Lesson 19 Accuracy and errors

# Episode 1

Reasoning Resources: Worksheet 1, Stopwatch, The Cat in the Hat Comes Back (optional)
Whole class preparation
Decimal measures of time are free from the ambiguity of units.

There are difficulties for the teacher in making connections and analogies to other contexts for decimals. Money values are not continuous in daily usage, and the place value of tenth coincides with the value of 10p, so that tenths and tens get mixed up. The metric system in general suffers from similar confusion caused, in measuring lengths for example, by the tendency in everyday life to focus on the separate measurements of metres, centimetres and millimetres rather than on a single unit represented by a decimal used in formal or technical writing. The common problem with the measurement system is that we have given names to the smaller subdivisions as units in their own right.
You could use the context of a recent sports event, where speed is important, to ask how to decide winners. This would lead the pupils to discuss the use of timers, including in watches and mobile phones, and how accurate they are.

Most classes would benefit from a short experiment on estimating time. Do you know how long a second is, without using a timer? Bang twice, on the table with an interval of 7-8 seconds. How long between the two bangs? Most estimates based on counting habits are likely to be between 5-10 seconds. Compare their estimates with the real time. Most pupils would recognise that their estimates differ by only 1 or 2 seconds. A brief discussion on whether that is a‘good enough’ estimate or not (it is good!) sets the scene well for the lesson. You may extend to discuss how near the average or mode of all the estimates is to the actual time, whether we may even sense ‘half-a-second’, and reasons why a second is an easy measure to estimate, e.g. in relation to using words, or heartbeats or normal walking.

Give out Worksheet 1 to pairs, and start work on it in a whole class mode until it is appropriate to leave them to work independently.
Paired work and whole class sharing on questions 1 and 2
Valid analogies, if needed, are in measuring lengths of strings, weights and volume of liquids. Many pupils will have some understanding of skiing so the setting is real and helpful. Allow the pupils to struggle as they use the decimals on the number line. Encourage them to talk to other groups for question 2, as they think about how they can both ‘un-round’ 4.6 seconds and show that on the number line. Many pupils may gain insights into the fact that the time may not be exact but near enough to the mark, and how enlarging the interval is helpful.

In whole class sharing, allow each group time to explain their ideas and challenge them further with questions such as: What could 4.6 seconds actually mean? How can you show that on the number line? Prod indirectly for further enlargement.

Focus the pupils upon the fact that we can keep enlarging and subdividing the number line and inventing ever more precise units, but there will always be more markers we could add, because time is continuous and countable only when we put in the markers.
Small group work and whole class sharing on question 3

The concrete model of the number line with markers for discrete numbers and semi-discrete halves and quarters is extended in depth to locate an infinite number of values, hence linking to a more advanced mental model of continuous quantity. Allow pairs, then pairs of pairs, to talk for a few minutes to rehearse their ideas clearly before presenting them to the rest of the class orally, or on the board. Some pupils will initially think that there are only nine numbers (or lines) between any two markers. This may provoke useful debates among pupils who may hold very strong and differing opinions, which you orchestrate. The aim is to explain how there can be an infinite number of numbers between any two markers. Better to accept a ‘compromise’ in which there are indeed ten smaller markers between any two larger markers, and when you enlarge these then you can draw more markers. Some pupils would need many such experiences to accept infinite subdivision or continuity. A ‘real-life’ example might be useful: Think of different sports when two competitors have the same times. How do they decide the winner?

Many 12 year olds appreciate the chance to play around and explore the concept of infinity, whether subdivision or extension, with their peers. The Cat in the Hat Comes Back is useful here as under his hat is a cat called A who is half his size and so on until we get to Z. Under his hat is...?