ࡱ>  %bjbjVV ;x<<:8?<{hR"R"R"qhshshshshshshjxmhshR""!0R"R"R"shh"%%%R"qh%R"qh%%c b1hp{f$Xe:]hh<h#fm$mt1hm1h,R"R"%R"R"R"R"R"shsh%R"R"R"hR"R"R"R"mR"R"R"R"R"R"R"R"R" : Title: Bear Hugs 1 Achievement Objectives: S5-1B Considering sources of variation S5-1D Using multiple displays, and re-categorising data to find patterns, variations, relationships, and trends S5-1E Comparing sample distributions visually, using measures of centre, spread, and proportion Key Competencies: Thinking: Investigate, predict/envision outcomes Relating to Others: Interpret statistical information Using Language, Symbols and Texts: Communicate thinking Managing Self: Seek understanding Participating and Contributing: Build on prior knowledge This activity explores the following key ideas: Focus on different statistical techniques needed to answer different types of questions (summary, relationship, comparison) Using the PPDAC cycle to investigate the question Explore and explain error/variability in measurement Resources Student work sheets For each group: Copy of the data set Graph paper CalculatorsPrior Knowledge: (or develop) Dot plots Draw box plots over median, quartile, max/min Percentages (calculation) Describing shape of distributions Majority/skew/centre descriptions of graphs Scatter plots Adapting for ICTIntroduction/Background Allow 60 minutes for this lesson. The focus of this lesson is on investigating which statistical techniques might help us answer a summary type question, in particular drawing graphs and calculating statistics. The activity leads on from the Nosey Parker activity. The same data set is used. Teachers may prefer to use data collected from their own class or use a particular group from census at school and pose a suitable summary question posed by their class. Problem What is the average arm span length of students? The goal of this lesson is for students to have more than a single figure answer to this question. They will be able to support their answer using statistical analysis. Plan Ask the class to make a prediction of the average arm span. Can they predict what the graph of these data would look like? Record their predictions in a safe place so they can be referred to later. Data Discuss where these (Nosey Parker) data came from. For the purposes of answering our question: What is the average arm span of students? the arm span measurement is all that is required. Analysis  Construct dot plot and circle approximately the middle 50% of the data. Draw box plot above. Use I notice and I wonder starters to separate discussion. Ask students to form small groups and brainstorm what they would do to answer the question. Collect ideas from the class and present on the white board. Group similar ideas together. Two main strategies should emerge: Draw graphs and calculate summary statistics. Other possible graphs: Histograms and Bar Graphs A bar graph (for discrete data) or a Histogram (for continuous data or grouped data), stem and leaf graph. These graphs may be useful when answering a Summary type question. To draw a histogram or bar graph these data need to be reorganised. Students are often reluctant to do this and may need to be guided to find suitable intervals. For these data intervals of 10 cm would be useful. Arm span (cm) Frequency 110 - 1 120 - 2 130 - 2 140 - 5 150 - 6 160 - 5 170 - 2 180 - 1  See Teacher Support Sheet for info on: discrete and continuous data grouped data bar graphs histograms Stem and leaf graph A stem and leaf graph is a useful graph as we dont need to sort the data first. We can still see each data value and also can see the overall shape of the data. A stem and leaf graph is also useful to quickly sort the data so that we can calculate the median, LQ and UQ.  Arm span (cm) 11 8 12 5 6 13 0 8 14 0 4 4 7 15 0 4 4 5 5 6 9 16 2 3 3 4 5 17 5 5 18 2 See Teacher Support Sheet for info on: Uses and advantages Reading and interpreting Constructing a stem and leaf graph Sample Statistics: mean, median, mode, range, upper quartile (UQ), lower quartile (LQ), interquartile range (IQR). These sample statistics may be useful when answering summary type questions. Discuss with the class how each value gives us information about the average arm span. Determine whether your class needs specific teaching relating to calculating these statistics. See Teacher Support Sheet for info on: Location of median (and of LQ and UQ) Common mistakes when using a stem and leaf graph Using class line up to show quartiles. Which is better: mean or median? The range is the difference between the minimum and maximum values (Range = Max value Min value). Since these are the 2 most extreme values the range gives only a crude measure of the spread of the data. The IQR is a better indicator of spread. It gives the range of the middle 50% of the data. Once the graphs are drawn and the summary statistics calculated stand back: use the wonder I notice starters to generate discussion on what the graphs show and how they help to answer our question. Conclusion Shape Common value (mode) Centre (media/mean) Middle 50% Encourage students to write a conclusion to the question posed. Model writing a conclusion on the white board. Refer to the graphs and the statistics. Comment on: the shape of the graph, the most common value or interval, the values of the mean and median (are they the same/different, why?) Encourage the students to explore the distribution of arm spans: Is there a central bump or are there two bumps (bi-modal)? Is the data reasonably symmetrical or is it skewed? What span length(s) do the middle 50% of students have? Locate the mean and median on the graph. Is this where you expected the mean and median to be? What percentage of students have arms long enough to hug the bear? How can we use the graphs to determine this? If a new student joined the class what arm span do you predict s/he might have? Whats the probability that s/he could hug the bear? See Teacher Support Sheet for info on: Distribution Skewed data Common errors in making conclusions Double check that the students have answered the question: Does the average student have arms long enough to give a bear with a circumference of 160cm a hug? Here is an example of a conclusion for these (Nosey Parker) data and question: The distribution of arm spans for this sample is approximately bell shaped. It has a tail to the left (slightly skewed to the left). Most students have an arm span length between 140 and 170cm. The mean arm span length is 152cm and the median is 155cm. They are similar but the mean is less than the median because the distribution is slightly skewed to the left. Eight out of the sample of 24 students have arm spans longer than 160cm. Based on these data it seems that approximately one third of students will be able to give the bear a bear hug. Reflection How do our findings compare with the predictions we made at the beginning of the lesson? How similar would the results for our class be? What other questions or issues arise during this investigation? e.g. Would the results be the same for our class? What about for our school? What about for the whole of New Zealand? Does age have anything to do with arm span length? To extend this activityMore lines: generate several sets of data for each question and compare conclusions between groups. E.g. did all groups conclude similar armspan for boys and girls?      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