Assessed mathematical skills and measurement conventions

A linear conversion must be raised to match area or volume. Because $1\text{ cm}=10\text{ mm}$, $1\text{ cm}^2=100\text{ mm}^2$ and $1\text{ cm}^3=1000\text{ mm}^3$. Likewise $1\text{ dm}^3=1000\text{ cm}^3$.

A rate is quantity divided by time, so its unit must preserve both: breathing rate is breaths min$^{-1}$, oxygen uptake may be mm$^3$ min$^{-1}$, and mass-specific respiration may be mm$^3$ min$^{-1}$ g$^{-1}$. For a ratio, check whether units cancel or remain.

Alveolar ventilation with tidal volume 500 cm$^3$, dead space 150 cm$^3$ and 12 breaths min$^{-1}$ is $(500-150)\times12=4200$ cm$^3$ min$^{-1}=4.2$ dm$^3$ min$^{-1}$. Per hour this is $4.2\times60=2.5\times10^2$ dm$^3$ h$^{-1}$ to two significant figures.

Keep unrounded calculator values through intermediate steps, then round the final answer to the required significant figures. Decimal places describe position after the decimal point; significant figures begin at the first non-zero digit. Trailing zeros can be significant: 0.0050 records more precision than 0.005.

Standard form makes organelle dimensions, concentrations and population sizes auditable. After using $\text{magnification}=\text{image size}/\text{actual size}$, check that the exponent gives a plausible cell or organelle scale.

A fraction can be converted to a proportion or percentage: 18 affected plants out of 120 gives $18/120=0.15=15\%$. Keep observed ratios separate from theoretical ratios and compare them only after defining categories in the same order.

An estimate gives the expected scale of a result before exact calculation. It is a reasonableness check, not a replacement for showing the measured or calculated value.

Round inputs to one useful figure, keep powers of ten visible and compare the exact answer with the estimate; investigate a large mismatch.

A 2.0 cm² leaf area with a flux near 5 units cm⁻² h⁻¹ should produce a total near 10 units h⁻¹, not 10,000.

A culture starts with 500 bacteria and doubles every 20 minutes. After 2 hours, $n=120/20=6$, so $N=500\times2^6=32{,}000$. This estimate assumes constant exponential growth without resource limitation or death.

A logarithm reverses a power: if $\log_{10}N=5.30$, then $N=10^{5.30}\approx2.0\times10^5$. On a base-10 logarithmic axis, equal vertical steps represent equal multiplication; an increase of 1 means tenfold, while a doubling changes $\log_{10}N$ by $\log_{10}2\approx0.301$.

Enter brackets explicitly, retain intermediate precision and test the output by substituting back. Use the calculator's $10^x$, $e^x$, $\log$ or $\ln$ function that matches the model; changing log base without changing the equation changes the result.

SI prefixes change the scale of a unit: milli is 10⁻³, micro 10⁻⁶, nano 10⁻⁹ and kilo 10³. Convert to one common unit before comparing or substituting.

Write the prefix as a power of ten, cancel units algebraically and check whether the direction of the conversion makes sense.

250 μm = 250 × 10⁻⁶ m = 2.50 × 10⁻⁴ m; it is smaller than 250 mm, not larger.

Significant figures communicate the precision supported by a measurement or calculation. Zeros between non-zero digits count; leading zeros only locate the decimal point.

Keep guard digits during working, then round the final result to match the least precise relevant measurement. State a value such as 2.40 to show its precision.

A mean based on readings recorded to 0.1 s should not be reported as 2.437891 s; 2.4 s or 2.44 s may be appropriate depending on the data.

The arithmetic mean is the total of the observations divided by their count. It summarises repeated measurements when they represent the same quantity and are on the same scale.

Inspect the raw values first, calculate the mean with full precision and report spread or anomalous values rather than hiding them.

Rates 9, 10 and 11 units min⁻¹ have mean 10; if one reading was taken at a different temperature, combining it may be misleading.

Tables keep exact values visible; bar charts compare categories; histograms show how continuous measurements are distributed; diagrams clarify structure or process.

Label axes and units, choose equal scales, include a key when needed and make the display match the variable type. Keep raw data available behind any summary.

Use a histogram for a distribution of cell diameters, but a bar chart for separate treatment groups; joining category bars can falsely imply continuity.

For mutually exclusive alternatives, add probabilities. For independent events that must both occur, multiply them. Independence must come from the biological model—for example, separate meioses or independently assorting genes—not from convenience.

In an $Aa\times Aa$ cross, the probability of an $aa$ child is $1/4$. Each birth is a new independent event, so two $aa$ children in succession have probability $(1/4)(1/4)=1/16$. For independently assorting $AaBb\times AaBb$, the probability of $aabb$ is $(1/4)(1/4)=1/16$.

Observed frequency estimates probability: 7 cases among 800 000 people gives $7/800000=8.75\times10^{-6}$, about $0.000875\%$ or 1 in 114 000. Sampling variation means the observed proportion is not an exact law.

From sampled species counts, $D=\dfrac{N(N-1)}{\sum n(n-1)}$, where $N$ is total individuals and $n$ is each species count. Use the same effort, method and identification rules when comparing habitats; a larger $D$ indicates greater diversity under this index.

Increase independent sampling units, distribute them across time/space where appropriate and report uncertainty. A small sample of 23 chicks with an 8:15 sex ratio may not represent the adult population or a stable 1:1 population ratio.

The mean uses every value, the median is the middle after ordering, and the mode is the most frequent value. The best summary depends on the measurement and its distribution.

Use the median when an outlier would distort the mean; use the mode for common categories or repeated discrete values. Report the raw context and sample size.

Income-like values 2, 2, 3, 3 and 20 have mean 6 but median 3, so the median better represents a typical observation.

A scatter diagram pairs two measured variables. Direction, strength and form of the pattern describe association; they do not by themselves identify a causal mechanism.

Plot the independent variable consistently, inspect outliers and restricted ranges, and use a correlation measure only with the assumptions and data type it requires.

Light intensity and photosynthetic rate may rise together before a plateau; the pattern suggests a relationship, while temperature or CO₂ could also influence both.

A leaf section measures 50 mm in a photograph at $\times100$. Its actual thickness is $50/100=0.50$ mm $=500$ µm $=5.0\times10^2$ µm. The unit conversion happens after or before division only if it is applied consistently.

Write sizes in standard form to compare scale. $2\times10^{-5}$ m is one order of magnitude larger than $2\times10^{-6}$ m because their powers differ by 1; a power difference of 3 means a thousandfold difference. Coefficients near a power boundary must be considered when rounding to the nearest order.

A measured scale bar can recover actual size even if an image is resized, provided the object and bar were resized together. A printed magnification label may become invalid after resizing.

Calculate the statistic with unrounded data, determine degrees of freedom where required and compare its magnitude with the critical value at the stated significance level. If the result exceeds the relevant critical threshold, reject the null; otherwise do not reject it. Use the course-provided table and state the biological conclusion.

Chi-squared uses frequency counts with suitable expected values; t-tests require quantitative samples and the specified distribution/design assumptions; correlation requires paired observations and tests association, with Spearman based on ranks.

For $n$ sample values, $s=\sqrt{\dfrac{\sum(x-\bar{x})^2}{n-1}}$. Calculate the mean, subtract it from each value, square deviations, sum, divide by $n-1$, then take the square root. Retain calculator precision until the final step.

For 14, 16 and 18 g, $\bar{x}=16$ g, range $=18-14=4$ g, and $s=\sqrt{(4+0+4)/2}=2.0$ g. The SD means observations typically vary around the mean; it is not an error bar for the mean unless explicitly defined that way.

SD is usually more informative than range because it uses all values, but it can still be affected by an outlier. Compare groups using both centre and spread and consider sample size and measurement scale.

Percentage uncertainty is $\dfrac{\text{absolute uncertainty}}{\text{measured value}}\times100$. Percentage error compares a measured value with an accepted/reference value: $\dfrac{|\text{measured}-\text{accepted}|}{\text{accepted}}\times100$. They answer different questions.

A diameter $2.0\pm0.1$ mm has 5% uncertainty. For area proportional to $d^2$, the simple propagated percentage uncertainty is about $2\times5=10\%$. Report the area to precision consistent with that uncertainty.

From $y\propto x$, write $y=kx$ only after introducing the constant $k$. If $y\propto x^2$, doubling $x$ multiplies $y$ by four, provided other conditions remain fixed.

From $M=I/A$, where $M$ is magnification, $I$ image size and $A$ actual size: multiply by $A$ to get $MA=I$, so $A=I/M$; alternatively $I=MA$. Image and actual size must use common units.

For $BMI=m/h^2$, multiply by $h^2$: $m=BMI\times h^2$. With $BMI=36$ kg m$^{-2}$ and $h=1.55$ m, $m=36(1.55)^2=86.5$ kg to three significant figures.

Use dimensions to test the rearrangement: $(\text{kg m}^{-2})(\text{m}^2)=\text{kg}$. A form that leaves the intended subject on both sides or gives incompatible units is not finished.

For $D=\dfrac{N(N-1)}{\sum n(n-1)}$, $N$ is the total individuals and each $n$ is one species count. With counts 50, 30 and 20, $N=100$, numerator $=100\times99=9900$, denominator $=50\times49+30\times29+20\times19=3700$, so $D=2.68$.

Evaluate repeated terms in a table to avoid missing a category. Keep numerator and denominator separate, then divide once. For a supplied rate, ratio or odds equation, preserve category order exactly and place each observed count in the defined position.

Cardiac output $CO=SV\times HR$, so $HR=CO/SV$. Convert stroke volume to dm$^3$ if output is dm$^3$ min$^{-1}$. For swimmers, $HR=4.43/0.07440=59.54$ beats min$^{-1}$; for controls, $HR=4.21/0.05840=72.09$ beats min$^{-1}$. The difference is $12.55$ beats min$^{-1}$.

If the equation contains powers, fractions or several terms, undo outer operations first and keep both sides balanced. Reject extraneous or negative solutions when the biological quantity cannot take them, but state the domain reason.

If exponential growth follows $N=N_0\times10^{gt}$, then $\log_{10}N=\log_{10}N_0+gt$. A plot of $\log_{10}N$ against time is therefore linear during constant exponential growth, with intercept $\log_{10}N_0$ and gradient $g$.

If the log-count gradient is 0.602 h$^{-1}$, a doubling is a log increase of $\log_{10}2=0.301$, so doubling time $=0.301/0.602=0.500$ h. A rise from $10^3$ to $10^6$ cells is three log units but a thousandfold increase.

Use a logarithmic axis when positive values span several orders of magnitude or when multiplicative change is the relationship of interest. Label the actual values or log-transformed quantity clearly.

The same relationship can appear as prose, a table, an equation or a graph. Translate one representation at a time and preserve variables, units and direction of change.

Name the independent and dependent variables, identify constants and check a point from one form against another before inferring a trend.

A table showing rate increasing with substrate can become a graph of rate against concentration; a plateau in the graph means the increase is no longer proportional.

Label both axes with quantity and unit, choose a linear or specified log scale that occupies most of the grid, plot every pair accurately and include error bars or a key when supplied. Do not extrapolate a fitted line to the origin unless the evidence or model requires it.

Draw a straight best-fit line only across a supported linear region. Choose two widely separated points on that line, calculate the signed gradient with a large triangle and read the intercept from the fitted line. Substitute both into the equation and test it against another point.

With excess enzyme and substrate well below saturation, rate may be proportional to substrate concentration, giving a line through or near the origin over that range. A cooling line with gradient $-1.32$ °C h$^{-1}$ and intercept 99.0 °C is $y=-1.32x+99.0$; the negative sign shows cooling.

An intercept is where a fitted relationship crosses an axis. It can represent a threshold, baseline or model parameter only when the variables and units make that interpretation meaningful.

Read the axis scale carefully, distinguish measured from extrapolated intercepts and report uncertainty when the crossing is not directly observed.

A compensation point where net photosynthesis is zero can be estimated from a graph, but it should not be read beyond the measured light range without qualification.

A rate is change in the measured quantity per unit time or another independent variable. On a graph, the gradient is rise divided by run, with units that reveal the rate.

Use two points on the relevant line or curve, keep the time interval explicit and avoid mixing a secant average with an instantaneous rate.

If oxygen increases from 12 to 20 cm³ over 4 min, the average rate is 2 cm³ min⁻¹; a tangent at one moment may give a different instantaneous rate.

The gradient of a tangent gives the instantaneous rate at one point on a curve. The tangent should touch the curve locally, not simply connect distant data points.

Draw a small, well-positioned tangent, choose two far-apart points on that tangent, calculate rise/run and include the correct units.

A respiration curve that flattens has a smaller tangent gradient later, showing that the instantaneous rate is falling even if total oxygen uptake still rises.

A chloroplast image has diameter 96 mm at $\times16{,}000$. Convert $96$ mm to $96{,}000$ µm, then actual diameter $=96{,}000/16{,}000=6.0$ µm and radius $=3.0$ µm. Approximating it as a sphere, $SA=4\pi(3.0)^2=113$ µm$^2$ to a whole number.

Surface-area-to-volume ratio requires separate calculations followed by division. For similar shapes, area scales with length squared and volume with length cubed, so enlarging a cell lowers its SA:V even when its shape is unchanged.