Hello, Guest!
 Instructional Focus DocumentMathematical Models with Applications
 TITLE : Unit 11: Research and Marketing SUGGESTED DURATION : 12 days

#### Unit Overview

This unit bundles student expectations that address types of research, analysis of claims made in marketing, and estimation of population mean and population proportion for sample data. Concepts are incorporated into both mathematical and real-world problem situations. According to the Texas Education Agency, mathematical process standards including application, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace.

Prior to this unit, in Grade 2, bar graphs were introduced and were revisited throughout elementary and middle school math. In Grades 3–7, students learned and used dot plots to organize and analyze data. In Grade 5, they were introduced to scatterplots and in grade 6, to box plots, histograms, and stem-and-leaf plots. In Grade 7 students were introduced to circle graphs. In Grades 6 – 8 students were introduced to mean, median, mode and variance, including range and interquartile ranges. In Grade 8, mean and absolute mean were compared.

During this unit, students define different types and methods of research. Students identify and differentiate purposes for surveys, experiments, and observational studies. Students predict the appropriate type of research to answer questions. Students generate questions that would require research, define the purpose of the research, and choose a method of research appropriate to the problem situation. Students define population mean and population proportion and discuss how to use data to find each. Students use either given data sets or collected data sets to find population means and proportions. Students are introduced to methods marketing professionals use to influence consumers. Students analyze examples of graphs and sets of statistics to discuss the validity of the claims. Students use the internet or print media to find examples of marketing using statistics or graphs and analyze the graphs and statistics to justify or refute the validity of the marketing claims.

After this unit, in Unit 12, students will connect concepts and extend their knowledge of statistical analysis to formulate a question, determine information needed, collect and analyze data, use mathematical models to solve problems to answer the question and draw conclusions, and present their results. In AP Statistics, students will study research methods and both population mean and population proportion.

This unit is supporting the development of the Texas College and Career Readiness Standards (TxCCRS): V. Probabilistic Reasoning A1; VI. Statistical Reasoning A1, C4; VIII. Problem Solving and Reasoning; IX. Communication and Representation; X. Connections.

According to the Connections Standard for Grades 9-12 from the National Council of Teachers of Mathematics (NCTM), “Instructional programs from pre-kindergarten through grade 12 should enable students to:

• recognize and use connections among mathematical ideas;
• understand how mathematical ideas interconnect and build on one another to produce a coherent whole;
• recognize and apply mathematics in contexts outside of mathematics.

When students can see the connections across different mathematical content areas, they develop a view of mathematics as an integrated whole. As they build on their previous mathematical understandings while learning new concepts, students become increasingly aware of the connections among various mathematical topics. As students' knowledge of mathematics, their ability to use a wide range of mathematical representations, and their access to sophisticated technology and software increase, the connections they make with other academic disciplines, especially the sciences and social sciences, give them greater mathematical power” (NCTM, 2000, p. 354).

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics: Connections standard for grades 9-12. Reston, VA: National Council of Teachers of Mathematics, Inc.
Education Policy Improvement Center (2009), Texas College and Career Standards, Austin, TX, University of Texas Printing.

#### OVERARCHING UNDERSTANDINGS and QUESTIONS

Statistical data are collected, analyzed graphically and numerically, and interpreted to determine the reliability of the data, make predictions, and draw conclusions.

• What is the purpose of analyzing statistical data?
• Why is it important to understand the analysis and interpretation of statistical data?
• What are the purposes and differences among types of research, including surveys, experiments, and observational studies?
• Why is it important to use appropriate data collection methods?
• How does the type of data determine the type of graphical analysis?
• How does the type of data determine the type of numerical analysis?
Performance Assessment(s) Overarching Concepts
Unit Concepts
Unit Understandings
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

Statistical Reasoning

• Conclusions/Predictions
• Data
• Research Methods
• Summary Statistics

Associated Mathematical Processes

• Application
• Tools and Techniques
• Problem Solving Model
• Communication
• Representations
• Relationships
• Justification

Types of research include surveys, experiments, and observational studies.

• What are the different types of research?
• What are the purposes of surveys, experiments and observational studies?
• What are the differences in surveys, experiments and observational studies?
• How can the appropriate type of research be selected to address a particular question?

Population mean and population proportion are used to find or estimate the portion of a population that fits a chosen attribute.

• How are the ratios and proportions of population mean and population proportion different?
• How are population mean and population proportion determined?
• How can estimations of population mean and population proportion be applied in problem situations?
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

Statistical Reasoning

• Conclusions/Predictions
• Data
• Research Methods
• Summary Statistics
• Statistical Representations

Associated Mathematical Processes

• Application
• Tools and Techniques
• Problem Solving Model
• Communication
• Representations
• Relationships
• Justification

Marketing claims made using graphs and statistics must be analyzed and justified to determine the validity of the claims.

• What characteristics and details can be used in analyzing marketing claims for validity?
• What are some ways to make graphs and statistics imply invalid conclusions?
• How can marketing graphs and statistics be presented so that they are valid?
• How can marketing graphs and statistics be presented so that they are misleading or not valid?

#### MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may not consider all factors and differences when analyzing marketing claims, graphs, and statistics. They may take graphs and statistics to be compared as “equal” or more similar than they actually are.
• Some student may think that either the mean or median can be used to accurately describe the central tendency of the data rather than understanding that is there are outliers in the data, the median is more robust and not as influenced by outliers as the mean.

#### Unit Vocabulary

• Experimental research – a research method in which the researcher determines a question to be answered, develops a hypothesis, and collects data to determine the validity of the hypothesis
• Observational study – a method of research in which the researcher collects data by observing behavior or conditions and activities without influencing or interfering with that behavior
• Population – total collection of all elements of a set of data
• Population mean – mean of total collection of all elements of a set of data
• Population proportion – fraction of the number of desired outcomes in the collection of elements divided by the total number of elements in the collection
• Sample mean – mean of a sample used to estimate the mean of a large population
• Sample of a population – subset of a population
• Sample proportion – proportion of a sample used to estimate the proportion of a large population
• Survey – a method to sample a portion of a population by asking a question, etc. in order to collect information

Related Vocabulary:

 Hypothesis Marketing Marketing claims Numerical statistics Random sampling Validity
Unit Assessment Items System Resources Other Resources

Show this message:

Unit Assessment Items that have been published by your district may be accessed through Search All Components in the District Resources tab. Assessment items may also be found using the Assessment Creator if your district has granted access to that tool.

System Resources may be accessed through Search All Components in the District Resources Tab.

Texas Higher Education Coordinating Board – Texas College and Career Readiness Standards (select CCRS from Standard Set dropdown menu)

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – STAAR Mathematics Resources

Texas Education Agency Texas Gateway – Revised Mathematics TEKS: Vertical Alignment Charts

Texas Education Agency Texas Gateway – Mathematics TEKS: Supporting Information

Texas Education Agency Texas Gateway – Interactive Mathematics Glossary

Texas Education Agency Texas Gateway – Resources Aligned to Mathematical Models with Applications Mathematics TEKS

Texas Instruments – Graphing Calculator Tutorials

TEKS# SE# TEKS Unit Level Specificity

• Bold black text in italics: Knowledge and Skills Statement (TEKS)
• Bold black text: Student Expectation (TEKS)
• Strike-through: Indicates portions of the Student Expectation that are not included in this unit but are taught in previous or future unit(s)
• Blue text: Supporting information / Clarifications from TCMPC (Specificity)
• Blue text in italics: Unit-specific clarification
• Black text: Texas Education Agency (TEA); Texas College and Career Readiness Standards (TxCCRS)
M.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
M.1A Apply mathematics to problems arising in everyday life, society, and the workplace.

Apply

MATHEMATICS TO PROBLEMS ARISING IN EVERYDAY LIFE, SOCIETY, AND THE WORKPLACE
Including, but not limited to:

• Mathematical problem situations within and between disciplines
• Everyday life
• Society
• Workplace

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• X. Connections
M.1B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.

Use

A PROBLEM-SOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEM-SOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION
Including, but not limited to:

• Problem-solving model
• Analyze given information
• Formulate a plan or strategy
• Determine a solution
• Justify the solution
• Evaluate the problem-solving process and the reasonableness of the solution

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VIII. Problem Solving and Reasoning
M.1C Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.

Select

TOOLS, INCLUDING REAL OBJECTS, MANIPULATIVES, PAPER AND PENCIL, AND TECHNOLOGY AS APPROPRIATE, AND TECHNIQUES, INCLUDING MENTAL MATH, ESTIMATION, AND NUMBER SENSE AS APPROPRIATE, TO SOLVE PROBLEMS
Including, but not limited to:

• Appropriate selection of tool(s) and techniques to apply in order to solve problems
• Tools
• Real objects
• Manipulatives
• Paper and pencil
• Technology
• Techniques
• Mental math
• Estimation
• Number sense

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VIII. Problem Solving and Reasoning
M.1D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

Communicate

MATHEMATICAL IDEAS, REASONING, AND THEIR IMPLICATIONS USING MULTIPLE REPRESENTATIONS, INCLUDING SYMBOLS, DIAGRAMS, GRAPHS, AND LANGUAGE AS APPROPRIATE
Including, but not limited to:

• Mathematical ideas, reasoning, and their implications
• Multiple representations, as appropriate
• Symbols
• Diagrams
• Graphs
• Language

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
M.1E Create and use representations to organize, record, and communicate mathematical ideas.

Create, Use

REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS
Including, but not limited to:

• Representations of mathematical ideas
• Organize
• Record
• Communicate
• Evaluation of the effectiveness of representations to ensure clarity of mathematical ideas being communicated
• Appropriate mathematical vocabulary and phrasing when communicating mathematical ideas

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
M.1F Analyze mathematical relationships to connect and communicate mathematical ideas.

Analyze

MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS
Including, but not limited to:

• Mathematical relationships
• Connect and communicate mathematical ideas
• Conjectures and generalizations from sets of examples and non-examples, patterns, etc.
• Current knowledge to new learning

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• X. Connections
M.1G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

Display, Explain, Justify

MATHEMATICAL IDEAS AND ARGUMENTS USING PRECISE MATHEMATICAL LANGUAGE IN WRITTEN OR ORAL COMMUNICATION
Including, but not limited to:

• Mathematical ideas and arguments
• Validation of conclusions
• Displays to make work visible to others
• Diagrams, visual aids, written work, etc.
• Explanations and justifications
• Precise mathematical language in written or oral communication

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
M.9 Mathematical modeling in social sciences. The student applies mathematical processes and mathematical models to analyze data as it applies to social sciences. The student is expected to:
M.9C Distinguish the purposes and differences among types of research, including surveys, experiments, and observational studies.

Distinguish

THE PURPOSES AND DIFFERENCES AMONG TYPES OF RESEARCH, INCLUDING SURVEYS, EXPERIMENTS, AND OBSERVATIONAL STUDIES

Including, but not limited to:

• Population – total collection of all elements of a set of data
• Sample of a population – subset of a population
• Survey – a method to sample a portion of a population by asking a question, etc. in order to collect information
• Purposes
• To collect data from randomly selected populations answering specific questions
• To influence the public through collection of responses from random populations
• Characteristics
• Collection of information from a chosen sample
• Information collected by several different methods: phone, online, television, mail, personal contact
• Types may be cross sectional and longitudinal
• Ability to gather information on one particular question
• Validity of collected data depends highly on the methods for choosing the sample population and the way questions are asked
• Experimental research – a research method in which the researcher determines a question to be answered, develops a hypothesis, and collects data to determine the validity of the hypothesis
• Purposes
• To study specific factors and results of treatments on variables while controlling other variables
• To determine relations between variables
• Characteristics
• Only type of research that uses two variables
• Control (treatment) and results
• Only type of research that may attempt to influence a particular variable to attempt to prove the hypothesis
• Allows the study of many variables at once to understand interactions among them
• Control of conditions
• Control of conditions is often difficult
• Observational study – a method of research in which the researcher collects data by observing behavior or conditions and activities without influencing or interfering with that behavior
• Purposes
• To discover conclusions based on unbiased observation; to compare numerous variables
• To document information to be used in the future; to describe change
• Characteristics
• Observations done in natural environment or lab
• Observer may have direct or indirect contact with subject(s)
• Observation may be behavioral or non-behavioral
• Provides information in a pure sense without intervention or influence
• Observational details are fixed and cannot be modified

Note(s):

• Mathematical Models with Applications introduces the study of research methods.
• AP Statistics will study research methods.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• V. Probabilistic Reasoning
• A1 – Determine the nature and the number of elements in a finite sample space.
• VI. Statistical Reasoning
• A1 – Plan a study.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
M.9D Use data from a sample to estimate population mean or population proportion.

Use

DATA FROM A SAMPLE

Including, but not limited to:

• Population – total collection of all elements of a set of data
• Sample of a population – subset of a population
• Sampling manipulatives
• Coin toss
• Toss of a die or dice
• Drawing objects from a bag
• Simulations with technology
• Random surveys

To Estimate

POPULATION MEAN OR POPULATION PROPORTION

Including, but not limited to:

• Population mean – mean of total collection of all elements of a set of data
• Sample mean – mean of a sample used to estimate the mean of a large population
• Formula for sample mean: , where n is the number of sample elements and ai represents each sample element in the set. Verbally this is read as the sum of all the elements in the sample set divided by the number of elements in the sample set
• Population proportion – fraction of the number of desired outcomes in the collection of elements divided by the total number of elements in the collection
• Sample proportion – proportion of a sample used to estimate the proportion of a large population
• When a population is too large to count every element, a sample can be used to estimate the mean and the proportion of the population.

Note(s):

• Mathematical Models with Applications introduces population mean and population proportion.
• AP Statistics will study both population mean and population proportion.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS
• V. Probabilistic Reasoning
•  A1 – Determine the nature and the number of elements in a finite sample space.
• VI. Statistical Reasoning
• A1 – Plan a study.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
M.9E Analyze marketing claims based on graphs and statistics from electronic and print media and justify the validity of stated or implied conclusions.

Analyze

MARKETING CLAIMS BASED ON GRAPHS AND STATISTICS FROM ELECTRONIC AND PRINT MEDIA

Including, but not limited to:

• Newspapers
• Magazines
• Online marketing
• Journals, print and internet
• Mobile apps

Justify

THE VALIDITY OF STATED OR IMPLIED CONCLUSIONS

Including, but not limited to:

• Percentages versus actual numbers
• Missing information or partial information

Note(s):