K-8 Mathematics Standards Content Training Data Displays Grades 3-8

Debi DePaul, Math Coordinator, ESD 123 2

 Develop participants’ skills and conceptual understanding related to data displays.  Experience activities that help students develop these skills and concepts.  Determine the skills and conceptual understanding required for data displays at your grade level based on the state standards. 3

 Allow ourselves and others to be seen as learners.  Monitor own airtime and sidebar conversations.  Allow for opportunities for equitable sharing.  Presume positive intentions.  Be respectful when giving and receiving opinions, ideas and approaches. 4

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For all students to learn significant mathematics, content should be taught and assessed in meaningful situations. 6

 With a partner match the name of the data display with it’s definition and graph. 7

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 Conceptual Understanding ◦ Making sense of mathematics  Procedural Proficiency ◦ Skills, facts, and procedures  Mathematical Processes ◦ Using mathematics to reason and think 9

 At each grade level: ◦ 3-4 Core Content areas ◦ Additional Key Content ◦ Core Processes (reasoning, problem solving, communication)  For each of these: ◦ Overview paragraph ◦ Performance Expectation ◦ Comments/Examples 10

 Use either your Standards Document or Strands Document to find all K-8 Performance Expectations that have students make or use some sort of data display or representation (e.g., graphs, tables, etc.)  Note the expectations for the grade level above and below yours and reflect on the following questions: 11 What should your students already know? What do you need to teach this year? What do they need to know for next year? What should your students already know? What do you need to teach this year? What do they need to know for next year?

A diagram that uses pictures or symbols to compare data. Each picture may represent one or more data value. 3.5.E 4.4.H 6.6.G 3.5.E 4.4.H 6.6.G

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 A title that tells WHO and WHAT the graph is about  A key for the picture  All the data  A label for the horizontal axis and vertical axis  A label for each category  A picture to show the number for each category

A table that shows the category names used to sort a data set, a tally of the data set by category, and a number for the total number of occurrences in each category E 4.4.H 6.6.G 3.5.E 4.4.H 6.6.G

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 A title that tells WHO and WHAT the graph is about  All the data  Tallies and/or numerals that show the distribution of data  A label for each column 17

A plot that shows the frequency of data on a number line. 3.5.E 4.4.H 6.6.G 8.3.B 3.5.E 4.4.H 6.6.G 8.3.B

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 A title that tells WHO and WHAT the graph is about  A scale that fits the data  A label for the axis  A mark (e.g. “+”, “x” or “o”) for each data point  An accurate display of data

A graph that uses the length of solid bars to represent numbers and compare data. 3.5.E 4.4.H 6.6.G 3.5.E 4.4.H 6.6.G

Categories Numbers as Categories 22

 A title that tells WHO and WHAT the graph is about  A scale that fits the data  All the data  A label for the horizontal axis and vertical axis  A label for each category  Separated bars (not symbols) to show the number for each category

A graph that uses one or more lines to show changes in data when there is a numerical value associated with equally spaced points along a continuous number scale. 5.5.C 6.6.G 7.6.G 8.5.G 5.5.C 6.6.G 7.6.G 8.5.G

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 A title that tells WHO and WHAT the graph is about  Appropriate and consistent scales on the axes that fit the data  Labels for the x-axis and y-axis  An accurate display of data

 Easy to read  Has minimum and maximum close to the minimum and maximum of the data (e.g., if data lies from 6-28 the appropriate scale would be from 0-30 by 2’s not by 5’s or 10’s)  Spans over half the space provided

 Same spacing that is established with the first two numbers.  Must have a zero (except horizontal axis of histogram) but it does not have to be written.

A plot that organizes data from least to greatest using the digits of the greatest place value to group data. 7.4.D 7.6.G 8.3.B 8.5.G 7.4.D 7.6.G 8.3.B 8.5.G

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 A title that tells WHO and WHAT the graph is about  A consistent stem  A key that explains how to read the stem and leaves  An accurate display of data

A bar graph that uses two scales, one for equally spaced intervals (replaces categories) and one for frequencies. 7.4.D 7.6.G 8.3.B 8.5.G 7.4.D 7.6.G 8.3.B 8.5.G

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 A title that tells WHO and WHAT the graph is about  An appropriate scale (vertical & horizontal may be different).  A consistent scale.  A label for each interval.  Labels for horizontal and vertical axes.  Contiguous bars to fit data (bars must start at the vertical axis, don’t need to start at 0).

A graph that uses a divided circle to show pictorially how a total amount is divided. 7.4.D 7.6.G 8.3.B 8.5.G 7.4.D 7.6.G 8.3.B 8.5.G

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 A title that tells WHO and WHAT the graph is about  All of the categories  Labels for sections of the circle with the category and number or percentage (or fractions)  Appropriately sized sections

A plot that displays the median, quartiles, and outliers of a set of data. Also called a box-plot. 8.3.B 8.5.G 8.3.B 8.5.G

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 A title that tells WHO and WHAT the graph is about  An appropriate and consistent scale that fits the data  A label for the axis  Labels for maximum, minimum, lower quartile, median and upper quartile (optional)  An accurate display of data

A graph consisting of points, one for each item being measured. The two coordinates of a point represent the measures of two attributes of each item. 8.3.C 8.5.G 8.3.C 8.5.G

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 A title that tells WHO and WHAT the graph is about  Appropriate and consistent horizontal and vertical scales that fit the data  Time as the horizontal axis when time is the controlled variable  Labels for the horizontal and vertical axes  Data point plotted with a circle  An accurate display of data

A diagram that shows grouping of people or objects in overlapping categories G 8.5.G 8.3.G 8.5.G

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 A title that tells WHO and WHAT the graph is about  A label for each circle or ring to show all the possible groupings of categories  Circles or rings may overlap, or intersect, to show objects that can be classified in more than one way. 46

 Data are gathered, organized and displayed in order to answer questions we have about our world.  Just because students construct graphs it doesn’t mean they know how to interpret graphs or vice versa. Each of these concepts have to be carefully developed.

 Circle Graph (“Candy Bar Graph” extension)  Histogram  Box-and-Whisker Plot

 Select a miniature candy bar. You may eat it! Keep the wrapper, you will need it.  Form a circle by standing next to people who have chosen the same candy bar as you have.  Use a hundredths circle to determine the size of the sector for each category.  Recreate the circle graph on paper.  Extension: Tape wrapper to bar graph. 49

 Determine your height in inches.  Line up behind the number that matches your height.  Notice: ◦ What happens when the “bin” size is changed? 50

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 Line up in order of your height in inches (side-by-side).  Determine the 5-Number Summary for the data.  Have those who represent those numbers step forward and hold sign.  Place onto a number line (this is REALLY important).  Use rope to mark off quartiles. 52

 Determine the 5-Number Summary of the data ◦ Minimum, Lower Quartile, Median, Upper Quartile, Maximum 53

 Finding the median ◦ Order the numbers ◦ Odd number of data values – one middle ◦ Even number of data values – two middles that need to be averaged  Finding the Lower Quartile ◦ Find the median of the bottom half of the data values ◦ DO NOT include the median in the half  Finding the Upper Quartile ◦ Repeat the same steps for the upper half of the data values 54

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30 minutes

 Stem-and-Leaf Plot  Histogram  Box-and-Whisker Plot 58

 Are a combination of a table and a graph.  Using grid paper can be VERY helpful to students.  Steps: 1.First make the stem. 2.Write in the leaves directly from the data. 3.Rewrite each leaf in numerical order. 59

60 Step 1 make stem Step 2 make leaves Step 3 rewrite leaves

61  Construct a double stem-and-leaf plot of the data  Notice: ◦ How does the ‘shape’ of the data look? Is it different for men than women?

 Sometimes it’s helpful to construct a stem-and-leaf plot of the data first. When turned on their sides they resemble a histogram. 62

 Work from your stem plot to create a histogram for either the men’s or the women’s data.  Determine the bin width that you will use. (Don’t chose a bin width of one. )  Compare with a neighbor who has a different bin width than yours. Make sure you used the same data.  Compare with a neighbor who has the same bin width but used different data.  Notice: ◦ How does the ‘shape’ of the data change when the bin widths and data are different? 63

 Determine the 5-Number Summary of the data ◦ Minimum, Lower Quartile, Median, Upper Quartile, Maximum 64

 Finding the median ◦ Order the numbers ◦ Odd number of data values – one middle ◦ Even number of data values – two middles that need to be averaged  Finding the Lower Quartile ◦ Find the median of the bottom half of the data values ◦ DO NOT include the median in the half  Finding the Upper Quartile ◦ Repeat the same steps for the upper half of the data values 65

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 With a partner construct a data set with eight data values for the graph.  Exchange your data set with another pair’s data set. Construct the graph to see how close you are to the original graph.  Make sure you know whether to make a histogram or a box-and-whisker plot. 67

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 Match the graphs that represents the same data. ◦ Elementary – pink set ◦ Secondary – yellow set 70

 Bar and Circle Graph Match ◦ a – iii ◦ b – i ◦ c – iv ◦ d – ii  Histogram and Box Plot Match ◦ a – 2 ◦ b – 3 ◦ c – 4 ◦ d – 1 71

 Reminder: The skill of constructing graphs doesn’t necessary transfer to the skill of interpreting graphs.  Ask LOTS of questions about the graphs that students see and construct. The questions should generate answers that require students to: ◦ Determine facts ◦ Make inferences  3 Facts and 1 Inference 72

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 Teaching Student-Centered Mathematics, John A. Van de Walle ◦  OSPI 3-8 Item Writing Guidelines, Sept ◦  Navigating through Data Analysis 6-8, NCTM ◦  Discovering Algebra, Key Curriculum Press ◦  Activity-Based Statistics, Scheaffer, Gnanadesikan, Watkins, Witmer