Mathematics Year 6: Measuring instead of counting: questions and collecting continuous data
Mathematics, Year 6 (Portugal): Heights, times and weights are measured, not counted: learn to frame the question, decide what, where and whom to collect from, and keep every measured value in a list.
Counting gives whole numbers, measuring gives almost any value
Until now you counted things: brothers and sisters, books, goals, and you got 0, 1, 2, 3… Now you will measure: the weight of a school bag, a student's height, the time it takes to get to school. These are continuous quantitative characteristics: between 4.1 kg and 4.2 kg there is still room for 4.15 kg, and for more values if the scale is finer. One question about them is “How much does each student's bag in our class weigh?” The answer changes from student to student, so you need data.
Before you measure, decide
Collecting data without a plan gives numbers you cannot compare. Decide first: what to measure and how precisely (the weight of the bag, in kilograms, to one decimal place); where to collect (primary source: you measure it yourself; or a credible website, with a table that others have already organised); whom to ask or what to observe (the 20 bags in the class). And use the same method for everyone: the same scale, the same day, the bag as it arrives at school. If each person weighs in their own way, the difference may lie in the method and not in the bags.
From curiosity to a list of 20 weights
Curiosity: “Are our bags heavy?” Step 1: statistical question: “How much does each student's bag in the class weigh?” Step 2: continuous characteristic, measured in kg to one decimal place. Step 3: observe the 20 bags, on Monday morning, on the same scale. Step 4: record each value just as the scale shows it, in weighing order: 4.6, 3.5, 5.2, 6.3, 3.8, 4.1, 5.0, 3.2, 5.4, 4.8, 3.6, 5.1, 2.9, 3.9, 5.5, 4.3, 3.4, 6.8, 5.3, 3.7. Check: 20 values, one per student.
Recording straight into classes closes doors
To save time, a team writes only “between 4 and 5 kg” instead of 4.6 or 4.1. It looks the same, but the values are lost for ever. With the list from the worked card, the four students weighing 4.1, 4.3, 4.6 and 4.8 kg can be grouped in [4, 5) (4 students) or split into [4, 4.5) (2) and [4.5, 5) (2). With the record “between 4 and 5” you no longer know how many weigh 4.5 kg or more. Keep the measured values and group them later, when you know how.
From the canteen to the shower: everything that is measured
Measuring to answer a question turns up everywhere: how much food is left on the canteen plates each day, how many minutes the bus takes to arrive, how tall the plants in the school garden are, how many litres of water a shower uses. A class study can start from any of them. You can also look for the data on a credible website, such as a table from a statistics institute, and always ask who collected them and how.
Test what you learned
Which of these questions is about a continuous quantitative characteristic?
- a) How many brothers and sisters does each student in the class have?
- b) How long, in minutes, does each student take to get to school?
- c) What is each student's favourite sport?
- d) How many students are in the class?
Correct answer: b) How long, in minutes, does each student take to get to school? — Right: time is measured with a clock, and between 12 and 13 minutes there are values such as 12.5. It is measured, not counted.
You recorded, in kg, the weights of 20 bags: 4.6, 3.5, 5.2, 6.3, 3.8, 4.1, 5.0, 3.2, 5.4, 4.8, 3.6, 5.1, 2.9, 3.9, 5.5, 4.3, 3.4, 6.8, 5.3, 3.7. How many bags weigh 4.5 kg or more?
- a) 12
- b) 8
- c) 10
Correct answer: c) 10 — Right: 4.6, 4.8, 5.0, 5.1, 5.2, 5.3, 5.4, 5.5, 6.3 and 6.8 make 10, half of the 20. Only the list of values allows this answer.
A team noted each bag only as “between 4 and 5 kg”, “between 5 and 6 kg”, and so on. What is the problem with this record?
- a) The measured values are lost: you can no longer tell how many weigh 4.5 kg or more, nor choose other classes.
- b) None: it is the same information and saves time.
- c) There is only a problem if there are more than 20 bags.
- d) All the bags now weigh 4.5 kg.
Correct answer: a) The measured values are lost: you can no longer tell how many weigh 4.5 kg or more, nor choose other classes. — Right: grouping straight into the record cannot be undone. With the list you can group in several ways; without it, in only one.
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