Help your students apply their maths knowledge to solving science questions

Students meet standard form in maths, but this form of numerical notation is critical for calculations in chemistry for the 14–16 and post-16 age groups. Generally, chemistry teachers presuppose that students have mastered decoding standard form notation by the time they need to use it in chemistry contexts, but this is not always the case. In maths, standard form is typically localised in questions in which students’ ability to manipulate quantities in standard form is the assessment criteria. In chemistry, using standard form accurately is a prerequisite for solving problems and may not independently carry any method marks. This means that standard form adds a level of cognitive challenge to a problem rather than being a problem in itself. As a result, a student able to use standard form correctly in maths will not always get it right in chemistry.

Maths lessons tend to focus on the ‘moving the decimal point’ (or ‘moving the digits’) method of converting numbers into and out of standard form.

**For example**

To convert 0.00357 into standard form, the decimal point is moved three spaces to the right (or three digits jump left over the decimal point) to give 3.57 (so the only integer before the decimal point is a number between 1 and 9), and the 10^{-3} shows how far and in which direction the decimal point moved (or the numbers jumped), giving an answer of 3.57 x 10^{-3}.

Maths lessons tend to focus on the ‘moving the decimal point’ (or ‘moving the digits’) method of converting numbers into and out of standard form. For example,

To convert 0.00357 into standard form, the decimal point is moved 3 spaces to the right (or 3 digits jump left over the decimal point) to give 3.57 (so the only integer before the decimal point is a number between 1 and 9), and the 10^{-3} shows how far and in which direction the decimal point moved (or the numbers jumped), giving an answer of 3.57 x 10^{-3}.

While this method is incredibly useful for computing these conversions, it hides the purpose of using standard form in the first place: expressing magnitude. Standard form allows scientists to compare the order of magnitude of different quantities in a way that helps them to explain phenomena and make useful predictions.

Essentially, there are two challenges we face from students when using standard form in chemistry classrooms: ‘can’t’ and ‘won’t’. The ‘can’t’ students struggle to interpret significant figures and so make mistakes in their use: they have not yet attained mastery of the principles. The ‘won’t’ students don’t see the point of using standard form as it takes time for them to convert in and out of the notation and/or they can just use a calculator. They have not yet understood the relationship between standard form and orders of magnitude. There is often a significant overlap between the ‘can’ts’ and ‘won’ts’.

### Interpreting standard form

Part of why students struggle with standard form is that they cannot ‘read’ numbers in standard form as fluently as numbers written out in full as they are not exposed to them so regularly. But, mastery of converting in and out of standard form is absolutely essential for students to deal with the increased cognitive challenge of using standard form in a chemistry context.

### Download this

Diagnostic exercises to assess your students’ fluency with standard form from the *Education in Chemistry* website: **rsc.li/3cgv7Sq**

Before introducing your students to chemistry topics involving orders of magnitude, assess their ability to deal with it. Download the diagnostic exercises and use the first to help you determine if your students are struggling with fluently decoding the standard form of notation. Depending on how students respond, you will need to follow one of these courses in your lesson:

**A.** If most/all students get the right answer, then seat any who struggled with a student who answered confidently and correctly and continue with the lesson using standard form throughout to reinforce notational fluency. Start the following lesson in a similar way.

**B.** If students are split 50/50, put them in mixed attainment pairs and ask the stronger students to explain their reasoning to the weaker student. Then ask each pair to design a new diagnostic question and rotate these around the room for students to have a second attempt. If there’s not sufficient improvement, move on to C.

**C.** If most/all students get the wrong answer, you’ll need to do some whole-class work on standard form to teach students how to decode the notation more explicitly. Here are some strategies to get you started.

### Teaching strategies

One of the weaknesses of the ‘move the decimal point’ method students meet in their maths lessons is that they never relate the numbers they are using to physical quantities. A simple exercise to carry out with 11–14 and 14–16 age groups is to plot a number line for the orders of 10: 1 mm, 10 mm, 100 mm, 1000 mm, 10,000 mm (across the lab) and 100,000 mm (on the playing field), and then compare these to standard form notation. Students experiencing the changes in magnitude themselves helps challenge the misconception that the orders of 10 form a linear scale and demonstrates to students how much the power of 10 matters to the answer.

Another strategy to familiarise students with decoding is to put a numberline on the board and the orders of 10 in a line along it (discussing scale with students), eg 0.001, 0.01, 0.1, 1, 10, 100, 1000. Model writing 1 and 10 in standard form (1 x 10^{0} and 1 x 10^{1}) and show your students how these two give the original numbers (10^{0} = 1 and 10^{1} = 10). Challenge students to spot the pattern and complete the number line in standard form. Finally, ask them to contribute as many different ways of writing the numbers on the number line as they can to assess their progress. If your students need more practice, use a different number line and repeat.

Lastly, expose students to numbers in standard form as often as possible. This means editing questions to express quantities in standard form on a regular basis, even when standard form is not one of the target skills of the lesson. Giving students worked answers (using standard form) and asking them to spot the errors is an excellent way to build in and assess this skill without overdoing calculations.

### Why is standard form so useful?

Another significant challenge for chemistry teachers is convincing our students that standard form actually makes problem solving easier. Essentially, students have a misconception that using standard form takes more effort than writing out numbers in their entirety – demonstrating the efficiency of standard form in problem solving is key to engaging the ‘wont’s’. For many of the problems students face in maths, the original number and the standard form notation can both be written out fairly easily: not so in chemistry (6.02 x 10^{23} vs 602000000000000000000000). This means chemistry teachers have an opportunity to reinforce students’ mathematical understanding of magnitude in a way that is difficult for maths teachers.

The second diagnostic exercise is designed to help determine how comfortable post-16 students are with multiplying and dividing base 10 indices in a chemistry context to determine the order of magnitude of a quantity. None of the problems are easy, and so pairing the students up or working in small groups may be more productive, circulating to check all students are contributing.

#### After the pairs/groups have their answers, try these follow-ups:

- Ask groups to come up and present their solutions to one of the problems, explaining any approximations they made, and their choice of notation.
- Ask groups to pick a problem and calculate their answer using standard form and by writing out all numbers in full. Ask groups to demonstrate this on the board, and ask them how standard form makes the problem easier or harder.
- Ask students to repeat the problems using a calculator and see which group made the most accurate initial approximations.
- Ask students to read ‘A Mole of Moles’ from the book
*What if?*and compare their methodology and the kinds of approximations in the article. - Ask groups to design their own estimation problem. They will need internet access or data books. Offer them bonus points for designing a problem with an answer in similar orders of magnitude to the problems in the diagnostic question.

### Mastery

Mastery of standard form is partly about repeated exposure to the skill, building fluency and automaticity. But it is also about being able to understand what standard form is and why it matters in chemistry. As chemistry teachers, we have the opportunity to deepen students’ mathematical understanding, as we can expose them to extremely large and small numbers in contexts which are familiar to them, in order to solve concrete, tangible problems.

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