Standards - Mathematics

2019 - Mathematics Course of Study Document
Mathematics (2019) Grade(s): 09-12 - Algebra II with Statistics

MA19.A2.18

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Extend to polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions.

Unpacked Content

Knowledge

Students know:

  • Techniques for graphing functions.
  • Techniques for determining the domain of a function from its context.

Skills

Students are able to:

  • Interpret the domain from the context.
  • Produce a graph of a function based on the context given.

Understanding

Students understand that:

  • Different contexts produce different domains and graphs.
  • Function notation in itself may produce graph points which should not be in the graph as the domain is limited by the context.

Vocabulary

  • Function
  • Quantitative
  • Domain

Aligned Learning Resources

Applying the Quadratic Formula (Part 1): Algebra 1, Episode 24: Unit 7, Lesson 17 | Illustrative Math
Mathematics (2019) Grade(s): 09-12 - Algebra II with Statistics

MA19.A2.19

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. Extend to polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions.

Unpacked Content

Knowledge

Students know:

  • Techniques for graphing.
  • Techniques for finding a rate of change over an interval on a table or graph.
  • Techniques for estimating a rate of change over an interval on a graph.

Skills

Students are able to:

  • Calculate rate of change over an interval on a table or graph.
  • Estimate a rate of change over an interval on a graph.

Understanding

Students understand that:

  • The average provides information on the overall changes within an interval, not the details within the interval (an average of the endpoints of an interval does not tell you the significant features within the interval).

Vocabulary

  • Average rate of change
  • Specified interval
Mathematics (2019) Grade(s): 09-12 - Algebra II with Statistics

MA19.A2.20

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Extend to polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions.

Unpacked Content

Knowledge

Techniques for graphing.

  • Key features of graphs of functions.

    • Skills

    Students are able to:

    • Identify the type of function from the symbolic representation.
    • Manipulate expressions to reveal important features for identification in the function.
    • Accurately graph any relationship.
    • Find the inverse of a function algebraically and/or graphically.

    Understanding

    Students understand that:

    • Key features are different depending on the function.
    • Identifying key features of functions aid in graphing and interpreting the function.
    • A function and its inverse are reflections over the line y = x.

    Vocabulary

    • Polynomial function
    • Logarithmic function Trigonometric (sine and cosine) function
    • Reciprocal function
    • Radical function
    • Period
    • Midline
    • Amplitude
    • End Behavior
    • Intervals
    • Maximum
    • Minimum
    • Horizontal Asymptote
    • Vertical Asymptote
    • Inverse functions
    Mathematics (2019) Grade(s): 09-12 - Algebra II with Statistics

    MA19.A2.21

    Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle, building on work with non-right triangle trigonometry.

    Unpacked Content

    Knowledge

    Students know:
    • Trigonometric ratios for right triangles.
    • The appropriate sign for coordinate values in each quadrant of a coordinate graph.

    Skills

    Students are able to:
    • Accurately find relationships of trigonometric functions for an acute angle of a right triangle to measures within the unit circle.
    • Justify triangle similarity.
    • Find the reference angle for any angle found by a revolution on a ray in the coordinate plane.
    • Relate the trigonometric ratios for the reference angle to those of the original angle.
    • Determine the appropriate sign for trigonometric functions of angles of any given size.

    Understanding

    Students understand that:
    • Trigonometric functions may be extended to all real numbers from being defined only for acute angles in right triangles by using the unit circle, reflections, and logical reasoning.

    Vocabulary

    • Unit circle
    • Radian measure
    • Quadrantal
    • Traversed

    Aligned Learning Resources

    Unit Circle: Exploring Unit Circle Trigonometry With Desmos
    Unit Circle: Unit Circle with Everything by CalcWorkshop
    Unit Circle: Making Your Own
    Coloring With the Unit Circle
    Cutting Up With the Unit Circle
    Closing Up the Unit Circle
    Mathematics (2019) Grade(s): 09-12 - Algebra II with Statistics

    MA19.A2.22

    Use the mathematical modeling cycle to solve real-world problems involving polynomial, trigonometric (sine and cosine), logarithmic, radical, and general piecewise functions, from the simplification of the problem through the solving of the simplified problem, the interpretation of its solution, and the checking of the solution’s feasibility.

    Unpacked Content

    Knowledge

    Students know:

    • When the situation presented in a contextual problem is most accurately modeled by a polynomial, exponential, logarithmic, trigonometric (sine and cosine), radical, or general piecewise functional relationship.

    Skills

    Students are able to:

    • Accurately model contextual situations.

    Understanding

    Students understand that:

    • There are relationships among features of a contextual problem and a created mathematical model for that problem.
    • Different contexts produce different domains and feasible solutions.

    Vocabulary

    • Mathematical modeling cycle
    • Feasibility

    Aligned Learning Resources

    Introduction to Functions and Equations for Functions
    Mathematics (2019) Grade(s): 09-12 - Algebra II with Statistics

    MA19.A2.23

    Use mathematical and statistical reasoning about normal distributions to draw conclusions and assess risk; limit to informal arguments.

    Example: If candidate A is leading candidate B by 2% in a poll which has a margin of error of less than 3%, should we be surprised if candidate B wins the election?

    Unpacked Content

    Knowledge

    Students know:

    • Properties of a normal distribution.
    • Empirical Rule

    Skills

    Students are able to:

    • Draw accurate conclusions and assess risk using their knowledge of the normal distribution.

    Understanding

    Students understand that:

    • For a normal distribution, nearly all of the data will fall within three standard deviations of the mean.
    • The empirical rule can be broken down into three parts: 68% of data falls within the first standard deviation from the mean, 95% fall within two standard deviations. and 99.7% fall within three standard deviations.

    Vocabulary

    • Normal distribution
    • Margin of error
    Mathematics (2019) Grade(s): 09-12 - Algebra II with Statistics

    MA19.A2.24

    Design and carry out an experiment or survey to answer a question of interest, and write an informal persuasive argument based on the results.

    Example: Use the statistical problem-solving cycle to answer the question, Is there an association between playing a musical instrument and doing well in mathematics?

    Unpacked Content

    Knowledge

    Students know:

    • Techniques to design an experiment or survey

    Skills

    Students are able to:

    • Develop a statistical question.
    • Design and carry out an experiment or survey.
    • Accurately interpret the results of an experiment or survey.

    Understanding

    Students understand that:

    • A statistical question is one that can be answered by collecting data and where there will be variability in that data.
    • An experiment is a controlled study in which the researcher attempts to understand cause-and-effect relationships. Based on the analysis, the researcher draws a conclusion about whether the treatment ( independent variable ) had a causal effect on the dependent variable.
    • Statistical surveys are collections of information about items in a population and may be grouped into numerical and categorical types.

    Vocabulary

    • Experiment
    • Survey
    Mathematics (2019) Grade(s): 09-12 - Algebra II with Statistics

    MA19.A2.25

    From a normal distribution, use technology to find the mean and standard deviation and estimate population percentages by applying the empirical rule.

    Unpacked Content

    Knowledge

    Students know:
    From a normal distribution,
    • Techniques to find the mean and standard deviation of data sets using technology.
    • Techniques to use calculators, spreadsheets, and standard normal distribution tables to estimate areas under the normal curve.

    Skills

    Students are able to:
    • From a normal distribution, accurately find the mean and standard deviation of data sets using technology.
    • Make reasonable estimates of population percentages from a normal distribution.
    • Read and use normal distribution tables and use calculators and spreadsheets to accurately estimate the areas under a normal curve.

    Understanding

    Students understand that:
    Under appropriate conditions,
    • The mean and standard deviation of a data set can be used to fit the data set to a normal distribution.
    • Population percentages can be estimated by areas under the normal curve using calculators, spreadsheets, and standard normal distribution tables.

    Vocabulary

    • Normal distribution
    • Population Percentages
    • Empirical Rule
    • Normal curve
    • Mean
    • Standard deviation

    Aligned Learning Resources

    Central Tendency | Targeted Math Instruction
    What Is Statistics? | Against All Odds: Unit 1
    Measures of Center | Against All Odds: Unit 4
    Statistics: Using Sampling to Count Trees
    Is This Normal? The Normal Distribution Explained
    Is This Normal? Extending the Normal Distribution
    Mathematics (2019) Grade(s): 09-12 - Algebra II with Statistics

    MA19.A2.26

    Describe the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

    Examples: random assignment in experiments, random selection in surveys and observational studies

    Unpacked Content

    Knowledge

    Students know:
    • Key components of sample surveys, experiments, and observational studies.
    • Procedures for selecting random samples.

    Skills

    Students are able to:
    • Use key characteristics of sample surveys, experiments, and observational studies to select the appropriate technique for a particular statistical investigation.

    Understanding

    Students understand that:
    • Sample surveys, experiments, and observational studies may be used to make inferences made about the population.
    • Randomization is used to reduce bias in statistical procedures.

    Vocabulary

    • Sample surveys
    • Experiments
    • Observational studies
    • Randomization

    Aligned Learning Resources

    Central Tendency | Targeted Math Instruction
    The Economic Impact of a Changing Middle Class
    What Is Statistics? | Against All Odds: Unit 1
    Statistics: Using Sampling to Count Trees
    Stemplots | Against All Odds: Unit 2
    Human Performance & Sampling
    P-Value as a Benchmark in Experimental Research | Prediction by the Numbers
    Understanding a Crowd's Predictive Ability | Prediction by the Numbers
    Histograms | Against All Odds: Unit 3
    Mathematics (2019) Grade(s): 09-12 - Algebra II with Statistics

    MA19.A2.27

    Distinguish between a statistic and a parameter and use statistical processes to make inferences about population parameters based on statistics from random samples from that population.

    Unpacked Content

    Knowledge

    Students know:
    • Techniques for selecting random samples from a population.

    Skills

    Students are able to:
    • Accurately compute the statistics needed.
    • Recognize if a sample is random.
    • Reach accurate conclusions regarding the population from the sample.

    Understanding

    Students understand that:
    • Statistics generated from an appropriate sample are used to make inferences about the population.

    Vocabulary

    • Population parameters
    • Random samples
    • Inferences

    Aligned Learning Resources

    Understanding a Crowd's Predictive Ability | Prediction by the Numbers
    Mathematics (2019) Grade(s): 09-12 - Algebra II with Statistics

    MA19.A2.28

    Describe differences between randomly selecting samples and randomly assigning subjects to experimental treatment groups in terms of inferences drawn regarding a population versus regarding cause and effect.

    Example: Data from a group of plants randomly selected from a field allows inference regarding the rest of the plants in the field, while randomly assigning each plant to one of two treatments allows inference regarding differences in the effects of the two treatments. If the plants were both randomly selected and randomly assigned, we can infer that the difference in effects of the two treatments would also be observed when applied to the rest of the plants in the field.

    Unpacked Content

    Knowledge

    Students know:

    • Techniques for selecting random samples from a population.
    • Techniques for randomly assigning subjects to experimental treatment groups.

    Skills

    Students are able to:

    • Recognize if a sample is random.
    • Reach accurate conclusions regarding the population from the sample.
    • Reach accurate conclusions regarding the cause and effect of an experimental treatment.

    Understanding

    Students understand that:

    • Random selection is essential to external validity, or the extent to which the researcher can generalize the results of the study to the larger population.
    • Random assignment is central to internal validity, which allows the researcher to make causal claims about the effect of the treatment.

    Vocabulary

    • Randomly
    • Non-randomized
    • Inference
    • Treatments
    • Cause and effect
    Mathematics (2019) Grade(s): 09-12 - Algebra II with Statistics

    MA19.A2.29

    Explain the consequences, due to uncontrolled variables, of non-randomized assignment of subjects to groups in experiments.

    Example: Students are studying whether or not listening to music while completing mathematics homework improves their quiz scores. Rather than assigning students to either listen to music or not at random, they simply observe what the students do on their own and find that the music-listening group has a higher mean quiz score. Can they conclude that listening to music while studying is likely to raise the quiz scores of students who do not already listen to music? What other factors may have been responsible for the observed difference in mean quiz scores?

    Unpacked Content

    Knowledge

    Students know:

    • Differences between random and non-random assignment.
    • The definition of independent and dependent variables.

    Skills

    Students are able to:

    • Conclude whether a causal relationship exists in an experiment based on the type of assignment of subjects in the experiment.
    • Identify uncontrolled variable that may be responsible for observed difference when non-randomized assignment is used in an experiment.

    Understanding

    Students understand that:

    • Uncontrolled variables are characteristic factors that are not regulated or measured by the investigator during an experiment or study, so that they are not the same for all participants in the research.
    • Randomized selection of subjects to groups in experiments is the only type of study able to establish causation.

    Vocabulary

    • Uncontrolled variables
    • Non-randomized
    Mathematics (2019) Grade(s): 09-12 - Algebra II with Statistics

    MA19.A2.30

    Evaluate where bias, including sampling, response, or nonresponse bias, may occur in surveys, and whether results are representative of the population of interest.

    Example: Selecting students eating lunch in the cafeteria to participate in a survey may not accurately represent the student body, as students who do not eat in the cafeteria may not be accounted for and may have different opinions, or students may not respond honestly to questions that may be embarrassing, such as how much time they spend on homework.

    Unpacked Content

    Knowledge

    Students know:

    • Techniques for conducting surveys.
    • Techniques to identify bias

    Skills

    Students are able to:

    • Given the description of a survey,
      • Evaluate bias that may occur in the survey.
      • Determine whether a bias precludes results of the survey from being generalized to the population.

    Understanding

    Students understand that:

    • Bias is the intentional or unintentional favoring of one group or outcome over other potential groups or outcomes in the population.
    • A common cause of sampling bias lies in the design of the study or in the data collection procedure, both of which may favor or disfavor collecting data from certain classes or individuals or in certain conditions.
    • Response bias (also called survey bias) is the tendency of a person to answer questions on a survey untruthfully or misleadingly.
    • Nonresponse bias is the bias that results when respondents differ in meaningful ways from nonrespondents.

    Vocabulary

    • Bias
    • Sampling
    • Response bias
    • Nonresponse bias
    Mathematics (2019) Grade(s): 09-12 - Algebra II with Statistics

    MA19.A2.31

    Evaluate the effect of sample size on the expected variability in the sampling distribution of a sample statistic.

    Unpacked Content

    Knowledge

    Students know:

    • Techniques to find the mean and standard deviation.

    Skills

    Students are able to:

    • Accurately compute the statistics needed.
    • Reach accurate conclusions regarding the population from the the sampling distribution of a sample statistic.

    Understanding

    Students understand that:

    • The center is not affected by sample size. The mean of the sample means is always approximately the same as the population mean.
    • As the sample size increases, the standard deviation of the means decreases. and as the sample size decreases, the standard deviation of the sample means increases.

    Vocabulary

    • Sample size
    • Variability
    • Sampling distribution
    • Standard deviation
    Mathematics (2019) Grade(s): 09-12 - Algebra II with Statistics

    MA19.A2.32

    Produce a sampling distribution by repeatedly selecting samples of the same size from a given population or from a population simulated by bootstrapping (resampling with replacement from an observed sample). Do initial examples by hand, then use technology to generate a large number of samples.

    Unpacked Content

    Knowledge

    Students know:
    • Techniques for producing a sampling distribution.
    • Properties of a normal distribution.

    Skills

    Students are able to:
    • Produce a sampling distribution.
    • Reach accurate conclusions regarding the population from the sampling distribution.
    • Accurately create and interpret a confidence interval based on observations from the sampling distribution.

    Understanding

    Students understand that:
    • The central limit theorem in statistics states that, given a sufficiently large sample size, the sampling distribution of the mean for a variable will approximate a normal distribution regardless of that variable's distribution in the population.
    • A 95% confidence interval is a range of values that you can be 95% certain contains the true mean of the population. With large samples, you know that mean with much more precision than you do with a small sample, so the confidence interval is quite narrow when computed from a large sample.

    Vocabulary

    • Bootstrapping
    • Population mean
    • Approximately normal
    • Standard deviation
    • Confidence interval

    Aligned Learning Resources

    Statistics: Using Sampling to Count Trees
    Mathematics (2019) Grade(s): 09-12 - Algebra II with Statistics

    MA19.A2.33

    Use data from a randomized experiment to compare two treatments; limit to informal use of simulations to decide if an observed difference in the responses of the two treatment groups is unlikely to have occurred due to randomization alone, thus implying that the difference between the treatment groups is meaningful.

    Example: Fifteen students are randomly assigned to a treatment group that listens to music while completing mathematics homework and another 15 are assigned to a control group that does not, and their means on the next quiz are found to be different. To test whether the differences seem significant, all the scores from the two groups are placed on index cards and repeatedly shuffled into two new groups of 15 each, each time recording the difference in the means of the two groups. The differences in means of the treatment and control groups are then compared to the differences in means of the mixed groups to see how likely it is to occur.

    Unpacked Content

    Knowledge

    Students know:
    • Techniques for conducting randomized experiments.
    • Techniques for conducting simulations of randomized experiment situations.

    Skills

    Students are able to:
    • Design and conduct randomized experiments with two treatments.
    • Draw conclusions from comparisons of the data of the randomized experiment.
    • Design, conduct, and use the results from simulations of a randomized experiment situation to evaluate the significance of the identified differences.

    Understanding

    Students understand that:
    • Differences of two treatments can be justified by a significant difference of parameters from a randomized experiment.
    • Statistical analysis and data displays often reveal patterns in data or populations, enabling predictions.

    Vocabulary

    • Randomized experiment
    • Significant
    • Parameters

    Aligned Learning Resources

    Statistics: Using Sampling to Count Trees
    Human Performance & Sampling
    P-Value as a Benchmark in Experimental Research | Prediction by the Numbers
    Understanding a Crowd's Predictive Ability | Prediction by the Numbers
    Mathematics (2019) Grade(s): 09-12 - Algebra II with Statistics

    MA19.A2.34

    Define the radian measure of an angle as the constant of proportionality of the length of an arc it intercepts to the radius of the circle; in particular, it is the length of the arc intercepted on the unit circle.

    Unpacked Content

    Knowledge

    Students know:

    • The circumference of any circle is 2πr and therefore, the circumference of a unit circle is 2π.

    Skills

    Students are able to:

    • Translate between arc length and central angle measures in circles.

    Understanding

    Students understand that:

    • Radians measure angles as a ratio of the arc length to the radius.
    • The unit circle has a circumference of 2π which aids in sense making for angle measure as revolutions (one whole revolution measures 2π radians) regardless of radius.
    • Use of the unit circle gives a one-to-one ratio between arc length and the measure of the central angle, putting the angle in direct proportion to the arc length, and that the circle can then be divided up to find the radian measure of other angles.

    Vocabulary

    • Radian measure
    • Constant of proportionality
    • Unit circle
    • Intercepted arc
    Mathematics (2019) Grade(s): 09-12 - Algebra II with Statistics

    MA19.A2.35

    Choose trigonometric functions (sine and cosine) to model periodic phenomena with specified amplitude, frequency, and midline.

    Unpacked Content

    Knowledge

    Students know:
    • Key features of trigonometric functions (e.g., amplitude, frequency, and midline).
    • Techniques for selecting functions to model periodic phenomena.

    Skills

    Students are able to:
    • Determine the amplitude, frequency, and midline of a trigonometric function.
    • Develop a trigonometric function to model periodic phenomena.

    Understanding

    Students understand that:
    • Trigonometric functions are periodic and may be used to model certain periodic contextual phenomena.
    • Amplitude, frequency, and midline are useful in determining the fit of the function used to model the phenomena.

    Vocabulary

    • Trigonometric functions
    • Periodic phenomena
    • Amplitude
    • Frequency
    • Midline
    Mathematics (2019) Grade(s): 09-12 - Algebra II with Statistics

    MA19.A2.36

    Prove the Pythagorean identity $\sin^2 (\theta) + \cos^2 (\theta) = 1$ and use it to calculate trigonometric ratios.

    Unpacked Content

    Knowledge

    Students know:
    • Methods for finding the sine, cosine, and tangent ratios of a right triangle.
    • The Pythagorean Theorem.
    • Properties of equality.
    • The signs of the sine, cosine, and tangent ratios in each quadrant.

    Skills

    Students are able to:
    • Use the unit circle, definitions of trigonometric functions, and the Pythagorean Theorem to prove the Pythagorean Identity sin2 (θ) + cos2(θ) = 1.
    • Accurately use the Pythagorean identity sin2 (θ) + cos2(θ) = 1 to find the sin(θ), cos(θ), or tan(θ) when given the quadrant and one of the values.

    Understanding

    Students understand that:
    • The sine and cosine ratios and Pythagorean Theorem may be used to prove that sin2 (θ) + cos2 (θ) = 1.
    • The sine, cosine, or tangent value of an angle and a quadrant location provide sufficient information to find the other trigonometric ratios.

    Vocabulary

    • Pythagorean Identity
    Mathematics (2019) Grade(s): 09-12 - Algebra II with Statistics

    MA19.A2.37

    Derive and apply the formula $A = \frac{1}{2} \cdot ab \cdot \sin(C)$ for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side, extending the domain of sine to include right and obtuse angles.

    Unpacked Content

    Knowledge

    Students know:
    • The auxiliary line drawn from the vertex perpendicular to the opposite side forms an altitude of the triangle.
    • The formula for the area of a triangle (A = 1/2 bh).
    • Properties of the sine ratio.

    Skills

    Students are able to:
    • Properly label a triangle according to convention.
    • Perform algebraic manipulations.

    Understanding

    Students understand that:
    • Given the lengths of the sides and included angle of any triangle the area can be determined.
    • There is more than one formula to find the area of a triangle.

    Vocabulary

    • Auxiliary line
    • Vertex
    • Perpendicular

    Aligned Learning Resources

    Triangle Area: No Height?--Use the Sine
    Mathematics (2019) Grade(s): 09-12 - Algebra II with Statistics

    MA19.A2.38

    Derive and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles. Extend the domain of sine and cosine to include right and obtuse angles.

    Examples: surveying problems, resultant forces

    Unpacked Content

    Knowledge

    Students know:

    • The auxiliary line drawn from the vertex perpendicular to the opposite side forms an altitude of the triangle.
    • Properties of the Sine and Cosine ratios.
    • Pythagorean Theorem.
    • Pythagorean Identity.
    • The Laws of Sines and Cosines can apply to any triangle, right or non-right.
    • Laws of Sines and Cosines.
    • Vector quantities can represent lengths of sides and angles in a triangle.
    • Values of the sin(90 degrees) and cos(90 degrees).

    Skills

    Students are able to:

    • Label triangles in context and by convention.
    • Perform algebraic manipulations.
    • Find inverse sine and cosine values.

    Understanding

    Students understand that:

    • The given information will determine whether it is appropriate to use the Law of Sines or the Law of Cosines.
    • Proof is necessary to establish that a conjecture about a relationship in mathematics is always true and may provide insight into the mathematics being addressed.
    • Proven laws allow us to solve problems in contextual situations.

    Vocabulary

    • Law of Sines
    • Law of Cosines
    • Resultant Force
    Mathematics (2019) Grade(s): 09-12 - Mathematical Modeling

    MA19.MM.1

    Use the full Mathematical Modeling Cycle or Statistical Problem-Solving Cycle to answer a real-world problem of particular student interest, incorporating standards from across the course.

    Examples: Use a mathematical model to design a three-dimensional structure and determine whether particular design constraints are met; to decide under what conditions the purchase of an electric vehicle will save money; to predict the extent to which the level of the ocean will rise due to the melting polar ice caps; or to interpret the claims of a statistical study regarding the economy.

    Unpacked Content

    Knowledge

    Students know:
    • how to approach and solve real-world problems using mathematical models and the mathematical modeling cycle.

    Skills

    Students are able to:
    • Read a real-world problem and distinguish between relevant and non-relevant information.
    • Create and apply a mathematical model to solve a real-world problem.
    • Analyze their work and their findings.
    • note restrictions or limits that arise when using a mathematical model.
    • Share their findings with others.
    • Apply their work to similar situations.

    Understanding

    Students understand that:
    • real-world problems can be solved using a mathematical model.
    • There is a defined process to follow to create a mathematical model for problem solving.
    • Mathematical models sometimes have limitations or need to be restricted in order to produce valid answers.

    Vocabulary

    • Mathematical Modeling Cycle
    • Statistical Problem Solving Cycle
    • Mathematical Model

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