Probability and Percentage increases and decreases

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Following our assessment task yesterday it is clear that some students need to revise certain areas of the work we have done in term 1 and 2:

Probability – Relative frequency with ten questions to complete.

Percentages – How to convert fractions and percentages to a pie chart (360 degrees) with ten questions to complete.

Percentage increases and decreases – Worked examples and five problems to solve. 

Stem and Leaf plotsWorked example and thirteen questions to solve. 

Problem #1: You have a list of 7 numbers. The average of the numbers is 9. If you take away one of the numbers, the average of the numbers is 8. What number did you take away?

Problem #2: Martin has completed five Maths tests and received an average score of 80%. What is the highest average he could have after the next test?

Problem #3: A Year 7 class was asked “How many goals did you shoot at lunchtime?”. The lowest answer was 5 and the highest answer was 20. The total of all the answers was 60. What is the smallest number of students who could have been asked?

What are the Chances?

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You will be familiar with hearing discussions about probability in terms of words, if not quantities – “It isn’t likely to rain on the washing today”, “The chances of winning the lotto are very small” “It is equally likely that the other team will win”. For the remainder of this term we are going to use numerical representations of probability, including tree diagrams, two-way tables and Venn diagrams.

Graphs and Data

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Image created in Create-A_Graph

Last week you were working on a poster showing the results of a class survey in table and graphical form. This data is just a small sample of the school and state data. When governments and businesses want to plan for the future they need to know information about the whole population – for example, where roads, schools and hospitals need to be built. This information is obtained using a census. The national census is conducted every four years, when the Australian Bureau of Statistics asks every household to complete a survey.

Some of the data obtained in the 2011 census is recorded here. Choose one of the categories that you are interested in and create a graph of the data using “Create-A-Graph”. Email your graph to me and a copy to yourself.

Year 7 Favourites

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Learning Intention: Students will understand how to collect data using a tally and create a frequency table and bar graph using the data. They will understand how to convert fractions to decimals and percentages. They will create a pie chart using this data by converting 100% to 360 degrees.

Success Criteria: Each student will produce a poster that includes a frequency table (including fractions, decimals and percentages), bar graph and pie chart of their chosen data, collected from the Year 7 Maths Survey.

  1. First collect your data in tally form.
  2. Add each category and find the total.
  3. Represent each category as a fraction.
  4. Convert to a decimal (2/25 = 16/100)
  5. Convert to a percentage 16/100 = 16%
  6. Create a bar graph using this data
  7. Remember to add SALT to your graph – Scale, Axes, Labels, Title
  8. Turn your bar chart into a pie chart (multiply percentage by 3.6 because 100% = 360 degrees)
  9. Make sure you have a key to interpret your data.
  10. Add a beautiful title and colour to present your poster.
  11. Go to Create-A-Graph and use your data to check that your graphs are correct.
  12. Print out the computer generated graphs to add to your poster.

Barbie figures and ratio

Learning Intention: Students will understand how to calculate and simplify ratios using body measurements as an example.

Success Criteria: Students will be able to correctly estimate the ratio of various body proportions, using average class data.

Do you think that “Barbie” dolls portray a realistic image of a human figure? What about “Bratz” dolls and Manga figures? What is it about these figures that make them so appealing to children? We are going to investigate the proportions of “Barbie” dolls and other figures to compare their ratios with the average year 7 student from Hawkesdale college. We now have the data that gives us some average measurements of height, head circumference, arm and leg length and chest, waist and hip measurements of the average student in our class.

  • As you can see from the figure above, “Barbie” has a head to height ratio of 4 to 28cm or 1:7.
  • Her leg to height ratio is 14cm to 28cm or 1:2.
  • Her neck to height ratio is 2cm to 28cm or 1:14.
  • Her head circumference is larger than her waist circumference.
  • Her foot length to leg length is 2cm to 14cm, so, 1:7.

Calculate the same ratios for the average Year 7 student, using our data. What can you conclude?

Bill Genereux has written a terrific post about the proportions of superheroes, from a book that teaches people how to draw comic book heroes. Malyn Mawby has also written a great post about using Da Vinci’s “Vitruvian Man” to learn about ratio and proportion.

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Our survey results

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Learning Intention: Students will understand the meaning of mean, median, mode and range. They will understand how to calculate an average from given data and be able to work out the range of that data. They will then use the data to explore and compare ratios.

Success Criteria: Students will correctly calculate the average and range of the given data and use the data to make some comparisons of ratio.

Statistical values, such as mean, median, mode and range, help us to determine trends in the data. They are used in a variety of ways in business, education and government organisations. For example,  a company might want to find out about which customers spend the most money in their shops, so they can target their advertising better. Governments might need to find out about the age groups in their community, so they can distribute funding to health care, aged care, kindergartens, schools, universities, hospitals and other services.

  • The mean (or average) of a set of numbers is calculated by adding all the numbers together and dividing by the number of values. It will always be somewhere between the smallest and largest value. What is the average height of students in Year 7 at Hawkesdale P12 College? Is there a height difference between males and females?
  • The mode is the most frequently represented score. A shoe shop will need to order more pairs of popular sizes – 8, 9 and 10 – than smaller or larger sizes, for example.
  • The median is the middle value – halfway between the smallest and largest value.
  • The range is calculated by subtracting the smallest number from the largest number. What is the range of arm lengths in Year 7?

 

Interpreting Graphs

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Graph created in Excel by Jakob Linke, Year 7

Learning Intention: This week you will continue to work with line graphs, bar charts, pie charts and scatter plots. Your learning intention is to understand how to describe various graphs in words. You need to be able to interpret the information given.

Success Criteria: You will be able to look at the scale, axes, labels and title of different graphs and be able to describe what they mean. You will be able to match a graph to it’s description and draw a sketch graph from a sentence. These are the activities planned:

1. Find a graph with data in a magazine or newspaper. Cut it out and paste it into your books. Look carefully at the title, axes, labels and scale. Write a few sentences describing what the graph shows.

2. “Which graph is which” worksheet. You will be given nine small graphs and have to match scenarios to the shape of the graph.

3. Maths300: Temperature Graphs. Match the average maximum and minimum temperatures to Australia’s capital cities.

4. Match the graph to the Olympic record time/distance.

5. Which kind of graph would best represent the following situations?

  • Height of a plant growing over time?
  • The various heights of different plants of the same species in a greenhouse, over time?
  • Thousands of plants in a crop to determine which genotype was the fastest growing?
  • Percentage of different species of plants in an area of forest?

6. “Purchasing pantyhose” and “Blood Bank” graphs

7. Go to the Melbourne Grand Prix map and note the speed and distance from the start as each car makes it’s way around the track. Draw a graph that shows distance from the starting line on the horizontal axis and speed on the vertical axis.

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Stem-and-Leaf Plots and Scatter plots

Learning Intention: Students will understand what data is suitable for graphing on a scatter plot and be able to describe the significance of a “line of best fit”.

Success Criteria: You will draw a correctly labelled scatter plot from our arm span and height data and determine if there is a relationship between these measurements.

Last week you learnt the definitions for mean, median, mode and range and created a stem-and-leaf plot using the height of students in Year 7. You also measured the length of seven leaves and calculated the mean, median, mode and range of this data. This week we will investigate another type of graph, the scatter plot. Use the data we collected from our Year 7 Maths Survey to graph arm span against height (in centimeters).

This week we may also get the chance to do other activities with scatter plots:
1. Barbie Bungee
How many rubber bands are needed for Barbie to safely jump from a height of 400 cm?
What is the minimum height from which Barbie should jump if 25 rubber bands are used?
How do you think the type and width of the rubber band might affect the results?
Do you think age of the rubber bands would affect the results–that is, what would happen if you used older rubber bands?
If some weight were added to Barbie, would you need to use more or fewer rubber bands to achieve the same results?
State a possible relationship between the amount of weight added and the change in the number of rubber bands needed.
(thanks to Mrs Jirkovsky at North Adams Public School for writing about this activity on her blog!)
2. Be an actuary – distance vs earthquake intensity.

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“Lies, Damn Lies and Statistics!”

Learning Intention: Students will understand that data can be displayed in various ways and they will be able to interpret different types of graphs, including bar graphs, pie charts, stem-and-leaf plots and scatter plots.

Success Criteria: Students will complete the following activites and be able to explain what the graphs tell us about the data collected.

The above quote, popularised by Mark Twain, refers to the ways that politicians can sometimes “manipulate the data” to suppport decision-making. The same data can often be used to support or disprove a theory, depending on the emphasis. Over the next couple of weeks we will use the data from the CensusAtSchool questionnaire to learn more about summary statistics and graphing. Some of the activities we will do are:

You can use the random data sampler to display the data relevant to each question.