Seventh Grade

 

Data Analysis and Probability

 

Standard 7-6: Through the process standards students will demonstrate an understanding of relationships between two populations or samples through data analysis and probability.

 

The indicators for this standard are grouped by the following major concepts:

 

The indicators that support each of those major concepts and an explanation of the essential learning for each major concept follows.

 

 

Collection and Representation

 

Indicator

7-6.2    Organize data in box plots or circle graphs as appropriate.

 

                Since kindergarten, students have been collecting and representing data.  The representations used include:

                Kindergarten;            drawings and pictures

                First Grade;               picture, object, bar graphs and tables

                Second Grade;          charts, pictographs, and tables

                Third Grade;              tables, bar graphs, and dot plots

                Fourth Grade;           tables, line graphs, bar graphs, and double

                                                bar graphs (scale increments greater than or

                                                equal to one)

                Sixth grade;              frequency tables, histograms, and stem-and-

                                                leaf plots

 

          In seventh grade, students should learn to use box-and-whisker plots as well as circle graphs to organize data.  Box-and-whisker plots are sometimes called a box plot for short.  A box plot is a visual way to easily see the median and range of a data set and can be drawn vertically or horizontally.

          A box-and-whisker plot displays the median, the quartiles, and outliers of a set of data, but does not display any other specific values.  There are five key parts.  The easiest way to construct a box-and-whisker plot is to write the data in order from least to greatest, draw a number line to show the data in equal intervals, mark the median, mark the medians of the upper and lower half and then mark the upper and lower extremes.  Be certain to stress that the quartile refers to one-fourth of the number of items, not one-fourth of the range.

          In circle graphs, students organize data by using knowledge of fractions, percents, and degrees of a circle.  Students can relate data as fractional parts of a circle, percentage of degrees or compare to a clock (1 minute =  of the clock face = 6°/360°).  Circle graphs are used to compare parts of a whole and show relationships at a glance.  The graph makes it easy to see the relationship of the categories to each other, as well as, the relationship of each category to the whole.

          Students should have practice using a protractor and compass to create circles and measure angles.  The knowledge of equivalent ratios is crucial for this activity.  For example, 6 students out of 20 students is , which is equivalent to .  This ratio is then converted to degrees,  = 108°/360°, therefore the measure of the central angle of the circle graph corresponding to these 6 students is 108°. It should also be noted that  = 30% so this section of the graph is 30% of the whole.  Students should realize that the sum of the percents for each category is 100%.

          Students should be engaged in a discussion about the steps needed to convert tabular data into a circle graph. Methods that might be suggested:

 

Have students develop the steps in constructing a circle graph as well as carrying out the steps. Students should be able to describe the purposes for using circle graphs or dot plots and discuss comparisons that can be made with each. 

 

Connections to:

 

Other Seventh Grade Indicators

7-2.9         Apply an algorithm to multiply and divide fractions and decimals.

7-3.6    Represent proportional relationships with graphs, tables, and equations.

 

Teacher Notes:  Box-and-whisker plots (or just box plots) are an easy way to display information about the ranges and distributions of a data set.  Box plots can provide effective comparisons between two data sets because they make descriptive characteristics such as median and interquartile range readily apparent. (Principles and Standards, NCTM)

          Students apply percentages to make and interpret circle graphs.

 

 

 

 

Data Analysis

 

Indicators

7-6.1    Predict the characteristics of two populations based on the analysis of sample data.

7-6.3    Apply procedures to calculate the interquartile range.

7-6.4    Interpret the interquartile range for data.

 

 

          Third grade students were asked to find the range of a data set as well as analyze dot plots and bar graphs to make predictions about populations. Fourth grade students interpreted data in tables, line graphs, bar graphs, and double bar graphs with scale increments greater than or equal to 1. Fifth grade students applied procedures to calculate to calculate the measures of central tendency as well as interpreted the meaning and applications of the measures. Sixth grade students predicted the characteristics of one population based on the analysis of sample data and analyzed which measure of central tendency (mean, median, or mode) was the most appropriate for a given purpose.

          In grade 7, students extend their predictions of characteristics of one population to two populations.  Emphasis should be placed on linearity and proportionality. 

          Students should plan and design experiments to collect and compare relevant two population data.  Students should be expected to gather reliable information with well-written questions.  Experiments should be used to make inferences and predictions.  For example, based on data regarding sneaker prices at three different stores, “What would you expect to pay for a pair of sneakers?”  Box plots are useful when making comparisons between populations.

          Students should describe observed relationships mathematically and discuss whether the conjectures that they drew from the sample data might apply to a larger population.  In deciding whether two data sets are similar or different consists essentially in deciding whether the distributions of the data sets are similar or different.  Students need to understand that differences in the variability, or spread, in two data sets are an important part of deciding whether the data sets are similar or different.  Students should describe the variability by characterizing the shape(s) of the distributions: bell curve, u-shape distribution, rectangle-shaped (flat or uniform), and J-shaped or backward J-shaped.

          In the middle grades, not only is the range a measure of the spread in a data set, but the use of quartile is another measure.  Quartiles are the three values that divide an ordered set of data into four equal-sized subsets.  Of the data, 25% fall between two successive quartiles.  The number of data elements between successive quartiles depends on the size of the data set, but the percent of the data between the successive quartiles is always 25%. 

Comparing quartiles involves comparing concentrations of data in two data sets.  This comparison is an example of multiplicative reasoning (i.e., ratios of numbers of elements).

          To find the interquartile range, students need to find the difference between the upper and lower quartiles (third and first or Q3 – Q1).  Half of the data is in that range.  The “box” of the box plot denotes the middle 50% of the data and the difference between the two ends of the box is the interquartile range.

          Students need to understand that data analysis is a sophisticated process.

 

Connections to:

 

Other Seventh Grade Indicators

7-3.2         Analyze tables and graphs to describe the rate of change between and among quantities.

7-3.6    Represent proportional relationships with graphs, tables, and equations.

 

Teacher Notes: Students use proportions to make estimates relating to a population on the basis of a sample.

          Students need an understanding of percents (percent as a rate) to use multiplicative reasoning effectively. 

 

 

Probability

 

Indicators

7-6.5   Apply procedures to calculate the probability of mutually exclusive simple or compound events.

7-6.6   Interpret the probability of mutually exclusive simple or compound events.

7-6.7   Differentiate between experimental and theoretical probability of the same event.

7-6.8         Use the fundamental counting principle to determine the number of possible outcomes for a multistage event.

 

          Beginning with grade two, students were asked to predict on the basis of data whether events are more likely or less likely to occur

          In grades 3-5, students learned how to quantify the likelihood of single-stage events as likely, unlikely, certain, impossible, or equally likely.  Third grade students understood when the probability of an event would be 0 or 1.  Fourth grade students analyzed possible outcomes for a simple event.

          Sixth grade students used theoretical probability to determine the sample space and probability for one- and two-stage events such as tree diagrams, models, lists, charts, and pictures and applied procedures to calculate the probability of complementary events.

          Since sixth students calculated the probability of complementary events, they were introduced to events that were mutually exclusive.  In seventh grade students will use prior knowledge to calculate the probability of mutually exclusive simple or compound events.  Mutually exclusive events can not happen at the same time (the events have no common elements in the sample space).  For example, when rolling 2 dice, getting an even sum and getting an odd sum are mutually exclusive.  When two outcomes are mutually exclusive, student can use this formula to find the probability; Probability (A or B) = Probability (A) + Probability (B).  Students should be given ample opportunities to calculate and interpret the probability of mutually exclusive simple or compound (combination of at least two simple events) events.  Students should be able to decide whether or not given events are mutually exclusive and describe how they know.

          When finding a probability by tossing a coin 100 times or throwing free throws on the basketball court 50 times are examples of finding an experimental probability (simulation).  It is considered experimental because the probability is based on the results of an experiment rather than theoretical analysis.  The experimental probability of an event is the ratio of number of observed occurrences of the event to the total number of trials:

 

Experimental Probability = Number of favorable outcomes

                                               Number of Trials

 

The more trials the more confident one is that the experimental (simulation) probability is close to the actual probability.  There are advantages to an experimental approach:   

 

An experimental (simulation) approach should be used in the classroom whenever possible. (Elementary and middle School Mathematics, Teaching Developmentally 3rd Edition, John A. Van de Walle, p410) 

 

(Adapted from Navigating Through Probability, Grades 6 – 8, NCTM, p.73 – 76)

          Students need to know that a simulation is a procedure for answering questions about a real problem by conducting an experiment that closely resembles the real situation.  As students think about and discuss the real problem and the factors that make the real problem more complex, remind them a simulation is an approximation of the real problem and that with a simulation some of the factors are eliminated such as any possible danger, complexity of the problem, or length of time necessary to solve the problem.

          Have students examine when a simulation’s results appear to be the closest to theoretical probability.  (Results of a simulation are more precise when it is run repeatedly.  Remember that probability deals more with long term trends than with outcomes of individual events.)

          When designing a simulation, students need to keep the following 5 tasks in mind:

  1. Identify the essential components and assumptions of the problems.

                                                              i.      Problem stated clearly.

  1. Select a random device for the essential components.

                                                              i.      Device must be able to generate chance outcomes with probabilities that match those of the problem.

  1. Define the trial.

                                                              i.      One trial must be clearly specified.  A trial may require several flips, spins, or draws for example.

  1. Conduct a large number of trials and record the information.

                                                              i.      There is no magic number of trials.  Rule of thumb, 25 to 30 trials are appropriate for most of the problems for middle school students.

  1. Use the data to draw conclusions.

                                                              i.      Data must be summarized in some meaningful way such as computing an average, making a graph, or noting trends in data.

          A simulation is a model of a problem that occurs in real life.  The model is designed so that it reflects the probabilities of the real situation.

Students use simulations to conduct experiments whose results represent experimental probability.

Discuss how results of simulations can be deceptive. (Not enough trials, very few attempts, etc.)

          Theoretical probability is the ratio of the number of favorable outcomes to the total number of possible outcomes.  Theoretical probability is based on a logical analysis of the experiment.  For example, the theoretical probability of a coin landing on tails is ½. 

 

P(heads) = number of sides with heads = ½

  number of sides

 

To use theoretical probability to predict how many times a coin would land on heads if tossed 30 times:

          Multiply its probability by the number of attempts.

 

                   P(heads) x number of tosses

                        ½       x    30    = 15

          You would toss heads 15 times out of 30.

 

          Students should discuss the differences between the probability of an event found through experimental/simulation and the theoretical probability of that same event

 

Theoretical Probability = Number of favorable outcomes

Number of possible outcomes

 

Experimental Probability = Number of favorable outcomes

Number of Trials

 

          Students should understand that when all outcomes of an experiment are equally likely, the theoretical probability of an event is the fraction of the outcomes in which the events occurs   Students should use theoretical probability and proportions to make approximate predictions.

          The Fundamental Counting Principle states that if a first event can occur in a ways, and a second event can occur in b ways, then the two events can occur together in a times b ways.  For example the school shirt comes in three colors and in short or long sleeves.  How many choices of shirts are there?  So, 3 x 2 = 6 choices. 

 

Connections To:

 

Other Seventh Grade Indicators

7-3.6         Represent proportional relationships with graphs, tables, and

equations.

 

Teacher Notes: In the middle grades, the groundwork is laid to help students know when events are or are not mutually exclusive.  This is important for the study of more complex situations in grades 9 – 12. 

          When a probability experiment has very few attempts or outcomes, the result can be deceptive. Computer simulations may help students avoid or overcome erroneous probabilistic thinking.  Simulations afford students access to relatively large samples that can be generated quickly and modified easily.” (NCTM 2000, p254)  Using large samples, the distribution is more likely to be close to the actual distribution.  When simulations are used, you will need to help students understand what the simulation data represent and how they relate to the problem situation.