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## Rearranging Before Substituting Does Represent Better Practice

In my mind solving algebraically before calculation is better because

- Letters are easier to rearrange than numbers
- Some terms may cancel and are not needed
- Copying of numbers may lead to transcription errors
- Errors are propogated
- Calculators need to be used multiple times, multiplying the opportunities for mistakes

Some people may disagree with these points so using NoMoreMarking, I set up a comparative judgement task for physics teachers and other physicists to judge a selection of answers I wrote to a GCSE question. The question required equating two formulae and a change in unit from mg to kg.* All answers were correct, so worthy of full marks but I varied the ordering of the working to provide 17 different permutations. 24 helpful tweachers completed 377 judgements with a reliability of 0.91.

The results are in! The top five solutions all rearranged before substituting.

Other points to note, physics teachers and physicists preferred answers that converted to SI units at the start and answers where variables were listed explicitly. These points are matched in AQAs published examiners report from last years AS exams

Lower performing students can improve their performance in a number of areas. It is good practice to summarise all data by writing it in the answer space along with the relevant formula to be used. The formula should have a subject and each step in the calculation should be presented.

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## Rearranging Physics Equations Worksheet

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## Where Is This Used In The Geosciences

To be honest, equation manipulation occurs in almost every aspect of the geosciences. Any time you see a P or T or or x , there is an equation that you could manipulate. Because equations can be used to describe lots of important natural phenomena, being able to manipulate them gives you a powerful tool for understanding the world around you!

See the Practice Manipulating Equations page for just a few examples.

## Question Type : Formulas Involving Squares

Sometimes the required subject can appear as a square. For example, if we are asked to rearrange the formula for the area of a circle to make r the subject. Firstly, recall that the formula for the area of a circle is,

A=\pi \textcolor

**Example:** For the equation A=\pi \textcolor, make \textcolor the subject.

**Step 1:**Dividing both sides of the equation by \pi, we get,

\dfrac=\textcolor

Now, r is on its own on one side but its not technically the subject, since it is squared.

**Step 2: I**f we now square root both sides of the equation, and since r is always positive, we get,

\textcolor=\sqrt}

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## Question Type : Formulas Involving Square Roots

Conversely, some questions may have the subject appear in a square root.

**Example:** Rearrange the following formula d=\sqrt}} to make \textcolor the subject.

**Step 1:**Squaring both sides of the equation, we get,

d^2=\dfrac}

**Step 2:** Next, multiply both sides by 2 to get,

2d^2=3\textcolor

**Step 3:** Finally, dividing both sides by 3 gives us,

\textcolor=\dfrac

## Question Type : Formulas Involving Fractions

Some questions will include making a letter that appears in a fraction the subject of the formula.

d=\dfrac

This formula describes the relation that distance, d, initial velocity, u, final velocity v and time t given constant acceleration. We can manipulate this formula to make u the subject.

**Example:** For the equation d=\dfrac+v)t}, make \textcolor the subject.

**Step 1:**Multiplying both sides by 2,

\begin\,\,\,\,\,\,\,\,\, d & =\dfrac+v)t} \,\,\,\,\,\,\,\,\, \\ 2d & =t \end

**Step 2: **Dividing both sides of the equation by t,

\begin\,\,\,\,\,\,\,\,\, 2d & =t \,\,\,\,\,\,\,\,\, \\ \dfrac & =\textcolor+v\end

**Step 3: **Finally subtractingv and rewriting so \textcoloru is on the left hand side,

\begin\,\,\,\,\,\,\,\,\, \dfrac & =\textcolor+v \,\,\,\,\,\,\,\,\, \\ \textcolor & = \dfrac -v\end

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## A Review Of Important Rules For Rearranging Equations

You probably learned a number of rules for manipulating equations in a previous algebra course. It never hurts to remind ourselves of the rules. So let’s review:

## Best Practice In Physics Calculations

Earlier this week I posed the following question on twitter

AQA have justified the mark given to student B but not student A with the following response

If the student had substituted into the equation before rearrangement, they could have scored one mark. As it is, they would score zero marks. If we were to credit for substitution into wrong equations then we would also need to credit substitution into any equation with the right letters in it. This would not be fair to students who are able to correctly substitute.

Therefore, students should be encouraged to write the equation down, substitute values and then rearrange.

I will make it clear at this point that I dont mind student B achieving their mark. I also dont mind this approach to questions. However my concern was that AQA encourage this ** over**student As response . I pointed out that I found this concerning because I believe student As response demonstrates better practice that student B. This was met with some disagreement.

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## How To Derive The Equations

The first step will be to calculate the slope of the diagonal line. In this case, since the slope will be a change in velocity divided by a change in time , the slope will equal the acceleration.

To keep things simple, rewrite t2 t1 as t.

This is **Equation 1**. Rearrange **Equation 1 **to get v2 on the left side of the equation. This expresses the equation in the slope-intercept form of a line, y = mx + b.

To get the next equation, derive an expression for the displacement of the object during the time interval, t. The displacement for an object traveling at a constant velocity can be found by multiplying the objects velocity by the time the object travels at that velocity.

The object in this activity, however, is not traveling at a constant velocity. How can the distance be calculated for an object that is not traveling at a constant velocity?

Consider an object moving with constant velocity, v1, from time t1 to t2. The displacement of the object is represented by **s**. The absolute value of the displacement is the distance traveled. The displacement can be calculated by:

Where t is the time interval t2 t1. Consider a graph of the motion of this object, as in Fig. 2.figure 2

Area A1 is a rectangle. The length, l, is v1. The width, w, is t2 t1, .

Area of rectangle A1 = l x w

Area A2 is a triangle with base t and height v2 v1.

Area of triangle A2 = ½ b x h

Adding areas A2 and A1 gives the total displacement of the object during the time interval.

Simplifying:

**Equation 3.**

## How Do You Rearrange Equations At Physics Gcse

Take the equation **W=Fs**. We want to make one thing the subject of the equation by clearing away all the stuff we dont want on that side of the equation. Say we want to find **s**. To do this draw a line under the equation like so: ** W=Fs**. This helps us to remember how to move things from one side of the equation to the other. The rule when moving a value to the

*other side*of the equation to leave one value by itself is:

*if the value you want to move is on top of the line then it moves to below the line, and if it is below the line, then it moves to above the line.*With

**, we want to leave**

__W__=__Fs__**s**by itself. The

**F**moves to the other side and because it is above the line on the right, it must go beneath the line on the left, under

**W**.

**=>**

__W__=__Fs__**By knowing the values of**

__W/F__=__s__**W**and

**F**, we can now work out

**s**by dividing

**W**by

**F**.

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## How To Esign An Attachment In Gmail

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