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Algebra I Module 5, Topic B: Completing the Modeling Cycle

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Algebra I Module 5, Topic B: Completing the Modeling Cycle


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Type: Lesson/Unit Plan


Tables, graphs, and equations all represent models. We use terms such as “symbolic” or “analytic” to refer specifically to the equation form of a function model; “descriptive model” refers to a model that seeks to describe or summarize phenomena, such as a graph. In Module 5, Topic B, students expand on their work in Topic A to complete the modeling cycle for a real-world contextual problem presented as a graph, a data set, or a verbal description. For each, they formulate a function model, perform computations related to solving the problem, interpret the problem and the model, and then, through iterations of revising their models as needed, validate, and report their results.

Students choose and define the quantities of the problem (N-Q.A.2) and the appropriate level of precision for the context (N-Q.A.3). They create 1- and 2-variable equations (A-CED.A.1, A-CED.A.2) to model the context when presented as a graph, as data and as a verbal description. They can distinguish between situations that represent a linear (F-LE.A.1b), quadratic, or exponential (F-LE.A.1c) relationship. For data, they look for first differences to be constant for linear, second differences to be constant for quadratic, and a common ratio for exponential. When there are clear patterns in the data, students will recognize when the pattern represents a linear (arithmetic) or exponential (geometric) sequence (F-BF.A.1a, F-LE.A.2). For graphic presentations, they interpret the key features of the graph, and for both data sets and verbal descriptions, they sketch a graph to show the key features (F-IF.B.4). They calculate and interpret the average rate of change over an interval, estimating when using the graph (F-IF.B.6), and relate the domain of the function to its graph and to its context (F-IF.B.5).

Content Standard(s):
MA2015 (2016)
Grade: 9-12
Algebra I
4 ) Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. [N-Q1]

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.Q.HS.4- Using real world models, express quantities of measurement to the given precision. (limited to measurements of length (inch, 1/2 inch, 1/4 inch), weight (pounds, kilograms (tenth of a unit), volume (cup, 1/2 cup, 1/4 cup, 1/3 cup, liter), temperature (degree), velocity (mph, kmph).

MA2015 (2016)
Grade: 9-12
Algebra I
5 ) Define appropriate quantities for the purpose of descriptive modeling. [N-Q2]

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.Q.HS.5- Recognize units of weight (ounces, pounds, grams, kilograms), length (inch, foot, mile, centimeter, meter, kilometer), area (square inches in^2, square feet ft^2, square centimeters cm^2, square meters m^2) and capacity (cubic inches in^3, cubic feet ft^3, cubic centimeters cm^3, cubic meters m^3).

MA2015 (2016)
Grade: 9-12
Algebra I
6 ) Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. [N-Q3]

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.Q.HS.6- Estimate to the nearest 1, 10, and 100 when adding, subtracting, multiplying, or dividing; include units with estimates.

MA2019 (2019)
Grade: 9-12
Algebra I with Probability
11. Create equations and inequalities in one variable and use them to solve problems in context, either exactly or approximately. Extend from contexts arising from linear functions to those involving quadratic, exponential, and absolute value functions.
Unpacked Content
Evidence Of Student Attainment:
  • Given a contextual situation that may include linear, quadratic, exponential, or rational functional relationships in one variable.
  • Model the relationship with equations or inequalities and solve the problem presented in the contextual situation for the given variable.
Teacher Vocabulary:
  • Variable
  • Equation
  • Inequality
  • Solution Set
  • Identity
  • No solution for a given domain
  • Approximate solutions
Students know:
  • When the situation presented in a contextual problem is most accurately modeled by a linear, quadratic, exponential, or rational functional relationship.
Students are able to:
  • Write equations in one variable that accurately model contextual situations.
Students understand that:
  • Features of a contextual problem can be used to create a mathematical model for that problem.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
ALGI.11.1: Solve the equation represented by the real-world situation.
ALGI.11.2: Set up an equation to represent the given situation, using correct mathematical operations and variables.
ALGI.11.3: Given a contextual situation, interpret and defend the solution in the context of the original problem.
ALGI.11.4: Define equation, expression, variable, equality and inequality.
ALGI.11.5: Create inequalities with one variable (Exponential, Quadratic, Linear).
ALGI.11.6: Create equalities with one variable (Exponential, Quadratic, Linear).
ALGI.11.7: Solve two-step equations and inequalities.
ALGI.11.8: Solve one-step equations and inequalities using the four basic operations.
ALGI.11.9: Compare and contrast equations and inequalities.
ALGI.11.10: Recognize inequality symbols including greater than, less than, greater than equal to and less than equal to.

Prior Knowledge Skills:
  • Test the found number or number set for accuracy by substitution.
  • Set up equations and inequalities to represent the given situation, using correct mathematical operations and variables.
  • Define equation, inequality, and variable.
  • Convert mathematical terms to mathematical symbols and numbers.

Alabama Alternate Achievement Standards
AAS Standard:
M.A.AAS.11.11 Select an equation or inequality involving one operation (limit to addition or subtraction) with one variable that represents a real-world problem. Solve the equation.

Tags: arithmetic sequence, descriptive modeling, domain, equations, Exponential function, expression, formulas, geometric sequence, graph, inequality, input, linear function, units
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There are six lessons in this topic.

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Author: Hannah Bradley