutorial
Algebra Lab online tutorial allowing students to practice solving for a variable.
Algebra Lab Tutorial Practice
(CCSS G-GPE.5)
Have students watch this Video Tutorial on graphing parallel and perpendicular lines. / (CCSS F-LE.2)
Exponential Functions - Learn about exponential functions from RegentsPrep.
(CCSS G-GPE.5)
Explore the slope of a line using Calibri Software. This can be modified using GeoGebra. / (CCSS G-GPE.5.)
Finding Slope by Working Backward
Students may use a variety of different methods to construct a parallel or perpendicular line to a given line and calculate the slopes to compare the relationships.
Students may replicate the concept of the “Is Slope Legal” project, and observe the school for illegal gradation of handicapped ramps.
Multimedia Activities: / Arithmetic/Geometric Sequences from iXL
Number sequences: Arithmetic sequences (Algebra - P.2)
Number sequences: Number sequences: mixed review (Algebra - P.7)
Solving One-Variable Equations from Purplemath
Using the Math Widget
Linear Functions Review from iXL
Linear functions
Identify linear functions
Find the slope of a graph
Slope-intercept form: find slope and y-intercept / Arithmetic/Geometric Sequences from NCTM Illuminations
Counting the Trains
More Trains
Recursive and Exponential Rules
The Devil and Daniel Webster
Finding the Slope of a Line By Investigation from Math Warehouse
Interactive Slope
Linear Functions Review from iXL
Slope-intercept form: graph an equation
Slope-intercept form: write an equation from a graph
Slope-intercept form: write an equation
Linear function word problems
Slopes of parallel and perpendicular lines / Arithmetic/Geometric Sequences from NCTM Illuminations
Trains, Fibonacci, and Recursive Patterns
Understanding the Least Squares Regression Line with a Visual Model: Measuring Error in a Linear Model
Linear Functions Review from iXL
Slope-intercept form: graph an equation
Slope-intercept form: write an equation from a graph
Slope-intercept form: write an equation
Linear function word problems
Slopes of parallel and perpendicular lines
Revised: 7.1.2013