# Recognise expressions in decimal and standard forms

Many of the numbers that occur in science data sets and calculations are complex either by being very large (e.g. sizes of the scale of the universe) or very small (e.g. size of cell or electron). Similarly, most of the data collected in experiments by students may well involve multiple decimal places. As such, students need to be familiar and confident with working with both decimal and standard form as well as performing calculations with them without errors.

Decimals can also be described as fractions where the denominator is a power of 10. We write decimal fractions with a decimal point.

Based on the fact that ^{1}/_{10} is 0.1 pupils may wrongly generalise that for example ^{1}/_{6 }is 0.6.

When converting between ordinary and standard form some pupils may incorrectly connect the power to the number of zeros; e.g. 4 × 10^{5} = 400 000 so 4.2 × 10^{5} = 4 200 000

Similarly, when working with small numbers (negative powers of 10) some pupils may think that the power indicates how many zeros should be placed between the decimal point and the first non-zero digit.

It is almost inevitable in most calculations across the three sciences that decimals will be needed however the most likely places where standard form will occur are:

- In the
**Cell Biology**topic when the size of cells are described. - In the
**Chemical Analysis**topic in Chemistry where stoichiometric calculations and performed. Calculations involving the mole and Avogadro’s constant are known to be challenging for manty students. - Speed and motion calculations in physics such as in the
**Forces and Motion**and**Light and Electromagnetic**waves topics. - When describing the sizes and distances involved in the
**Space Physics**topic. - When describing the quantities and sizes of atoms in the
**Atomic Structure**topic in physics, particularly the section on ionizing radiations.

## Links and Resources

### Estimating Length Using Standard Form N4

This Standards Unit resource is designed to help learners to interpret decimals using metric units, estimate lengths and interpret standard form.

Begin the session by explaining that the distance from your nose to your finger tip is about one metre. Ask learners to name objects or everyday distances that have lengths that are

approximately 10 m, 100 m, 1 000 m, 10 000 m and then 0.1 m, 0.01 m, 0.001 m and 0.0001 m, using questions provided in the resource.

You can then produce a table as demonstrated in the resource and fill in the standard form version of the estimate. Students then match images to standard measurement and the same measurement expressed in standard form. Students can be challenged using card set C to make comparisons between different pairs of items.

### Number

Produced by the Learning and Skills Improvement Service (LSIS), these resources are aimed at mathematics practitioners but could be adapted for use in a Science classroom. The resource includes a unit on **Standard form - exposing and discussing common misconceptions**. The resource can be used to review the notation for standard form and expose common misconceptions associated with standard form notation.

You could start by writing a very large and a very small quantity on the whiteboard. For example, the mass of the Earth is 5980000000000000000000000kg and the diameter of a human red blood cell is 0.0000075m.

There is an opportunity for overlap with some of the resources mentioned in the the Orders of Magnitude list.

Use open questions to review learners’ understanding of standard form and justify the use of the alternative representations 5.98 × 10^{24} and 7.5 × 10^{–6}

You then use a set of handout that show one learner’s attempts to answer questions about standard form. Students then work in pairs to act as ‘critical friends’ for the person who has written the answers. If they agree with the answer given, then they must justify their decision by providing an example that confirms the explanation given by the fictional learner. If the learners disagree with the answer given, then they need to identify the mistake, correct it and write a brief but helpful comment for the person who submitted the work.

There is clear guidance on the main misconceptions included in the unit and it could be used as a short activity to check students understanding prior to using Standard form in your classroom.

### Indices and Molecules

From the Nuffield Foundation, the purpose of this module is to introduce students to index notation and standard form on the way to the determination of the size of a molecule. It builds up from simple rice-counting and paper-cutting to experiments in chemistry and physics which give a good approximation to molecular size.

This module suggests joint working by a mathematics teacher and a science teacher and that at least some of the work is done in a science laboratory. The resource includes a brief overview of the content of the work cards and references to other resources.

There are 17 work cards and associated teachers' notes which include pre-requisite knowledge as well as a list of material needed for the science laboratory experiments. Each card has detailed ‘Notes for the teacher’ and most cards have ‘Notes for the pupil’.

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Last updated | 15 December 2015 |

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URL | https://www.stem.org.uk/lxq6a |

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