Sixth Grade

 

Data Analysis and Probability

 

Standard 6-6:      The student will demonstrate through the mathematical

processes an understanding of the relationships within one population or sample.

 

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.

 

 

Data Collection and Representation

 

Indicator

6-6.2         Organize data in frequency tables, histograms, or stem-and-leaf plots 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)

 

                In sixth grade, students should learn to organize data in frequency tables, histograms, and stem-and-leaf plots.  This is the first time students are introduced to those forms of data representation.  As a result, students will need sufficient experiences so that they are able to make a determination as to which form of representation is appropriate for different purposes.  For example, a stem-and-leaf plot displays all data while a frequency table or histogram shows a range of data.  Therefore, the type of representation that would be most appropriate would depend on the question that needed to be answered by the data.

 

Teacher Notes:  Stem-and-leaf plots are a popular form of representation in which numeric data are plotted by using the actual numerals in the data set to form a graph.  In stem-and-leaf plots, all of the data are maintained.  Thus, it is an efficient method of ordering data and individual elements of data can be easily identified. Students need to understand the importance of putting the stems and leaves in numeric order.  If students are comparing a double stem and leaf plot make sure they know how to read the numbers correctly.  In addition, stress the importance of creating a key when creating a stem-and-leaf plot.

          A histogram is a form of a bar graph in which the categories are consecutive, equal intervals along a numeric scale.  The height or length of each bar is determined by the number of data elements falling into that particular interval and the bars are drawn without any spaces between them.

          Teachers need to help students to connect data analysis with content outside mathematics – science and social studies.  They need to guide students to the understanding that data analysis is a process that helps make sense of a situation.

 

Data Analysis

 

Indicators

6-6.3    Analyze which measure of central tendency (mean, median, or mode) is the most appropriate for a given purpose.

6-6.1    Predict the characteristics of one population based on the analysis of sample 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 the measures of central tendency as well as interpreted the meaning and applications of the measures. Sixth grade students should now analyze which measure of central tendency is the most appropriate for a given situation.

          Students are often just told how to find the measures of central tendency and directed when to use each one.  Students should be given different situations and determine which of these measures best communicate information in the given situation.  Students need to look at each measure and determine how they compare to each other.  Students should also be able to determine which measure best represents the data with the respect to the context in which it is presented.  This indictor lends itself for students to apply reasoning skills to determine which measure or measures best represent the data as well as communication skills to explain and defend the reasoning used.

          In third grade students analyzed dot plots and bar graphs to make predictions about populations. Now in sixth grade students will use their knowledge about the shape of data to make predictions about the characteristics of a population. It is sometimes difficult to predict characteristics of one population based on the analysis of sample data by simply looking at the numbers.  By drawing a graph or picture of the data, this allows students to see immediately how close the data sets are. On the other hand, simply knowing the measures of center can aid in making predictions about the characteristics of a population. For example, if the question is, “Based on the data what predictions can you make about the age of the people at the movie?  The mode is 22 and the range is 10.” Students might predict that attendees were under the age of 33 because the range is 10 and the mode is 22. For examples like this, all predictions should be deemed reasonable or unreasonable by viewing the actual data.

 

Teacher Notes:  Sets of data have really different looks. 

 

Probability

 

Indicators

6-6.4    Use theoretical probability to determine the sample space and probability for one- and two-stage events such as tree diagrams, models, lists, charts, and pictures.

6-6.5    Apply procedures to calculate the probability of complementary events.

 

          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 fourth grade students analyzed the possible outcomes for a simple event. Fifth grade was the first time students were formally introduced to the concept of probability by representing the probability of a single-stage event in words and fractions. Now in sixth grade students relate the terminology “sample space” to the terminology “possible outcomes” they have used in the past. Also, learning experiences now include the use of tree diagrams, models, lists, charts, and pictures to determine the sample space for one- and two-stage events. It should be noted that sixth grade is the first time students have been introduced to two-stage events.

          Also, sixth grade us the first time students have been introduced to the concept of complementary events – if two events are complementary the sum of their probabilities is 1. (An example – You have a spinner with odd and even numbers. The probability of spinning an odd or an even number are complementary events because all numbers on the spinner are either odd or even. There are no other type numbers on the spinner. Those are two mutually exclusive events. So in this example students should calculate the probability of spinning each even number and each odd number and the sum of those probabilities should equal one.)

 

Teacher Notes:  In sixth grade, teachers should establish a problem-solving approach to probability.  Students should develop probability-based thinking by performing actual experiments, recording and discussing the results and using the results as evidence for drawing conclusions and making decisions. Students should create the sample space (the set of all possible outcomes) for one-and two-stage events and represent them in the form of a list, chart, picture, model, or tree diagram.  Such activities help students understand relationships among the available data and enable students to make decisions about what form of representation is best for the situation.

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

P(heads) = number of sides with tails =

number of sides

 

          In fifth grade students explained why the sum of the probabilities of the outcomes of an experiment must equal one and represented the probability of a single-stage event in words and fractions (ratios).  In sixth grade that concept is extended and students should be able to identify and describe complementary events.  Since these events cannot happen at the same time, they are mutually exclusive.  They are the only possible events that can occur so they are called complementary events.  They encompass the entire sample space.  The sum of the probabilities of all the outcomes in a sample space is 1; therefore, the sum of the complementary events is 1.  It can also be stated that if the two events are complementary, the sum of their probabilities is 1.  For example “sum = 5” and “sum ≠ 5” are complementary events for a roll of two dice.

          Students should understand and use commonly accepted appropriate terms in their discussions.

          Many students have misconceptions about the outcomes of real events in life, basing predictions on what they believe should happen, rather than on real data. Studying probability will help students develop critical thinking skills and interpret the probability of events that happen in their lives.

            Helpful terminology -

·         Tree diagrams are diagrams used to show all of the possible outcomes (combinations) in a sample space.