This unit encourages students to assess the validity of a set of statements and then plan and carry out investigations to find out whether the statements are true. Students then carry out statistical investigations of their own.
Specific Learning Outcomes
- Plan a statistical investigation to assess the reasonableness of a statement.
- Carry out a statistical investigation to assess the reasonableness of a statement.
- Check the reasonableness of answers obtained using a calculator by calculating mentally and using approximations.
Description of Mathematics
In order to assess the validity of statements students will need to identify the assumptions underlying the statements and the accuracy of the statements. Some criteria for assessing the statements are needed. For example, what measure will be used as an average lifespan?
Students will also need an understanding of the term average. The word average is used in two ways and students need to understand the distinction between these. In statistics an average is a measure of centre for a set of data. There are three common measures of centre, mean, median and mode. The mean is calculated by adding all the numbers together and dividing by the number of numbers. The median is the middle score when the data is arranged in ascending order and the mode is the most common score.
The word average is also be used in everyday language, to mean something that is not out of the ordinary or not very good. For example, to say “I am feeling a bit average today,” means that you are not feeling wonderful.
Required Resource Materials
- Begin a discussion about the tasks students complete, to get ready for school in the morning.
What do you need to do to get ready for school in the morning?
How long does it take you to get ready?
How long do you spend…
- in the shower?
- brushing your hair?
- making and eating breakfast?
- Does it take you the same amount of time every morning or does it vary a bit?
- What is the reason for the variation?
- Introduce the McDonalds advertisement contained in Copymaster 1.
- Introduce / review the concept of averages with the group.
What is an average? (measure of the middle or centre, sometimes assumed to be the normal)
What measures of the middle do we use? (mean, median, mode)
How do we calculate those measures?
Ensure students understand the difference between the use of ‘average’ in everyday speech and its particular meaning in statistics.
- Ask a group of students to estimate how many minutes they spend each morning preparing and eating breakfast. Display the data as a dot plot or stem and leaf graph. Many online tools exist to create the displays.
- Give the students calculators and calculate the mean together. Add all the times and divide by the number of times. The median and mode can be ‘eyeballed’ from the graphs. Close in from top and bottom to find the median (middle time) and look for the highest frequency to get the mode.
Are the averages all the same? Why or why not?
- Return to Copymaster 1. Encourage the students to consider the validity of the statements.
Do you think these statements are accurate?
How would we check their accuracy?
- Split the class into groups and assign each group one of the statements. Tell the groups their task is to check whether the statement is accurate. They may need to collect data from their fellow students. As the students work ask them to record the assumptions they are making. These will include:
- the estimated length of the activity each day
- the variations that occur in the length of the activity
- whether the activity occurs for the same amount of time on weekends and week days
- the length of an average lifetime
- Once the groups have finished checking the accuracy of the statements, gather together as a class to discuss the results.
Do you think the statement is accurate?
How did you work that out?
What averages did you use? Why did you choose those measures?
- Make a list of the assumptions each group has made.
Over the next few days have the groups of students carry out their own statistical investigations about the average time spent on various activities:
- Discuss the average amount of time spent on a variety of activities. Encourage students to identify activities they would like to find out about.
How long in a lifetime do you think you would spend watching TV?
How long do you think you would spend riding your bike?
How long do you spend sleeping?
- Discuss the assumptions that will need to be made. As a class decide the assumptions that will be used by all groups and list these clearly. For example, the average lifespan is 83 years for females, and 79 years for males. Some assumptions will be specific to a statement, so groups will have to make some assumptions themselves. For example, average time spent on watching television might include only time viewing rather than time the television was used as background noise. Time spent sleeping may involve assumptions about when people slept and woke up, as opposed to surfing the internet, checking their phone, reading books or watching television.
- Place the students into groups.
- Groups list questions for investigation. For example:
In an average lifetime how long will be spent reading?
In an average lifetime how long will be spent eating?
- Show students the worked example of the investigation, Copymaster 2 and explain the process they will follow to answer their question.
- Groups work to answer their questions using the investigation template, Copymaster 3. This will take some time with some and you may need to have several small group / whole class discussions to help students through this process.
As students work, encourage them to use mental strategies to estimate answers before they are calculated, and confirm the results of calculations. Discuss the strategies students are using and compare strategies used by different students.
What do you think the answer will be? How did you work that out?
Does the answer seem reasonable? How do you know?
Do you think the answer Jack got using the calculator is about right? How could you check?
That’s a good way to work that out. Did anybody use a different method?
You may also need to support your students to convert between units of time. For example, 242 300 minutes is not as meaningful as 168.5 days. Since time measures are mostly based on non-decimal units, such as 24 hours per day and 60minutes per hour, the calculations can be difficult. Allow students access to calculators for the complex calculation.
Once groups have answered their questions, have them swap questions (without the assumptions, calculations and answer) and work out the answers to each other’s questions. Ask the review group to consider the assumptions that were made and to follow the template to get their own answers.
What answer did you get to the question?
How close was your answer to the conclusion of the investigating group?
Why do you think the answers you got were same/different?
What assumptions did each group make that were the same? Why?
What assumptions did each group make that were different? Why?
Dear Family and Whānau,
This week at school we are carrying out investigations to find out the amount of time spent on various activities in an average lifetime. Please help us time activities around the home to help us with our investigations.