Lesson 19 Accuracy and errors

# Episode 3

Reasoning Resources: Worksheet 3
Analysing the errors further.

The model for the appropriateness of measurement is essentially that of proportionality. And this can be quantified as limits of acceptable errors.
This extension seeks to develop a link between pupils’ formulations for reasonable degrees of accuracy (raised in the questions in the previous episode) to a whole variety of different contexts. Give out Worksheet 3and as in the first two episodes, short mini-episodes around each question may be best.

Question 1 focuses upon the social and practical basis for rounding i.e. to make sense of many-digit numbers in order to make comparisons. Pupils suggest two possible roundings (e.g. to one and two significant figures) for each number given. They calculate the error ratio as a percentage and discuss how useful the approximations are. They could arrive at the generalisation that rounding to one significant figure, e.g. ‘nearly 6 appropriateness of million’ people for population, is what is needed most of the time. measurement is essentially.

You could use question 2 to demonstrate using the number line with two types of markings. that of proportionality. And for the population example place 5 million and 6 million on the larger markers and involve this can be quantified as the class in finding the position of the seven digit number. Pupils could then use the extra limits of acceptable errors.Using the smaller markers for rounding should emerge. The fact that each smaller interval is 10% of the larger one should explain why halfway between small markers is so small in comparison to the full amount. This leads to the question of quantifying percentage errors. Take rounding to the two digits with the highest place values.

• What is the range of the error ratios in percentages?
• How do you explain that on the number line?

Rounding to two significant figures can cause maximum errors between about 5 and 0.5%, depending upon the values given.