Chapter 9 Exploring Graphs, Data, Statistics and Probability

Objectives Draw and interpret eight types of graphs and analyze misleading graphs Describe how to incorporate collecting and displaying data Compute and use the statistical measures: mean, median, mode and range Calculate probabilities using appropriate rules and terminology Describe experiments and activities that can be used in the early childhood classroom to introduce children to the concepts and processes of probability

Common Core Standards CCSS.Math.Content.2.MD.D.10 Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put- together, take-apart, and compare problems 1 using information presented in a bar graph. CCSS.Math.Content.2.MD.D.10 CCSS.Math.Content.3.MD.B.3 Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step “how many more” and “how many less” problems using information presented in scaled bar graphs. For example, draw a bar graph in which each square in the bar graph might represent 5 pets. CCSS.Math.Content.3.MD.B.3

Graphs Due to their visual nature, graphs are a very important part of the mathematics curriculum and needs to begin at the very earliest grade levels The reason is most elementary textbooks include graphs Graphs and statistical analysis may be easily incorporated into other subject areas and are commonly used Think-Pair-Share Time

Which Graph (pre-assessment) Think-Pair-Share Time A table shows the favorite activities of 200 students. What type of graph would be most appropriate to show the data as parts of a whole? A table shows the kinds of pets the students have in a class. What type of graph would be most appropriate for the data? Identify the graph that is most appropriate to represent a data that denotes the amount of money spent on military over a period of time?

Foundation for Gathering Data To introduce probability vocabulary use statements such as “we’ll probably have recess this afternoon”, “it’s most likely going to be sunny today” Also can build an understanding of chance by doing fun experiments such as choosing color objects from a bag Any other ideas?

Eight Types of Graphs Think-Pair-Share Pictographs Tally Graphs Line Plots Histograms Stem-and-Leaf Plot Line Graphs Pie Charts Misleading Graphs

Eight Types of Graphs to Represent Data Pictographs:Uses pictures Tally Graphs:Tally marks Line Plots:Uses X or a similar mark Histograms:Bars are used Stem-and-Leaf Plot: The Stem shows the tens and the leaf shows the ones Line Graphs:Points joined with line segment Pie Charts:Circle chart Misleading Graphs:Misleading data from graph display

Pictographs Pre-K to 1 st grade: one- on-one correspondence 2 nd Grade a key is used

Tally Graphs Simple Displays K-2Next:Skip counting by five

Line Plots Students Keep TrackAdd to computer

Histograms Great For displaying amounts Chart positive pin-pointing for prizes Keep Track of Weather

Stem-and-Leaf Plot 4 th grade to review place value

Line Graphs Great for displaying changes overtime Also to compare data

Pie Charts To introduce percents and fractional or decimal amounts (Parts of a whole)

Misleading Graphs The Size Amount spaced on the Y axis

A table shows the favorite activities of 200 students. What type of graph would be most appropriate to show the data as parts of a whole? Circle Graph A table shows the kinds of pets the students have in a class. What type of graph would be most appropriate for the data? Bar Graph Identify the graph that is most appropriate to represent a data that denotes the amount of money spent on military over a period of time? Line Graph Displaying our smartness (post-assessment)

Collecting and Interpreting Data Another Important part of the graphing process is gathering data for a graph. It is good practice for children to learn how to collect data as well as how to graph and analyze it.

Teaching Connections Have you ever heard the statement “four out of five dentist prefer sugarless gum”? The figure four out of five is an average based on the results of a survey. In a survey situation the population is the set about which information is desired and the sample is the subset of the population that is studied. This is because it would be imposable to survey the entire population. In the above example, the population is all dentists, and the sample is the set of dentist that were surveyed. Using a sample to make a conclusion regarding a population that is larger than the sample is an excellent exercise in estimation and ratios or multiplication.

Statistical Analysis of Data Average value of a set of data: In early grade levels students can use concrete objects or drawings to determine an average. **Example of this would be; Jim has 4 marbles, Joe had 8, and Jerry has 3. If they share the marbles so that each gets the same number of marbles, how many marbles will each boy get? Solution: At the first or second grade level, students could use actual objects (or draw a picture) to find that there are 15 marbles. They could then distribute the marbles equally to three sets to find that each boy gets 5 marbles. Use this example or will allow students to gain insight that to evenly distribute or share sets of objects, to find the average of a set of given values, you must add all of the values and then divide by the total number of values. This “average” of a set of data is called the Mean. (Median is usually presented for the first time at the third or fourth grade level.) The formula for this quantity is; Sum of values in a data set X= Number of values in the data set

Statistical Analysis of Data cont. The alternate measure of the “middle value” of a data set is the Median. To find the Median, the value in the data set must be listed in increasing order. 1. Data set is odd; the median is the middle value in the list. 2. Data set is even; the median is the average of the two values in the middle positions of the list. The Mode of the set of data is the value (or values) that occur most. (Example: Determine the mode of the ages in the classroom.) If each data value occurs exactly once in a set of data, we say there is no mode. Range of a set of data is the difference between the smallest and the largest values in the set. (This definition is usually learned around the second or third grade.)

Determining Empirical and Theoretical Probability Terminology and definitions Probability: an event is the measure of how likely the event will occur. (Like flipping a coin. 50/50 chance to guess right.) Empirical Probability of an event is the probability of the event as determined from the results of experiments that are performed. Theoretical Probability of an event is the probability of the event determined by logical mathematical analysis and the consideration of all possible outcomes. **Empirical results may not always coincide with Theoretical results.** Law of Large Numbers: If an experiment is repeatedly performed, the empirical probability of any particular outcome of the experiment will approach a fixed value. Special Note: When determining theoretical probabilities, it is important that the set of all outcomes is correctly identified. Sample Space: set of all possible outcomes of an experiment.

Probability Rules In general, the following probability rules exist. 1. An event has probability of 0 if and only if it is an impossible event. 2. An event has probability 1 if and only if it is a certain event. 3. For any event E, 0 < P (E) < 1. (Between impossible and certain. This means all rules are between 0 and 1.) 4. The sum of the probabilities of all outcomes in a sample space is equal to The probability that event E will not occur is P (not E) = 1 – p (E).

Illustrating All Combinations Tree Diagram: Are useful in some probability situations.

Illustrating All Combinations Venn Diagrams: can be useful in counting problems or probability situations.

Illustrating All Combinations Graphing: Like Chad mentioned earlier due to their visual nature, graphs are a very important part of the mathematics curriculum and needs to begin at the very earliest grade levels and can be can be useful in counting problems or probability situations.

To conclude let’s take a quick moment (30 seconds) to Think-Pair-Share with a partner a way you might use any of these illustrating ideas in your own class room with a lesson.