# Representing Data Using Frequency Graphs

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1 Lesson 25 Mathematics Assessment Project Formative Assessment Lesson Materials Representing Data Using Graphs MARS Shell Center University of Nottingham & UC Berkeley Alpha Version If you encounter errors or other issues in this version, please send details to the MAP team c/o 11 MARS University of Nottingham

2 Representing Data Using Graphs Teacher Guide Alpha Version August Representing Data Using Graphs Mathematical goals This lesson unit is intended to help you assess how well students are able to: Interpret data on a single count or measurement variable using frequency graphs and box plots. Understand how a large number of data points may result in the graph being approximated by a continuous distribution. Common Core State Standards This lesson involves mathematical content in the standards from across the grades, with emphasis on: S-ID Summarize, represent, and interpret data on a single count or measurement variable. This lesson involves a range of mathematical practices, with emphasis on: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. Introduction The unit is structured in the following way: Before the lesson, students complete an assessment task individually that is designed to reveal their current understanding. A whole class introduction provides students with guidance on how to work through the task. Students work in pairs or threes on a collaborative discussion task, matching written interpretations and graphs as they begin to link these two representations. After a whole-class plenary discussion, students revise their solutions to the assessment task. This lesson can be taught in conjunction with the lesson Representing Data Using Box Plots or independently. Materials required Each student will need a copy of the Assessment task: Cell Phones 1, a mini-whiteboard, a pen and an eraser. Each small group of students will need the following resources: Card Set: Graphs, and Card Set: Interpretations. Cut-up both card sets before the lesson. One large sheet of paper and a glue stick for making posters. There are also some projector resources to help with whole class discussions. Time needed Approximately 15 minutes before the lesson, a one hour lesson and a follow-up homework. Exact timings will depend on the needs of the class MARS University of Nottingham 1

4 Representing Data Using Graphs Teacher Guide Alpha Version August 11 Common issues Limited or incorrect description of the graph For example: Student mentions just the lowest and highest scores. Student assumes the median value is equal to the mode or student assumes the median value is exactly half way between the maximum and minimum value Student does not contextualize the data For example: The student states the mode is 23, maximum and minimum values are 5 and 44 respectively, but does not refer to the context. Suggested questions and prompts What can you tell me about other mathematical measures? What can you tell me about the monthly spending of most of the students? What proportion of students spends less than the median value? How can you show this on your graph? [The value at the point where the area under the curve is split in half.] What does the median represent? What do these figures represent? MARS University of Nottingham 3

5 Representing Data Using Graphs Teacher Guide Alpha Version August Suggested lesson outline Throughout this lesson, encourage students to use the correct mathematical language. Whole class introduction: Interpreting Graphs ( minutes) Give each student a mini-whiteboard, an eraser and a pen. Show Slide 1 of the projector resource: Discrete and Continuous Representations Bar Graph Line Graph Explain to students that the bar chart represents the scores of students in a test for which the maximum score was. When there are many bars close together, the data can be represented as a continuous line, that is a frequency graph. This makes it a little easier to read off the values. Now show Slide 2 of the projector resource: Score Alpha May MARS, University of Nottingham Projector resources: 1 Line Graph Score Alpha May MARS, University of Nottingham Projector resources: 2 Students are to write on their mini-whiteboards all the information they can derive from the graph. After a couple of minutes ask students to show you their answers. Ask one or two students to justify their answers. Even if their explanations are incorrect or only partially correct, write them next to the graph. Encourage students to challenge these interpretations and then replace them with new ones. You may want to ask students a selection of the following questions in turn: What can you say about the test? Did the students find it difficult or easy? How can you tell? What is the range of scores? What is the range of scores for most students? What is the mode score? Do you think the median score is higher or lower than the mode score? How can you tell? Do you think the mean score is higher or lower than the mod? How can you tell? Roughly how many candidates took the test? [About 1,.] How do you know? 1 11 MARS University of Nottingham 4

9 Representing Data Using Graphs Teacher Guide Alpha Version August 11 Solutions Assessment task: Cell phones 1 1. Here is a possible description of the graph; however, the measures are all approximate: The graph shows the students spent between \$6 and \$44 a month, a range of \$38. The mode amount spent is about \$23. The median is about \$18. The majority of students spend between \$23 and \$24 a month. Very few students spend more than \$3 a month. Collaborative activity: Graph Interpretation Graph Interpretation A 8 E 3 B 1 F 5 C 2 G 6 D 7 H 4 11 MARS University of Nottingham 8

10 Amount spent (\$) Representing Data Using Graphs 16 Student 7 Materials Alpha Version June Cell 35 Phones Here is a frequency graph that shows the monthly 1 spending of a group of students on their cell phones: (!" '!"!"#\$%#&'() &!" %!" \$!" #!" Amount spent (\$)!" 6 8!" &" 2)" #\$" #(" \$!" \$&" \$)" %\$" %(" &!" *+,%&-)./#&-)12) The graph shows the students 24 spent between 32 \$5 and \$ a month, a range of \$35. The modal 28 8 amount spent is about \$31. The median is about \$29. Most students spent over \$25 each month Here is another frequency graph that shows the monthly spending of a second group of students on 44 1 their cell phones:!"#\$%#&'() '!" &#" &!" %#" %!" \$#" \$!" #"!"!" '" (" \$%" \$)" %!" %'" %(" &%" &)" '!" ''" *+,%&-)./#&-)12) Describe what this graph shows: &#" &!"!"#\$%#&'() %#" %!" \$#" \$!" #"!"!" '" 11 MARS University of Nottingham (" \$%" \$)" %!" %'" %(" &%" &)" '!" ''" S-1 *+,%&-)./#&-)12)

11 Representing Data Using Graphs Student Materials Alpha Version June 11 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 11 MARS University of Nottingham S-2

12 Representing Data Using Graphs Student Materials Alpha Version June 11 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 sat the test. One group had studied the work for two years. The other group had only just begun. 5. In this test, the mean score was greater than the modal score. 4. This test resulted in a huge range of scores. Everyone could do something but nobody could do it all. 6. In this test, the mean score was smaller than the modal score. 7. In this test, the median and the modal scores were the same. There was a very big range of scores. 8. This test was much too difficult for most people. 11 MARS University of Nottingham S-3

13 Discrete and Continuous Representations Bar Graph Line Graph Score Alpha Version August MARS, University of Nottingham Projector Resources: 1

14 Line Graph Score Alpha Version August MARS, University of Nottingham Projector Resources: 2

15 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 cards or the poster. You may find some of Word cards match two graphs. This problem will be resolved as you match more cards. Be prepared to change your mind about the matches. Alpha Version August MARS, University of Nottingham Projector Resources: 3

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