• # Middle School Mathematics Grade 7

Subject: Mathematics
Timeline: 6 days
Unit Title: Samples and Populations

Unit Overview:

Statistics is a tool for representing and analyzing data that may then be used to describe a population.  Probability is a tool for understanding sampling issues in statistics.  The problems in Samples and Populations Investigation 2 help students make connections between probability and statistics.  Students explore what samples are, how they are related to populations, and ways to select samples, including random samples.  In this investigation, data is provided.  It is assumed that students have had prior experience collecting data as part of statistical investigations.  If they have not, it is recommended to have the students collect their own data as time and situations permit.  The problems can be explored using either the data provided or data collected by students (Samples and Populations, pg. 3).

Unit Objectives:

At the end of this unit, all students must:
• Use the process of statistical investigation to explore problems
• Use information from samples to draw conclusions about populations
• Explore the influence of sample size on the variability of the distribution of sample means or medians
• Evaluate sampling plans
• Use probability to select random samples from populations
• Compare sample distributions using measures of center (mean and median), measures of variability (range, minimum, and maximum data values, percentiles) and data displays that group data (line plots and box-and-whisker plots)

Focus Standards:

PA.CCSS.Math Content.CC.2.4.7.B.1 Draw inferences about populations based on random sampling concepts. (7.SP.1, 7.SP.2)
PA.CCSS.Math Content.CC.2.4.7.B.2 Draw informal comparative inferences about two populations. (7.SP.3, 7.SP.4)

Mathematical Practice Standards:

#1 Make sense of problems and persevere in solving them.

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends.

#3 Construct viable arguments and critique the reasoning of others.

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

#5 Use appropriate tools strategically.

Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, or a calculator. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

#7 Look for and make use of structure.

Mathematically proficient students look closely to discern a pattern or structure. Mid-level students will see 7 × 8 equals the well-remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property.  Structure can be found in tables, lists, area models and other diagrams.

Concepts - Students will know:
• The difference between a sample and a population
• How to use samples to make predictions about a population
• The various ways to develop a sampling plan
• How to select a random sample from a population
• How to use sampling distributions, measures of center, and measures of variability to describe and compare samples
• How to apply elementary probability with spinners or calculators to choose from random samples
• How to use data from a sample to estimate a characteristic of a population
Competencies -Students will be able to:
• Identify populations, samples and method of sample collection from a given survey
• Calculate percent
• Estimate populations by scaling up results of a survey
• List reasons why certain data collection is inaccurate
• List the advantages and disadvantages of four common types of sampling methods
• Collect random samples from students using spinners or graphing calculators
• Make line plots
• Describe and compare variability
• Make and compare box plots
• Calculate mean and find median
• Compare distributions of line plots
• Discuss how well samples of different sizes predict the mean and median for the entire population

Assessments:

Formative Assessments:
• Informal assessments on learning targets
• Partner Quiz
Summative Assessment:
• Common Core Unit Assessment- Samples and Populations

Elements of Instruction:

In Grade 6, instructional time focused on four critical areas: (1) connecting ratio and rate to whole number multiplication and division and using concepts of ratio and rate to solve problems; (2) completing understanding of division of fractions and extending the notion of number to the system of rational numbers, which includes negative numbers; (3) writing, interpreting, and using expressions and equations; and (4) developing understanding of statistical thinking.

Samples and Populations builds on collecting and organizing data into different contexts.  This began during two previous books, How Likely Is It? and What Do You Expect?, which apply the process of statistical investigation to pose questions, identify ways data is collected, determine strategies for analyzing data, and interpret the analysis in order to answer the questions posed in the lessons. This previous work ties together work with statistics and probability.  In this unit, students explore ways to select samples, including the use of random sampling techniques.

Also, in the Grade 6 book, Data About Us, students find and use range and the shape and distribution of graphs to make inferences and predictions about a data set.  These skills are extended in the book Samples and Populations to explain variability in both categorical and numerical data.  Finding measures of center in Data About Us is deepened be deciding when to use the mean and median to describe a distribution.

Finally, finding percents in the Grade 6 books, Bits and Pieces I, Bits and Pieces III, and Comparing and Scaling, grows during this unit to use counts or percents to report frequencies of occurrence data.

Students consider samples and populations, and also use results of analyses of data from samples, to make estimated about population characteristics or behaviors.  First, students consider a survey that raises issues about projecting the analysis results of its sample to the entire population.  Next, students consider the difference among convenience samples, voluntary-response samples, and random samples.  They explore techniques for randomly choosing samples from a population- such as using spinners, number cubes and random-number generations on graphing calculators- and think about why random samples are often preferable.  Students then investigate the idea that sample size affects the accuracy of population estimates.  Through sampling and determining mean and median statistics for each sample, students learn that the statistics of larger samples are more reliably predictive of the population than statistics from smaller samples (Samples and Populations, pg. 3).

Major misconceptions by and struggles for students in this unit include:
1. Confusing the definition of “sample” with “population”
2. Estimating larger population results using proportions and scaling up from sample results
3. Confusing random sampling techniques with other common sampling methods
4. Synthesizing conclusions about and describing variability of samples
5. Describing the parts and usage of box plots
6. Confusing the definitions of “mean” and “median”

Differentiation:

Each lesson has differentiation options for each portion of the lesson. Additional differentiation options are listed with directions and student masters in the Teacher’s Guide.
• Special Needs Handbook
• Unit Projects
• Spanish Additional Practice and Skills guide
• Strategies for English Language Learners Guide
• “Extension” homework questions
• District-created notebooks

Interdisciplinary Connections:
• Mathematical Reflections
• “Did You Know?” sections
• phschool.com and web codes
• “Connections” homework questions
• The real-world context embedded in lesson problems