The formula sheet and the cheat sheet: how to make one that teaches you while you build it

A formula sheet copied from the textbook is a list you will read in the exam and not understand. A formula sheet built from memory, five columns wide, is a study method: the formula, when it applies, the units, the mistake you make with it, and one tiny example you computed. The research on crib sheets is blunt: having the sheet in the exam does not raise marks on its own. Building it does, if the building is recall. Here is the five-column method and a physics sheet made with it.

One word first. "Cheat sheet" here means the one page a teacher allows into the exam, and nothing else. Nothing on this page is about getting around a rule.

What the research on cheat sheets found

Teachers have tested the sheet, because students have asked for it for decades.

Dickson and Miller (2005) let 54 students in two sections of a psychology course bring a prepared crib card to two of their four exams. The students expected the card to raise their marks and lower their nerves. It did neither: scores on the card exams were no higher, on either the factual questions or the harder ones, and the students were no less anxious.

Dickson and Bauer (2008) then asked whether making the card teaches anything. Students prepared crib sheets expecting to use them, and were first given a test without them. The students who had expected to lean on the sheet had not learned the material as well as the ones who had not; the authors' reading is that a student who knows the sheet is coming puts the facts on the page instead of in their head.

Erbe (2007), writing as a teacher rather than reporting an experiment, argues the other side: a one-page sheet structures the student's study time and lowers anxiety in the room, and the learning happens while the page is made. The two positions are not in conflict. The sheet in the exam is a comfort. The sheet you build is the method, if the building is done as recall.

Why recall? Slamecka and Graf (1978) named the generation effect: a word you produce yourself from a cue is remembered better than the same word read. Roediger and Karpicke (2006) found the same for whole passages, with recall tests beating rereading a week later. A sheet written from memory is a page of generated answers. A sheet copied is a page of read ones. They look identical and do opposite things, and the rest of this page is how to make the first kind. It is the four moves applied to a single page: answer first, then check.

Step 1: The list of formulas from the outline, not the textbook

One row per formula or procedure the unit uses. The list comes from the course outline or the unit's headings, not from the textbook's summary page, so that nothing the course tests is missed and nothing the course does not test is copied in. The textbook's summary belongs to the textbook's chapter, which is never quite your unit. The outline-first rule for a study guide is the same rule; this page applies it to formulas alone.

Write the list before you open the book. A unit test usually has five to ten rows; a term exam twenty to thirty.

Step 2: The five columns

Each row gets five short entries, one line each. The sheet is landscape, one unit per side.

1. The formula. As you will write it in the exam, with the symbols the course uses.
2. When it applies. The question's words that call for it: "work done by a force along a displacement", "any moving mass". This column is the one students leave out, and its absence is the commonest formula-sheet failure in an exam: the formula is on the page and the student does not know that this is the question it answers.
3. Units of every symbol. Not only the answer's unit; every symbol's. Half of physics marking is units.
4. The trap. The mistake you make with this formula, from your own error log. The error log that names the wrong line is where this column comes from; if you have none yet, the trap is the one the worked example below names.
5. A tiny example you computed. Two or three numbers, one line, the answer. Small enough to redo in thirty seconds, computed by you, never copied.

Step 3: Fill it from memory first, then check

Close the book. Write every row you can, all five columns, from memory. Some rows will have a formula and no trap; some will have a "when" and no formula. Leave the gaps.

Then open the book and fill the gaps in a second colour. The second colour is tonight's list: those are the rows you do not yet have, and the attempt-and-check you just did is the honest diagnostic that found them. A sheet with no second colour was either already known or was copied; you know which.

Step 4: Study the sheet by covering, not reading

Reading the sheet is rereading. The sheet is studied by covering columns.

• Cover the formula column. Read the "when" column, write the formula on scrap paper, uncover, mark.
• Cover the trap column. Say the trap for each row before you look.
• Cover the example's answer. Redo the tiny example cold, then check.

Three passes on three separate days. Spacing the passes is what moves a formula from tonight's memory to the exam's; the research on spacing is clear that the same three passes on one evening do far less. If the exam allows no sheet, this is how the formulas are learned. If it allows one, you will barely look at it in the room, which is the point: the sheet was the method, and the exam copy is the comfort.

Worked example: a physics energy sheet, five rows

SPH4U ideas 6 to 8: work and kinetic energy, conservation of mechanical energy, power and efficiency. Five rows, g = 9.8 m/s² stated, every example to two significant figures as the data are given.

Formula	When it applies	Units	The trap	Tiny example
W = F·d·cos θ	Work done by a force along a displacement	F in N, d in m, W in J	θ is the angle between the force and the displacement, not the incline's angle	F = 20 N, d = 3.0 m, θ = 0°: cos 0° = 1, so W = 20 × 3.0 × 1 = 60 J
Ek = ½mv²	Any moving mass	m in kg, v in m/s, Ek in J	Forgetting to square v	m = 2.0 kg, v = 3.0 m/s: v² = 9.0, so Ek = ½ × 2.0 × 9.0 = 9.0 J
Eg = mgh	Height above a chosen zero	m in kg, g in m/s², h in m, Eg in J	Using two different zero levels in one problem	m = 2.0 kg, h = 3.0 m: Eg = 2.0 × 9.8 × 3.0 = 58.8 J, which is 59 J to two significant figures
P = W/t	The rate of doing work	W in J, t in s, P in W	Using the wrong time interval (the whole trip for one stage's work)	W = 60 J in t = 4.0 s: P = 60 / 4.0 = 15 W
Efficiency = (useful energy out ÷ energy in) × 100 %	Any machine or process that loses energy	Both energies in J, efficiency in %	Dividing the wrong way round (in over out gives more than 100 %)	45 J out from 60 J in: 45 / 60 = 0.75, so 75 %

The next evening's cover test, as one student scored it: five formulas written from the "when" column, 5 of 5; five traps said before uncovering, 4 of 5 (the zero-level trap missed, so Eg's row is the one that comes back tomorrow); five tiny examples redone cold, 5 of 5. The sheet is now a 14-of-15 sheet with one row named, which is a thing you can act on. A copied sheet is never a score.

Math and chemistry sheets: what changes

Math. Many rows are procedures rather than formulas, and the trap column does most of the work. One row: completing the square. Formula: x² + bx = (x + b/2)² − (b/2)². When: a quadratic to be put in vertex form, or a circle's equation to be put in standard form. Units: none. Trap: forgetting to subtract the (b/2)² you added. Tiny example: x² + 6x + 5 = (x + 3)² − 9 + 5 = (x + 3)² − 4; check by expanding, (x + 3)² − 4 = x² + 6x + 9 − 4 = x² + 6x + 5. Another: the quotient rule, where the trap is the order of the two terms in the numerator, because the minus makes it matter.

Chemistry. Rows carry a unit conversion, and the tiny example is where the conversion is practised. One row: moles from mass. Formula: n = m / M. When: a mass is given and the question wants moles, or the reverse. Units: m in g, M in g/mol, n in mol. Trap: using the atomic mass of one element for a compound. Tiny example: 29 g of NaCl, with M = 22.99 + 35.45 = 58.44 g/mol, so n = 29 / 58.44 = 0.50 mol to two significant figures. Chemistry studied as procedures has the rest of the method; the sheet is where the conversions get their thirty-second redo.

If the exam gives you the sheet

Many exams supply a formula list, and many students see it for the first time in the room. Get a copy from your teacher weeks before, and study it the same way: cover it and test yourself on what is on it, so you know what the exam carries for you and what you still have to carry in your head. Learn its notation, because the exam's sheet may write a formula differently from your textbook. This page reproduces no exam's official sheet and will not; ask for yours.

The mistakes

• Copying the textbook's summary page. It is the book's unit, not yours, and copying is reading.
• Tiny handwriting that fits everything. A sheet you cannot read under time teaches nothing and helps less.
• No "when" column. The formula without its question is the commonest exam failure: it is on the page and you do not know that this is the moment for it.
• Never covering it. A sheet that is only read is a sheet that is only reread.
• Building it the night before. The three passes need three days; the night before, there is one. The week before a test has where the three days sit.

The method on one page

1. The list from the outline, one row per formula or procedure.
2. Five columns: formula · when it applies · units · the trap · a tiny example you computed.
3. From memory first, then the gaps in a second colour.
4. Study by covering, three passes on three days.

The sheet in the exam is a comfort. The sheet you built is the method. Chapter 10 of the free playbook is the one-page cheat sheet for studying itself, built the same way.