Representing Data with Frequency Graphs

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1 CONCEPT DEVELOPMENT Mathematics Assessment Project CLASSROOM CHALLENGES A Formative Assessment Lesson Representing Data with Graphs Mathematics Assessment Resource Service University of Nottingham & UC Berkeley For more details, visit: MARS, Shell Center, University of Nottingham May be reproduced, unmodified, for non-commercial purposes under the Creative Commons license detailed at - all other rights reserved

2 Representing Data with Graphs MATHEMATICAL GOALS This lesson unit is intended to help you assess how well students: Are able to use frequency graphs to identify a range of measures and make sense of this data in a real-world context. Understand that a large number of data points allow a frequency graph to be approximated by a continuous distribution. COMMON CORE STATE STANDARDS This lesson relates to the following Standards for Mathematical Content in the Common Core State Standards for Mathematics: S-ID: Summarize, represent, and interpret data on a single count or measurement variable. This lesson also relates to all the Standards for Mathematical Practice in the Common Core State Standards for Mathematics, with a particular emphasis on Practices 1, 2, and 7: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. INTRODUCTION The unit is structured in the following way: Before the lesson, students work alone to complete an assessment task designed to reveal their current understanding. After a whole-class introduction, students work in pairs or threes on a collaborative discussion task, matching written interpretations and graphs as they begin to link these two representations. There is a whole-class discussion to end the lesson. In a follow-up lesson, students work alone on a similar task to the assessment task. This lesson can be taught in conjunction with the lesson Representing Data with Box Plots or independently. MATERIALS REQUIRED Each student will need a copy of Cell Phones 1, Cell Phones 1 (revisited), and Drawing Graphs, a mini-whiteboard, pen, and eraser. Each small group of students will need a cut-up copy of Card Set: Graphs and Card Set: Interpretations, a large sheet of paper for making a poster (large enough to accommodate three different sets of cards), and a glue stick. TIME NEEDED Approximately 2 minutes before the lesson, a 9-minute lesson (or two 5-minute lessons), and 2 minutes in a follow-up lesson. Exact timings will depend on the needs of the class. Teacher guide Representing Data with Graphs T-1

4 Common issues Provides an incorrect range For example: The student writes 5-4. Or: The student assumes the range is the difference between the maximum and zero. Uses the frequency readings for the measures of central tendency For example: The student writes that the mode is 48. Does not understand the term frequency For example: The student assumes frequency represents a percentage. Does not understand what median represents or how to use the graph to figure out the median For example: The student assumes the median value is equal to the mode. Or: The student assumes the median value is exactly half way between the maximum and minimum value. Does not contextualize the data For example: The student states the mode is 31, the minimum and maximum values are 5 and 4 respectively, but does not refer to the context. Suggested questions and prompts What is the definition of the term range? What does the vertical axis represent? What measure is the question asking for? What does the term frequency mean? What does the axis marked frequency represent on the graph? What does the median represent? Can the mode and median values be different? How can you show the median value on the graph? [Draw a vertical line to divide in half the area under the graph. The median amount is the value at the point this line intersects the x-axis.] Roughly what proportion of students spend less than the median value? What do these figures represent? Complete this sentence 5% of students spend less than \$ _ each month. Explain how you arrived at the figure. Teacher guide Representing Data with Graphs T-3

5 SUGGESTED LESSON OUTLINE Throughout this lesson, encourage students to use the correct mathematical language to not just provide the figure for a measure, but to place the figure in the context. For example, rather than The median is 2, encourage students to say The median score is 2 out of 1. Whole-class interactive introduction: Drawing and Interpreting Graphs (2 minutes) Give each student a copy of the Drawing Graphs sheet. Maximize participation in the introduction by asking all students to show you their graphs once sketched. This introduction will provide students with a model of how they should justify their matching of cards in the collaborative activity. It will also help students understand how to identify a range of measures from a graph and in particular the median value. Often students think that the median should be the middle score of the range of scores. The first task in this introduction helps demonstrate that this is not always the case. Display Slide P-1 of the projector resource: Bar Charts Eleven students take a test. The test is out of 1. All students scored more than 5. At least one student scored each mark between 6 and 1. The mode is 9 out of 1. Sketch on the Drawing Graphs handout two bar charts that could represent the results of the test. On the first graph make sure the median is equal to the mode. On the second graph make sure the median is different from the mode. Projector Resources Representing Data with Graphs P-1 Sketch on your sheet two bar charts that could represent the test results for the eleven students. On the first make sure the median is equal to the mode. On the second the median should be different from the mode. If students struggle, encourage them to discuss the task with a neighbor. It may help if they write down the values of the eleven scores. After a few minutes, ask students to show you their bar charts and select two or three students with different graphs to justify them to the class. Ask the rest of the class if they agree with the explanations. You may want to use Slide P-2 of the projector resource, Mode and Median, to support the discussion. How can you check that eleven students took the test? How many students achieved the median score or less? How do you know? What is meant by the statement the median is the middle score? [The middle score of a sorted list of scores.] Could the median score ever be equal to the minimum or maximum score? [Yes, but the mode would not be 9 out of 1.] How do you know? Teacher guide Representing Data with Graphs T-4

6 Show Slide P-3 of the projector resource: Discrete Representation Explain to students that the bar chart represents the scores of students in a test in which the maximum Projector Resources Representing Data with Graphs P-3 score was 1. Ask the following questions in turn: Did anyone achieve the maximum score of 1? [No.] How can you tell? What can you say about the test? Did students find it difficult or easy? How can you tell? How could you work out how many students took the test? [Add up the total heights of the bars.] Show Slide P-4 of the projector resource: Continuous Representation of Discrete Data Bar Chart 2!"#\$%#&'(),-")."-/) 1 Projector Resources Line Graph &!" When there are many bars close together the data can be represented as a continuous line, that is, a frequency graph 1 %!". This makes it a little easier to read off the values. Display Slide P-5 of the projector resource:!"#\$%#&'() (!" '!" \$!" #!" Score Representing Data with Graphs!" "!"""""""""#!""""""""\$!"""""""%!""""""&!"""""""'!""""""""(!""""""+!""""""""*!"""""""")!""""#!!" #" #)" \$*" %+" &(" ''" (&" +%" *\$" )#" #!!" Continuous *'+"#) Representation P-4!"#\$%#&'()12&#)."-/)!"#\$%#&'() (!" '!" &!" %!" \$!" #!"!" "!"""""""""#!""""""""\$!"""""""%!""""""&!"""""""'!""""""""(!""""""+!""""""""*!"""""""")!""""#!!" #!!" *'+"#) 1 Projector Resources Representing Data Using Graphs P-5 Strictly speaking, the vertical axis on the frequency line graph should be labeled frequency density rather than frequency. As each bar in the bar chart represents a single score and the data is ungrouped, the frequency density is equal to the frequency and so the label of frequency has been used. In this case, the area under the graph represents the total number of students who took the test. Teacher guide Representing Data with Graphs T-5

11 SOLUTIONS Assessment task: Cell phones 1 All answers are approximate. The range is about \$35 a month. I know this from the graph because the range represents the difference between the maximum (\$4 a month) and minimum (\$5 a month) amount spent each month. The mode is about \$31 a month. I know this from the graph because the maximum frequency is about \$31 a month. The median is about \$29 a month. I know this from the graph because if I drew a vertical line from this point on the x-axis it would divide in half the area under the graph. This means about half the students spent less than \$29 a month and half spent more than \$29 a month. Most students spend over \$24 a month. I know this from the graph because the area under the graph up to \$24 is a lot less than the area under the graph between \$24 and \$4. Collaborative activity: A Scor e C 2. This test was much too difficult for most people. B Sco re D 7. This was the sort of test where you could either do everything or you couldn t get started Score This test did not sort out the stronger students from the weaker ones. They all got similar scores Scor e In this test, the median and the mode scores were the same. There was a very big range of scores. E 3. F Scor e Two groups of students took the test. One group had studied the work for two years. The other group had only just begun Score In this test, the median score was greater than the mode score. G 6. H Score In this test, the median score was smaller than the mode score Scor e This test resulted in a huge range of scores. Everyone could do something but nobody could do it all. Teacher guide Representing Data with Graphs T-1

12 Assessment task: Cell phones 1 (revisited) All answers are approximate. The range is about \$38 a month. I know this from the graph because the range represents the difference between the maximum (\$44 a month) and minimum (\$6 a month) amount spent. The mode is about \$23 a month. I know this from the graph because the maximum frequency is about \$23 a month. The median is about \$18 a month. I know this from the graph because if I drew a vertical line from this point on the x-axis it would divide in half the area under the graph. This means about half the students spent less than \$18 a month and half spent more than \$18. Very few students spend over \$3 a month. I know this from the graph because the area under the graph from \$3 to \$44 is a lot less than the area under the graph between \$6 and \$3. Teacher guide Representing Data with Graphs T-11

13 Cell Phones Here is a frequency graph that 4 shows the monthly 1 spending of a group of students on their cell phones: (!" '!"!"#\$%#&'() &!" %!" \$!" #!"!"!" &" )" #\$" #(" \$!" \$&" \$)" %\$" %(" &!" *+,%&-)./#&-)12) The graph shows: A range of spending of about I know this from the graph because The mode is about I know this from the graph because The median is about I know this from the graph because Most students spend over I know this from the graph because Student materials Representing Data with Graphs S MARS, Shell Center, University of Nottingham

14 Drawing Graphs &\$%'(%)"* &\$%'(%)"*!!!* ) ( ' & % \$ # "! *!!!* ) ( ' & % \$ # "! *! " # \$ % & ' ( )!*!"#\$%! " # \$ % & ' ( )!*!"#\$% Student materials Representing Data with Graphs S MARS, Shell Center, University of Nottingham

15 Card Set: Graphs Graph A Scor e Graph B Sco re Graph C Score Graph D Scor e Graph E Scor e Graph F Score Graph G Score Graph H Scor e Student materials Representing Data with Graphs S MARS, Shell Center, University of Nottingham

16 Card Set: Interpretations This was the sort of test where you could either do everything or you couldn t get started. This test did not sort out the stronger students from the weaker ones. They all got similar scores. 3. Two groups of students took the test. One group had studied the work for two years. The other group had only just begun. 4. This test resulted in a huge range of scores. Everyone could do something but nobody could do it all. 5. In this test, the median score was greater than the mode score. 6. In this test, the median score was smaller than the mode score. 7. In this test, the median and the mode scores were the same. There was a very big range of scores. 8. This test was much too difficult for most people. Student materials Representing Data with Graphs S MARS, Shell Center, University of Nottingham

17 Cell Phones 8 1 (revisited) Here is a frequency graph that 44 shows the 1 monthly spending of a group of students on their cell phones:!"#\$%#&'() '!" &#" &!" %#" %!" \$#" \$!" #"!"!" '" (" \$%" \$)" %!" %'" %(" &%" &)" '!" ''" *+,%&-)./#&-)12) The graph shows: A range of spending of &#" I know this because &!" %#" The mode is!"#\$%#&'() I know this because %!" \$#" \$!" #"!" The median is I know this because!" '" (" \$%" \$)" %!" %'" %(" &%" &)" '!" ''" *+,%&-)./#&-)12) Very few students spend over I know this because Student materials Representing Data with Graphs S MARS, Shell Center, University of Nottingham

18 Eleven students take a test. The test is out of 1. Bar Charts All students scored more than 5. At least one student scored each mark between 6 and 1. The mode is 9 out of 1. Sketch on the Drawing Graphs handout two bar charts that could represent the results of the test. On the first graph make sure the median is equal to the mode. On the second graph make sure the median is different from the mode. Projector Resources Representing Data with Graphs P-1

19 Mode and Median!!!!!*!* ) ) ( ( ' ' &\$%'(%)"* & % \$ # &\$%'(%)"* & % \$ # " "!! *! " # \$ % & ' ( )!* *! " # \$ % & ' ( )!*!"#\$%!"#\$% Projector Resources Representing Data with Graphs P-2

20 Discrete Representation Projector Resources Representing Data with Graphs P-3

21 Continuous Representation of Discrete Data Bar Chart Line Graph Score Projector Resources Representing Data with Graphs P-4

22 Continuous Representation Score Projector Resources Representing Data with Graphs P-5

23 Matching Cards 1. Take turns at matching pairs of cards that you think belong together. 2. Each time you do this, explain your thinking clearly and carefully. 3. Your partner should either explain that reasoning again in his or her own words, or challenge the reasons you gave. 4. You both need to be able to agree on and explain the placement of every card. 5. Write your reasons for the match on the poster. You may find some of the Word cards match two graphs. This problem will be resolved as you match more cards. Be prepared to change your mind about the matches. Projector Resources Representing Data with Graphs P-6

25 Representing Data Using Graphs Student Materials Beta Version February 212 Card Sets Representing Data Using Graphs Student Materials Beta Version February 212 Card Set: Graphs Card Set: Interpretations Graph A Graph B This was the sort of test where you could either do everything or you couldn t get started. This test did not sort out the stronger students from the weaker ones. They all got similar scores. Graph C Graph D 3. Two groups of students took the test. One group had studied the work for two years. The other group had only just begun. 4. This test resulted in a huge range of scores. Everyone could do something but nobody could do it all. Graph E Graph F In this test, the median score was greater than the mode score. In this test, the median score was smaller than the mode score. Graph G Graph H In this test, the median and the mode scores were the same. There was a very big range of scores. This test was much too difficult for most people. Projector Resources Representing Data with Graphs P MARS University of Nottingham S MARS University of Nottingham S-4

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