Linear Equations, Inequalities and Systems
Lesson 3
Objective
Determine if a function is linear based on the rate of change of points in the function presented graphically and in a table of values.
Common Core Standards
Core Standards
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F.IF.B.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. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
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F.IF.C.7.A — Graph linear and quadratic functions and show intercepts, maxima, and minima.
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F.IF.C.9 — Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.
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F.LE.A.1.A — Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
Foundational Standards
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8.F.A.2
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8.F.A.3
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8.F.B.4
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8.F.B.5
Criteria for Success
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- Given a table of values, identify the rate of change between two points.
- Describe the difference between subsequent $$y$$-values, given equal intervals on the $$x$$-values.
- Write an equation from a table of values.
- Compare a table of values with a graph of a linear equation, and describe which has the larger rate of change.
- Identify graphs that appear linear but based on the non-constant rate of change are not linear.
Anchor Problems
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Problem 1
The following table of values shows some solutions to a linear equation.
Write the linear equation that contains these solutions.
Guiding Questions
Problem 2
Jonas wants to borrow money from his parents to buy a phone and has two options for a payment schedule: one presented in a table and one presented as an equation.
Payment Schedule A:
Payment Schedule B: $${4x + 25 = y}$$ where $$y$$ is cost of the phone, and $$x$$ is the number of weeks required to pay Jonas’s parents back.
Which one has the greater rate of change? What does this mean in context?
Guiding Questions
Problem 3
Describe why this graph is not linear.
Guiding Questions
Problem Set
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The following resources include problems and activities aligned to the objective of the lesson that can be used to create your own problem set.
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Illustrative Mathematics Equal Differences over Equal Intervals 1
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Illustrative Mathematics Equal Differences over Equal Intervals 2
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Open Middle Writing Linear Equations
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Open Middle Line Builders
Target Task
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Write two linear equations: one that is represented in a table of values and has a negative rate of change, and one that is represented in an equation and has a lower rate of change than the first function.